The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields...

•The System of Equations of Interacting Electromagnetic,

Scalar Gravitational and Spinor Fields

•

Anatoliy N. SERDYUKOVFrancisk Skorina Gomel State University

Gomel, Byelorussia

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

jAc

/cμcΛ1

1 222 v2222 1 U/cμcΛ v

ec

e/cmcL Av222 1 v 2222 1 U/cmcL v

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

jAc

/cμcΛ1

1 222 v2222 1 U/cμcΛ v

ec

e/cmcL Av222 1 v 2222 1 U/cmcL v

t

LL

L

dt

d

LL

dt

d

vv

rv 2222μ 1 U/cmc

dt

dP v

22

22

22

2

/1

/1

c

Umc

c

i

c

Um

P

v

v

v

ec

mc

c

i

c

e

c

m

P

22

2

22

/1

/1

v

vA

v

Avc

edt

dP 1μ

LL

c

iLP

vv

v,

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)

ec

e/cmcL Av222 1 v 2222 1 U/cmcL v

2222μ 1 U/cmcdt

dP v

Avc

edt

dP 1μ

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)

uBc

e

dτ

dpμ ugugum

dτ

dp

22

2

22 /1,

/1 c

mc

c

i

c

mp

vv

v

AAB Ucg ln2 2

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)

22 cu ugugum

dτ

dp

0E

EB

i

iB

gugumA

AA

uAdτ

dpμ

02

2

dτ

dum

dτ

dpu μ

Bc

eA

uBc

e

dτ

dpμ

mup ),()( iηg g

uBc

e

dτ

dpμ ugugum

dτ

dp

),()( iηg g

0E

EB

i

iB

Lor.

221F

v

/c

m

dt

d

v

BvEF

ce

1Lor.

grav

221F

v

/c

m

dt

d

v

η

cc/c

mvgvvgF

11

1222

grav

v

The Classical Particle in the Electromagnetic and Scalar Gravitational Fields

(Comparative analysis) Including of GravitationalInteraction

(By Means of Multiplication)

Including of ElectromagneticInteraction

(By Means of Addition)

22

2

22 /1,

/1 c

mc

c

i

c

mp

vv

v

The Gauge-Invariance of Gravitational Field

Uc ln2 2

22/ ceU

λ' g

22/ cUeUU

22

22

/1 c

UmcE

v

2222 1 U/cμcΛ v

2/ ceEE'E

2/ ceΛΛ'Λ

Here is an arbitrary real constant

ggg

η

cc/c

mvgvvgF

11

1222

grav

v

) , ()( iηg g

Ucg ln2 2

U

Λ

U

Λ

The Closed Classical Systems: Particles and Fields

2/ ceΛΛ'Λ

2222 1 U/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

The Massive Particles and the Gravitational Field

A

Λ

A

Λ

jc

B4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB

(Comparative analysis)

U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB

(Comparative analysis)

U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AAB

2222

/142

1cGμg

cg v

Ucg ln2 2

(Comparative analysis)

U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

jc

A4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

0 A

Ucg ln2 2

AAB

2222

/142

1cGμg

cg v

(Comparative analysis)

U

Λ

U

Λ

0/12 22

2

Ucμc

Gv

A

Λ

A

Λ

jc

B4

jc

A4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

0 A

Ucg ln2 2

AAB

2222

/142

1cGμg

cg v

(Comparative analysis)

0/12 22

2

Ucμc

Gv

jc

B4

jc

A4

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

2222

16

11

1

AAjAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

2222

/142

1cGμg

cg v

(Comparative analysis)

AAB

0 Be

g

0 gg

The Closed Classical Systems: Particles and Fields

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

0 Be 0 gg

The Closed Classical Systems: Particles and Fields

jc

B4

uBc

e

dτ

dpμ ugugum

dτ

dp

0/12 22

2

Ucμ

c

Gv

jc

A4

2222

/142

1cGμg

cg v

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

Three-dimensional Form of Equations

0 Be 0 gg

./142

1 2222

cGμgc

g v

jc

B4

uBc

e

dτ

dpμ ugugum

dτ

dp

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

Three-dimensional Form of Equations

0 Be 0 gg

./142

1 2222

cGμgc

g v

jc

B4

0

1

B

BE

tcη

tc

g

g

1

0

4

41

E

jE

Bctc 2222

2/14

2

11cGμη

ct

η

cv

gg

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

Three-dimensional Form of Equations

uBc

e

dτ

dpμ ugugum

dτ

dp

BvE

v

ce

/c

m

dt

d 1

1 22v

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

Three-dimensional Form of Equations

BvE

v

ce

/c

m

dt

d 1

1 22v

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv

0

1

B

BE

tcη

tc

g

g

1

0

4

41

E

jE

Bctc 2222

2/14

2

11cGμη

ct

η

cv

gg

The Charged Particles and the Electromagnetic Field

The Massive Particles and the Gravitational Field

(Comparative analysis)

Newtonian Limit of Relativistic Equations

The Massive Particlesand the Gravitational Field

η

ccc

m

c

m

dt

dvgvvg

v 11

/1/122222 vv

ηtc

g

g

1

0

22222

/142

11cGμη

ct

η

cv

gg

0

0

t

g

g

01

tc

η

Gμ 4gc

22/2

21, ,

2

ceUcc c

v

gv mmdt

d

cv

Gμ 40/12 22

2

Ucμc

Gv

Newtonian Limit of Relativistic Equations

The Massive Particlesand the Gravitational Field

rc

GMeU c

22/

21

2 0/1

2 222

Ucμc

Gv

22222

/142

11cGμη

ct

η

cv

gg

dVG

UUμc

cM

8

1 2222

2

g

02

22

Uμc

G

Gμc

42

1 22

gg

r

rc

GMr

GM rrg

22

21

rc

GMeU c

22/

21

2

r

rc

GMr

GM rrg

22

21

2/2

8ce

GW

g

2/

22

2

1

ce/c

mcE

v m

mmcE

2

22 v

GW

8

2g2

/

2

2

22

1

21

1

1

2

ce

c/c

c

v

v

12

c

r

GM

rr

GM rrg

2

22/

21

2

ce c

2c

GMr

Newtonian Limit of Relativistic Equations

The Massive Particlesand the Gravitational Field

2)ef(

16

1

AAΛ 2

4)gf(

2U

G

cΛ

The Energy-Momentum Tensor: Particles and Fields

The Gravitational FieldThe Electromagnetic Field

2

222

16

1

1 1

AA

jAc

/cμcΛ v

24

2222

2

1

UG

c

U/cμcΛ

v

AA

ΛΛT

)(

)ef()ef(can U

U

ΛΛT

)(

)gf()gf(can

(Comparative analysis)

The Energy-Momentum Tensor: Particles and Fields

AA

ΛΛT

)(

)ef()ef(can U

U

ΛΛT

)(

)gf()gf(can

2can

16

1

4

1BBAT

HA4

1

2Bel

16

1

4

1BBBT

canBel TT

22

can

2

1

4ggg

G

UT

24

can

2

1UUU

G

cT

Ugc

U22

1

The Gravitational FieldThe Electromagnetic Field(Comparative analysis)

The Energy-Momentum Tensor: Particles and Fields

2Bel

16

1

4

1BBBT

22

can

2

1

4ggg

G

UT

8

1 22 BEW

4

BES

c

4

1BEπ

c

8

1 222 ηUG

W

g gS ηUG

c 2

4

gπ ηUcG

2

4

1

The Gravitational FieldThe Electromagnetic Field(Comparative analysis)

Once more about the Gauge-Invariance of Gravitational Field

λ'

22/ cUeUU

22

22

/1 c

UmcE

v

2222 1 U/cμcΛ v

2/ ceEE'E

2/ ceΛΛ'Λ

ggg

2/ ceT'TT

22

2

1

4ggg

G

UT

Once more about the Gauge-Invariance of Gravitational Field

./142

1 2222

cGμgc

g v

ugugum

dτ

dp

0 gg

0/12 22

2

Ucμ

c

Gv

The System of Gauge-Invariant Equations

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

jAc

/cμcΛ1

1 222 v

216

1

AA

2222 1 U/cμcΛ v

24

2U

G

c

2/ cΛeΛΛ 22/

λ

cUeU

?

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

To take into account the presence of gravitation field of system (in the frame of scalar model of gravitation) it is necessary to ensure the next transformation law of complete Lagrangian at gauge transformation Ю '= of gravitation potential.

jAc

/cμcΛ1

1 222 v

216

1

AA 2

4

2U

G

c

2/ cΛeΛΛ

2222 1 U/cμcΛ v

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v

2/ cΛeΛΛ ? 2222 1 U/cμcΛ v 2

4

2U

G

c

jAc

1

22/

λ

cUeU

jAc

/cμcΛ1

1 222 v

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

22/

λ

cUeU

2UAA

2UAA

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

22/

λ

cUeU

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

216

1

AA 2

4

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

2/ ceAAA

jAc

1

22/

λ

cUeU

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

24

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

216

1

AA

216

1

AA

? 22/

λ

cUeU

2/ ceAA 2/222 ceAAAA

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

24

2U

G

c

2222 1 U/cμcΛ v jAc

/cμcΛ1

1 222 v

2/ cΛeΛΛ

2222 1 U/cμcΛ v 24

2U

G

c

jAc

1

216

1

AA

216

1

AA 2U

2/ ceAA

22/

λ

cUeU

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

2/ ceAA

22/

λ

cUeU

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

aa

LL

dt

d

rv

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

AB

AE

1

tc

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

0E

EB

i

iB

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

The movement equations of classical particles

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU

The movement equations of classical particles

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

0E

EB

i

iB

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU

The movement equations of classical particles

0D

DH

i

iH

2 UBH

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

0E

EB

i

iB

2/ ceBBB

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

fAAA

2/ ceAA

22/

λ

cUeU

a

aaa

a

aa ec

eU/ccmL Av2222 1 v

BHED 22 UU

The movement equations of classical particles

HHH

0D

DH

i

iH

2 UBH

AB

AE

1

tc

AAB

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

BvEgv

c

eecmU

c

Um

dt

d 222

22

2

/1/1

vv

HvDvgvvgv

c

eeη

ccc

m

c

m

dt

d 11

/1

/122222 vv

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

A

Λ

A

Λ

jc

H4

U

Λ

U

Λ

)(

016

1/1

2 22

222

UH

ccμ

c

Gv

The equations of interacting electromagnetic and gravitational fields

2/ ceBBB

fAAA

2/ ceAA

22/

λ

cUeU

HHH

2222

22

/1442

1cGμH

c

Gg

cg v

0 Be 0 gg

2 UBH

ggg

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

The Extended System of Charge Particles, Electromagnetic and Gravitational Fields

2224

2222

16

11

21

UAAjAc

UG

cU/cμcΛ v

2

2

/

/

c

c

e

e

BBB

EEE

BH

ED2

2

U

U

0

1

B

BE

tc

4

41

D

jD

Hctc

Three-dimensional form of field equations

22/

λcUeU

22222

222

/1422

11cGμ

c

Gη

ct

η

cv

HDgg

ηηη ggg

t

f

c

f

1

AAA

2

2

/λ

/λ

φφφ c

c

e

e

AAA

02

ψmcUA

c

ei

The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields

ψψmcψAc

eiψcΛ 2

222

4

16

1

2

UAAUG

cΛ

222224

16

1

2ψUψmcψA

c

eiψcUAAU

G

cΛ

02

mcUA

c

eiψ

016

12 222

UH

cψψm

c

G

jc

H4

0 Be

UHB

ψψcej

AABfcieeψψ )/(

fAA 22/ ceUU

2/λ ce

2/ ceAA

ψψ 2/λ ceΛΛ

02

ψmcUA

c

ei

The System of Equations of Interacting Electromagnetic, Scalar Gravitational and Spinor Fields

222224

16

1

2ψUψmcψA

c

eiψcUAAU

G

cΛ

02

mcUA

c

eiψ

016

12 222

UH

cψψm

c

G

jc

H4

0 Be

UHB