Gauge Equivalence Classes Of Connections Research Articles

Towards non-perturbative quantization and the mass gap problem for the Yang–Mills field

In this paper, we reduce the problem of quantization of the Yang–Mills field Hamiltonian to a problem for defining a probability measure on an infinite-dimensional space of gauge equivalence classes of connections on [Formula: see text]. We suggest a formally self-adjoint expression for the quantized Yang–Mills Hamiltonian as an operator on the corresponding Lebesgue [Formula: see text]-space. In the case when the Yang–Mills field is associated to the abelian group [Formula: see text], we define the probability measure which depends on two real parameters [Formula: see text] and [Formula: see text]. This yields a non-standard quantization of the Hamiltonian of the electromagnetic field, and the associated probability measure is Gaussian. The corresponding quantized Hamiltonian is a self-adjoint operator in a Fock space the spectrum of which is [Formula: see text], i.e. it has a gap.

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

Algebro-geometric analogues of Donaldson's polynomials

Donaldson [5] has introduced polynomials on the second homology of any simply connected closed four-manifold, M, with b~odd (at least 3) which, up to + , are invariant under orientation preserving diffeomorphisms. For every c > Ibm+ there is a polynomial 7c(M)~ Syma(~ H2(S), where d(c) = 4c Z2(b~ + 1); to define it one considers J'/c, the set (which is in a natural way a manifold) of Gauge equivalence classes of connections on the SU(2)-bundle on M with c2 = c, antiself-dual with respect to a generic metric on M. There is a natural map /~: H2(M) ~ H2(j/c), namely slant product with 88 where P is the universal SO(3) principal bundle on M x J/c. One way of defining 7c(M) [10] is to construct a compactification of .//r Xc, carrying a fundamental class [Xc]. One proves that # extends to a map #: H2(M) ~ H2(Xc), then one defines