Analyticity properties of trilinear SL(2, C) invariant forms in elementary representation parameters

Trilinear invariant forms over spaces transforming under the so-called elementary representations of SL (2, C) (obtained from the principal series by analytic continuation in the representation parameters) are studied with regard to their analyticity properties in the representation parameters. The method is based on a natural one-one correspondence between the invariant forms and invariant separately homogeneous distributions (called kernels of the forms) in three complex two-dimensional non-zero vectors. There exists a family Ψ of kernels of forms with analytic dependence on the representation parameters (Ψ being unique up to a family of complex multiples dependent on the parameters). Also associated kernels obtained by differentiating Ψ in the parameters are studied.     

On the covariant structure of the two-point function

The spectral representation of the two-point function for arbitrary fields proposed recently [1] is rigorously proved and analyzed. The problem is treated in momentum space where the covariant structure is simpler because of the spectrum conditions. For finite-component fields the explicit matrix structure is found in coordinate space too and is applied to the definition of time-ordered Green functions for arbitrary spin. The decomposition of the two-point function into kernels of definite spin is carried out in the general case, a necessary and sufficient condition for the growth of the coefficients in this decomposition being given. The positive-definiteness condition (in the case of Hermitian conjugate fields) is fulfilled automatically by the elementary kernels.The formalism of homogeneous distributions in two dimensional complex domain [2] is used throughout the paper.On leave of absence from Joint Institute for Nuclear Research, Dubna, USSR and from Physical Institute of the Bulgarian Academy of Sciences, Sofia, Bulgaria.     

The “free” quantum brownian particle as a non-fock linear bosonic system

We quantize the Langevin equation for a “free” Brownian particle. The corresponding linear bosonic system possesses infrared singularities and is therefore non-Fock. We construct the physical representations of fields using the generalized stationary states. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116. No. 2, pp. 201–214. August, 1998.     

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR ⁿ , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR ⁿ . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.     

Quantization of stationary Gaussian random processes and their generalizations

We consider quantization of stationary Gaussian random processes whose physical counterparts are states of open systems in equilibrium with the environment. For this, we propose a formalism and its physical interpretation in accordance with the concept of Hamiltonian modeling. The method is universal and includes the known models as particular cases. We also consider extending the method applicability domain to linear systems with infrared singularities of two-point functions. In particular, fractal Brownian motions constitute a family of reference models in this class.