Abstract
We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extension of the free theory. It is verified that the evolution equations implied by the resulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations. The modifications and requirements for the inclusion of matter are discussed in detail.
Highlights
BRST quantization is a pivotal tool in developing theories of the fundamental interactions, where the acronym BRST refers to Becchi, Rouet, Stora [1] and Tyutin [2]
The purpose of this paper is to show in the context of Yang-Mills theories how all the above facets can be handled entirely within the Hamiltonian approach, where explicit constructions on a suitable Fock space allow for a maximum of intuition
A new representation of ghost-field operators in terms of canonical creation and annihilation operators has been introduced in Eqs. (35)–(38)
Summary
BRST quantization is a pivotal tool in developing theories of the fundamental interactions, where the acronym BRST refers to Becchi, Rouet, Stora [1] and Tyutin [2]. CLASSICAL YANG-MILLS THEORY Yang-Mills theory introduces antisymmetric fields Faμν that are defined in terms of four-vector potentials Aaμ, Faμν 1⁄4 ∂μAaν − ∂νAaμ − gfabcAbμAcν; ð1Þ where the superscripts a, b, c label the generators of the underlying symmetry group and the indices μ, ν label the space-time components; the parameter g is the strength of the interaction and the set of numbers fabc are the structure constants of the underlying Lie group. − g2fabsfcdsAb · AcAd; ð20Þ together with the relation (15) These equations define classical Yang-Mills theories in a formulation that does not manifestly exhibit Lorentz or gauge symmetry. These equations imply a conservation law with source term,.
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