On the Connection between Operators in the Heisenberg and Interaction Representations

Giancarlo Calvanese Strinati

This chapter introduces the Hamiltonian operator in the language of second quantization, which is associated with the many-particle system to be considered throughout, including its time-dependent part. An expression is derived for the corresponding time-evolution operator, which depends only on the Hamiltonian and not on the initial preparation of the system before the time-dependent part begins to act. The connection between the Schrödinger and Heisenberg representations is discussed.

Stephen P. Martin

Alexander D. Sukhanov

Chia-Yi Ju, Adam Miranowicz, Jacob L. Barnett +2 more

The equivalence between the Schr\"odinger and Heisenberg representations is a cornerstone of quantum mechanics. However, this relationship remains unclear in the non-Hermitian regime, particularly when the Hamiltonian is time dependent. In this paper, we address this gap by establishing the connection between the two representations, incorporating the metric of the Hilbert space bundle. We not only demonstrate the consistency between the Schr\"odinger and Heisenberg representations but also present a Heisenberg-like representation grounded in the generalized vielbein formalism, which provides a clear and intuitive geometric interpretation. Unlike the standard Heisenberg representation, where the metric of the Hilbert space is encoded solely in the dual states, the Heisenberg-like representation distributes the metric information between both the states and the dual states. Despite this distinction, it retains the same Heisenberg equation of motion for operators. Within this formalism, the Hamiltonian is replaced by a Hermitian counterpart, while the ``non-Hermiticity'' is transferred to the operators. Moreover, this approach extends to regimes with a dynamical metric (beyond the pseudo-Hermitian framework) and to systems governed by time-dependent Hamiltonians.

ELIZABETH A. RAUSCHER

V. A. Andreev, A. B. Klimov, P. B. Lerner

We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.