Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space

Abstract

The Reggeon field theory is governed by a non-self adjoint operator constructed as a polynomial in A, A * , the standard Bose annihilation and creation operators. In zero transverse dimension, this Hamiltonian acting in Bargmann space is defined by H λ ′ , μ = λ ′ A * 2 A 2 + μ A * A + i λ A * ( A * + A ) A , where i 2 = − 1 , λ ′ , μ and λ are real numbers and the operators A , A * satisfy the commutation relation [ A , A * ] = I . As the quantum mechanical system described by H λ ′ , μ has a velocity-dependent potential containing powers of momentum up to the fourth, the problem of existence of Hamiltonian path integral for the evolution operator e − t H λ ′ , μ of this theory is of interest on its own. In particular, can we express e − t H λ ′ , μ as a limit of “integral” operators? In this article one considerably reduces the difficulty by studying the Trotter product formula of H λ ′ , μ to reach two objectives: • The first objective is to prove a very specific error estimate for the error in a Trotter product formula in trace-norm for H viewed as the sum of the operators λ ′ A * 2 A 2 and μ A * A + i λ A * ( A * + A ) A . • The second objective of this work is to give a approximation of the semigroup generated by H λ ′ , μ when H λ ′ , μ is split in the sum of λ ′ A * 2 A 2 + μ A * A and i λ A * ( A * + A ) A . We note that this case is entirely different. In fact, the usual Trotter product formula is not defined, because the interaction operator A * ( A * + A ) A is not the infinitesimal generator of a semigroup on Bargmann space. For λ ′ > 0 and ɛ > 0 , we choose an approximation operator θ ɛ = [ I − ɛ i λ A * ( A * + A ) A ] e − ɛ ( λ ′ A * 2 A 2 + μ A * A ) and we give a connection between θ ɛ and e − ɛ H λ ′ , μ . This choice allows us to give in [A. Intissar, Note on the path integral formulation of Reggeon field theory, preprint] a “generalized Trotter product formula” for T μ = μ A * A + i λ A * ( A + A * ) A , i.e., for limit case as λ ′ = 0 and answers to the above question.