Space-time reflexions, isobaric spin and the mass ratio of bosons By H. Frohlich, F.R.S.
148 H. Frohlich for 0 = 0, say, yields the first frame and for = the reflected one. This brings I formal comparison with a Pauli spin to mind, which has definite values (± only for two directions, say </> = 0 and (j> = n.The interpretation of intermediate angles would then have to be given in terms of a mixture of the original triangle, and of the reflected one. Thus right from the beginning, the present description considers the possibility of both these triangles. The co-ordinate <j> decides which one is realized.
Space-time ,reflexions isobaric spin and mass ratio of bosons 149 where cycl. means that the equation holds for cyclic permutations of the suffixes; a represents a vector with the three ctk elements as components, and x indicates the outer products. Further six ekl, and three Akl defined as eki — eik — ^k^i + ^ k — i$ik>-e- efj = e|2 = e|3 = 1,| ^kl — ~^lk = KkQl + Ql&k’ J 2 4) From (2-3) and (2*2) it is found that the qk satisfy <li = -ifaffl-fta)* cycl., i.e. iq = qxq, (2-5) as well as (2*3).
150 H. Frohlich Rimming indicates that the qk{4:b) and ekk(4:b) are reducible into 3x3 and (vanishing! lx l matrices. This will be described by = <?&(3 + 1); 6^(4) = efcfc(3+1). (2*13) It follows immediately that the qk3() represent the generating elements of an irreducible three-dimensional representation, a*(3) = ?*( 3) (2-14) because the qk satisfy (2*2). The second three-dimensional representation is ob tained by afc(3~) = — (2*15) This representation is not equivalent with (2*14) because (2*15) leads to qk(3~) = 3) so that ak3(~) = — qk(3~), an algebraic relation which is different from (2*14). The one-dimensional representation clearly consists of the unit element only with «*(!) = (0).
Space-time reflexions, isobaric spin and mass ratio of bosons 151 representation according to (2-10) and (2-11), the required unitary transformation is obtained as D(9) AkD~\9), where 222 ( - ) and Dx is given in (6*26). Apart from the factor the diagonal elements of T3 and S3 are then identical with those given in table 1.
152 H. Frohlich The new description, obtained by the replacements (3*3) leads to the generalization (i-u2)ii/72Fi = - n sn 1vV3+in2v„. (3-7) These equations must be consistent with (3-6), as well as with the definitions of 77, and 772. In other words the right-hand sides of (3*7) must be equal to — and to i772( — vV3 4- V0), respectively. Hence the conditions Til = i773/72, 772 = - in zIIl (3-8) must hold. From 77| = 1 it follows that i773 = n iTI2 and n iII2 + Il2n i = 0, cycl. (3*9) It should be noted that the normalization of 77f and 77| according to (3*2) implies *hat s f| — f§ = {^v.+ u i.r^ if f* - zf|. (3-io> k The IIk satisfy the same algebraic rules as do Pauli matrices <rk. One might have thought at first that IJk = 1 should satisfy the consistency conditions, but this possi- 1 bihty was excluded by the replacement of the time component V0 by HI2V0 which in I view of 77| = 1 represents an anti-Hermitian operator. The significance of this will! be discussed in Frohlich (i960) (§§ 1 and 6). While conditions (3*9) are fulfilled by the 1 Pauli matrices (2*6) much more general expressions for the IIk can be obtained.! For this purpose angular co-ordinates, as anticipated in § 1, will be introduced.
Space-time reflexions, isobaric spin and mass ratio of bosons 153 tnd rx = ( cr1 cos x + ^2 sin X)008 sin. 6, (3* 15) r2 = - <rx sin % + <r2 cos x, (3-16) 172 = r3 = (o'! cos ^ + <r2 sin x) sin + <r3 cos 6, (3* 17) leading to TI1 = rt cos 0 + r2 sin 0; 172 = — rx sin 0 + r2 cos 96. (3* 18) nOomparison of (3*18), using (3*15) to (3*17) with (3*11) provides the components t >f the ufc in terms of Q. The transformation s has been chosen such that 0 = 0 Establishes a system rk which differs from the nk only by rotation around the 5 ,i3 axis such that r3 = 173 (3*19) 1 From the above developments it will be expected that reflexions of a vector K, now described by (TIxVk, in2V0) are obtained by certain changes of the values \ :>f the angles (0, x, 0). Furthermore, in view of the invariance properties of the IJk oiiuch changes must be identical with certain unitary transformations by say r, rnk(0, >x <f>) r~x = n k{d 72>0 ~ 7s)- (3’2°) dChus r = IJX, U2,173 leads to reflexions of time, space or time and space co-ordinates. dChe alternative angular changes are rotations from (0, x, 0) into — 0), %, in — 0, x — rr, n — 0) and ( 0 , 0 — )n ,respectively.
154 H. Frohlich The x are the space-time co-ordinates, and = d/dx^. The refer to Kemmer’s five-dimensional representation only and will be used in the form so that the wave equation becomes Pk dk + P*d* +K, 0. (4*4) ^ is a column matrix with components i}ro, fi = 1, 2 , 5 . The present paper deals with unquantized fields only so that the \jfg are c-numbers.
Space-time reflexions, isobaricspin and mass ratio of bosons 155 Before continuing the further development of (4* 11) it seems useful to show how a slirect application to (4*4) of the criticism of previous treatments of reflexions also Beads to the conclusion that each equation (4-4) should be replaced by four equa tions. In previous treatments, to prove invariance of (4*4) under reflexions—say space reflexion—co-ordinates (xk, a;0) are assumed in terms of a certain frame, say a right-handed one. Another, left-handed frame is then introduced by replacing .ck by x'k = —xk. A new function ft'(xk) = ft{ — xk) = ft(xk) can then be defined. It satisfies „ + & (xk) - °* (4‘12) plearly replacement of one system of co-ordinates by another is quite an arbitrary performance which does not influence the physical content of meaningful equations. Equation (4*12), therefore, is equal in content to (4*4) but different in form because >.t refers to xk and not to xk. In addition to the above the usual procedure postulates Further changes which do not refer to the co-ordinate frame but to properties of the field, namely replacement of fik by R2fikR2 — ~ i-e- and of ft'(xk) by iK^ie) ~ Rzfr'fak) = ftrfak)’ This clearly is the type of reflexion which according to jbhe discussion in § 1 cannot be given a simple physical meaning in terms of the un- reflected quantities Vk and ft. In fact, ftr(xk) should, thus be considered not as de fined by the (prohibited) reflexion of ft but as a new feature of the field. It satisfies — AA + /?404 + k, = 0, (4'13) an equation which is equal in form to (4*12), but different in content, as it refers to \ck and not to xk. Corresponding equations would be obtained from the remaining two reflexions, and as a result (4-4) would be replaced by a set of four equations giving all four + combinations in the terms ± fikdk and ± 94, as is provided in (4-11) by a particular representation of the ITkIIk.
156 H. Frohlich For this purpose it will be observed that by multiplication of various elements 17* V and II'k 1 (4 x 4 matrices) a total of sixteen independent matrices can be ob tained. They can all be expressed in terms of the 4x4 representation of the a. algebra in §2, by introducing a4k() = \( (4*16) For, as shown in connexion with (2*7), this holds for a particular representation of the 77*’s (by <rks) and, therefore, is also true after unitary transformations.