Commutation relations and relativistic wave equations

Nuclear

Physics, North-Holland Publishing Co,, Amsterdam, 1 (1956)

C O M M U T A T I O N R E L A T I O N S AND R E L A T I V I S T I C WAVE E Q U A T I O N S H. U M E Z A W A

Faculty of Sciences, Dept. of Physics, Tokyo University and A. V I S C O N T I t C E R N , Theoretical Study Division, Copenhagen R e c e i v e d 27 D e c e m b e r 1955 A b s t r a c t : C o v a r i a n t c o m m u t a t i o n r e l a t i o n s for particles of a n y s p i n are g i v e n in t h e cases w h e r e t h e particles h a v e o n l y one m a s s s t a t e or h a v e a m a s s s p e c t r u m .

1. General Theory In a recent paper 1), S. N. Gupta studied the quantization of particles of spin 3/2, b y starting from equations whose form is different from the Rarita-Schwinger ~) equations. We wish to give a simple general formulation which leads to Gupta's equations in the special case of spin 3/2. We assume here that the equation for the free field Q(x) is where Q (z) is a one column matrix of N elements Qa (x) : (A = 1 . . . N); 0, (/,=1,2,3,4) and /~ are N×N matrices. In order to determine the commutation relations for the field Q(z), we are going to solve the following problem: "How do the commutation relations change, when one changes the field variables?" The equation (1) m a y be derived from the following action:

~t=~O*(x)~laa,Aa,~(O)Q~(x)dx~yQ*(x)~TA(~O)Q(~)dz,

(2)

where the second expression for ~ has been obtained by integration b y parts and 0 acts on the left. The matrix ~7 is a non singular matrix which has been chosen such that the equations deduced t N o w a t I n s t i t u t H. Poincar6, Paris. 348

COMMUTATION RELATIONS AND RELATIVISTIC WAVE EQUATIONS

S49

from (2) by independent variations of Q and Q* are compatible. Therefore we have ~A(O) = (~IA(--O)) t, (3) where the symbol t means the hermitian conjugate. Let us change the wave functions Q(x) as follows:

{L (=)=~A,(0)q~,(=);

(4)

~ ( a ) is a matricial differential operator. Equation (1) then takes the following form:

-- [q*. (=)~:..,. (- O)n,,,AAjB (O)~B~, (a)q~, (=)d=.

(5)

The bracket in the first line of eq. (5) means that N * (a) acts only on q*(x), and the second expression for ~ has been deduced by integration by parts. Introducing the hermitian conjugate of the matrix ~ , the formula (5) takes the form

~ = f q * (x)~t ( - a)~/A (0)~ (0)q (z)dx

(6)

and it follows that q(x) satisfies S t (-- a)~A (0)2 (O)q (x) = 0. (7) We now suppose that the commutation (or anticommutation) relations for the field q(x) are known:

c,,~(=-=')= [q,, (=), q*(=')]~.

(s)

and we try to deduce the commutation relations for the field Q (x):

c.~(=-~') = [Q. (=),~2,,(~')],

(9)

where

~, (=) = g*, (=)~,.,. Using (4) we obtain the following formula:

c.~(=-=')=~..,(a)~*,~,.(a')c..~..(=-=')~.~,

(1o)

where O' acts on x'. Equation (10) can also be written in matrix form

c(=)--~(a)c(=)~*(-a)n.

(11)

Thus, the question which was asked before m a y be considered as answered, provided that the fields Q(x) and q(x) are compatible ? W e c a n define a differential o p e r a t o r F(5) b y its F o u r i e r t r a n s f o r m w i t h /(~)

an

/ ~ ( 5 ) / ( m ) = .rexp (ik#mlj) a r b i t r a r y derivable f u n c t i o n of

F(ikj,)t(ik#)dk, m.

350

H. UMEZAWA AND A. VISCONTI

(cf. §3 for supplementary conditions introduced b y the consideration of the spin of these two fields). These formulae can be simplified to a great extent if we take a field q(z) such that equation (7) reduces to the equation

(-a},TA where [(a) is a scalar differential operator ([~_~2 for a one masslevel field, I-1~.1([]--~) for a multimass field), and that c(x) is a diagonal matrix. We m a y then look for a differential operator D(a) s u c h that

A (a)D(a)=/(O)I.

(12)

It

can easily be seen b y using the scalar character of the right hand side that

hCO)=~(0)~*(--0)~?. Therefore, if one goes back to the expression i l l ) for

C(x)-:O(O)c(z).

(13)

C(x), one has (14)

Let us now show that the matrix/~ in (1) is a non singular matrix, if we assume the existence of the differential operator D (a). Indeed, we m a y write down an expression for this operator as follows:

oca)----=+~oa , + % o, ao, a , , + . . . . (15) where the ~, ~°i ~ ~,1°, are N x N matrices and the suffices a~ run from 1 to 4. We introduce this expression into equation (12) and identify terms of the same derivative order on both sides. If we suppose that the expression for ](a) contains a term M independent of a, we finally obtain ~ot=MI. (16) The matrix ~ is therefore non singular. Defining the matrices 0, as follows: ~-~--l~t~,

equation (1) can be written (/~,a,+}¢)Q (x)=0.

(17) (18)

2. One-mass Equation In all problems connected with particles with only one mass, we have to require that one possible form f(a) is [ ~ - - ~ , and q(x) then is a scalar field (with several components). Thus the canonical

COMMUTATmN~LATmSS AS. r~LAT~WSTXCWAW EgUATXOSS

S51

formalism leads to the well-known commutation relations for the field variable q~ (z):

[q~ (=), ~*(=')]~ =i~ ( ~ - = ' ) ~ ,

(19)

and by means of the transformation (14)

[(?~(~), ~B(=')]~=a)~Ca)a (x-=').

(20)

The commutation relations are therefore completely determined b y calculating D(a) from (13). Moreover, we m a y notice that only 2S-{-1 components QA(k) (Fourier amplitudes of Q,4(z)) are not zero when k~-{-~*=0, since a field of spin S is described by 2S-l-1 independent components. We conclude therefore from (20) that the rank of the matrix D(a)A@) is 2 3 + 1 . Let us write now the operator D(a) as in equation (15): D(O) = Z j _ 0 %...o, O o . . . 8o,. (21) We m a y suppose, without a n y loss of generality, that the %,...o, are symmetrical with respect to the exchange of their indices. With A~flf, O~,W~, the relation (12) leads to recurrence formulae which determine the coefficients %:

a) %=--~I.

b) %fl,-t-xa,=0.

c) S(fl,~.-[-~)=2O,..

(22)

d) S(flo ~o,...°,_ -F~ao,...o,)=0, p >2; S denotes a summation over all possible permutations with respect to the suffices. The solutions of the equations (22) can be easily worked out: ~,=~,, (23) 1 1 ~,,,=- ; 0,.-- ~ •

°

.

,

.

.

•

S~,~.. .

•

•

(24)

.

1 1 Thus, we have 1

DCa)=-=+~.a.+- (D-/~,,#.a,,a.)+... 1 +-~-

~°

. . .~o,_,([:]-~,,_,~°

a°,_ a°,)ao

. . .

ao,_. (26)

We can now prove, for a field of maximum spin S, that the expansion of (26) must stop at the term whose order is 2S in the

~9.

H. UMEZAWA AND A. VISCONTI

derivative operators. At least one of the coefficients a,,...,,s is different from-zero, and

• %...~ = 0

for ~ > 2 S .

(27)

The QA (A = 1 . . . N ) span a space yielding a representation of the 3-dimensional rotation group which can be reduced into irreducible representations Ds, Ds_1. . . . . Therefore, the representation whose basis is given by the direct product QAQs can be split into D~s, D~s-1, . . . (Clebsch-Gordan theorem). Similarly, since A (~) is a scalar, a ~ . . . a ~ A (z) corresponds to a set of representations D~, D~_I. . . . except when some of the a, .build up dalembertians. Thus, =o~...~ a , 1 . . , a% ( p > 2 S ) can appear only in the form

s~,...o,s ( D)~(p-~ a o . . . a~,s

(28)

and the recurrence formula (22d) gives

' o, S~x~...a:s~

(29)

which shows that a~....o = 0 for p>2S. We finally deduce from (27) that the /~ matrices for a field of m a x i m u m spin S satisfy the equations

s~,.., po~_~(~o~.,~÷-~o~ ~÷~) =o.

(3o1

Taking S = 1/2, we obtain the anticommutation relations of Dirac theory. If we take S = I , equation (30) admits, as a solution, the p, of the Duffin-Kemmer-Petiau theory. For S = 3 / 2 , one obtains the relation given by S. N. Gupta. Therefore, we can conclude that the/~, must satisfy (30) in order to give a quantized field theory of equation (7) and then, their commutation relations can be given by (20) and (26). HarishChandra 4) obtained an equation similar to (30) by means of purely algebraical methods. His equation is of highest order ~ in/~, where ,t is just the order of the minimal equation of ~,. Our discussion gives a physical meaning to this number ~t: ~t= 2S.-~-I,

(31)

by requiring the canonical quantization of fields. Indeed, the minimal equation derived from (30) is

~ - 1 (1--~)---- O.

(32)

As pointed out by the same author, (30) does not generate a finite

COMMUTATION R.BLATIONS A N D RIgLATIVISTIC W A V E E Q U A T I O N S

~5~

algebra. Some stronger tensor condition compatible with (30) is needed in order to make this algebra finite. An example of this is found in the Duffin-Kemmer-Petiau theory. However, whatever the new condition is, the commutation relations must be given by (20) and (26) with ib----2S.

3. M u l t i m a s s Equations We now extend our results to the case where the wave equation (1) leads to multimass equations t CE3-

) • • •

(33)

As in the former case, there must exist a differential operator D(a) defined as follows: DCO)A ( a ) = n ; . ~ C D - - ~ , ) . (34) We can prove, by using the same arguments as in section 2, that •the commutation relations are [Qa Ca), QB (z')]+ = i D ~ ( 0 ) A

(z--z'),

(35)

where A (x) is a scalar function satisfying t t (3o)

and

a(x)=0

(aT)

for any space like vector x,. Writing down the wave equation in the form (18) and substituting D(0) in (34), we have %-- (-- 1)I II~=l up, a,,--

(--1)'-1 ~2

'

l-Ij=l

~

(--1)t~1 (Xo a =

11.

(38a)

9. / 1~

b)

JPO' I us

c)

The discussion given in the former section about the highest derivative order in D(0) remains valid: this order must be 2S. ? As is well-known, the energy operator is not positive definite in such a c a s e . ?? An explicit form of this function has been given for instance by Y. Takahashi and H. Umezawa 5).

354

H. UMEZAW A AND A. VISCONTI

As an example consider the c a s e / = 2 S , S integer; n~----~//'. Since ~ol..-o~s+~ is zero,

o = %...o~+ = ~s (~o, ~o.-~o,o,)(~o, ~o - (2)%,°,)

x . . . (~o~_,&s-S~Q~-,,o,~)~+,,

(39)

where a is a constant (c-number). This relation leads to the following minimal equation for the ft,:

( ~ - 1)(~.'-4)...

(~,'-s,)~,=o,

(40)

which has been given b y Bhabha e). In the case of half integer S, if we choose l----S4-{,

(41)

and 2~

~J= 2i+ 1'

(42)

we have for the minimal equation:

(~-¼) (~-~,)... (~-s,)=o.

(43)

This is again a relation given by Bhabha t. The authors thank Professor L. Rosenfeld and Professor C. Meller for their kind hospitality and interest and the late Dr. J. Podolanski for several helpful discussions. References 1) 2) 3) 4) 5) 6) 7)

S. N. Gupta, Phys. Rev. 95 (1954) 1334 P. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61 H. Umezawa and Y. Takahashio Prog. Theor. Phys. 9 (1953) 501 Har/sh-Chandra, Phys. Rev. 71 (1947) 793 Y. Takahashi and H. Umezawa, Prog. Theor. Phys. 9 (1953) 14 H. J. Bhabha, Rev. Mod. Phys. 17 (1945) 200 K. K. Gupta, Prod. Ind. Acad. Sc. 35 (1951) 255

t The generating operator St~ of the rotation group has been given by K. K. Gupta 7) for the special case 5 = 3 / 2 . The expression differs from Sw,=g[~,,~,]because this last leads to a multimass theory.