INTERNAL SYMMETRY OF RELATIVISTIC WAVE EQUATIONS

V. A. Pletyukhov UDC 539.12:530.145

The properties are considered of the internal symmetry of first-order relativ- istic wave equations on the basis of the Gel'fand-Yaglom approach. A suffi- cient condition is established for the existence of such a symmetry for free fields.

The possibility of describing additional (besides spin) internal degrees of freedom of particles by means of the first-order RWE

(7~0.+• ~ = 0 (1 )

(~ i s a wave f u n c t i o n , yU i s a q u a d r u p l e o f s q u a r e m a t r i c e s , ~ i s a r e a l c o n s t a n t ) i s t i g h t l y connected with the properties of the internal symmetry of these equations, by which is under- stood the existence of a collection of operators {Q}, not reducing to scale-phase transfor- mations and satisfying the relations

[Q, 7.]-----0. (2 )

In [!] was formulated a sufficient condition, from which it follows that the commutation re- lations (2) are satisfied for all Dirac-like RWE, with the exception of the Dirac equation. In this paper a theorem is proved which allows one to establish a wider class of equations possessing internal symmetry, and some corollaries flowing from it are also considered.

We will use the Gel'fand-Yaglom formalism [2], in the framework of which the matrix y~, playing the basic role for Eq. (i), has the structure

~C ~ (3) l

where C s is the spin block corresponding to the spin g; I2s is the identity matrix of di- mension 2s + i. The possible values of the (rest) mass of a particle in a given spin state are determined according to the formula m i = </iXi[, where h i are the nonzero eigenvalues (roots) of the block C s

THEOREM. For the existence of an internal symmetry for a first-order RWE of the form (i), it is sufficient that at least one spin block of the matrix 74 (3) have multiple (re- peating) nonzero roots.

Let us take the case when one of the roots I (~ # 0) of the characteristic polynomial of some spin block (let us denote it C s, s is fixed) has a multiplicity equal to two, and all the remaining nonzero roots of the spin blocks of the matrix y~ are different from I

A and single. Let us introduce into consideration the operators of the square of the spin S 2

A and its projections S n mutually conunuting and with Y4. Let us choose a basis in which these operators have a diagonal form, and the spin blocks under the preservation of the structure (3) of the matrix y~ are reduced to normal Jordan form. As was shown in [3], the nonzero roots in normal Jordan form of the matrix y~ correspond to the scalar cells. In this manner, in the considered basis the components having a physical sense of the wave function ~ are the set of "pure" states {~_+m,s in the rest system of the particle (the index k charac- terizes the projection of the spin of the particle, the signs + and - pertain to the posi- tive-frequency and negative-frequency regimes). The presence of a double root % for the block C s denotes that inherent to the particle are two sets of states ~• and ~•

A. S. Pushkin Brest Pedagogical Institute. Zavedenii, Fizika, No. 7, pp. 16-18, July, 1991. 1990.

Translated from Izvestiya Vysshikh Uchebnykh Original article submitted December I0,

0038-5697/91/3407-0575512.50 �9 1992 Plenum Publishing Corporation 575

not differing in mass, absolute magnitude and projection of spin, that is one can write

/ '~• s. ~h

=" I o, '.>, .,, , . ) . \ }

Now let us consider the transformations Q in the space of the wave function ~ (4), which, not affecting the space-time coordinates, have in the basis used the structure

(4)

Q=(q~/2t2~.+t) ) ]'

where q is an arbitrary unitary matrix of dimension 2 x 2. mations commute with the matrix u

(5)

It is clear that these transfer-

[Q. ~ d = o (6)

and intermix the states ~• and ~'+~ s k" Let us perform a pure Lorentz transformation implementing the transition from the res~ system to an arbitrary reference system. The transformation T induced by it in the space of the representation of the wave function can be written in the form T--~-ex!)(kfjJi4). where Jj, (j = i, 2, 3) are infinitesimal operators.

Since the separation of the states according to mass have a relativistically invariant char- acter, Lorentz transformations do not intermix states with different masses. This signifies that the matrix TQT -z preserves the structure (5), that is

TQT-'~{Q} (7)

[by {Q} is understood the set of all operators of the form (5)]. Let us rewrite condition (7) for infinitely small Lorentz transformations: (|-/i~]f,)Q(l--h{Jj~) ~{Q}. We hence obtain

Q+iq'j[Jj~, Q] ~ {Q}. (8)

Since any transformation from the set {Q} commutes by definition with the matrix 72, then in accordance with (8) the operators [Jj,, Q] also commute with 74

i;~, [4.,, Q]]=0 It is known (see, for example, [2]) that the matrices ~j of Eq. bythe relations

(9) (1) are connected with ~

?~=144, ~'d. (10) With the help of (10) the commutant [Q, ~j] reduces uncomplicatedly to the form [Q, ~j] = [44, [ d j , , Q]], whence i t fol lows due to (9) t ha t

[Q, ~. j] =0, (11)

Uniting conditions (6) and (Ii), we obtain that the set of transformations {Q} form a group of internal symmetry of the RWE considered.

The generalization of the proof presented to the cases when the multiplicity of the root ~ of the block C s is greater than two or when the nonzero multiple roots, including those coincident with X, occur for several spin blocks, bears a clear character, while in the last case the internal symmetry transformations can intermix states with different Val- ues of spin.

The theorem is proved.

COROLLARY1. From the proof of the theorem follows the existence of two types of in- ternal symmetry for an RWE of the form (i). Actually, the relations (~) automatically hold if

[4,, Q]=o. (12)

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However, in principle the satisfiability of (9) is assumed when

[I~4, Q] ~o . (13) Case (i2) signifies that the internal symmetry transformations commute with the Lorentz transformations. This is the "usual" symmetry which is characteristic for equations decom- posing in the relativistically covariant sense, for example, for the SU(n)-invariant Dirac theory. But case (13) corresponds to the internal symmetry transformations not commuting with the Lorentz transformations. A typical example is the "dyal" symmetry of the Dirac- Kahler equation [i, 4, 5] (a vector field of general type according to the terminology of [4]).

COROLLARY 2. For the class of equations considered in the theorem the transformations A A

(5) commute not only with the matrices T~, but also with the spin operators S 2, S n. One of the generators of these transformations can be included along with the 4-impulse and the spin variables in the complete set of commuting variables and associated with an additional internal degree of freedom of the particle. Generalizing the given result, one can make the deduction: the internal symmetry of an RWE of form (I) caused by the multiplicity of the nonzero roots of the matrix y~ "inside" the spin blocks C s corresponds to the additional (in the sense indicated above) degrees of freedom. Along with this it is necessary to em- phasize that the theorem does not exclude the possibility of the existence of an internal symmetry for equations not embraced by its formulation. As an example one can present a P-noninvariant Dirac-like RWE constructed on the basis of the set of representations of the

Lorentz group [(0, 0)~(~, ~) ~(0, |)] [6]. The spin blocks C O , C i of the matrix proper T~

of this equation have identical eigenvalues • with unit multiplicity in each block. The SU(2)-sy~netry peculiar to it is not connected with an additional degree of freedom, since

A the transformations of this syrametry do not commute with the operator S 2.

LITERATURE CITED

i. V. I. Strazhev and V. A. Pletyukhov, Acta Phys. Pol., BI___22, 651-664 (1981). 2. I.M. Gel'fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of the Rotation Group

and the Lorentz Group [in Russian], Fizmatgiz, Moscow (1958). 3. F. I. Fedorov, Dokl. Akad. Nauk SSSR, 79, 787-790 (1951). 4. V. I. Strazhev, Acta Phys. Pol., B_~9, 449-458 (1978). 5. S. I. Kruglov and V. I. Strazhev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 77-81 (1978). 6. S. I. Kruglov, V. I. Strazhev, E. A. Tolkachev, and P. L. Shkol'nikov, Preprint No. 212,

B I. Stepanov Physics Institute of the Academy of Sciences of the Belorussian SSR, Minsk (1980).

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