•
Anatoliy N. SERDYUKOV
Francisk Skorina Gomel State University
Gomel, Byelorussia
•The System of Equations of
Interacting Electromagnetic,
Scalar Gravitational
and Spinor Fields
The Classical Particle in the Electromagnetic
and Scalar Gravitational Fields
Including of Electromagnetic
Interaction
(By Means of Addition)
Λ   μc
2
(Comparative analysis)
1
1  v /c  A j
c
2
2
e
L  mc2 1  v 2 /c 2  Av  e
c
Including of Gravitational
Interaction
(By Means of Multiplication)
Λ   μc 2 1  v 2 /c 2U 2
L  mc2 1  v 2 /c 2U 2
The Classical Particle in the Electromagnetic
and Scalar Gravitational Fields
Including of Electromagnetic
Interaction
(By Means of Addition)
Λ   μc
2
(Comparative analysis)
1
1  v /c  A j
c
2
Λ   μc 2 1  v 2 /c 2U 2
2
e
L  mc2 1  v 2 /c 2  Av  e
c
1

 e   Av   
dt
c

dPμ


m
v
e

 A  

2
2
 1v / c
c 
P   

2

 i  mc

 e  
 
2 2

 c  1v / c
L  mc2 1  v 2 /c 2U 2
d L L

dt v r
d  L
L

 L  
v
dt  v
t

dPμ
dt
P    Lv , ci  v Lv  L  

Including of Gravitational
Interaction
(By Means of Multiplication)


 mc2 1  v 2 /c 2 U 2
 mv U 2

 1  v 2 / c2
P   
2 2
 i mc U
c
2
2
 1v / c







The Classical Particle in the Electromagnetic
and Scalar Gravitational Fields
Including of Electromagnetic
Interaction
(By Means of Addition)
L  mc
2
(Comparative analysis)
e
1  v /c  Av  e
c
2
2
1

 e   Av   
dt
c

dPμ
Including of Gravitational
Interaction
(By Means of Multiplication)
L  mc2 1  v 2 /c 2U 2
dPμ
dt
 mc2 1  v 2 /c 2 U 2

mv
i
mc2
 p    2 2 ,
2
2
 1 v / c c 1 v / c
dpμ
e
 B u
dτ c
B   A    A




dp
dτ
 mu g  u g u
g  2c 2 ln U
The Classical Particle in the Electromagnetic
and Scalar Gravitational Fields
Including of Electromagnetic
Interaction
(By Means of Addition)
dpμ
e
 Bu
dτ c
B    iBE 0iE 


e
A  B
c
(Comparative analysis)
u2  c 2
dp
dτ
p  mu
Including of Gravitational
Interaction
(By Means of Multiplication)
 mu g   u g  u
( g )  (g, iη)
2
m du
u

0
dτ
2 dτ
dp μ
dpμ
dτ
 Au
A   A
A  mu g   u g  
The Classical Particle in the Electromagnetic
and Scalar Gravitational Fields
Including of Electromagnetic
Interaction
(By Means of Addition)
dpμ
dτ

e
B u
c
  B   iE 
B    iE 0 


d 
mv
dt  1  v 2 /c 2
F
Lor.
(Comparative analysis)
dp
dτ
Including of Gravitational
Interaction
(By Means of Multiplication)
 mu g  u g u
2

m
v
i
mc
 p   
,
2
2 c
1  v 2 / c2
 1v / c
( g )  (g, iη)
d 
mv
dt  1  v 2 /c 2

  F Lor.


1


 e E  v  B 
c






Fgrav 

  F grav


1
1 

g

v

v

g

vη 

2
2 2 
c 
c
1  v /c
m
The Gauge-Invariance of Gravitational Field
Fgrav 
1
1 

g

v

v

g

vη 

2
c 
c
1  v 2 /c 2 
m
g  2c 2 ln U
g   
  2c 2 ln U
U e
 / 2c 2
Λ   μc 2 1  v 2 /c 2U 2
E
mc2U 2
1v / c
2
2
( g )  (g , iη)
Here  is
  '    λ an arbitrary
real constant
g  g  g
U  U   Ue
 / 2c 2
Λ  Λ'  Λ e
 / c2
E  E'  E e
 / c2
The Closed Classical Systems: Particles and Fields
Λ   μc 2 1  v 2 /c 2U 2
Λ  Λ'  Λ e
 / c2
Λ
Λ


 U  U
The Massive Particles
and the Gravitational Field
Λ   μc2 1  v 2 /c 2U 2 
c4


U 2
2G
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
Λ
Λ


   A  A
B   A    A
 B  
4
j
c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
Λ
Λ


   A  A
Λ
Λ


 U  U
2G

2
2


μ
1

v
/
c

U  0
2
c


B   A    A
 B  
4
j
c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
Λ
Λ


   A  A
Λ
Λ
g  2c 2 ln U


 U  U
2G

2
2


μ
1

v
/
c

U  0
2
c


B   A    A
4
 B   j
c
1 2
  g   2 g   4Gμ 1  v 2 / c 2
2c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
Λ
Λ


   A  A
Λ
Λ
g  2c 2 ln U


 U  U
4
A  
j
c
B   A    A
4
 B   j
c
2G

2
2


μ
1

v
/
c

U  0
2
c


  A  0
1 2
  g   2 g   4Gμ 1  v 2 / c 2
2c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
Λ
Λ


   A  A
Λ
Λ
g  2c 2 ln U


 U  U
4
A  
j
c
B   A    A
4
 B   j
c
2G

2
2


μ
1

v
/
c

U  0
2
c


  A  0
1 2
  g   2 g   4Gμ 1  v 2 / c 2
2c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
Λ   μc2 1  v 2 /c 2 
Λ   μc2 1  v 2 /c 2U 2 
1
1

 A j 
  A    A 2
c
16
c4


U 2
2G
4
A  
j
c
2G

2
2


μ
1

v
/
c

U  0
2
c


4
 B   j
c
1 2
  g   2 g   4Gμ 1  v 2 / c 2
2c
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
B   A    A
e  B  0
g   
  g     g  0
The Closed Classical Systems: Particles and Fields
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
  g     g  0
e B  0
 B  
4
j
c
dpμ
e
 B u
dτ
c
4
A  
j
c
  g 
1 2
2
2
g


4

Gμ
1

v
/
c

2c 2
dp
dτ
 mu g  u g u
2G

2
2
   2 μ 1  v / c U  0
c


Three-dimensional Form of Equations
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
  g     g  0
e B  0
 B  
dpμ
4
j
c
e
 B u
dτ
c
  g 
1 2
2
2
g


4

Gμ
1

v
/
c
.
2 
2c
dp
dτ
 mu g  u g u
Three-dimensional Form of Equations
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
  g     g  0
e  B  0
E  
1 B
c t
g  0
1 g
 η
c t
B  0
4
 B   j
c
1 E 4

j
c t
c
E  4
B 
1 2
  g  2 g  4Gμ 1  v 2 / c 2 .
2c
g 


1 η
1
 2 g 2  η2  4Gμ 1  v 2 / c 2
c t 2c
Three-dimensional Form of Equations
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
dpμ
e
 B u
dτ
c
d  mv  
1


e
E

v

B


dt  1  v 2 /c 2  
c

dp
dτ
 mu g  u g u
d
mv
m
1
1 


g

v

v

g

vη 

2
2
2
2
2
dt 1  v / c
c 
1v / c  c
Three-dimensional Form of Equations
(Comparative analysis)
The Massive Particles
The Charged Particles
and the Gravitational Field
and the Electromagnetic Field
g  0
1 B
E  
c t
B  0
1 E 4

j
c t
c
E  4
B 
d  mv   1


e
E

v

B


dt  1  v 2 /c 2   c

1 g
 η
c t
g 


1 η
1
 2 g 2  η2  4Gμ 1  v 2 / c 2
c t 2c
d
mv
m
1
1 


g

v

v

g

vη 

2
2
2
2
2
dt 1  v / c
c 
1v / c  c
Newtonian Limit of Relativistic Equations
The Massive Particles
and the Gravitational Field
g  0
g  0
1 g
 η
c t
g 


1 η
1
 2 g 2  η2  4Gμ 1  v 2 / c 2
c t 2c
g  4Gμ
c 
d
mv
m
1
1 


 g  2 v  v  g  vη 
2
2
2
2
dt 1  v / c
c 
1v / c  c
2G

2
2
   2 μ 1  v / c U  0
c


g
0
t
1 
η
0
c t
d
mv  mg
dt
v  c
  4Gμ
v  c,   c ,U  e
2
 / 2c 2
1

2c 2
Newtonian Limit of Relativistic Equations
The Massive Particles
and the Gravitational Field
U  e / 2c
2
GM
1 2
2c r
GM
r
gr   

 GM  r
r 2 1  2 
 2c r 
1
g  2 g 2  4Gμ
2c
 2 2G 
   2 μ U  0
c


1
M 2
c
g 

2G

2
2
   2 μ 1  v / c U  0
c


 2 2 g 2U 2 
 μc U 
dV

8G 



1 η
1
 2 g 2  η2  4Gμ 1  v 2 / c 2
c t 2c
Newtonian Limit of Relativistic Equations
The Massive Particles
and the Gravitational Field
E
mc2
1  v /c
2
2
e
 / c2
U e
gr   
 / 2c 2
1
v2
 1 2
2c
1  v 2 /c 2
2

e / c  1  2
c
1
g2  / c2
W
e
8G
GM
2c 2r
GM
r

 GM  r
r 2 1  2 
 2c r 
mv 2
E  mc 
 m
2
2

 1
c2
g2
W
8G

e / 2c  1 
2
r 
GM
c2

2c 2
gr   
GM
r
GM r

2
r
r
The Energy-Momentum Tensor: Particles and Fields
(Comparative analysis)
The Electromagnetic Field
1
Λ   μc2 1  v 2 /c 2  A j 
c
1


  A    A 2
16 
1
( ef )

Λ 
 A    A 2
16
can
T
Λ
( ef )

Λ(ef )

  A
( A )
The Gravitational Field
Λ   μc2 1  v 2 /c 2U 2 
c4


U 2
2G
4
c

Λ(gf )  
U 2
2G
can
T
 Λ(gf )
Λ(gf )

 U
(U )
The Energy-Momentum Tensor: Particles and Fields
(Comparative analysis)
The Electromagnetic Field
can
T
Tcan
Λ
( ef )

Λ(ef )

  A
( A )
The Gravitational Field
can
T
Λ
( gf )

Λ(gf )

 U
(U )
1 2
1
 can c4 
1

2
   A  B 
B  T 



U


U


U

 


 
16
G 
2
 4


1
     A  H  TBel  Tcan  
4
1 2

Bel  1
T   B B 
B 
16
 4

1
 U   2 Ug
2c
2
U
1 2 

can
T 
 g  g  g 
4G 
2

The Energy-Momentum Tensor: Particles and Fields
(Comparative analysis)
The Electromagnetic Field
The Gravitational Field
2
U
1 2 

can
T 
 g  g  g 
4G 
2

1 2

Bel  1
T   B B 
B 
16
 4


1 2 2
W  E B
8

c
S  EB
4
1
π
EB
4c
W

1 2 2 2
U g η
8G

π
S
c
U 2ηg
4G
1
U 2ηg
4cG
Once more about the Gauge-Invariance
of Gravitational Field
  '    λ
g  g  g
Λ   μc 2 1  v 2 /c 2U 2
E
mc2U 2
1v / c
2
2
U2 
1

2


T 
g

g

g

  

 
4G 
2

U  U   Ue
 / 2c 2
Λ  Λ'  Λ e
 / c2
E  E'  E e
 / c2
T  T'  T e
 / c2
Once more about the Gauge-Invariance
of Gravitational Field
The System of Gauge-Invariant Equations
dp
dτ
 mu g  u g u
  g     g  0
1 2
  g  2 g  4Gμ 1  v 2 / c 2 .
2c
2G

2
2


μ
1

v
/
c

U  0
2
c


The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16

?
λ
 / 2c 2
U  Ue
Λ  Λ  Λe
 / c2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

1

 A    A 2
16
Λ  Λ  Λe
Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G


 / c2
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 ' =  of gravitation potential.
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2
4
c
21A j
2 2 2

 U 
1  v /c U 
 
c
2G
λ
 / 2c 2
U  Ue

Λ  Λ  Λe
 / c2
?
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2

4
c
2 1
2 2 2
 A j




U
1  v /c U

c
2G
λ
 / 2c 2
U  Ue
Λ  Λ  Λe
 / c2
A  AU 2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2

4
c
2 1
2 2 2
 A j




U
1  v /c U

c
2G
λ
 / 2c 2
U  Ue
Λ  Λ  Λe
 / c2
A  AU 2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2

4
c
2 1
2 2 2
 A j




U
1  v /c U

c
2G
λ
U  Ue / 2c
2
Λ  Λ  Λe  / c
2
A  A  A  e / c
2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2

4
1
c
2
2 1
2 2 2

A
j




A


A




U
1  v /c U
 
 
 

c
16
2G
λ
 / 2c 2
U  Ue
A  A  e
 / c2
Λ  Λ  Λe
 / c2
 A   A 
2
?
  A    A  e
2 2 / c 2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
1
Λ   μc2 1  v 2 /c 2  A j 
c

Λ   μc 2 1  v 2 /c 2U 2 
c4

 U 2
2G

1

 A    A 2
16
Λ   μc 2

4
1
c
2 2
2 1
2 2 2

A
j




A


A




U
1  v /c U
 
 
  U

c
16

2G
λ
 / 2c 2
U  Ue
A  A  e
 / c2
Λ  Λ  Λe
 / c2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
Λ   μc
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
2
λ
U  Ue / 2c
2
A  A  e
 / c2
2
2
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a

a
 ea

 Av a  ea 
c

The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d L L

dt v a ra
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
B   A    A
B    iBE

 iE 

0 
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
B   A    A
B    iBE

d  mv 
m
1 
e
 1

g

v

v

g

v
η

e
D

vH


2


2
2
2
2
dt  1  v / c  1  v / c  c
c 
c
 iE 

0 
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
B   A    A
B    iBE

d  mv 
m
1 
e
 1

g

v

v

g

v
η

e
D

vH


2


2
2
2
2
dt  1  v / c  1  v / c  c
c 
c
D  U 2E
H  U 2B
 iE 

0 
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
B   A    A
B    iBE
 iE 

0 
H    iHD 
 iD 

0 

d  mv 
m
1 
e
 1

g

v

v

g

v
η

e
D

vH


2


2
2
2
2
dt  1  v / c  1  v / c  c
c 
c
D  U 2E

H   BU 2
H  U 2B
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The movement equations of classical particles
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
L
2
2

mac 2 1  v a2 /c 2U 2 
a
d  mvU 2
dt  1  v 2 / c 2

a
 ea

 Av a  ea 
c


  mU 2 1  v 2 / c 2 g  eE  e v  B

c

E
1 A
 
c t
B  A
B   A    A
B    iBE
 iE 

0 
H    iHD 
 iD 

0 

d  mv 
m
1 
e
 1

g

v

v

g

v
η

e
D

vH


2


2
2
2
2
dt  1  v / c  1  v / c  c
c 
c
A  A  e / c
λ
U  Ue / 2c
2
2
A  A  A   f
  B e / c
B  B
  H
H   H 
2
D  U 2E

H   BU 2
H  U 2B
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
The equations of interacting electromagnetic and gravitational fields
Λ   μc
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
2
2
  g     g  0
e  B  0

Λ
Λ

  A  A
 H   
4
j
c
  g 
2
A  A  A   f
  B e / c
B  B
Λ
Λ

(U ) U
1 2 G 2
2
2
g

H

4

Gμ
1

v
/
c


2c 2
4c 2
2G 
1

2
2
2 


μ
1

v
/
c

H
U 0


2
2  
c 
16c


H   BU 2
A  A  e / c

2
  H 
H   H 
λ
U  Ue / 2c
2
g  g  g
The Extended System of Charge Particles,
Electromagnetic and Gravitational Fields
Three-dimensional form of field equations
Λ   μc
E  
2
c4
 U 2  1 A j  1   A    A 2U 2
1  v /c U 
2G
c
16
2
2
2
1 B
c t
B  0
1 D 4
H 

j
c t
c
D  4
D  U 2E
2
H U B
A  A  A  f
1 f
     
c t
g 




1 η
1
G
 2 g 2  η2   2 D2  H 2  4Gμ 1  v 2 / c 2
c t 2c
2c
λ
U  Ue / 2 c
g  g  g
η  η  η
2
E  E  Ee
 / c2
A  A  Ae λ / c
 / c2
φ  φ  φe λ / c
B  B  Be
2
2
The System of Equations of Interacting
Electromagnetic, Scalar Gravitational and Spinor Fields
c4
1


Λ
 U 2 
 A    A 2U  2
2G
16
e 

Λ  cψ    i  A ψ  mc2ψ ψ
c 

c4
1
e 



Λ
U 2 
 A    A 2U  2  cψ    i  A ψ  mc2ψ ψU 2
2G
16
c 

e 
 
2

i


A

mcU




 
ψ  0 
c
2G 
1

 

2 
H
U 0
  2  mψ ψ 
2  
c 
16c


e 
 

ψ   i  A   mcU 2   0
c 
 

U U e
 / 2c 2
Λ  Λ  eλ / c
2
A  A  e / c
ψ ψ
e
λ / c2

2
e  B  0
 H 
4

j
c
B  UH
A  A   f
j   ceψ ψ
ψ  ψ  e(ie / c ) f
B   A    A
The System of Equations of Interacting
Electromagnetic, Scalar Gravitational and Spinor Fields
c4
1
e 



Λ
U 2 
 A    A 2U  2  cψ    i  A ψ  mc2ψ ψU 2
2G
16
c 

e 
 
2

i


A

mcU




 
ψ  0 
c
2G 
1

 

2 
H
U 0
  2  mψ ψ 
2  
c 
16c


e 
 

ψ   i  A   mcU 2   0
c 
 

e  B  0
 H 
4

j
c
B  UH