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Volume 8 Number 25
EJTP Electronic Journal of Theoretical Physics
ISSN 1729-5254
This picture taken from http://mathpages.blogspot.com under Attribution 3.0 Unported (CC BY 3.0)
Editors
José Luis Lopez-Bonilla Ignazio Licata Ammar Sakaji http://www.ejtp.com May, 2011 E-mail:[email protected]
Volume 8 Number 25
EJTP Electronic Journal of Theoretical Physics
ISSN 1729-5254
This picture taken from http://mathpages.blogspot.com under Attribution 3.0 Unported (CC BY 3.0)
Editors
José Luis Lopez-Bonilla Ignazio Licata Ammar Sakaji http://www.ejtp.com May, 2011 E-mail:[email protected]
Editor in Chief
Ignazio Licata
Foundations of Quantum Mechanics, Complex System & Computation in Physics and Biology, IxtuCyber for Complex Systems , and ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy
José Luis Lopez-Bonilla Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, National Polytechnic Institute, SEPI-ESIME-Zacatenco, Edif. 5, CP 07738, Mexico city, Mexico Email: jlopezb[AT]ipn.mx lopezbonilla[AT]ejtp.info
Ammar Sakaji
Theoretical Condensed Matter, Mathematical Physics ISEM, Institute for Scientific Methodology, Palermo, Sicily – Italy International Institute for Theoretical Physics and Mathematics (IITPM), Prato, Italy Naval College, UAE And Tel:+971507967946 P. O. Box 48210 Abu Dhabi, UAE Email: info[AT]ejtp.com info[AT]ejtp.info
Editorial Board
Gerardo F. Torres del Castillo Mathematical Physics, Classical Mechanics, General Relativity, Universidad Autónoma de Puebla, México, Email:gtorres[AT]fcfm.buap.mx Torresdelcastillo[AT]gmail.com
Leonardo Chiatti Medical Physics Laboratory AUSL VT Via Enrico Fermi 15, 01100 Viterbo (Italy) Tel : (0039) 0761 1711055 Fax (0039) 0761 1711055 Email: fisica1.san[AT]asl.vt.it chiatti[AT]ejtp.info
Francisco Javier Chinea Differential Geometry & General Relativity, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Spain, E-mail: chinea[AT]fis.ucm.es
Maurizio Consoli
Non Perturbative Description of Spontaneous Symmetry Breaking as a Condensation Phenomenon, Emerging Gravity and Higgs Mechanism, Dip. Phys., Univ. CT, INFN,Italy
Email: Maurizio.Consoli[AT]ct.infn.it
Sergey Danilkin Instrument Scientist, The Bragg Institute Australian Nuclear Science and Technology Organization PMB 1, Menai NSW 2234 Australia Tel: +61 2 9717 3338 Fax: +61 2 9717 3606 Email: s.danilkin[AT]ansto.gov.au
Avshalom Elitzur Foundations of Quantum Physics ISEM, Institute for Scientific Methodology, Palermo, Italy Email: Avshalom.Elitzur[AT]ejtp.info
Elvira Fortunato Quantum Devices and Nanotechnology:
Departamento de Ciência dos Materiais CENIMAT, Centro de Investigação de Materiais I3N, Instituto de Nanoestruturas, Nanomodelação e Nanofabricação FCT-UNL Campus de Caparica 2829-516 Caparica Portugal
Tepper L. Gill Mathematical Physics, Quantum Field Theory Department of Electrical and Computer Engineering Howard University, Washington, DC, USA
Email: tgill[AT]Howard.edu tgill[AT]ejtp.info
Alessandro Giuliani
Mathematical Models for Molecular Biology Senior Scientist at Istituto Superiore di Sanità Roma-Italy
Email: alessandro.giuliani[AT]iss.it
Richard Hammond
General Relativity High energy laser interactions with charged particles Classical equation of motion with radiation reaction Electromagnetic radiation reaction forces Department of Physics University of North Carolina at Chapel Hill, USA Email: rhammond[AT]email.unc.edu
Arbab Ibrahim Theoretical Astrophysics and Cosmology Department of Physics, Faculty of Science, University of Khartoum, P.O. Box 321, Khartoum 11115, Sudan
Kirsty Kitto Quantum Theory and Complexity Information Systems | Faculty of Science and Technology Queensland University of Technology Brisbane 4001 Australia
Email: kirsty.kitto[AT]qut.edu.au
Hagen Kleinert Quantum Field Theory Institut für Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany
Email: h.k[AT]fu-berlin.de
Wai-ning Mei Condensed matter Theory Physics Department University of Nebraska at Omaha,
Omaha, Nebraska, USA Email: wmei[AT]mail.unomaha.edu physmei[AT]unomaha.edu
Beny Neta Applied Mathematics Department of Mathematics Naval Postgraduate School 1141 Cunningham Road Monterey, CA 93943, USA Email: byneta[AT]gmail.com
Peter O'Donnell General Relativity & Mathematical Physics, Homerton College, University of Cambridge, Hills Road, Cambridge CB2 8PH, UK E-mail: po242[AT]cam.ac.uk
Rajeev Kumar Puri Theoretical Nuclear Physics, Physics Department, Panjab University Chandigarh -160014, India Email: drrkpuri[AT]gmail.com rkpuri[AT]pu.ac.in
Haret C. Rosu Advanced Materials Division Institute for Scientific and Technological Research (IPICyT) Camino a la Presa San José 2055 Col. Lomas 4a. sección, C.P. 78216 San Luis Potosí, San Luis Potosí, México Email: hcr[AT]titan.ipicyt.edu.mx
Zdenek Stuchlik Relativistic Astrophysics Department of Physics, Faculty of Philosophy and Science, Silesian University, Bezru covo n´am. 13, 746 01 Opava, Czech Republic Email: Zdenek.Stuchlik[AT]fpf.slu.cz
S.I. Themelis Atomic, Molecular & Optical Physics Foundation for Research and Technology - Hellas P.O. Box 1527, GR-711 10 Heraklion, Greece Email: stheme[AT]iesl.forth.gr
Yurij Yaremko
Special and General Relativity, Electrodynamics of classical charged particles, Mathematical Physics, Institute for Condensed Matter Physics of Ukrainian National Academy of Sciences 79011 Lviv, Svientsytskii Str. 1 Ukraine Email: yu.yaremko[AT]gmail.com yar[AT]icmp.lviv.ua
yar[AT]ph.icmp.lviv.ua
Nicola Yordanov Physical Chemistry Bulgarian Academy of Sciences, BG-1113 Sofia, Bulgaria Telephone: (+359 2) 724917 , (+359 2) 9792546
Email: ndyepr[AT]ic.bas.bg ndyepr[AT]bas.bg
Former Editors:
Ammar Sakaji, Founder and Editor in Chief (2003-2010)
Table of Contents
No Articles Page 1 Editorial Notes
Ignazio Licata
i
2 Bogoliubov's Foresight and Development of the ModernTheoretical Physics A. L. Kuzemsky
1
3 Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions Hagen Kleinert
15
4 Hubbard-Stratonovich Transformation:Successes, Failure, and Cure Hagen Kleinert
57
5 A Clarification on the Debate on ``the Original Schwarzschild Solution'' Christian Corda
65
6 Entropy for Black Holes in the Deformed Horava-Lifshitz Gravity Andres Castillo and Alexis Larra
83
7 Canonical Relational Quantum Mechanics from Information Theory
93
Joakim Munkhammar
8 On the Logical Origins of Quantum Mechanics Demonstrated By Using Clifford Algebra: A Proof that Quantum Interference Arises in a Clifford Algebraic Formulation of Quantum Mechanics Elio Conte
109
9 The Ewald-Oseen Extinction Theorem in the Light of Huygens' Principle Peter Enders
127
10 Market Fluctuations -- the Thermodynamics Approach S. Prabakaran
137
11 Magnetized Bianchi Type VI_{0} Bulk Viscous Barotropic Massive String Universe with Decaying Vacuum Energy Density \Lambda Anirudh Pradhan and Suman Lata
158
12 Position Vector Of Biharmonic Curves in the 3-Dimensional Locally \phi-Quasiconformally Symmetric Sasakian Manifold Essin Turhan and Talat Körpinar
169
13 A Study of the Dirac-Sidharth Equation Raoelina Andriambololona and Christian Rakotonirina
177
14 Physical Vacuum as the Source of Standard Model Particle Masses
183
C. Quimbay and J. Morales
15 Quantum Mechanics as Asymptotics of Solutions of Generalized Kramers Equation E. M. Beniaminov
195
16 Application of SU(1,1) Lie algebra in connection with Bound States of Pöschl-Teller Potential Subha Gaurab Roy Raghunandan Das Joydeep Choudhury Nirmal Kumar Sarkar and Ramendu Bhattacharjee
211
17 Algebraic Aspects for Two Solvable Potentials Sanjib Meyur
217
18 Bound State Solutions of the Klein Gordon Equation with the Hulthén Potential Akpan N. Ikot Louis E. Akpabio and Edet J. Uwah
225
19 Chaotic dynamics of the Fractional Order\\ Nonlinear Bloch System Nasr-eddine Hamri and Tarek Houmor
233
20 A Criterion for the Stability Analysis of Phase Synchronization in Coupled Chaotic System Hadi Taghvafard and G. H. Erjaee
245
21 Synchronization of Different Chaotic Fractional-Order Systems via Approached Auxiliary System the Modified Chua Oscillator and the Modified Van der Pol-Duffing Oscillator
253
T. Menacer and N. Hamri
22 A Universal Nonlinear Control Law for the Synchronization of Arbitrary 4-D\Continuous-Time Quadratic Systems Zeraoulia Elhadj and J. C. Sprott
267
23 On a General Class of Solutions of a Nonholonomic Extension of Optical Pulse Equation Pinaki Patra, Arindam Chakraborty and A. Roy Chowdhury
273
24 Schwinger Mechanism for Quark-Antiquark Production in the Presence of Arbitrary Time Dependent Chromo-Electric Field Gouranga C. Nayak
279
25 Relic Universe M. Kozlowski and J. Marciak-Kozlowska
287
26 Halo Spacetime Mark D. Roberts
299
27 C-field Barotropic Fluid Cosmological Model with Variable G in FRW space-time Raj Bali and Meghna Kumawat
311
28 Two-Fluid Cosmological Models in Bianchi Type-III Space-Time K. S. Adhav S. M. Borikar, M. S. Desale, and R. B. Raut
319
29 Shell Closures and Structural Information from
Nucleon Separation Energies C. Anu Radha V. Ramasubramanian and E. James Jebaseelan Samuel
327
30 Calculating Vacuum Energy as a Possible Explanation of the Dark Energy B. Pan
343
31 Some Bianchi type-I Cosmic Strings in a Scalar --Tensor Theory of Gravitation R.Venkateswarlu, J.Satish and K.Pavan Kumar
354
32 Gravitons Writ Large; I.E. Stability, Contributions to Early Arrow of Time, and Also Their Possible Role in Re Acceleration of the Universe 1 Billion Years Ago? A. Beckwith
361
33 Dimensionless Constants and Blackbody Radiation Laws Ke Xiao
379
Electronic Journal of Theoretical Physics 8, No. 25 (2011) i
WELCOME TO EJTP AND 25th ISSUE!
Ignazio Licata
ISEM, Institute for Scientific and Methodology, Palermo, Italy
Here β = 1/kBT is the reciprocal temperature and Ω is the normalization factor.
It is known [16] that the averages 〈A〉 are unaffected by a change of representation. Themost important is the representation in which ρ is diagonal ρmn = ρmδmn. We then have
〈ρ〉 = Trρ2 = 1. It is clear then that Trρ2 ≤ 1 in any representation. The core of the
problem lies in establishing the existence of a thermodynamic limit [19] (such as N/V =
const, V → ∞, N = number of degrees of freedom, V = volume) and its evaluation for
the quantities of interest.
The evolution equation for the density matrix is a quantum analog of the Liouville equa-
tion in classical mechanics. A related equation describes the time evolution of the expec-
tation values of observables, it is given by the Ehrenfest theorem. Canonical quantization
yields a quantum-mechanical version of this theorem. This procedure, often used to de-
vise quantum analogues of classical systems, involves describing a classical system using
Hamiltonian mechanics. Classical variables are then re-interpreted as quantum operators,
while Poisson brackets are replaced by commutators. In this case, the resulting equation
is∂
∂tρ = − i
�[H, ρ] (7)
where ρ is the density matrix. When applied to the expectation value of an observable,
the corresponding equation is given by Ehrenfest theorem, and takes the form
d
dt〈A〉 = i
�〈[H,A]〉 (8)
where A is an observable. Thus in the statistical mechanics the average 〈A〉 of anydynamical quantity A is defined in a single-valued way [16, 18].
In the situations with degeneracy the specific problems appear. In quantum mechanics, if
two linearly independent state vectors (wavefunctions in the Schroedinger picture) have
the same energy, there is a degeneracy. In this case more than one independent state
of the system corresponds to a single energy level. If the statistical equilibrium state
of the system possesses lower symmetry than the Hamiltonian of the system (i.e. the
situation with the spontaneous symmetry breakdown), then it is necessary to supplement
the averaging procedure (6) by a rule forbidding irrelevant averaging over the values of
macroscopic quantities considered for which a change is not accompanied by a change in
energy.
This is achieved by introducing quasiaverages, that is, averages over the Hamiltonian Hν�e
supplemented by infinitesimally-small terms that violate the additive conservations laws
Hν�e = H + ν(e · M), (ν → 0). Thermodynamic averaging may turn out to be unstable
with respect to such a change of the original Hamiltonian, which is another indication of
degeneracy of the equilibrium state.
According to Bogoliubov [13, 14], the quasiaverage of a dynamical quantity A for the
system with the Hamiltonian Hν�e is defined as the limit
� A �= limν→0〈A〉ν�e, (9)
where 〈A〉ν�e denotes the ordinary average taken over the Hamiltonian Hν�e, containing the
small symmetry-breaking terms introduced by the inclusion parameter ν, which vanish
as ν → 0 after passage to the thermodynamic limit V → ∞. Thus the existence of de-
generacy is reflected directly in the quasiaverages by their dependence upon the arbitrary
unit vector e. It is also clear that
〈A〉 =∫
� A � de. (10)
According to definition (10), the ordinary thermodynamic average is obtained by ex-
tra averaging of the quasiaverage over the symmetry-breaking group [13, 17]. Thus to
describe the case of a degenerate state of statistical equilibrium quasiaverages are more
convenient, more physical, than ordinary averages [16, 13]. The latter are the same quasi-
averages only averaged over all the directions e.
It is necessary to stress, that the starting point for Bogoliubov’s work [13] was an in-
vestigation of additive conservation laws and selection rules, continuing and developing
the approach by P. Curie for derivation of selection rules for physical effects. Bogoliubov
demonstrated that in the cases when the state of statistical equilibrium is degenerate, as
in the case of the Heisenberg ferromagnet, one can remove the degeneracy of equilibrium
states with respect to the group of spin rotations by including in the Hamiltonian H an
additional noninvariant term νMzV with an infinitely small ν. Thus the quasiaverages
do not follow the same selection rules as those which govern the ordinary averages. For
the Heisenberg ferromagnet the ordinary averages must be invariant with regard to the
spin rotation group. The corresponding quasiaverages possess only the property of co-
variance. It is clear that the unit vector e, i.e., the direction of the magnetization M
vector, characterizes the degeneracy of the considered state of statistical equilibrium. In
order to remove the degeneracy one should fix the direction of the unit vector e. It can
be chosen to be along the z direction. Then all the quasiaverages will be the definite
numbers. This is the kind that one usually deals with in the theory of ferromagnetism.
The value of the quasi-average (9) may depend on the concrete structure of the addi-
tional term ΔH = Hν−H, if the dynamical quantity to be averaged is not invariant withrespect to the symmetry group of the original Hamiltonian H. For a degenerate state
the limit of ordinary averages (10) as the inclusion parameters ν of the sources tend to
zero in an arbitrary fashion, may not exist. For a complete definition of quasiaverages it
is necessary to indicate the manner in which these parameters tend to zero in order to
ensure convergence [16]. On the other hand, in order to remove degeneracy it suffices, in
the construction of H, to violate only those additive conservation laws whose switching
lead to instability of the ordinary average. Thus in terms of quasiaverages the selection
rules for the correlation functions [16] that are not relevant are those that are restricted
by these conservation laws.
By using Hν , we define the state ω(A) = 〈A〉ν and then let ν tend to zero (after passingto the thermodynamic limit). If all averages ω(A) get infinitely small increments under
infinitely small perturbations ν, this means that the state of statistical equilibrium under
consideration is nondegenerate [16]. However, if some states have finite increments as
ν → 0, then the state is degenerate. In this case, instead of ordinary averages 〈A〉H , oneshould introduce the quasiaverages (9), for which the usual selection rules do not hold.
where the number of creation operators Ψ† may be not equal to the number of annihilationoperators Ψ. We fix times and split the arguments (t1, x1, . . . tn, xn) into several clusters
(. . . , tα, xα, . . .), . . . , (. . . , tβ, xβ, . . .). Then it is reasonably to assume that the distances
between all clusters |xα − xβ| tend to infinity. Then, according to the cluster property,the average value (11) tends to the product of averages of collections of operators with
The system of bosons is contained in the cube A with the edge L and volume V . It
was assumed that it satisfies periodic boundary conditions and the potential Φ(q) is
spherically symmetric and proportional to the small parameter. It was also assumed
that, at temperature zero, a certain macroscopic number of particles having a nonzero
density is situated in the state with momentum zero.
The operators Ψ(q), and Ψ†(q) are represented in the form
Ψ(q) = a0/√V ; Ψ†(q) = a†0/
√V , (16)
where a0 and a†0 are the operators of annihilation and creation of particles with momen-
tum zero. To explain the phenomenon of superfluidity, one should calculate the spectrum
of the Hamiltonian, which is quite a difficult problem. Bogoliubov suggested the idea of
approximate calculation of the spectrum of the ground state and its elementary excita-
tions based on the physical nature of superfluidity. His idea consists of a few assumptions.
The main assumption is that at temperature zero the macroscopic number of particles
(with nonzero density) has the momentum zero. Therefore, in the thermodynamic limit,
the operators a0/√V and a†0/
√V commute
limV→∞
[a0/√V , a†0/
√V]=
1
V→ 0 (17)
and are c-numbers. Hence, the operator of the number of particles N0 = a†0a0 is a c-number, too. The concept of quasiaverages was introduced by Bogoliubov on the basis of
an analysis of many-particle systems with a degenerate statistical equilibrium state. Such
states are inherent to various physical many-particle systems. Those are liquid helium in
the superfluid phase, metals in the superconducting state, magnets in the ferromagneti-
cally ordered state, liquid crystal states, the states of superfluid nuclear matter, etc.
From the other hand, it is clear that only a thorough experimental and theoretical inves-
tigation of quasiparticle many-body dynamics of the many-particle systems can provide
the answer on the relevant microscopic picture [20]. As is well known, Bogoliubov was
first to emphasize the importance of the time scales in the many-particle systems thus
anticipating the concept of emergence of macroscopic irreversible behavior starting from
the reversible dynamic equations.
More recently it has been possible to go step further. This step leads to a deeper under-
standing of the relations between microscopic dynamics and macroscopic behavior on the
basis of emergence concept [21, 22, 23]. There has been renewed interest in emergence
within discussions of the behavior of complex systems and debates over the reconcilability
of mental causation, intentionality, or consciousness with physicalism. This concept is
also at the heart of the numerous discussions on the interrelation of the reductionism and
functionalism.
A vast amount of current researches focuses on the search for the organizing principles re-
sponsible for emergent behavior in matter [23, 24], with particular attention to correlated
matter, the study of materials in which unexpectedly new classes of behavior emerge in
response to the strong and competing interactions among their elementary constituents.
[18] D. N. Zubarev, Nonequilibrium Statistical Thermodynamics. Consultant Bureau,New York, 1974.
[19] N. N. Bogoliubov, D.Ya. Petrina, B.I. Chazet, Mathematical Description ofEquilibrium State of Classical Systems Based on the Canonical Formalism. Teor.Mat. Fiz. 1 251-274 (1969).
[20] A.L. Kuzemsky, Statistical Mechanics and the Physics of Many-Particle ModelSystems. Physics of Particles and Nuclei, 40,949-997 (2009).
[21] R. B. Laughlin, A Different Universe. Basic Books, New York, 2005.
[22] R. B. Laughlin, The Crime of Reason: And the Closing of the Scientific Mind. BasicBooks, New York, 2008.
[23] R. D. Laughlin, D. Pines. Theory of Everything. Proc. Natl. Acad. Sci. (USA). 97,28 (2000).
[24] D. L. Cox, D. Pines. Complex Adaptive Matter: Emergent Phenomena in Materials.MRS Bulletin. 30, 425 (2005).
[25] A.L. Kuzemsky, Theory of Transport Processes and the Method of NonequilibriumStatistical Operator. Int.J. Mod. Phys., B21,2821-2949 (2007).
[26] N.N. Bogoliubov, Jr., D. P. Sankovich, N. N. Bogoliubov and Statistical Mechanics,Usp. Mat. Nauk., 49, 21 (1994).
[27] A.L. Kuzemsky, Works on Statistical Physics and Quantum Theory of Solid State.JINR, Dubna, 2009.
[28] A.L. Kuzemsky, Symmetry Breaking, Quantum Protectorate and Quasiaverages inCondensed Matter Physics. Physics of Particles and Nuclei, 41 1031-1034 (2010).
[29] A.L. Kuzemsky, Bogoliubov’s Quasiaverages, Broken Symmetry and QuantumStatistical Physics, e-preprint: arXiv:1003.1363 [cond-mat. stat-mech] 6 Mar, 2010.
[30] A.L. Kuzemsky, Quasiaverages, Symmetry Breaking and Irreducible Green FunctionsMethod. Condensed Matter Physics (http://www.icmp.lviv.ua/journal), 13 43001:1-20 (2010).
Fig. 2 Typical Ω-dependence of Nth approximations WN at T = 0 for increasing orders N .The coupling constant has the value g/4 = 0.1. The dashed horizontal line indicates the exactenergy.
Even to lowest order, the result is surprisingly accurate. For N = 1, the energy EN
we has the linear dependence
E1 = ω
(1
2+
3
16
g
ω3
). (11)
After the replacement (7) and the reexpansion up to power g at fixed r we find
W 1 = Ω
(1
4+ω2
4Ω+
3
16
g
Ω4
). (12)
In the strong-coupling limit, the minimum lies at Ω ≈ c(g/4)1/3 where c is some constant
and the energy behaves like
W 1 ≈(g4
)1/3(c
4+
3
4c2
). (13)
The minimum lies at c = 61/3 where W 1 ≈ (g/4)1/3 (3/4)4/3≈ (g/4)1/3 × 0.681420. The
treatment can easily be extended to 40 digits [11] starting out like
E1= (g/4)1/3 × 0.667 986 259 . . . .
The result is shown in for g/4 = 0.1 in Fig. 3. If we plot the minimum as a function
of g we obtain the curve shown in Fig. 3. The curve has the asymptotic behavior
2 4 6 8
0.5
1
1.5
2
g
E1
min W 1
Fig. 3 First-order perturbative energy E1 and the variational-perturbative minimum of W 1.The exact result follows closely the curve min W 1.
(g/4)1/3×0.68142. This grows with the exact power of g and has a coefficient that differsonly slightly from the accurate value 0.667 986 259 . . . found by other approximation
The important critical behavior is seen in the correlation function which have the limiting
form
〈φi(x)φj(x′)〉 ∼ e−|x−x
′|/ξ(T )
|x− x′|D−2+η. (16)
where η is the anomalous field dimension, and ξ is the coherence length which diverges
near Tc like ξ(T ) ∼ (T − Tc)−ν .
3.1 Critical Behavior in D − ε Dimensions
The field fluctuations cause divergencies which can be removed by a renormalization of
field, mass and coupling constant to φ, m, and g. This is most elegantly done by assuming
the dimension of spacetime to be D = 4− ε, in which case the renormalization factor are
g0 = Zg(g, ε)Zφ(g, ε)−2 με g, (17)
m20 = Zm(g, ε)Zφ(g, ε)
−1m2, (18)
φ20 = Zφ(g, ε)φ
2. (19)
The factors have weak-coupling expansions:
Zg(g, ε) = 1 +n+ 8
3εg +
{(n+ 8)2
9ε2− 5n+ 22
9ε
}g2 + . . . , (20)
Zφ(g, ε) = 1− n+ 2
36εg2 + . . . , (21)
Zm(g, ε) = 1 +n+ 2
3εg +
{(n+ 2)(n+ 5)
9ε2− n+ 2
6ε
}g2 + . . . .
The dependence of these on the scale parameter μ defines the renormalization group
functions
β(g, ε) = μdg
dμ
∣∣∣∣0
= −ε{∂
∂gln[gZg(g, ε)Zφ(g, ε)
−2]}−1 , (22)
γm(g) =μ
m
dm
dμ
∣∣∣∣0
= −β(g, ε)2
∂
∂gln[Zm(g, ε)Zφ(g, ε)
−1] , (23)
γ(g) = −μφ
dφ
dμ
∣∣∣∣0
=β(g, ε)
2
∂
∂glnZφ(g, ε). (24)
At the phase transition g0 goes to the strong-coupling limit g0 → ∞. In this limit the
renormalized coupling g tends to a constant g∗, called the fixed point of the theory.From the renormalization group functions in the strong-coupling limit one finds the
Fig. 7 Strong-coupling values for the critical exponent ν−1(x) as a function of x(L) = e−cL1−ω
For the critical exponent α characterizing the behavior of the specific heat C ≈|T −Tc|−α of superfluid helium near the critical temperature Tc, the strong-coupling limit
is [15].
α ≈ 2− 3× 0.6712 ≈ −0.0136. (39)
If we extrapolate the asymptotic behavior expansion coefficients of ν up to the 9th order
according using the theoretically known large-order behavior this result can be improved
to α ≈ −0.0129 [24] (see Fig. 8). This value agrees perfectly with the space shuttle
value [8] α = −0.01285 ± 0.00038. The experimental result extracted from Fig. 1 and
Fig. 8 Strong-coupling limits of α as a function of x = e−cL1−ω
for 7th and 9th order inperturbation theory. The latter limit α ≈ −0.0129 agrees well with the satellite experiment [8].
the various theoretical numbers obtained from the divergent perturbation series for α are
Fig. 9 Survey of experimental and theoretical values for α. The latter come from resummedperturbation expansions of φ4-theory in 4 − ε dimensions, in three dimensions, and from high-temperature expansions of XY-models on a lattice. The sources are indicated below.
4. Shift of the Critical Temperature in Bose-Einstein Conden-
sate by Repulsive Interaction
A free Bose gas condenses at a critical temperature
T (0)c =
2π
M
[n
ζ(3/2)
] 23
, (40)
where n is the particle density. A small relative shift of Tc with respect to T(0)c can be
calculated from the general formula
ΔTc
T(0)c
= −23
Δn
n(0), (41)
where n(0) is the particle density in the free condensate and Δn its change at Tc caused
by a small repulsive point interaction parametrized by an s-wave scattering length a. For
small a, this behaves like [25, 26]
ΔTc
T(0)c
= c1an1/3 + [c′2 ln(an
1/3) + c2]a2n2/3 +O(a3n). (42)
where c′2 = −64πζ(1/2)/3ζ(3/2)5/3 � 19.7518 can be calculated perturbatively, whereas
c1 and c2 require nonperturbative techniques since infrared divergences at Tc make them
basically strong-coupling results. The standard technique to reach this regime is based
on a resummation of perturbation expansions using the renormalization group [27, 18],
two values are fitted by the same inverse power of L, we find c1 ≈ 0.83 − 14/L6. From
the extrapolations to infinite order we estimate c1,∞ ≈ 0.92± 0.13.
1 1.5 2 2.5 3 3.5
0.5
1
1.5
2
2.5
3
L
c1 ≈ 1.053 + 2/L6
c1 ≈ 0.830− 14/L6
Fig. 11 The three approximants for c1 plotted against the order of variational approximationL ≡ L− 1 = 1, 2, 3, and extrapolation to the infinite-order limit.
This result is to be compared with latest Monte Carlo data which estimate c1 ≈1.32 ± 0.02 [36, 37]. Previous theoretical estimates are c1 ≈ 2.90 [38], 2.33 from a 1/N -
expansion [39]), 1.71 from a next-to-leading order in a 1/N -expansion [40], 3.059 from
an inapplicable δ-expansion [41] to three loops, and 1.48 from the same δ-expansion to
five loops, with a questionable evaluation at a complex extremum [29] and some wrong
expansion coefficients (see [31]). Remarkably, our result lies close to the average between
the latest and the first Monte Carlo result c1 ≈ 0.34± 0.03 in Ref. [42].
As a cross check of the reliability of our theory consider the result in the limit N →∞.
Here we must drop the first term in the expansion (45) which vanishes at the critical
point (but would diverge for N →∞ at finite m). The remaining expansion coefficients
whose optima yield the approximations c1 ≈ 1.886 and 2.017, converging rapidly towards
the exact large-N result 2.33 of Ref. [39], with a 10% error.
Numerically, the first two 1/N -corrections found from a fit to large-N results ob-
tained by using the known large-N expression for ω′ = 1 − 8(8/3π2N) + 2(104/3 −9π2/2)(8/3π2N)2 [43] produce a finite-N correction factor (1− 3.1/N + 30.3/N2 + . . . ),
to be compared with (1− 0.527/N + . . . ) obtained in Ref. [40].
Since the large-N results can only be obtained so well without the use of the first
term we repeat the evaluations of the series at the physical value N = 2 without the first
whereKν(z) is the modified Bessel function. For small g, the function Z(g) has a divergent
Taylor series expansion, to be called weak-coupling expansion:
Z(L)weak(g) =
L∑l=0
al gl, with al = (−1)l Γ(2l + 1/2)
l!√π
. (73)
For g < 0, this is non-Borel-summable. For large |g| there exists a convergent strong-
coupling expansion:
Z(L)strong(g) = g−l/4
L∑l=0
bl g−l/2, with bl = (−1)l Γ(l/2 + 1/4)
2l!√π
. (74)
As is obvious from the integral representation (72), Z(g) obeys the second-order differ-
ential equation
16g2Z ′′(g) + 4(1 + 8g)Z ′(g) + 3Z(g) = 0, (75)
which has two independent solutions. One of them is Z(g), which is finite for g > 0 with
Z(0) = a0. The weak-coupling coefficients al in (73) can be obtained by inserting into
(75) the Taylor series and comparing coefficients. The result is the recursion relation
al+1 = −16l(l + 1) + 3
4(l + 1)al. (76)
A similar recursion relation can be derived for the strong-coupling coefficients bl in
Eq. (74). We observe that the two independent solutions Z(g) of (75) behave like Z(g) ∝gα for g →∞ with the powers α = −1/4 and −3/4. The function (72) has α = −1/4. Itis convenient to remove the leading power from Z(g) and define a function ζ(x) such that
Z(g) = g−1/4 ζ(g−1/2). The Taylor coefficients of ζ(x) are the strong-coupling coefficientsbl in Eq. (74). The function ζ(x) satisfies the differential equation and initial conditions:
4ζ ′′(x)− 2xζ ′(x)− ζ(x) = 0, with ζ(0) = b0 and ζ ′(0) = b1. (77)
The Taylor coefficients bl of ζ(x) satisfy the recursion relation
bl+2 =2l + 1
4(l + 1)(l + 2)bl . (78)
Analytic continuation of Z(g) around g =∞ to the left-hand cut gives:
The Lth variational approximation to Z(g) is given by (see [15, 18, 32, 33])
Z(L)var (g,Ω) = Ωp
L∑j=0
( g
Ωq
)j
εj(σ), (91)
with
σ ≡ Ωq−2(Ω2 − 1)/g , (92)
where q = 2/ω = 4, p = −1 and
εj(σ) =
j∑l=0
al
((p− lq)/2
j − l
)(−σ)j−l . (93)
In order to find a valley of minimal sensitivity, the zeros of the derivative of Z(L)var (g,Ω)
with respect to Ω are needed. They are given by the zeros of the polynomials in σ:
P (L)(σ) =L∑l=0
al(p− lq + 2l − 2L)
((p− lq)/2
L− l
)(−σ)L−l = 0, (94)
since it can be shown [13, 15] that the derivative depends only on σ:
dZ(L)var (g,Ω)
dΩ= Ωp−1
( g
Ωq
)L
P (L)(σ) . (95)
g−.8
Z(g) Z(g)
−.4 0 g−.5 0 .5
−.2
−.4
1
.8
Fig. 15 Plot of the 1st- and 2nd-order calculation for the non-Borel-summable region of g < 0,where the function has a cut with non-vanishing imaginary part: imaginary (left) and real parts
(right) of Z(1)var(g) (dashed curve) and Z
(2)var(g) (solid curve) are plotted against g and compared
with the exact values of the partition function (dotted curve). The root of (92) giving the optimalvariational parameter Ω has been chosen to reproduce the weak-coupling result near g = 0.
Consider in more detail the lowest non-trivial order with L = 1. From Eq. (94) we
obtain
σ =5
2, corresponding to Ω =
1
2
(1±
√1 + 10g
). (96)
In order to ensure that our method reproduces the weak-coupling result for small g, we
have to take the positive sign in front of the square root. In Fig. 15 we have plotted Z(1)var(g)
(dashed curve) and Z(2)var(g) (solid curve) and compared these with the exact result (doted
curve) in the tunneling regime. The agreement is quite good even at these low orders [51].
Next we study the behavior of Z(L)var (g) to higher orders L. For selected coupling values
in the non-Borel-summable region, g = −.01, −.1, −1, −10, we want to see the erroras a function of the order. We want to find from this model system the rule for selecting
systematically the best zero of P (L)(σ) solving Eq. (94), which leads to the optimal value
of the variational parameter Ω. For this purpose we plot the variational results of all
zeros. This is shown in Fig. 16, where the logarithm of the deviations from the exact
value is plotted against the order L. The outcome of different zeros cluster strongly near
the best value. Therefore, choosing any zero out of the middle of the cluster is reasonable,
in particular, because it does not depend on the knowledge of the exact solution, so that
this rule may be taken over to realistic cases.
g = −.01 g = −.1
g = −1 g = −10
10 20 30 L
10 20 30 L
10 20 30 L
10 20 30 L
−20
−30
−40
−10
−20
−10
−20
−10
−20
Fig. 16 Logarithm of deviation of the variational results from exact values log |Z(L)var − Zexact|
plotted against the order L for different g < 0 in the non-Borel-summable region. All complexoptimal Ω’s have been used.
Δ(L)
L10 20 30
−10
−20
−30
Fig. 17 Logarithm of deviation of variational results from exactly known value Δ(L) =
log |Z(L)var − Zexact|, plotted against the order L for g = 10 in Borel-summable region. The real
positive optimal Ω have been used. There is only one real zero of the first derivative in everyodd order L and none for even orders. There is excellent convergence Δ(L) � 0.02 exp (−0.73L)for L→∞.
We wish to emphasize, that for the Borel-summable domain with g > 0, variational
perturbation theory has the usual fast convergence in this model. In fact, for g = 10,
probing deeply into the strong-coupling domain, we find rapid convergence like Δ(L) �0.02 exp (−0.73L) for L → ∞, where Δ(L) = log |Z(L)
(exact)l | on the right, plotted for some strong-coupling coefficients
bl with l = 0, 4, 8, 12, 16, 20 against the order L.
as a function of the order L. This is shown in Fig. 17. Furthermore, the strong-coupling
coefficients bl of Eq. (74) are reproduced quite satisfactorily. Having solved P (L)(σ) =
0 for σ, we obtain Ω(L)(g) by solving Eq. (92). Inserting this and (93) into (91), we
bring g1/4 Z(L)var (g) into a form suitable for expansion in powers of g−1/2. The expansion
coefficients are the strong-coupling coefficients b(L)l to order L. In Fig. 18 we have plotted
the logarithms of their absolute and relative errors over the order L, and find very good
convergence, showing that variational perturbation theory works well for our test-model
Z(g).
A better selection of the optimal Ω values comes from the following observation. The
imaginary parts of the approximations near the singularity at g = 0 show tiny oscillations.
The exact imaginary part is known to decrease extremely fast, like exp (1/4g), for g → 0−,practically without oscillations. We can make the tiny oscillations more visible by taking
this exponential factor out of the imaginary part. This is done in Fig. 19. The oscillations
differ strongly for different choices of Ω(L) from the central region of the cluster. To
each order L we see that one of them is smoothest in the sense that the approximation
approaches the singularity most closely before oscillations begin. If this Ω(L) is chosen
as the optimal one, we obtain excellent results for the entire non-Borel-summable region
g < 0. As an example, we pick the best zero for the L = 16th order. Fig. 19 shows
g−.014 −.012 −.01 −.008
−.75
−.7
−.65
−.6AB
CD
EF
Fig. 19 Normalized imaginary part Im[Z(16)var (g) exp (−1/4g)] as a function of g based on six
different complex zeros (thin curves). The fat curve represents the exact value, which isZexact(g) � −0.7071 + .524g − 1.78g2. Oscillations of varying strength can be observed nearg = 0. Curves A and C carry most smoothly near up to the origin. Evaluation based on eitherof them yields equally good results. We have selected the zero belonging to curve C as our bestchoice to this order L = 16.
the normalized imaginary part calculated to this order, but based on different zeros from
the central cluster. Curve C appears optimal. Therefore we select the underlying zero
as our best choice at order L = 16 and calculate with it real and imaginary part for
the non-Borel-summable region −2 < g < −.008, to be compared with the exact values.Both are shown in Fig. 20, where we have again renormalized the imaginary part by
the exponential factor exp (−1/4g). The agreement with the exact result (solid curve) isexcellent as was to be expected because of the fast convergence observed in Fig. 16. It
is indeed much better than the strong-coupling expansion to the same order, shown as
a dashed curve. This is the essential improvement of our present theory as compared to
previously known methods probing into the tunneling regime [51].
This non-Borel-summable regime will now be investigated for the quantum-mechanical
anharmonic oscillator.
log (−g)0 −2 −4 log (−g)0 −2 −4
.9
1.0
1.1
−.7
−.6
−.5
Fig. 20 Normalized imaginary part Im[Z(16)var (g) exp (−1/4g)] to the left and the real part
Re[Z(16)var (g)] to the right, based on the best zero C from Fig. 19, are plotted against log |g|
as dots. The solid curve represents the exact function. The dashed curve is the 16th order of
the strong-coupling expansion Z(L)strong(g) of equation (74).
6.2 Tunneling Regime of Quantum-Mechanical Anharmonic Oscillator
The divergent weak-coupling perturbation expansion for the ground state energy of the
anharmonic oscillator in the potential V (x) = x2/2 + g x4 to order L
E(L)0,weak(g) =
L∑l=0
al gl , (97)
where al = (1/2, 3/4, −21/8, 333/16, −30885/128, . . . ), is non-Borel-summable forg < 0. It may be treated in the same way as Z(g) of the previous model, making use as
before of Eqs. (91)–(94), provided we set p = 1 and ω = 2/3, so that q = 3, accounting
for the correct power behavior E0(g) ∝ g1/3 for g → ∞. According to the principle
of minimal dependence and oscillations, we pick a best zero for the order L = 64 from
the cluster of zeros of PL(σ), and use it to calculate the logarithm of the normalized
imaginary part:
f(g) := log[√−πg/2 E(64)
0,var(g)]− 1/3g . (98)
This quantity is plotted in Fig. 21 against log(−g) close to the tip of the left-hand cutfor −.2 < g < −.006.
Fig. 21 Logarithm of the imaginary part of the ground state energy of the anhar-monic oscillator with the essential singularity factored out for better visualization, l(g) =
log[√−πg/2 E
(64)0,var(g)
]− 1/3g, plotted against small negative values of the coupling constant
−0.2 < g < −.006 where the series is non-Borel-summable. The thin curve represents thedivergent expansion around a critical bubble of Ref. [52]. The fat curve is the 22nd order ap-proximation of the strong-coupling expansion, analytically continued to negative g in the slidingregime calculated in Chapter 17 of the textbook [6].
Comparing our result to older values from semi-classical calculations [52]
Fig. 22 Logarithm of the normalized imaginary part of the ground state energy
log (√−πg/2 E
L)0,var(g))− 1/3g, plotted against log (−g) for orders L = 4, 8, 16, 32 (curves). It
is compared with the corresponding results for L = 64 (points). This is shown for small negativevalues of the coupling constant −0.2 < g < −.006, i.e. in the non-Borel-summable critical-bubbleregion. Fast convergence is easily recognized. Lower orders oscillate more heavily. Increasingorders allow closer approach to the singularity at g = 0−.
the derivation of a strong-coupling expansion of the type (74) from the divergent weak-
coupling expansion, and an analytic continuation of the strong-coupling expansion to
negative g. This method was applicable only for large enough coupling strength where
the strong-coupling expansion converges, the so-called sliding regime. It could not invade
into the tunneling regime at small g governed by critical bubbles, which was treated in [6]
by a separate variational procedure. The present work fills the missing gap by extending
variational perturbation theory to all g arbitrarily close to zero, without the need for a
separate treatment of the tunneling regime.
It is interesting to see, how the correct limit is approached as the order L increases.
This is shown in Fig. 22, based on the optimal zero in each order. For large negative g,
even the small orders give excellent results. Close to the singularity the scaling factor
exp (−1/3g) will always win over the perturbation results. It is surprising, however, howfantastically close to the singularity we can go.
6.3 Dynamic Approach to the Critical-Bubble Regime
Regarding the computational challenges connected with the critical-bubble regime of
small g < 0, it is worth to develop an independent method to calculate imaginary parts
in the tunneling regime. For a quantum-mechanical system with an interaction potential
g V (x), such as a the harmonic oscillator, we may study the effect of an infinitesimal
increase in g upon the system. It induces an infinitesimal unitary transformation of
the Hilbert space. The new Hilbert space can be made the starting point for the next
infinitesimal increase in g. In this way we derive an infinite set of first order ordinary
differential equations for the change of the energy levels and matrix elements (for details
Fig. 23 Logarithm of the normalized imaginary part of the ground state energy of the anhar-monic oscillator as solution of the coupled set of differential equations (102), truncated at theenergy level of n = 64 (points), compared with the corresponding quantity from the L = 64thorder of non-Borel-summable variational perturbation theory (curve), both shown as functionsof the coupling constant g.
This system of equations holds for any one-dimensional Schroedinger problem. Individ-
ual differences come from the initial conditions, which are the energy levels En(0) of
the unperturbed system and the matrix elements Vnm(0) of the interaction V (x) in the
unperturbed basis. For a numerical integration of the system a truncation is necessary.
The obvious way is to restrict the Hilbert space to the manifold spanned by the lowest N
eigenvectors of the unperturbed system. For cases like the anharmonic oscillator, which
are even, with even perturbation and with only an even state to be investigated, we may
span the Hilbert space by even basis vectors only. Our initial conditions are thus for
Table 2 Perturbation coefficients up to order B6 for the weak-field expansions of the variationalparameters and the binding energy in comparison to the exact ones of Ref. [55].
n 0 1 2 3
ηn 1.0 − 405π2
7168 ≈ −0.5576 16828965π4
1258815488 ≈ 1.3023 − 3886999332075π6
884272562962432 ≈ −4.2260
Ωn329π ≈ 1.1318 99π
224 ≈ 1.3885 − 1293975π3
19668992 ≈ −2.03982 524431667187π5
27633517592576 ≈ 5.8077
εn − 43π ≈ −0.4244 9π
128 ≈ 0.2209 − 8019π3
1835008 ≈ −0.1355 256449807π5
322256764928 ≈ 0.2435
εn [55] −0.5 0.25 − 53192 ≈ −0.2760 5581
4608 ≈ 1.2112
Expanding the variational parameters into perturbation series of the square magnetic
field B2,
η(B) =∞∑n=0
ηnB2n, Ω(B) =
∞∑n=0
ΩnB2n (119)
and inserting these expansions into the self-consistency conditions (118) and (118) we
obtain order by order the coefficients given in Table 2. Inserting these values into the
expression for the binding energy (115) and expand with respect to B2, we obtain the
perturbation series
ε(1)(B) =B
2−
∞∑n=0
εnB2n. (120)
The first coefficients are also given in Table 2. We find thus the important result that
the first-order variational perturbation solution possesses a perturbative behavior with
respect to the square magnetic field strength B2 in the weak-field limit thus yielding the
correct asymptotic. The coefficients differ in higher order from the exact ones but are
Note that the prefactor 1/π of the leading ln2B-term differs from a value 1/2 obtained
by Landau and Lifschitz [56]. Our different value is a consequence of using a harmonic
trial system. The calculation of higher orders in variational perturbation theory would
improve the value of the prefactor.
At a magnetic field strength B = 105B0, which corresponds to 2.35× 1010T = 2.35×1014G, the contribution from the first six terms is 22.87 [2Ryd]. The next three terms
suppressed by a factor ln−1B contribute −2.29 [2Ryd], while an estimate for the ln−2B-terms yields nearly −0.3 [2Ryd]. Thus we find
ε(1)(105) = 20.58± 0.3 [2Ryd]. (135)
This is in very good agreement with the value 20.60 [2Ryd] obtained from an accurate
numerical treatment [58].
Table 3 lists the values of the first six terms of Eq. (134). This shows in particular
the significance of the second-leading term −(4/π)lnB lnlnB, which is of the same order
of the leading term (1/π)ln2B but with an opposite sign. In Fig. 24, we have plotted the
expression
εL(B) =1
2ln2B (136)
from Landau and Lifschitz [56] to illustrate that it gives far too large binding energies
even at very large magnetic fields, e.g. at 2000B0 ∝ 1012G.
This strength of magnetic field appears on surfaces of neutron stars (1010−1012G). Arecently discovered new type of neutron star is the so-called magnetar. In these, charged
particles such as protons and electrons produced by decaying neutrons give rise to the
giant magnetic field of 1015G. Magnetic fields of white dwarfs reach only up to 106−108G.All these magnetic field strengths are far from realization in experiments. The strongest
magnetic fields ever produced in a laboratory were only of the order 105G, an order of
magnitude larger than the fields in sun spots which reach about 0.4× 104G. Recall, for
comparison, that the earth’s magnetic field has the small value of 0.6G.
The nonleading terms in Eq. (134) give important contributions to the asymptotic
behavior even at such large magnetic fields, as we can see in Fig. 24. It is an unusual
property of the asymptotic behavior that the absolute value of the difference between the
Landau-expression (136) and our approximation (134) diverges with increasing magnetic
field strengths B, only the relative difference decreases.
8. Appendix A: Modification of Principle of Minimal Sensitivity
The naive quantum mechanical variational perturbation theory has been used by many
authors under the name δ-expansion. This name stems from the fact that one may write
Fig. 24 Ground state energy E(B) of hydrogen in a strong magnetic field The dotted figureon the left is Landau’s old upper limit. On the right-hand side our curve is compared withthe accurate values (dots [57, 58]). It also shows various lower-order approximations withinour procedure. The quantity ε(B) is the binding energy defined by ε(B) ≡ B/2 − E(B). Allquantities are in atomic natural units � = 1, M = 1, e = 1, energies in units of 2 Ryd= e4M2/�3.
the Hamiltonian of an anharmonic oscillator
H =p2
2M+M
2ω2x2 +
g
4x4 (137)
alternatively as
H =p2
2M+M
Ω2
2x2 + δ
[M
2
(ω2 − Ω2
)+g
4x4], (138)
and expand the eigenvalues systematically in powers of δ. Each partial sum of order L is
evaluated at δ = 1 and extremized in Ω. It is obvious that this procedure is equivalent
the re-expansion method in Section 2..
As mentioned in the text and pointed out in [16], such an analysis is inapplicable in
quantum field theory, where the Wegner exponent ω is anomalous and must be determined
dynamically. Most recently, the false treatment was given to the shift of the critical
temperature in a Bose-Einstein condensate caused by a small interaction [29, 41, 50]. We
have seen in Section 4. that the perturbation expansion for this quantity is a function
of g/μ where μ is the chemical potential which goes to zero at the critical point, we are
faced with a typical strong-coupling problem of critical phenomena. In order to justify the
application of the δ-expansion to this problem, BR [50] studied the convergence properties
of the method by applying it to a certain amplitude Δ(g) of an O(N)-symmetric φ4-field
theory in the limit of large N , where the model is exactly solvable.
Their procedure must be criticized in two ways. First, the amplitude Δ(g) they
considered is not a good candidate for a resummation by a δ-expansion since it does not
possess the characteristic strong-coupling power structure [15] of quantum mechanics and
field theory, which the final resummed expression will always have by construction. The
power structure is disturbed by additional logarithmic terms. Second, the δ-expansion
is, in the example, equivalent to choosing, on dimensional grounds, the exponent ω = 2
in [15], which is far from the correct value ≈ 0.843 to be derived below. Thus the δ-
expansion is inapplicable, and this explains the problems into which BR run in their
We now explain the second criticism. Suppose we ignore the just-demonstrated fun-
damental obstacle and follow the rules of the δ-expansion, defining the Lth order ap-
proximant Δ(δ,∞) by expanding (139) in powers of δ up to order δL, setting δ = 1, and
defining z ≡ g. Then we obtain the Lth variational expression for b0:
b(L)0 (ω, z) =
L∑l=1
alzl
(L− l + l/ω
L− l
), (144)
with ω = 2, to be optimized in z. This ω-value would only be adequate if the approach
to the strong-coupling limit behaved like A+B/h2 + . . . , rather than (143). This is the
reason why BR find no real regime of minimal sensitivity on z.
0.5 1 1.5 2.5-0.5
0
0.5
1
Fig. 25 Plot of 1 − b(L)0 (ω, z) versus z for L = 10 and ω = 0.6, 0.843, 1, 2 . The curve
with ω = 0.6 shows oscillations. They decrease with increasing ω and becomes flat at aboutω = 0.843. Further increase of ω tilts the plateau and shows no regime of minimal sensitivity.At the same time, the minimum of the curve rises rapidly above the correct value of 1− b0 = 0,as can be seen from the upper two curves for ω = 1 and ω = 2, respectively.
Let us attempt to improve the situation by determining ω dynamically by making the
plateau in the plots of Δ(L)(ω, h) versus h horizontal for several different ω-values. The
result is ω ≈ 0.843, quite far from the naive value 2. This value can also be estimated
by inspecting plots of Δ(L)(ω, h) versus h for several different ω-values in Fig. 25, and
selecting the one producing minimal sensitivity.
It produces reasonable results also in higher orders, as is seen in Fig. 26. The
approximations appear to converge rapidly. But the limit does not coincide with the
known exact value, although it happens to lie numerically quite close. Extrapolating the
successive approximations by an extremely accurate fit to the analytically known large-
order behavior [15] with a function b(L)0,plateau(ω = 0.843) = A+B L−κ, we find convergence
to A = 1 − 0.001136, which misses the correct limit A = 1. The other two parameters
are fitted best by B = −0.002495 and κ = 0.922347 (see Fig. 27).
We may easily convince ourselves by numerical analysis that the error in the limiting
value is indeed linked to the failure of the strong-coupling behavior (143) to have the power
structure of [15]. For this purpose we change the function f(x) in equation (140) slightly
into f(x)→ f(x) = f(x) + 1, which makes the integrals for bm in (141) convergent. The
exact limiting value 1 of Δ remaines unchanged, but b(L)0 acquires now the correct strong-
coupling power structure of [15]. For this reason, we can easily verify that the application
Fig. 26 Left-hand column shows plots of 1 − b(L)0 (ω, z) for L = 10, 17, 24, 31, 38, 45 with
ω = 2 of δ-expansion of BR, right-hand column with optimal ω = 0.843. The lower row enlargesthe interesting plateau regions of the plots above. Only the right-hand side shows minimalsensitivity, and the associated plateau lies closer to the correct value 1− b0 = 0 than the minimain the left column by two orders of magnitude. Still the right-hand curves do not approach theexact limit for L→∞ due to the wrong strong-coupling behavior of the initial function.
20 40 60
-0.00125
-0.00126
-0.00127
Fig. 27 Deviation of 1 − b(L)0,plateau(ω = 0.843) from zero as a function of the order L. Asymp-
totically the value −.001136 is reached, missing the correct number by about 0.1%.
of variational theory with a dynamical determination of ω yields the correct strong-
coupling limit 1 with the exponentially fast convergence of the successive approximations
for L→∞ like b(L)0 ≈ 1− exp (−1.909− 1.168 L).
It is worthwhile emphasizing that an escape to complex zeros which BR propose to
remedy the problems of the δ-expansion is really of no help. It has been claimed [53] and
repeatedly cited [49], that the study of the anharmonic oscillator in quantum mechanics
suggests the use of complex extrema to optimize the δ-expansion. In particular, the
use of so-called families of optimal candidates for the variational parameter z has been
suggested. We are now going to show, that following these suggestions one obtains bad
resummation results for the anharmonic oscillator. Thus we expect such procedures to
lead to even worse results in field-theoretic applications.
In quantum mechanical applications there are no anomalous dimensions in the strong-
coupling behavior of the energy eigenvalues. The growth parameters α and ω can be
directly read off from the Schrodinger equation; they are α = 1/3 and ω = 2/3 for the
anharmonic oscillator (see Appendix A). The variational perturbation theory is applicable
for all couplings strengths g as long as b(L)0 (z) becomes stationary for a certain value of
z. For higher orders L it must exhibit a well-developed plateau. Within the range of the
plateau, various derivatives of b(L)0 (z) with respect to z will vanish. In addition there will
be complex zeros with small imaginary parts clustering around the plateau. They are,
however, of limited use for designing an automatized computer program for localizing the
position of the plateau. The study of several examples shows that plotting b(L)0 (z) for
various values of α and ω and judging visually the plateau is by far the safest method,
showing immediately which values of α and ω lead to a well-shaped plateau.
Let us review briefly the properties of the results obtained from real and complex
zeros of ∂zb(L)0 (z) for the anharmonic oscillator. In Fig. 28, the logarithmic error of b
(L)0
is plotted versus the order L. At each order, all zeros of the first derivative are exploited.
To test the rule suggested in [53], only the real parts of the complex roots have been used
to evaluate b(L)0 . The fat points represent the results of real zeros, the thin points stem
from the real parts of complex zeros. It is readily seen that the real zeros give the better
result. Only by chance may a complex zero yield a smaller error. Unfortunately, there is
no rule to detect these accidental events. Most complex zeros produce large errors.
0 20 40 60 80-40
-30
-20
-10
0
Fig. 28 Logarithmic error of the leading strong-coupling coefficient b(L)0 of the ground state
energy of the anharmonic oscillator with x4 potential. The errors are plotted over the order Lof the variational perturbation expansion. At each order, all zeros of the first derivative have
been exploited. Only the real parts of the complex roots have been used to evaluate b(L)0 . The
fat points show results from real zeros, the smaller points those from complex zeros, size isdecreasing with distance from real axis.
We observe the existence of families described in detail in the textbook [6] and redis-
covered in Ref. [53]. These families start at about N = 6, 15, 30, 53, respectively. But
each family fails to converge to the correct result. Only a sequence of selected members in
each family leads to an exponential convergence. Consecutive families alternate around
the correct result, as can be seen more clearly in a plot of the deviations of b(L)0 from their
L → ∞ -limit in Fig. 29, where values derived from the zeros of the second derivative
of b(L)0 have been included.These give rise to accompanying families of similar behavior,
Fig. 29 Deviation of the coefficient b(L)0 from the exact value is shown as a function of pertur-
bative order L on a linear scale. As before, fat dots represent real zeros. In addition to Fig. 28,
the results obtained from zeros of the second derivative of b(L)0 are shown. They give rise to own
families with smaller errors by about 30%. At N = 6, the upper left plot shows the start of two
families belonging to the first and second derivative of b(L)0 , respectively. The deviations of both
families are negative. On the upper right-hand figure, an enlargement visualizes the next twofamilies starting at N = 15. Their deviations are positive. The bottom row shows two moreenlargements of families starting at N = 30 and N = 53, respectively. The deviations alternateagain in sign.
deviating with the same sign pattern from the exact result, but lying closer to the correct
result by about 30.
9. Appendix B: Ground-State Energy from Imaginary Part
We determine the ground state energy function E0(g) for the anharmonic oscillator on
the cut, i.e. for g < 0 in the bubble region, from the weak coupling coefficients al of
equation (97). The behavior of the al for large l can be cast into the form
al/al−1 = −L∑
j=−1βj l
−j . (145)
The βj can be determined by a high precision fit to the data in the large l region of250 < l < 300 to be
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[6] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, andFinancial Markets, World Scientific, Singapore, 2009(http://www.physik.fu-berlin.de/~kleinert/b5).
[7] H. Kleinert, Annals of Physics 266, 135 (1998)(http://physik.fu-berlin.de/~kleinert/255/255.pdf).
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[17] R. Seznec and J. Zinn-Justin, J. Math. Phys. 20, 1398 (1979). This paper hasdeveloped important techniques for understanding the convergence mechanism ofvariational perturbation theory.
[18] H. Kleinert and V. Schulte-Frohlinde, Critical Phenomena in Φ4-Theory, WorldScientific, Singapore, 2001 (http://www.physik.fu-berlin.de/~kleinert/b8).
[19] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon, Oxford,1989.
[20] See Section 19.4 in [18].
[21] F.J. Wegner, Phys. Rev. B 5, 4529 (1972); B 6, 1891 (1972).
[22] B.G. Nickel, D.I. Meiron, and G.B. Baker, Univ. of Guelph preprint 1977(unpublished). The preprint is readable on the WWW athttp://www.physik.fu-berlin.de/~kleinert/nickel/guelph.pdf;The results are cited and used in Chapters 19 and 20. They were axtended to sevenloops byD.B. Murray and B.G. Nickel, Univ. of Guelph preprint 1991.The additional g7 coefficients of the renormalization group functions are listed inSection 20.4 of the textbook [18].
[23] See Section 20.2 in [18].
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[25] M. Holzmann, G. Baym, J.-P. Blaizot and F. Laloe, Phys. Rev. Lett. 87, 120403(2001).
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[31] These are the results of [30]. They differ from those of Refs. [29] which are a3 =0.644519, a41 = 0.87339, a42 = 3.15905, a43 = 1.70959, a44 = 4.4411, a45 = 2.37741. Thecoefficients of the series (57) to be resummed differ mainly in the last term: f−1 =−126.651 10−4, f0 = 0, f1= −4.04857 10−4, f2= 2.40587 10−4, f3= −2.06849 10−4.
[32] H. Kleinert, Strong-Coupling Behavior of Phi4-Theories and Critical Exponents,Phys. Rev. D 57 , 2264 (1998); Addendum: Phys. Rev. D 58 , 107702 (1998) (cond-mat/9803268); Seven Loop Critical Exponents from Strong-Coupling φ4-Theory inThree Dimensions , Phys. Rev. D 60 , 085001 (1999) (hep-th/9812197); Theory andSatellite Experiment on Critical Exponent alpha of Specific Heat in Superfluid HeliumPhys. Lett. A 277, 205 (2000) (cond-mat/9906107).
[33] H. Kleinert, Strong-Coupling φ4-Theory in 4− ε Dimensions, and Critical Exponent ,Phys. Lett. B 434 , 74 (1998) (cond-mat/9801167); Critical Exponents without beta-Function, Phys. Lett. B 463, 69 (1999) (cond-mat/9906359).
[34] See [29, 30, 41] and references cited there.
[35] With standard normalization conditions used in the 3-dimensional φ4-theory, theapproach to scaling is governed by Wegner’s exponent ω (see [32]). The presentdefinition of m differs from the inverse correlation length m = ξ−1 by a factor:m = mZ−1φ ∝ mm−η/2 for m → 0. This changes the exponent of approach toω′ = ω/(1− η/2). I thank B. Kastening for noting this.
[36] P. Arnold and G. Moore, Phys. Rev. Lett. 87, 120401 (2001); Phys. Rev. E 64,066113 (2001). The authors derive a 1/N correction factor (1 − 0.527/N) to theleading N →∞ result.
[37] V.A. Kashurnikov, N.V. Prokof’ev and B.V. Svistunov, Phys. Rev. Lett. 87, 120402(2001).
[38] G. Baym, J.-P. Blaizot M. Holzmann, F. Laloe and D. Vautherin, Phys. Rev. Lett.83, 1703 (1999).
[39] G. Baym, J.-P. Blaizot and J. Zinn-Justin, Europhys. Lett. 49, 150 (2000).
[40] P. Arnold and B. Tomasik, Phys. Rev. A62, 063604 (2000).This paper starts out from the 3+1-dimensional initial theory and derives from it thethree-dimensional effective classical field theory, the field-theoretic generalization ofthe quantum-mechanical effective classical potential ofR.P Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986). This reduction programwas started for the Bose-Einstein gas byA.M.J. Schakel, Int. J. Mod. Phys. B 8, 2021 (1994); J. Mod. Phys. B 8, 2021(1994); Boulevard of Broken Symmetries , Habilitationsschrift, FU-Berlin, (cond-mat/9805152) (1998).Unfortunately, Schakel did not go beyond the one-loop level so that he was happyto have found a positive shift ΔTc/Tc, and did see the cancellation at the two-looplevel. See his recent paper in J. Phys. Stud. 7, 140 (2003) (cond-mat/0301050).
[41] F.F. de Souza Cruz, M.B. Pinto and R.O. Ramos, Phys. Rev. B 64, 014515 (2001).
[42] P. Grueter, D. Ceperley, F. Laloe, Phys. Rev. Lett. 79, 3549 (1997) (cond-mat/9707028).
[43] See Eq. (20.23) in the textbook [18] or S.E. Derkachov, J.A. Gracey, and A.N.Manashov, Eur. Phys. J. C 2, 569 (1998) (hep-ph/9705268).
[44] W. Janke and H. Kleinert, Phys. Lett. A 117, 353 (1986)(http://www.physik.fu-berlin.de/~kleinert/133); Phys. Rev. Lett. 58, 144(1986). H. Kleinert, Phys. Lett. A 257, 269 (1999) (cond-mat/9811308); M.Bachmann, H. Kleinert, A. Pelster, Phys. Lett. A 261, 127 (1999) (cond-mat/9905397); Physical Review E 63, 051709/1-10 (2001) (cond-mat/0011281); seealso B. Kastening, Phys.Rev. A 68, 061601 (2003) (cond-mat/0303486); Phys.Rev.A 69, 043613 (2004) (cond-mat/0309060); Phys. Rev. E 73, 011101 (2006) (cond-mat/0508614); Phys. Rev. A 70, 043621 (2004) (cond-mat/0406035).
[45] J.S. Langer, Ann. Phys. 41, 108 (1967).
[46] C.M. Bender and T.T. Wu, Phys. Rev. 184, 1231 (1969);
[48] B.Bellet, P.Garcia, A.Neveu, Int. J. of Mod. Phys. A 11, 5587(1997)
[49] J.-L. Kneur, D. Reynaud, (hep-th/0205133v2). See also [50], [29], [41].
[50] E. Braaten, E. Radescu, (cond-math/0206186v1).
[51] The low-order results were first obtained byH. Kleinert, Phys. Lett. B 300, 261 (1993)(http://www.physik.fu-berlin.de/~kleinert/214),and extended by R. Karrlein and H. Kleinert, Phys. Lett. A 187, 133 (1994) (hep-th/9504048).
[52] J. Zinn-Justin, J. Math Phys. 22(3), 511 (1981). The first 10 coefficients of expansion(99) are calculated.
[53] B.Bellet, P.Garcia, A.Neveu, Int. J. of Mod. Phys. A11, 5587(1997). The familystructure of optimal variational parameters emphasized in this paper was discussedin great detail earlier in Chapter 5 of the textbook [6], but with correct applicationrules.
[54] M. Bachmann, H. Kleinert, and A. Pelster, Phys. Rev. A 62, 52509 (2000) (quant-ph/0005074), Phys. Lett. A 279, 23 (2001) (quant-ph/000510).
[55] J.E. Avron, B.G. Adams, J. Cızek, M. Clay, M.L. Glasser, P. Otto, J. Paldus, andE. Vrscay, Phys. Rev. Lett. 43, 691 (1979).
[57] J.E. Avron, I.W. Herbst, B. Simon, Phys. Rev. A 20, 2287 (1979). See also theresummation treatment in J.-C. Le Guillou, J. Zinn-Justin, Ann. Phys. 147, 57(1983).
[58] H. Ruder, G. Wunner, H. Herold, and F. Geyer, Atoms in Strong Magnetic Fields(Springer-Verlag, Berlin, 1994).
[59] G. Ahlers, Phys. Rev. A 3, 696 (1971); K.H. Mueller, G. Ahlers, F. Pobell, Phys.Rev. B 14, 2096 (1976);
[60] L.S. Goldner, N. Mulders and G. Ahlers, J. Low Temp.Phys. 93 (1992) 131.
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[62] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977); Phys. Rev. B 21,3976 (1980); J. de Phys. Lett 46, L137 (1985).
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[64] D.B. Murray and B.G. Nickel, unpublished.
[65] A. Pellisetto and E. Vicari, preprint IFUP-TH 52/97, cond-mat/9711078.
[66] F. Jasch and H. Kleinert, Berlin preprint 199 (cond-mat/9907214).
[67] W. Janke, Phys.Lett. A148 (1990) 306.
[68] H.G. Ballesteros, L.A. Fernandez, V. Martin-Mayor and A. Munoz Sudupe, Phys.Lett. B 387, 125 (1996).
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Then we can extremize A1Δ,ρ with respect to Δ and ρ, and find that, to this order, the field
expectation values (17) are given by the free-field propagators (21) at equal arguments:
Δx = g[GΔ]x,x, ρx = g[Gρ]x,x. (22)
Thus we see that at the extremum, the action A1Δ,ρ is the same as the extremal action
A1[Δ, ρ] = A0[Δ, ρ]−1
g
∫x
(|Δ|2 + ρ2). (23)
Note how the theory differs, at this level, from the collective quantum field theory derived
via the HST. If we assume that ρ vanishes identically, the extremum of the one-loop action
A1[Δ, ρ] gives the same result as of the mean-field collective quantum field action (6),
which reads for the present δ-function attraction A1[Δ] = A0[Δ]− 1g
∫x|Δ|2. On the other
hand, if we extremize the action A1Δ,ρ at Δ = 0, we find the extremum from the expression
A1[Δ, ρ] = A0[Δ, ρ]− 1g
∫xρ2. The extremum of the first-order collective classical action
(23) agrees with the good-old Hartree-Fock-Bogolioubov theory.
The essential difference between this and the new approach arises in two ways:
• First when it is carried to higher orders. In the collective quantum field theory
based on the HST the higher-order diagrams must be calculated with the help of
the propagators of the collective field such as 〈ΔxΔx′〉. These are extremely compli-cated functions. For this reason, any loop diagram formed with them is practically
impossible to integrate. In contrast to that, the higher-order diagrams in the present
theory need to be calulated using only ordinary particle propagators GΔ and Gρ of
Eq. (21) and the interaction (12). Even that becomes, of course, tedious for higher
orders in g. At least, there is a simple rule to find the contributions of the quadratic
terms 12
∫xfTxMxfx in (11), given the diagrams without these terms. One calculates
the diagrams from only the four-particle interaction, and collects the contributions
up to order gN in an effective action AN [Δ, ρ]. Then one replaces AN [Δ, ρ] by
AN [Δ− εgΔ, ρ− εgρ] and re-expands everything in powers of g up to the order gN ,forming a new series
∑Ni=0 g
iAi[Δ, ρ]. Finally one sets ε equal to 1/g [15] and obtains
the desired collective classical action AN [Δ, ρ] as an expansion extending (23):
AN [Δ, ρ] =N∑i=0
Ai[Δ, ρ]− (1/g)
∫x
(|Δ|2 + ρ2). (24)
Note that this action must merely be extremized. There are no more quantum
fluctuations in the classical collective fields Δ, ρ. Thus, at the extremum, the action
(24) is directly the grand-canonical potential.
• The second essential difference with respect to the HST approach is that it is now
possible to study a rich variety of possible competing collective fields without the
danger of double-counting Feynman diagrams. One simply generalizes the matrix
Mx subtracted fromAint and added toAint in (11) in different ways. For instance, we
may subtract and add a vector field ψ†σaψSa containing the Pauli matrices σa and
study paramagnon fluctuations, thus generalizing the assumption (??) and allowing
for a spontaneous magnetization in the ground state. Or one may do the same thing
with a term ψ†σa∇iψAia + c.c. in addition to the previous term, and derive the
Ginzburg-Landau theory of superfluid He3 as in [6].
An important property of the proposed procedure is that it yields good results in
the limit of infinitely strong coupling. It was precisely this property which led to the
successful calculation of critical exponents of all φ4 theories in the textbook [12] since
critical phenomena arise in the limit in which the unrenormalized coupling constant
goes to infinity [18]. This is in contrast to another possibility, in principle, of carrying
the variational approach to higher order via the so-called higher effective actions [19].
There one extremizes the Legendre transforms of the generating functionals of bilocal
correlation functions, which sums up all two-particle irreducible diagrams. That does
not give physically meaningful results [20] in the strong-coupling limit, even for simple
quantum-mechanical models.
6. The mother of this approach, Variational Perturbation Theory [11], is a systematic
extension of a variational method developed some years ago by Feynman and the author
[16]. It converts divergent perturbation expansions of quantum mechanical systems into
exponentially fast converging expansions for all coupling strength [17]. What we have
shown here is that this powerful theory can easily be transferred to many-body theory,
if we identfy a variety of relevant collective classical fields, rather than a fluctuating
collective quantum field suggested by the HST. This allows us to go systematically beyond
the standard Hartree-Fock-Bogoliubov approximation.
Acknowledgement
I am grateful to Flavio Nogueira, Aristieu Lima, and Axel Pelster for intensive discussions.
References
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[2] These identities were first employed in relativistic quantum field theory by P. T.Mathews, A. Salam, Nuovo Cimento 12, 563 (1954), 2, 120 (1955), and later instudies of the large-N limit of various model field theories, such as Gross-Neveu andnonlinear σ models.
[3] H. Kleinert, On the Hadronization of Quark Theories, Lectures presented at the EriceSummer Institute 1976, in Understanding the Fundamental Constituents of Matter,Plenum Press, New York, 1978, A. Zichichi ed., pp. 289-390 (klnrt.de/53/53.pdf).
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[13] For more details see klnrt.de/b8/crit.htm).
[14] Note that the hermitian adjoint Δ∗↑↓ comprises transposition in the spin indices, i.e.,Δ∗↑↓ = [Δ↓↑]
∗.
[15] The alert reader will recognize her the so-called square-root trick of Chapter 5 in thetextbook Ref. [11].
[16] R.P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080 (1986) (klnrt.de/159).
[17] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast ConvergentStrong-Coupling Expansions, Lecture presented at the Summer School on”Approximation and extrapolation of convergent and divergent sequences and series”in Luminy bei Marseille in 2009 (arXiv:1006.2910).
[18] H. Kleinert, Phys. Rev. D 57, 2264 (1998); Phys.Rev. D 60, 085001 (1999). (See alsoklnrt.de/critical).
[19] C. De Dominicis, J. Math. Phys. 3, 938 (1962); C. De Dominicis and P.C. Martin, J.Math. Phys. 5, 16, 31 (1964); J.M. Cornwall, R. Jackiw, and E.T. Tomboulis, Phys.Rev. D 10, 2428 (1974); H. Kleinert, Fortschr. Phys. 30, 187 (1982) (klnrt.de/82);Lett. Nuovo Cimento 31, 521 (1981) (klnrt.de/77).
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where f1 and f1 are arbitrary functions of the new coordinates r′ and t′. At this point,if one wants the “standard Schwarzschild solution”, r and t have to be chosen in a way
that a(r, t) = 0 and k(r, t) = −r2 [42]. In particular, the second condition implies thatthe standard Schwarzschild radial coordinate is determined in a way which guarantees
that the length of the circumference centred in the origin of the coordinate system is 2πr
[42].
In our approach we will suppose again that a(r, t) = 0, but, differently from the
standard analysis, we will assume that the length of the circumference centred in the
origin of the coordinate system is not 2πr. We release an apparent different physical
assumption, i.e. that arches of circumference are deformed by the presence of the mass
of the central body M. Note that this different physical hypothesis permits to circum-
navigate the Birkhoff Theorem [4] which leads to the “standard Schwarzschild solution”
[3]. In fact, the demonstration of the Birkhoff Theorem starts from a line element in
which k(r, t) = −r2 has been chosen, see the discussion in paragraph 32.2 of [1] and, in
particular, look at Eq. (32.2) of such a paragraph.
Then, we proceed assuming k = −mr2, wherem is a generic function to be determined
in order to obtain that the length of circumferences centred in the origin of the coordinate
system are not 2πr. In other words, m represents a measure of the deviation from 2πr
of circumferences centred in the origin of the coordinate system.
The line element (2) becomes
ds2 = hdr2 −mr2(sin2 θdϕ2 + dθ2) + ldt2. (5)
One puts
X ≡ 13r3
Y ≡ − cos θ
Z ≡ ϕ.
(6)
In the X, Y, Z coordinates the line-element (5) reads
Thus, the match works for all time during the collapse if and only if
R =(a30 sin
3 χ0 − r3g) 1
3
rg = a0 sin3 χ0.
(49)
By inserting the first of Eqs. (49) in Eq. (39) one gets
r =1
2
{[(a0 sinχ0)(1 + cos η)]3 − r3g
} 13 . (50)
Eq. (50) represents the run of the collapse for both the external and internal solu-
tions for 0 ≤ η ≤ 2rg
(R3+r3g)13−1
. When η = 2rg
(R3+r3g)13−1
it is r = 0 and particles reach the
Schwarzschild sphere which is the origin of the coordinate system. For η > 2rg
(R3+r3g)13−1
Eq. (50) represents only the trend of the internal solution and the r coordinate becomes
negative (this is possible because the origin of the coordinate system is the surface of the
Schwarzschild sphere). The r coordinate reaches a minimum r = −rg for η = π. Thus,
we understand that at this point the collapse terminates and the star is totally collapsed
in a singularity at r = −rg. In other terms, in the internal geometry all time-like radial
geodesics of the collapsing star terminate after a lapse of finite proper time in the ter-
mination point r = −rg and it is impossible to extend the internal space-time manifold
beyond that termination point. Thus, the point r = −rg represents a singularity basedon the rigorous definition by Schmidt [43].
Clearly, as all the particle of the collapsing star fall in the singularity at r = −rgvalues of r > −rg do not represent the internal geometry after the end of the collapse,
but they will represent the external geometry. This implies that the external solution
(30), i.e. “the original Schwarzschild solution” to Einstein field equations which has been
derived for the first time by Karl Schwarzschild in [3] can be analytically continued for
values of −rg < r ≤ 0 and it results physically equivalent to the solution (1) that is
universally known like the ”Schwarzschild solution”. In fact, now the transformation
(31) can be enabled and the origin of the coordinate system, r = 0, θ = 0, ϕ = 0, which
is the surface of a sphere having radius rg in the r, θ, ϕ coordinates, results transferred
in a non-dimensional material point r = 0, θ = 0, ϕ = 0 in the r, θ, ϕ coordinates. Such
a non-dimensional material point corresponds to the point r = −rg, θ = 0, ϕ = 0 in the
original r, θ, ϕ coordinates.
Then, the authors who claim that “the original Schwarzschild solution” leaves no
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As is noted by Myung [5] an isolated black hole like Schwarzschild black hole is never
in thermal equilibrium because it decays by the Hawking radiation. This can be seen
from the negative value of its heat capacity by doing α = 0 in (30), CS = −2πr2+.In the case of the Horava-Lifshitz black hole, expression (30) shows that the heat
capacity can be negative but also positive, depending on the value of the parameter α.
In Figure 4 is easily seen that the heat capacity have positive values for different values
of α. The value r+ = rm at which the heat capacity blows is given by
rm ==
√5
2
√33√α. (31)
Black holes with r+ < rm are local thermodynamically stable while those with r+ > rmare unstable.
Finally, another interesting question is whether there exists the Hawking-Page phase
transition associated with the Horava-Lifshitz black hole. In order to discuss the Hawking-
Page transition, we have to calculate the Euclidean action or free energy for the black
hole. The Euclidean action is related with the free energy by
I =1
TF, (32)
where T is the temperature of the black hole and the free energy F is given by
F =M − TS. (33)
Using equations (21), (23) and (28), we find
F =r4+ + 7αr2+ + 4α2 + 4α2 ln (r+)− 4r2+α ln (r+)
4r+(r2+ + 2α). (34)
Note that the free energy is negative only for small enough horizon radius, which
means that large black holes in Horava-Lifshitz gravity are thermodynamically unstable
globally.
5. Conclusion
We studied the entropy of black holes in the deformed Horava-Lifshitz gravity with cou-
pling constant λ. It has been shown that in the case λ = 1, the black hole resembles the
Reissner-Nordstrom black hole when it is noted that the geometric parameter α = 12ω
in
the horizon radius assumes a similar role as that the electric charge. The entropy of the
Horava-Lifshitz black hole is calculated by assuming that the first law of thermodynamics
is valid for this geometry. The obtained expression reduces to Schwarzschild’s entropy
in the limit α = 0 but differs for other values. Finally we studied the heat capacity
and Hawking-Page phase transition, to show that Black holes with r+ < rm are glob-
ally thermodynamically stable, while large black holes are thermodynamically unstable
[2] T. Takahashi and J. Soda, arXiv:0904.0554 [hep-th]. G. Calcagni, arXiv:0904.0829.E. Kiritsis and G. Kofinas, arXiv:0904.1334 [hep-th]. S. Mukohyama, arXiv:0904.2190[hep-th]. R. Brandenberger, arXiv:0904.2835 [hep-th].
[3] H. Lu, J. Mei and C. N. Pope, arXiv:0904.1595 [hep-th].
[4] R. G. Cai, L. M. Cao and N. Ohta, arXiv:0904.3670 [hep-th]. arXiv:0905.0751 [hep-th]
[5] Y. Myung. arXiv:0905.0957 [hep-th]; Physics Letters B 684, Issues 2-3, Pages 158-161,(2010)
which is compatible with the knowledge constraints [12]. By varying the Lagrange
multipliers we enforce the two constraints, giving λ and α. Especially one gets: e−1−λ = 1Z
where Z is the partition function on the form:
Z =∑paths
e−αS[path] =∫Dqe−αS[q] (9)
and the parameter α is determined by solving:
〈S〉 =∫DqS[q]p[q] =
1
Z
∫DqS[q]e−αS[q] = − ∂
∂αlogZ. (10)
In order to fit the purpose Lisi concluded that the Lagrange multiplier value; α ≡1i�. Lisi concluded that this multiplier value was an intrinsic quantum variable directly
related to the average path action 〈S〉 of what he called the universal action reservoir.
In similarity with Lisi’s approach we shall also assume that the arbitrary scaling-part of
the constant α is in fact 1/�. Lisi also noted that Planck’s constant in α is analogous
to the thermodynamic temperature of a canonical ensemble, i� ↔ kBT ; being constant
reflects its universal nature - analogous to an isothermal canonical ensemble [12]. This
assumption along with (9) brings us to the following partition function:
Z =∑paths
eiS[path]
� =
∫Dqei
S[q]� . (11)
By inserting (11) into (2) we arrive at the following expectation value for any physical
quantity Q:
〈Q〉 =∑paths
p[path]Q[path] =
∫DqQ[q]p[q] =
1
Z
∫DqQ[q]ei
S[q]� , (12)
This suggests that a consequence of the incomplete information regarding the studied
system is that physics is inevitably based on a probabilistic framework. Conversely, had
physics not been probabilistic in the situation of incomplete information then information
of the system could be inferred. But a process of inferring results from existing limited
information does not provide more information regarding that system than the limited
information had already provided. That would have required, as we previously argued, an
additional principle of perfect information. Instead it is only interaction that can provide
new information. We may conclude that by the principle of information covariance
physics is local and based only on the available information in the local information-based
frame of reference. In turn this this creates an ensemble of possible states with a definite
and assigned expectation value for each physical quantity in the studied system according
to (12). This formalism, which might be called information covariant, is then directly
compatible with the general principle of relativity wherein All systems of reference are
equivalent with respect to the formulation of the fundamental laws of physics.
Fig. 1 This illustration shows the path of a particle from one point to another in a completeinformation frame of reference K (essentially a particle that is observed along its path). It alsoshows some of the possible paths a particle takes in the incomplete information frame of referenceK ′.
3. Connections to Quantum Mechanics
3.1 Path Integral Formulation
The path integral formulation, originally proposed by Dirac but rigorously developed by
Feynman [6], is perhaps the best foundational approach to quantum mechanics avail-
able [12]. It shows that quantum mechanics can be obtained from the following three
postulates assuming a quantum evolution between two fixed endpoints [6]:
1. The probability for an event is given by the squared length of a complex number
called the probability amplitude.
2. The probability amplitude is given by adding together the contributions of all the
histories in configuration space.
3. The contribution of a history to the amplitude is proportional to eiS/�, and can be
set equal to 1 by choice of units, while S is the action of that history, given by the
time integral of the Lagrangian L along the corresponding path.
In order to find the overall probability amplitude for a given process then one adds up
(or integrates) the amplitudes over postulate 3 [6]. In an attempt to link the concept of
information-based frames of reference - developed in this paper - to quantum mechanics
we shall utilize Lisi’s approach wherein the probability for the system to be on a specific
path is evaluated according to the following setup (see [12] for more information). The
probability for the system to be on a specific path in a set of possible paths is:
p(set) =∑paths
δsetpathp[path] =
∫Dqδ(set− q)p[q]. (13)
Here Lisi assumed that the action typically reverses sign under inversion of the pa-
This implies that the probability for the system to pass through configuration q′ atparameter value t′ is:
p(q′, t′) =∫Dqδ(q(t′)− q)p[q] =
(∫ q(t′)=q′
Dqpt′[q]
)(∫q(t′)=q′
Dqpt′ [q]
)= ψ(q′, t′)ψ†(q′, t′),
(15)
in which we can identify the quantum wave function:
ψ =
∫ q(t′)=q′
Dqpt′[q] =
1√Z
∫ q(t′)=q′
Dqe−αSt′=
1√Z
∫ q(t′)=q′
DqeiSt′� . (16)
The quantum wave function ψ(q′, t) defined here is valid for paths t < t′ meeting at q′
while its complex conjugate ψ†(q′, t′) is the amplitude of paths with t > t′ leaving from q′.Multiplied together they bring the probability amplitudes that gives the probability of the
system passing through q′(t′), as is seen in (15). However, just as Lisi points out [12], thisquantum wave function in quantum mechanics is subordinate to the partition function
formulation since it only works when t′ is a physical parameter and the system is t′
symmetric, providing a real partition function Z. Indeed, the postulate of an information
covariant setup on the laws of physics according to the previous section suggests that
physics is ruled by the general complex partition function (9):
Z =∑paths
eiS[path]
� =
∫Dqei
S[q]� . (17)
How does this relate to the path integral formulation? The sum in the partition func-
tion (17) is a sum over paths. Let us take the common situation when the path is that
of a particle between two points. We can then conclude that each term is on the form
eiS[path]/� which is equivalent to postulate 3. Furthermore all paths are added, thus postu-
late 2 is also checked. Also, at least for the situation where p(q′, t′) = ψ(q′, t′)ψ†(q′, t′) thesum adds up to the probability density, checking postulate 1 as well. Thus we may con-
clude that the information covariant approach is equivalent to the canonical path integral
formulation of quantum mechanics under the circumstances provided for it.
3.2 Quantum Properties
The path integral formulation is canonical for quantum mechanics and covers the wide
variety of special features inherent to quantum mechanics [6, 12]. Since the approach in
this paper is equivalent to the path integral formulation in most aspects, some properties
are be worth discussing. A pivotal component of quantum mechanics is the canonical
commutation relation which gives rise to the Heisenberg uncertainty principle [3, 6]. For
Fig. 2 This illustration shows on the left hand side the uncertainty of path of a particle fromone point to another and on the right hand side that the particle takes all possible paths fromone point to another. These two interpretations are equivalent under the general interpretationthat information is incomplete. Uncertainty in path means in practice that it takes any possiblepath until we observe it, a superposition of states is inevitable when information is incomplete.
example the famous commutation relation between position x and momentum p of a
particle is defined as:
[x, p] = i�. (18)
This can be obtained through the path integral formulation by assuming a random
walk of the particle from starting point to end point [6]. This works with this theory as
well under the same considerations since a random walk is equivalent to a walk with no
information about direction. In the path integral formulation it is also possible to show
that for a particle with classical non-relativistic action (where where m is mass and x is
position):
S =
∫mx2
2dt, (19)
that the partition function Z in the path integral formulation turns out to satisfy the
following equation [6]:
i�∂Z
∂t=[− 1
2∇2 + V (x)
]Z. (20)
This is the Schrodinger equation for Z = ψ and where V (x) is a potential [3]. It
is also possible to show the conservation of probability from the Schrodinger equation
(20) [3]. Here we can see that the traditional usage of operators on a Hilbert space in
quantum mechanics is a useful tool when information is incomplete. Another interesting
aspect of quantum mechanics is the superposition principle which states that a particle
occupies all possible quantum states simultaneously [3]. That the dynamics of a system
is fundamentally unknown or occupying all states simultaneously are both parts of the
same concept that information is incomplete regarding the system. The popular quantum
superposition thought experiment Schrodinger’s cat in which the alive/dead state of a
cat in a hazardous closed box is also evidently based on the lack of information regarding
the state of the cat. The superposition is intuitively equivalent to the lack of information.
The resolution of this problem in this theory is that the state of the cat is fundamentally
Together (25), (26), (27) and the fact that the partition function Z is path-independent
brings the following general expression for the entropy of the system:
H = −k( ∑
paths
p[path]iS[path]
�−
∑paths
p[path] logZ
)= −k
(i〈S〉�− logZ
). (28)
Let us now further assume the special case when the following identity holds:
ψ = Z. (29)
This identity holds at least when ψ = ψ(q′, t′) and t′ is a symmetric physical param-eter, just as described in section 3.1. Let us also assume that the structure of the wave
function is as follows:
ψ = ReiSc� (30)
where R = |ψ| and Sc is the classical action [3]. This brings the following expression:
logψ = log |ψ|+ iSc
�. (31)
Together (28) and (31) amounts to the following special case of the entropy:
H = −k(i〈S〉�− i
Sc
�− log |ψ|
). (32)
If we assume the equivalence between the classical action SC and the expected action
〈S〉, which is in accordance with the Ehrenfest theorem [3], then we get the following
expression for entropy:
H = k · log |ψ|. (33)
An expression similar to (33) was suggested as a basis for the holographic approach
to gravity [18] in a somewhat more speculative paper recently [13]. In that approach
the constant was suggested to be k = −2kB, where kB was Boltzmann’s constant. The
expression for entropy (33) is strikingly similar to Boltzmann’s formula for entropy in
thermodynamics:
H = kB · log(W ), (34)
where H is the entropy of an ideal gas for the number W of equiprobable microstates
[14]. The suggested entropy (33) and it’s more general version (28) are, up to a scalable
constant, measures of the lack of information in the information-based frame of reference.
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[6] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, New York,McGraw-Hill (1965).
[7] R.B. Griffiths, Consistent histories and the Interpretation of Quantum Mechanics, J.Stat. Phys. 36 (1984).
[8] E.T. Jaynes, Information Theory and Statistical Mechanics, Phys. Rev. 106: 620(1957).
[9] E.T. Jaynes, Information Theory and Statistical Mechanics II, Phys. Rev. 108: 171(1957).
[10] J-W. Lee, Quantum mechanics emerges from information theory applied to causalhorizons, Found. Phys. DOI 10.1007/s10701-010-9514-3 (2010) : arXiv:1005.2739v2.
[11] J-W. Lee, Physics from information, arXiv:1011.1657v1 [hep-th] (2010).
[12] G. Lisi, Quantum mechanics from a universal action reservoir,arXiv:physics/0605068v1 [physics.pop-ph] (2006).
[13] J.D.Munkhammar, Is Holographic Entropy and Gravity the result of QuantumMechanics?, arXiv:1003.1262 (2010).
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[15] C. Rovelli, Relational Quantum Mechanics, arXiv:quant-ph/9609002 (1997).
[16] C.E. Shannon, The Mathematical Theory of Communication, Univ. Illinois Press,(1949).
[17] J.B.Marion, S.T.Thornton, Classical Dynamics of Particles and Systems, Harcourt(1995).
[18] E.Verlinde, On the Origin of Gravity and the Laws of Newton, arXiv:1001.0785v1[hep-th] (2010).
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On the Logical Origins of Quantum MechanicsDemonstrated By Using Clifford Algebra: A Proofthat Quantum Interference Arises in a CliffordAlgebraic Formulation of Quantum Mechanics
Elio Conte∗
Department of Pharmacology and Human Physiology – TIRES – Center for InnovativeTechnologies for Signal Detection and Processing, University of Bari- Italy;
School of Advanced International Studies for Applied Theoretical and Non LinearMethodologies of Physics, Bari, Italy
Received 6 July 2010, Accepted 10 February 2011, Published 25 May 2011
Abstract: We review a rough scheme of quantum mechanics using the Clifford algebra.
Following the steps previously published in a paper by another author [31], we demonstrate
that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In
1932 J. von Neumann showed that projection operators and, in particular, quantum density
matrices can be interpreted as logical statements. In accord with a previously obtained result
by V. F Orlov , in this paper we invert von Neumann’s result. Instead of constructing logic
from quantum mechanics , we construct quantum mechanics from an extended classical logic. It
follows that the origins of the two most fundamental quantum phenomena , the indeterminism
and the interference of probabilities, lie not in the traditional physics by itself but in the logical
The content of this theorem is thus established: given three abstract basic elements
as defined in (a) and (b), an algebraic structure is established with four generators
(e0, e1, e2, e3).
Of course, as counterpart, the (11) are well known also in quantum mechanics and the
isomorphism with Pauli’s matrices at various orders is well known and discussed in detail
in [34]. Here, they have been derived only on the basis of two algebraic assumptions,
given respectively in (a) and (b).
We may now add some comments to the previous formulation.
Let us attempt to identify the phenomenological counterpart of the algebraic structure
given in (1), (2), and (11) with
e21 = 1, e22 = 1, e23 = 1 (12)
A generic member of our algebra is given by
x =3∑
i=0
xiei (13)
with xi pertaining to some field �or C. The (12) evidences that the ei are abstract
potential entities, having the potentiality that we may attribute them the numerical
values, or ±1. Admitting to be p1(+1)the probability to attribute the value (+1)to e1 andp1(−1) the probability to attribute (−1), considering the same corresponding notationfor the two remaining basic elements, we may introduce the following mean values:
B, and it may be either reflected and detected by the detector D1 or transmitted and
detected by the detector D2.The particles arriving from path L2, impinge on the opposite
side of to be either transmitted reaching the detector D1 or reflected to reach the counter
D2. As it is well known we are considering here the interference pattern of a beam of
particles passing through a Mach Zender interferometer.
The considered random variable A assumes the value a = +1in the case of reflection
and the value a = −1 in the case of transmission. The random variable Bassumes the
value b = +1in the case of reflection and the value b = −1 in the case of transmission.We have a third variable C = ABthat is determined by the product of the values of A
and B.
In analogy with the rough quantum scheme previously developed we call still write
the mean value of A by < A > and
< A >= (a = +1)pab + (a = −1)pab (39)
the mean value of Bby < B > and
< B >= (b = +1)pab + (b = −1)pab (40)
and the mean value of C by < C > and
< C >= (ab; a = +1, b = +1)pab + (ab; a = +1, b = −1)pab+(ab; a = −1, b = +1)pab + (ab; a = −1, b = −1)pab (41)
Let us follow directly the argument as it was recently developed in [31]. According to
this interesting paper , we may write easily the expression of the probability for the
corresponding four alternatives (a = ±1, b = ±1) in the following manner
author in [31] concluded that typical objects of the required kind are Hermitean matrices
with eigenvalues (±1).We may now take a step on.
Rather recently [32] we gave proof of two theorems on existing two Clifford algebras,
the A(Si) that has isomorphism with that one of Pauli matrices, and the Ni,±1 whereNi stands for the dihedral Clifford algebra. The salient feature was that by using such
two theorems, we showed that the Ni,±1 algebra may be obtained from the A(Si) algebra
when we attribute a numerical value (+1 or –1) to one of the basic elements (e1, e2, e3) of
the A(Si). The arising physical model was that the A(Si)−Clifford algebra refers to therepresentation of the general situation in quantum mechanics where the observer has no
right to decide on the state of a two-state system while instead, through the operation
represented by Ni,±1 algebra, he finally specifies which state is the one that will be or
is being observed. The A(Si)− algebra has as counterpart the description of quantum
systems that in standard quantum mechanics are considered in absence of observation
and quantum measurement while the Ni,±1 attend when a quantum measurement is
performed on such system with advent of wave function collapse. There is another salient
feature that needs to be outlined here. As said, under a Clifford algebraic profile, the
quantum measurement with wave function collapse induces the passage in the considered
quantum system from the A(Si) to Ni,+1or to the Ni,−1 algebras: it is of interest froma mathematical and physical view points to observe that in the passage from A(Si) to
N1,±1, each N1,±1 algebra has now its proper rules of commutation that are new and
different respect to standard ones calculated in A(Si). Under the profile of a quantum
measurement, wave function collapse is thus characterized, at least from an algebraic
view point, just from such transition from standard to new commutation rules for the
basic algebraic elements. This is an important feature that deserves careful physical
consideration.
In [32] we re-examined also the well known von Neumann postulate on quantum
measurement, and we gave a proper justification of such postulate by using such two
theorems. In detail, we studied some application of the above mentioned theorems to
some cases of interest in standard quantum mechanics, analyzing in particular a two
state quantum system, the case of time dependent interaction of such system with a
measuring apparatus and finally the case of a quantum system plus measuring apparatus
developed at the order n=4 of the considered Clifford algebras and of the corresponding
density matrix in standard quantum mechanics. In each of such cases, we found that
the passage from the algebra A(Si) to Ni,±1 actually describes the collapse of the wavefunction. We concluded that the actual quantum measurement has as counterpart in the
Clifford algebraic description, the passage from the A(Si) to the Ni,±1, reaching in thismanner the objective to reformulate von Neumann postulate on quantum measurement
and proposing at the same time a self-consistent formulation of quantum theory.
The aim of the present paper has been to propose a step on.
As it is well known, quantum mechanics runs about two basic foundations that are
the indeterminism and the quantum interference. It is also well known that in 1932 J.
von Neumann [34] gave proof that the projection operators and, in particular, quantum
density matrices represent logical statements. We may say that he constructed a matrix
logic on the basis of quantum mechanics. In the present paper we have re-constructed
the two basic foundations of quantum mechanics starting from logic and thus arriving
to explain that quantum mechanics has logical origins. Not logic deriving from quantum
mechanics as in von Neumann but quantum mechanics having logical origins. This is
to say that the two basic foundations of quantum mechanics, the indeterminism and the
quantum interference, may be explained on a purely logical basis. In the development
of the paper we have used our Clifford rough scheme of quantum mechanics including
the theorems shown in [32]. No element of physics has been evoked by us but only
the idempotents of Clifford algebra , given in the (22) once again one has admitted the
necessary existence of the Clifford basic elements given in the (72). Such idempotents
represent of course in quantum mechanics the projection operators that were introduced
by von Neumann as logical statements. Therefore , a conclusion seems to be unavoidable.
We have to consider the basic foundations of quantum mechanics as basic framework
representing conceptual entities [33].
Acknowledgment
The author is indebted with the friend and colleague Alessandro Giuliani (Istituto Su-
periore di Sanita – Rome) for the continuous and stimulating discussions held during
the elaboration of the present paper about the fundamental theme on the possibility to
represent cognitive processes of mind by the classical and quantum profiles of the physics.
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Recently, Bali et al. [53-57], Pradhan et al. [58-60], Yadav et al. [61] and Pradhan [62]
have investigated Bianchi type I, II, III, V, IX and cylindrically symmetric magnetized
string cosmological models in presence and absence of magnetic field. Pradhan and
Bali [50] have investigated some solutions for Bianchi type V I0 cosmology in presence
and absence of magnetic field. In this paper we have derived some Bianchi type V I0string cosmological models for bulk viscous fluid distribution in presence and absence of
magnetic field and discussed the variation of Λ with time. This paper is organized as
follows: The metric and field equations are presented in Section 2. In Section 3, we deal
with the solution of the field equations in presence of magnetic field. In Section 4, we
have described the solution of the field equations in presence of bulk viscous fluid and
some geometric and physical behaviour of the model. Section 5 includes the solution in
absence of magnetic field. In Section 6, we hav discussed the bulk viscous solution of the
field equations in absence of magnetic field. In the last Section 7, concluding remarks are
given.
2. The Metric and Field Equations
We consider the Bianchi Type V I0 metric in the form
Thus the energy conditions ρ ≥ 0, ρp ≥ 0 are satisfied under conditions given by (53)
and (54).
The model (36) starts with a big bang at t = 0. The expansion in the model decreases
as time increases. The proper volume of the model increases as time increases. Sinceσθ= constant, hence the model does not approach isotropy. Since ρ, λ, θ, σ tend to
infinity and V 3 → 0 at initial epoch t = 0, therefore, the model (36) for massive string
in presence of magnetic field has Line-singularity (Banerjee et al. [47]). For the condi-
tion coth2 (2√kt) < 3, the solution gives accelerating model of the universe. It can be
easily seen that when coth2 (2√kt) > 3, our solution represents decelerating model of the
universe.
5. Solutions in Absence of Magnetic Field
In absence of magnetic field, i.e. when b→ 0 i.e. K → 0, we obtain
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metric may be so chosen that the following relations hold
φξ = 0, η (ξ) = 1, η ◦ φ = 0, (3)
φ2X = −X + η (X) ξ, (4)
g (X, ξ) = η (X) , (5)
g (φX, φY ) = g (X, Y )− η (X) η (Y ) , (6)
for any vector fields X, Y . IfM is a Sasakian manifold, then besides (3)-(6), the following
relations hold
∇Xξ = −φX, (∇Xη)Y = g (X,φY ) , (7)
Φ (X, Y ) = (∇Xη)Y, (8)
Φ (X, Y ) = −Φ (Y,X) , (9)
Φ (X, Y ) = 0, (10)
R (X, Y ) ξ = η (Y )X − η (X)Y, (11)
R (ξ,X)Y = (∇Xφ)Y, (12)
S (X, ξ) = (n− 1) η (X) . (13)
Lemma 3.1. A 3-dimensional Sasakian manifold M is locally φ-quasiconformally
symmetric if and only if the scalar curvature r is constant.
3. Biharmonic Curves in Locally φ-Quasiconformally Symmet-
ric Sasakian Manifold M
Let us consider biharmonicity of curves in 3-dimensional locally φ-quasiconformally sym-
metric Sasakian manifold. Let (T,N,B) be the Frenet frame field along γ . Then, the
Frenet frame satisfies the following Frenet–Serret equations:
∇TT = κN,
∇TN = −κT+ τB,
∇TB = −τN,
where κ = |T (γ)| = |∇TT| is the geodesic curvature of γ and τ its geodesic torsion.A helix is a curve with constant geodesic curvature and geodesic torsion. In particular,
curves with constant nonzero geodesic curvature and zero geodesic torsion are called
(Riemannian) circles. Note that geodesics are regarded as helices with zero geodesics
curvature and torsion.
Biharmonic equation for the curve γ reduces to
∇3TT−R (T,∇TT)T = 0, (14)
that is, γ is called a biharmonic curve if it is a solution of the equation (14).
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Keywords: Particle Mass Generation; Physical Vacuum; Standard Model without Higgs Sector;
Self-Energy, Polarization Tensor
PACS (2010): 12.60.Fr; 14.80.Bn; 12.15.Ff
1. Introduction
The Standard Model (SM) is a gauge theory based on the SU(3)C × SU(2)L × U(1)Ygauge group. In this model particles acquire masses by means of implementation of the
electroweak symmetry spontaneous breaking using Higgs mechanism. This mechanism
is based on the fact that the potential must be such that one of neutral components
of the Higgs field doublet acquires spontaneously a non-vanishing vacuum expectation
value. Since the vacuum expectation value of the Higgs field is different from zero, the
Higgs field vacuum can be interpreted as a medium with a net weak charge. On this
way the SU(3)C × SU(2)L × U(1)Y gauge symmetry is spontaneously broken into the
It is straight to show that if we take central experimental values for the strong
constant at the MZ scale as αs(MZ) = 0.1184, the fine-structure constant as αe =
7.2973525376×10−3 and the cosine of the electroweak mixing angle as cos θw =MW/MZ =
80.399/91.1876 = 0.88168786 [17], then gs = 1.21978, gw = 0.641799 and ge = 0.343457.
Substituting the values of gs, gw and ge and the values for the experimental masses of
the electrically charged fermions, given by [17] mu = 0.0025 GeV, md = 0.00495 GeV,
mc = 1.27 GeV, ms = 0.101 GeV, mt = 172.0 ± 2.2 GeV, mb = 4.19 GeV, me =
0.510998910 × 10−3 GeV, mμ = 0.105658367 GeV, mτ = 1.77682 GeV, into the expres-
sions (33) and (34), and assuming neutrinos as massless particles, mνe = mνμ = mντ = 0,
we obtain that theoretical masses of the W and Z electroweak gauge bosons are given by
M thW± = 79.9344± 1.0208GeV (42)
M thZ = 90.6606± 1.1587GeV . (43)
These theoretical masses are in agreement with theirs experimental values given by
M expW = 80.399 ± 0.023 GeV and M exp
Z = 91.1876 ± 0.0021 GeV [17]. Central values
for parameters A1, A2, A3 and A4 in expressions (33) and (34) are A1 = 1.32427× 10−5,A2 = 15478, A3 = 34.0137 and A4 = 0. We observe that A2 is very large respect to
A3 and A1. Taking into account the definition of parameter A2 given by (36) we can
conclude that masses of electroweak gauge bosons coming specially from top quark mass
mt and strong running coupling constant gs. Notwithstanding neutrino masses are not
known, direct experimental results show that neutrino masses are of order 1 eV [17], and
cosmological interpretations of five-year WMAP observations find a limit on the total
mass of neutrinos of Σmν < 0.6 eV (95% CL) [18]. These results assure us that values
of left-handed lepton chemical potentials obtained of taking neutrinos to be massless will
change a little if we take true small neutrinos masses.
We have presented an approach of mass generation for Standard Model particles in which
we have extracted some generic features of the Higgs mechanism that do not depend on
its interpretation in terms of a Higgs field. On this approach the physical vacuum has
been assumed to be a medium at zero temperature which is formed by fermions and an-
tifermions interacting among themselves by exchanging gauge bosons. The fundamental
effective model describing the dynamics of this physical vacuum is the SMWHS. We have
assumed that every fermion flavor in physical vacuum has associated a chemical potential
μf in such a way that there is an excess of antifermions over fermions. This fact implies
that physical vacuum can be understood as a virtual medium having an antimatter finite
density.
Fermion masses are calculated starting from fermion self-energy which represents fun-
damental interactions of a fermion with the physical vacuum. The gauge boson masses
are calculated from the charge fluctuations of physical vacuum which are described by
a vacuum polarization tensor. Using this approach for particle mass generation we have
generated masses for the electroweak gauge bosons in agreement with their experimental
values.
A further result of this approach is that left-handed neutrinos are massive due to that
they have weak charge. Additionally our approach has established a strong restriction to
the existence of a new fermion family in the SMWHS. We have also predicted that top
quark mass is mtht = 173.0015± 0.6760 GeV. Finally we have obtained the highest limit
for the summing of squares of neutrino masses given by Σm2ν < 0.06213 GeV2
Acknowledgments
We thank Vicerrectoria de Investigaciones of Universidad Nacional de Colombia by the
financial support received through the research grant ”Teorıa de Campos Cuanticos apli-
cada a sistemas de la Fısica de Partıculas, de la Fısica de la Materia Condensada y a la
descripcion de propiedades del grafeno”. C. Quimbay thanks to Rafael Hurtado, Rodolfo
Dıaz and Antonio Sanchez for stimulating discussions.
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the superposition principle, and the density of probability ρD(x, p) in a bounded domain
of the phase space (x, p) ∈ D ⊂ R2n, corresponding to the wave function ϕ(x, p), is given
by the standard formula
ρD(x, p) = |ϕ(x, p)|2/∫D
|ϕ(x, p)|2dxdp. (1)
In quantum mechanics, the time evolution of the wave function can be defined by the
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t = t0 a wave function ϕ(x0, p0, t0) is given, then the value of the wave function at the point
(x, p) at the moment t = t1 is defined by the integral over all paths {x(t), p(t)} joiningthe points (x0, p0, t0) and (x, p, t1), of the quantity exp
(− i
�
∫ t1t0[V (x(t))− p2(t)/(2m)]dt
),
where � is the Planck constant, with respect to certain “measure” on paths defined by
Feynman.
In contrast to Feynman’s assumption, in the present paper we study the model in
which the Feynman measure on paths is replaced by the probability measure of the
diffusion process (the heat Brownian motion) given by the Kramers equation [4], [5]:
∂f
∂t=
n∑j=1
(∂V
∂xj
∂f
∂pj− pjm
∂f
∂xj
)+ γ
n∑j=1
∂
∂pj
(pjf + kTm
∂f
∂pj
), (2)
where f(x, p, t) is the density of probability distribution of the particle in the phase space
at the moment of time t; m is the mass of the particle; V (x) is the potential function
of the external forces acting on the particle; γ = β/m is the resistance coefficient of
the medium per unit of mass; k is the Boltzmann constant; T is the temperature of the
medium.
This is the classical Kramers equation describing the diffusion motion of a particle in
the phase space under action of external forces defined by the potential function V (x),
the heat medium with temperature T , and the medium resistance per unit of mass γ.
Consider the following modified Kramers equation for the wave function ϕ(x, p, t):
∂ϕ
∂t= Aϕ+ γBϕ, (3)
where Aϕ =n∑
j=1
(∂V
∂xj
∂ϕ
∂pj− pjm
∂ϕ
∂xj
)− i
�
(mc2 + V −
n∑j=1
p2j2m
)ϕ (4)
Bϕ =n∑
j=1
∂
∂pj
((pj + i�
∂
∂xj
)ϕ+ kTm
∂ϕ
∂pj
).
Equation (3) is obtained from the Kramers equation (2) by adding to the right hand side
the summand of the form −i/�(mc2 + V − p2/(2m))ϕ and by replacing multiplication of
the function ϕ by pj in the diffusion operator by the action of the operator (pj+ i�∂/∂xj)
on the function ϕ.
Adding the summand−i/�(mc2+V−p2/(2m))ϕ is related with the additional physicalrequirement that the wave function at the point (x, p) oscillates harmonically with the
The operator of the right hand side of this equation is well known (see, for example,
[6]). This operator has a full set of eigenfunctions in the space of functions tending to
zero as |p′| tends to infinity. The eigenvalues of this operator are nonpositive integers.The eigenfunctions corresponding to the eigenvalue 0 have the form
ϕ0(s′, p′) =
1
(2π)n/2ψ(s′)e−
(p′−s′)22 ,
where ψ(s′) is an arbitrary complex valued function of s′ ∈ Rn .
The rest of eigenfunctions are obtained as derivatives of the functions ϕ0(s′, p′) with
respect to p′, and have eigenvalues −1,−2, ..., respectively, depending on the degree of
derivative, and the projector P0 to the subspace of eigenfunctions with eigenvalue 0 has
the form
ϕ0(s′, p′) = P0ϕ =
1
(2π)n/2ψ(s′)e−
(p′−s′)22 , where ψ(s′) =
∫Rn
ϕ(s′, p′)dp′. (12)
Hence, considering equation (11) in the basis of these eigenfunctions, we obtain that each
solution ϕ(s′, p′, t′) of this equation tends exponentially in time with exponent −1 to astationary solution of the form ϕ0. Therefore, taking into account the presentation (9) of
the function ϕ(x′, p′, t) via ϕ(s′, p′, t′), we obtain that “stationary” solutions ϕ0(x′, p′) of
equation (6) look as follows:
ϕ0(x′, p′) =
1
(2π)n
∫Rn
ψ(s′)e−(p′−s′)2
2 eis′x′ds′.
Let us present the function ψ(s′), in its turn, as the Fourier integral:
ψ(s′) =1
(2π)n/2
∫Rn
ψ(y′)e−is′y′dy′.
Substituting this presentation into the preceding expression and integrating over s′, weobtain
ϕ0(x′, p′) =
1
(2π)3n/2
∫R2n
ψ(y′)e−(p′−s′)2
2 eis′(x′−y′)ds′ dy′
=1
(2π)n
∫Rn
ψ(y′)e−(x′−y′)2
2 eip′(x′−y′)dy′
or, taking into account (12),
ϕ0 = P0ϕ =1
(2π)n
∫Rn
ψ(y′)e−(x′−y′)2
2 eip′(x′−y′)dy′, where ψ(y′) =
∫Rn
ϕ(y′, p′)dp′. (13)
Thus, if ε is small, then at the time t′ of order 1, a solution of equation (6), starting
from an arbitrary function ϕ, will become close to a function of the form ϕ0 which, in
Consider equation (3) in the dimensionless system of variables (5). In these variables,
the equation takes the form (6), and “stationary” solutions, to which arbitrary solutions
of equation (6) tend at time t′ of order 1, have the form (13).
Let ϕ0(x′, p′, t′) be a function of the form (13) corresponding to the function ψ(y′, t′).
Let us substitute this expression into equation (6), and let us take projection of both
parts of this equation to the space of functions ψ(y′, t′) by formula (13). We have:∫Rn
∂ϕ0
∂t′dp′ =
∫Rn
(kT
γ�A′ + B′
)ϕ0dp
′
or, taking into account that B′ϕ0 = 0, after substitution of expression ϕ0(x′, p′, t′) in the
form (13), we obtain:
1
(2π)n
∫R2n
∂ψ(y′, t′)∂t′
e−(x′−y′)2
2 eip′(x′−y′)dy′dp′
=kT
γ�
1
(2π)n
∫R2n
A′ψ(y′, t′)e−(x′−y′)2
2 eip′(x′−y′)dy′dp′.
Let us integrate the right hand side of this equality over p′ and over y′. Noting that inthe right hand side of the equality we have the delta function, we obtain:
∂ψ(x′, t′)∂t′
=kT
γ�
1
(2π)n
∫R2n
A′ψ(y′, t′)e−(x′−y′)2
2 eip′(x′−y′)dy′dp′.
Taking into account expression (7) for operator A′, we deduce from the latter equality
Since ελk1 is imaginary, each summand in the sum giving the function ϕ0(t′) in ex-
pression (31), decreases in time t′ = γt proportionally to
exp(ε2Re(λk2)t′) = exp(γε2Re(λk2)t),
where Re(λk2) is the real part of the number λk2. This implies that after the time t ∼
1/(γε2) this sum will be determined by the summand with the maximal number Re(λk2)
among the summands with the valuable coefficients ck(0).
Thus, Theorem 4 is proved.
References
[1] Beniaminov E. M. Diffusion processes in phase spaces and quantum mechanics //Doklady Mathematics (Proceedings of the Russian Academy of Sciences), 2007,vol.76, No. 2, 771–774.
[2] Beniaminov E. M. Quantization as asymptotics of a diffusion process in phase space//http://beniaminov.rsuh.ru/ExpandedDAN.pdf (in Russian; English translation:http://arXiv.org/abs/0812.5116v1). (2008)
[3] Feynman R., Hibbs A. Quantum mechanics and path integrals, New York: McGraw-Hill, 1965.
[4] Kramers H.A. // Physica. 1940. Vol. 7. P. 284-304.
[5] Van Kampen N.G. Stochastic Processes in Physics and Chemistry. North Holland,Amsterdam, 1981.
[6] Kamke E., Differentialgleichungen: Losungsmethoden und Losungen, I, GewohnlicheDifferentialgleichungen, B. G. Teubner, Leipzig, 1977.
[7] Wigner E. On the Quantum Correction For Thermodynamic Equilibrium // Phys.Rev. 1932. V. 40. P. 749-759.
[8] Beniaminov E.M. A Method for Justification of the View of Observables in QuantumMechanics and Probability Distributions in Phase Space http://arxiv.org/abs/quant-ph/0106112 (2001).
[9] Zeh H.D. Roots and Fruits of Decoherence. // In: Quantum Decoherence, Duplantier,B., Raimond, J.-M., and Rivasseau, V., edts. (Birkhauser, 2006), p. 151-175(arXiv:quant-ph/0512078v2).
[10] Zurek W. H. Decoherence and the transition from quantum to classical - REVISITEDarXiv:quant-ph/0306072v1. 2003 (An updated version of PHYSICS TODAY, 44:36-44 (1991)).
[11] Menskij M. B. Dissipation and decoherence of quantum systems. Physics-Uspekhi(Advances in Physical Sciences), 2003, vol.173, 1199-1219.
[12] Shlyajh V. P. Quantum optics in the phase space, Fizmatlit, Moscow, 2005. 760 pp.(in Russian).
[13] Ibort A., Man’ko V.I., Marmo G., Simoni A., Ventriglia F. On the TomographicPicture of Quantum Mechanics. Phys.Lett.A374:2614-2617, 2010. arXiv:1004.0102v1.
[14] Khrennikov A. Quantum Randomness as a Result of Random Fluctuations at thePlanck Time Scale? Int. J. Theor. Phys. 2008. 47, N 1, P.114-124. arXiv:hep-th/0604011v3.
[15] Maslov V. P. Kolmogorov–Feller equations and the probabilistic model of quantummechanics. // Itogi nauki i tehniki. Probability, Mathematical Statistics andCybernetics, 1982, vol. 19, p. 55–85 (in Russian).
[16] Maslov V. P. Quantization of thermodynamics and ultrasecondary quantization,Moscow: Institute for Computer Studies, 2001, 384 pp. (in Russian).
Then the basis states correspond to bound states of the Poschl-Teller potential. For
a given potential (i.e., m fixed positive integer or half-integer) the bound state spectrum
is given by Ej = – (j+ 1/2)2.
On the other hand if we consider ‘j’ is a negative integer or half integer i.e.
j = – 1/2, –1, – 3/2, –2 . . . . . . . . . . . .
and m is unbounded as
m = j, j–1, j–2 . . . . . . .. . . . . . .
Then also it describes the same physical state (bound state) as the potential retains
the same symmetry as that before.
Conclusion
In this work the bound state of Poschl-Teller potential are described using a new realiza-
tion of the SU(1,1) algebra. This approach has the advantage that it can be generalised
to cases where the Hamiltonian is specified in terms of the generators of the group rather
than as a differential Schrodinger operator. Here the energy spectrum is obtained without
solving the Schrodinger equation. Also in all earlier works, SU(2) algebra was used to
describe the bound states but her we have shown that how one can explain the bound
state employing the algebra of only SU(1,1) group.
Acknowledgement
We would like to thank Mr. Pritibhajan Byakti, Dr. Uday Shankar Chakraborty, Dr.
Himadri Sekhar Das, Dr. B. I. Sharma, Dr. Sudip Choudhury and Mr. Saurav Shome for
their useful discussions and support. Subha Gaurab Roy is thankful to Assam University,
Silchar for the grant of UGC Ph.D. fellowship. Subha Gaurab Roy is also extremely
grateful to DST, New Delhi for the grant of INSPIRE fellowship.
References
[1] A. Arima and F. Iachello, Ann. Phys., 99, 253 (1976); 111, 201 (1978), and 123, 581(1981).
[2] F. Iachello, Chem. Phys. Lett., 78, 581 (1981); F. Iachello and R. D. Levine, J. Chem.Phys., 77, 4046 (1982), and O. S. van Roosmalen, A. E. L. Dieperink and F. Iachello,Chem. Phys. Lett., 85, 32 (1982).
[3] N. K. Sarkar, J. Choudhury and R. Bhattacharjee, Mol. Phys., 104, 3051 (2006);106, 693-702(2008), and Indian J. Phys., 82, 767 (2008).
[4] S. R. Karumuri, N. K. Sarkar, J. Choudhury and R. Bhattacharjee, Mol. Phys., 106,1733 (2009).
[5] J. Choudhury, S. R. Karumuri, N. K. Sarkar and R. Bhattacharjee, Pramana J.Phys., 71(3), 439 (2008); 73(5), 881 (2009); 72(3), 517 (2008) and 74, 57 (2010).
In recent times the Nikiforov-Uvarov (NU) method has been used successfully in solving
the Schrodinger, Dirac, Klein-Gordon, and Duffin-Kemmer-Petiau wave equations in the
presence of some well known potential [1-5]. In relativistic mechanics, the solution of the
Klein-Gordon and Dirac equation with some physical potential play a significant role in
nuclear physics and other areas [6,7]. These relativistic equations contain two objects,
the vector V(r) and scalar potential S(r).
The Klein-Gordon equation with the vector and scalar potentials can be written as
follows: [−(i∂
∂t− V (r)
)2
−∇2 + (S(r) +M)2]ψ(r, θ, ϕ) = 0 (1)
where M is the rest mass and for the case S(r) = ±V (r) has been studied recently [8,9].However, the analytical solutions of the Klein-Gordon equations are possible only
Fig. 3 Bifurcation diagram with parameter q increasing form 0.7 to 1.
0.70.8
0.91
−0.20
0.20.4
0.6−0.2
−0.1
0
0.1
0.2
0.3
z(t)x(t)
y(t)
q=0.85
0.50.6
0.70.8
0.9
−0.20
0.20.4
0.6−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.86
0.40.5
0.60.7
0.80.9
−0.20
0.20.4
0.6−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.93
0.20.4
0.60.8
1
−0.20
0.20.4
0.6−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.94
0.20.4
0.60.8
1
−0.20
0.20.4
0.6−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.947
Fig. 4 Projections of original cycle, period two cycle, period four cycle, period eight cycle andFeigenbaum attractor in the fractional order nonlinear Bloch system.
The ordering n � k in (14) means that the existence of a cycle of period k implies the
existence of all cycles of period n. So, if the system (13) has a stable limit cycle of period
three then it has also all unstable cycles of all periods in accordance with the Sharkovskii
order (14).
The Sharkovskii complete subharmonic cascade of bifurcations of stable cycles is proved
by existence of a limit cycle of period 6 for the parameter value q = 0.948, a limit cycle
of period 5 for q = 0.955 and a limit cycle of period 3 lying in the interval [0.965, 0.979]
which with further increase of the parameter q goes through a cascade of period doubling
bifurcations. Thus, for q = 0.98 we observe a doubled cycle of period 3. The subharmonic
cascade also terminates with the formation of an irregular attractor.
Some cycles of this cascade and a subharmonic singular attractor are shown in Fig.5.
To demonstrate the chaotic dynamics, the largest Lyapunov exponent should be the
first thing to be considered, because any system containing at least one positive Lyapunov
exponent is defined to be chaotic [16]. Measuring the largest Lyapunov exponent (LLE) is
always an important problem whatever in a fractional order system or in an integral-order
Fig. 5 Projections of period six cycle, period five cycle, period three cycle, doubled period threecycle and more complex subharmonic singular attractor in the fractional order nonlinear Blochsystem.
system. Wolf and Jacobian algorithms are the most popular algorithm in calculating the
largest Lyapunov exponent of integer-order system. However, Jacobian algorithm is not
applicable for calculating LLE of a fractional order system, since the Jacobian matrix
of fractional order system is hard to be obtained. As to Wolf algorithm [21] which is
relatively difficult to implement. Therefore, in this paper, the small data sets algorithm
developed by Michael T. Rosenstein etc [14] is chosen to calculating LLE of the Fractional
order nonlinear Bloch system, the diagram is plotted in Fig.6.
0.7 0.75 0.8 0.85 0.9 0.95 1−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
q
Max
imal
Lya
puno
v E
xpon
ents
Fig. 6 Maximal Lyapunov Exponents versus q form 0.7 to 1 step with 0.01.
E = (0.02730, 0.00429, 0.99847) : λ1 = −0.19971, λ2,3 = 5.6155± 35.685j. Hence, the
fixed point E is a saddle point of index 2. The necessary condition to remain chaotic for
the fractional order nonlinear Bloch system with this set of parameters is q > 0.90.
At q ≈ 0.90 a Hopf bifurcation gives birth to an orbitally stable limit cycle. For cer-
tain parameter values, this limit cycle co-exists with another limit cycle with different
period, each with its basin of attraction. Fig.7 shows the bifurcation diagram against the
parameter q and the coexisting limit cycles for different initial condition
0.88 0.9 0.92 0.94 0.96 0.98 1−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
q
x
−0.10
0.10.2
0.3 −0.4−0.2
00.2
0.4
−0.4
−0.2
0
0.2
0.4
x(t)z(t)
y(t)
q=0.975
Fig. 7 (a) bifurcation diagram vs q, (b)coexisting limit cycles. Initial conditions: (0.1, 0.1, 0.1)for the thick line and (0.01, 0.01, 0.01) for the thin line.
Fixing the initial conditions at (0.1, 0.1, 0.1) and increasing the parameter q, the initial
period one limit cycle will disappear suddenly and is replaced by a period four limit cycle
at q = 0.99, the two limit cycles goes through a cascade of period doubling bifurcations
which terminates with the formation of irregular attractors as shown in Fig.8.
−0.20
0.20.4
0.6
−0.2−0.1
00.1
0.2−0.2
−0.1
0
0.1
0.2
z(t)x(t)
y(t)
q=0.98
−0.10
0.10.2
0.3
−0.4−0.2
00.2
0.4−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.99
−0.10
0.10.2
0.3
−0.4
−0.2
0
0.2
0.4−0.4
−0.2
0
0.2
0.4
y(t)
q=0.9915
−0.2−0.1
00.1
0.20.3
−0.4−0.2
00.2
0.4−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.9923
−0.050
0.050.1
0.150.2
−0.2−0.1
00.1
0.2
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
z(t)x(t)
y(t)
q=0.994
−0.2−0.1
00.1
0.20.3
−0.4−0.2
00.2
0.4−0.4
−0.2
0
0.2
0.4
z(t)x(t)
y(t)
q=0.995
Fig. 8 Projections of the period doubling bifurcations and irregular attractors.
The largest Lyapunov exponents are calculated numerically with q ∈ [0.85, 1] for an
Fig. 9 Maximal Lyapunov Exponents versus q form 0.85 to 1 step with 0.01 .
Conclusion
In this paper, we have studied the dynamics of the fractional-order nonlinear Bloch system
by means of the bifurcation diagram and largest Lyapunov exponents. A numerical
algorithm is used to analyze the fractional-order system. In this study the fractional
order is the explore direction. Through these, Period-doubling and subharmonic cascade
routes to chaos were found in the fractional-order nonlinear Bloch equations. Especially,
a period-3 window is presented in bifurcation diagram. Moreover, coexisting limit cycles
were also found. We calculate the largest Lyapunov exponent by using the small data
sets instead of wolf algorithm, which was used frequently in preview research. The results
show the validity of the algorithm.
References
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[7] Hartley TT, Lorenzo CF, Qammer HK. Chaos in a fractional Chua’s system. IEEETrans Circ Syst Theory Appl 1995;42(8):485-490.
[8] Li CP, Peng GJ. Chaos in Chen system with a fractional order. Chaos SolitonsFractals 2004;20:443-450.
[9] Lu JG, Chen G. A note on the fractional-order Chen system. Chaos Solitons Fractals2006;27:685-688.
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[11] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974.
[12] Petras I. A note on the fractional-order Chua’s system. Chaos Solitons Fractals2008;38:140-147-147.
[13] Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.
[14] Rosenstein Michael T, Collins James J, De Luca Carlo J. A practical method forcalculating largest Lyapunov exponents from small data sets. Physica D 1993;65:117-134.
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Fig. 2 Phase synchronization for the coupled system (8) in the text.
see that the eigenvalues of this matrix are -1 and zero with multiplicity 2. Since μ∞(A) =0, the existence of phase synchronization in system (6) is globally asymptotically stable
by Theorem 1.
As the second example, consider our new four dimensional system defined as follows.⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩x = −ax− by + w,
y = −cy − axz,
z = −z + axy + d.
w = −fw − exz.
(7)
This system is chaotic for the parameter values a = 3, b = 2, c = 0 and f = 1. Applying
the nonlinear coupling feedback function method to this system yields⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
x1 = −ax1 − by1 + w1,
y1 = −cy1 − ax1z1 + ra(x1z1 − x2z2),
z1 = −z1 + ax1y1 + d+ ra(x2y2 − x1y1),
w1 = −fw1 − ex1z1 + re(x1z1 − x2z2),
x2 = −ax2 − by2 + w2,
y2 = −cy2 − ax2z2 + ra(x2z2 − x1z1),
z2 = −z2 + ax2y2 + d+ ra(x1y1 − x2y2),
w2 = −fw2 − ex2z2 + re(x2z2 − x1z1).
(8)
Figure 2 shows different states of phase synchronization in system (8). Here the eigenval-
So, if the collector synchronized movements inX⊕Y ⊕Zis linearly stable forz(t)−y(t),then it is linearly stable for ζy(t) = y(t)− φ(x(t)) and vice versa
The study of synchronization goes back to the study of stability in the vicinity of the
origin of a new system that gives it the name of “system error”. The latter represents
the disturbance that may exist between the transmitting and receiving system.
To study the stability of the system error we will use the criterion of Routh-Hurwitz
generalized to fractional order [1]
3.2 Some Stability Conditions
Let (xe, ye, ze)be an equilibrium solution of the following three dimensional fractional-
order systems: ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩dqx(t)dtq
= f(x, y, z)
dqy(t)dtq
= g(x, y, z)
dqz(t)dtq
= h(x, y, z)
(7)
Where q ∈ (0, 1].the eigenvalues equation of the equilibrium point (xe, ye, ze) is given by
the following polynomial:P (λ) = λ3 + a1λ2 + a2λ+ a3 = 0
And its discriminant D (P) is given as:
D(P ) = 18a1a2a3 + (a1a2)2 − 4a3(a1)
3 − 4(a2)3 − 27(a3)
2
(1) If D(P ) 0, then the necessary and sufficient condition for the equilibrium point
(xe, ye, ze), to be locally asymptotically stable, is :
a1 0, a2 0, a1a2−a3 0.
(2) If D(P ) ≺ 0, a1 0, a2 0, a3 0. then (xe, ye, ze) is locally asymptotically stable
for,q ≺ 2/3, However, if D(P ) ≺ 0, a1 ≺ 0, a2 ≺ 0 , q 2/3, then all roots of
equation satisfy the condition |arg(λ)| ≺ qπ/2.
(3) if D(P ) ≺ 0, a1 0, a2 0,a1a2 − a3 = 0. then (xe, ye, ze) is locally asymptotically
stable for all q ∈ (0, 1).(4) the necessary condition for the equilibrium point (xe, ye, ze) To be locally asymptot-
ically stable, is a3 0
Synchronization of Modified Chua’s System and MAVPD System
Let us take in this paragraph the two preceding studies, the first is the fractional modi-
fied Chua system, the second is the fractional MAVPD system and we will detect their
Fig. 5 Graphs of the time variation of the synchronization errors e1 = x3 −x2, e2= y3−y2, e3= z3−z2 (a) and (b) the error system converge to zero if k = 90.685
φ(x, y, z) =
⎛⎜⎜⎜⎜⎝−m(x33 − x32)
0
0
⎞⎟⎟⎟⎟⎠is a nonlinear function satisfies the Lipschitz condition, so locality to zero it converges to zero
To study the stability of system (12) we use the conditions of criterion Routh-
Hurwitz generalized to fractional order [1]
The characteristic polynomial of matrix A is given by:
Fig. 6 Graphs of the time variation of the synchronization errors e1 = x3 −x2, e2= y3−y2, e3= z3−z2 (a) and (b) the error system converge to zero if k = 22 (c) the errorsystem is instable if k = 2
Synchronization of Fractional Chen System and Fractional LU
System
In this paragraph we replied this method of synchronization for two well known systems,
the first is frictional Chen system and the second is fractional Lu system with auxiliary
system
For this we assume the fractional Chen system as transmitter (master):
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩dqx1
dtq = a(y1 − x1),
dqy1
dtq = (c− a)y1 − x1z1 + cy1,
dqz1
dtq = x1y1 − bz1,
(14)
Where (a, b, c) = (35, 3, 28) Consequently, the lowest fractional order qfor which the
fractional-order Chen system (14) demonstrates chaos using the above mentioned param-
eters is given by the inequality.q 0.82
And we assume the fractional Lu system as receiving (slave).
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩dqx2
dtq= α(y2 − x2),
dqy2dtq
= δy2 − x2z2 − k(y2 − y1),
dqz2dtq
= x2y2 − βz2,
(15)
(α, β, δ) = (36, 3, 20) and k is coupling parameter Consequently, the lowest fractional
order q for which the fractional-order LU system (15) demonstrates chaos using the above
mentioned parameters is given by the inequality.q 0.91605
The master system is coupled with the slave system only by the y(t) scalar
We choose the auxiliary system that is identical to the slave system (15) (with different
In this paper we have studied the synchronization between different chaotic fractional
–order systems, with the auxiliary system, we have applied this method on both systems:
The modified Chua oscillator as transmitter system and Van der Pol-Duffing modified
(MVDPD) as receiver system; we have also used it again on two well know systems Chen
and LU.Using the criterion of routh Hurwitz to study the stability of the system error.
Numerical results show the effectiveness of the theoretical analysis.
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formulae (38) and (42) the Hubble parameter H, and the age of our Universe can be
calculated
υ = HR,H = 12Mτp
= 5 · 10−18s−1,
T = 2Mτp = 2 · 1017s ∼ 1010years,(18)
which is in quite good agreement with recent measurement [13, 14, 15].
As is well known in de Sitter universe the cosmological constant Λ is the function of
R, radius of the Universe,
Λ =3
R2. (19)
Substituting formula (38) to formula (47) we obtain
Λ =3
πN2L2p
, N = 0, 1, 2.... (20)
The result of the calculation of the radius of the Universe, R, the acceleration of the
spacetime, a, and the cosmological constant, Λ are presented in Figs. 1, 2, 3, 4 for
different values of number N . As can be easily seen the values of a and R are in very
good agreement with observational data for present Epoch. As far as it is concerned
cosmological constant Λ for the firs time we obtain, the history of cosmological constant
from the Beginning to the present Epoch.
Conclusions
In this paper the diffusion model of the Universe expansion is developed. Considering the
anthropic argument Universe temperature :ImT( r,t) = 0 the quantization of the space-
time is obtained. The radius, velocity of the Universe expansion, the acceleration and
cosmological parameter as the function of the discreteness parameter N is obtained. For
N=10 60 the age Universe= 1017 s = the present Epoch. The present day Universe is
the relic of the primordial point Universe which expands in discrete steps N=1,2 ,. . . .
1060. . . .
References
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k = 2.5 × 10−3, there does not seem to be a critical value for the mass function which
always goes negative for large enough r. For k = −1 the pressure is negative, the criticaldensity for which positive pressure is restored is around k = −2.5× 10−10.
The null convergence condition is RCC ≡ Rabnanb ≥ 0, for na a null vector p.95[9],
the weak energy condition is TCC ≡ TabVaV b, for Va a timelike vector p.89[9], the
dominant energy condition is the timelike convergence condition and T abVa is a non-
spacelike vector p91[9], NSV is the size of T abVa. In figure two 103×RCC and 106×TCC
and −105×NSV are plotted for vc = 0.05 and k = 0. For k = 1 the null convergence
condition is violated, the critical value around which it seems to be restored is k = 10−6.For k = −1 the weak energy condition is violated and this seems always to be the case forlarge enough r. The curvature invariant RiemSq is defined by RiemSq = RabcdR
abcd, with
similar definitions for WeylSq and RicciSq. In figure three 104×RiemSq, 104×WeylSq,
104×RicciSq and 102×Ricciscalar (not squared) are plotted for vc = 0.05 and k = 0. It
does not seem to be possible numerically to determine whether the divergence happens
at r = exp(−1/(2v2c )) ≈ 10−86 or r = 0 or both. A surprising feature of figure three is
that WeylSq is large compare to RicciSq, this has the interpretation, see [9]p.85, that
more of the curvature is due to gravity as opposed to matter. For k = ±1 the RicciSq islarge compared to WeylSq, the critical value seems to be around k = ±10−4, but seemsto highly dependent on r.
5. Asymptotics and Units
Constant velocity curves are only observed over a certain region, how short and long
radial distances fit to this region is a problem. There seem to be three approaches to
this: the first is to adjust things at the last moment and produce an onion model in which
shows that both the density and mass function do not tend to zero asymptotically. Thus
galactic rotation requires field modifications as opposed to fluid modifications of Einstein’s
equations, such a field model has been given [21].
Acknowledgements.
I would like to thank Jakob Bekenstein, James Binney, Antony Fairall, Alex Feinstein,
Gerry Gilmore, Tom Kibble, Andrew Liddle, David Matravers, Michael Merrifield and
Andrew Taylor for discussion on various aspect of this paper, and the referee for his
comments.
References
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[9] Hawking, S. W. and Ellis, G. F. R. (1973)The Large Scale Structure of Space-time.Math.Rev.http://www.ams.org/mathscinet-getitem?mr=54:12154 54 #12154,Cambridge University Press.
[10] Dan Hooper & Graham D. KribsKaluza-Klein dark matter and the positron excess.http://arxiv.org/abs/hep-ph/0406026 hep-ph/0406026
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[12] V.V. KislevGhost condensate model of flat rotation curves.http://arxiv.org/abs/gr-gc/0406086 gr-qc/0406086
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provide evidence for shell closures. It is important to look for new shell closures or the
disappearance of existing shell closures from the separation energy calculation [4, 5]. The
origin of the unusual stability of nuclei with nucleon numbers 2, 8, 20, 28, 50, 82 and
126, commonly called to as “magic numbers”, is explained to be due to nuclear shell
structure.At present there is a proliferation of new magic or rather quasi-magic numbers
[6, 7]. At the same time some magic numbers are demoted and seem to lose their magicity.
In the simple shell model these are due to shell or sub-shell closures. Shell closure may
be demonstrated by a large drop in separation energies. Such phenomena can be simply
explained by the simple shell model. The single- and two-nucleon separation energies are
fundamental properties of the atomic nucleus [8]. It is a challenge for nuclear many-body
theories to derive the shell model out of complex calculations. Systematic of proton and
neutron separation energies can be powerful tools to study the nuclear structure at and
even beyond the drip lines [9]. It can be used to predict masses and separation energies
of nuclei beyond the neutron and proton drip lines.
This paper is organized as follows. Section 1 is introductory. Section 2 deals with the
theoretical framework used in the study of shell closures and nuclear structure effects.
The found shell gaps, new magicity and shape transitions obtained in the sample case
of fp shell region nucleus titanium isotope are discussed in Sec. 3. The evolution of the
shapes in the rotating titanium isotopes is also traced in Sec.3. Finally, Sec. 4 contains
a summary and conclusion.
2. Theoretical Formalism
An important question in nuclear structure physics is the nature of shape evolution taking
place at critical angular momenta near the limit of stability. In order to know the shape of
the nucleus before fission, this work involves two formalisms. The first framework shows
the nucleon separation energy calculations for various Z and N values in detail. The second
formalism depicts the shape transition in the fp shell region nucleus for evolution of shapes
in the β−γ plane at zero temperature using cranked Nilsson Strutinsky calculations withtuned to fixed spins. The shape transitions details are predicted by using potential energy
surface calculations.
2.1 Separation Energy for sp, sn, s2p, s2n for Different Isotopes and Iso-
tones
Separation energy values for single proton, diproton, single neutron and di neutron are
calculated to show the magicity prevailing in their numbers. The separation energies are
The second section gives the theoretical framework for obtaining potential energy surfaces
of the considered nuclei as a function of deformation β and nonaxiality γ parameters at
different spins by the Strutinsky method.
For a non rotating nuclei (zero spin) shell energy calculations assumes a single particle
field
H0 = Σ h0 (3)
where h0 is the triaxial Nilsson Hamiltonian given by
h0 =p2
2m+1
2m
3∑i=1
ω2i x
2i + Cls+D
(l2 − 2
⟨l2⟩). (4)
By Hill-Wheeler parameterization the three oscillator frequencies ωi are given as include
the energy term first
Ek = hωk = hωGDR exp [ -√5/4π β cos (γ – 2/3 Ti K)]
ωx = ω0 exp
(−√
5
4πβ cos
(γ − 2
3π
))
ωy = ω0 exp
(−√
5
4πβ cos
(γ − 4
3π
))
and ωz = ω0 exp(−√
54πβ cos γ
)with the constraint of constant volume for equipotentials
ωxωyωz =◦ω30 = cons tan t (5)
The values [11] for the Nilsson parameters κ and μ are chosen as
κ = 0.093 and μ = 0.15
The value for �ω0 is taken as
�ω0 =45.3 MeV
(A1/3 + 0.77)
. (6)
The same values are used for both protons and neutrons.
The factor 2 in front of 〈l2〉 [Eq. (4)] has been used to obtain better agreement betweenthe Strutinsky-smoothed moment of inertia and the rigid rotor value. The parameter D
which is plotted in Fig. (2). When mH < 140GeV , the dominant channels is the H → bb,
and the total decay width is in the 10MeV range.
The solid line results in Fig. (2) is arguably for its applicability. When mH is less
than 2mW or 2mZ , we may research the ’sub-threshold decay’, such as H → ZZ∗, whereone Z boson is on-shell and the second Z boson’s mass is off-shell. Simulation from CMS
(Physics TDR) for the H → ZZ∗ →4 leptons final state result is also shown as dots in
Fig. (2) [6] [7]. According to equ. (23), H → WW channel’s width should be a little
greater than H → ZZ. But experimentally, it’s very hard to measure the energy and
momentum of the neutrinos in the H → WW →2 charged leptons + 2 neutrinos case;
compares to the easy be detected one in the H → ZZ →4 charged leptons channel.
Fig. (2) shows the CMS simulation result that, for the Higgs mass around 130GeV ,
ZZ∗ channel sub-threshold decay has Γ � 2GeV . If we think the WW ∗ channel is aboutsame, then the total width for mH = 130GeV is ΓE ≈ 4GeV . That means, from (22),
Γp =√3ΓE ≈ 6.8GeV for mH = 130GeV .
5. Transition Probability of the Vacuum States
The lowest order S matrix involves the transition amplitude of the φ4 interaction, which
made of two incoming vacuum unbroken states φi1φi2 in equ. (16), and two out-going
The approximate sign in equ. (24) means we replace the square of tanh function with
tanh2 ≈ 1 as in (10). So only the Gaussian integrations of ϕ are left. The phase term
does not contribute to the following probability calculation. So we just omit it. The
displacement term reflects the effect that the two out-going broken states centered at
different points b1 = (b1x, b1y, b1z) and b2 = (b2x, b2y, b2z). We just let them to be at same
points, so the displacement term reaches its maximum value, which is 1. Other values
we used are, υ = 246GeV , λ = m2H/2υ
2 ≈ 1/8, mH = 130GeV , momentum square
p2 = m2H/2 is in equ. (19), Γp =
√3ΓE � 6.8GeV . The time integral gives out a term, T ,
which is the age of the universe, equals to about 1.3×1010years = 6.2×1041GeV −1. Thenwe get the probability that the vacuum spontaneously transfer from unbroken states to
the broken states,
probability = |Sif |2 = 10−52. (25)
6. Summation over Possible States with Probability
Vacuum does not like matter, in which matter has so many possible states that can
oscillate in many frequencies. The vacuum energy can not contain contributions from
all arbitrary wavelengths, except those physically existed states that are permitted by
the Hamiltonian, e.g. To calculate the summation of states, we made the following
assumptions:
1) The only contribution to the vacuum energy comes from the spontaneous symmetry
breaking, the Higgs mechanism; which means we do not care any other possible sources.
Vacuum has only two states: unbroken state and broken state, which will be the only
two permitted states that appeared in the summation. Because the vacuum unbroken
state has the energy E = 0, so the only state left in the summation is the vacuum broken
state.
2) The summation needs some modulation factor. For instance, to avoid diverge in
the thermal radiation expression, in the derivation of the Planck’s formula of blackbody
radiation, we not only add up the energy of photons, but also times the Bose-Einstein
distribution factor 1/(ehνkT − 1). In this paper, we chose the probability of the transition
from unbroken state to the broken state as the suppression factor.
Finally we sum the energy in phase space and get the vacuum energy density,
ρ =4πp2dp · E(2π)3
× probability = 10−47GeV 4, (26)
in which the momentum and energy are in equ. (19) and (20), ≈ (1 to 2) × 102GeV ;
dp ≈ Γp; probability term is in equ. (25). It is amazing to see that our result in (26) is
exact same as the value that observed in the experiments.
Gravitons Writ Large; I.E. Stability, Contributionsto Early Arrow of Time, and Also Their PossibleRole in Re Acceleration of the Universe 1 Billion
Years Ago?
A. Beckwith∗
Institute of Theoretical Physics,Department of Physics, Chongquing University, ChinaAmerican Institute of Beamed Energy Propulsion†, Seculine Consulting, USA
Received 6 July 2010, Accepted 10 February 2011, Published 25 May 2011
Abstract: This document is due to a question by Debasish of the Saha institute of India
asked in the Dark Side of the Universe conference, 2010, in Leon, Mexico, and also is connected
with issues as to the initial configuration of the arrow of time brought up in both Rudn 10, in
Rencontres de Blois, and Fundamental Frontiers of Physics 11, in Paris, in July 2010. Further
reference is made as to how to reconcile early inflation with re acceleration, partly by dimensional
analysis and partly due to recounting a suggestion as by Yurov, which the author thinks has
merit and which ties into, to a point with using massive gravitons as a re acceleration of the
Keywords: cosmology; Early Universe; Arrow of Time; Gravitons
PACS (2010): 98.80.-k; 98.80.Cq; 14.70.Kv
1. Introduction
The supposition advanced in this article is that relic energy flux initially is central to
making predictions as to verifying Sentropy ∼ nf [1,2,6,7], where nf is a ‘particle count’
per phase space ‘volume’ in the beginning of inflation. Having said that, is nf due
to gravitons in near relic conditions? Or is Sentropy ∼ nf due to coherent clumps of
gravitons? If so, can the gravitons/ coherent clumps of gravitons carry information ? The
author in previous manuscripts [1,2] identified criteria as to Sentropy ∼ nf |start−of−inf ∝105 ⇔ initial information ∝ 107 bits of ‘information’ in line with G. Smoot’s Paris (2007)
1.1 Defining the Graviton Problem and Using Visser’s (1998) Inputs into
Tuv
We begin our inquiry by initially looking at a modification of what was presented by R.
Maartens [15] , as done by Beckwith [12,13]
mn(Graviton) =n
L+ 10−65grams (4)
On the face of it, this assignment of a mass of about 10−65 grams for a 4 dimensional gravi-ton, allowing for m0(Graviton − 4D) ∼ 10−65grams[‘12,13] violates all known quantummechanics, and is to be avoided. Numerous authors, including Maggiore [9] have richly
demonstrated how adding a term to the Fierz Lagrangian for gravitons, and assuming
massive gravitons leads to results which appear to violate field theory, as we can call it .
Turning to the problem, we can examine what inputs to the Eqn. (1) above can tell us
about if there are grounds for m0(Graviton− 4D) ∼ 10−65grams [12,13] , and what thissays about measurement protocol for both GW and gravitons as given in Eqn. (2) above.
Visser [10] , in 1998 came up with inputs into the GR stress tensor and also , for the
perturbing term huv which will be given below. We will use them to perform a stability
analysis of the consequences of setting the value of m0(Graviton − 4D) ∼ 10−65grams[10,12,13], and discuss how T’Hooft’s [12,13,14] supposition of deterministic QM, as an
embedding of QFT, and more could play a role if there are conditions for stability of
m0(Graviton− 4D) ∼ 10−65grams [10,12, 13]
1.2 Visser’s treatment of the Stress Energy Tensor of GR, and its Appli-
cations
Visser[10] in 1998, stated a stress energy treatment of gravitons along the lines of
Tuv|m �=0 =
[(�
l2Pλ2g
)·(GM
r
)· exp
(r
λg
)+
(GM
r
)2]×
⎡⎢⎢⎢⎢⎢⎢⎢⎣
4 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎥⎦(5)
Furthermore, his version of guv = ηuv + huv can be written as setting
huv ≡ 2GM
r·[exp
(−mgr
�
)]· (2 · VμV ν + ηuv) (6)
If one adds in velocity ‘reduction’ put in with regards to speed propagation of gravitons
As well as setting (MG/r) ≈ 1/5for reasons which Visser[10] outlined, one can obtain a
real value for the square of frequency > 0, i.e.
�2ω2 ∼= m2
gc4 · [1/(1− A)] > 0 (8)
A =
{1− 1
6mgc2
(�2
l2Pλ2g
· exp[− r
λg+mg · r�
]+
(MG
r
)· exp
(mgr
�
))}2
(9)
According to Jin Young Kim [16] , if the square of the frequency of a graviton, with
mass, is >0, and real valued, it is likely that the graviton is stable, at least with regards
to perturbations. Kim’s article [16] is with regards to Gravitons in brane / string theory,
but it is likely that the same dynamic for semi classical representations of a graviton with
mass.
1.3 Conditions Permitting Eqn (8) to have Positive Values
Looking at Eqn. (1.8) is the same as looking at the following, analyzing how
A =
{1− 1
6mgc2
(�2
l2Pλ2g
· exp[− r
λg+mg · r�
]+
(MG
r
)· exp
(mgr
�
))}2
< 1 (10)
I.e. setting
0 <1
6mgc2
(�2
l2Pλ2g
· exp[− r
λg+mg · r�
]+
(MG
r
)· exp
(mgr
�
))< 1 (11)
Note that Visser [10] (1998) writes mg < 2 × 10−29eV ∼ 2 × 10−38mnucleon, and a wave
length λg ∼ 6 × 1022meters. The two values, as well as ascertaining when one can useMGr∼ 1/5, with r the usual distance from a graviton generating source, and M the mass’
of an object which would be a graviton emitter put severe restrictions as to the volume
of space time values for which r could be ascertained. If, however, Eq. (10) had, in most
cases, a setting for which, then in many cases, Eq. (1.8) would hold.
0 ≤ exp
[− r
λg+mg · r�
]<< 1 (12)
The author believes that such a configuration would be naturally occurring in most
generation of gravitons at, or before the Electro Weak transition point in early cosmology
evolution.
1.4 Review of if there is a nf ≈ 105to · 106 Initial Production of CoherentGroups of Gravitons in Relic Conditions. And its effect on the arrow
of time question
The author, Beckwith, believes, that satisfying Eqn. (12) would allow to predict a particle
count behavior along the lines where Beckwith[1,2,3] obtained nf ≈ 105to·106. This value
of nf ≈ 105to · 106 as given by Beckwith[1,2,12,13] would be put into Eqn. (2) above,
which would have implications for what to look for in stochastic GW generation. The
question to raise, is what “particle” is being counted, in nf ≈ 105to · 106. Conceivably,it could be coherent packets of gravitons. The reasons for raising this question will be
spelled out in the following analysis.
Recently, Beckwith asked [1,2,3] if the following could occur, S ≡ [E − μN ]/T → S ∝T 3 by setting the chemical potentialμ → 0with initial entropy S ∼ 105 at the beginning
of inflation . Conventional discussions of the arrow of time states that as the Universe
grows its temperature drops, which leaves less energy available to perform useful work
in the future than was available in the past. Thus the Universe itself has a well-defined
thermodynamic arrow of time. The problem of the initial configuration of the arrow of
time, however, is not brought up. This paper is to initiate how to set up a well defined
initial starting point for the arrow of time. Specifically re setting the degrees of freedom
of about g∗ ∼ 100−120[1,2,3] of the electro weak era, to g∗ ∼ 1000at the onset of inflation
[1] , may permit Sinitial ∝ T 3. If the initial temperature of an emerging universe were
very low, scaling S ∝ T 3may be a way to get an arrow of time, with respect to thermal
temperatures, alone, with the graviton count a later, emergent particle phenomenon.
2. What can be Said Initially about Usual Arrow of Time For-
mulations of Early Cosmology?
Usual treatments of the arrow of time, i.e. the onset of entropy . The discussion below
makes the point that expansion of the universe in itself does not ‘grow’ entropy
The entropy density s of a radiation field of temperature T is s ∼ T3. The entropy
S in a given comoving volume V is S = sV . Since the commoving volume V increases
as the universe expands, we have V ∼ R 3. And since the temperature of the microwave
background goes down as the universe expands: T ∼ 1/R, we have the result that the
entropy of a given comoving volume of given space S ∼ R −3 * R3 = constant. Thus
the expansion of the universe by itself is not responsible for any entropy increase. There
is no heat exchange between different parts of the universe. The expansion is adiabatic
and isentropic: dS expansion = 0. I.e. a process has to be initiated in order to start
entropy production
This discussion above is to emphasize the importance of an initial process for the
onset and the growth of entropy. We will initiate candidates for making sense of the
following datum
To measure entropy in cosmology we can count photons. If the number of photons in
a given volume of the universe is N, then the entropy of that volume is S ∼ kN where k
is called here Boltzmann’s constant
Note that Y. Jack Ng. has [6] , from a very different stand point derived S ∼ nbased
upon string theory derived ideas , with n a ‘particle’ count , which in Y. Jack Ng’s
procedure is based upon the number of dark matter candidates in a given region of phase
space..Y. Jack Ng’s idea was partly based upon the idea of quantum ‘ infinite ‘ statistics,
This counting procedure is different from traditional notions . To paraphrase them,
one can state that “The reason why entropy is increasing is because there are stars in
that “box” ( unit of phase space used for counting contributions to entropy). Hydrogen
fuses to helium and nuclear energy is transformed into heat.” I.e. the traditional notion
would be akin to heat production due to, initially start BBN nucleosynthesis, and then,
frankly , star production/ nuclear burning. I.e. one would need to have nuclear processes
to initiate heat production. This idea of heat production is actually similar to setting
S ∝ T 3, with heat production due to either BBN/ hydrogen burning leading to an increase
in temperature, T. In this manuscript, we make use of, if S ≡ [E − μN ]/T → S ∝ T 3
by setting the chemical potentialμ → 0with initial entropy S ∼ 105 at the beginning of
inflation. This entails, as we will detail , having increased number of degrees of freedom,
initially, with re setting the degrees of freedom of about g∗ ∼ 100 − 120of the electro
weak era, to g∗ ∼ 1000at the onset of inflation, I.e. what will be examined will be the
feasibility of the following: S ≡ [E − μN ]/T −→μ→0
S ∝ T 3 ≈ n, with n an initial ‘quantum
unit’ count in phase space of Planckian dimensions, where S ∼ 105 at the beginning of
inflation. Let us now look at how to initiate such a counting algorithm if one is looking
at , say, highly energized gravitons , initially, as part of a counting ‘algorithm’.
2.1 Estimating the Size of Contribution to Energy in S ≡ E/T , Assuming
a Peak Frequency ν ∼ 1010 Hertz for Relic Gravitons, if the Standard
Chemical Potential is Effectively μ = 0 at the Onset of Creation
As suggested earlier by Beckwith [12,13], gravitons may have contributed to the re-
acceleration of the universe one billion years ago. Here, we are making use of refining the
following estimates. In what follows, we will have even stricter bounds upon the energy
value (as well as the mass) of the graviton based upon the geometry of the quantum
bounce, with a radii of the quantum bounce on the order of lP lanck ∼ 10−35meters [1], [5].
mgraviton|RELATIV ISTIC < 4.4× 10−22h−1eV/c2
⇔ λgraviton ≡ �
m·gravitonc< 2.8× 10−8meters
(13)
For looking at the onset of creation, with a bounce; if we look at ρmax ∝ 2.07 · ρplanck forthe quantum bounce with a value put in for when ρplanck ≈ 5.1 × 1099grams/ meter3,
Then, taking note of this , one is obtaining having a scaled entropy of S ≡ E/T ∼105 when one has an initial Planck temperature T ≈ TPlanck ∼ 1019GeV . One needs,
then to consider, if the energy per given graviton is, if a frequency ν ∝ 1010Hz and
Having said that, the [Egraviton−effective ∝ 2 · hv ≈ 5× 10−5eV ] is 1022greater than the
rest mass energy of a graviton if E ∼ mgraviton [red− shift ∼ .55] ∼ (10−27eV )grams istaken when applied to Eq. (2) above.
2.2 The Electro Weak Generation Regime of Space Time for Entropy and
Early Universe Graviton Production before Electro Weak Transitions
A typical value and relationship between an inflation potential V [φ], and a Hubble pa-
rameter value, H is [1]
H2 ∼ V [φ]/m2
Planck (16)
Also, if we look at the temperature T ∗ occurring about the time of the Electro weak
transition , if T ≤ T ∗ when T ∗ = Tcwas a critical value, (of which we can write v(Tc) /Tc
>1 , where v(Tc) denotes the Higgs vacuum expectation value at the critical temperature
Tc., i.e. v(Tc)/Tc >1 according to C. Balazc et al (2005) [17] and denotes that the
electro weak transition was a ‘strongly first order phase transition’) then one can write ,
by conventional theory that
H ∼ 1.66 ·[√
g∗]· [T 2
/m2
Planck] (17)
Here, the factor put in, of g∗ is the number of degrees of freedom. Kolb and Turner [18]put a ceiling of about g∗ ≈ 100 − 120 in the early universe as of about the electro weak
transition. If , however, g∗ ∼ 1000 or higher for earlier than that, i.e up to the onset of
inflation for temperatures up toT ≈ TPlanck ∼ 1019GeV , it may be a way to write, if we
also state that V [φ] ≈ Enet that if [1]
S ∼ 3m2
P lank
[H = 1.66 ·
√g∗ · T 2
/mplanck·
]2T
∼ 3 ·[1.66 ·
√g∗]2T 3 (18)
Should the degrees of freedom hold, for temperatures much greater than T ∗, and with
g∗ ≈ 1000 at the onset of inflation, for temperatures, rising up to , say T ∼ 1019GeV,
from initially a very low level, pre inflation, then this may be enough to explain how and
why certain particle may arise in a nucleated state, without necessarily being transferred
from a prior to a present universe.
Furthermore, if one assumes that S ∝ T 3 [5] when g∗ ≈ 1000 or even higher even if T
∼ 1019GeV >> T ∗, then there is the possibility that S ∝ T 3 when g∗ ≈ 1000 could also
hold, if there was in pre inflationary states very LOW initial temperatures, which rapidly
built up in an interval of time, as could be given by 0 < t < tPlanck ∼ 10−44seconds [1]
2.3 Justification for Setting g∗ ≈ 1000 Initially
H. de La Vega, in conversations with the author in Colmo, Italy, 2009 [7]. flatly ruled
out having g∗ ≈ 1000 initially. What will be presented here will be a justification for
taking this step which H. de La Vega says is not measurable and possible. The author
2.5 Making an Argument for DM/ DE, if there is a Small Rest Graviton
Mass a Billion Years Ago
Either there is clumping of gravitons into coherent GW states, as may be the resolution
of the 1038 factor in Eq. (3) , and the GW frequency drops dramatically a billion years
ago , to take into account having, instead of the energy associated with relic gravitons
of value ≈ 5 × 10−5eV , as assumed in Eq (1) , or else Y. J. Ng’s S ≈< n > will only
work for particles with Erelic−particles−effective ≈ 100 ·GeV which is the energy-mass valueof WIMP DM. Needless to say, if the coherent GW state interpretation is correct, for
relic GW, as clumped to make S ≈< n > correct, then if there is a drop in frequency
a billion years ago, for existing Gravitons, with an effective rest mass per graviton , one
may have an explanation for Beckwith’s re acceleration graph when Beckwith found at
z ∼ . 423, a billion years ago, that acceleration of the universe increased, as shown in
Figure 1 which uses a de celebration parameter defined by
q = − aaa2≡ −1− H
H2≈ −1 + 2
2 + δ (z)(23)
Fig. 1 Reacceleration of the universe based on Beckwith’s Dark Side of the Universe lecture(note that q < 0 if z< .423
If a modification of DM along the lines of Eq. (1.4) can be proved, i.e. a small rest
graviton mass, instead of treating Eq. (23) as a purely 4 dimensional construction as was
done by Alves, then one has to consider the following as far as how to get appropriate de
celebration parameter behavior.
Beckwith[12,13] used a version of the Friedman equations as inputs into the deceler-
ation parameter using Maarten’s[15]
a2 =
[(κ2
3
[ρ+
ρ2
2λ
])a2 +
Λ · a23
+m
a2−K
](24)
Maartens [12,13,15] also gives a 2nd Friedman equation, as
H2 =
[−(κ2
2· [p+ ρ] ·
[1 +
ρ2
λ
])+Λ · a23
− 2m
a4+K
a2
](25)
Also, if we are in the regime for which ρ ∼= −P, for red shift values z between zero to
1.0-1.5 with exact equality, ρ = −P, for z between zero to .5 and using a ≡ [a=0 1]/(1 + z).
consider. From [25] we wrote for how to isolate the effects of a 4 dimensional graviton
with rest mass.
If one looks at if a four dimensional graviton with a very small rest mass included [23]
we can write how a graviton would interact with a magnetic field within a GW detector.
1√−g ·∂
∂xν· (√−g · gμαgνβFαβ) = μ0J
μ + Jeffective (45)
where for ε+ �= 0 but very small
F[μν,α] ∼ ε+ (46)
The claim which A. Beckwith made [23] is that
Jeffective∼= ncount ·m4−D−Graviton (47)
As stated by Beckwith, in [23],m4−D−Graviton grams, while ncount is the number of
gravitons which may be in the detector sample. What would be needed to do would be to
try to isolate out an appropriate stress energy tensor contribution due to the interaction
of gravitons with a static magnetic field T μν assuming a non zero graviton rest mass.
The details of the ncount would be affected by the degree of the graviton mass, the
frequency range and a whole lot of other parameters. This requires obtaining a stable
graviton.
Acknowledgements
The author wishes to thank Dr. Fangyu Li for his hospitality in Chongquing, PRC, as
well as Stuart Allen, of international media associates whom freed the author to think
about physics, and get back to his work.
References
[1] A. W. Beckwith,” Inquiry as to if Higher Dimensions Can be Used to Unify DMand DE, if Massive Gravitons Are Stable”, accepted contribution to Dark Side of theUniverse, 2010, Leon, Mexico meeting, http://vixra.org/pdf/1006.0027v2.
[2] A. W. Beckwith, “Can a Massive Graviton be a Stable Particle ?”, sent to the Journalof Modern physics for evaluation, http://vixra.org/abs/1006.0022
[3] A. W. Beckwith, “Configuration of the Arrow of Time, in Initial Start ofInflation?”, accepted contribution to the 2010 rencontres de Blois conference,http://vixra.org/abs/1008.0055.
[4] A. W. Beckwith, “Massive Gravitons Stability , and a Review of How Many GravitonsMake up a Gravity Wave Detectable / Congruent with B.P. Abbott, Et.al., Nature460, 991 (2009)”. http://vixra.org/abs/1006.0051.
[5] A. Yurov, arXiv : hep-th/028129 v1.pdf, 19 Aug, 2002
[6] Y. Ng, Entropy 2008, 10(4), 441-461; DOI: 10.3390/e10040441 Y. J. Ng,”Article:SpacetimeFoam: from Entropy and Holography to Infinite Statistics and Nonlocality”Entropy 2008, 10(4), 441-461; DOI: 10.3390/e10040441 Y. J. Ng,” Quantum Foamand Dark Energy”, International work shop on the Dark Side of the Universe,http://ctp.bue.edu.eg/workshops/Talks/Monday/QuntumFoamAndDarkEnergy.pdfY. J. Ng, Entropy 10(4), pp. 441-461 (2008); Y. J. Ng and H. van Dam, Found.Phys. 30, pp. 795–805 (2000); Y. J. Ng and H. van Dam, Phys. Lett. B477 pp.429–435 (2000);
[7] A.W. Beckwith, http://vixra.org/abs/1002.0056
[8] G. Smoot; 11th Paris Cosmology Colloquium, August 18th, 2007 with respectto Smoot, G, “CMB Observations and the Standard Model of the Universe”’D.Chalonge’ school, http://chalonge.obspm.fr/Programme2007.html
[9] M. Maggiore, Gravitational Waves , Volume 1 : Theory and Experiment, OxfordUniv. Press(2008).
[10] M. Visser, “Mass for the graviton”, Gen.Rel.Grav. 30 (1998) 1717-1728.http://arxiv.org/pdf/gr-qc/9705051
[11] R. Durrer, Massimiliano Rinaldi , “Graviton production in non-inflationarycosmology “, Phys.Rev.D79:063507,2009, http://arxiv.org/abs/0901.0650
[12] A. Beckwith, “Applications of Euclidian Snyder Geometry to the Foundations ofSpace-Time Physics”,EJTP 7, No. 24 (2010) 241–266 http://www.ejtp.com/articles/ejtpv7i24p241.pdfhttp://vixra.org/abs/0912.0012, v 6 (newest version).
[14] G. ’t Hooft, http://arxiv.org/PS cache/quant-ph/pdf/0212/0212095v1.pdf (2002);G. ’t Hooft., in Beyondthe Quantum, edited by Th. M. Nieuwenhuizen et al. (World Press Scientific 2006),http://arxiv.org/PS cache/quant-ph/pdf/0604/0604008v2.pdf,(2006).
[15] R. Maartens, Brane-World Gravity, http://www.livingreviews.org/lrr-2004-7 (2004).;R, Maartens Brane world cosmology, pp 213-247 from the conference The physicsof the Early Universe , editor Papantronopoulos, ( Lect. notes in phys., Vol 653,Springer Verlag, 2005).
[16] J. Y. Kim,” Stability and fluctuation modes of giant gravitons with NSNS B field“,Phys.Lett. B529 (2002) 150-162, http://arxiv.org/pdf/hep-th/0109192v3
[17] S. Lloyd, “Computational Capacity of the Universe”,Phys. Rev. Lett. 88, 237901(2002).
[18] E. Kolb, and S. Turner The Early Universe, Westview Press, Chicago, USA, 1994 .
[19] E. Alves, O. Miranda. and J. de Araujo, arXiv: 0907.5190 (July 2009).
[20] L. Glinka SIGMA 3, pp. 087-100 (2007) arXiv:0707.3341[gr-qc]
[21] L. Glinka AIP Conf. Proc. 1018, pp. 94-99 (2008) arXiv:0801.4157[grqc];arXiv:0712.2769[hep-th]; Int. J. Phys. 2(2), pp. 79-88 (2009) arXiv:0712.1674[gr-qc];arXiv:0711.1380[gr-qc]; arXiv:0906.3827[gr-qc]
[22] A.W. Beckwith., L. Glinka, “The Arrow of Time Problem: Answering if Time FlowInitially Favouritizes One Direction Blatantly”, Prespacetime Journal | November2010 | Vol. 1 | Issue 9 | pp. 1358- 1375, http://vixra.org/abs/1010.0015
[23] A. W. Beckwith, F.Y. Li, et al.,”Is Octonian Gravity relevant near the Planck Scale”,accepted for publication by Nova Book company , http://vixra.org/abs/1101.0017
[24] R. Clive Woods , Robert M L Baker, Jr., Fangyu Li, Gary V. Stephenson , EricW. Davis and Andrew W. Beckwith (2011), “A new theoretical technique for themeasurement of high-frequency relic gravitational waves,” submitted for possiblepublication . http://vixra.org/abs/1010.0062
Planck and Einstein each noted respectively in 1905 and 1909 that e2/c ∼ h havethe same order and dimension.[1, 2] This was before Sommerfeld’s introduction of thefine structure constant α = e2/�c in 1916.[3] Therefore, the search for a mathematicalrelationship between e2/c ∼ h began with blackbody radiation.[4] The Stefan-Boltzmannlaw states that the radiative flux density or irradiance is J = σT 4 [erg · cm−2 · s−1] in CGSunits. From the Planck law, the Stefan-Boltzmann constant σ = 5.670400(40) × 10−5
The Stefan-Boltzmann law can be expressed as the volume energy density of a blackbodyεT = aRT 4 [erg · cm−3], where the radiation density constant aR is linked to the Stefan-Boltzmann constant
aR = 4σ
c= 43π5
5!k4
B
(hc)3 (2)
In 1914, Lewis and Adams noticed that the dimension of the radiation density con-stant divided by the 4th power Boltzmann constant aR/k4
B is (energy × length)−3,while e2 is (energy × length). However, they obtained an incorrect result equivalentto α−1 = �c/e2 = 32π (π5/5!)1/3 = 137.348.[5] In 1915, Allen rewrote it as α = e2/�c =(15/π2)1/3/(4π)2.[6]
In CGS units, e2 = (4.80320427(12) × 10−10)2 [erg·cm], aR = 7.56576738 × 10−15
[erg · cm−3K−4], and k4B = (1.3806504(24) × 10−16)4 [erg4K−4]. We get the experimental
dimensionless constant[7]
αR = e2(
aR
k4B
)1/3
= 1157.5548787 (3)
= 0.00634699482
This is the dimensionless blackbody radiation constant αR.2
2. Relationship to the Fine Structure Constant
The dimensionless blackbody radiation constant αR is on the same order of the finestructure constant α= e2/�c, and equal to
αR = 0.8697668 · α (4)
= 2π
(π5
5!
)1/3
α =(
Γ(4)ζ (4)π2
)1/3
α =(
π2
15
)1/3
α
Therefore, αR �= α, both α and αR are experimental results incapable of producing theα math formula. Physically, the fine structure constant α is obtained from the atomicdiscrete spectra, while the blackbody radiation constant αR is obtained from the thermalradiation of a 3D cavity in the continuous spectra. However, their relationship can begiven by the Riemann zeta-function or by the modification of Euler’s product formula2 Not to be confused with the Stefan-Boltzmann constant σ or hc/k (blackbody radiation constant)
where the Euler product extends over all the prime numbers. In other words, the finestructure constant and the blackbody radiation constant can be linked by the primenumbers.
The pattern of Planck spectra is given by f(x) = x3/(ex − 1) where the photonhν is hidden in the argument x = hν/kBT . The photon integral in (1) is equal to adimensionless constant (Fig. 1)
Figure 1 Photon integral is a dimensionless number Γ(4)ζ (4) = π4
15 = 6.4939394
∞
0
x3dx
ex − 1 = Γ(4)ζ (4) = π4
15 = 2 · 3 ·∏p
(p4
p4 − 1
)(6)
= 2 · 3 · 24
(24 − 1)34
(34 − 1)54
(54 − 1)74
(74 − 1) · · ·
where the Euler product extends over all the prime numbers. The photon distributionintegral (6) yields a zeta-function that is linked to the Euler prime products. (5) and (6)clearly show how the fine structure constant α for the discrete spectra in (4) is convertedto the blackbody radiation constant αR for the continuous spectra by multiplying a di-mensionless constant (Fig. 2). (5) and (6) also indicate that this dimensionless constantcan be expressed as an Euler infinite prime number product.
Figure 2 α and αR from the discrete and continuous spectra.
From (5), the Stefan-Boltzmann law written as the volume energy density of a black-body εT is related to the fine structure constant α and the oscillating charge e2 withdifferent resonating frequencies in a cavity εT = aRT 4 = 4σ
cT 4
εT = ζ (4)ζ (2)
(α
e2
)3k4
BT 4 =(
αR
e2
)3k4
BT 4 (7)
and the radiative flux density is J = σT 4
J = ζ (4)ζ (2)
(α
e2
)3 c
4k4BT 4 = c
4
(αR
e2
)3k4
BT 4 (8)
and the total brightness of a blackbody is B = J/π
B = ζ (4)ζ (2)
(α
e2
)3 c
4πk4
BT 4 = c
4π
(αR
e2
)3k4
BT 4 (9)
and the inner wall pressure of the blackbody cavity is P = 4σ3c
T 4
P = ζ (4)ζ (2)
(α
e2
)3 13k4
BT 4 = 13
(αR
e2
)3k4
BT 4 (10)
According to the Bose-Einstein model of photon-gas,[8] the free energy in the thermody-namics is F = −PV = −4σ
3cV T 4
F = −ζ (4)ζ (2)
(α
e2
)3 V
3 k4BT 4 = −V
3
(αR
e2
)3k4
BT 4 (11)
and the total radiation energy is E = −3F = 3PV = 4σc
V T 4
E = ζ (4)ζ (2)
(α
e2
)3V k4
BT 4 = V(
αR
e2
)3k4
BT 4 (12)
where the photon gas E = 3PV is the same as the extreme relativistic electron gas, andthe entropy is S = −∂F
∂T= 16σ
3cV T 3
S = ζ (4)ζ (2)
(α
e2
)3 4V
3 k4BT 3 = 4V
3
(αR
e2
)3k4
BT 3 (13)
and the specific heat of the radiation is CV =(
∂E∂T
)V
= 16σc
V T 3
CV = ζ (4)ζ (2)
(α
e2
)34V k4
BT 3 = 4V(
αR
e2
)3k4
BT 3 (14)
We have
NkBT = ζ (4)ζ (2)
(α
e2
)3 V
3 k4BT 4 = V
3
(αR
e2
)3k4
BT 4 (15)
From PV = NkBT , the total number of photons in blackbody radiation is
Landau assumed that the volume V in (11)∼(16) must be sufficiently large in orderto change from discrete to continuous spectra.[7] Planck’s law is violated at microscopiclength scales. Experimentally, solids or dense-gas have the continuous spectra, and hotlow-density gas emits the discrete atomic spectra. The photon hν is hidden in f(x) =x3/(ex − 1) where x = hν/kBT , therefore, there is no hν in (7)∼(16). In (7)∼(16),the charged oscillators α/e2 = 1/�c or αR/e2 play a critical role in the electromagneticcoupling on a 3D surface (Fig. 3). Therefore, the traditional 3D box (or sphere) modelis not necessarily composed of solid walls; the plasma gas photosphere layer of a star canhave the same effect.
Figure 3 Blackbody radiation is related to α and e2 in a 3D cavity.
4. Planck’s Law and the Stefan-Boltzmann law
Planck’s original formula is a experimental fitting result. There are many derivationsto explain the blackbody radiation law, including Planck in 1901, Einstein in 1917, Bosein 1924, Pauli in 1955.[9, 10, 11] We are not reinventing the blackbody radiation law, butinstead pointing out that the surface charge is Planck’s oscillator, and it is related to thefine structure constant. Planck’s radiation law is derived as the result of the 3D interfaceinteraction between photons and charged particles.
Using the 3D surface charge model in Fig. 3 and the energy quanta ε = hν, Planck’slaw in terms of the spectral energy density in [erg · cm−3 · sr−1 · Hz−1] can be rewrittenas
where 1/π is related to a solid angle in fractions of the sphere [1sr = 1/4π fractional area];and the interaction ratio of photon hν and charge e2 is regulated by fine structure constantα, the cubic term involving the closed 3D cavity wall; Under the statistic thermodynamicequilibrium, Bose-Einstein distribution can be derived as
∑∞n−0 ne−nhν/kBT∑∞
n−0 e−hν/kBT= 1
ehν/kBT − 1 (18)
The frequency of photons in ε = hν is constantly shifted into a continuous spectra throughphoton-electron scattering, such as the Compton effect
ν ′ = ν
1 + (hν/mec2)(1 − cos θ) (19)
Fig. 3 shows that the scattering angle θ varies during each photon-electron interactioninvolving the fine structure constant. For εT = aRT 4, using dν = (1/h)dε
εT =∞
0
u(ε, T )dε
h=( 1
π
)2 ( α
e2
)3(kBT )4
∞
0
x3dx
ex − 1 (20)
= Γ(4)ζ (4)π2
(α
e2
)3(kBT )4 =
(αR
e2
)3(kBT )4
This links the quantum theory to the classical theory of blackbody radiation with orwithout using the Planck constant
aR = ζ (4)ζ (2)
(α
e2
)3k4
B =(
αR
e2
)3k4
B = ζ (4)ζ (2)
(2π
hc
)3k4
B (21)
=(
α
e2
)3k4
B · 45 · 9
10 · 2526 · 49
50 · 121122 · 169
170 · 289290 · 361
362 · · ·
Planck’s law in terms of the spectral radiative intensity or the spectral radiance in[erg · s−1 · cm−2 · sr−1 · Hz−1] has a electromeganetic radiation with lightspeed c and σ =caR/4, therefore, we multiply c/4 on u(ν, T ) in (17)
The Planck constant h with the revolutionary concept of energy quanta is a bridgebetween classical physics and quantum physics. Einstein’s proposal of the light quantahν in 1905 was based on the Planck constant. In QED, the photon is treated as a gaugeboson, and the perturbation theory involves the finite power series in α. The discrete-continuous spectra is bridged by the Bose-Einstein distribution, and the prime sequenceslink the fine structure constant α to the blackbody radiation constant αR.
5. Wien’s Displacement Law
Wien’s frequency displacement law is
νmax = bν · T (25)
where bν = 5.878933(10) × 1010 [Hz · K] in CODATA-2006. It has the numerical solutionfrom
The blackbody radiation constant is a new method to measure the fine structure constant.It links the fine structure constant to the Boltzmann constant.
Acknowledgment
The Author thanks Bernard Hsiao for discussion.
References
[1] M. Planck, letter to P. Ehrenfest, Rijksmuseum Leiden, Ehrenfest collection(accession 1964) , July (1905)
[2] A. Einstein, Phys. Zeit. , 10 , 192 (1909)
[3] A. Sommerfeld, Annalen der Physik 51 (17), 1-94 (1916)