Appendix A Symrne try Rcpresentat ion Theorem 9 1 reference to symmetry principies, hecat~sewhatever one thinks the syrnrnelry group of nature map be, there is alwcsys another group whose colasequences are iden~icalexcept Jhr the rrbsence uf supersrlr~ctionrules. Appendix A The Symmetry Representation Theorem This appendix presents the proof of the fundamental theorem of wiper2 that any symmetry transformation can be represented on the Hilbert space of physical states by an operator that is either linear and unitary or antilinear and antiunitary. For our present purposes, the property of symmetry transformations on which we chiefly rely is that they are ray transformations T that preserve transition probabilities, in the sense that if and Y!2 are state-vectors belonging to rays 21 and 92then any state-vectors Yi and Y!; belonging to the transformed rays T.@]and T9f2 satisfy We also require that a symmetry transformation should have an inverse that preserves transition probabilities in the same sense. To start, consider some complete orthonormal set of state-vectors YJk belonging to rags &, with and let YL be some arbitrary choice of statevectors belonging to the transformed rays T9k. From Eq. (2.A.11, we have I{'Y;,T;)I~= ~(vkry!)~~ = ski. But (Yk,Y'k) is automatically real and positive, so this requires that it should have the value unity, and therefore it is easy to see that these transformed states VTi also form a complete set, for if there were any non-zero state-vector Yfthat was orthogonal to all of the Yb, then the inverse transform of the ray to which Yf belongs would consist of non-zero state-vectors Y" for which, for all k: which is impossible since the Vk were assumed to form a complete set. We must now establish a phase convention for the states YL. For this purpose, we single out one of the Yk,say Y1, and consider the state-vectors 92 2 Relativistic Quantum Mechalalcs belonging to some ray Yk,with k f 1. Any state-vector Ti belonging to the transformed ray TYkmay be expanded in the state-vectors Yi, From Eq. (2.A.1) we have 1 and for 1 # k and l + 1 : For any given k, by an appropriate choice of phase of the two state- vectors Yk and Yi we can clearly adjust the phases of the two non-zero coefficients ckk and ckl so that both coefficients are just 118. From now on, the state-vectors TL and Y; chosen in this way will be denoted UTk and UYk. As we have seen, 1 1 U-[vk + vI]= UYk = -[UTk + Uyl] . (2.A.5) Js $ However, it still remains to define UY for general state-vectors Y. Now consider an arbitrary state-vector Y! belonging to an arbitrary ray 9,and expand it in the Yk: Any state Y" that belongs to the transformed ray TW may similarly be expanded in the complete orthonormal set UYk: The equality of (Yk,y)12 and I(UYLk,yf)12 tells us that for all k (including k = 1): 1412= lcLl2, (2.A.8) while the equality of i(Yk, Y)1 and LIT^, Y!')12 tells us that for all k # 1: The ratio of Eqs. (2.A.9)and (2.A.8) yields the formula which with Eq. (2.A.8) also requires Appendix A Symmetry Representation Theorem and therefore either or else Furthermore, we can show that the same choice must be made for each k. (This step in the proof was omitted by Wigner.) To see this, suppose that for some k, we have Ck/C1= Ci/C;, while for some I # k, we have instead CI/CI = (C;/C;)*. Suppose also that both ratios are complex, so that these are really different cases. (This incidentally requires that k # 1 and I # 1, as well as k # I.) We will show that this is impossible. Define a state-vector (D [V1 Yk Yi]. Since all the ratios of the = ,. 3 + + coefficients in this state-vector are real, we must get the same ratios in any state-vector @' belonging to the transformed ray; where a is a phase factor with lcll = 1. But then the equality of the transition probabilities I(@, CY)I and I(@',lyl)l requires that and hence This is only possible if or, in other words, if Hence either Ck/C1 or CI/C1 must be real for any pair k, 1, in contra- diction with our assumptions. We see then that for a given symmetry transformation T applied to a given state-vector CkYlk,we must have either Eq. (2.A.12) for all k, or else Eq. (2.A.13) for all k. Wigner ruled out the second possibility, Eq. (2.A.13), because as he showed any symmetry transformation for which this possibility is realized would have to involve a reversal in the time coordinate, and in the proof he presented he was considering only symmetries like rotations that do 94 2 Relativistic Quantum Mechanics not affect the direction of time. Here we are treating symmetries involving time-reversal on the same basis as all other symmetries, so we will have to consider that, for each symmetry T and state-vector CkCkVk, either Eq, (2.A.12) or Eq. (2.A.13) may apply. Depending on which of these alternatives is realized, we will now define UY to be the particular one of the state-vectors Y' belonging to the ray TB with phase chosen so that either C1 = C; or CI = c;', respectively. Then either or else It remains to be proved that for a given symmetry transformation, we must make the same choice between Eqs. (2.A.14) and (2.A.15) for arbitrary values of the coefficients Ck. Suppose that Eq. (2.A. 14) applies for a state-vector CkAkYk while Eq. (2.A.15) applies for a state-vector CkBkVk. Then the invariance of transition probabilities requires that or equivalently 1m (A;Al)lm (B,'BI)= 0 . (2.A.16) .k I We cannot rule out the possibility that Eq. (2.A.14)may be satisfied for a pair of state-vectors CkAkYk and Ck&YE; belonging to different rays. However, for any pair of such state-vectors, with neither Ak nor Bk all of the same phase (so that Eqs. (2.A.14) and (2.A.15) are not the same), we can always find a third state-vector '& Ck Yk for which* and also * If Eor some pair k,l both A; A, and B; Br arc complex, then ct~onscall C s to vanish excepl Tor Ck and Cl,and chouse these twn coefficients to hsvc dineren1 phases. If Aidr is cumplex but B'BI is real for some pair k,l, then there must be some othcr pair m,n (possibly ahh either m or n bu~ nnl both equal to k or I) tor which BiB, ic complex. If ahAiA, is complex, then choose all Cs to vanish except for I;, and Cn, and choose these two coefficients Lo have diffcrcnt phase. If AkA, is rcal, Lhen choose all Cs to vanish except for CL,Cf ,C'm, and C,, and choose thcsc four coefficients all to have djfTercnt phases. The case whcrc BiB1 is complex but A;Al is red is handled in just the samc way. Appendix A Symmetry Representation Theorem 9 5 As we have seen, it follows from Eq. (2.A.17)that the same choice between Eqs. (2.A. 14) and (2.A.15) must be made for CkAkYk and CkCkYk, and it follows from Eq. (2.A.18) that the same choice between Eqs. (2.A.14) and (2.A.15) musl be made for CkBkYk and CkCkYk, SO the same choice between Eqs. (2.A.14) and (2.A.15) must also be made for the two state- vectors CkAkVk and CkBkYk with which we started. We have thus shown that for a given symmetry transformation T either a11 state-vectors satisfy Eq. (2.A.14) or else they all satisfy Eq. (2.A.15). It is now easy to show that as we have defined it, the quantum mechan- ical operator U is either linear and unitary or antilinear and antiunitary. First, suppose that Eq. (2.A.14) is satisfied for all state-vectors CkCkYk. Any two stale-vectors Y and @ may be expanded as and so, using Eq. (2.A.141, Using Eq. (2.A.14) again, this gives so U is linear. Also, using Eqs. (2.A.2) and (2.A.3), the scalar product of the transformed states is and hence so U is unitary. The case of a symmetry that satisfies Eq. (2.A.15) for all state-vectors may be dealt with in much the same way. The reader can probably supply the arguments without help, but since antilinear operators may be unfamiliar, we shall give the details here anyway. Suppose that Eq. (2.A-15) is satisfied for all state-vectors CkYlk.Any two state-vectors Y and @ may be expanded as before, and so: 96 2 Relativistic Qualatuna Mechanics Using Eq. (2.A.15) again, this gives so U is antilinear. Also, using Eqs. (2.A.2) and (2.A.31, the scalar product of the transformed states is and hence (UY, UO) = (Y!, a))', Appendix B Group Operators and Homotopy Classes In this appendix we shall prove the theorem stated in Section 2.7, that the phases of the operators U(T)for finite symmetry transformations T may be chosen so that these operators form a representation of the symmetry group, rather than a projective representation, provided (a) the generators of the group can be defined so that there are no central charges in the Lie algebra, and (b) the group is simply connected.