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Origins of Global Anomalies in Quantum Mechanics I Nuclear Physics B273 (1986) 93-108 North-Holland, Amsterdam ORIGINS OF GLOBAL ANOMALIES IN QUANTUM MECHANICS S. ELITZUR1 and E RABINOVICI 1 Racah Institute of Physics, Hebrew University, Jerusalem, Israel Y. FRISHMAN t Department of Physics, Weizmann Institute of Science, Rehovot, Israel A. SCHWIMMER2 Laboratoire de Physique Thborique, Ecole Normale Supbrieure, Paris, France Received 13 January 1986 Simple and tractable examples of abelian and non-abelian gauge systems with global anomalies are presented in quantum mechanics. Explicit calculations are done both in the path-integral and hamiltonian formalism. Algebraic criteria are given for the existence of global gauge anomalies. These criteria are applied for every gauge group and many representations. The inconsistency of theories with a global gauge anomaly is discussed. I. Introduction Local anomalies are old companions in studying quantum field theories [1]. Global anomalies, on the other hand, are rather recent, appearing in gauge [2] and gravitation [3] field theories as well as in string [4] theories. Anomalies occur when symmetries of the classical system cannot be maintained in the quantum theory. Starting with the path-integral formulation of gauge theory we first obtain an effective action by integrating out the fermionic fields using some regularization. It may then turn out that certain symmetries cannot be jointly incorporated into the regulated action. In a hamiltonian formulation this freedom in choosing a regularization scheme shows up as an ordering ambiguity, when classical variables are replaced by their quantum counterparts. This ambiguity may not be enough to allow a quantum realization of all classical symmetries. Both these ingredients appear already in quantum mechanical systems. Due to the simple i Supported in part by the Israel Commission for Basic Research and the US-Israel Binational Science Foundation. 2 On leave from the Weizmann Institute, Rehovot, Israel. 0550-3213/86/$03.50©Elsevier Science Publishers B.V, (North-Holland Physics Publishing Division) 94 S. Elitzur et aL / Global anomalies structure of these models, the mechanism leading to the anomalies can be easily identified. Global anomalies arise when the effective action fails to be invariant under a large gauge transformation, one that is disconnected from the identity. The anomaly can take two forms. Either the gauge invariance is inherently broken, or that the gauge invariance can be restored at the expense of breaking some global classical symmetry. Quantum mechanical examples exist for both cases. Moreover, it will be shown that in the case of a dynamical gauge field, a system with a global gauge anomaly is not consistent. We start our analysis with a few general remarks. Consider the 0-space + time euclidean action S=fdt [-0,-~t8, _ O + OkX,tOzA (t) , (1.1) with 0 k fermionic variables transforming under some representation of the gauge group whose generators are the anti-hermitian matrices ~'~. When the fermions 8 k are complex or pseudoreal, namely/~ 4: 0, there will be no gauge anomaly. This is a consequence of the fact that one can always regularize the system by adding Pauli-Villars fields Pk in the same representation with the gauge-invariant mass term MEPkP,. (1.2) k Although this term may induce an anomaly in some global symmetry, gauge anomalies can be avoided. An anomaly in the gauge symmetry may occur only for a real system with the action S = ~'f0 T dt -~,~+/+ ~b,~,AbtAm m(t) , (1.3) with ffz being real ferrnions transforming under the real representation with the antisymmetric generators )m. AS a corollary of this fact it follows that in any case the gauge anomaly could manifest itself at most by a change of sign of the generating functional when a topologically non-trivial gauge transformation is performed. Indeed, generally, the system (1.3) does not allow for a gauge-invariant regularization, and has to be regularized by a possibly gauge-non-invariant regu- larization scheme A, giving a generating function z~l)= f l-I dq~,exp[SA]. (1.4) Following Alvarez-Gaum6 and Witten [5], one can add another set of real fermions ~ in the same representation and obtain a new system which can be S. Elitzur et al. / Global anomalies 95 regularized in two ways. First one can use the previous non-invariant regularization A on both species of fermions q, and ~. The generating function for the new doubled system is, then, Z+., = Z~. (1.5) On the other hand, the same doubled system can be described by the complex fermionic variables 0k = f~-5(1+k + ithk) (1.6) and may be regularized by the gauge-invariant Pauli-Villars mass term of (1.2), giving the generating function Zo(A). For the doubled system we have then two partition functions Z 0 and Z~,, resulting from two different regularizations of the same system. In the limit of a very high regulator scale the two functions differ at most by an exponential of a local functional of the background field A Z,l, , ¢, = ZoeF( A) . By adding the local expression -½F(A) to the original ~b action (1.3) as a counterterm, one gets for the modified system ZA2= Z0, which is gauge invariant by construction. Since Z 2 is invariant, Z~ can change at most by a sign under a globally non-trivial gauge transformation. For a non-trivial topology to exist we let t in (1.3) vary in the interval [0, T] which is taken to be a circle S 1. Possible candidate representations of the fermions in (1.3) for producing anomalies are real representations which are simply connected and whose ~rx group contains elements of even order, i.e. elements which give the identity only when raised to an even power (elements of odd order cannot induce a change of sign in the generating functional). Since there is a correspondence between the elements of the ~rx group of a representation and the elements of the center of the original covering group we have to look for real representations of groups with even order elements in their center. In sect. 2 the U(1) case will be studied using the path integral method. An anomaly will be found in either some discrete global symmetry or in a topologically non-trivial gauge transformation. In sect. 3 the same partition function is calculated in the hamiltonian formulation. In both sections we demonstrate that the system cannot be consistently quantized with dynamical gauge fields in the presence of a global gauge anomaly. Sect. 4 deals with the non-abelian case using a hamiltonian approach. The O(N) case was treated by Witten [4] in the lagrangian formalism, where it was related to 96 S. Elitzur et al. / Global anomalies the problem of the existence of spin structures on manifolds. For groups and representations satisfying the conditions discussed above, one may find genuine Z 2 gauge anomalies which cannot be traded for a global symmetry, in which case the gauge field cannot be consistently quantized. For every relevant group we shall study the anomaly properties of at least one real representation for all possible structures of the ~r1 group. We shall also give rules on how to calculate the anomaly of a general representation. The search for global anomalies has been turned into an algebraic problem. 2. Anomalies in an abelian system-lagrangian formulation Consider the lagrangian .~,a= ~(t)[i 0 t + a(/)] ~(1). (2.a) Note that in (0 + 1) dimensions "t0 = 1, and thus f = ~p*. In addition to its gauge invariance under ~(t)--*eiA(t)~(t), ~b(t)'-*~(t)e -ia(t) , A(t)---~A(t)+OtA(t), (2.2) with e iA(t) a periodic function in [0, T], the lagrangian is invariant under a discrete symmetry, charge conjugation, under which D ~(t)~ g'(t), A(t)~ -A(t). (2.3) Corresponding to any given background gauge field A(t) there is a gauge invariant quantity By a small gauge transformation, i.e. one which is continuously connected to the identity, A can be brought into a time independent form .'t= ~ SordtA(t). (2.4) A large gauge transformation A(t)= (2~rn/T)t then shifts .4 by 2~rn/T leaving only the fractional part of ,~T/2~r gauge invariant. In particular, if .4T/2~r is not an integer, the background field cannot be gauged to zero. The euclidean partition function is, in the gauge (2.4), Z(A)= fd+(t)d~(t)exp[ Sdt ~(t)(-Ot+ iA)+(t)]. (2.5) S. Elitzur et al. / Global anomalies 97 The potential A is, at this stage, taken to be a classical background. Z(.~) = det[- 0, + i.4]. (2.6) To give a meaning to (2.6) we have to specify a regularization scheme. Let us choose, first, a gauge-invariant Pauli-Villars regularization. It turns out that to maintain Z(.4) finite, several Pauli-Villars ghosts are needed. Specifically, replace (2.6) by Zpv(A) = lim [det(-Ot+i.4)][det(-O,+iA+M1)] ~1 M1,M2..-~ X [det(-0 t + i.4 + 3'/2) ] -2 (2.7) Zpv is finite if 1+al+a2=O, aiM t + a2M 2 = 0. (2.8) Explicitly, Z~v(d) = lim lim I-I n + ½) + .4 n + 1) + .~ + iM 1 MI. M2 -'°° N1. N2 ''°° n = - N1 ×T 2~r (n+ ½) +.~ + iM2 (2.9) Antiperiodic boundary conditions have been chosen in (2.9) such that Z(.4) represents Tr e-rH. The resultant anomaly is, however, insensitive to the choice of boundary conditions. We have Zav('4) = m,-*oolimN,-.oolim [F(_Nl+½+a)j[F(_N,+a+a+iml[ 1[ ) ]°' x[ F(N2+½+a+im2) ~: LF(-~, ~- ~ ~ ¥,~--~) , where a = AT/2~r, m i = MiT/2¢¢.
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