C, P, T and general first and second order partial differential relativistic wave equations

ANNALS

OF PHYSICS:

50,

323-354 (1968)

C, P, T and General First and Second Order Partial Differential Relativistic Wave Equations* YUK-MING

P. LAM

The Enrico Fermi Institute, and the Department of Physics, The University of Chicago, Chicago, Illinois 60637

The extended Lorentz group, which contains space and time inversions, is extended by the two element group Z, ; so that representations of the resulting group, called the full Lorentz group 2, describe integral or half integral spin states. Some of these representations are finite-dimensional, while others have infinite dimensions. The full Poincare group @ is constructed as the semiproduct of 2 with the translational group y. The main hypothesis is that single particle systems form representations of @. First or second order partial differential relativistic wave equations are used to project out those subspaces to whose vectors single physical particle states may be assigned. Under a set of hypotheses originating from heuristic arguments, solutions of wave equations are shown to possess the transformation properties of free particle states under @, so that the assignment of physical states to solutions is meaningful. Since parity and time-inversion are elements of 8, wave equations are automatically invariant under them. Parities of particles are found to be &l or ii. A “Poincare invariant” exists so that distinct particle states are orthogonal. In all those equations having particle anti-particle pairs, a charge conjugation operator can always be defined.

1. INTRODUCTION

In usual c-number theory, elementary particles are described by the solutions of a wave equation; e.g., Dirac’s equation for spin l/2 particles, Klein-Gordon’s equation for spin 0 particles, and many others. Such an equation is usually constructed so that it is invariant under the restricted Lorentz group P0 and the translational group 9. Then on the equation, the operations of time-inversion, spaceinversion, and possibly charge conjugation are defined, so that a solution of the equation remains a solution under these operations. It is the purpose of this paper to pursue a different line of attack, ending up with a wide classof first and second order partial differential wave equations that describe single particle states of any spin or of any group of neighbouring spins, according to choice. Some equations have infinite components, while others have a finite number of components. * This work was supported in part by the U.S. Atomic Energy Commission under Contract No. AT(1 l-1)-264.

323

324

Y. M. P. LAM

Interaction between particles shall not be attempted here, and, since the success of an equation is measuredby comparing its prediction with experiments on physical processes(that is, interactions), successof these equations are beyond the scope of this paper. Only when each of these equations has been properly quantised can they be selectedto describe the physical world. The concept in deriving the wave equations is as follows: It is well-known that particle states form representations not of the restricted Lorentz group so, but of its covering group gO, commonly called X(2, C). -& is a double covering of go and so is an extension of so by the two element group 2,. Since space-inversion and time-inversion are part of the extended Lorentz group 9 and are well-defined physical operations, it is natural to postulate that physical systemsare not only representations of 6p0, but of 9 also. However, z0 has to be extended by Z, ; hence physical systemsmust be representations of an extension of 9 by Z, , which will be called the,full Lorentz group 2’. To find 8, one faces the possibilities that the squares of space-inversion and time-inversion can be 1 or -1:

s2= fl t2 = fl,

(1.1) (1.2)

and the possibility that space inversion and time inversion either commute or anti-commute: st = *ts. (1.3) Wigner (Z), while discussingthe consequencesof (1.2) and (1.3), restricted (1.1) to the positive sign only by saying that its phaseis so normalized that s2 = 1. On the other hand, s2 = -1 was discussedand interesting conclusions were drawn by Yang and Tiomno (2), and the author seesno theoretical grounds on which to exclude this case. Therefore all the possibilities in (1.1) to (1.3) will be included in determining the extensions of A’. Instead of 9, 9 will be used to define the full Poincare group 9. Now the fundamental hypothesis is that the set of all physical single particle states are isomorphic to a representation of the full Poincare group g, and that the time-inversion element t is representedby an anti-linear operator1 while all operators not involving time-inversion are linear. Those representations X of 9 that admit a set of 4-current operators r, will be constructed from irreducible representations of PO in Sections 3 to 8. Then a relativistically invariant wave equation is nothing more than an operator equation in X: e+=o, IcrEX (1.4) 1 This is the generalization spin l/2 particles.

from nonrelativistic

quantum mechanics and Dirac’s theory of

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where the operator Q involves I’, in some way and is invariant under g. Such simple equations will be postulated in Section 9, and shown to be first and second order partial differential equations. A scalar number (#, v) will be defined for any two solutions Z/J,v of (1.4) and this scalar, though will be called a Poincare scalar, is in fact not invariant under p (but nearly invariant enough to be useful). In Section 10, it will be shown that there exists naturally a charge conjugation operator C for electrically charged particles. However, there is a freedom of choice in the definition of C. One can choose C such that the current expectation values either reverse themselves or not. In Dirac’s theory, they do not, but Lee (3) and Horwitz and Neeman (4) have considered the opposite property. Hence these two possibilities will be taken up simultaneously. In order that solutions of (1.4) may describe particle states consistent with very simple physical experience, a set of propositions will be postulated in Section 12, and these will be shown self-consistent, so that the entire mathematical set up is appropriate for the description of elementary particles, and is a study of first and second order differential wave equations in general. The wave equations will be solved, and their time-like solutions define mass spectra as functions of spin. Some equations have solutions with eightfold mass degeneracy, others have one, two, or fourfold degeneracy. Since all presently known physical particles belong to one or twofold mass degeneracy, all equations yielding such degeneracy will be enumerated and their particular properties studied. It is then found that the Dirac’s equation falls naturally among this list. Finally it will be mentioned here that the 4-momentum of both particle and antiparticle states will be defined in such a way that the energy is always positive. Although this is predicted by Dirac’s hole theory, the author feels that this can be regarded as a definition of 4-momentum and is completely logical. Hence, the hole theory is discarded altogether, and frees us from any commitment on statistics assignment when this c-number theory is quantised. Quantisation will not be carried out in this note.

2. THE

LORENTZ

AND

POINCARE

GROUPS

It is well-known that the extended Lorentz Group 2 consists of all 4 x 4 matrices 4 that satisfy +g4’ = g where g is the metric (which is diagonal) whose elements are -g,, = 1. go0 = -&1 = -g22 -(2.1) It is also well-known

that J? is the union of four sheets

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Y. M.

P. LAM

where zO , generally called the proper Lorentz Group, is continuous, and i, , it are the space-inversion and time-inversion elements respectively: i, = -it

= g.

(2.2)

gO has six generators &IV . The generator of an active rotation going from the p-axis to the v-axis has matrix elements RL”>,~ = kx,p

in the direction

- &&““>

(2.3)

and they satisfy [q,

31 = -ieij&

[g , .%J = -icijlcXk [& , 41 = ieij&$

(2.4) ,

where zi = &ijk9jk ) (2.5) xi = 2&, i, j, k = 1, 2, 3. To extend 9 by Z, , consider the following group (depending on three parameters a, b, d = &l):

where PO is the famous SL(2, C) with generators F and 2, following relations:

t

and S, satisfy the

s2 = b t2 = d st = ats

IT Iexp(i(e * 2

+ E’ * 2))

Now define the homomorphism

(2.6)

= exp(i(e * F - E’

k: 8 + 9 by

k(exp(i(c * 2 + E’ * 57))) = exp(i(e * 9 + E’ * 37)) k(f1) = 1 k(s) = i, k(t)

(2.7)

= it .

k has kernel Z, = { 1, - l}; so that L?, called the full Lorentz group, is the required

extension of 9.

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The extended Poincare group is the famous semi-product B = 9 x 9, where F is the translational group, an element of which is denoted by four real numbers a = {a,}. Multiplication law in 9 is well-known: (a, #(a’, 4’) = (a + $a’, +$‘>.

w3)

Similarly, one defines the full Poincare group g = 8 x F with the obvious generalization of (2.8):

(a,tL)(a’,P’) = (a + k(p) a’, w’)

(2.9)

for all a, a’ E F and p, p’ E 8. Both PO E go x r

(2.10)

and Ps = (Lg u &)

x 9-

(2.11)

are subgroups of g.

3. IRREDUCIBLE

REPRESENTATIONS

OF -r”,

This is again a well-known subject, and a large number of articles treat this most thoroughly (5). However, Y. Nambu’s notation (6) is most suitable for the purpose of this note. The following is a brief account. Let the orthonormal basis vectors of a unitary complex vector space Y be j j, s) = [(j + s)! (j - s)!]-l12 (a,+)i+s(a2+)j-s10) (j, s / j’, s') = S,,~S,,~ ,

where j = 0, l/2, 1, 312,...

and

s = -j,

-j

+ l,..., j.

The

(3.1) (3.2) operators

ai ,

a,+(i = 1, 2) satisfy ai IO) = 0 [a,+, aj+] = [ai , aj] = 0 [ai , aj+] =

A representation

Sij

of p0 can be defined in Y by specifying2 F+ L = -++aa %--fK=

(3.3)

.

-[X+F+FX-LG],

(3.4) (3.5)

* Notice the minus sign in L and K. This is different from Reference 6 because active transformations are considered here. o are the Pauli’s matrices.

328

Y.

M.

P.

LAM

where X = $aC+oa C E io, = -CT = -C+

(3.6) = C* = -C-l

(3.7)

cat-1

= -&

(3.8)

JW~s)

=Wlisj

G/As)

= WI.As>

(3.9) (3.10)

and F(j), G(j) are scalar functions ofj depending on two parameters therefore specify the representation: FcF(j) = 2 (

[(j + 1)2 - k,“][(j + 1)” - c”] lj2 (2j + l)(j + 1)” (2j + 3) - 1

W)

4ik,c + 1)” _ 1 .

=

(2j

To avoid confusion, the square-root function understood to be the single-valued function

(k, , c) which

(3.11)

(z)lj2 in this entire note shall be

(z)lP = (I z 1@)l12 = (I z l)lP &O/2 where -Vi-
(3.13)

The range of k, is 0, &l/2, &l, &3/2 ,...; and c is an arbitrary complex number. In each case, the representation is spanned by vectors 1j, s) withj = I k, I,1 k, I + l,... I c I - 1 if c is real and I c 1 - I k, I is a positive integer; otherwise the range of j extends to infinity. When there are more than one irreducible representations involved, the notation 1j, s, h) is used to denote the basis vector belonging to the irreducible representation h = (k, , c). Correspondingly, operators will be labelled also by A. From the definitions, it is obvious that (k, , c) and (-k, , -c) are identical. Symbolically, (k,, c) = (-k,, -c). (3.14)

4. REPRESENTATIONS

OF s AND gs

Let s be represented by an operator 71= R(s) in a representation of L. Then it follows from (2.6) and the requirement that 7 is linear that qLq-1 = L

(4.1)

7Kv-l = -K

(4.2) (4.3)

T2 = b.

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Before trying to satisfy these conditions, define an operator 71 quite arbitrary as follows: Let the set {A} denote a set of irreducible representations S(h) of PO, and let A, be a permutation on A. Now define in S = Co S(X) the operator 7 by 7 Ij, 25 A> = i%

I j, % b),

(4.4)

where u,+are arbitrary non-zero complex numbers. The proposition is: The irreducible representations of A?‘$= g0 u sPO fall into two classes (in each of which s is represented by 7): Class (i) S = (k, , c) 0 (k, - c) when k, , c are both nonzero. (k, , c), S(2) = (k, , -c) and XP = (i I).

Here S(1) =

Class (ii) S = (k, , c) when either k, or c are zero. To prove this statement is to show that equations (4.1) and (4.2) are satisfied for L = CA L(h) and K = CA K(h). For this purpose, all that one requires is to show that $44 17-l = L(b), yX+(X) 7j-1 = -x+(hp),

$w)

7-l = -X(b),

@v)

q-l = WFG,

and VW

77-l = -G(b),

and then the rest follows easily from equations (4.3) is to be regarded as a constraint on uA’s.

5. REPRESENTATIONS

OF

(3.5), (3.9) and (3.10). Equation

t AND

2?

Let t be represented by an operator 19= R(t). From equations requirement that 8 is antilinear, it follows that

(2.6) and the

BLB-1 = -L

(5.1)

BK8-l = K

(5.2)

9” = d.

(5.3)

and It is not as simple to find such an operator 0 as it is with 7. Since 0 is antilinear, B can be written as a product of a linear operator with an anti-linear operator. This anti-linear operator shall be chosen to be the complex conjugation operator K defined by K ; j, s, A) = / j, s, A). (5.4)

330

Y. M. P. LAM

To find the other part of 9, two linear operators E and T need to be defined as follows. If h is an irreducible representation of g0 such that P(j, h) < 0 for some j, then by inspection on formula (3.1 l), one sees that there exists a number II,, (half integer or integer as k, is) such that P(j,

A) < 0

for

j < nh,

mj,

4 2 0

for

j > 7zA.

and In other words,

F*(j,4 = ztm 4

for

j 2 nh .

(5.5)

The linear operator E is defined in any representation by (5.6)

where 1 (_ l)+wl

I

4.d =

.i > no j < nA, if n,, is defined;

and c(j) = I for those representations whose P(j, h) is never real and negative. Let hr be a permutation over the set {h). Define a linear operator 7 by T 1j, s, A) = (- l>j-S Pw, /j, -s, AT) TT* = d.3

(5.7) (5.8)

where We are arbitrary nonzero complex numbers. PROPOSITION:The smallestrepresentationsof L& v tpO fall into three classes,in each of which t is representedby e = ETK; ClassA:

k, f

0, c* f

c; or k, = 0, c* # fc: (kc,, c> 0 (k, , c*>.

Here the permutation is obviously

3 For any operator 0, its complex conjugate is defined to be 0* = KOK.

(5.9)

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Class B:

k, = 0 3 c* = -c

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331

5 0; or c* = c:* (k, , cl.

For where

both

classes

A and B, G(j, h) = -G*(j,

hr), F(j, X) = d(j) F*(j, hr),

nA , for those representations where n,, is dejned, O(j) = 1,“; or th’ose9’representations whose F2(j, h) is never negative. The proof of the proposition first of all:

is straight forward, though tedious. A hint is to obtain

4)

43 4j + 1) = 1 TL*(h) 7-r = -L(hr) BL(h) G(h) 8-l = L(h,) G(h,) TX+*(X) T-1 = x+(x,) .x*(x)

7-l = X(h,)

OX+(h)F(A) 13-l = X+(X,) F(&) OF(h) X(h) 8-l = F(A,) X(h,), and the rest is obvious. Representations of 2 can now be easily obtained by combining this proposition with that in the last section. They fall into five cases,which will be stated in the following. In each case, the component spaces S(h) will be understood to be enumerated by X = 1, 2,... in order of appearance in the expression S = C@ S(h). 1. k, f

0, c* # fc: S = (k, , c> 0 6% , -cl h=l234 P (2 14 3’1 h=1234 T (3 4

II.

0 (k, , c*) 0 (k, , -*I,

12’ )

k, = 0, c* # &-c: s = (0, c> 0 (0, c*j, hP

=

(;

“,),

AT

=

(;

;)*

4 It is only when c* = c and I c 1 > I kO j + 1 that P(j) tion.

is negative for some j in the representa-

332

Y. M.

III.

I’.

LAM

k, f 0, c* = -c f 0:

S = (kc,,c>0 (k, >-1, A*=&= k, f 0, c*

IV.

V.

( ; ;.1

c # 0:

k, = 0, c* = &cf

0; or c = 0: S = (k,,c).

To complete the story, recall that (2.6) imposes and v: r# = aOq,

a further

relation

between ~9 (5.10)

or, equivalently, ~7 = amj*.

6. LORENTZ

(5.11)

SCALAR

In order that transition amplitudes be described in a representation S of 8, there must exist a scalar functional of two vectors in S. Any linear operator [ defines a scalar functional (# I 4 1 y), linear in 1 y) and anti-linear in ( J, I . It would be ideal if ( # I 5 1y ) is invariant under 8. However, under time inversion, this ideal property would require that5

(4 I 4 I F> -5

(16I (et@> I y>* = (4 I E I v>.

Since I3 is antilinear, B+@ and 5 are both linear. Thus the right hand side of the above equation is linear in I v) while the left hand side is anti-linear in j y). Hence this equation cannot be true. Thus there does not exist a 4 such that (# j 5 1 y) is invariant under 8. Since the cause of this unpleasant situation is the fact that 0 is anti-linear, it 5 To obtain two antilinear

the correct transformation under operators A, and A,, (<$ / &)(I&

t or C, it is necessary to remember 1 v,>) = <# I( q>*.

that

for any

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333

may be possible that ($ / e I v) is invariant not under 8, but gs, and that it just transforms into its complex conjugate under t. Thus, define an operator f by: (9 (ii) (iii)

5 is linear;

(3

(914l~>-t(~l5l~>*undert.

<# I5 I#>* = ($1 f I #>; (# 1.$I y) is invariant under Ps ;

(iii) and (iv) imply that I(# I l I CJI)]is invariant under 8. (# I e I y) is called the Lorentz scalar, although it is not invariant under 2. Such a f then must satisfy: t+ = 5

(6.1)

and %4+ WI*)

= c

(6.2)

where R(p) is the operator representing p in 8. The last equation is equivalent to the set 5L”P = LL” rl+‘h = t e+&l = 5

(6.3) (6.4) (6.5)

To find this operator 6, let 5 be a linear operator such that:

5l.L&X> = oAIL%M 5’ = 5 7’5q = 5

(6.6)

(6.7) (6.8)

and 7+<7 = c*

(6.9)

As = Gbh

(6.10)

where is a permutation implies that

over {A}, and (T,,are arbitrary nonzero complex numbers. (6.6) CL(h) 5-1 = L(X,) = L(h,)+ t;X@) 5-l = xw

and OX’

5-l = X+(h,).

(6.11)

Then it is not difficult to show that in general (6.12) 595/P/2-10

334

Y.

M.

P.

LAM

is the required linear operator satisfying equations (6.1) and (6.3) through (6.5). Incidentally, 5 is proportional to the unit operator for case III, and for the following parts of case V: k, = 0, c* = -c;

k, = 0,

c* = c,

1c 1 < 1;

and k, f 0, c = 0, as should have been expected from the unitarity representations. 7. CURRENT

OPERATOR

AND REPRESENTATIONS

of these

OF L?

Both Dirac’s equation and Majorana’s infinite component wave equation (7) are first order differential equations involving a current operator. Since it is a similar wave equation that will be constructed, those representations of 8 that have such current operators will be determined here. First of all, the notion of current operator will be made clear. A set of four linear operators r,(X = 0, I, 2, 3) in a representation S of 8 will be called a 4-current operator if

R-YP) rJW R-l(t) r&t)

= M4ln” r, = [i$

v P Ez- ,

r, .

(7.1)

Under an element TVE 9, the four scalar numbers (# / fr, 1v) are transformed as

(3LI WA I P> L

(<# I R+(P)) SrAm4 I TJ)).

(7.2)

Equations (7.1) define the transformation completely under 2’. Therefore (# I fI’,, 1 v) transforms like a 4-vector under gs, and under time inversion t, (see footnote 5)

c+ I ui I y) -5

-f+ I tri I d*

and

c+ I w, I F) -2+ (+ I w, I e. Hence if ($ I [I’, ) $> is . real (as will be so postulated in Section 12.2),
mw

= bwlA~ c

eh

= [islAvc

e-lr,e = [ispr, . These three equations will be studied in turn.

VP4

(7.3) (7.4) (7.5)

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EQUATIONS

Expanding p = exp(ie@“9PV) for all p in L& , R(p) = exp(iPl,,), k(p) = exp(iP&), so that (7.3) reduces to exp( -iPLUy)

r, exp(iFL,,)

= (exp(i&PY9Uy))n~F,

which is equivalent to

In terms of L, K, it is ro ,&I

= 0

(7.7) (7.8)

i[r, , &] = ri i[r. 7Lj] = EijJk

(7.9)

(7.10)

i[& ) Kj] = aijro.

Y. Nambu (6) has shown that the simplest representation of g0 admitting r, satisfying (7.7)-(7.10) (neglecting 8 and 9 for the moment) consists of the pair w, , cl 0 e;,

4,

where (7.11) (7.12) To is given by r. I j, S, (k,, 4) = yw) I j, S, (6 9~7) r. I j, S, (PO, 0) = y’B*(j) I j, 4 Way 4)

(7.13)

where for for

(7.11) (7.12).

(7.14) (7.15)

ri are given in terms of r. and K according to (7.8). Because of equations (4.2) and (5.2), (7.8) implies that (7.4) and (7.5) are each equivalent to only one equation. They are respectively, h, roi = 0, 7p+0 = r 0’ 6 6 This is a consequence of the obvious commutation

[E, I’,] = 0.

(7.16) (7.17)

336

Y. M.

P. LAM

Therefore the smallest representation S of 2 that admits the existence of r, is obtained by doubling the spaces in each case of Section 5 according to (7.11) or (7.12). If (ki , c’) happens to be the same as one of those component spaces occuring in the original representation of 2, then the doubling is unnecessary. The component spaces can then be numbered again, and the permutations hp , AT (and hence A,) can be naturally extended. In order to describe r,, in a compact form, let hr be a permutation of {A} such that hr denotes (kh , c’), if X denotes (k, , c). For example, the representation obtained by doubling. Case I is s = (kl 2 c> 0 (kl 2 -cl

0 (k, 3 c*> 0 (kl , -c*1 -c’) @ (kl, ) c’*) @ (k;, ) -c’*). 0 (kl, 3 4 0 (kl, ,

(7.18)

If these spaces are numbered in this order by h running 1 to 8, then the permutations are 12345678 A’-(2 1 4 3 6 5 8 71 “=(3

12345678 4 1 2

7 8

5 61

“=(4

12345678 3 2 1 8 7

6 51

Ar=(5

12345678 6 7 8

3 4’1

1 2

In general (7.13) is generalized to roi j, $2 A) = mm where B(j, A) are given by the appropriate It also follows that

expressions

(7.13), and (7.14) or (7.15).

B(j, A) = B(j, A,) = B*(j, AT) = B”(j,

8. REPRESENTATIONS

(a,

(7.19)

3 I .A s, b-h

OF

A,).

(7.20)

g

Let G-P be the space of all complex functions of x, , p = 0, 1,2, 3. For any p) in g’, an operator U(a, p) in X is defined by (9

U is a homomorphism:

g --+ operators in 2,

(ii)

Wa, 1-4f(x)

= f((a, WY

(iii)

U(0, t) f(x)

= K’f(i,-lx)

-4

V(a,p.)E~s,

(8.1) (8.2)

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where K’f(x)

= f(x)*.

Thus, 2 is a representation (reducible) of 9. Note that U(a, p) is linear for (a, p) in gs and U(0, t) is anti-linear. For (a, p) in :pO , U(a, CL)can be exponentiated: U(a, exp(z#P,,))

= exp(--ia,

$li) exp(kr”L;,),

(8.3)

where (8.4) and LZ” = X&j,

-

x$,

.

(8.5)

Since U(0, s)” = U(0, t)2 = 1 and U(0, s) U(0, t) = U(0, t) U(0, s), only those 9 with a = b = d = 1 can be represented on 2. Identically from the definition of U(a, p):

w,Pw?3s

U-Y4 dhAw% PL)= MayA P, and U-l(O,

t) jAU(O,

2) = [i,],” p.

(8.6)

Comparing them with (7.1), jn is a 4-current operator. The fact that only those J? with a = b = d = 1 can be representedon z&’ shows that such representations are not very useful. However, the homomorphism U is in fact a useful tool for constructing representations of g’, not restricted to a = b = d = 1. Let R(a) be a representation of 2? (with arbitrary a, b and d) on a space S. The isomorphism: UG P) = NPL) WG P>

v (a, p) E P

(8.7)

defines a representation of g on the product space [email protected].

c3.8)

If the notations P = R(s) U(0, s) = qU(O, s) T = R(t)

U(0, t) = eU(O, t)

are used, then it follows from (4.3), (5.3) and (5.10) that P2 = b

(8.9)

T2 = d

(8.10)

PT = aTP.

(8.11)

338

Y. M.

P. LAM

Thus any P can be represented on X in this manner. P and 71will be called the parity operators and their eigenvalues will be called parity. The space S appearing in (8.8) is called the auxiliary space. To each 4 E X, there is associated with it a vector function in S; for each 4 E X can be expanded as (8.12) and

is a variable vector in S depending on the parameters x,, . This defines a one-one correspondence between each # E X and a 1 #(x)) E S. The correspondence induces a representation of g on the set of vector functions in S: (8.14) To every two vectors #, y E X there is defined a function of x, : (8.15) Its transformation

under $ is:

(+, vX4 @A (A v)((a, WkW

4

V (a, PL)E E

and ($4 d(x) -L

[($, y)(i;‘x)]*

(see footnote 5).

(8.16)

Therefore (#, v)(x) transforms like a function in &’ under g and hence is called the scalar density between #, y E X. It’s complex conjugate can be shown to be (8.17) so that (#, #)(x) is real and hence is truly an invariant scalar density. 9. RELATIVISTIC

WAVE

EQUATION

A relativistic wave equation is just an operator equation in X: Q# = 0 for # E X, where the operator Q must be invariant under 8: V-l(a, r-L>QV(a, r-L>= Q, V (a, CL>E 8. Obviously, there are many such operators; but this paper will be

RELATIVISTIC

WAVE

339

EQUATIONS

confined to the following choice for Q: CW~W~+ ~fi,$ is the Pauli-Lubanski vector’

$ 2xI’&’

- K where w,

w, = - &&&LY$~ = (L - 6, $,,L + K x ii),

(9.1)

and the constants 01,& x and K are reaLa Thus the relativistic wave equation in Xis: (crW,W” + /3j,+ This equation 4, = -23,):

+ 2xr3”

can be written as a partial

- /c)~ = 0.

differential

(cd?W,~, + /33,@ + 2ixr,S

P-2)

equation in 5’ (recalling

+ K) 1 +4(x)) = 0,

(9.3

where UUy, defined by w,ws = o~“j,j,,

(9.4)

is a symmetric tensor of second rank under 8 and satisfies

q”t = ‘i$&” .

(9.5)

The adjoint equation is9 WP

+ ~)<&4

I + ~%U~(X)

I P = 0,

I oy - 2ixwp(x)

(9.6)

and (9.7)

($Rx) I = (VW I 8.

Let the linear space of all solutions of the wave equation (9.2) be denoted by W. Then for each pair #, y E W the four quantities

a

du c-4 = m

’ The Pauli-Lubanski

I wz

- ~wcy + kkd

5 I d4h

vector does not transform exactly like a vector, for

U-Y4 1.4w@Xa,cl) = LWlpy WYV(a, P) Ego, P-‘WfiP = -[id]@”Iv”, T-‘WT = - [i#, Iv”. However, its square wW,, is clearly invariant under p. 8 These constants are chosen real so that the Lagrangian

Jqx> = G%wwu”

density

+ Isg,,l@” I SW>) -
is real. 8 For this purpose, equation (12.2) is to be anticipated.

+ XII 4(x>>

P-8)

340

Y. M. P. LAM

where use has been made of the definition

<$tx> I%I 9Jw= l> I dx-I>, satisfy, by virtue of the wave equations, the zero-divergence condition: @(3L, FL t-4 = 0. Their transformation

(9.9)

under L? is described by

(A P),(4 (@h[WP)ILII\ ($4V>A ((4ew 4

v (4 PIEe

and (A F&A(4 -5

(9.10)

PSI; ($7 rp);r e4-

For this reason (I/, #)U (x ) is called the 4-current density of #. From these properties, a Poincare scalar can be defined for any pair #, q E W: ($3 F> E J, ~u”t4(+,

(9.11)

v), (x>,1O

where u is any space-like surface. It is a simple task to prove the following properties of ($, y), justifying its name “Poincare scalar”:

and (see footnote 9). 10. CHARGE

CONJUGATION

Putting electromagnetic interaction into the wave equation particle with electric charge e, the resulting equation is Kc@‘““+ /$T)(&-

e4Ax))tB,--

(9.13)

e-h(x)) + 2xJWu-

4L4)

(9.2) describing a - ~l# = 0

(10.1)

where the principle of minimal coupling to the electromagnetic potential A,(x) has been assumed. Does there exist an operator C on X such that Clfr satisfies a similar equation but with e replaced by -e: [(a@” + &%ju

+ e4M>(A

+ e&(x)) + 2xWA

+ eA,(x)) - ~1 W = O? (10.2)

lo The absence of 0 in the notation is justified by the independence of the right hand side of equation (9.11) on the choice of space-like surface 0. An analogous proof for Dirac’s equation is given by S. Schweber, An Introduction to Quantum Field Theory, 1961 (p. 81).

RELATIVISTIC

Comparing

WAVE

EQUATIONS

341

the above equations, C must have properties Cl-$-1

= -r,

)

(10.3)

C&C-l

= - j, )

(10.4)

C&WC-l

=

(10.5)

(p

Since this transformation does not induce a space-time reflection x, --f - x, , and since K’$,K’-1 = - 6, , C must contain K’ and is, hence, anti-linear. In other words, there must exist an anti-linear operator C’ on S such that (10.6)

C = C’K’.

Let a linear operator @?on S be defined by % I j, s, A) = (- l>j-s VA1j, -s, A,) w,*+F

= 4,

)

(10.7) (10.8)

where v,,‘s are arbitrary non-zero complex numbers. Then it is straight forward to show that the anti-linear operator C’ = &K

anti-commutes

(10.9)

with the Lorentz generators: CL,,

(10.10)

= -L,,C’.

Equations (7.8), (10.8) and (10.10) imply that

cfr,c’-1 = -r, ,

(10.11)

C'&Jwc'-1

(10.12)

=

&l/L"

,

so that C = C’K’ does satisfy (10.3) through (10.5). This operator C is usually called the charge conjugation operator. It follows from the correspondence #t, 1 Z/(X)) that the charge conjugation operator on ) +(x)> is just c’:

I VW> -L

C’ I #(x)>.

(10.13)

342 11. REST

Y. M. VECTORS

AND

TIME-LlKE

P. LAM SOLUTIONS

OF

THE

WAVE

EQUATION

Since [olw,wU + pj,& + 2xI’& - K, $J = 0, there exist solutions (of the wave equations) that are also eigenvectors of 6, . Those solutions of the form I +!J)exp(ip,x”) with p,, = (m, 0), where m is a nonzero real number, are called rest vectors, and m is called rest muss. Substituting this into the wave equation, {[-aL2

+ B] WP + 2xFom -

K)

( $4)

=

0.

(11.1)

Hence I #) is an eigenvector of r,, , and L2. By the definition of r,, its eigenvector is a linear combination of 1j, s, A) and 1j, s, A,}. Therefore, it is labelled among others by indicesj, s. Let the other indices be denoted collectively by q. Then the eigenvectors are 1 q, j, s} and let them be so chosen that they form an orthonormal basis in S: (q’, j’, s’ I 4, j, s> = &Q8j~j%s .

(11.2)

Thus the rest vectors are:

M4 4, j, s) = I 4, j, s> e’mt,

(11.3)

where the “0” appearing in the argument of # is inserted there to denote that it is a rest vector, and m is obtained from solving (11.1) so that m is a function of q and j. If $J is a rest vector, then, because C contains K’, C# and J$ have opposite rest masses. Other time-like solutions are obtained from rest vectors as follows. Let p be any set of three real numbers. Then there exists a set of 3 real numbers ~(1 m I, p) such that the Lorentz matrix

4 m I, PI z exp(4 m I, PI * XI has the transformation

(11.4)

property

41 m I9P)

Im I = 0

!

0 0

PO Pl P2 ij P3

1

(11.5)

where pa = (I m I2 + p2F2.

(11.6)

Let the image of A under Y be denoted by WU

m I, PI>

=

L(I m I, P>

=

4 m I, p>B(l m I, PI,

(11.7)

RELATIVISTIC

WAVE

343

EQUATIONS

where A(1 m 1,~) = expP(l m I, p) * Kl

(11.8)

B(l m I, P> = ew[W

(11.9)

and m I, P)

* K”l.

Invariance of the wave equation under g guarantees that the vector Y% 4, J s) = L(I m I, I-9 #(O, 4, i, s>

(11.10)

is also a solution. Since B(I m I, P> eimt = exp[iW)

m/l m

II,

S,$(P, 9, i 4 = WI m I>P,$@, 4, -A 3).

(11.11) (11.12)

pu will be called the energy-momentum

of the state $(p, q, j, s). Note that the relations between L(I m 1,p) or A([ m 1,p) and ~7, 8, C’, P, T or C are (11.13) 774 m I, P>= AtI m 1,-Ph

e&l m I, PI = 4 m I, --PI0

[C, 4 m I, PII = 0

(11.15)

PUI m I, PI = JW m I, 3-W

(11.16)

WI m I, P> = L(I m I, -PIT

(11.17)

[C, UI m I, 1-91= 0. 12. PHYSICAL

(11.14)

MOTIVATIONS

(11.18)

AND

POSTULATES

Physical motivations will now be investigated leading to a set of postulates such that solutions of the wave equations have the correct kinematic properties for describing free particles. 12.1.

MASSES

OF PARTICLES

ARE

REAL

Looking at equation (1 l.l), it is seen that to achieve real masses it is necessary to postulate that r,, is hermitian: r,t = r,.

(12.1)

344 12.2. ~-CURRENT

Y. M. P. LAM DENSITIES ARE REAL

This meansthat (#, #), (x ) are real, from which it follows that

r,+t = gr, .

(12.2)

Becauseof (6.12), (7.8) and (12.1), this is equivalent to a single equation

[To2111= 0,

(12.3)

which is therefore the postulate to make ($, $), (x) real. 12.3. ~-CURRENT

DENSITY AND CHARGE CONJUGATION

It is physically sound to expect that the current density (#, #)I>,(x) of a state # is reversed in all four directions under charge conjugation. However, this is not so in Dirac’s c-number theory; while this apparent difficulty is the very reason for the 4-current density to possessthe desired physical property under C when the theory is quantised according to Fermi-Dirac’s statistics. Thus, it may seem plausible at this point to postulate that (#, #)>y(x) transforms just like $(x) r&(x) under charge conjugation. However, Lee (3) has considered a charge conjugation under which current densities transform oppositely. These two possibilities will be considered together, thus it is postulated that

where Making use of equations (10.1l)-(10.13) and (12.2), one calculates that

Therefore, c+&Y

= -co.&

(12.5)

or, equivalently, SF?+@? = -0Jp.

(12.6)

From this it follows that ($4 VW4 -5

-4~~

$4x)-

(12.7)

Thus the transformation property of the scalar density under charge conjugation is intimately related to that of the 4-current densities.

RELATIVISTIC

12.4. ORTHOGONALITY

WAVE EQUATIONS

345

OF PHYSICAL STATES

To define orthogonality, a scalar quantity is required, and the Poincare scalar defined in Section 9 is the appropriate choice. Then the concept of orthogonality is best stated as (#(P’, 4’, j’, s’), $(P, 9, j, s)) = 0 unlessp’ = p, q’ = q, j’ = j and s’ = s. For p’ = p this implies (4, j’, .s’I 5Px~o + (-aj(j

+ 1) + #m’

+ m)] I q, “i, s> = 0

unless q’ = q, j’ = j and s’ = s (m’ and m are the rest masses).Since 1q, j, s) are eigenvectors of TO, they must be also eigenvectors of f, and hence of 5. Thus in order that distinct statesare orthogonal, it is necessaryto postulate that / q, j, s) are eigenvectors of 5 and that

rll >il = 0.

(12.3)

12.5. PARITY AND PARTICLE STATES Parity operation reversesthe positions and motions of an entire physical system. Therefore PI,,!J(P, q, j, s) is proportional to z/(-p, q, j, s) so that #(O, q, j, s) are eigenstatesof P. To achieve this property of parity operation, it is therefore necessary to postulate that 1q, j, s) are eigenvectors of 77and

h TOI= 0 h, 51= 0.

(7.16)

(12.8)

It is now clear that q collectively denotes the eigenvalues with respect to TO, 5 and q.

12.6. TIME-INVERSION

AND PARTICLE

STATES

Under time-inversion, the spatial configuration of a physical system is not changed, but all motions are reversed. Therefore T#(p, q, j, s) is proportional to #(-p, q, j, -s) so that the eigenvalues q are not changed under time-inversion. In view of (5.7) and (11.17), T#(p, q, j, s) does indeed have 3-momentum -p and spin --s. But in order that # and T$ describe the sameparticle (in different states, of course) 0 1q, j, s) and j q, j, s> must have the sameeigenvalueswith respect to TO, 5 and v. Since the eigenvaluesof r,, and 5 are real, this implies equation (7.17) and r<* = 57. (12.9)

346

Y. M.

P. LAM

As for v, its eigenvalues are real when b = 1 and purely immaginary Therefore

when b = - 1.

$9 = b&j

(12.10)

yr = by*.

(12.11)

or, equivalently,

12.7.

CHARGE

CONJUGATION

AND

PARTICLE

STATES

It is here postulated that the charge conjugation operator takes a particle into its anti-particle affecting neither its 3-momentum nor spin. In view of (11.18), the 3-momentum is indeed unchanged. However, the definition of V in (10.6) does show that the spin is reversed. This apparent difficulty will be resolved later in Section 15. Applying this postulate to rest vectors, it is seen that C’ 1 q, j, s) are also eigenvectors of F’, , 5 and 7. Consequences of this fact will be explored in Section 13.

13. CONSEQUENCES

OF

CONDITIONS

ON

r, , <,v,

7 AND

V

A large collection of properties have been defined for the operators r,, , 5, q, T and ‘X. This Section will deal with their implications and consistency. These properties are equations (4.3), (5.8), (5.11), (6.7), (6.8), (6.9), (7.16), (7.17), (10.8), (12.1), (12.3), (12.6), (12.8), (12.9), (12.11), and Section 12.7. The above equations are nothing but relationships between u,, , vh , w,+, yh , (TV, B(j, X), a, b, d, w and j, (the smallest j in the auxilliary space). Here are their consequences (the derivation of which is simple): (i) r] is unitary: qt = q-1.

(13.1)

g = 7-l.

(13.2)

(iii)

a=b

(13.3)

(iv>

d = (-l)Qo.

(13.4)

(ii) 7 is unitary: Therefore T and 0 are anti-unitary.

Therefore d alone determines whether the auxilliary or half-integral spin states. (v)

space consists of integral

1yA 1 is a constant, which can then be conveniently

chosen to be unity.

(vi) B(j, X) is real. Therefore B(j, X) is independent of A (see equation (7.20)) and the argument h can now be dropped. The eigenvalues of F, are &B(j).

RELATIVISTIC

WAVE EQUATIONS

347

(vii) 1uA 1 = 0, a constant. There are then only two eigenvalues of 5: &a, and they are real. (viii) As a consequence of this and Section 12.7, it follows that for every / q, j, s) there exists a number u~,~,~(= &l) such that

Comparing

this with (12.5), c+c I 4, “L s> = -wu,,j.s

I 4, j, s>,

so that expanding I q, j, s) in terms of I j, s, A), one finds C’+C’ = c+c = 1,

(13.5)

and U 0.i.s

=

--w.

(ix) Since the eigenvalues of p are real when a = b = + 1 and purely imaginary when a = -1, and because of Section 12.7, there exists a number 8Q,j,S(= 51) for every / q, j, s) such that

Then it can be shown that 8Q,j,s is independent of q, j, s, so that r$’ = aGC’q,

(13.6)

PC = aGCP.

(13.7)

and 6 is the particle-antiparticle relative parity. The consequence of Section 12.7 is therefore to supplement the equations mentioned earlier with the anti-unitary of C and equation (13.6). It is appropriate to remark here that this set of equations can be shown to be self-consistent and solutions of Us, We, yA , u,, exist. The solutions are not unique, and the arbitrariness corresponds to independent choice of phase in the operators.ll Solutions of uAalso exists, again not unique, as long as h, is not the identity permutation. If X, is the identity permutation, then vA = 0. 14. SIMULTANEOUS EIGENVECTORS OF r, , 5, AND 7 AND PROPERTIES OF C, P, T

Explicit calculation of / q, j, s) for all types of auxilliary spaces is too tedious and not rewarding. However their common properties are very interesting and I1 The relation between interactions and the arbitrary choice of phase of C, P, T has been investigated in detail by Feinberg and Weinberg (8).

348

Y. M.

I’.

LAM

simple to discuss. It has already been noted that q denotes the eigenvalues with respect to I’,, , 5 and 7, and their eigenvalues are known. Thus let the simultaneous eigenvectors be denoted by 1r, t, p, j, s) with eigenvalues given by

r. I r, t, p, .i, s> = rW> I r, t, p, j. s> 5 I r, t, P, .A s> =

fa I r, 6 P, j,

77 I r, 4 p, j, s> = PC-lY-‘o +

(14.2)

J>

x I r, t, P, j, s>

L2 I r, 4 p, j, s> =

Aj

L2 I r, 4 p, j, s> =

s I r, 4 P, j, s>,

(14.5)

Their transformation

C’ I r, t, p, j, s) = (-l>i-”

tp j -r,

--wt,

Sp, j, -s)

t, p, j, s) = (- l>j-8 i2jtpv I r, t, p, j, -s),

where TVand v are constants of modulus Further they can be normalised to

unity depending

properties,

under C (14.6) (14.7)

only on vh and We.

(r’, t’, p’, j’, s’ I r, t, p, j, s) = S,~,St~t6,~,6j~jS,~, . From these transformation

(14.3) (14.4)

1) I r, t, P, j, s>

where r, t, p = f I and x = -i”jo[(-l)%]““. and Tare:

e 1r,

(14.1)

(14.8)

one obtains

(74 = C’2 = (- l)%+l w c’e = (- lp+l

(14.9)

wsp2v-2ec

(14.10)

and CT = (- l)2io+1 wS~~V-~TC. 15. PARTICLE MOMENTUM

AND

(14.11)

ANTI-PARTICLE STATES, AND SPIN, C, P AND T

According to Section 11, the rest masses m of the rest vectors 1r, t, p, j, s) exp(imt) are obtained by solving the following equation derived from (11.1): [-aj(j

+ 1) + /3]m2 + 2rxB(j)m

- K = 0.

(15.1)

For a second order wave equation, the roots are

NE, r, j) =

-rxNj>

+ 4x2B(.02 + i-d -aj(j

+ 1) + /3

+ 1) + 81 ~1~‘~ (15.2)

RELATIVISTIC

WAVE

349

EQUATIONS

where E = * 1. Thus the rest vectors are labelled by E as well: t&O, E, r, t, p, j, s) =

I r, t, p, j, s) ei7’l(c*r*j)t.

(15.3)

Their boosted states are, of course, #(p, 5 r, 4 1-3 .A s> == ~3 m(e, r, j)l, P> W,

5 r, t, P, .A ~1.

(15.4)

Then the Poincare scalar with respect to these states is (yW, E’, r’, t’, p’, j’, s’>, #(P, 6, r, 4 P, j, SN = (2~)~ S3(p) S,~,St~tS,~,Sj~jS,~, exp{i[m(e, r, j) - m(d, r’, j’)]t)

x 0, t, p, .i s I WrxW) + [m(c’,r, j) + 45 r, Al x [-4(j + 1) + PI>I r, t, P, j, s>. But WZ-•E, r, j) $ ITZ(E,r, j) is the sum of two roots of equation (15.1), so that the expression inside the curly bracket on the right hand side is proportional to S,,, . By effecting a pure Lorentz transformation, the above equation is generalized to (VW,

E’, r’, t’, P’, j’, ~‘1, #(P, E, r, t, P, .i, 4)

=2(2rr)3S3(p’-p)p,jm(~,r,j)l--1S~S~S~S~S~~S~ EETT tt ll,B 33 ss

x 43 t47Aj)

+ m(~,r, j)[--4(j + 1) + PI>-

(15.5)

The first order wave equation (a = /3 = 0) is less complicated and yields the massspectrum: m(r, j) = rfc/2xB(j). (15.6) Its solutions are

#(P, r, t, p, j, s> = L(l m(r, j)l, PI I r, t,p, .As> ei”“(~~jJt,

(15.7)

and they are orthogonal: (QW, r’, t’, P’, j’, s>, 9% r, 4 p, .A sN = 2(2~)~ S3(p’ - p)po I m(r, j)i-’

S,,,S,,tS,,,S,,jS,,,&(j)

rtqB(j).

(15.8)

For the purpose of separating the space W into particle states and anti-particle states,the usual convention, that rest vectors with positive rest massesare identified with particle states at rest and those with negative rest masseswith anti-particle states at rest, shall be adopted. In order that solutions of first and second order equations may be treated simultaneously, (i) let a function c(r, j) be defined for a second order wave equation so that m(r, j) = m(c(r, j), r, j) > 0; (ii) let it be understood from here onwards that whenever first order equations are dealt with

350

Y.

M.

P.

LAM

the parameter r only takes that one value (either 1 or - 1) such that m(r, j) > 0. Therefore, hereafter m(r, j) > 0, whether a first or second order equation is concerned. Without any confusion, the particle states are u(p, r, t, p, j, s) = L(m(r, j), p) j r, t, p, j, sj eiv’(rB’)t

(15.9)

= A(m(r, j), p) / r, 2, p, j, s) eiDx. To look for anti-particle states, note that Cu(0, r, t,p, j, s) have negative rest masses. Moreover, because C commutes with L(I m 1, p), Cu(p, r, t, p, j, s) and u(p, r, t, p, j, s) are obtained by a single booster L(m(r, j), p) on their respective rest states. Therefore Cu and u describe the same linear motion, and it is tempting to identify Cu(p, r, t, p, j, s) as the anti-particle states of physical momentum p. However, as noted in Section 12.7, Cu and u have opposite spin. Furthermore, they have opposite eigenvalues with respect to j, . The latter discrepancy is not serious because energy is positive definite so that the energy-momentum operator has to be defined as j,, for particle states and - j, for anti-particle states. The question of spin is not so simple. However, since C’ is anti-linear and anti-commutes with L,, , for each pair x and x’ E S, ((x 1 C’+) exp(iP’L,,)(C’ / x’)) = (x I exp(iEUYLU,) I x’)*. Th us Cu transforms contragrediently to u under g0 . In the contragredient representation, the Lorentz generators are not L,, , but -L,*y . Therefore the spin operators can be defined to be -L* for anti-particle states, while for particle states they remain as L. Then ~(0, r, t, p, j, s) and Cu(0, r, t, p, j, s) have same spin. Having resolved all difficulties, the vectors 4~~ r, t, P, .i, s> = WP,

r, t, P, j, s>

(15.10)

are truly the anti-particle statesof momentum p and spin j, s.12 To express the definition of energy-momentum and spin in a compact form, let W+( W-) denote the linear spaceof all particle (anti-particle) states.Then the energy momentum operators P, are defined in W+ @ W-,13 by pu* = =k4&*

for

$ E Wk.

(15.11)

Similarly the spin operators S are defined in W+ @ W- by for for

$ E W+; * E W-.

I2 Although often not explicitly mentioned, this concept of interpreting the anti-particle is standard with Dirac’s equation. IS IV+ @ W- is the linear space of all time-like solutions of a given wave equation.

(15.12) states

RELATIVISTIC

The transformation properties 5, P, C, Tare easily calculated:

r, t, P, j, s) = rW>

ro4t

r, 4 P, .L s> =

r, t, P, j, 8) = t40,

and anti-particle

states under r,, ,

r, f, P, j, s>

(15.13)

40, r, t, P, j, s>

(15.14)

40,

-rBW

351

EQUATIONS

of particle

r440,

540,

WAVE

(15.15)

r, f, P, j, 3)

540, r, 4 P, j, s> = --wMO, Pu(p, r, t, P, j, s> = PC-l>j-‘0

r, 6 P, .A s>

(15.16)

=4--P,

(15.17)

Wp,

r, 4 P, .i s> = a~P(--lYo

Wp,

r, 1, P, j, s> = 4P, r, t, P, .A $1

Wh

r, t, P, j, s> = (- lYjo+l

Tu(p, r, t, p, j, s) = (-l)j+

r, t, p, i, s)

x+-P,

wdp,

r, 4 P, .i, s>

(15.18) (15.19)

r, 4 P, j, ~1

i2jfpvu(-p,

r, t, p, j, -s)

Tv(p, r, t, p, j, s) = (- l)j-S+l P~t8ppv~-~~~(-p, r, t,p, j, -s)

(15.20) (15.21) (15.22)

PCT~P, r, t, P, j, s> = C-1) ‘+’ i2’oxt8v-1z;(p, r, I, p, j, -s)

(15.23)

PCTv(p, r, t, p, j, s) = (-l)‘-’

(15.24)

i2’oxt8v-1~2u(p, r, t, p, j, -s).

The values of the Poincare scalar readily follow from (15.5) and (15.8): MP’,

r’, t’, 8, j’, s’), 4P, r, t, P, j, sN = 4v(p’,

r’, t’, p’, j’, ~‘1, v(p, r, t, P, j, s>>

= 2(2~+)~a3(p - p) pom(r, j)-l S,~,St~t6,~,6j~i6,~,

x 4j) t4rxNd + m(r, j)[bdj Mp’,

+ 1) + PI>,

(15.25) (15.26)

r’, f, P’, j’, ~3, u(rh r, 6 P, j, ~1) = 0.

Equations (15.17) through (15.22) show that u(p, r, t, p, j, s) and v(p, r, t, p, j, s) have the correct kinematic properties of single free particle and anti-particle. (15.25) and (15.26) show that they are mutually orthogonal. Thus their use for describing single free particle states have been justified.

16. EQUATIONS

WHOSE

PARTICLE

STATES

HAVE

MASS

DEGENERACY

1 OR

2

Let N be the number of irreducible representations of p0 appearing in the auxilliary spaceS.Then it is easy to seefrom equation (11.1) that N is just the number of rest states with the same absolute value of rest mass. Possible values of N are 1, 2, 4 and 8.

352

Y. M. P. LAM

Particles in nature seemto fall into two groups. Some particles, like yr” and photon, do not have their anti-particle partners and are therefore characterized by massdegeneracy N = 1. Others, like proton and kaon, form particle-anti-particle pairs and therefore are characterized by N = 2. All equations with N = I or 2 will be listed here. They are completely determined by specifying their auxilliary spaceS. For each of them the constants w and 6 are no longer arbitrary. Their values will be stated without proof. N = 2 auxiliary spacesare

(1)

6 c>0 (ii,-cl

(2)

hl-=hp=hT= i; ; ) w=s=--1. (4 (3,c>0 (4,-cl for c* = c f 0; and @I

(ko

74) 0

(ko

for

9 -4)

c* = -c f

for

0:

k, # 0:

w=-S==l. (3)

(4 (0,c>0 (0,c - 1) (b)

(ko,O)@(ko+

for

O
LO):

The following conclusions for N = 2 at once follow: (i) The combination w = S = 1 does not occur at all. (ii) w = 6 = -1 occurs only for half-integral spin representations, (iii) 6 takes both values (* 1) for both integral and half-integral spin representations. I4 The combination (0, c) @ (0, c + 1) fails because it does not yield real B(j). For the same reason the range c > 1 and c < -1 is excluded.

RELATIVISTIC

WAVE EQUATIONS

353

Dirac’s theory of spin l/2 particles can be shown to follow directly from (2a) with c = 3/2, taking CL= /3 = 0. In fact, even the anticommutation relations [r, , TV]+ = 2g,, and the expression for F,+: r,,+ = rJ’,J’, are obtained automatically. However, these properties are not shared by all equations and must be regarded as accidental. N = 1 auxiliary spacesare (a) (0, l/2) and (b) (l/2, O).r5Of course, for this type of equations, w and 6 have no meaning (becausethere are no anti-particle states).

17. DISCUSSION

The aim of constructing relativistic equations whose solutions are capable of describing single free particle states have been achieved in the preceding Sections. It now remains to make a few comments. (i) Since a = b, not all Poincare groups are possible space-time symmetries. Only those with a = b are possible candidates. As yet, it seemsthat there is no theoretical grounds on which to exclude those Cg with a = - 1. This will yield particle states of parity &i. However, very few discussions(2) on this possibility are known to the author. (ii) The number N does not only specify the massdegeneracy, but also determines the constants w and 6 to someextent. When N = 8, w and 6 are completely free, as an actual solution to the set of equations named in Section 13 will reveal. But for other values of N, there are fewer constants uA, u,,, etc., (becauseX = l,..., N and N < 8) so that they are over determined. Therefore, consistency of the set of equations will force w or 6 to take definite values, as had been demonstrated in the last section for N = 2. Since the particle-antiparticle relative parity 6 = 1 or -1 for both integral and half-integral spin representations, it is natural to ask if all such representations are realized in nature. In other words, are there integral spin particles with 6 = - 1 or half-integral spin particles with 6 = 1? The answer to this question lies in the investigation of parity. All presently known particles are characterized by N = 1 or 2. Perhaps the future may reveal particles with higher N. (iii) One of the consequencesof the set of postulates of Section 12 is that the function B(j), appearing in the definition of current operators r, , is real. This places a further restriction on the choice of the auxilliary space: If (k,,, c) and (kk , c’) are connected by (7.12), c must be real, and 1k, 1 > min(l c 1, j c’ I}. Thus this caseinvolves infinite component wave equations. I5 The first order equations built on these auxiliary spaces are precisely Majorana’s See Reference 7.

equations.

354

Y. M.

P. LAM

(iv) For fixed r, t, and p, the set of particle states {u(p, r, t, p, j, s)) form a spin tower and the parity alternates with increasing j. The same holds for antiparticle states. (v) Since B(j) N j for large j, all four branches of the mass spectrum of equation (15.2) approach zero for large j if 01is nonzero. If 01is zero, then two of the four branches still approach zero for large j. This is a very unphysical feature. The same is true with all the two branches of the mass spectrum of a first order wave equation (see (15.6)). A slightly more physical spectrum is obtained by setting 01 = K = 0. Then

mtr,j) = 2WWP, and, because B(j) is a monotonically ascending mass spectrum.

increasing function

of j, this gives an

(vi) The solutions of wave equations considered in this note are time-like. Needless to say, space-like ( p2 < 0) and light-like ( p2 = 0) solutions flourish abundantly for most equations. While the interpretation of light-like solutions is definitely the assignment of massless particles, spacelike solutions may describe faster-than-light particles. Properties of such fictitious particles have been discussed by Feinberg (9). ACKNOWLEDGMENT The author is most sincerely and gratefully indebted to his supervisor, Professor Y. Nambu, for his constant interest, suggestions and encouragement at all times. RECEIVED: June, 9, 1968 REFERENCES 1. E. P. WIGNER, Unitary Representations of the Inhomogeneous Lorentz Group including Reflections, Group Theoretical Concepts and Methods in Elementary Particle Physics, (Lectures of Istanbul Summer School of Theoretical Physics, 1962), edited by F. Gursey, Gordon and Breach, Science Publishers. 2. C. N. YANG AND J. TIOMNO, Phys. Rev. 79, 495 (1950). 3. T. D. LEE, Phys. Rev. 140, B959 (1965). 4. L. HORWITZ AND Y. NEEMAN, Physics Letters 22, 699 (1965). 5. For example: M. A. NAIMARK, “Linear Representations of the Lorentz Group.” Pergamon Press, Oxford, 1964. [English translation.] 6. Y. NAMBU, Dalhousie Summer School Lectures, University of Delhi, 1967. 7. E. MAJORANA, Nuovo Cimento 9, 335 (1932). 8. G. FEINBERG AND S. WEINBERG, NUOVO Cimento 14, 571 (1959). 9. G. FEINBERG, Phys. Rev. 159, 1089 (1967).