ñòð. 61 |

H > H? = V1 HV1?1 ,

? > ? = V1 ?, (24)

where

1 ?a pa 1 ?a pa

V1?1 = 1 +

V1 = 1 ? (25)

?, ?.

2 E2 2 E2

The Hamiltonian H? is Hermitian in respect of the following scalar product:

d3 x ?† (t, x)(V1?1 )† V1?1 ?2 (t, x). (26)

(?1 , ?2 ) = 1

To draw the correct conclusion about the C, P , T propertios equation (22) it is

necessary to write the algebra (5) in the ?-representation. We shall not do this here.

We shall remark only that due to the invariance of equation (1) under P (1) , T (2) , C

transformations equation (22) is invariant with respect to the transformations

P? = V1 P (1) V1?1 , C? = V1 CV1?1 , T? = V1 T (2) V1?1 .

(1) (2)

(27)

One can show in an analogous manner that the two-component equation of the

type [5]

??(t, x)

i = (?a pa + B)?(t, x),

?t (28)

B?a pa = ??a pa B, B 2 = 0,

can be obtained from the Weyl equation with the help of the operator

1 ? a pa 1 ? a pa

V2?1 = 1 +

V2 = 1 ? (29)

B, B.

2 E2 2 E2

1. Fushchych W.I., Nucl. Phys. B, 1970, 21, 321; Theor. Math. Phys., 1971, 9, 91 (in Russian).

2. Fushchych W.I., Grishchenko A.L., Lett. Nuovo Cimento, 1970, 4, 927.

3. Simon M.T., Lett. Nuovo Cimento, 1971, 2, 616.

4. Santhanam T.S., Tekumalla A.R., Lett. Nuovo Cimento, 1972, 3, 190.

5. Tekumalla A.R., Santhanam T.S., Lett. Nuovo Cimenro, 1973, 6, 99.

6. Seetharaman T.S., Simon M.T., Mathews P.M., Nuovo Cimetro A, 1972, 12, 788.

7. Fushchych W.I., Grishchenko A.L., Nikitin A.G., Theor. Math. Phys., 1971, 8, 192 (in Russian).

8. Fushchych W.I., Lett. Nuovo Cimento, 1973, 6, 133.

W.I. Fushchych, Scientific Works 2000, Vol. 1, 274–277.

On the additional invariance of the Dirac

and Maxwell equations

W.I. FUSHCHYCH

In this note we show that there exists a new set of operators {Q} (this set is different

from the operators which satisfy the Lie algebra of the Poincare group P1,3 ) with

respect to which the Dirac and Maxwell equations are invariant. We shall give the

detailed proof of our assertions only for the Dirac equation, since for the Maxwell

equations all the assertions are proved analogously.

The Dirac equations [1]

??(t, x)

= H?(t, x), H = ?0 ?a pa + ?0 ?4 m (1)

i

?t

is invariant with respect to such a set of operators {Q} which obey the condition

?

? H, Q ?(t, x) = 0, ? Q ? {Q}. (2)

i

?t

It is well known that there are two sets of operators which satisfy the condition (2).

The first set has the form [2]

?

? (1)

?P0 = p0 = i ? , ?

Pa = pa = ?i

(1)

, a = 1, 2, 3,

?t ?xa

{Q1 } = (3)

? (1)

?J = x p ? x p + S , µ, ? = 0, 1, 2, 3,

µ? ?µ µ?

µ?

where

i

(?µ ?? ? ?? ?µ ), [xµ , p? ] = ?igµ? .

Sµ? =

4

The second set has the form [3]

? (2)

?P0 = H = ?0 ?a pa + ?0 ?4 m, (2)

P a = pa ,

?

?

? (2)

{Q2 } = Jab ? Jab = xa pb ? xb pa + Sab , a, b = 1, 2, 3, (4)

?

? (2)

?

?J = x0 pa ? 1 (xa H + Hxa ),

0a

2

We shall prove the followimg assertion.

Theorem 1. The eq. (1) is invariant with respect to such two sets of operators

? (3) (3) (2)

?P0 = p0 , Jab ? Jab ? Jab ,

(3)

? P a = pa ,

{Q3 } = (5)

?J (3) = x0 pa ? xa p0 ? i 1 ? v0 H ? Hp

? ?

v a ? 0 v a p0 ;

? 0a

H2 H2 H2 H2

2

Lettere al Nuovo Cimento, 1974, 11, ¹ 10, P. 508–512.

On the additional invariance of the Dirac and Maxwell equations 275

? (4) (4)

?P0 = H, Jab = Jab = xa pb ? xb pa + Sab ,

(4)

P a = pa ,

{Q4 } = (6)

?J (4) = x p ? 1 (x H + Hx ),

0a a a

0a

2

where

?0 H ? Hp

i ?

v a ? 0v a

1? v (7)

xa = xa + .

H2 H2 H2 H2

2

Proof. It may be shown by an immediate verification that the invariant condition

(2) is satisfied for the operators (5) and (6). However, a more easy and elegant way

is the following. Let us perform a unitary transformation [1] over eq. (1) and the

operators (5) and (6)

?0 H

1

U=v 1+ v (8)

.

H2

2

Under the transformation eq. (1) and the operators (5), (6) will have the form

??(t, x)

= Hc ?(t, x), Hc = ?0 E, ? = U ?, E = p2 + m2 , (9)

i

?t

?

?P (3) = U P (3) U ?1 = p0 , Pa = U Pa U ?1 = pa ,

(3) (3)

0 0

{Q3 } = (10)

?J (3) = U J (3) U ?1 = J , (3)

J0a = x0 pa ? xa p0 ,

ab

ab ab

? (4)

?P0 = U HU ?1 = Hc = ?0 E, (4)

P a = pa ,

{Q4 } = (11)

?0

?J (4) = U J (4) U ?1 = Jab , (4)

= x0 pa ? (xa E + Exa ).

J0a

ab ab

2

Now it may be readily verified that the invariant condition (2) in the new represen-

tation

?

? Hc , Q ?(t, x) = 0 (12)

i

?t

is satisfied if the operators {Q} have the form (10) and (11). This proves the theorem.

Remark 1. The operators (10), (11) (this means that also the operating (5), (6))

satisfy the relations

(j) (j)

Pµ , J?? = i gµ? P? ? gµ? P? ,

(j) (j) (j) (j)

(13)

P µ , P? = 0, j = 3, 4.

(j) (j) (j) (j) (j)

[Jab , Jcd ]? = i gcd Jbc ? gac Jbd + gbc Jad ? gbd Jac ,

(j)

(14)

(j) (j) (j)

= ?i ? Sab ,

J0a , J0b Jab a, b, c, d = 1, 2, 3; j = 3, 4.

?

From (14) it follows that if the matrices Sab are added to the operators (10), (11),

(j) (j)

then the set of operators Pµ , Sµ? , Sab form the Lie algebra.

276 W.I. Fushchych

Remark 2. From the above considerations it follows that the wave function ? in

passing from one inertial frame of reference to another which is moving with velocity

?V may be transformed by four nonequivalent ways

(j)

?(j) (t, x) = exp iJ0c ?c ?(j) (t, x), j = 1, 2, 3, 4,

(15)

U J0c U ?1 ,

(j) (j)

tgh ? = |V |.

J0c =

It is to be emphasized that by the transformation (15) the time does not change if

(j)

J0c ? {Q2 } or {Q4 }:

(4) (4) (2) (2)

x0 = exp iJ0c ?c x0 exp ?iJ0b ?b = exp iJ0c ?c x0 expn ?iJ0b ?b = x0 ,

(4) (4)

xa = exp iJ0c ?c xa exp ?iJ0b ?b .

Such transformations xa , are not equivalent to the conventional Lorentz transforma-

(j)

tions. If in these formulae J0a ? {Q3 }, the xa and x0 transform in the conventional

Lorentz way. We thus find that, if the energy of a free particle is defined as usually

E = p2 + m2 , then this does not mean in general that the theory must be invariant

with respect to the Lorentz transformations.

Theorem 2. The Hamiltonian H in eq. (1) commutes with the operators

i

(?a ?b ? ?b ?a ),

Sab = a, b = 1, 2, 3,

4

(16)

i

= (?4 ?a ? ?a ?4 ),

S4a

4

where

?0 H (?a ?c ? ?c ?a )pc + 2?a ?4 m

1

1? v v

?a = ?a + ,

H2 H2

2

?b pb + ?4 m ?4 ?c pc

v v

?4 = ?4 + 1 ? .

H2 H2

Proof. If we perform the transformation (8) over the operators (16), we obtain

i

Skl = U Skl U ?1 = Skl = (?k ?l ? ?l ?k ), (17)

k, l = 1, 2, 3, 4.

4

From (17) it follows [Hc , Skl ] = 0 and

[Skl , Snr ]? = i(gkr Sln ? gkn Slr + gln Skr ? glr Skn ), k, l, n, r = 1, 2, 3, 4. (18)

The analogous theorem is valid for any arbitrary relativistic equation in the cano-

nical form describing free particle motion with spin s [1].

Remark 3. The operators (16) serve as an example of the nonlocal generators (in

configuration space) which satisfy the Lie algebra of the group O4 . Previously it was

known that the Hamiltonian had only the group O3 symmetry since the spin of a

particle was the integral of the motion.

On the additional invariance of the Dirac and Maxwell equations 277

Following Good [4, 5], the Maxwell equations may be written in the Hamiltonian

form

??(t, x)

= H1 ?(t, x), H1 = Bp,

ñòð. 61 |