It is an old dream of theoretical physicists to base the description of Yang-Mills (gauge) theories on the so-called Wilson-Loop observables. These are simply (products of) traces of holonomies around closed loops in the given spacetime. Also general relativity, when formulated as a dynamical theory of connections, can be explored via Wilson loop variables [1]. The advantage of these observables is that they provide for an overcomplete set of coordinates for the gauge invariant information that is contained in the connection [2], also called the moduli space of (spacetime) connections modulo gauge transformations, A/G. There are two major disadvantages : 1) The space A/G is nonlinear. Therefore, all the mathematical physics techniques that have been developed for field theories whose underlying space of fields is linear are not available. A solution out of this is to fix a gauge and to work with Schwinger functions of connections in that gauge but that comes at a price : manifest gauge invariance is lost and we have the problem of the annoying Gribov copies. Also from a geometrical viewpoint it just does not seem to be right to enforce linearity by brute force. 2) If one keeps manifest gauge invariance, and the only way to do this as far as we know is to work with Wilson-Loop functionals1, then the the connection is smeared with a loop function. There is then an immediate problem : it is well-known that linear quantum fields are rather distributional and need to be smeared in all spacetime directions, however, a loop only smears in one such direction. That either means that we have to give up this approach or that YM quantum fields are simply better behaved in the precise sense that the (vacuum expectation value of the) Wilson observables exists. There is a chance that this is true, at least in the non-Abelian case, since the physically relevant phase of, say, QCD is not described by the Fock Hilbert space. In this article we take advantage of the existence of new integration techniques developed in [4, 5, 6] in order to set up a system of Osterwalder-Schrader (OS) axioms that are tailored to A/G [7]. Our axioms are imposed directly on the measure rather than on the associated Schwinger distributions [8] and is thus more in the fashion of [9]. This will enable us to circumvent all the problems that are connected with these earlier approaches. Our approach is as rigorous the the ones in [9] for the linear case or [8] for the YM case. Furthermore, we prove non-triviality of these axioms by showing that they have a non-trivial solution, namely we verify them for two-dimensional pure YM theory for any semi-simple compact gauge group which is known to be an integrable, finite dimensional model. The paper is organized as follows : In section 2 we review the relevant notions from calculus on A/G. In section 3 we motivate and introduce a new set of axioms tailored to quantum gauge field theory. In section 4 we derive the general form for the generating functional of the Yang-Mills measure on A/G for any compact semi-simple gauge group for the twodimensional spacetimes of the topology of the plane and the cylinder which are the ones of physical relevance. In section 5 we explicitly verify the new axioms for the model analyzed in section 4 and give the relation to the Hamiltonian approach. inspirehep.net/record/402337
Posted on: Tue, 02 Sep 2014 11:42:00 +0000
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