Russian Federal Nuclear Center

All-Russian Research Institute of Experimental Physics

The Isotopic Foldy-Wouthuysen

Representation and Chiral Symmetry.

The Standard Model in the Isotopic Foldy-

Wouthuysen Representation without Higgs

Bosons in the Fermion Sector.

V.P.Neznamov

Publications Publications V.P.Neznamov. Voprosy Atomnoi Nauki I Tekhniki. Ser.: Theoretical and Applied Physics, issue 2, 21 (1988) (in

Russian).

V.P. Neznamov. Physics of Particles and Nuclei 37 (1), 86; Pleadas Publishing, Inc. (2006) (in Russian: Physics of

Particles and Nuclei 37 (1), 152 (2006)).

V.P.Neznamov, A.J.Silenko. J. Math. Phys. 50, 122302 (2009).

V.P. Neznamov. arxiv:0412047 hep-th.

V.P.Neznamov. Voprosy Atomnoi Nauki I Tekhniki. Ser.: Theoretical and Applied Physics, issue 1-2, 3 (2010) (in

Russian);arxiv: 0605215 hep-th.

V.P.Neznamov. Physics of Particles and Nuclei (2012) (in press) (in Russian),

arxiv: 1107.0691 (physics. gen-ph).

V.P.Neznamov. Physics of Particles and Nuclei (2012) (in press) (in Russian),

arxiv: 1107. 0693 (physics. gen-ph).

2

3

ContentContent1. Basic features of the Foldy-Wouthuysen representation

2. Isotopic Foldy-Wouthuysen representation and chirally symmetric

equations of motion of massive fermion fields

2.1 Chirally symmetric equations for massive fermions in the Dirac

representation

2.2 Isotopic Foldy-Wouthuysen representation for massive fermions

2.3 Chiral invariance in the isotopic Foldy-Wouthuysen representation

2.4 Physical content of chirally symmetric equations and their

Hamiltonians in the isotopic Foldy-Wouthuysen representation

3. Continuity equation. Conservation of vector and axial current of massive

fermions in the isotopic Foldy-Wouthuysen representation

4. Possibility of spontaneous breaking of parity in the isotopic Foldy-

Wouthuysen representation

5. The Standard model in the isotopic Foldy-Wouthuysen representation

4

1. Basic features of the Foldy-Wouthuysen representation1. Basic features of the Foldy-Wouthuysen representation

As we known, the Foldy-Wouthuysen (FW) transformation is performed

by means of the unitary operator FWU (L.L.Foldy and S.A.Wouthuysen, 1950).

The wave function (Dirac field operator) and the Hamiltonian of the

Dirac equation in this case transform in the following way:

0 Dp H ;

;FW FWU

†† .FW

FW FW D FW FW

UH U H U iU

t

(1)

0 FW FW FWp H

In expressions (1) and below, the system of units is 1с ; metrics of

the Minkowski space is taken in the form of 1, 1, 1, 1vg diag ;

p ix

; ( )x , ( )FW x are four-component wave functions (field

operators) in the Dirac and Foldy-Wouthuysen representations.

5

For free motion with the matrices i , in the Dirac-Pauli

representation,

0( )DH m αp ;

†0 0 0 0( ) ( ) ( ) ( )FW FW D FWH U H U E ;

0( ) 1 12FW

E mU R L

E E m

αp

; (2)

2 2 , ,2

E mE m R L

E E m

αp

p .

After the FW transformation, the Dirac equation

0 ( ) ( )p x m x αp (3)

transforms into the Foldy-Wouthuysen equation

0 0( ) ( ) ( ) ( )FW FW FW FWp x H x E x (4)

In the Foldy-Wouthuysen equation one can see clear asymmetry in

space and time coordinates, although it is Lorentz-invariant in itself.

6

For the chiral representation of the Dirac matrices, expressions for the

transformation operator chirFWU and Hamiltonian

chirFWH are deduced from the

expressions for FWU , FWH with the matrices 5, ,i in the Dirac-Pauli

representation with replacements of m Σp , 5 (V.P.Neznamov, 2006).

In particular, the Foldy-Wouthuysen equation for free motion in the chiral

representation takes the form

50 0( ) ( ) ( ) ( )chir

FW FW FW FWp x H x E x (5)

Solutions of free equations (4), (5) are plane waves of both positive and

negative energy

( ) ipxFW s3 / 2

( ) ipxFW s3 / 2

1( x,s ) U e ;

( 2 )

1( x,s ) V e ;

( 2 )

(6)

0s

s ss

; ; p0

U V0

E.

In expressions (6), x, p are four-vectors, and s are normalized two-

component Pauli spin functions.

7

In the presence of interaction with a common boson field, there is no

closed Foldy-Wouthuysen transformation.

For stationary external fields, the general form of the exact FW

transformation has been found by Eriksen (E.Eriksen, Phys. Rev. 111, 1011

(1958)).

Another direct way of deriving the Foldy-Wouthuysen transformation for

the case of interaction of fermions with arbitrary (including time-dependent)

boson fields has been developed in studies by the author.

The transformation matrix and the relativistic Hamiltonian have been obtained in

the form of a series in powers of the coupling constant

0 1 2 31 ....FW FWU U (7)

1 2 3 ....FWH E K K K

In equalities (7) 0 FWU is the FW transformation matrix for free Dirac particles;

2 2E m p ; 2 31 1 2 2 3 3, ; , ; , ;K q K q K q q is the coupling constant.

The Foldy-Wouthuysen transformation was used by the author previously for

constructing the Standard Model and, in particular, for construction of quantum

electrodynamics.

8

Let us formulate the basic features of the Foldy-Wouthuysen

transformation and representation:

1. In the Foldy-Wouthuysen representation, the Hamiltonian FWH is

block-diagonal with respect to the upper and lower components of the

wave function (field operator).

2. An obligatory condition for the transition to the FW representation

for free motion and motion of fermions in static external boson fields is

that either upper or lower components of FW should be equal to zero

(condition of Dirac wave function reduction).

3. Because of the form of the basis wave functions (field operators) in

the FW representation with zero upper or lower components the Foldy-

Wouthuysen representation constricts the space of possible states of a

Dirac particle. Special measures (modification of the FW representation)

are required for getting back to the Dirac space of states.

9

Let us consider the Hamiltonian density of a Dirac particle with mass m ,

which interacts with an arbitrary vector boson field ( )B x

ℋD = † †L R L Rm q B P P m q B P P

αp αp

† † † †L L R R L R R Lq B q B m m

αp αp (8)

In equalities (8) q is the coupling constant; 1 0

1,2,3i i

;

5 51 1,

2 2L RP P

are the left and right projection operators;

,L L R RP P are the left and right components of the Dirac field operator .

The Abelian case for the field ( )B x is considered for simplicity. If we

consider the general case of a Dirac particle interacting with a non-Abelian

boson field, the conclusions and results obtained in this study will be the

same.

2. Isotopic Foldy-Wouthuysen representation and chirally 2. Isotopic Foldy-Wouthuysen representation and chirally symmetric equations of motion of massive fermion fieldssymmetric equations of motion of massive fermion fields

10

The Hamiltonian density ℋD allows obtaining the motion equations for

L and R

0

0

L L R

R R L

p q B m

p q B m

αp

αp (9)

One can see that both Hamiltonian density (8) and Eqs. (9) have such a

form that the presence of mass in fermions results in mixing the right and

left components of the field operator , and chiral symmetry is therefore

preserved only for massless fermions.

11

2.1 Chirally symmetric equations for massive fermions in 2.1 Chirally symmetric equations for massive fermions in the Dirac representationthe Dirac representation

Now, let us question ourselves, whether it is possible to write chirally

symmetric equations and their Hamiltonians for massive fermions.

It follows from Eqs. (9) that

1

0

1

0

L R

R L

p q B m

p q B m

αp

αp (10)

By substituting (10) to the right-hand part of Eqs. (9) proportional to

m , we obtain integro-differential equations for R and L

10 0

0 0

10 0

0 0

0

0

L

R

p q B m p q B m

p q B m p q B m

αp αB αp αB

αp αB αp αB (11)

One can see that equations for R and L have the same form, and, in

contrast to Eqs. (9), the presence of mass m does not lead to mixing the

right and left components of .

12

Eqs. (11) can be written as

1 20 0 , 0L Rp q B p q B m

αp αp (12)

In expression (12), designation ,L R shows that equations for L and R

have the same form; 1

i

.

If we multiply Eq. (12) on the left by the term 0p q B αp , we obtain

second-order equations with respect to p

20 0 , 0L Rp q B p q B m

αp αp (13)

For the case of quantum electrodynamics, ,q e B A , Eq. (13)

take the form of

2 2 20 0 , 0L Rp eA e m e ie p A ΣH αE (14)

13

In Eq. (14) rot H A is magnetic field, and 0At

A

E is

electrical field.

Eq. (14) coincide with the second-order equation obtained by Dirac in

the 1920s for the bispinor . However Eq. (14) are written for the left and

right components of the bispinor and contain no “excess" solutions.

The operator 5 commutes with Eq. (13), (14). Consequently,

25 1; 1 . The case of 1 corresponds to the solutions of

Eqs. (13), (14) for L , and 1 corresponds to the solutions of Eqs. (13),

(14) for R .

Thus, expressions (12), (13) show that equations of motion for initially

massive fermions interacting with boson fields can be written in the

chirally symmetric form.

14

2.2 Isotopic Foldy-Wouthuysen representation for massive 2.2 Isotopic Foldy-Wouthuysen representation for massive fermionsfermions

Eqs. (12), (13) are nonlinear with respect to the operator 0p it

.

The linear form of the chirally symmetric equations of fermion fields with

respect to 0p can be derived using the Foldy-Wouthuysen transformation

in a specially introduced isotopic space.

Let us introduce an eight-component field operators, 1R

L

,

R2 1 1

L

and isotopic matrices, 4 4 4 43 1

4 4 4 4

0 0,

0 0

E E

E E

affecting the four upper and four lower components of the operator 1 .

Now, Eq. (9) can be written as

0 1 1 1p m q B αp (15)

0 2 1 2.p Ф m q B Ф αp (16)

15

Finally, let us introduce a sixteen-component spinor,

1

2

R

L

L

R

(17)

and generalization of the isotopic matrices 1 3, :

16 16 1 1 16 16 3 3 1 1 1; ; ;I I I IE E (18)

8 8 8 81 3 2 3 1

8 8 8 8

0 0; ; .

0 0I I I I IE E

iE E

In expressions (18), the symbol " "I shows that the introduced

matrices, similarly to the matrices 1 3, , act in the isotopic space without

affecting the internal structure of the fields ,R L .

The Dirac equation for ( )x accounting for (18) can be written as

0 1 16 16 1

1( ) ( )

2I Ip x m E q x

αp (19)

16

Let us find the Foldy-Wouthuysen transformation in the isotopic space for

“free” Eqs. (15), (16), (19) using the Eriksen transformation.

1

23 3 1 3

0

33 1

3

1 11

2 2 2

11

2

ErIFW

mU U

E E

Em

E E

αp αp

αp

αp

(20)

Transformation (20) is unitary †

0 0 1IFW IFW

U U , and

†

0 0 1 0 3IFW IFW IFWH U m U E αp (21)

Thus, Eqs. (15), (16) in the isotopic Foldy-Wouthuysen representation

have the form

0 1,2 3 1,2IFW IFWp E (22)

17

By applying the transformation matrix 0 IFWU (20) to 1 2, we

obtain

( )

( )

1 0 1

( )2

1 ( ), , ;

2 00

0

RiEt iEtIFW IFW

Et U t e eE

x

x xx x Σp

1 0 1 ( )

( )

00

01, , ;2

( )2

( )

iEt iEtIFW IFW

L

t U t e eE

E

x xxx

Σpx

(23)

( )

( )

2 0 2

( )2

1 ( ), , ;

2 00

0

LiEt iEtIFW IFW

Et U t e eE

x

x xx x Σp

2 0 2 ( )

( )

00

02 1, , .

( )2

( )

iEt iEtIFW RIFW

Et U t e e

E

x x xxΣp

x

18

In the presence of boson fields ( )B x interacting with fermion fields

1 2( ), ( ),x x the isotopic Foldy-Wouthuysen transformation and the form of

the Hamiltonians of Eqs. (15), (16), (19) in the IFW representation can be

obtained as a series in powers of the coupling constant.

1,2 1,2;IFWIFWФ U Ф

1 2 3 01 ... ( ) ;IFW IFWU U (24)

;IFWIFW

Ф U Ф

1 2 3 01 ... ( ) .IFW IFWU U (25)

19

By applying the same transformation (24) to each of Eqs. (15), (16), we

obtain the following equations and Hamiltonian densities in the isotopic

Foldy-Wouthuysen representation:

0 3 1 2 3 1... ( ) ;I FW IFWp E K K K (26)

ℋ †1 3 1 2 3 1( ) ... ( ) ;I

IFW IFW IFWE K K K (27)

0 2 3 1 2 3 2( ) ... ( ) ;IFW IFWp E K K K (28)

ℋ †2 3 1 2 3 2( ) ... ( ) .II

IFW IFW IFWE K K K (29)

Similarly, by applying transformation (25) to Eq. (19), we obtain

0 16 16 3 1 2 3 ... ;IFW IFWp E E K K K (30)

ℋ †16 16 3 1 2 3( ) ... ( ) .III

IFW IFW IFWE E K K K (31)

Sixteen-dimensional operators nK in (30), (31) constitute generalized

expressions nK with the following replacements:

1 16 16 1 3 16 16 3 16 16 1

1, ,

2IE E q q E q

(32)

20

Expressions for the operators C , N , C , N forming the basis for writing

the interaction Hamiltonian in the FW representation, are written for the

isotopic Foldy-Wouthuysen representation as follows:

† 0 00 0

even

IFW IFWC U q B U R qB LqB L R R q Lq L R αB αB

† 0 00 0

odd

IFW IFWN U q B U R LqB qB L R R Lq q L R αB αB (33)

33 1

3

1;

2

ER L m

E E

αp

αp

Expressions for the sixteen-dimensional operators , ,C N in (30), (31)

are derived from expressions (33) with replacement (32).

Thus, the analysis shows that the equations of fermion fields (26), (28), (30)

and their Hamiltonians in the isotopic Foldy-Wouthuysen representation

for the case of the considered interaction are written in the chirally

symmetric form irrespective of whether fermions possess mass or are

massless.

21

2.3 Chiral invariance in the Foldy-Wouthuysen representation2.3 Chiral invariance in the Foldy-Wouthuysen representation

In the isotopic Foldy-Wouthuysen representation Hamiltonians (27),

(29), (31) and Eqs. (26), (28), (30) are invariant with respect to the chiral

transformation

5

1,2 1,2i

IFW IFWe . (34)

Accordingly, because of the unitary property of the IFW

transformation, equivalent Dirac equations (15), (16), (19) with the

interaction ( )q B x are invariant with respect to the transformations

5( )1,2 1,2

DiФ e Ф (35)

5 † 5 .IFW IFWDU U (36)

22

For the free case, expression (36) equals

†5 5 5 3 3 10 0 .

FW FWD

mU U

E E

αp (37)

In equality (37), the operator 1

2 2 2E m p is non-local with an

infinite number of differentiation operators , 1,2,3ii

p i ix

.

In case of interaction, operator (36) is a complex expression

depending on the boson fields .

As far as the author knows, invariance of the Dirac equation with

respect to transformation (35) has not been considered in literature before.

23

2.4 Physical content of chirally symmetric equations and 2.4 Physical content of chirally symmetric equations and their Hamiltonians in the isotopic Foldy-Wouthuysen their Hamiltonians in the isotopic Foldy-Wouthuysen

representationrepresentation

Now, let us consider physical content of Hamiltonians and Eqs. (26) –

(31) by means of symbolic patterns in Figs 1, 2, 3.

Let us first consider Eq. (30) and the Hamiltonian ℋ IIIIFW (31). The

corresponding physical diagram is shown symbolically in Fig. 1. Fig. 1

0E 0E

3

1

2T

3

1

2T

1

2 1

, 0,

,

0

R

IFW iEtIFW

IFW L

A

tt e

t A

x

xx

x x

1 1

2

0

,,

0,

IFW LiEtIFW

IFW

R

t At e

t

A

x xx

x

x

IIIIFW

ℋ

24

The left half-plane in Fig. 1 represents the states of basis functions

(23) with the isotopic spin 13 2T and positive energy 0E ; the right half-

plane shows the states with 13 2T and negative energy 0E . The

Hamiltonian contains the states of left and right fermions and left and right

antifermions; particles and antiparticles interact with each other in a real

(solid arrow in Fig 1) and virtual (dashed arrow in Fig. 1) manner. The

physical diagram in Fig. 1 corresponds to the world immediately around

us.

Let us proceed to considering the Hamiltonians ℋ IIFW (27) and ℋ II

IFW (29)

with respective Eqs. (26) and (28). The symbolically representation is

shown in Fig. 2.3.

25

2

0, iEt

IFW

R

t eA

x

x

IIFW

0E

3

1

2T

1

0, iEt

IFW

L

t eA

x

x

0E

3

1

2T

1

2 ,0

iEt LIFW

At e

xx

0E

3

1

2T

2

0, iEt

IFW

R

t eA

x

x

IIIFW

Fig 2.

Fig 3.

3

1

2T

0E

ℋ

ℋ

26

It follows from Fig.2, 3 that Hamiltonians ℋ IIFW , ℋ II

IFW provide

existence either right fermions and left antifermions, or left fermions and

right antifermions. In both cases there is not an interaction of real particles

and antiparticles.

Hamiltonian densities (27), (29), (31) and equations of fermion fields (26),

(28), (30) make it possible to formulate Feynman rules for calculating

specific physical processes of the quantum theory of interacting fields

using the methods of the perturbation theory .

27

3. Continuity equation. Conservation of vector and axial 3. Continuity equation. Conservation of vector and axial current of massive fermions in the isotopic Foldy-current of massive fermions in the isotopic Foldy-

Wouthuysen representationWouthuysen representationThe continuity equation for free motion can be derived using Eqs. (26),

(28), (30) and Hermitian-conjugated equations.

The analysis below will be provided for Eqs. (26), (28) with eight-

component spinors 1,2 IFW . The analysis for Eq. (30) with sixteen-

component spinors IFW leads to the same results and conclusions.

1,2

3 1,2

†

†1,2

1,2 3( )

IFW

IFW

IFW

IFW

i Et

i Et

(38)

† † †

1,2 1,2 1,2 3 1,2 1,2 3 1,2IFW IFW IFW IFW IFW IFWi E E div

t

j (39)

† †1,2 3 1 1,2 3 1,2

† †21,2 3 1,2 1,2 3 1,23

† †2 2 21,2 3 1,2 1,2 3 1,25

1

21

81

16

i i iIFWIFW IFW IFW

i iIFW IFW IFW IFW

i iIFW IFW IFW IFW

j p pm

p pm

p pm

p

p p p

(40)

28

Similarly to (39), (40), one can derive equations of axial current

conservation with the matrix 5 between the eight-component spinors

†

1,2 IFWФ , 1,2 IFW

Ф .

† † †5 5 51,2 1,2 1,2 3 1,2 1,2 3 1,2 AIFW IFW IFW IFW IFW IFW

Ф Ф i Ф E Ф E Ф Ф divt

j (41)

The axial current is preserved in the isotopic Foldy-Wouthuysen

representation irrespective of whether fermions possess mass or not.

Recall that in the Dirac representation, the axial current is preserved only

for massless fermions.

5 † 5 5 † 52 ; ( ) ( ) ( ).j im j x x x (42)

29

4. Possibility of spontaneous breaking of parity in the 4. Possibility of spontaneous breaking of parity in the isotopic Foldy-Wouthuysen representationisotopic Foldy-Wouthuysen representation

In the previous section, equations (26), (28), (30) and their Hamiltonians

were shown to be invariant with respect to chiral symmetry

transformations (34).

Vacuum or the ground state of free Hamiltonian (31) contains a Dirac sea

of negative energy states of left and right fermions.

Vacuum of Hamiltonian (27) is a Dirac “sea” of left fermions with 0E , and

that of Hamiltonian (29) is a Dirac “sea” of right fermions with 0E .

Since Eqs. (26), (28), (30) were obtained by unitary IFW transformations

from equivalent Eqs. (15), (16), (19), it is clear that there exists degenerate

vacuum and conditions for spontaneous breaking of P - symmetry.

30

5. The Standard model in the isotopic Foldy-Wouthuysen 5. The Standard model in the isotopic Foldy-Wouthuysen representationrepresentation

Based upon the formalism of the transition to the isotopic Foldy-

Wouthuysen representation, we originally derive a Hamiltonian written in

terms of the left and right components of fermions of the first generation

ev ,e,u,d from Lagrangian of the Standard model:

e

† † † †i i i i i i i i i iL R R LL R L R

i v ,e,u ,d

0 0W W Z Z EM EML L L R L R

m m

g W J W J Z J Z J eA J eA J ,

αp αpH (43)

where

e

† †W v e u dL LL LL

1J ;

2

e

† †W e v d u LL L LL

1J ;

2

31

e e

† † 2Z v v e w e LL LL L

w

† 2u w u LL

† 2d w d LL

1 1J sin

cos 21 2

sin2 31 1

sin ;2 3

(44)

† 2 † 2Z e w e u w uR RR R R

w

† 2d w d RR

1 2J sin sin

cos 31sin ;

3

† † †EM e e u u d dL L LL L L L

† † †EM e e u u d dR R RR R R R

2 1J 1 ;

3 32 1

J 1 .3 3

Then, for each fermion field, we introduce eight-component spinors

i Ri e1

i L

, i v ,e,u,d

(45)

and a special isotopic space with matrices 1 2 3, , affecting only the four

upper and four lower components of the spinors i 1 (45).

32

The introduced isotopic space is in no way connected with the weak-

isospin space in the Standard Model and, in particular, in the GWS model.

Now, Hamiltonian (43) can be written as

e

e e

e e

i 1 i i i i i1 11 1i v ,e ,u ,d

3 2 0i 1iw w i i 11

w

3 3 3 3v 1v w 1ew e u 1uw 1dw d1 111

3 3 3 3e 1ew 1v w v d 1dw 1uw u 11 11

m e Q A

gT sin Q Z

cosg

2T W 2T 2T W 2T2g

2T W 2T 2T W 2T .2

αpH

(46)

In expression (46) 3

3 iR1iw 3

iL

T 0T

0 T

is an eight-component weak-isospin

matrix with 3iRT 0 for right particles and

e

3 3 3 3v eL uL dL

1 1 1 1T ,T ; T ; T

2 2 2 2 for

left particles in accordance with the GWS model. The quantity iQ , which

equals ev e u d

2 1Q 0, Q 1; Q ; Q

3 3 is the same for right and left

particles.

33

The following equations for fermion fields i 1 are deduced from

Hamiltonian (46):

e e e e e

3 0 3 30 v 1 v 1v w v 1v w 1ew e 11 1

w

g gp m T Z 2T W 2T

cos 2

αp (47)

e e

3 2 00 e 1 e 1ew w e1 1

w

3 31ew 1v w v 1

gp m e A T sin Z

cosg2T W 2T

2

αp (48)

3 2 00 u 1 u 1uw w u1 1

w

3 31uw 1dw d 1

2 g 2p m e A T sin Z

3 cos 3g2T W 2T

2

αp (49)

3 2 00 d 1 d 1dw w d1 1

w

3 31dw 1uw u 1

1 g 1p m e A T sin Z

3 cos 3g2T W 2T

2

αp (50)

In each of Eqs. (47) – (50), the only summand that mixes up right and left

field components is the fermion mass term 1 im .

34

In each Eqs. (47) – (50), the interaction between the left fermion field and

W bosons results in the presence of another left fermion field from the

corresponding SU 2 doublet.

For right fermion fields, there is no interaction with W bosons in the

GWS model.

The fermion fields

i Li 1 i e2 1

i R

, i v ,e,u,d

(51)

are also solutions to Eqs. (47) – (50) with a modified weak-isospin matrix

33 3 iL2iw 1 1iw 1 3

iR

T 0T T .

0 T

(52)

35

If we introduce 16-component fermion fields

i R

i i1 Li e

i i2 L

i R

, i v ,e,u,d

(53)

with the replacement of

I16 16 1

I16 16 1

1e e E ,

21

g g E ,2

(54)

it will be possible to write equations similar to Eqs. (47) – (50) for the

fields i .

The equations for i i i1 2, , are equivalent to each other.

However, in the IFW representation, these equations represent different

physical patterns.

36

Let us perform the isotopic Foldy-Wouthuysen transformation for the

case of the fermion fields i 1 governed by Eqs. (47) – (50).

i i i1 IFW 1IFWU .

(55)

Each of Eqs. (47) – (50) is then multiplied on the left by the transformation

matrix

i 1i 2i 0iIFW IFWU 1 ... U . (56)

The matrix IFWU is unitary, and expansion in (56) is performed in powers

of corresponding coupling constants.

The IFW transformation gives the following equations:

or0 i 3 i i1IFW 1IFW

j e1IFW

p E K K ...K K ... ; i, j v ,e u,d; i j.

1i 2i' '1i j 2i j

(57)

37

Eq. (57) contain equations for the lepton and quark SU 2 left doublets

and right singlets, respectively.

In Eq. (57), in the expressions for K ,K ...1i 2i describing the electromagnetic

interaction and weak interactions with exchange of 0Z bosons, the q B

interaction should be replaced as follows:

3 2 0i 1iw w i

w

gq B eQ A T sin Q Z .

cos

(58)

In Eq. (57), the summands K ,K ,... ' '1i j 2i j are responsible for the weak

interaction involving charged W bosons. Let us consider the structure

of expressions for K 'ni j by the example of the transformation of Eq. (47).

38

After the IFW transformation, the last summand in Eq. (47) takes the

following form:

e e

e e e e e

e e e e e

3 3v 1v w 1ew e e 1IFWIFWIFW

3 31v 2v 0e 1v w 1ew 0e 1v 2v e 1IFWIFW IFW

3 3 30v 1v w 1ew 0e 1v 0v 1v w 1eIFWIFW IFW

gU 2T W 2T U

2g

1 ... U 2T W 2T U 1 ...2

g gU 2T W 2T U U 2T W 2T

2 2

e e e e

3w 0e IFW

3 30v 1v w 1ew 0e 1e e v e 2v e e1IFW 1IFWIFWIFW

U

gU 2T W 2T U ... K K ... .

2

' '1

(59)

In expression (59), the operators e1v and 1e are defined within the

formalism with replacement (58).

In Eq. (57), the summands with 3 i niE ,K , by definition, are even with

respect to the mixing-up of upper (right) and lower (left) components

i 1IFW . The summands with

env eK ' are formally odd, but they actually can

be treated as even because only left fermions participate in the

interaction with charged bosons W, which is reflected in the definition of

31ewT (see (46)).

39

For the fermion fields i 2 IFW , one can write equations similar to (57) with

replacement of 3 31iw 2iwT T (see (52)). Finally, one can derive Foldy-

Wouthuysen equations in the IFW representation for Dirac fields i (53)

with replacement (54) in expressions (58), (59).

Let us write these equations together with Eq. (57) and their

Hamiltonian densities.

0 i 3 i i1IFW 1IFW

j 1IFW

p E K K ...K K ... ;

1i 2i' '1i j 2i j

(60)

'IFW i 3 i i1IFW 1IFW

i

i j1IFW 1IFWi , j

E K K ...

K K ... ;

1i 2i

' '1i j 2i j

H (61)

0 i 3 i i2 IFW 2IFW

j 2IFW

p E K K ...K K ... ;

1i 2i' '1i j 2i j

(62)

40

''IFW i 3 i i2 IFW 2IFW

i

i j2 IFW 2IFWi , j

E K K ...

K K ... ;

1i 2i

' '1i j 2i j

H (63)

0 i 3 i iIFW IFW

j IFW

P E K K ...

K K ... ;

1i 2i' '1i j 2i j

(64)

'''IFW i 3 i iIFW IFW

i

i jIFW IFWi , j

E K K ...

K K ... .

1i 2i

' '1i j 2i j

H (65)

In expressions (60) – (65), ei, j v ,e u,d;i j and ; in expressions (60), (61) 3

3 iR1iw 3

iL

T 0T

0 T

; in expressions (62), (63)

33 iL2iw 3

iR

T 0T

0 T

; in expressions (64),

(65), we use the matrix 31iwT for the eight-component fields i 1IFW

and the

matrix 32iwT for the fields i 2 IFW

.

To ensure complete compliance with the GWS model, one should exclude the states with the right neutrino from (64) - (65). At this stage, this can be

done easily by taking ev R

0 in the basis functions

e e ev v v1IFW 2IFW IFW

, , .

41

1. Equations and Hamiltonians (22) - (27) are SU 2 -invariant, whether

fermions have mass or not.

2. The SU 2 -invariant formulation of the GWS model in the isotopic

Foldy-Wouthuysen representation exists for both massless and massive

fermions. This conclusion is not changed neither by accounting for two

other generations of particles in the Standard Model v , ,c,s and

v , ,t ,b in the IFW representation, nor by accounting for SU 3 -

symmetric quantum chromodynamics.

3.Thus, we demonstrated that it is not necessary for ensuring of SU 2 -

invariance of theory to introduce of Yukawa interaction between Higgs

bosons and fermions in the isotopic Foldy-Wouthuysen representation.

Thank you for attention

42