On the physical meaning of gauge and super-gauge in general-relativistic field theories

The physical meaning of gauge groups in bimetrical, Riemannian, and Hermitian theories of gravitation is discussed. In Hermitian relativity, Einstein's A-invariance means a super-gauge group which characterizes the Einstein-Schrödinger equations as the only nondegenerate general-relativistic field theory.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

On the connection between spin glasses and gauge field theories

A simple connection between Ising spin glasses and the Z₂ lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.     

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems. The text was submitted by the author in English.     

On nonlinear gauge theories

In this note, we study non-linear gauge theories for principal bundles, where the structure group is replaced by a Lie groupoid. We follow the approach of Moerdijk–Mr?un and establish its relation with the existing physics literature. In particular, we derive a new formula for the gauge transformation which closely resembles and generalizes the classical formulas found in Yang Mills gauge theories.     

Instantons and monopoles in Yang-Mills gauge field theories

This is a survey article on instantons and monopoles and is intended for those who have no prior knowledge of Yang-Mills gauge field theories. With minimal amount of physical motivation and mathematical apparatus, the basic field equations and their solutions, wherever known, are presented. Particular emphasis is put on those problems which are as yet unsolved.     

Non-abelian gauge theories in a background field

Green functions of on-mass-shell transverse gauge fields in a background gauge are proved to be obtained from Green functions in the perturbative vacuum by a mere translation.     

On gauge theories of mass

The classical Einstein–Hilbert action in general relativity extends naturally to a blow-up (in the sense of algebraic geometry) of the usual space of pseudo-Riemannian metrics; this presents the metric tensor _gik$g_{ik}$ as a kind of Goldstone boson associated to the real scalar field defined by its determinant. This seems to be quite compatible with the Higgs mechanism in the standard model of particle physics.     

Self-consistent normal ordering of gauge field theories

Mean field theories with a real action of unconstrained fields can be self-consistently normal ordered. This leads to a considerable improvement over standard mean-field theory. This concept is applied to lattice gauge theories. We construct first an appropriate real action mean-field theory. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean-field theory are derived.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

The energy-momentum tensor in scalar and gauge field theories

A previous study of the energy-momentum tensor in ?⁴ theory and spontaneously broken non-Abelian gauge field theories is extended here to show finiteness to all orders in perturbation theory. Divergences of Green's functions Γ_μν^(j) (q; p₁, …, p_j) involving the energy-momentum tensor θ_μν and j particle fields are removed by counterterms of the ordinary Lagrangian plus a renormalization of the coefficient of the Callan-Coleman-Jackiw improvement term in θ_μν. Physically the extra renormalization means that the mean square “mass radius” of elementary spin zero particles must be specified from experiment.     

Bianchi identities and the automatic conservation of energy-momentum and angular momentum in general-relativistic field theories

Automatic conservation of energy-momentum and angular momentum is guaranteed in a gravitational theory if, via the field equations, the conservation laws for the material currents are reduced to the contracted Bianchi identities. We first execute an irreducible decomposition of the Bianchi identities in a Riemann-Cartan space-time. Then, starting from a Riemannian space-time with or without torsion, we determine those gravitational theories which have automatic conservation: general relativity and the Einstein-Cartan-Sciama-Kibble theory, both with cosmological constant, and the nonviable pseudoscalar model. The Poincaré gauge theory of gravity, like gauge theories of internal groups, has no automatic conservation in the sense defined above. This does not lead to any difficulties in principle. Analogies to 3-dimensional continuum mechanics are stressed throughout the article.     

Renormalization group approach to lattice gauge field theories

The fluctuation field integral, constructed in Part I, is represented by the exponentiated cluster expansion. It is proved that the terms of the expansion satisfy the inductive assumptions. This completes the construction of the sequence of effective actions in the small field approximation.Work supported in part by the Air Force under Grant AFOSR-86-0229 and by the National Science Foundation under Grant DMS-86-02207     

Propagating modes in gauge field theories of gravity

The particle content of the most general quadratic field Lagrangian for Poincaré gauge field theories is examined and restrictions on the coupling constants for absence of ghosts and tachyons are derived. Our final field Lagrangian contains three coupling constants, the usual gravitational constant in front of an Einsteinian part and two other constants governing pure torsion terms.     

Renormalization group approach to lattice gauge field theories

We study four-dimensional pure gauge field theories by the renormalization group approach. The analysis is restricted to small field approximation. In this region we construct a sequence of localized effective actions by cluster expansions in one step renormalization transformations. We construct also -functions and we define a coupling constant renormalization by a recursive system of renormalization group equations.     

On zero modes and gauge fixing in light-cone quantized gauge theories

It is shown that regularistation by dimensional reduction is a viable alternative to dimensional regularisation in non-supersymmetric theories.

     

Non-perturbative scale anomaly and composite operators in gauge field theories

In non-asymptoticall free gauge theories with a non-trivial ultraviolet fixed point, scale symmetry breaking (the scale anomaly) caused by the non-perturbative PCAC dynamics is studied. In the two-loop approximation the analytical expression for the gluon condensate is obtained. It is shown that the form of the anomaly depends on the type of phase of the theory to which it relates. The hypothesis about the soft behaviour at small distances of composite operators in such theories is confirmed.     

Symmetry restoration and the background field method in gauge theories

Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a first-order phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants $\text{F}{}_1\text{=}\text{1}\text{2}\text{(B}{}^2\text{? E}{}^2\text{)}$ and F₂ = E · B are discussed. In a simple SU(2) model the critical field strength is found to be g_R²(F₁)_crit = 0.057 m_w⁴, m_w being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge, is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter α. Further applications of these methods are discussed.     

Propagators for lattice gauge theories in a background field

We prove regularity and decay properties for propagators connected with the renormalization group method in lattice gauge theories. These propagators depend on an external gauge field configuration, called a background field.Research supported in part by the National Science Foundation under Grant PHY-82-0369