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: