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

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 Hλ ,μ = λ A ∗2A2 + μA ∗ A+ iλA ∗ A ∗ +A A, where i2 =−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 Hλ ,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 −tHλ ,μ of this theory is of interest on its own. In particular, can we express e −tHλ ,μ as a limit of “integral” operators? In this article one considerably reduces the difficulty by studying the Trotter product formula of Hλ ,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 ∗2A2 and μA ∗ A+iλA ∗ × (A ∗ + A)A. • The second objective of this work is to give a approximation of the semigroup generated by Hλ ,μ when Hλ ,μ is split in the sum of λ A ∗2A2 + μA ∗ A and iλA ∗ (A ∗ + A)A. We notethat 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 andε >0, we choose an approximation operator θε = [I −εiλA ∗ × (A ∗ + A)A]e −ε(λ A ∗2A2+μA ∗ A) and we give a connection between θε and e −εHλ ,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 Tμ = μA ∗ A + iλA ∗ (A + A ∗ )A, i.e., for limit case as λ = 0 and answers to the above question