Zeta-function method in field theory at finite temperature

Engineering-Physics Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 3, pp. 444–457, March, 1989.     

Boundary conditions and confinement in the gauge field theory at finite temperature

A Hamiltonian formulation of a non-Abelian gauge theory confined in a finite domain is constructed in a generalized three-dimensional Fock-Schwinger gauge in the presence of surface terms. The dependence of the partition function on the boundary value of the longitudinal electric-field component, which because of the Gauss law, coincides with the electric-field flow through an infinitesimal boundary-surface element in this gauge, is investigated. This dependence is related to the confinement—deconfinement phase transition. In the confinement phase, the chromoelectric current through any boundary element is zero, because all observable quantities are singlet w.r.t. the remaining gauge-transformation group, i.e., color objects are unobservable at spatial infinity. In addition to the non-Abelian theory, a simpler example of quantum electrodynamics with an external-charge density in a spherical domain is considered in which the effective partition function is exactly calculable. With a feeling of deep sadness and irrecoverable loss, I had to complete this paper on my own. In N. A. Sveshnikov, theoretical physics has lost an extremely talented scientist, a great educator, and a very nice person. No matter how big an influence Nikita Alexeevich's ideas had on his students, which I am privileged to be one of, and on colleagues who knew him well, it would be only fair to say that future generations will judge their true merit. This paper is a logical conclusion to our years of joint research on the confinement problem. Although, in my view, it sheds light on one of the most difficult problems in elementary particle physics, it is really only a small step forward. It was Sveshnikov's gift to see the hidden mathematical beauty of the physical world, the beauty that expresses the essence of all things. Deceased Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 206–220, November, 1998.     

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naïve category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.     

Finite rank perturbations and distribution theory

Perturbations of a selfadjoint operator by symmetric finite rank operators from to are studied. The finite dimensional family of selfadjoint extensions determined by is given explicitly.

     

Initial conditions in quasi-classical field theory

We investigate the problem of divergences and renormalizations in the Hamiltonian formalism of quasiclassical field theory. This approach is known to involve divergences in the leading term of the expansion. Proposals have been made to eliminate the divergences by using nonequivalent representations of the canonical commutation relations at different moments of time. In this paper, we consider the Schrödinger equation with ultraviolet and infrared cutoffs. In order to remove the cutoffs, conditions are imposed on the initial state of the regularized theory in addition to the conditions imposed on the counterterms in the Hamiltonian. In the leading order of the quasi-classical expansion, we give the explicit form of these conditions, which is invariant under the evolution. This allows us to show that this approximation does not require the introduction of nonunitary evolution transformations.     

Quantum field theory meets Hopf algebra

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S ‐matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and leads to a correspondence between Feynman diagrams and semi‐standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S (V) to V. In many cases, noncommutative analogues are derived (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Newton's second law in field theory

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic k-vector field, analogous to the ordinary geodesic field and which describes the inertial motions (i.e., evolution in the absence of forces). From this generalized Newton's law, the corresponding Hamilton's canonical equations of field theory (Hamilton-De Donder-Weyl equations) are obtained by a simple procedure. It is shown that solutions of generalized Newton's equation also hold the canonical equations. However, unlike the ordinary case, Newton equations determined by different forces can define equal Hamilton's equations.