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Gauge Theory and Gravity in the Loop Formulation

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Gauge Theory and Gravity in the Loop Formulation 6 Cb 3 Li I } ` VO Center for Gravitational Physics and Geometry The Pennsylvania State University University Pant, PA 16802-6300 ,. ,_ 3 (F V9 ;m8eio uyemusrpiroo etmrm CGPG-94/ 1-2 Gauge Theory and Gravity in the Loop Formulation Renate Lcll cam Lieeemi-;s. cassava Center fg; Ggavitatignal and Geometry, Pennsylvania State University, F‘@B@El 125 University Psi-k, PA 16802-62:00, USA 1 Introduction This article introduces the basic notions and mathematical structures of the so-called loop approach to gauge theory and gravity. The two theories will be treated in parallel as far as this is possible. Emphasis will be put on those aspects of the loop formulation that make it conceptually different from other, local formulations. This contribution is meant to be complementary to other articles in this volume, but some overlap has been unavoidable to ensure a coherent presentation. Many more mathematical details and references for those who want to learn more about loops are contained in a related review (Loll 1993d). Another useful reference focussing on the gravitational application is the review article by Pullin (1993). The- following Section 2 summarizes the classical and quantum description of both gauge theory and gravity as theories on connection space, to set the stage for Sections 3-5 which deal with specific aspects of the loop formulation. In Section 3, I define paths, loops and holonomies, and briefly review past applications in physics. Section 4 is devoted to the iden tities satisfied by the holonomies and their traces, the latter also known as the Mandelstam constraints. It is shown how a group structure arises on the space of loops. Section 5 discusses the equivalence between the connection and the loop formulation which lies at the heart of the loop approach, and describes the content of the underlying reconstruction theorems. ln Section 6, I introduce some ideas from lattice gauge theory which have proved useful in a regularized loop approach and enabled us to do certain calculations explicitly. To round off the discussion, Section 7 describes some features of quantized loop theories, and the final outlook contains a list of mathematical and physical inputs for the loop formulation which still require a more complete understanding. To appear in the Proceedings of the Workshop on Classical and Quantum General Relativity, Bad Honnef, Germany, 1993, Lecture Notes in Physics, ed. H. Friedrich, Springer, Berlin OCR Output This section summarizes the Lagrangian and Hamiltonian formulation of both Yang-Mills gauge theory and gravity, defined on a four—dimensional manifold M = E3 >< IR of Lorentzian signature. The gravitational theory will be described in the Ashtekar form, in which it most closely resembles a gauge theory. We will consider only the pure theories, without any matter coupling. The classical actions defining these theories are (up to overall constants) SyM[4A] = — / d4x Tr FM,,F"" : —Tr(F/\/ M =•=F) (YM1) and .S'GR[4A,e] = / d4x eeQ’e§F,,,,=eI_;KLU `/I M ef /\ eJ /\ FKL (GRI) for Yang-Mills theory and general relativity respectively. ln (YM1), the basic variable A is a g—valued connection one-form on M, where G denotes the compact and semi-simple Lie structure group of the Yang—Mills theory (typically G : S U (N )), and g its Lie algebra. In (GR1), which is a first—order form of the gravitational action, A is a self-dual, s0(3,1)C valued connection one—form, and e a vierbein, defining an isomorphism of vector spaces between the tangent space of M and the fixed internal space with the Minkowski metric nu. The connection A is selfdual in the internal space,N AT:MN‘] —%eUA£, with the totally antisymmetric 6-tensor. As usual, F denotes in both cases the curvature associated with the four—dimensional connection 4A, Favor) = 3uAv($) ' 3¤Au(”#`) 'l' lAu($)» Av($)l· Note that the action (GR1) has the unusual feature of being complex. I will not explain here why this still leads to a well-defined theory equivalent to the standard formulation of general relativity. Details may be found in the overview articles by Ashtekar (1991, 1993). Our main interest will be the Hamiltonian formulations associated with (YMI) and (GR1). Assuming the underlying principal fibre bundles P(E,G) is trivial, we can iden tify the relevant phase spaces with cotangent bundles over the affine spaces A of g—valued OCR Output connections on the three-dimensional manifold E. (A connection space A is affine since only the difference, but not the sum of two of its elements lies again in A; hence it is "not quite a vector space”). We assume the spatial manifold E to be orientable and compact, in order not to have to worry about boundary terms. The cotangent bundle T*A is coordinatized by canonically conjugate pairs (A, E), where A is a g-valued, pseudo—tensorial one-form ("gauge potentia1") and E a g-valued vector density ("genera]ized electric field") on E. We have G = S U (N ), say, for gauge theory, and G = SO(3)qg for gravity. The standard symplectic structure on T*A is expressed by the fundamental Poisson bracket relations {A;<x>.i5§<y>} = 636:6% — yi cw ln (2.2), a and b are three-dimensional spatial indices and i and j internal gauge algebra indices. However, points in T*A do not yet correspond to physical configurations, because both theories possess gauge symmetries, which are related to invariances of the original Lagrangians under certain transformations on the fundamental variables. At the Hamiltonian level, this manifests itself in the existence of so-called first-class constraints. They are sets C of functions C on phase space, which are required to vanish, C = 0, for physical configurations and at the same time generate transformation between physically indistinguishable configurations (for more details, see the article by Wipf in this volume). In our case, the first—class constraints are CyM = {D,,E‘”} (YM2) and cm, Z {parse, 0,, E F,,*E;,>j’=* 0 E J1¤,,,,EgE,;}. (em) The physical interpretation of these constraints is as follows: the so·called Gauss law con straint D,,E‘" = 0 restricts to configurations whose covariant divergence vanishes, and at the same time generates local gauge transformations in the internal space. The well—known Yang Mills transformation law corresponding to a finite group element g E G, the set of G—valued functions on E, is OCR Output (A. F) ~»"1`1`1 (gAy + ydy»yEy) (2-3) The second set, {Ca}, of constraints in (GR2) generates three-dimensional diffeomorphisms of E and the last expression, the so—called scalar or Hamiltonian constraint C, generates phase space transformations that are interpreted as corresponding to the time evolution of E in M in a space—time picture. Comparing (YM2) and (GR2), we see that in the Hamiltonian formulation, pure gravity may be interpreted as a Yang-Mills theory with gauge group G = SO(3)q·;, subject to four additional constraints in each point of E. However, the dynamics of the two theories is different; we have HYM(A»E) = / d‘¢ (EZE? + B?BZ) (YM3) as the Hamiltonian for Yang-Mills theory (where for convenience we. have introduced the generalized magnetic field B? := —%e°Fbc,).“° Note the explicit appearance of a Rieman nian background metric g on E, to contract indices and ensure the integrand in (YM3) is a density. On the other hand, no such additional background structure is necessary to make the gravitational Hamiltonian well-defined. The Hamiltonian HGH for gravity is a sum of a subset of the constraints (GR2), 11GR(A,E) zi [ da (N¤F,,,,iE$- §,¢y€F,.kE3E§),ijk (ons) and therefore vanishes on physical configurations. and N “ in (GR3) are Lagrange mul tipliers, the so-called lapse and shift functions.) This latter feature is peculiar to generally covariant theories, whose gauge group contains the diffeomorphism group of the underlying manifold. A further difference is the need for a set of reality conditions for the gravitational theory, because we are using the complex coordinate A on phase space. The space of physical configurations of Yang-Mills theory is the quotient space A/G of gauge potentials modulo local gauge transformations. It is non-linear and has non-trivial topology. By excluding the reducible connections from A (roughly speaking, these are the connections with some additional symmetry properties), and slightly restricting the gauge group Q, this quotient can be given the structure of an infinite·dimensional manifold for compact and semi-simple G (Arms et al. 1981). The corresponding physical phase space is then its cotangent bundle T* (A/G Elements of T* (A/Q) are called the classical observables of the Yang-Mills theory. OCR Output With Ashtekar’s reformulation of general relativity, a remarkable unification has been achieved in the description of the fundamental interactions of physics, which are all described as theories on spaces of connections, subject to a local gauge principle. Moreover, a similar formulation exists for a whole class of so—ca.lled topological field theories, examples of which are Chern—Simons theory and 2+1 gravity. The non—linearities of both the equations of motion and the physical configuration / phase spaces, and the presence of gauge symmetries for these theories lead to numerous difficulties in their classical and quantum description. For example, in the path integral quantization of Yang—Mills theory, a gauge-fixing term has to be introduced in the "snm over all configura tions” / [d]A{SYM , A (2.4) to ensure that the integration is only taken over one member 4A of each gauge equivalence class (see Cheng and Li (1984) for an illustration of this phenomenon).
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