The various formulations of gauge theories characterized by the parameter θ are constructed for the lattice version of these theories. We do not rely on the existence of topologically stable solutions of the classical equations. These constructions are based on the existence of inequivalent representations of the canonical commutation relations. © 1979

Bander, M

Bander, Myron

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UC IrvineUC Irvine Previously Published WorksTitleMultiple vacua for non-abelian lattice gauge theoriesPermalinkhttps://escholarship.org/uc/item/6b5087mqJournalNuclear Physics B, 149(2)ISSN0550-3213AuthorBander, MyronPublication Date1979-03-01DOI10.1016/0550-3213(79)90237-2Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaNuclear Physics B149 (1979) 211-219 © North-Holland Publishing Company MULTIPLE VACUA FOR NON-ABELIAN LATTICE GAUGE THEORIES * Myron BANDER California Institute of Technology, Pasadena, California 91125 and Department of Physics, University of California, &vine, California 92717 **, USA Received 13 July 1978 (Revised 24 October 1978) The various formulations of gauge theories characterized by the parameter 0 are con- structed for the lattice version of these theories. We do not rely on the existence of topologically stable solutions of the classical equations. These constructions are based on the existence of inequivalent representations of the canonical commutation relations. 1. Introduct ion The Hilbert space spanned by the eiegnstates of  a Hamiltonian invariant under a discrete set o f  transformations, T n, may be decomposed into a sum of  subspaces, such that in each subspace the states transform with a definite phase, Tla;O) = e 2~/° l a ; 0 ) .  (1.1) As long as all allowed operators (observables) commute with T, the different 0 sec- tors do not  communicate with each other. On the classical level such a per iodici ty  is associated with topologically stable solutions of  the corresponding Euclidean the- ory. The discovery of  such solutions [ 1 ] for Yang-Mills theories has focussed atten- tion on the existence of  such 0 sectors in these models [2]. A different approach to the study of  these theories has been to work in a space- time or space lattice [3,4], rather than in a continuum. We address ourselves to the question how to generalize the expressions for topological quantities to the lattice and thus introduce the different 0 theories. It is important  to maintain invariance under local gauge transformations. Our approach will be based on a sequence of  observations. In cont inuum situa- tions the quantum mechanics of  a particular 0 sector of  a theory specified by an * Work supported in part by US Department of Energy under Contract EY76-C-03-0068. ** Permanent address. 211 212 M. Bander / Multiple vacua action So is equivalent to the 0 = 0 sector of another theory specified by art action So - OS1. S~ is related to topological invariants; a property which we shall not use here. The property of  interest is that S~ is a time integral over a total time deriva- tive. On a classical level S o and So - OS] are related by canonical transformations. On a quantum level, such a transformation is achieved by different realizations of canonical commutat ion relations. It is this property we shall use in obtaining differ- ent 0 theories on a lattice. This observation, that reference to Euclidean solutions or topological invariants is not necessary for the understanding of multiple vacua, has been made previously in the case of two-dimensional Abelian theories [5]. The difficulty in extending these ideas to lattice non-Abelian theories is that it is not immediately possible to write the generator of the aforementioned contact  transformations, or the different realizations of  the quantum mechanical commuta- tion relations, in a form invariant under local gauge transformations. The generator of the contact transformations in the continuum case must first be cast into a form manifestly invariant under local gauge transformations, with terms affected only by global transformations appearing explicitly as surface integrals. A lattice analogue can then be obtained immediately.  The fact that differing physical systems may have seemingly identical Hamilton- ians, as an operator is defined only when its domain is likewise specified, is illus- trated by the simple example of  a one-dimensional periodic potential and continu- um gauge theories. No new results are obtained as this serves as an introduction to our methods. A crucial identity permitting the extension of continuum methods to the lattice is discussed in subsect. 3.2. 2. One-dimensional periodic potential Consider a one-dimensional system specified by the Hamiltonian H = ~p2 + V(~),  (2.1) with V(~ + 27r) = V(q~). The usual prescription, learned in elementary quantum me- chanics, is to replace p by 0 (2.2) p = - i  0~b and let all operators act on the Hilbert space of functions periodic in ~ with period 2rr. The spectrum o f p  consists of all integers. However, there is no reason for choos- ing periodic functions; we could as well take our Hilbert space to consist of func- tions acquiring a phase under a translation of  ~ by 27r. ff(~b + 2rr) = e 2~ri° ff(~b). (2.3) The spectrum o fp  now consists of  n + 0, where n is an integer. This new realization can be viewed in a somewhat different manner. The only a priori relation we know M. Bander / Multiple vacua 213 between p and ~b is the commutat ion relation [p, ~b] -- - i .  (2.4) Eq, (2.2) is not the only realization of such commutat ion relations, for we may make the identification p = - i ~ +  0 ,  (2.5) and operate on periodic functions. (0 could be a function of q~, but we will not be concerned with this more general situation.) Another way of obtaining eq. (2.5) is first to perform a contact transformation on the classical Hamiltonian, p ~ P = p + O  , q~ ~ q5 = q~, (2.6) and then replace p by the realization of eq. (2.2). The generating function for the above transformation is [6] F2(P,  (P) = c~P - 0 0 .  • (2.7) The action corresponding to the Hamiltonian acquires an additional term: - 0  f ~  d t .  (2.8) So ~ S 0 Our reason for belaboring these rather trivial points is that finding the analogues of  eq. (2.7) is a sure way of  finding different realizations of  the basic commutat ion relations without upsetting any of them. Bypassing the classical level, this new real- ization may be obtained by the transformation ° - - ' +  U-I(~)) i U(~b), U(~) = e x p ( - i 0 ¢ ) .  (2.9) - i  0¢ Despite appearances, U(40 is not an unitary transformation for it connects different Hilbert spaces. In order to fix the theory completely, we must specify 0 in addition to specifying the Hamiltonian. To determine some relations between the differing 0 worlds it is useful to study the evolution kernel for the various theories corresponding to a given Hamiltonian. Let us first consider the situation where the position variable ¢ is unconstrained and allowed to vary between plus and minus infinity. For this case let G(¢',  ¢; t) = (4)' le-iHtl4~) • (2.10) It is easy to check that for the periodic case where -~r < q5 < lr and a given 0 we ob- tain Go(dp', dp; t )  = ~ e  2"in° G(¢ '  + 21rn, ¢; t )  . (2.11) 214 M. Bander / Multiple vacua It was noted in sect. 1 that as long as we are interested in operators having the peri- odicity of  the potential, the different 0 worlds do not communicate with each other. If  on the other hand, our algebra of  operators includes non-periodic ones, the Hil- bert space of states consists of  a sum of the 0 spaces. Eq. (2.11) may be inverted, 1 a((p', c~; t)  = f dOao(c~', O; t) . o (2.12) Finally, note that the operator U(q~) of eq. (2.9) is not  periodic; part of  the diffi- culty in the construction of lattice gauge theories will be in the determination of such aperiodic variables. 3. SU(2) gauge theory 3.1. Cont inuum theory As the easiest formulation of  a lattice gauge Hamiltonian is in the Ao = 0 gauge, we will, likewise, discuss the continuum theory in this gauge. With goa,4~ A i - 2i o~Ti  Ei- 2ig' Bi : F j k  = ~jAk - ~kAi + [A / ,Ak]  , lEft(x), A]~0,) ] = - i 8 i ] 6 ~ 8  (x - y )  , (3.1) the Hamiltonian is This Hamiltonian commutes with the generators of  time-independent gauge transfor- mations f d r  T r E "  D A .  (3.3) t[A] ~ g  A is a position dependent matrix and D is the covariant derivative D i A  = OiA + [Ai, A] . (3.4) Integration by parts yields = - g / ' d r  Tr m o .  E + g T r / ' d S "  E A .  (3.5) t[A] d d M. Bander / Multiple vacua 215 In order to recover the usual equations of  motion we require D • E = 0 as a condi- tion on the states. The effect of  the boundary term is open. In fact it is the specifica- tion of  the behavior of  states under gauge transformations on large surfaces that specifies the various 0 sectors. Specifically, let a(r)  be a smooth function of  the radius vector with a(0)  = 0 and a(oo) = 21r and in analogy with one-dimensional translations by 21r define T = exp{t[a(r)½~" f]} , (3.6) with t defined in eq. (3.3). The 0 Hilbert spaces consist of  states defined as in eq. (1.1). In a field representation this space is made up of  functionals if0 [Ai] with ¢o [T-1Ai  T] = e2~i° gTo [Ai] • (3.7) Next, we wish to find the analogue of the operator U of eq. (2.9) which connects the different 0 spaces. Let 1 f dr [AiaiAk + , (3.8) W[A] : ~ 2  £ilk Wr 2aih iZk]  then U[A] = exp{OW[A]} ; (3.9) namely if ~0 [Ai] is periodic under the transformation of  eq. (3.7) then U[A] ~o [A] obtains the phase 2~r0 under the same transformation. Choosing a definite 0 restricts the values of  the operator  7r f dSiEfff a to n + 0, with n integral. As in the one-dimen- sional potential  problem we may insist on working in the sector of  periodic states (0 = 0) at the price of  changing the operators for the electric fields E. With the aid of  U[A ] of eq. (3.9) we transform E --)'E' = U -1 Jill EU[AI, (3.10) which gives the same results as the classical contact  transformation 0 E ~ E'  = E - -~2 B . (3.11) Summarizing; a theory with the Hamiltonian of  eq. (3.2) in a 0 sector of  Hilbert space (as defined in eq. (3.7)) is equivalent to one defined on the 0 = 0 sector with E in the Hamiltonian replaced by E '  of eq. (3.11). Note that D • E = 0 and D • E '  = 0 are equivalent as, by the Bianchi identities, D • B = 0. The new Hamiltonian may be obtained from an action o fdt dg[Ai~]A k +fliAiAk] (3.12) S = S O - ~ eij k Tr which up to surface terms is the same as 0 f d a x  T r F F  (3.13) S =So - 1-~2 216 M. Bander/Multiple vacua 3.2. Reformulation of  W[A ] All the previous discussion is standard and served as an introduction to the proce- dure for lattice theories. To follow the previous steps we will need the lattice anal- ogue of W[A]. Besides requiring that it has the correct continuum limit, we also insist that it be invariant under local lattice gauge transformations. The form of IliA] given in eq. (3.8) does not have an obvious lattice version satisfying the above criteria. In this section we shall obtain an expression for IV[A] consisting of  a volume term which is manifestly invariant under all gauge transformations and a surface term which changes appropriately under global gauge transformations. Let us separate the coordinates of  each point into its z-component and a two- dimensional vector ri, perpendicular to z. Define v-l(r±,z)= P exp j A3(h,z '  ) dz , (3.14) _ _ o o  where P denotes a path ordering. It is now a matter of  integration by parts to show that 87r2W[ A ] = !e"2 ~1 Tr fd3ri dz dz'e(z' - z) X { V - 1 @1, z' ) F3i (r±, z') V(r i, z') V-  1 (r±, z ) F3/(r±, z ) V(r±, z ) } + Tr f d z  fdr..A3(r±, z)[Ai(ri, z) - V(r±, z) Oiv -1 @1, z)l IIr±t-~= +eij Tr f d2r±{aW(r±, oo) v-l(r±, oo)Aj(r± ' oo) + V-l(r., oo) a;V(rl, °°)Ai(r±, - ~ )  + v-l(r . ,  oo) A,.(ri, oo) V(r~, oo) XAI(r±, _oo)} _ ei j Tr f d2rz dz{(V-'(r±, z) aiV(r±, z)) X (V- 1 (r±, z) O/V(rj_, z))(V- 1 (rl, z) 0 3 V(E±, z))} . (3.15) The first term, a volume integral is manifestly gauge invariant. The next two terms are surface integrals while the last, although at first glance appears to depend on all of space, can be brought to the form of a surface integral. This is obvious in that this term has the form of  the winding number of  a pure gauge field and therefore depends only on the value of  its variables on the surface. More specifically, note that V is a unitary matrix and therefore may be written as V(r) = exp [i~" p(r)] , (3.16) with 13@) regular. Evaluating the last term in eq. (3.1 5) yields a surface integral ei] Tr f dr (g -~ iv ) (g - l~ jg ) (g - l~3g)  M. Bander / Multiple vacua 217 = fdSkeki](l[3 - ¼ sin 2t3)~" (8i~ ^ 0/~). (3.17) As in the one-dimensional case we have to consider functions of "angles" rather than of  periodic variables. 3.3. Lattice theory Hamiltonian lattice gauge theories may be viewed as a collection of  interacting symmetric tops defined on the links of  the lattice [4]. At each lattice point r we associate a set of  right-handed coordinate axes and denote the respective links by (r, i). On each link we define the matrix u(r, i) = exp aAi(r) ,  (3.18) with a denoting the lattice constant and Ai(r) defined in eq. (3.1). Further, on each link we define two sets of  angular momenta La(r, 8 and Ea(r, 8 with the following commutation relations [L~(r, 8, u(r',i)] = 6r,r'6ij~o~u(r, 8 ,  [[,~(r, 8 ,  u(r',i)] = 6r,r'6i~[u(r, 8 ½o~. (3.19) The L's and/~ 's  have the usual angular momentum commutation relations among themselves and commute with each other. The Hamiltonian of  the theory is 2 H = ~ g - L 2 ( r , i ) +  V r,i 2a V = -  ~ 1 Xr[u(r,r)u(r +i,])u_~(r + l, Su_ l ( r ,h]  (3.20) I r,i~j ag- The generator of  gauge transformation which commutes with the Hamiltonian is t[A(r)] = ~ A( r ) .  [L(r,i) - L-(r, 81 . (3.21) r,i In the above it should be understood that some of  the link variables emanating from points on the surface of  the lattice do not exist. In the interior of the lattice we again choose Gauss' law as a condition on the states ~. [L~(r, 8 - £~(r, i)l = 0 .  (3.22) I As in the continuum theory we choose the A(r)'s on the surface to be rotations by 27r with the isotopic axis of rotation normal to the lattice surface. The L's  and/T's along links connected to surface points and pointing along the rotation axes may be diagonalized and the eigenstates of  H will be invariant under these rotations. This corresponds to a 0 = 0 sector. 218 M. Bander / Multiple vacua In order to construct theories corresponding to other 0 sectors we need the lat- tice analogue of  W[A ] of  eq. (3.8). We require that it be invariant under all gauge transformations restricted to interior points of  the lattice. The W[A] in the form of eq. (3.8) does not inspire an immediate lattice version. However, the form of eq. (3.15) does. We define Fi] = o~½ Tr[o~u(r, i~ u(r, i , j)  u-a (r + ], i) u - l  (r,])] , VL(r,z) = ~ I  u(r±,z';[~) = exp[ ie"  ]~(r)] . (3.23) 2 < Z  The transcription of  the volume terms in eq. (3.15) is immediate and manifestly gauge invariant. As the surface terms are not gauge invariant anyway, any lattice version with a correct continuum limit will do. Define B~(r, i) and/~a(r,/') as the link magnetic fields: i 1 _ [ E ~ ( r ,  i), 14/] = - ~ B~(r, ~ .  (3.24) The explicit forms of  the B's and B ' s  are extremely complicated and related in a very non-local way to the link variables u(r, ~. They do, however, have two nice properties. They transform under gauge rotations as their corresponding L's or L ' s  and they satisfy a lattice version of the Bianchi identities ..~ IBm(r,/) - B~(r,/')l = 0 .  (3.25) I Thus, the 0 theory is constructed by replacing L '  = L + (0/47r 2) B for L in the Ha- miltonian, eq. (3.20), while maintaining that the states are invariant under all trans- formations generated by the t[A] of eq. (3.21) with the original angular momentum variables and not the transformed ones. As both the old and transformed angular momenta have the same spectra, namelyj ( /+  1), the question as to which 0 sector we are in is determined by which operators we use to define the gauge transforma- tions. Due to the identities of  eq. (3.25) this is irrelevant for the interior points of  the lattice but makes all the difference on the surface. Finally, it might betempting to introduce the 0 theories on a lattice by adding a lattice version of  Tr FF, using only local forms o f F ,  as in eq. (3.21), to a lattice Lagrangian. This would have the correct continuum form, but for any finite lattice constant this term would not be a time derivative, or even a lattice finite time differ- ence. We have no answer to the question whether the continuum limits o f  the two quantum theories are the same. As a last point we mention that, as in eq. (2.12), we may integrate over 0 in the evolution kernel and obtain a theory with no periodicity at all. M. Bander / Multiple vacua 219 Discussions on 0 vacua with Dr. B. Hasslacher, Dr. P. Raroond and Dr. G. Ross are grateful ly acknowledged.  References  [1] A. Belavin, A. Polyakov, A. Schwartz and Y. Tyupkin, Phys. Lett. 59B (1975) 85; A.M. Polyakov, Nucl. Phys. B121 (1977) 429. [2] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8; R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; C. Callan, R. Dashen and D. Gross; Phys. Lett. 63B (1976) 334. [3] K. Wilson, Phys. Rev. D10 (1974) 2445. [4] J. Kogut and L. Susskind, Phys. Rev. D l l  (1975) 395. [5] H. Quinn and M. Weinstein, Phys. Rev. D17 (1978) 1063. [6] H. Goldstein, Classical mechanics (Addison-Wesley, Cambridge, Mass., 1950) p. 241. 

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Multiple vacua for non-abelian lattice gauge theories

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Multiple vacua for non-abelian lattice gauge theories

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