Jan Rzewuski

On entire functionals in quantum field theory

Abstract The paper contains an investigation of convergence and asymptotic behaviour of the generating functionals on mass shell in the Quantum Field Theory of one real, scalar, selfinteracting field. Also the relations between the Hilbert space fo entire functionals and the various spaces of exponentially bounded entire functionals are treated in some detail. There result various estimates for the functional series and double series representing vectors, operators and their functional derivatives.

On generating functionals for antisymmetric functions and their application in quantum field theory

An isomorphism between certain subspaces of the Hilbert spaces of symmetric and antisymmetric n -point functions (or, more generally, symmetric and antisymmetric tensor products of a Hilbert space) is described. It permits a construction of generating functionals for sets of antisymmetric functions. In this way the theory of Hilbert spaces of functional power series as described in [7] and [8] can be extended to the case of antisymmetric coefficients. As an application, the functional representation for the anticommuntation relations is derived. It enables to obtain a functional formulation of quantum field theory also in the antisymmetric case without the use of Grassman algebras.

Hilbert spaces of functional power series

The paper concerns linear spaces of functional power series over arbitrary linear spaces with a given bilinear form and involution. The domain of convergence, growth and type or convergence radius resp. of functional power series belonging to certain Hilbert spaces are estimated. Sufficient conditions (concerning growth and type) for a functional to belong to a given Hilbert space are determined. The connection of Hilbert spaces of functional power series and Fock spaces is discussed. The paper contains estimates for functional power series and their derivatives in certain multiplets consisting of topological spaces invariant with respect to differentiation on certain domains. The results are applied to quantum field theory, yielding an estimation of the convergence domain and asymptotic behaviour of the generating functionals for the S -matrix on and off mass shell, for the field and for the derivatives of these quantities with respect to elements of the basic space.

Some geometrical consequences of physical symmetries

Invariant submanifolds of the linear representation space C4m of the physical symmetry group SU(2,2)×SU(m) and its subgroup P×SU(m) are studied in some detail. It is shown that there exists only one such manifold admitting unique projection onto Minkowski space. The structure of this manifold is investigated by using proper local coordinate systems.

On the projections of representation spaces of the symmetry group on the Minkowski space

In this paper we generalize the projection of the representation space of the symmetry group SU(2,2)×U(2) on the Minkowski space to arbitrary internal symmetries U(m). The procedure involves certain restrictions on the coordinates of the representation space. Representations of the symmetry group in the restricted space and in the corresponding restricted Hilbert space are constructed.

On some geometrical consequences of physical symmetries

It is shown that there exists only one submanifold O(4,m)2 of the representation space C4m of the group GL(4,C)×GL(m,C) which admits a unique projection onto Minkowski space, consistent with the group. We describe the decomposition of this manifold O4,m)2 when the group is restricted to the physical symmetry group SU (2,2)× ×SU(m) or P×SU(m). We consider also representations of SU(2,2)×SU(m) in the resulting submanifolds and in the Hilbert space of functions over these manifolds.

Structure of Matrix Manifolds and a Particle Model

The decomposition of matrix manifolds into homogeneous spaces of certain groups is studied in some detail. The results are applied to the derivation of the internal structure of SU(2,2)×SU(m)‐ and P4×SU(m)‐invariant particle models where the first (second) factor in the direct product represents external (internal) symmetry.

Matrix manifolds and symmetry breaking

We investigate systematically the process of decomposition of the representation spaceCN, N = ∏i=1m ni, of the group GL(N, C) into invariant submanifolds when the symmetry is reduced according to the scheme GL(N, C) → ∏i=1m GL(ni, C) → ∏i=1m SU (pi, qi), pi,+qi = ni. The submanifolds are described analytically, group-theoretically and geometrically. The general method is applied to the SU (2, 2) SU (3) SU (2) SU (1) symmetry, where SU (2, 2) represents the external conformal or Poincare symmetry and SU (3) SU (2) SU (1) represents the internal symmetry as appearing in grand unified theories.

quantum field theory as group algebra

Abstract A group theoretical approach to dynamical quantization in general, and quantum field theory in particular, is developed. This approach opens possibilities of new quantization schemes. Some of these schemes are discussed in detail. They offer certain advantages such as relaxation of the conventional principles of unitarity and causality on the one hand and the possibility to attach some meaning to the formal solutions of the equations of unitarity and causality in terms of functional integrals on the other.

Generalized Fock constructions in dual structures

Abstract Certain topological dual structures are described and applied to a generalization of Fock constructions and to the theory of functional power series.