06002000 Particle Physics

O. M. Boyarkin

Boyarkin

Volume II

Advan

ced

Particle Physics

Physics

The Standard Model and BeyondVolume II

The Stan

dard

Model

and Beyo

nd

www.c rcp r e s s . c om

an informa business

6000 Broken Sound Parkway, NWSuite 300, Boca Raton, FL 33487270 Madison AvenueNew York, NY 100162 Park Square, Milton ParkAbingdon, Oxon OX14 4RN, UK

ISBN: 978-1-4398-0416-2

9 781439 804162

90000

K10230

Helping readers understand the complicated laws of nature, Advanced Particle Physics Volume II: The Standard Model and Beyond explains the calculations, experimental procedures, and measuring methods of particle physics, particularly quantum chromodynamics (QCD). It also discusses extensions to the Standard Model and the physics of massive neutrinos.

Divided into three parts, this volume begins with QCD. It explains the quantization scheme using functional integrals and investigates renormalization problems. The book also calculates cross sections of basic hard processes and covers nonperturbative methods, such as the lattice approach and QCD vacuum. The next part focuses on electroweak interactions, in which the author describes the GlashowWeinbergSalam theory and presents composite models and a left-right symmetric model as extensions to the Standard Model. The book concludes with chapters on massive neutrino physics that cover neutrino properties, neutrino oscillation in vacuum and matter, and solar and atmospheric neutrinos.

Features Presents the technique of renormalization in QCD Discusses the status of current QCD experiments Explores physics beyond the Standard Model, including composite

models and a left-right model Describes how solar and atmospheric neutrinos are detected and

analyzed Includes all necessary derivations and calculations Offers numerous application examples so that readers gain a hands-on

understanding of the material Contains a host of references at the end of each section

K10230_COVER_final.indd 1 1/25/11 3:40 PM

Advanced Particle Physics Volume II

This page intentionally left blankThis page intentionally left blank

The Standard Modeland Beyond

O. M. Boyarkin

A TAYLOR & FRANC IS BOOK

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

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2011 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

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To allthose whom

I sweetlywas deceived by.

Contents

I Quantum chromodynamics 1

1 Canonical quantization 31.1 Fundamental relations of chromodynamics . . . . . . . . . . . . . . . . . . 31.2 Gauge invariance in QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Quantization of the free gluon field . . . . . . . . . . . . . . . . . . . . . . 121.4 Feynman rules in QCD with a covariant gauge . . . . . . . . . . . . . . . 14

2 Formalism of functional integration 212.1 Functional integral in quantum mechanics . . . . . . . . . . . . . . . . . . 212.2 Boson fields quantization by means of the functional integrals method . . 282.3 Fermion fields quantization by means of the functional integrals method . 332.4 Functional formulation of QCD . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Renormalization and unitarity 493.1 Primitive divergent diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 SlavnovTaylor identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Ghosts and S-matrix unitarity . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Renormalization of one-loop diagrams . . . . . . . . . . . . . . . . . . . . 60

4 Asymptotical freedom 774.1 CallanSymanzik equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Evolution of the effective coupling constant in QCD . . . . . . . . . . . . 83

5 Chiral symmetries 915.1 Currents algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Weak and electromagnetic currents of hadrons . . . . . . . . . . . . . . . . 955.3 Adlers sum rules for neutrino reactions . . . . . . . . . . . . . . . . . . . 1015.4 Goldstone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.5 Hypothesis of the partially conserved axial current . . . . . . . . . . . . . 1155.6 U(1)-problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Anomalies 1216.1 AdlerBellJackiw anomaly (perturbative approach) . . . . . . . . . . . . 1216.2 AdlerBellJackiw anomaly (nonperturbative approach) . . . . . . . . . . 1296.3 0 decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.4 Chiral anomalies and chiral gauge theories . . . . . . . . . . . . . . . . . . 1346.5 Anomalous breaking of scale invariance . . . . . . . . . . . . . . . . . . . . 138

7 Hard processes in QCD 1417.1 ee+ annihilation into hadrons . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Deep inelastic scattering of leptons by a proton . . . . . . . . . . . . . . . 1487.3 Parton distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . 154

vii

viii Contents

7.4 Quark-gluon dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.5 Scaling violation. AltarelliParisi equation . . . . . . . . . . . . . . . . . . 1727.6 Fragmentation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817.7 DrellYan process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.8 Gauge bosons production at hadron collisions . . . . . . . . . . . . . . . . 1917.9 Higgs bosons production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8 Lattice QCD 2198.1 Lattice approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2198.2 Scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.3 Fermion fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2268.4 Geometric interpretation of gauge invariance . . . . . . . . . . . . . . . . . 2298.5 QCD action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2338.6 Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2408.7 Hadron masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

9 Quark-gluon plasma 2539.1 Aggregate states of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 2539.2 Looking for QGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2559.3 QGP within holographic duality . . . . . . . . . . . . . . . . . . . . . . . 2669.4 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

10 QCD vacuum 27310.1 Kinks and breathers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27310.2 Topology and vacuum in gauge theory . . . . . . . . . . . . . . . . . . . . 27710.3 Vortices and monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29010.5 Instanton physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

11 QCD experimental status 309

References 315

II Electroweak interactions 323

12 GlashowWeinbergSalam theory 32512.1 Choice of gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.2 Lagrangian in generalized renormalizable gauge . . . . . . . . . . . . . . . 32812.3 CabibboKobayashiMaskawa matrix . . . . . . . . . . . . . . . . . . . . . 33512.4 Four-fermion invariants and Fierz identities . . . . . . . . . . . . . . . . . 33912.5 Muon decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34412.6 (g 2) anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35912.7 Effects of the interference of weak and electromagnetic interactions . . . . 36712.8 Neutrino-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 37312.9 Neutrino-nucleon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 379

13 Physics beyond the standard model 38313.1 Models, models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38313.2 Multipole moments of gauge bosons . . . . . . . . . . . . . . . . . . . . . 38913.3 Left-right model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39613.4 Left-right model in the making . . . . . . . . . . . . . . . . . . . . . . . . 410

Contents ix

Appendix: Feynman rules for the GWS model 431

References 439

III Neutrino physics 443

14 Neutrino properties 44514.1 Neutrino mass in models with the SU(2)L U(1)Y gauge group . . . . . 44514.2 Multipole moments of neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 454

15 Neutrino oscillation in a vacuum 46115.1 Description in the formalism of de Broglies waves . . . . . . . . . . . . . 46115.2 Oscillations and the uncertainty relation . . . . . . . . . . . . . . . . . . . 46515.3 Wave packet treatment of neutrino oscillations . . . . . . . . . . . . . . . 46715.4 Evolution equation for the neutrino . . . . . . . . . . . . . . . . . . . . . . 469

16 Neutrino oscillation in matter 47316.1 Neutrino motion in condensate matter . . . . . . . . . . . . . . . . . . . . 47316.2 Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47916.3 Nonadiabatic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48216.4 Neutrino oscillations in the magnetic field . . . . . . . . . . . . . . . . . . 48816.5 Neutrino oscillations in the left-right model . . . . . . . . . . . . . . . . . 489

17 Solar neutrinos 49717.1 Some information about Sun structure . . . . . . . . . . . . . . . . . . . . 49717.2 Sources of solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . 50017.3 Detection of solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 50217.4 Analysis of neutrino observation data . . . . . . . . . . . . . . . . . . . . . 50617.5 Neutrino propagation through the Earth . . . . . . . . . . . . . . . . . . . 511

18 Atmospheric neutrinos 51518.1 Cosmic rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51518.2 Atmospheric neutrino production . . . . . . . . . . . . . . . . . . . . . . . 51718.3 Atmospheric neutrino detection . . . . . . . . . . . . . . . . . . . . . . . . 519

19 Results and perspectives 523

Instead of an epilogue 543

References 547

Index 551

Part I

Quantum chromodynamics

1

1Canonical quantization

Thank God, that he has created the worldsuch, that everything, that is important in it,is simple, and that is unimportant is difficult.

G. Skovoroda, wandering Ukrainianphilosopher of Eighteenth Century

1.1 Fundamental relations of chromodynamics

In Section 10 we have already gained some insight of the modern theory of the stronginteraction. Before proceeding to the quantization procedure, it is worthwhile to systematizeand expand our knowledge relative to this theory. This chapter is devoted to this task. Inso doing it will be assumed that we deal with classical field theory.The chromodynamics symmetry group SU(3)c is described by generators T

a to obey thecommutation relations

[T a, T b] = ifabcT c, (1.1)

where fabc are structural constants (a, b, c = 1, . . . , 8). Generators in the associate repre-sentation (dimensionality equals eight) are expressed in terms of the structural constants

(T aadj

)bc= ifabc,

T aadjTaadj = 3I, Sp

(T aadjT

badj

)= 3ab.

(1.2)

In its turn, in the fundamental representation (dimensionality equals three) generators aregiven by the set of the GellMann matrices

T afund =1

2a, T afundT

afund =

4

3I, Sp

(T afundT

bfund

)=

1

2ab. (1.3)

The quark field qj belongs to the fundamental representation of the SUc(3), that is, it hasthree components in a color space (j = 1, 2, 3). The gluon field falls into the associaterepresentation (a = 1, . . . , 8). Gluon field strengths Ga are expressed by potentials G

a in

the following way

Ga = Ga Ga + gsfabcGbGc . (1.4)

Sometimes it is more convenient to use the matrix writing down for quantities entering intothe theory rather than the componentwise one. Thus, for example, we have for the gluonfields potentials and strengths:

G = TaadjG

a, G = T

aadjG

a . (1.5)

3

4 Advanced Particle Physics: Standard Model and Beyond

By virtue of the commutation relation (1.1), we have

G = G G igs[G, G ]. (1.6)

Acting the covariant derivative D on the gluon fields G is defined as follows:

DG = G igs[G, G ],

or in the componentwise written as

DGa = [ac + gsf

abcGb]Gc . (1.7)

On the other hand, for the quark field qj we have

Dqj = [jk igs(T afund)jkGa]qk, (1.8)

where j, k are color indices. Under the covariant differentiation Jacobis identity is fulfilled

[[D, D ], D] + [[D , D], D] + [[D, D], D ] = 0.

Reasoning from the covariant derivative definition it is easy to check that

[D, D ]G = igs[G , G].

For the gluon field strengths Bianchis identity takes place

DG +DG +DG = 0. (1.9)

At the gauge transformation with parameters a(x) the gluon field tensor and potentialsare transformed according to the formulae

G eiGei,G eiGei ig1s

(e

i) ei,}

(1.10)

where = aT aadj. In the case of the infinitesimal transformations Eqs. (1.10) take theform

G G i[,G ],G G i[,G] g1s = G g1s D.

}(1.11)

In the componentwise writing down the infinitesimal gauge transformations will look like:

Ga Ga + fabcbGc g1s a, (1.12)

for the gluon field and

qj [jk + i(T afund)jka]qk, (1.13)for the quark field. The total Lagrangian describing the gluon and quarks fields is given bythe expression

LQCD = i2{q(x)[q(x)] [q(x)]q(x)} + gsq(x)T afundGa(x)q(x)

mq(x)q(x) 14Ga(x)G

a(x). (1.14)

Canonical quantization 5

Using the least action principle, we obtain the following equations of motion

iq(x) + gsG

(x)q(x) mq(x) = 0,i[q(x)] + gsq(x)G(x) mq(x) = 0.

}(1.15)

for the quark field andD(x)G

(x) = gsJ(x), (1.16)

whereJa(x) = q(x)T

afundq(x) (1.17)

is a fermion current, for the gluon field. Taking into account Eq. (1.16) one may show thatthe current Ja satisfies the so-called compatibility condition

D(x)J(x) J(x) igs[G(x), J(x)] = 0. (1.18)

From Eq. (1.16) follows that J(x) is determined by the expression

J(x) = i[G(x), G(x)]. (1.19)

Now we can define the conserved total color current

Jc (x) J(x) + i[G(x), G(x)] (1.20)and the corresponding charge

Qc = gs

d3xJ0c (x) =

d3xiG

0i(x) =

S

dSiG0i(x). (1.21)

Note that Qc is not changed under the gauge transformations to vanish at infinity.If we wish to consider external sources Ja of the YangMills fields being given in the space-

time (for example, heavy quarks), then we cannot set their color components arbitrarily.The compatibility condition demands orientations of Ja in the color space to be consideredas dynamical observables connected with the field Ga .By analogy with electrodynamics we introduce chromoelectric and chromomagnetic fields

Eai = Ga0i = Gi0a , Hai =

1

2ijkGjka , (1.22)

that is,

G =

0 E1 E2 E3E1 0 H3 H2E2 H3 0 H1E3 H2 H1 0

. (1.23)

The electric and magnetic fields are expressed in terms of the potentials in a nonlinear way:

Ea = 0Ga G0a + gsfabcGbG0c ,Ha = [Ga] 12gsfabc[Gb Gc].

}(1.24)

Let us rewrite the motion equations for the gluon field in terms of chromoelectric andchromomagnetic fields

Ea + gsfabcGb Ec = gsJ0a ,0Ea [Ha] + gsfabcG0bEc + gsfabc[Gb Hc] = gsJa.

}(1.25)


These equations are analogous to the pair of the Maxwell ones to involve sources. One moreequations pair corresponds to the Bianchi identity (1.9)

Ha gsfabcGb Hc = 0,0Ha + [Ea] + gsfabcG0bHc gsfabc[Gb Ec] = 0.

}(1.26)

There is a fundamental distinction between the chromodynamics and Maxwell equations.First, the chromodynamics equations are nonlinear. It leads to a self-action of the gluonfield, that is, different field components are interacting with each other. Second, the chromo-dynamics equations being gauge invariant nevertheless involve, along with the field tensor,the potentials Ga, which depend on a particular gauge choice. Thus, in chromodynam-ics not only does the field tensor has physical meaning, but the potentials being the fieldequation solutions are physical observables as well.In theory a gluon field pseudotensor G that is dual to the tensor G :

G =1

2G

is also used. It is easy to verify that components of G are expressed in terms of thestrengths E and H by the following way:

Gi0 = G0i = Hi, Gij = ijkEk, Ei =

1

2ijkGjk,

that is, just as in the case of electrodynamics they follow from the gluon field tensor com-ponents by means of the dual transformation

E H, H E.

Multiplying the Bianchi identity by and contracting the umbral indices, we arrive at

DG = 0.

With the help of G one may define an important type of gauge fields. The gauge fieldis called self-dual (antiself-dual) if the components of its tensor and the ones of the dualtensor are connected by the condition

Ga = iGa (G

a = iGa ). (1.27)

The validity of Eqs. (1.27) follows from the obvious relation

G = G .

When we address the evident writing down of the tensors Ga and Ga , the condition (1.27)

is reduced to the demandHa = iEa (Ha = iEa). (1.28)

Appearance of the imaginary unit is caused by using the pseudo-Euclidean metric of theMinkowski space. That, by the way, explains why self-dual fields are not usually consideredin electrodynamics where all fields are real.As is well known, electrodynamic fields are classified by means of values of their invariants

FF and FF . In non-Abelian SU(3)c-theory quantities

Ga Ga = 2(H2a E2a), Ga Ga = 4(Ea Ha) (1.29)


are their analogues. It should be stressed that in parallel with (1.29) there is a reach varietyof other invariants in the chromodynamics. As a result, the total classification of fields isvery complicated.The energy-momentum tensor is a dynamical characteristic of classical fields

T =L

(Qi)Qi L, (1.30)

where Qi = Ga(x), q(x), q(x). Substituting (1.30) into the Lagrangian (1.14), we get

T = Ga Ga +1

4gGaG

a +i

2[q(q) (q)q]. (1.31)

Since the spins of the gluon and quark fields do not equal zero, the obtained canonicalenergy-momentum tensor appears to be asymmetric. However, the energy-momentum ten-

sor is defined ambiguously. We may always add to it a total divergence of the kind f[] ,

where f[] is the third-rank tensor antisymmetric with regard to indices and . To make

the expression (1.30) symmetric and obtain the right relation between the angular momen-

tum and energy-momentum tensors, we should define the quantity f[] with the help of

the spin-moment tensor (see Section 4.3 of Advanced Particle Physics Volume 1). Carryingout corresponding computations, we arrive at the following expression for the symmetrized(metric) energy-momentum tensor (Belinfantes form):

T (metr) = T +1

2[

(j)

(j)Qj (j) (j)Qj +(j) (j)Qj ] (1.32),

where summing over all the fields Qj = Ga, q, q is implied and

(j) =

L (Qj)

,((G)

)

= gg gg, (q) =1

4[, ].

By means of the motion equations the symmetrized energy-momentum tensor T(metr) may

be represented in the form

T (metr) = T(metr)G + T

(metr)q , (1.33)

where

T (metr)G = GaGa +1

4gG

aG

a ,

T (metr)q =i

4[q(q) (q)q + q(q) (q)q] + gs

2(qGq + qGq) .

The energy-momentum tensor of the gluon field T(metr)G has the track equal to zero.

Having expressed its components in terms of the electric and magnetic strengths, we obtainthe expressions for the gluon energy density:

T(metr)G00 =

1

2

(E2a +H

2a

), (1.34)

for the Poynting vector components:

T(metr)G0i = [E

a Ha]i , (1.35)and for the stress tensor:

T(metr)Gij =

(Hai H

aj + E

ai E

aj

)+1

2ij(E2a +H

2a

). (1.36)


We see that the expressions obtained are completely analogous to the corresponding onesof the electrodynamics.The quark part of the energy-momentum tensor is manifestly gauge invariant because it

is written in terms of the fields q, q, and the covariant derivatives to act on these fields

T (metr)q =i

4[q(Dq) (Dq)q + q(Dq) (Dq)q] , (1.37)

Using the motion equation, one could show that the track of T(metr)q is proportional to a

quark mass(T (metr)q) = mqq. (1.38)

Hereon, we complete the formulae summary of chromodynamics and proceed to the quan-tization of this theory.

1.2 Gauge invariance in QCD

All quantum field theories to successfully describe the world surrounding us are non-Abeliangauge theories. Those, in turn, are based on gauge-invariance principles generalizing theordinary gauge U(1)-invariance of quantum electrodynamics (QED). The significance ofthese principles lies in the fact that the main properties and even the gauge field existenceitself can be deduced from the invariance of a theory with respect to gauge transformations.Thus, for example, in quantum chromodynamics (QCD) the gauge invariance (GI) guaran-tees the following: First, as follows from the covariant derivative definition, the GI demandsuniversality of a coupling constant, that is, one and the same coupling constant gs governsthe interaction of quarks with gluons and self-interaction gluons. Second, as was shown inRef. [1], the GI ensures the renormalizability. Third, only non-Abelian gauge theory canpossesses the property of the asymptotic freedom [2].However, under quantization, the GI gives rise to considerable troubles. The GI of a

theory accounts for the fact that couplings are imposed on dynamical variables of a systemunder consideration, that is, among these variables are those not to be associated with truedynamical degrees of freedom. Before proceeding to consideration of QCD, it is useful toaddress the simpler construction, namely, to QED where troubles connected with the GIare being met in a simple form.The gauge invariant Lagrangian of the free electromagnetic field has the form

L = 12F

(1

2F A + A

). (1.39)

Varying L over A and F leads to the equations

L

(A) LA

= 0. (1.40)

L

(F) L(F)

= 0, (1.41)

From (1.39) (1.41) it follows the Maxwell equations and definition of the electromag-netic field tensor. Within the canonical formalism the canonical variables A(x) and the


conjugated momenta

(x) =L

[0A(x)]

are considered as operators to obey the commutation relations

[A(r, t), (r, t)] = ig(3)(r r). (1.42)

Subjecting the electromagnetic field potentials A(x) to the gauge transformation

A(x) A(x) = A(x) + (x),

where (x) is an arbitrary function, we obtain already other commutation relations. Ithappens because photons possessing zero mass have only two degree of freedom while A(x)is a four-dimensional vector. And what is more, it is impossible to fulfill the relations (1.42)for all components of the potential A(x) since there exist several constraints in the theory.Really, using (1.39), we find

0(x) =L

[0A0(x)]= F00(x) = 0. (1.43)

This restriction is called a primary constraint inasmuch as it follows directly from theLagrangian structure. At = 0, Eq. (1.40) gives the second constraint for that quantity tobe fixed by the primary constraint:

F0(x) = ii(x) = L

A0= 0. (1.44)

Now the quantities 0(x) and (x) vanish. As a result, the operators A0(x) and A(x)commute with all canonical operators, that is, they are actually c-numbers. Therefore, thecanonical commutation relations for transverse fields A(x) and (x), to be in line withtwo independent freedom degrees, must be determined so that they will be compatible withthe aforementioned constraints. Fictitious degrees of freedom that are present due to thegauge invariance must be removed by means of an appropriate choice of a condition fixinga gauge. And at the same time, two possibilities exist. The first consists in a choice of sucha gauge to ensure the absence of the nonphysical degrees of freedom. Since, in so doing, theLorentz invariance is manifestly violated, these gauges are said to be noncovariant. Amongthese are

A(x) = 0 Coulomb, or radiation gauge,A3(x) = 0 axial gauge,

A0(x) = 0 temporal, or timelike gauge.

The second possibility resides in the fact that all the field components A(x) are consideredon an equal footing (covariant gauges). Since nonphysical degrees of freedom are retained,then the necessity of introducing an indefinite metric space appears.Needles to say the canonical quantization method is not unique in quantum field theory.

A quantization formalism with the help of functional integrals is very convenient for gaugetheories. We shall become acquainted with it in Chapter 22 and now we consider using thecovariant gauges in QED.Quantizing a free electromagnetic field showed that there is no way to impose the Lorentz

gauge

A(x) = 0 (1.45)


and at the same time to preserve the commutation relations. A consistent theory will beobtained only in the event that a physical state satisfies the condition

< |A(x)| >= 0 (1.46).In Section 17.3 of Advanced Particle Physics Volume 1 in order to take into account gaugeambiguity of potentials, we introduced the gauge-fixing term to the electromagnetic fieldLagrangian

LGF = 12A

A . (1.47)

Such a modification does not lead to any physical consequences since on the strength ofthe condition (1.46) matrix elements of (1.47) over the physical vectors vanish. Now allthe momenta to be canonical conjugated to the fields A(x) do not equal to zero and therelations (1.42) are fulfilled for all values of and . In this case the photon propagator isdefined by the expression

Dc(k) = [g + ( 1)kk

k2

]1

k2. (1.48)

( = 1 is Feynman gauge, = 0 is transverse gauge or Landau one).Let us investigate consequences of introducing the term LGF . We consider an infinitesimal

gauge transformation for the potential A

A A e1(). (1.49)At that the action variation is given by the expression

S =

(L+ LGF )d4x =

LGFd4x = 1

2

(A

e1)

(A e1)d4x 12

A

Ad4x =

1

e

(A

)d4x =

=1

e

(

A)d4x = 0. (1.50)

From (1.50) follows the equation

(x) = 0, (1.51)

where (x) = A(x), defining the behavior of massless scalar -particles which, as is easy

to see, do not interact with any other particles. There is nothing to prevent a choice ofinitial conditions in the view

(x)

t=0

= 0,(x)

t

t=0

= 0. (1.52)

Then, at any instants of time, we shall always have (x) = 0.However, introducing LGF violates manifest gauge invariance that is very undesirable.

To restore this important property of the theory one should append the term

L = 12(x)

(x)

corresponding to a free neutral scalar massless field (x), to the total QED Lagrangian

LQED = q(x)(iD m)q(x) +1

2F(x)

[1

2F(x) A(x) + A(x)

]


12A

(x)A(x). (1.53)

The field (x) is nonphysical and the particles connected with it will be called ghosts. Let usgeneralize the gauge transformations in such a way as to include the ghost fields. Defininginfinitesimal transformation parameters in the form (x) = (x), we obtain for q(x) andA(x)

q(x) q(x) + ie(x)q(x), A(x) A(x) (x). (1.54)If one demands (x) to be transformed by the law

(x) (x) A

(x), (1.55)

the total LagrangianLQED = LQED + L , (1.56)

will be invariant relative to the transformations (1.54) and (1.55) with a precision of aninessential four-dimensional divergence. Thanks to the fact that photons are neutral anddo not interact with each other, the fields may be chosen in the form of free real fields.One could show [3], that the gauge transformations (1.54) and (1.55) generate all the Wardidentities. The S-matrix unitarity, that is, the absence of transitions in the Hilbert spacebetween the physical and nonphysical states (states with the opposite metric parity), is oneof consequences of these identities. In order to check this circumstance it is enough to provethat the photon propagator is transverse in all orders of the perturbation theory. We shalldemonstrate that such is the case.Invariance of the quantity < 0|T (A(x)(0))|0 > with respect to the gauge transforma-

tions (1.54) and (1.55) leads to the relation

1

< 0|T (A(x)A(0))|0 >=< 0|T ((x)(0))|0 > . (1.57)

Passing on to Fourier images in the right and left sides of (1.57), we get

1

< 0|T (A(x)A(0))|0 > exp[ikx]d4x =

=ik

< 0|T (A(x)A (0))|0 > exp[ikx]d4x = ik

Dc(k) (1.58)

and< 0|T ((x)(0))|0 > exp[ikx]d4x = ik

< 0|T ((x)(0))|0 > exp[ikx]d4x =

=k

k2 + i0, (1.59)

where it has been taken into account that (x) describes a scalar field. Equating (1.58)with (1.59), as well as allowing for separating the photon propagator on the transverse andlongitudinal parts

Dc(k) =

(g + kk

k2

)D(tr)(k2) +

kkk2

D(l)(k2), (1.60)

we obtain

D(l)(k2) = ik2 + i0

. (1.61)


So, the longitudinal part of the photon propagator has the same view as in the case of afree electromagnetic field. In other words, if one expands the propagator Dc(k) as a powerseries in the interaction constant

Dc(k) = Dc(0) (k) +

e2

4Dc(2) (k) + ..., (1.62)

then all the quantities Dc(n) (k) (n = 2, 4, ...) will satisfy the transversality condition

kDc(n) = 0. (1.63)

Taking into consideration that the photon polarization operator (n) (k) is connected with

Dc(n) by the relation

Dc(n) (k) = Dc(0) (k)

(n)(k)Dc(0) (k), (1.64)

we also find the transversality condition of the electromagnetic field polarization tensor inall orders of the perturbation theory

k(n) (k) = 0.

Thus, the aforementioned procedure allows us to carry out quantization in the QED andat the same time to conserve the manifest gauge invariance and the S-matrix unitarity.

1.3 Quantization of the free gluon field

Using canonical formalism, we shall quantize a free gluon field. The Lagrangian of thetheory under investigation is as follows:

LG = 12Ga(x)

[1

2Ga (x) Ga(x) + Ga(x) gsfabcGb (x)Gc (x)

]. (1.65)

Simultaneous commutation relations are written in the form

[Ga(r, t), b (r

, t)] = iabg(3)(r r), (1.66)where the momenta b (x) canonically conjugated to the fields G

b (x) are given by the

expressionb (x) = G

0b (x). (1.67)

Since zero components of the momenta 0b (x) are identically equal to zero, then we shouldfix a gauge to remove contradictions. We shall use the covariant gauge. Let us introducethe gauge-fixing term

LGF = 12

a

(Ga)(G

a) (1.68)

into the Lagrangian (1.65) and demand the physical state vectors to satisfy the relation

< 0||Ga)(x)| >= 0. (1.69)Now the momenta b (x) will look like

b (x) = G0b (x)

1

g0G

b (x). (1.70)


Since none of the components b (x) turns into zero, the canonical quantization schemeis free from contradictions. However, the Hilbert space of the theory under considerationpossesses the indefinite metric. In order to be certain that such is the case, we examine therelation (1.66) when = 0:

1

[G

a(r, t), G

b (r

, t)] = iabg0(3)(r r). (1.71)

In the momentum space we introduce canonical tetrads () (k) (k2 = 0):

(i)0 = 0, (

(i) k) = 0, i = 1, 2,

(3) = k(k

0)1 0, (i) (j) = ij , i, j = 1, 2, 3, (0) = 0

(1.72)

and choose the gauge parameter as = 1. The components (i) (i = 1, 2) are associated

with the physical freedom degrees, (3) represent the longitudinal component, while the

component (0) describes the scalar particle. Then, expansion over the production and

destruction operators for the gluon field becomes

Gb (x) =1

(2)3/2

d3k2k0

[()(k)a(b, k) exp (ikx)+

+()(k)a(b, k) exp (ikx)]. (1.73)

Substituting (1.73) into (1.66), we arrive at the following commutation relations:

[a(b, k), a(b

, k)] = bbg(3)(k k). (1.74)From (1.74) it follows that the vacuum expectation value of the scalar particles numberoperator

N0 = a0(b, k)a0(b, k)

in the gauge at hand is negative, that is, we deal with the indefinite metric theory.Let us pass on to the gluon field potentials into the Lagrangian (1.65) and represent the

sum LG + LGF = Lt in the form of two terms:Lt = L0(G) + Lself (G, gs). (1.75)

The former involves terms that are quadratic in the gluon field and do not include self-interaction

L0(G) = 14(Ga Ga)2 (1/2)(Ga)2, (1.76)

while the latter contains terms describing self-interaction of gluons

Lself (G, gs) = L(1)self + L(2)self , (1.77)

L(1)self = 1

2gsfabc(Ga Ga)GbGc = gsfabc(Ga)GbGc , (1.78)

L(2)self = 1

4g2sfabcfadeGbGcGdGe . (1.79)

Writing the Lagrangian as (1.75) allows us to quantize a free gluon field in the linearapproximation (Lt L0(G)) and then take into consideration the triple and quadruplevertices using the perturbation theory in powers of the coupling constant gs. As switching


on the gluon interaction with matter leads to the appearance of the quark-gluon verticesinvolving gs, this procedure is altogether lawful. As a result we have the perturbation theoryseries in powers of the same coupling constants gs.The expression L0(G) may be chosen in the form of the componentwise sum:

L0(G) =a

L0(Ga) (1.80)

which represents the sum of the quadratic nondegenerate forms to be analogous to theelectromagnetic field Lagrangian

L = 14(A A)2 1

2A

A (1.81)

Using the same methods as in the case of the electromagnetic field, we get the expressionfor the gluon propagator

Dcab(k) = ab[g + ( 1)kk

k2

]1

k2. (1.82)

The quadratic Lagrangian L0(G) is often called a free gluon field Lagrangian. In this casethe fields being described by the Lagrangian (1.75) are named by gluon fields in a vacuum.Further, in order to avoid confusion we choose the following terminology. The field toobey linear equations resulting from the quadratic Lagrangian L0(G) will be called a linearapproximation field or a linear gluon field. The gluon field being described by the LagrangianLt, which does not include the gluon interaction with matter fields, will be referred to as afree gluon field.Applying the ordinary quantization rules to Lt, one can build up the perturbation theory

in powers of the coupling constant gs. In the momentum representation the Feynman rulesfor the free gluon field involve the following elements:

1) triple gluon vertex (Fig. 1.1)

V3G = gsfabc[g(k p) + g(p q) + g(q k) ](2)4(4)(p+ k + q) (1.83)2) quadruple gluon vertex (Fig. 1.2)

V4G = ig2s{fablfcdl(gg gg) +

(b c

)+

(b d

)}(2)4(4)(

k)

(1.84)3) motion of virtual gluon with momentum k (Fig. 1.3)

P(G)ab (k) =

iab(2)4

[g + ( 1)kk

k2

]1

k2. (1.85)

1.4 Feynman rules in QCD with a covariant gauge

Because photons are not subjected to self-interaction, then using covariant gauges doesnot produce any additional troubles within QED. In the case of QCD, however, gluonself-interaction results in considerable complications.


(, c)

p

(, a) (, b)

k

q

FIGURE 1.1

Triple gluon vertex.

(, a) (, b)

(, c) (, d)

FIGURE 1.2

Quadruple gluon vertex.

(, a) (, b)

k

FIGURE 1.3

Gluon propagator.

Let us extensively investigate the consequences of introducing the gauge-fixing term intothe Lagrangian. To find an equation the scalar field a = G

a obeys, we subject a free

gluon field to an infinitesimal gauge transformation

Ga(x) = Ga(x) + Ga(x), (1.86)

with

Ga(x) = 1gsDa(x). (1.87)

Further, we compute the action variation of the free gluon field being described by theLagrangian LG + LGF . In so doing we take into account that LG = 0. From the least


action principle follows:

S =

d4xLGF = 1

d4x(G

a)(G

a) =

= 1

d4x (G

a)

[

(fabcb(x)G

c (x)

1

gsa(x)

)]=

=1

gs

d4x[a(x)

(Ga) + gsfabcG

c b(x)(G

a)] = 0. (1.88)

In such a way, for -particles we get the equation:

Da(x) = 0,

or in an explicit form[ab

gsfabcGc (x) ]a(x) = 0. (1.89)Unlike QED, this equation is not free, since it includes interaction with the field Gc . By thisreason, the field a will be generated by the field G

c even at zero initial conditions for a

and a/t. In quantum language it means that the field Gc excites quantum fluctuations

of the field a, that is, generates -particles that can be both in real and virtual states.Let us switch on the gluons interaction with the matter field. In this case the QCD

Lagrangian is given by the expression

LQCD = i2[qk(x)

qk(x) qk(x)qk(x)] mqk(x)qk(x)

gs[qk(x)Gkj(x)qj(x)]1

4(Ga(x))

2 12(G

a)

2, (1.90)

where

Gkj(x) =

1

2Ga(x) (a)kj

and k, j are color indices. It is evident that the Lagrangian (1.90) also generates Eq. (1.89)for -particles. Inasmuch as the equation (1.89) contains the source giving rise to -particles,then its solution cannot be expanded into positive and negative frequency parts ((+) and()) in a relativistically invariant way. Therefore, in QCD we cannot choose physical states| > in such a way as to

(+)| >= 0, (1.91)as it was possible in QED. Since -particles possess the negative metric parity, transitionsbetween the Hilbert space parts having the opposite metric exist, that is, the theory isnonunitarity. To put it differently, the Lagrangian Lt transfers physical states whose metricis positive into nonphysical states to possess a negative metric. For the first time thiscircumstance was mentioned in Ref. [4]. A solution of the problem for some particular caseswas proposed by Feynman [5] while Faddeev and Popov found a solution in the general case[6]. The idea is as follows. Along with the fields being present at the Lagrangian, additionalnonphysical fields, ghosts, are introduced in such a way that they turn nonphysical statesproduced by the Lagrangian Lt into zero. A multiplet of scalar fields a(x) described bythe Lagrangian

LFPG = (a(x))[ab gsfabcGc (x)]b(x) = (a(x))Da(x) (1.92)is used as ghosts. From LFPG, in its turn, follows the equations

[ab gsfabcGc (x) ]a(x) = 0,

[ab gsfabcGc (x) ]a(x) = 0,

}(1.93)


that is, in the linear approximation a(x) and a(x) obey the KleinGordon free equation for

the massless particle. Let us introduce a new quantum number ngh, the number of ghosts.It is equal: to +1 for the ghost field a, to -1 for the antighost field and to zero for all otherfields. Then, from the Lagrangian LFPG follows the ghosts number conservation law. Ghostlines enter into diagrams only in the form of loops. In parallel with every diagram involvinga closed loop of a gauge field there exists a topologically equivalent diagram with a closedghost line in the same place. Ghost fields are subjected to the FermiDirac statistics. Thisleads to the fact that a closed loop of the a(x) field has a redundant minus sign as comparedwith a loop of the -particles. Since the multiplet of the a(x) ghosts satisfies the sameequation as does the field a(x), then loops connected with - and -fields are mutuallycancelled. Thus, the aforementioned trick results in mutual cancellation of contributionscoming from nonphysical states and and allows us not to include these particles in realstates.The term LFPG causes the appearance of the following Feynman diagrams:

4) motion of virtual ghost with momentum p (Fig. 1.4)

P()ab (p) =

i

(2)4abp2

, (1.94)

pba

FIGURE 1.4

The line associated with the ghost as well as the fermion line, has a direction, that is, theghost differs from its antiparticle.

5) ghost-ghost-gluon vertex (Fig. 1.5)

VG = gsfabcp(2)4(

k). (1.95)

In the theory the following diagrams associated with quarks are also present:6) propagation of virtual quark with momentum p (Fig. 1.6)

P (q)(p) =1

(2)4i

pm =1

(2)4i(p+m)

p2 m2 (1.96)

7) quark-gluon vertex (Fig. 1.7)

VqqG =igs2(2)4 (a)kj

(k). (1.97)

To obtain the total set of Feynman rules in the QCD, we have to supplement (1.83)(1.85),(1.94)(1.97) with rules describing particles in the initial and final states. For quarks and

Since the fields a(x) do not appear in both the initial and final states, the inconsistency between theirspin and statistics must not give concern.


p

a c

b,

FIGURE 1.5

Ghost-ghost-gluon vertex.

p

FIGURE 1.6

Quark propagator.

q

p k

b,

FIGURE 1.7

Quark-gluon vertex.

gluons these rules follow from the corresponding QED ones under the replacementsleptonsquarks, photonsgluons.

In the Feynman gauge, the summation over gluon polarizations is fulfilled using the relation

()(k)()(k) = g . (1.98)

Note, that only the gauge field propagator depends on the gauge parameter .Introducing the ghosts into the QCD Lagrangian that is written in a covariant gauge was

stimulated by the demand of the S-matrix unitarity reconstruction. However, in view of thegauge invariance of the theory, the S-matrix unitarity property must be carried out in anygauge. It is evident that the given violation is connected with introducing the gauge-fixingterm that does not possess the gauge invariance property. In Section 1.3, we showed that inthe QED case introducing the ghosts may be interpreted as a manner of the gauge invariancereconstruction of the Lagrangian. Let us demonstrate that the analogous property takesplace in the QCD as well.For a non-Abelian theory BecchiRouetStora (BRS) transformations [7] are generaliza-

tions of the gauge transformations (1.45) and (1.55). Here, the ghost fields, as well as all


other fields, are subjected to gauge transformations and, as a result, LQCD becomes gaugeinvariant with a precision of a four-dimensional divergence. The BRS transformations re-sult in Slavnov[8]Taylor[16] identities that represent the analogue of the Ward identitiesin QED.It is more convenient to use real Grassmann fields and to be defined as

a =12(a + ia),

a =

12(a ia)

rather than the complex ghost fields a and a. Making use of the anticommutativity

property of Grassmann fields

2a = 2a = 0, ab = ba,

we obtain

LFPG = i

(a(x))[ab gsfabcGc (x)]b(x) = i

(a(x))D

a(x). (1.99)

Infinitesimal BRS transformations have the form

Ga Ga + (ab gsfabcGc )b,q q + igs(a/2)aq, a a (/2)gsfabcbc,

a a (i/)Ga ,

(1.100)

where an infinitesimal BRS transformation parameter is an anticommuting Grassmannvariable not dependent on x. It should be noted that the production a is an ordinarynumber. Thus, we have

2 = 0, a = a, q = q, Ga = Ga. (1.101)Using these transformation in just the same way as it was done in the QED, one may showthat the longitudinal propagator part of a gluon is defined by the expression

D(l)ab (k) =

iabkk/k2

k2, (1.102)

that is, has the same form as in the case of a free gluon field. Then, expanding Dcab (k)into series in square powers of the coupling constant gs

Dcab (k) = Dc(0)ab (k) +

(g2s4

)Dc(2)ab (k) + . . . , (1.103)

we get the transversality condition of a gluon field in all orders of the perturbation theory

kDc(n)ab (k) = 0, n = 2, 4, ... (1.104)

Since (1.104) is the direct consequence of the gauge invariance, we convince ourselves thatintroducing the FaddeevPopov ghosts really restores the gauge invariance of the theory.The gauge invariance of the total QCD Lagrangian could be demonstrated by the example

of the action invariance with respect to the BRS transformation. Because

Ga = (a gsfabcbGc ),

then the BRS transformation is actually a gauge transformation associated with a particularchoice of a gauge function

a(x) = gsa(x).


In that way, the action

SQCD =

d4xLQCD

is not changed under this transformation, that is, SQCD = 0 and proving the total actioninvariance

Seff = SQCD + Sgf + Sgh (1.105)

is reduced to proving the relation

S =

d4x[LGF + LFPG] = 0.

We have for S

S = 1

d4x(G

a)(G

a) i

d4x(a)(D

a) id4x(a)(D

a). (1.106)

First, we find variation of the covariant derivative of b:

(ab gsfabcGc )b =gs2fabc

(bc) + gsfabcb[(c gsfcdedGe)+

gsfabcGc(gs2fbdede

). (1.107)

Because a is a Grassmann quantity, then the derivative in the first term is

fabc(bc) = fabc[(

b)c + b(c)] = fabc[(

b)c (c)b] =

= 2fabc(b)c. (1.108)

Allowing for the structural constants fabc to obey the Jacobi identity

facbfbde + fadbfbec + faebfbcd = 0, (1.109)

we can rewrite the last term in Eq. (1.107) as

g2s

2facbfbdedeG

c =

g2s

2[fadbfbecdeG

c + faebfbcddeG

c ] =

= g2sfadbfbecdeGc . (1.110)Bringing together the results obtained, we arrive at

(a gsfabcGcb) = 0. (1.111)

With allowance made for Eqs. (1.100) and (1.111), the final result is as follows

S =

d4x(G

a)(D

a)

d4x(G

a)(D

a) =

=

d4x[(G

a)(D

a)] = 0. (1.112)

So, we have proved that Seff is invariant with regard to the BRS transformations.

2Formalism of functional integration

Way down below youll never meetThis magic beautya tenth of it!

V. Vysotsky

2.1 Functional integral in quantum mechanics

In this chapter we shall again obtain the same set of the QCD Feynman rules as in theprevious chapter. But now, our approach will be based on a formalism of a functional (orcontinual) integration. The paths integral idea introduced by R. Feynman [10] in quantummechanics lies at the heart of this formalism (detailed presentation of this method couldbe found in the book [11]). Later on this method received wide recognition in statisticalphysics from which it was extended to quantum field theory. The functional integrationmethod has a whole series of advantages over other quantization methods. It traces thelinkage with classical dynamics. Besides, under such an approach we are operating withordinary functions. This permits carrying out of any nonlinear transformations of funda-mental dynamical variables in a more simple style and to directly see effects appearing underthese transformations. The functional integration method is a general and self-containedquantization method that works for a wide class of systems including those with couplings.It easily deduces the Feynman rules and, at the same time, gives an opportunity for calcu-lating nonperturbative Green functions. The advantage of the method is that it establishesthe commonness of quantum field theory (QFT) with quantum statistical physics. This, inturn, permits the use of the quantum statistics technique in QFT. Formulation compactnessof fundamental relations of QFT enables us to easily trace the structure of one or anothermodel and its gauge invariance in particular.An analogue with Gauss integrals lies at the heart of the functional integration. In a

theory of the function of a single variable a Gauss integral has the form

dyexp (ay2) = 1

a, (2.1)

where a > 0. Let us consider a quadratic form of k variables

(y, Ay) k

i,j=1

yiAijyj 0. (2.2)

Diagonalizing this expression with the help of the orthogonal linear transformation, we caneasily ascertain the validity of the following value of the k-dimensional Gauss integral:

dy1 . . .

dyk()k/2

exp [(y, Ay)] = [detA]1/2. (2.3)

21


Further, we carry out generalization of the finite-dimensional case on infinite-dimensionalfunctional space. At that the index i is replaced with a continuous variable x (i x, , (2.7)

where states |q > obey the condition Q|q >= q|q >, symbols |q; t > label states in theHeisenberg picture, and |q; t >= exp (iHt)|q >, is a fundamental quantity in the NQM.There is need to bear in mind that states in the Heisenberg picture do not depend on timeand our designations mean the states |q; t > and |q; t > coincide with two Schrodingerstates |q(t) > and |q(t) > in the time moments t and t, respectively. We divide theinterval (t, t) into a great number of small periods

t tn1, tn1 tn2, . . . , tj tj1, . . . , t1 t.

Then the transition amplitude may be represented as

< q| exp [iH(t t)]|q >=

dq1...dqn1 < q| exp [iH(t tn1)]|qn1 >

< qn1| exp [iH(tn1 tn2)]|qn2 > ... < q1| exp [iH(t1 t])|q >, (2.8)where we have substituted total eigenstate sets of the operator QS in the Schrodingerpicture. For arbitrarily small tj = tj tj1 we have

< qj | exp [iH(tj tj1)]|qj1 >=< qj |[1 iH(P,Q)tj ]|qj1 > +O[(tj)2]. (2.9)

Formalism of functional integration 23

In Eq. (2.9) we replace the operator Q by its eigenvalue at a time tj and pass on to themomentum representation to get rid of the operator P . Assuming the symmetric orderingof operators in V (Q) as well, we get

< qj | exp [iHtj]|qj1 >< qj |qj1 > itj[< qj |P

2

2m|qj1 > +V

(qj + qj1

2

)

(qj qj1)]=

dpj2

exp [ipj(qj qj1)]{1 itj

[p2j2m

+ V

(qj + qj1

2

)]}

dpj2

exp [ipj(qj qj1)] exp [itjH(pj, (qj + qj1)/2)] (2.10)where H(pj , (qj + qj1)/2) is by now the classical Hamiltonian. Substitution of (2.10) into(2.8) gets

< q| exp [iH(t t)]|q >

dp12

...dpn2

dq1...dqn1

expi

nj=1

[pj(qj qj1) tjH

(pj,

qj + qj12

)]. (2.11)

With an allowance made for the obtained expressions, the transition amplitude can berepresented in the view

< q| exp [iH(t t)]|q >= limn

(dp12

)...

(dpn2

)dq1...dqn1

expi

nj=1

tj

[pj

(qj qj1

tj

)H

(pj ,

qj + qj12

)]

[

dpdq

2

]exp

{i

tt

dt[pq H(p, q)]}, (2.12)

where [dpdq

2

]=

ni=1

(dpi2

) n1j=1

dqj . (2.13)

In Eq. (2.12) we shall fulfill the momentum integration over[dp

2

]=

ni=1

dpi2

.

Since the integrand is an oscillating function, it may be analytically continued to the Eu-clidean space. Then, considering (itj) to be a real quantity, we find

dpj2

exp

[itj2m

p2j + ipj(qj qj1)]=( m2it

)1/2exp

[im(qj qj1)2

2t

], (2.14)

where we have taken into account (2.1) and set all tj to be equal to each other (tj = t).Gathering the obtained results, we get the final expression for the transition amplitude (2.7)in the functional integration formalism

< q; t|q; t >= limn

( m2it

)n/2 n1i

dqi exp

i

nj=1

t

[m

2

(qj qj1

t

)2 V (q)

] =


= N

[dq] exp

{i

tt

d

[mq2

2 V (q)

]}= N

[dq] exp

[i

tt

dL(q, q)], (2.15)

where N is a normalization factor and [dq] is the measure on the functional space of trajec-tories q(t). Restoring the Plank constant, we rewrite this expression in the form

< q; t|q; t >= N[dq] exp

[i

~S(t, t)

], (2.16)

where S is the action. So, we have written the matrix element of the transition betweenthe initial and final states in the form of the functional integral that represents the sumof contributions over all paths connecting points (q; t) and (q; t). These contributionsare taken with weights which are equal to the exponentials of the action multiplied bythe imaginary unit. The functional integration trick actually consists of the following. Atsufficiently small divisions of time, we get rid of operator quantities and can deal only withclassical variables, but at the price of introducing infinite large number of intermediate timevalues t and allowing for all possible values of coordinates qj at every instant of time tj .The superposition principle reflects a fundamental property of continual integrals. It lies inthe fact that, when t belongs to the interval t0, tf , then the equality

[dq] exp

[i

~S(tf , t0)

]=

dq(t)

[dq] exp

[i

~S(tf , t)

] [dq] exp

[i

~S(t, t0)

](2.17)

must be fulfilled.The formula (2.16) is of a mathematical expression of the Huygens principle and with its

help one can clearly understand the difference between classical and quantum mechanics or,what is the same, between geometrical and wave optics. In classical mechanics, in general,the action is very large as compared with ~. For this reason, exp (iS/~) extremely oscillateswhen we pass on from trajectory to trajectory; in so doing nearly a total cancellation of allcontributions takes place except the contribution coming from the region in the immediatevicinity to that trajectory on which the action S reaches an extreme value (minimum moreoften than not). Therefore, in classical mechanics, a trajectory associated with minimumaction is the only important one. In contrast, in quantum mechanics, the action S maybe comparable or less than ~, so that many various trajectories give more or less identicalcontributions. Therefore, the quantum-mechanical description may be interpreted as theinclusion of fluctuations close to a classical trajectory.Paths integration formalism allows us to calculate not only transition amplitudes like

< q; t|q; t >, but matrix elements between states < q; t| and |q; t > for products ofarbitrary operatorsA(P (t), Q(t)) as well. In so doing, we bear in mind one subtlety. Becausethe subdivision points tj in (2.8) correspond to an increase of tj from right to left (tj =tj tj1 > 0), then the functional integral actually represents a time-ordering product ofoperators (T product). In quantum field theory (QFT) one usually studies the T -productsof vacuum expectation values of field operators that are called n-point Green functions

G(t1, t2, ..., tn) =< 0|T ((t1)(t2)...(tn)|0 > .With their help, n-partial amplitudes of various processes are obtained. As in QFT, wavefunctions play a part of coordinates, averages over the ground system state |0 > of theT -product of coordinate operators taken at the time moments t1, . . . , tn

(t1, . . . , tn) =< 0|T (Q(t1) . . . Q(tn)|0 >are analogues of the Green functions. So then, our following task is to produce (t1, . . . , tn)in the form of functional integrals.


From the beginning we consider a quantity

(t1, t2) =< 0|T (QH(t1)QH(t2)|0 > . (2.18)

Substituting the total state sets into (2.18), we get

(t1, t2) =

dq

dq < 0|q; t >< q; t|T (QH(t1)QH(t2))|q; t >< q; t|0 >=

=

dq

dq0(q, t)0(q, t) < q

; t|T (QH(t1)QH(t2))|q; t >, (2.19)

where we have taken into account the definition of the wave functions of the ground state

< 0|q; t >= 0(q, t), < q; t|0 >= 0(q, t).

Further, assuming that t1 > t2 (i.e., t > t1 > t2 > t) and allowing for (2.12), we find

< q; t|T (QH(t1)QH(t2))|q; t >=< q| exp [iH(t t1)]QS exp [iH(t1 t2)]QS

exp [iH(t2 t)]|q >=

dq1

dq2 < q

| exp [iH(t t1)]|q1 >< q1|QS

exp [iH(t1 t2)]|q2 >< q2|QS exp [iH(t2 t)]|q >=< q|q1(t)q2(t)

exp [iH(t t)]|q >= [

dpdq

2

]q1(t1)q2(t2) exp

{i

tt

d [pq H(p, q)]}. (2.20)

It is evident, that the same formula holds true for the case t2 > t1 (i.e., t > t2 > t1 > t) as

well. Substituting (2.20) into (2.19) results in

(t1, t2) =

dq

dq0(q, t)0(q, t)

[dpdq

2

]q1(t1)q2(t2) exp

{i

tt

d [pq H(p, q)]}.

(2.21)The presence of the wave functions of the ground states is the disadvantage of Eq. (2.21)and our next task is to get rid of them. Examine a matrix element

< q; t|T (QH(t1)QH(t2))|q; t >=

dQ

dQ < q; t|Q; >

< Q; |T (QH(t1)QH(t2))|Q; >< Q; |q; t >, (2.22)t (t1, t2) t. We write the first factor in the right side of (2.22) as

< q; t|Q; >=< q| exp (iH(t )|Q >=n

< q|n >< n| exp (iH(t )|Q >=

=n

n(q)n(Q) exp (iEn(t ), (2.23)

where we have accepted

H |n >= En|n >, n(q) =< q|n > n(Q) =< n|Q > . (2.24)

Later we allow forEn > E0 n 6= 0


and pass to the limit t i. Then, the expression (2.23) becomes

limti

< q; t|Q; >= 0(q)0(Q) exp [E0|t|+ iE0 ]. (2.25)

Similar calculations give

limti

< Q; |q; t >= 0(q)0(Q) exp (E0|t| iE0). (2.26)

To substitute (2.25) and (2.26) into (2.22) results in

limtiti

< q; t|T (QH(t1)QH(t2))|q; t >=

dQ

dQ0(q

)0(Q)

< Q; |T (QH(t1)QH(t2))|Q; > 0(Q)0(q) exp (E0|t|+ iE0 E0|t| iE0) == 0(q

)0(q) exp (E0|t| E0|t|)(t1, t2), (2.27)where at the final stage we have taken into consideration Eq. (2.19). In turn, from Eqs.(2.25) and (2.26) follows

limtiti

< q; t|q; t >= 0(q)0(q) exp (E0|t| E0|t|). (2.28)

Making use of (2.27) and (2.28), we get the sought formula for (t1, t2)

(t1, t2) = limtiti

[< q; t|T (QH(t1)QH(t2))q; t >

< q; t|q; t >]=

= limtiti

1

< q; t|q; t > [

dqdp

2

]q(t1)q(t2) exp

{i

tt

d [pq H(p, q)]}, (2.29)

where the factor < q; t|q; t > being in the denominator is defined by the expression (2.15).The evident generalization of (2.29) for the case of n-products of chronologically orderedcoordinate operators has the view

(t1, . . . tn) =< 0|T (QH(t1, . . . QH(tn))|0 >=

= limtiti

1

< q; t|q; t > [

dqdp

2

]q(t1) . . . q(tn) exp

{i

tt

d [pq H(p, q)]}. (2.30)

To introduce the idea of a generating functional, we make a small mathematical digression.In the case of finite numbers of variables yj (j = 1, 2, . . . , k) it is defined as follows:

F [y] F (y1, . . . , yk) =n=0

kj1=1

. . .

kjn=1

1

n!Fn(j1, j2, . . . , jn)yj1 . . . yjn , (2.31)


where Fn(j1, j2, . . . , jn) is a coefficient symmetrical over indices j1, j2, . . . , jn. Thus, forexample, at k = 2 we have the Taylor series expansion:

F (y1, y2) = F (0) + y1

(F

y1

)0

+ y2

(F

y2

)0

+1

2y1y1

(2F

y21

)0

+

+1

2y1y2

(2F

y1y2+

2F

y2y1

)0

+1

2y2y2

(2F

y22

)0

+ . . . =

= F (0) +

2j=1

yj

(F

yj

)0

+

2j1=1

2j2=1

yj1yj2

(2F

yj1yj2

)0

+ . . . =

= F (0) +

2j=1

yjF1(j) +

2j1=1

2j2=1

yj1yj2F2(j1, j2) + . . .

It is clear, that

F1(1) =

(F (y1, y2)

y1

)0

, F2(1, 1) =

(2F (y1, y2)

y21

)0

and so on, that is, F (y1, y2) is the generating functional (in the given case, it is a function)allowing us to calculate all the expansion coefficients Fn(j1, j2, . . . , jn).Let us generalize the situation on a functional space. We introduce the functional

F [y(x)] =

n=0

1

n!

dx1 . . .

dxnFn(x1, . . . , xn)y(x1) . . . y(xn). (2.32)

Now we have to use functional derivatives /y(xj) rather than the partial ones. For/y(xj) the following relation takes place

y(xj)y(xk) = (xj xk).

Then, the coefficient functions in the functional (2.32) may be written in terms of thefunctional derivatives of F [y(x)]:

Fn(x1, . . . , xn) =

y(x1). . .

y(xn)F [y(x)]

y=0

. (2.33)

By this reason, F [y(x)] is referred to as the generating functional for the functions Fn(x1, . . . , xn).Let us return to the question at hand. We shall consider that during a finite time interval

a force J(t) (source) acts on a system. The corresponding Lagrangian

Lint = J(t)q(t)must be supplemented to the total one. Then, using the generating functional W [J ]

W [J ] = limtiti

1

< q; t|q; t > [

dqdp

2

]exp

{i

tt

d [pq H(p, q) + J()q()]}=

= N limtiti

1

< q; t|q; t >[dq] exp

{i

tt

d [L(q, q) + J()q()]}, (2.34)


where N is a constant factor, one can obtain all (t1, . . . , tn) with the help of the followingformula

(t1, . . . tn) = (i)n n

J(t1) . . . J(tn)W [J ]

J=0

. (2.35)

We shall demonstrate later that in QFT the factors < q; t|q; t > and N , which do notdepend on J() are inessential and can be omitted under the calculation of connectedGreen functions. We may choose the following normalization condition for the generatingfunctional:

W [0] = 1. (2.36)

To compare (2.35) with (2.30) leads to the key conclusion, the generating functional W [J ]is in line with the transition amplitude between the ground state at a time t and that at atime t that is computed under the addition of a source J()

W [J ] =< 0|0 >J . (2.37)All the aforesaid is directly extended on the case of n degrees of freedom. In doing so, it

is enough to understand q1(t) . . . qn(t) as a totality of generalized coordinates q1(t) . . . qn(t).In this case the generating functional is given by the expression

W [J1, . . . Jn] =

= N limtiti

1

< q; t|q; t > n

j=1

[dqj ] exp

i tt

d

nj

[L(qj , qj) + Jj()qj()]. (2.38)

It is well to bear in mind that functional integrals were defined as a limiting case of Gaussintegrals (converging!) when dimensions tend to infinity. By this reason the functionalintegrals entering into (2.29) and (2.30) would be well defined if they have the Gaussianform. It is needed for that to pass on from the Minkowski space with the pseudo-Euclideanmetric to the Euclid space, that is, to carry out the Wick rotation

(x0, x1, x2, x3) (x4 = ix0, x1, x2, x3).Just this condition is reflected by the presence of nonphysical limits t i, t i inEq. (2.38).

2.2 Boson fields quantization by means of the functional integralsmethod

The functional approach is immediately generalized on QFT to be considered as a quantum-mechanical system with infinite numbers of degrees of freedom. Now, rather than divide atime interval by small periods , we divide a space-time on four-dimensional cells having avolume 4. Besides, the previous trajectory definition such that a value xj is confronted withevery time moment tj should be changed too. We shall define a trajectory giving a valueof a field (x) = (r, t), which may be multicomponents, in every cell. The paths integralpresently involves a summation over all possible field values in every cell. We confrontobjects that were introduced in NQM with the following ones:

j=1

[dqj , dpj ] [d(x), d(x)],


L(qj , qj), H(qj , pj)

d3xL((x), (x)),

d3xH((x), (x)),

where (x) is a momentum canonically conjugate to a field.In QFT a physical process in question is defined by the numbers of initial and final

quanta with given values of spin and momentum. Therefore, our goal is as follows. Withinformalism under consideration we are going to build up such a mechanism that automaticallyproduces initial and final particles from a vacuum independently on a specific shape ofinteraction. This is realized by the introduction of terms with an arbitrary source intothe Lagrangian. Because in QFT a vacuum state is a ground state, then the generatingfunctionalW [J ] is nothing else than a vacuum-vacuum transition amplitude in the presenceof an external source J(x). Then the evident generalization of the formula (2.38) has theform

W [J ] = N

[d] exp

{i

d4x[L((x), (x)) + J(x)(x)]

}, (2.39)

where N is a normalization factor. In the expression obtained we have to do the Wicksrotation

t x0 = it, W [J ]WE [J ] (2.40)and, after that, WE [J ] may be used to calculate Green functions. When formulating theFeynman rules, we have to take into account that disconnected diagramsdiagrams de-scribing the propagation of groups of particles not connected with each otherare beyondour interest. Scattering amplitudes can be obtained by connected diagrams only. There-fore, we are interested in merely those Green functions that are associated with connecteddiagrams. To remove a disconnected part of the Green function we have to divide the expres-sion (2.34) into the vacuum-vacuum transition amplitude, WE [J ]. Thus, in the Euclideanspace, connected Green functions are defined in the following way:

G(n)E (x1, . . . xn) =

[1

WE [J ]

nWE [J ]

J(x1) . . . J(xn)

] J=0

=

[n lnWE [J ]

J(x1) . . . J(xn)

] J=0

. (2.41)

Such a definition has an important practical consequence, namely, the normalization factorof the quantity W [J ] being independent on J proves to be inessential under the subsequentcalculations of Green functions. All this could be demonstrated by the example of a realscalar field with the Lagrangian

L = L0 + Lint, (2.42)where

L0 = 12

(

m22) , Lint = 4!4.

We set Lint = 0 and compute the Euclidean generating functional for the noninteractingfield W 0E [J ]. After integration by parts, we obtain

W 0E [J ] =

[d] exp

{d4x

[12(x)(

+m2)(x) + J(x)(x)

]}, (2.43)

where

=

2

x202.

The integral in the exponential is the limit of the four-dimensional cells sum. Havingdenoted the field in a cell as , the one in a next cell as , and so on, one mayconsider 1/2(

+m2) to be a limit of symmetric matrix A connecting neighbor cells.


Further, allowing for symmetry of the matrix A and using Eq. (2.1), one may show thevalidity of the following formula:

Ni=1

d4xi exp (xiAijxj + 2Skxk) = N/2[detA]1/2 exp (SiA1ij Sj). (2.44)

Taking into account this result, setting Sk = J(x)/2 and throwing away the inessentialinfinite factor being independent on J(x), we arrive at the following expression for W 0E [J ]

W 0E [J ] = exp

{1

2

d4xd4yJ(x)[

+m2]1J(y)}= exp

{1

2

d4xd4yJ(x)(x y)J(y)

}.

(2.45)The quantity (x y) = (+m2)1 may be represented by way of the Fourier integral

(x y) = 1(2)4

d4k

k2 +m2exp [ik(x y)],

where k = (ik0,k) is a four-dimensional Euclidean momentum vector. Carrying out analyt-ical continuation into the Minkowski space, we lead to a well-known resultto the Feynmanpropagator of a scalar particle with a mass m.

(x y) ic(x y) = i(2)4

d4p

p2 m2 + i exp [ip(x y)] (2.46)

(c(x y) is a causal Green function of a free scalar field). The term i ensures theFeynman prescription of the propagator poles detour. In the Lagrangian, it corresponds tothe additional term +i2/2, which ensures convergence of the functional integral (2.43) inthe Minkowski space.The factor in the exponential of the right side of (2.45) has the following meaning: in

the point y the source J(y) produces a free scalar particle that propagates from y to x andthe source J(x) annihilates this particle in the point x. Expanding the exponential gives aseries of terms corresponding to 0, 1, 2, . . . free scalar particles.Now we switch on interaction (Lint 6= 0). To obtain a series of the perturbation theory

for WE [J ], we have to expand exp{

d4xLint}as a power series. Then, we get the sum of

terms, each representing the functional integral of the Gauss function

exp

{d4x[L0 + J(x)(x)]

},

multiplied by the expression of the following view

1

n!

[d4xLint

]n.

Such integrals are calculated when Lint n. It is easy to see that every power of (x)can be obtained by differentiating the exponential in

WE [J ] =

[d] exp

{d4x [L0 + Lint + J(x)(x)]

}= exp

{d4xLint

}W 0E [J ] (2.47)

with respect to the source J(x), that is, by the change (x) /J(x),

WE [J ] =

n=0

1

n!

[d4zLint

(

J(z)

)]n [d] exp

{d4x[L0 + J(x)(x)]

}=


= exp

{d4zLint

(

J(z)

)}[d] exp

{d4x[L0 + J(x)(x)]

}=

=

n=0

1

n!

{d4z

[ 4!

(

J(z)

)4]}nexp

{1

2

d4x

d4yJ(x)(x y)J(y)

}. (2.48)

In order to deeper realize the generating functional structure, we display WE [J ] in theform of graphs, Feynman diagrams. First and foremost we note that, in the absence of aninteraction, W 0E [J ] is represented as the diagrams sum shown in Fig. 2.1. Here, the solidline indicates (x y), the cross does J(x) and integrating over xj , yj is meant. When

+ + 13!

+ ...12! y3y2x2y1 x3x1y2y1x1

J(y1)

x2

J(x1)

c(x1 y1)

FIGURE 2.1

The Feynman diagrams for W 0E [J ].

differentiating, every term of interaction (2.48) destroys four sources pinching four pointsxj into one point z and ascribing the factor to this point. As a result, from the graphsof Fig. 2.1 follows additional set of graphs (Fig. 2.2) associated with the second and thirdterms in WE [J ]:

WE [J ] =W0E [J ]{1 + w1[J ] + 2w2[J ] + . . .}, (2.49)

where

w1[J ] = 14!(W 0E [J ])

1{

d4x

[

J(x)

]4}W 0E [J ] =

1

4![(x y1)(x y2)

(x y3)(x y4)J(y1)J(y2)J(y3)J(y4) + 3!(x y1)(x y2)(x x)J(y1)J(y2)],(2.50)

w2[J ] = 12(4!)2

(W 0E [J ])1{

d4x

[

J(x)

]4}W 0E [J ] =

= 12(4!)

W10 [J ]

{d4x

[

J(x)

]4}w1[J ] =

1

2w21 [J ]+

1

2(3!)2(x1y1)(x1y2)(x1y3)

(x1 x2)(x2 y4)(x2 y5)(x2 y6)J(y1)J(y2)J(y3)J(y4)J(y5)J(y6)+

+3

2(4!)(x1 y1)(x1 y2)[(x1 x2)]2(x2 y3)(x2 y4)J(y1)J(y2)J(y3)J(y4)+

+2

4!(x1y1)(x1x1)(x1x2)(x2y2)(x2y3)(x2y4)J(y1)J(y2)J(y3)J(y4)+

+1

8(x1y1)(x1x1)(x1x2)(x2x2)(x2y2)J(y1)J(y2)+1

8(x1y1)[(x1x2)]2

(x2x2)(x1y2)J(y1)J(y2)+ 112(x1y1)[(x1x2)]3(x2y2)J(y1)J(y2). (2.51)


2

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

J(x) J(y)

(z z)

z

+

+J(x3) J(x4)

J(x1) J(x2)+

+z

(x z)(z y)

FIGURE 2.2

The Feynman diagrams for WE [J ].

In Eqs. (2.50) and (2.51) terms independent on J are omitted and integration over locationpoints of vertices and crosses is again meant.When J = 0, the quantityW [0] gives the diagrams set of the first column in Fig. 2.2 that

describes vacuum-vacuum transitions. In QFT, when finding the Green functions, thesediagrams are excluded at the expense of division operation by the quantity W [0], which isequivalent to redefining a vacuum.In Eq. (2.51), the term w21[J ]/2 governs disconnected diagrams. Restricting ourselves

to the second order of the perturbation theory, we demonstrate this term not to actuallycontribute to connected Green functions. Using Eq. (2.47) with

Lint() Lint(

J(x)

),

we get

lnWE [J ] = lnW0E [J ] + ln{1 + (W 0E [J ])1(WE [J ]W 0E [J ])} = lnW 0E [J ]+


+ ln{1 + (W 0E [J ])1(eSint 1)W 0E [J ]}, (2.52)where

d4xLint = Sint.

Since at small values of the quantity (W 0E [J ])1(eSint 1)W 0E [J ] is small, we can expend

the logarithm in Eq. (2.52). The inclusion of (2.49) leads to the result

lnWE [J ] = lnW0E [J ] + (w1[J ] +

2w2[J ] + . . .) 12(w1[J ] +

2w2[J ] + . . .)2 + . . . =

= lnW 0E [J ] + w1[J ] + 2

(w2[J ] 1

2w21

)+ . . . (2.53)

to confirm cancellation of disconnected diagrams.Thus, in the Minkowski space the recipe of calculating the cross sections for n interacting

particles by means of the generating functional W [J ] is as follows. The connected Greenfunctions G(n)(x1, . . . , xn) are determined by the expression:

G(n)(x1, . . . , xn) = (i)n[

1

W [J ]

n

J(x1) . . . J(xn)W [J ]

] J=0

, (2.54)

which is to say that

lnW [J ] =n

in

n!

dx1 . . . dxnG

(n)(x1, . . . xn)J(x1) . . . J(xn). (2.55)

From (2.54) it follows that every differentiation with respect to J(xj) removes a cross in thediagrams associated with W [J ] and ascribes a coordinate xj to a free end of the field . Insuch a way the two-particle Green function G(2)(x1, x2) is derived from the second columndiagrams (see Fig. 2.2), the four-particle Green function G(4)(x1, x2, x3, x4)from thethird column diagrams and so on. Now, so as to pass on to the real process amplitude, it isnecessary to take the Green function G(n)(x1 . . . , xn) with n external lines, cut off n externallines (i.e., divide by n free Green functions corresponding to these lines), and multiply byn wave functions of initial and final states. After that, activity is reduced to computing arequired totality of Feynman diagrams. In what follows, unless otherwise stated, we shallwork with amputated one-particle irreducible Green functions (nG,nq)(ki; pj), the Greenfunctions that have no external lines and do not break up into disconnected parts undersplitting one of the diagram lines.

2.3 Fermion fields quantization by means of the functional inte-grals method

Fields subjected to the FermiDirac statistics could be described by the Feynman pathintegral language as well. But now, unlike the Bose-system, one has to introduce new kindof functionals to produce the Grassmann algebra. In this section we shall give the necessaryknowledge about this topic.

This operation is called amputation.


n-Dimensional Grassmann algebra Gn is specified by n generatrices y1, y2 . . . yn to obeythe conditions

{yj, yi} = 0, j, i = 1, 2, . . . n. (2.56)The arbitrary Grassmann algebra element F Gn can be represented in the forms of thefinite series

F (y) = F0 + F(1)i1

yi1 + F(2)i1i2

yi1yi2 + . . .+ F(n)i1...in

yi1 . . . yin , (2.57)

where every summation index takes the values from 1 to n. Thanks to the conditions(2.56) this expansion is broken off. As an example, we consider one-dimensional Grassmannalgebra G1

{y, y} = 0, y2 = 0.For any element F (y) G1, we have

F (y) = F0 + yF1. (2.58)

To be specific, let us hold F (y) to be an ordinary number. Then F0 and F1 are ordinary andGrassmann numbers, respectively. We introduce differentiation and integration operationsinto G1. Two kind of derivatives, left and right ones, exist in the Grassmann algebra owingto the anticommutation relations. The following designations for them are used:

d

dy d

L

dy

d

dy d

R

dy.

We have for the element F (y)

d

dyF (y) = F1

d

dyF (y) = F1, (2.59)

where the anticommutation of d/dy with F1 has been taken into account. The relation{d

dy, y

}= 1 (2.60)

takes place for any derivative as well. The integration operation is usually defined as aoperation being inverse with respect to the differentiation one. In the Grassmann algebra,however, there is the relation

d2

dy2F (y) =

d2

dy2F (y) = 0 (2.61)

which hampers the ordinary definition of the integration operation. As a consequence, theintegration definition is rather artificial. The simplest way of defining integration rules restson the assumption that the integration operation acts on a function in the same manner asthe differentiation one. This avoids the presented dilemma. We demand the integration tolead to the same result as does the left differentiation operation, that is,

dyF (y) =d

dyF (y) = F1. (2.62)

Hence, it follows that dy = 0,

ydy = 1. (2.63)

These two integrals prove to be sufficient for finding all integrals in the Grassmann algebra.


Now we investigate the problem of the integration variable replacement y y = a+ by,where a and b are Grassmann and ordinary numbers, respectively. Allowing for the relation(2.62), we get

dyF (y) =d

dyF (y) = F1. (2.64)

On the other hand, we have dyF (y) =

dyF1by = F1b (2.65)

to finally give dyF (y) =

dy

(dy

dy

)1F (y(y)). (2.66)

Thus, the formula for the replacement of anticommuting variables involves the quantitythat is inverse to Jacobian rather than Jacobian, as it is in the case of ordinary variables.Let us generalize the formulae obtained on a case of n-dimensional Grassmann algebra.

The formulae for the left and right derivatives have the view

L

yj(y1y2 . . . yn) = j1y2 . . . yn j2y1y3 . . . yn + . . .+ (1)n1jny1y2 . . . yn1 (2.67)

(y1y2 . . . yn)R

yj= jny1y2 . . . yn1 . . .+ (1)n1j1y2 . . . yn. (2.68)

Integration rules are assigned by the relationsdyj = 0,

ykdyj = jk, (2.69)

where {dyj, dyk} = 0. Multiple integrals are defined in the following way:yk1 . . . ykndyn . . . dy1 = k1...kn , (2.70)

where k1...kn is a n-dimensional LeviCivita tensor.Further we find generalization of the formula (2.66). To do this, we fulfill the replacement

of variables yj yj = ajkyk and compare the expressionsdyn . . . dy

1F (y

)

dyn . . . dy1F (y(y)).

In F (y) only the terms containing n factors yj contribute to these integrals

y1 . . . yn = a1j1 . . . anjnyj1 . . . yjn = a1k1 . . . anknk1...kny1 . . . yn = (det a)y1 . . . yn.

Then, for the relations (2.69) to be fulfilled it needs to have

dyn . . . dy

1F (y

) =

dyn . . . dy1

[det

(dy

dy

)]1F (y(y)), (2.71)

as distinguished from the ordinary rule of variables replacement.The integral of the Gauss type

f(A) =

dyn . . . dy1 exp

[1

2(y,Ay)

], (2.72)


where A is an antisymmetric matrix and (y,Ay) = yjAjkyk, is a fundamental integral tobe dealt with in the Grassmann variables theory. In the simplest case n = 2, the matrix Ais given by the expression

A =

(0 A12

A12 0)

and the integral (2.72) is taken without difficulty

f(A) =

dy2dy1 exp [y1y2A12] =

dy2dy1[1 + y1y2A12] = A12 =

detA. (2.73)

In the general case the matrix A represents an antisymmetric nn matrix with a even valuen (if n is odd, the integral (2.72) vanishes). With the help of a unitarity transformationUAU , the matrix A is reduced to the standard view

UAU = As =

a

(0 11 0

)

b

(0 11 0

).

..

, (2.74)

where all elements outside the main diagonal are equal to zero. To find the matrix U , weproceed as follows. Since the matrix iA is Hermitian, there exists a unitary transformationV that reduces it to the diagonal form

Ad = V (iA)V, (2.75)

where the diagonal elements of the matrix A is found from the secular equation

det |iA I| = 0. (2.76)

On the other hand,det |iA I|T = det | iA I| = 0. (2.77)

If is a solution of Eq. (2.76), then will be its solution too. Therefore, the matrix Adhas the form

Ad =

11

22

..

.

, (2.78)

where 1 = a, 2 = b . . .. Let us introduce two more n n-matrices, D and C

D =

B

B.

..

, C =

1/21

1/21

1/22

1/22

..

.

, (2.79)


where

B =12

(i 11 i

)

anddet(C1) =

detA.

Taking into account the matrix B to obey the relation

B(i)(1 00 1

)B =

(0 11 0

),

we findAs = D(iAd)D = (DV )A(DV ),

that is, U = DV . Collecting together the results obtained, we arrive at

C(UAU )C = CAsC As =

(0 11 0

)(

0 11 0

).

..

. (2.80)

Now, the Gauss integral (2.72) could be reported in the form

f(A) =

dyn . . . dy1 exp

[1

2(y, U C1AsC1Uy)

]. (2.81)

Passing on to new integration variables y = (C1U)y and allowing for Eq. (2.71), we get

f(A) =

dyn . . . dy

1 exp

[1

2(y, Asy)

] [det

(dy

dy

)]= det

(dy

dy

)=

= det(C1U) = det(C1) =detA. (2.82)

This formula is essentially distinguished from that for the Gauss integral in the case ofcommuting real variables (see Eq. (2.3)). It is easy to show that in the case of complexGrassmann variables the Gauss integral is given by

dyndyn . . . dy1dy

1 exp [(y

, Ay)] = detA, (2.83)

where yj and yj are independent generatrices of the Grassmann algebra.

To describe fermion fields, we have to pass on to infinite-dimensional Grassmann algebra.Transition from discrete variables

y1, . . . , yj, . . . , yn

to continuous ones y(x) is realized by n, and the replacement j x on a constrainedinterval of changing a variable y. As a consequence, the Grassmann function F (y) (2.57)turns to the Grassmann functional defined by the expression

F [y] = F0 +

dx1F

(1)(x1)y(x1) +

dx1dx2F

(2)(x1, x2)y(x1)y(x2) + . . .+


+

dx1 . . . dxnF

(n)(x1, . . . xn)y(x1) . . . y(xn), (2.84)

where the generators y(x) are subjected to the relations

{y(xi), y(xj)} = 0.

Differentiation of Grassmann functionals is defined as a functional generalization of therules (2.67) and (2.68)

L

y(x)y(x1) . . . y(xn) = (xx1)y(x2) . . . y(xn) (xx2)y(x1)y(x3) . . . y(xn)+ . . . (2.85)

The same is true for R/y(x). Generalization of the integrals over Grassmann variables(2.69) and (2.70) is realized by means of the transition to functional integrals. In the caseof complex Grassmann variables the fundamental integral used in the fermion fields theorywill look like:

[dy(x)]

[dy(x)] exp{

dx

dxy(x)A(x, x)y(x)} = det A, (2.86)

where A is a function being antisymmetric with respect to variables x and x.Now we shall consider classical fermion fields to be elements of the infinite-dimensional

Grassmann algebra. The generating functional of fermion fields is given by the expression:

W [, ] =

[d(x)][d(x)] exp

{i

d4x[L((x), ) + (x) + ]

}, (2.87)

where and are auxiliary Grassmann sources to obey the conditions

{(x), (x)} = 0, {(x), (x) = 0

and so on. The Green functions are obtained by the previous recipe:

G(n)(x1, . . . , xn) =< 0|T ((y1) . . . (x1) . . .)|0 >=

=

(i

(y1)

). . .

(i

(x1)

). . . lnW [, ]

==0

. (2.88)

In doing so, it has to take into consideration that the quantities , , , , /, / anti-commute. Since fermion fields usually enter into the Lagrangian quadratically, L = A,we must calculate the following integral

W [, ] =

[d(x)][d(x)] exp

{i

d4x[(x)A(x) + (x)(x) + (x)(x)]

}. (2.89)

After the replacement of variables

(x) (x) + (x), (x) (x) + (x),

where(x) = A1(x), (x) = (x)A1

the int

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