I ~LIIIIll l~_l LIml 5"dh~!(111 g| PROCEEDINGS Nuclear Physics B (Proc. Suppl.) 30 (1993) 224-227 SUPPLEMENTS North-Holland Lattice gauge theory in terms of independent Wilson loops R. Lolla * aPhysics Department, Syracuse University Syracuse, NY 13244-1130, U.S.A. We discuss the construction of a complete set of independent Wilson loops Tr P exp ~w A that allows one to formulate the physics of pure lattice gauge theory directly on the subspace of physical configurations, and report on progress in recasting the theory in terms of these variables. 1. THE NEW LOOP APPROACH set of gauge orbits in A (G is the group of local gauge transformations), the right-hand side de- The mathematical ingredients necessary in the notes the set of complex-valued functions on loop description of our general approach consist of a d- space, subject to a set of so-called Mandelstam dimensional manifold I3d, d = 2, 3, 4, a Yang-Mills constraints. These can be seen as deriving from gauge group G C Gl(n, C), and a representation identities satisfied by the traces of n × n-matrices. R of G on a linear space Vn. A path is a continu- They are algebraic equations non-linear in T, and ous mapping 3': [0, 1] ---* I] a. If 3'(0) = 3'(1), 3' is their form depends on G and R, in particular the called a loop. For each loop 3', we define on the dimensionality n of VR. configuration space .4 of pure Yang-Mills theory Our aim is to recast the tbeory, given in terms the path-ordered exponential of the local and gauge-covariant gauge poten- tials A(x), in terms of the non-local and gauge- U,~(7) := Pexpig/A~(7(t));ru(t)dt. (1) invariant variables T(3'). One hopes that such an approach may be better suited to deal with the This is a mathematically well-defined object, non-linearities of non-abelian gauge theory and called the holonom~ of % which can be inter- allow for a more direct physical interpretation, preted geometrically as the transformation un- since gauge-invariance is automatically taken care dergone by an element of Vn after parallel- of. Also it may be more amenable to a non- transporting it along the horizontal lift of the perturbative quautization, for example, by mak- curve 3" to the corresponding fibre bundle over ing use of certain algebraic structures on loop space. It is well-known that equivalence of the What we will call the "new loop approach" is classical formulations does not necessarily imply the attempt to reformulate gauge theory solely in the equivalence of the corresponding quantum terms of the gauge-invariant traced holonomies theories. Za(7) := Tr UA(7), the so-called Wilson loops. Throughout the last 30 years, starting with the The physical motivation for this procedure is the work of Mandelstam [8], these heuristically ap- expectation that all physics in the confined phase pealing ideas have led many physicists to incor- of Yang-Mills theory is expressible in terms of porating variables based on the holonomy (1) in gauge-invariant quantities only. Mathematically, their descriptions of gauge theory (c.f. the review this is justified by the classical equivalence theo- [7]). For a number of reasons, these attempts rem (which holds locally in I] d) [4] have been largely unsuccessful: A/O ~- {T(3')}/{Mandelstam constraints}. (2) the mathematics of the infinite-dimensional loop space is not sufficiently well devel- The left-hand side of (2) gives the usual descrip- opped, and we do not have a powerfid dif- tion of the physical configuration space as the ferential calculus as we have in the case of *Supported by Deutsche Forschungsgemeinschaft finite-dimensional analysis. Only recently a 0920--5632/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved. R. Loll I Lattice gauge theory in terms of independent W~lson loops 225 few rigorous mathematical results have be- to approximating the set of all loops in ~d come available, which however often take by the set of all closed contours that can be a somewhat different viewpoint from what formed from the links of a hypercubic lattice is required in physical applications. In any N d with periodic boundary conditions. For a case, we do not yet have a precise idea which set {li,i = 1,...,n} of contiguous lattice links, part of the loop space is physically the most set T(7) = Tr Uh...Ul., where the UI, are the relevant. link holonomies of the standard Kogut-Susskind- Wilson approach. Note that even on a finite lat- • Nobody has formulated an action principle tice, both the number of loop variables T(7) one in terms of loop variables; this leaves con- can construct in this way, and the number of siderable freedom for one's preferred deriva- Mandelstam constraints (3) are infinite. tion of classical or quantum equations of An important result in this context was ob- motion. tained in [5], where it was shown that for G = SU(2) and d = 2, 3,4 one can solve the highly • Since the loop variables are non-local, stan- coupled set of lattice Mandelstam constraints (3), dard quantization methods cannot be ap- and give an explicit local description of the (finite- plied straightforwardly. There are many dimensional) physical configuration space unresolved issues concerning a meaningful regularization and renormalization proce- Cphy.., = A/Gliai.i.i<-,~ = dure in the loop formulation. : {T(7), 7 a lattice loop}/{M.c.} (4) • Although the variables T(7) do not carry in terms of a set of independent loop variables. To any gauge-dependence, they still do not describe the result, it is convenient to perform a form a set of physical variables because of linear transformation to another set of loop vari- their overcompleteness and the restrictions ables, defined by on them implied by the Mandelstam con- straints. Their existence gives rise to many L,(~,,) = ~T(T,), non-trivial features in the loop approach. L~(Ti, 1'./) - ~-(T(Ti o 7j-i) - T(Ti o1'i)). (5) For example, for the case of SU(2) gauge theory It turns out that for a complete description in the fundamental, two-dimensional representa- of Cphy s it suffices to consider variables Ll(7;), tion, the non-linear Mandelstam constraints are where 7i is a plaquette loop, and L2(71,7j), where given by (7i,7j) is a (not necessarily co-planar) pair of neighbouring plaquette loops. T(Ti)T(72) - T(71 o= 1'2) - T(1"1 o= 1"~-i) -- 0, (3) The first step in the proof consists in deriving for any pair of loops 3'1 and 1'2 intersecting at a an algorithm for expressing traced holonomies of point z (where ox denotes the loop composition composite loops, T(TI o... o 70) as functions of T- in z, and 1"-1 the inverse of 7), together with a variables depending on fewer than n elementary set of inequalities which will be given in the next loops. This still does not lead to an algebraically section. In the continuum, we do not know of independent set of lattice loop variables, but to a systematic way of solving the constraints (3) an educated guess as to what such an independent in order to reduce the number of redundant loop set may be. variables. The second step involves the computation of determinants of certain large matrices, in or- der to show that the chosen subset of Wilson 2. LATTICE RESULTS loops does indeed span the space Cphy~ locally. Some of the above-mentioned problems can For this a simple FORTRAN program with a be tackled successfully in a regularized lattice MATH/LIBRARY subroutine was used (see [5] version of the loop formulation. This amounts for more details). This second step depends on 226 R. Loll/Lattice gauge theory in terms of independent Wilson loops the dimension of the lattice because of the differ- in [6], but the Jacobian or/ so far is known only ent lattice geometries. for the one-point model in d -=- 2. In a sense the solution for the independent de- grees of freedom is surprisingly simple: it is suf- 3. Outlook ficient to look at small lattice loops (hence the formulation is "not very non-local"). For exam- Only a few numerical results have been ob- ple, in d = 3, associated with each unit cube, tained in a pure loop approach, and all of them there are six independent Wilson loops, three of in a Hamiltonian framework. The only work we type L~ and three of type L~, which is exactly the are aware of is that of Furmanski and Kolawa [2] number of physical degrees of freedom per unit in d = 3 + 1, and that of Gambini, Leal and Trias cube in three dimensions (recall the T-variables [3] and Brfigmann [1], both in d = 2 + 1 dimen- for SU(2) are real-valued). sions. These are all small-scale calculations for Another remarkable fact is that the set of inde- G -- SU(2). They have in common that quantum pendent variables {Li} in all dimensions is invari- states are labelled by lattice loops, the Hamilto- ant under lattice translations, in contrast with the nian operator/I acts by fusion and fission of loop usual gauge-covariant and maximally gauge-fixed arguments, and the spectrum is calculated in a description in terms of link holonomies. Thus, truncated loop basis. the loop approach seems qualitatively different The main problem in these approaches is how from the standard covariant formalism, in which to deal efficiently with the overcompleteness of no complete gauge-fixing exists.