Unitary relativistic quantum theory. - … · Relativistic Quantum Principle, ... in section 2 we obtain the relativistic transform of the space- ... quantum particle in a field we

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Mater Sci Nanotechnol 2018 Volume 2 Issue 117

Review Article

IntroductionWe perceive the physical world as a collection of objects, in

a three-dimensional space of coordinates defined by a position vector zyx zyxr 111

++= , and in time, defined by a scalar t [1-5]. For a matter object we define an inertial property called mass, 0M and a dynamic quantity as the product of the mass

with the velocity, trv

dd

= , called momentum, vMp

0= . We

also define a conservative quantity, called energy,

( ) ( ) ( )rUpTprHE

+== ,0 (1.1)

as a sum of the kinetic energy ( )pT , which depends on momentum, and the potential energy ( )rU , which depends on the coordinates [6-8]. From the conservation condition:

0dd

dd

dd 00 =

∂∂

+∂∂

= ptp

Hrtr

HtE

(1.2)

we obtain the dynamic equations called Hamilton equations,

( )

( ) .Force - dd

dd

0

0

rUrr

Hpt

pTpp

Hrt

∂∂

−=∂∂

−=

∂∂

=∂∂

=

(1.3)

while the energy function of coordinates and momenta,

( ) ( )rUMpprH

+=0

2

0 2, (1.4)

is called Hamiltonian. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it

has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time, with a single quantum constant

[9-11].

( )( )

( ) ( )

( )( )

( ) ( )

303 2

30 3 2

1

2

1

2

i pr Et

E /

i pr Et

E/

r ,t p,t e d p

p,t r ,t e d r

−

− −

ψ = ϕπ

ϕ = ψπ

∫

∫

(1.5)

In this case, one can define a momentum operator

rp

∂∂

−= i (1.6)

and a Hamiltonian operator,

( ) ErUrMt

H =+∂∂

−=∂∂

=

2

2

0

2

0 2i

(1.7)

With these operators one obtains eigenvalue equations, for momentum,

( ) ( )trptrr EE ,,i

ψψ =∂∂

− (1.8)

and for energy, as a Schrödinger equation:

( ) ( ) ( )trEtrrUrM EE ,,

2 2

2

0

2

ψψ =

+

∂∂

− (1.9)

However, when the group velocities are calculated for the wave packets (1.5), which with (1.1) are of the form

We consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrödinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation.

Abstract

Unitary relativistic quantum theory.

Eliade Stefanescu*Advanced Studies in Physics Centre of the Romanian Academy, Academy of Romanian Scientists Bucharest, Romania

Keywords: Quantum particle, Wave packet, Group velocity, Lagrangian, Hamiltonian, Maxwell equations, Spin, metric tensor, Covariant derivation.

Accepted on February 05, 2018

Citation: Stefanescu E. Unitary relativistic quantum theory. Mater Sci Nanotechnol. 2018;2(1):17-28.

18Mater Sci Nanotechnol 2018 Volume 2 Issue 1

waves. In section 3, we consider a quantum particle in a field described by a scalar potential conjugated to time, and a vector potential conjugated to coordinates. We obtain the Lagrange equation as the velocity of the particle waves, the Lorentz force, and the Maxwell equations. In section 4, based on the Relativistic Quantum Principle, we obtain the dynamics of a quantum particle in electromagnetic field. In section 5, from the invariance of the time dependent phase of a quantum particle at an arbitrary change of coordinates, we obtain the relativistic transform of the electromagnetic field. In section 6, we obtain a relativistic equation for the quantum particle waves, show that the solution of the Schrödinger equation is only the amplitude of the wave function of a quantum particle, which also includes a rapidly varying factor depending on the particle rest mass, and derive the spin of a quantum particle wave. In section 7, we consider the inversion of two particles as a double rotation of these particles, and obtain the spin-statistic relation. In section 8, we consider a coordinate deformation according to the Theory of General Relativity, and obtain the dynamics of the particle waves in gravitational field. In section 9, we consider the particle waves as a distribution of matter described by a density and a velocity field, and find that these waves are perpendicular to the geodesic tracks. We derive an invariant for the matter density of a quantum particle, and an equation of conservation of this matter. Section 10 is for conclusions.

Relativistic Kinematics and DynamicsA realistic particle has a finite spectrum as a function of the wave

propagation velocity. A finite spectrum is obtained for a relativistic Lagrangian. According to (1.15), the invariance of the scalar time dependent phase variation ( )0L r dt

of a quantum particle wave function is equivalent to the invariance of the space-time interval ds. For such a particle, we obtain the momentum,

rM

cr

rMrLp

=

−

=∂∂

=

2

200

1 (2.1)

and the mass,

2

2

0

1cr

MM

−

= (2.2)

as functions of the particle velocity r and the cut-off velocity c. The invariance of the space-time interval means that a change of coordinates is in fact a rotation of the space-time coordinates (Figure 1).

By a well-known calculation, the relativistic transform of the coordinate intervals is obtained for the quantum particle waves. Thus, from the coordinate transform,

0 0 1

1 1 0

2 2

3 3

dx dx cos dx sindx dx cos dx sindx dxdx dx

′ ′

′ ′

′

′

= φ + φ

= φ− φ

=

= (2.3)

with the expressions of the rotation angle trigonometric functions,

( )( )

( )( ) ( )[ ]{ }

( )( )

( )( ) ( )[ ]{ }

,

d,2

1,

d,2

1,

3i

2/30

3i

02/3

∫

∫

+−−

+−

=

=

retrtp

petptr

trUpTrpE

trUpTrpE

ψπ

φ

φπ

ψ

(1.10)

we obtain an erroneous equation, contradictory to the corresponding Hamilton equation (1.3):

( )

( )

0

0

Hd r T p - OKdt p p

Hd p U r - Erroneous equationdt r r

∂ ∂= =

∂ ∂∂ ∂

= =∂ ∂

(1.11)

- a minus sign is missing [12-14]. We get back the minus sign only if instead the Hamiltonian ( ) ( ) ( )rUpTprH

+=,0 we consider the Lagrangian

( ) ( ) ( ) ( ) ( )rUvMrUpTrpHrprrL

−=−=−=2

,,2

00

(1.12)

In his case, the wave packets (1.10) take a form

( )( )

( )( ) ( )[ ]{ }

( )( )

( )( ) ( )[ ]{ }

,d,2

1,

d,2

1,

3i

2/30

3i

02/30

∫

∫

−−−

−−

=

=

retrtp

petptr

trUpTrpE

trUpTrp

ψπ

φ

φπ

ψ

(1.13)

with group velocities in agreement with the Hamilton equations (1.3):

( )

( ) .dddd

0

0

rHrU

rp

t

pHpT

pr

t

∂∂

−=∂∂

−=

∂∂

=∂∂

=

(1.14)

However, this description is still unrealistic, having an infinite spectrum of waves, as a function of the wave velocity r . A finite spectrum is obtained when the relativistic Lagrangian ( )rL

0 ,

( ) scMtcrcMtrL dd1d 02

22

00 −=−−=

(1.15)

is considered in the time dependent phase of a particle wave:

( )( )

( ) ( )

( )( )

( ) ( )

0

0

3 30 03 2

30 3 2

1

2

1

2

i Mrr L r ,r t

/

i Mrr L r ,r t

/

r ,t r ,t e M d r

r ,t r ,t e d r

−

− −

ψ = ϕπ

ϕ = ψπ

∫

∫

(1.16)

From (1.15), we notice that the invariance of the space-time interval of the Theory of Relativity is equivalent to the invariance of the time dependent phase variation of a quantum particle, and define a Relativistic Quantum Principle: The scalar tim-dependent phase variation of a quantum particle wave is an invariant for an arbitrary change of coordinates. On this basis, in section 2 we obtain the relativistic transform of the space-time coordinates, and the relativistic dynamics of the particle

Stefanescu

19 Mater Sci Nanotechnol 2018 Volume 2 Issue 1

2

2

2

2

0

1

1

cVi

sin

1

1cos

cViitan

cV

cV

tcx

xx

−

−=

−

=

−=−

=−

=

ϕ

ϕ

δδ

δδϕ

(2.4)

for the coordinates of a quantum particle wave, we obtain the transform:

2 2

2 21 1

2

Vdt dx dx Vdtcdt , dxV Vc c

dy dy , dz dz ,

′ ′+ ′ ′+= =

− −

′ ′= =

(2.5)

which, in this case does not refer to the coordinates of some classical particle as in the conventional Theory of Relativity, but to the coordinates of the quantum particle waves.

Electromagnetic FieldWhen a particle with a charge e is placed in a field of a vector

potential ( )trA ,

conjugated to the coordinates, and a scalar potential ( )rU conjugated to time, for the two wave packets of this particle

( )( )

( ) ( )

( )( )

( ) ( )

3 2

3 2

1

2

1

2

i P r L r ,r ,t t 3/

i P r L r ,r ,t t 3/

r ,t P ,t e d P

P,t r ,t e d r ,

−

− −

ψ = ϕπ

ϕ = ψπ

∫

∫

(3.1)

a time dependent phase variation arises, with terms proportional to the coordinate variations, and the time variation,

( ) 22

210

rc

L r ,r ,t dt M c dt eA r ,t dr eU r dt − − + = −

(3.2)

In this case, we get a canonical momentum,

( ) ( ) ( ),,,,,

2

21

0 trAeptrAerMtrrLr

P

cr

+=+=∂∂

=−

(3.3)

with a mechanical component depending on the particle mass and velocity, and an electromagnetic component as the product of the particle charge with the vector potential. For a quantum particle in a field we consider again the relativistic quantum principle: The time dependent phase variation of a quantum particle is the same in any system of coordinates.

We are in agreement with the Aharonov-Bohm effect [15]: the time-dependent phase of a quantum particle ( )tr ,ψ ′ ,

tV

m ∂′∂

=′−′∇−ψψψ

i2

22

in a magnetic field includes a term proportional to the space integral of the vector potential of this field:

( ) ( ) ( )

( ) ( )0

ig r

r

r ,t e r ,t

eg r A r dr

′ψ = ψ

′ ′= ∫

This effect has been experimentally put out into evidence (Figure 2).

Besides this variation, we consider a time-dependent phase variation with a term proportional to the scalar potential and time, and the invariance of this phase. From equation (3.3), we obtain the electric force:

( )trAt

ePt

pt

Fe ,dd

dd

dd

−== (3.4)

Figure 1. The rotation of the space-time system of coordinates for a velocity V

of the system ( )0 1x ct ,x′ ′′= over the system ( )0 1x ct ,x= .

Figure 2. Experimental evidence of the Aharonov-Bohm effect: between the two electron beams, a phase difference arises due to the vector potential A

[4].



while from a particle wave velocity in the momentum space we obtain the Lagrange equation:

( ) ( )[ ] ( )rUr

ertrAr

etrrLr

Pt

∂∂

−∂∂

=∂∂

= ,,,dd

(3.5)

With the vector formula:

( ) ( )[ ] ( )trAr

rtrArr

trAr

r ,,,

∂∂

−∂∂

=

×∂∂

× (3.6)

we obtain the Lorentz force:( ) ( )trBretrEeFe ,,

×+= (3.7)

with the electric field,

( ) ( ) ( )trAt

rUr

trE ,,

∂∂

−∂∂

−= (3.8)

and the magnetic field,

( ) ( )trAr

trB ,,

×∂∂

= (3.9)

Taking into account that the curl of the gradient is null, from (3.8) with (3.9) we obtain the Faraday-Maxwell law of the electromagnetic induction:

( ) ( )trBt

trEr

,,

∂∂

−=×∂∂

(3.10)

Taking into account that the divergence of the curl is null, from (3.9) we obtain the Gauss-Maxwell law for the magnetic induction flow:

( ) 0, =∂∂ trBr

(3.11)

With the Gauge condition

( ) 0, =∂∂ trAr

(3.12)

we obtain the Gauss-Maxwell law for the electric field flow:

( ) ( ) ( )0

2

2,

ερ rrU

rtrE

r

=∂∂

−=∂∂

(3.13)

with the Laplacian of the scalar potential as source of this divergence, considered as a ratio of the charge density ( )rρ and the dimensional constant 0ε called electric permittivity. Considering a current density under the action of the electric field,

( ) ( ) ( ) ( ) 0,,,, =×⇒== trErtrErtrtrj

σρ (3.14)

with the vector formula

( )[ ] ( ) ( )

( ) ( ),,,

,,,

0

0

trEr

rtrr

trEr

rtrEr

rtrErr

∂∂

−=

∂∂

−

∂∂

=××∂∂

ερ

(3.15)

we obtain the time derivative of the electric field of the form:

( ) ( ) ( )trEt

trjtrEt

,,,dd

00

∂∂

+= εε (3.16)

For the relation (3.10) of the two vector fields, asserting that a time variation of a magnetic field determines a curl of the electric field, we consider a symmetric relation, namely that a time variation of the electric field determines a curl of the magnetic field:

( ) ( ).,1,dd

00 trB

rtrE

t

×∂∂

=µ

ε (3.17)

From (3.16) and (3.17), the Ampère-Maxwell law of the magnetic circuit is obtained. In this way we obtain the whole system of Maxwell equations:

( ) ( ) ( )

( ) ( )

( )

( ) ( )

00

0

1

0

B r ,t j r E r ,tr t

E r ,t B r ,tr t

B r ,tr

r ,tE r ,t

r

∂ ∂× = + ε

µ ∂ ∂∂ ∂× = −

∂ ∂∂

=∂

ρ∂=

∂ ε

(3.18)

and the Lorentz force,

( ) ( )trBretrEeFe ,,

×+= (3.19)

This force acts on the particle wave functions (3.1), with the time-dependent phase coefficient:

( ) 22

210

rc

L r ,r ,t M c eA r ,t r eU r − − + = −

(3.20)

as a function of the two potentials of the electromagnetic field:

( ) ( ) ( )

( ) ( ).,,

,,

trAr

trB

trAt

rUr

trE

×∂∂

=

∂∂

−∂∂

−=

(3.21)

It is interesting that by the hypothetical relation (3.17), we obtain a field propagating in waves with a velocity 00/1 µε , in agreement with the electromagnetic theory. For the physical consistency, this velocity takes the limit value of the velocity of the quantum particle waves (Figure 3):

00

1µε

=c (3.22)

Otherwise, an electromagnetic field not interacting with a quantum particle, or a quantum particle not interacting with any electromagnetic field could exist, which is contradictory to our basic hypothesis of the particle-field interaction.

Stefanescu


Dynamics of a Quantum Particle in Electromagnetic Field

We consider the space-time interval,

21 ddd2

2

icr xtcs −== −

(4.1)

and the phase variation (3.2),

( ) ( ) ( )0dS L r ,r ,t dt M cds eA r ,t dr eU r dt= = − + −

(4.2)

as functions of the coordinate four-vector

( ) ( )ix x, y,z ,ict= (4.3)

and the field four-vector

( )

= U

cAAAA zyxi

i,,, (4.4)

For the wave velocity in the space of the momentum (3.3), from (3.1) we obtain a wave/group velocity of the form of the Lagrange equation:

( ) ( )trrLr

trrLrt

,,,,dd

∂∂

=∂∂

(4.5)

which means that the time-dependent phase of a quantum particle wave function is of the form of the action of the Lagrangian (3.20),

( ) ( )0 i iS L r ,r ,t dt M cds eA dx= = − +∫ ∫

(4.6)

In agreement with the principle of the least action,

0 0ii i i i i

dxS M c dx eA dx e A dx

ds

δ = δ + δ + δ = ∫ (4.7)

for the velocity four-vector

sxu i

i dd

= (4.8)

and the field four-tensor

k iik

i k

A AF

x x∂ ∂

= −∂ ∂

(4.9)

we obtain the dynamic equation

0i

ik k

duM c eF u

ds=

(4.10)

or

i kik

dx dxd M eFdt dt dt

= (4.11)

with the relativistic mass

0MM γ= (4.12)

which depends on the relativistic coefficient

2

21

1

cV

−

=γ

(4.13)

With the explicit expression of the field four-tensor

( )

0

0

0

0

z y x

z x y

ik

y x z

x y z

iB B EciB B EcFiB B Ec

i i iE E Ec c c

− − − −

= − − (4.14)

we obtain the Lorentz force,

( ) BreEerMt

×+=dd

(4.15)

for the interaction of a particle wave with an electromagnetic field.

Relativistic Transform of a Field Interacting with a Quantum Particle

For a coordinate four-vector transform, 1

i ij j j ji ix x , x x−′ ′= α = α (5.1)

with the transformation matrix,

( )

0 0

0 1 0 00 0 1 0

0 0 1

ij

Vic

Vic

γ − γ α = γ (5.2)

in the least action equation (4.7), we obtain an invariant including a mechanical term and an electromagnetic term:

0ddneticelectromagmechanical

0 =

+−= ∫

sxuFxucMS ikikii δδδ

(5.3)

Since the mechanical term is invariant to any change of coordinates,

Figure 3. Wave packet of a quantum particle, with a limit velocity c equal to the electromagnetic field velocity.



1i i ij ik j k ji ik j k jk j k j jdu x du x du x du x du x−′ ′ ′ ′ ′ ′ ′ ′δ = α α δ = α α δ = δ δ = δ (5.4)

the electromagnetic term must be also an invariant,

ik k i ik kl l ij j jl l jF u x F u x F u x′ ′ ′ ′ ′δ = α α δ = δ (5.5)

or1 1

jl l j jl lk k ji i jl kl k ij i ik k iF u x F u x F u x F u x− −′ ′ ′ ′ ′δ = α α δ = α α δ = δ (5.6)

On this basis, we find the transform for a change of coordinates of a field interacting with a quantum particle

jl ik ij klF F′ = α α (5.7)

orik ij kl jlF F ′= α α

(5.8)

with the explicit form:

2 2

2 2

2 2

2 2

2 2

1 1

1 1

y z z yx x y z

y x z y

z z y z

E VB E VBE E , E , E ,

V Vc c

V VB E B Ec cB B , B , B .V Vc c

′ ′ ′ ′′

′ ′ ′ ′

′

+ −= = =

− −

− += = =

− − (5.9)

Particle Wave Function and SpinFrom the Lagrange equation obtained as the group velocity

(3.5) with the momentum expression (3.3),

( ) ( )trrLr

trrLrt

Pt

,,,,dd

dd

∂∂

=∂∂

= (6.1)

and the Hamiltonian definition

( ) ( )trrLrPtrPH ,,,,

−= (6.2)

as a conservative function,

( ) ( ) ( )

tt

HttLr

rLr

rLrPrP

tt

HrtrPHr

PtrPHP

trPH

PPrP

dddddd

dd,,d,,,,d

0

d∂∂

=∂∂

−∂∂

−∂∂

−⋅+⋅=

∂∂

+∂∂

+∂∂

≡

(6.3)

we obtain the Hamilton equation

( )

( )

( ) ( ).,,,,

,,

,,

trrLt

trPHt

trPHr

P

trPHP

r

∂∂

−=∂∂

∂∂

−=

∂∂

=

(6.4)

At the same time, with the definition expression (6.2), the wave packets (3.1) take a form

( )( )

( ) ( )[ ]{ }

( )( )

( )( )[ ]{ }

,d,2

1,

d,2

1,

3,i

2/3

3,i

2/3

∫

∫

−−−

−−

=

=

retrh

tP

PetPh

tr

trPHrPrP

trPHrPrP

ψπ

φ

φπ

ψ

(6.5)

as solutions of a wave equation

( ) ( ) ( ) ( )i r ,t i r r ,t H P,r r ,tt r∂ ∂ψ = − ψ − ψ

∂ ∂

(6.6)

which, compared to the Schrödinger equation includes a term depending on the velocity.

On the other hand, from (3.1) we notice that, in a classical

approximation,

( ) ( ) ( )

( ) ( )

22

0 2

22

0

1

2

rL r ,r ,t M c eA r ,t r eU rc

MrM c eA r ,t r eU r

= − − + −

≈ − + + −

(6.7)

the Schrödinger wave function ( )trE ,ψ , we usually use in our studies, in fact is not the particle wave function, bat only an amplitude of a wave function including a rapidly varying factor,

( ) ( ) ( )2

0 2i M c eU r t

Er ,t e r ,t + ψ = ψ

(6.8)

We consider an electromagnetic field with a constant scalar potential for a stationary electric field, and a time-dependent vector potential for a coherent radiation field. From equation (6.2) with (3.3) and (3.20), we notice that although the vector potential ( )trA ,

depends on time, the Hamiltonian does not:

( ) ( )

( ) ( ) ( )2 2

200 22

2

1

1

H P,r ,t P r L r ,r ,t

M r reA r ,t r M c eA r ,t r eU rcr

c

= −

= + − − − + − −

(6.9)

With this Hamiltonian we, obtain the total energy as a function of the mechanical energy and the potential energy in electric field:

( ) ( ) ( )2

0

2

21

M cH P,r ,t eU r E r ,r

rc

= + =

−

(6.10)

With the mechanical energy as a function of the canonical momentum (3.3),

( )[ ] ,,11

220

2220

2

2

220

2

2

220 cMtrAePcM

crrM

crcM

+−=+−

=−

(6.11)

we obtain the canonical form of the Hamiltonian

( ) ( ) ( )22 20H P,r ,t c M c P eA r ,t eU r = + − +

(6.12)

According to Dirac’s well-known spin theory, we consider

Stefanescu


the Schrödinger equation

( ) ( )trHtrt EE ,,i

ψψ =∂∂

(6.13)

with the Hamiltonian

( ) ( )( ) ( )

2 2 20

1 1 2 2 3 3 4 0

H p,r ,t c p M c eU r cp p p M c eU r

= + + =

α +α +α +α +

(6.14)

which depends on Dirac’s spin operators

=

=

=

−=

00

,0

0,

00

,1̂0

01̂

3

33

2

22

1

110 σ

σα

σσ

ασ

σαα

(6.15)

as functions of Pauli’s spin operators,

−

=

−=

=

1001

,0ii0

,0110

321 σσσ (6.16)

These operators satisfy the commutation relations:

{ } 2i j i j j i ij,α α = α α +α α = δ (6.17)

{ } 2i j i j j i ij,σ σ = σ σ +σ σ = δ (6.18)

With the notation

( )321 ,, αααα = (6.19)

the Schrödinger equation (6.13) with the Hamiltonian (6.14) takes a form:

( ) ( ) ( ) ( )0 0 E Ec M c p eU r r ,t E r ,t α + α + ψ = ψ

(6.20)

with a wave function which can be written as a vector with two components, or a vector with four components,

=

=

4

3

2

1

2

1

φφφφ

ψψ

ψ E

(6.21)

We obtain the two-dimensional Schrödinger-Dirac equation,

( )( )

( )( ) ( ) ( )

( )( )( )

1 2 1 10

2 1 2 2

r r r rc M c p eU r E

r r r r ψ σψ ψ ψ

+ + = −ψ σψ ψ ψ

(6.22)

with two coupled Schrödinger-Dirac equations for the two

components ( )r1ψ and ( )r2ψ ,

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

20 1 2 1

20 2 1 2

M c eU r r c p r E r

M c eU r r c p r E r ,

+ ψ + σ ψ = ψ − + ψ + σ ψ = ψ

(6.23)

where

( )321 ,, σσσσ =

(6.24)

By eliminating the coupling terms, the two Schrödinger-Dirac equations take non-linear forms, which, in a non-relativistic approximation, small velocity, small electric potential, become linear:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

20

20

22 2 20 0 1 1

2

22 2 20 0 2 2

2

c

c

EM c

E M c

E M c eU r E M c eU r r c p r

E M c eU r E M c eU r r c p r

≈

≈

+ − − − ψ = σ ψ

− − + − ψ = σ ψ

(6.25)

In the non-relativistic approximation, we get Schrödinger-Dirac equations

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

1 10

2

2 20

2

2

c

c

peU r r E r

M

peU r r E r

M

σ + ψ = ψ σ + ψ = ψ

(6.26)

for the classical energy2

0cMEEc −= (6.27)

The Hamiltonian of these equations includes a kinetic term depending on the momentum and the Pauli spin operators,

( ) ( ) ( )ppppppp

×+=++= σσσσσ i22332211

2 (6.28)

With the mechanical momentum in a magnetic field,

( ) ( )trAetrAePp ,i,

−∇−=−= (6.29)

we obtain the vector product of equation (6.28) of the form

( ) ( ) ( )

( ) ( ) ( ) ( )

↑

×∇=∇×+×∇=∇×+×∇

=∇×+×∇=−∇−×−∇−=×

AAAAA

BeAAeAeAepp

φφφφ

.iiii

(6.30)

Thus, in the kinetic term of the Hamiltonian, besides the mechanical term, proportional to the square of the mechanical momentum, we obtain a magnetic potential, proportional to the magnetic field, as of a proper rotation, called spin:

( ) Bepp

σσ −= 22 (6.31)

We obtain Schrödinger-Dirac equations with a Hamiltonian including the kinetic energy, the electric potential energy, and a potential energy in magnetic field, due to the particle spin:

( ) ( ) ( )

( ) ( ) ( )

2

1 10

2

2 20

2

2

s c

s c

p B eU r r E rM

pB eU r r E r ,

M

−µ + ψ = ψ

−µ + ψ = ψ

(6.32)

with the spin magnetic moment

σµ

02Me

s = (6.33)

and the component

30

3 2σµ

Me

= (6.34)

in the direction of the magnetic field. Taking into account the commutation relation

[ ] [ ] 0,, 333 =+= slHjH (6.35)



of the Hamiltonian( )cMpppcH 04332211 αααα +++= (6.36)

with the total angular momentum, which includes the orbital angular momentum l3 and the spin angular momentum s3, we obtain a relation between the spin angular momentum and the orbital angular momentum which is known:

[ ] [ ]33 ,, lHsH −= (6.37)

With the commutation relations

[ ] kji plp ijki, δ= (6.38)

from (6.35) and (6.36) we obtain the commutation relations

1 3 2

2 3 1

3 3

4 3

00

,s i

,s i

,s,s

α = − α α = α

α = α =

(6.39)

These equations have a solution of the form

213 ααss = (6.40)with the coefficient

2i −=s

(6.41)

Thus, we obtain the spin angular momentum in the direction of the magnetic field

=−=

3

3213 0

022

iσ

σαα s

(6.42)

with the Eigen value equations

.

2

2

2323

1313

ψσψ

ψσψ

=

=

s

s

(6.43)

From (6.34) and (6.41) we obtain the gyromagnetic ratio

03

3Me

sgs ==

µ

(6.44)In this way, the spin is obtained from the relativistic quantum

principle, in the framework of a unitary relativistic quantum theory.

Spin-Statistic Relation

We consider a system of two particles, in the states 1i and 2i , with the position vectors 1r

and 2r (Figure 4).

For a two particle wave function

2121 ,, iirr

(7.1)we define an inversion operator I :

21212112 ,,,, iirrIiirr

= (7.2)

By applying two times this operator, 2121

221122121 ,,,,,, iirrIiirrIiirr

== (7.3)

we find the Eigen values

12

2

11

1I for Fermions

II for Bosons= −= = (7.4)

On the other hand, we notice that an inversion is equivalent with a double rotation with the angle π (Figure 4),

( ) ( )21ππ RRI = (7.5)

For a wave function rotation with a differential angle αδ

, ( ) ( ) ( )

( ) ( )

( )re

rr

rr

rr

rrrr

SSJ

ψ

ψαδψ

ψαδψαδψ

αδiii

=

∂∂

×⋅+=

∂∂

×+=×+

=

(7.6)

we find a rotation operator depending on the spin operator S

,αδ

αδ

SeR i= (7.7)

which, for a rotation with an arbitrary angleα

generatesα

α

SeR i= (7.8)

For πα = we obtain( ) ( ) SeRR π

ππi21 == (7.9)

With this expression, from (7.4) and (7.5) we obtain the inversion eigenvalues as a function of spin,

1

2

1 1

2 2

112

1 1

i2 Si2 S

i2 S

I e S -I e

I e S -

ππ

π

= = − ⇒ == = = ⇒ =

Fermions

Bosons

Figure 4. Two-particle system – a particle inversion equivalent to a double rotation with the angle π .

Stefanescu


Quantum Particle in Gravitational FieldWe consider the wave functions (1.16) in a system of

curvilinear coordinates ( ) ( )0 0ix x ,x , x ct ,α = =

( )( )

( ) ( ) ( )( )

( )( )

( ) ( ) ( )( )

0

0

1 3 1 2 303 2 1 2 3

1 1 2 33 2 1 2 3

1

2

1

2

i jij

i jij

i M c g x x dsi i 3/

i M c g x x dsi i/

x , y,zx ,t x ,t e M c dx dx dx

x ,x ,x

x, y,zx ,t x ,t e dx dx dx

x ,x ,x

− +

− − +

∂∫ψ = ϕ∂π

∂∫ϕ = ψ∂π

∫

∫

(8.1)

with the time-space differential interval ds,2ds g dx dxα β

αβ= (8.2)

and use Dirac’s formalism of General Theory of Relativity [16]. In (8.1), we used the notation:

ii dxx

ds= (8.3)

From (8.2), we notice that

1dx dxg x x gds ds

α βα β

αβ αβ= =

(8.4)

For a non-relativistic case, small particles compared with the non-uniformities of any gravitational field, small velocities, we consider:

002 0

00i j

ij

negligible

ds dxds g dx g dx ds dx cdt cdt dt

= + ≈ = ⇒ ≈ =

(8.5)

and linearize the wave phases:

( ) ( )( )

2 2

1

112

12

12

i j i j i jij ij ij

i j i jij ij

i j i jij ij

i jij

i jij

g x x ds ds g x dx c dt g dx dx

g x dx cdt g x x

g x dx cdt g x x

g x dx c g x x dt

g x x c g x x t

α βαβ

α βαβ

− + = − + +

= − + +

= − + +

= − +

= − +

∫ ∫∫

∫

∫

(8.6)

In this way, for a quantum particle we obtain wave packets with linear phases in space and time,

( )( )

( ) ( )( )

( )( )

( ) ( )( )

0

0

13 1 2 3203 2 1 2 3

11 2 32

3 2 1 2 3

1

2

1

2

i jij

i jij

i M c g x x c g x x ti i 3

/

i M c g x x c g x x ti i

/

x , y,zx ,t x ,t e M c dx dx dx

x ,x ,x

x, y,zx ,t x ,t e dx dx dx

x ,x ,x

α βαβ

α βαβ

− +

− − +

∂ψ = ϕ

∂π

∂ϕ = ψ

∂π

∫

∫

For this wave packet we obtain a self-consistent expression of the wave velocity,

( )( )

1

21 22

j jjj

iij

xdx dxv c g x x c cdt dsg x xg x

α βαβ α β

αβ

∂= = = =

∂

(8.7)

and the acceleration of a particle on a geodesic as a function of the Christoffel symbol j

µνΓ of the second kind,2

2 22

jj j jd d x dx dxa v c c

dt ds dsds

µ ν

µν= = = − Γ (8.8)

We introduce the Christoffel symbol of the first kind, and consider the expression of this symbol as a function of the metric tensor derivatives:

( )

2 2

2 12

j j j

j, , ,

dx dx dx dxa c c gds ds ds ds

dx dxc g g g gds ds

µ ν µ νλ

µν λµν

µ νλ

λµ ν λν µ µν λ

= − Γ = − Γ = −

+ −

(8.9)

For the non-relativistic case, considered here, we neglect the spatial coordinate derivatives compared to the time derivative, and take into account a stationary state, which means that the derivatives with time disappear. We obtain a particle accelerations proportional to the derivatives of the metric tensor element g00:

0 02 2

0 0 0 0 00 00

0 01 1

1 12 2

jj j j

, , , ,dv dx dxa c g g g g c g gdt ds ds

λ λλ λ λ λ

= = − + − = (8.10)

which means that this matrix element behaves as a potential. Considering the Newtonian potential V in this matric element,

00 1 2

1

g VmV , m m,r

= −

= − = (8.11)

we obtain( )2 2

2 2 1

1 1 22

1

j j j

j j

Va c g V c gx x

mc g mc gr r rx

λ λλ λ

λλ

∂ ∂= − = − =

∂ ∂∂ ∂ − = − ∂∂

(8.12)

With the Schwarzschild solution for the contravariant metric elements conjugated to the spatial coordinates, we obtain the three components of the acceleration,

1 21

2

2

3

21

00

m mcar r

aa

− = − −

=

= (8.13)

We notice that, for a rather small distance r, we obtain a correction of the Newtonian force, increasing the gravitational attraction.

Quantum Particle as a Distribution of MatterWe notice that, in this framework, a particle is conceived as

a distribution of matter with a matter velocity field:

jjj

j xcs

xct

xv ===d

dd

d (9.1)

From the relativistic quantum principle of the space-time interval invariance,

(8.1)



ds g dx dxµ νµν=

(9.2)

we obtain that the covariant derivative of velocity vector is perpendicular to this vector,

( ) ( )1

0 2: : ::

g x x

g x x g x x x x g x x ,

µ νµν

µ ν µ ν µ ν µ νµν µν σ σ µν σσ

=

= = + =

0: =σν

ν xx (9.3)

If, besides the acceleration on a geodesic track, we consider an additional component

µA ,

,dx x x x x Ads

µµ ν µ ν σ µν νσ= = −Γ +

(9.4)

we find that this component is perpendicular to the velocity vector:

( )

0

,

:

:

x x x A

x x A / x

x x x A x

µ µ σ ν µν νσ

µ ν µν µ

µ ν µµ ν µ

+ Γ =

=

=

0=µµ Ax

(9.5)

This means that any additional acceleration to the acceleration on a geodesic is perpendicular to the wave velocity, which is in agreement with wave propagation (Figure 5).

In particle propagation on a geodesic, other matter propagations are allowed only in perpendicular directions on the particle velocity.

We consider a quantum particle as a normalized matter distribution – a normalized integral of the matter density, in Cartesian coordinates,

( ) ( ) 21x, y,z ,t dxdydz x, y,z ,t dxdydzρ = ψ =∫ ∫ (9.6)

or

( ) ( ) ( )( ) 1ddd

,,,,,ddd, 321

321

2321 =∂∂

= ∫∫ xxxxxxzyxtxxxxtx ii ψρ

(9.7)

in curvilinear coordinates. We integrate the matter density

( ) ( ) ( )( )321

2

,,,,,,

xxxzyxtxtx ii

∂∂

= ψρ (9.8)

on a space-time volume V , in two systems of coordinates.

( ) ( )∫∫ =′′′′′

VV

32103210 dddddddd xxxxJxxxxxx µµ ρρ (9.9)

and remark that the elements of the Jacobian,

( ) ( ) ( )( )3210

3210

, ,,,,,,DetDetxxxxxxxxxJJ

∂∂

===′′′′

′′

µααµ

(9.10)

are also elements of a tensor transform, which, for the metric tensor is:

, ,

J J

g x x g′µ α ′ν β

′ ′µ ν′ ′αβ α β µ ν=

(9.11)

Calculating the determinants,

( ) 2g Det g J gαβ ′= = (9.12)

we obtain the Jacobian as a function of the determinants of the metric tensor in the two systems of coordinates,

gg

J′−

−=

(9.13)

Introducing this expression of the Jacobian in the integral equation (9.9), we obtain an invariant for the matter density:

( ) ( ) Invariant=′−=− ′ gxgx µα ρρ (9.14)

We consider a matter flow density µJ , as the product of the matter density with the velocity

µµ ρxJ = (9.15)

and the matter conservation as a null covariant divergence

0: , ,J J J J Jµ µ µ ν ν µ νµ µ νµ ν νµ= + Γ = + Γ = (9.16)

From the general expression of the Christoffel symbol of the

Figure 5. Plane wave – any additional particle acceleration µA to the acceleration on a geodesic is perpendicular to the velocity µx .

Stefanescu


second kind as a function of the metric tensor,

( )12 , , ,g g g g gµ µλ µλ

νσ λνσ λν σ λσ ν σν λΓ = Γ = + − (9.17)

we obtain

( ) ( ) ( )11 1 1 12 2 2 2, , , , , ,

g g g g g g g g gg

µ µλ µλ −νµ λν µ λµ ν µν λ λµ ν ν ν

Γ = + − = = = −− (9.18)

At the same time,

( ) ( )νν ,, 21 g

gg −

−=−

which is( )( )ν

ν

,

,

211

g

g

g −

−

−=

(9.19)

From (9.18) and (9.19), we obtain the Christoffel symbol of the second kind in the expression (9.16),

( )g

g

−

−=Γ νµ

νµ,

(9.20)

which takes the form of a divergence of the product of the matter flow density with the square root of the metric determinant,

( ) ( ) ( ) 0,,,,: =−=−=−+−=− µµ

νν

ννν

νµµ gJgJgJgJgJ (9.21)

Integrating this expression on a volume V of spatial coordinates,

( ) 0d3, =−∫

V

xgJ µµ

(9.22)and separating the time derivative from the coordinate

derivatives, we find a conservation law, as a matter variation in a volume V by the flow of this matter through the surface VΣ of the volume V:

( )

.3,2,1,d

dd

V

2

3,

0,

30

=−−=

−−=

−

∫

∫∫

Σ

mxgJ

xgJxgJ

mm

Vm

m

V

(9.23)

In the nonrelativistic case, cxm <<

, weak gravitational field, we obtain

0 0

m m

J xJ x= ρ ≈ ρ

= ρ

(9.24)

while the conservation equation (9.23) takes a more explicit form, of density and matter flow,

( )mmx ,0, ρρ −=

(9.25)

ConclusionWe discovered that a packet of Schrödinger wave functions,

representing a quantum particle, is not in agreement with the Hamilton equations which describe the dynamics of such a particles – such an agreement is obtained only when instead of the Hamiltonian we consider the Lagrangian function. For a realistic particle, which must have a finite spectrum, we considered the relativistic Lagrangian, with a cut-off velocity c. We defined a Relativistic Quantum Principle: The time-dependent phase of a quantum particle is invariant to any change of coordinates. We obtained a wave equation for a quantum particle, depending on velocity, while the conventional Schrödinger equation describes only the amplitude of the particle wave-function - it includes a rapidly varying factor, with a phase proportional to the particle rest mass. On this basis, the relativistic kinematics and dynamics, the electromagnetic field equations, the particle spin, and the Schwarzschild-Newtonian dynamics in a gravitational field have been obtained. A quantum particle has been described by a wave function representing a distribution of conservative matter in motion, according to the General Theory of Relativity.

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and Danvers (Massachusetts, USA): Bentham Science Publishers. 2014.

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*Correspondence to:Eliade StefanescuAdvanced Studies in Physics Centre of the Romanian AcademyAcademy of Romanian Scientists BucharestRomaniaTel: +4021-3188106/3138E-mail: [email protected]

mailto:[email protected]

Unitary relativistic quantum theory. - … · Relativistic Quantum Principle, ... in section 2 we obtain the relativistic transform of the space- ... quantum particle in a field we

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