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arXiv:hep-th/0001058 · AcPPB

The operator form of the effective potential governing the time evolution in n-dimensional subspace of states

Jarosław Piskorski

Published Jan 11, 2000• 0 citations
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Abstract

This paper presents the operator form of the effective potential V governing the time evolution in 2 and 3 and n dimensional subspace of states. The general formula for the n dimensional case is considered the starting point for the calculation of the explicit formulae for 2 and 3 dimensional degenerate and non-degenerate cases. We relate the 2 and 3 dimensional cases to some physical systems which are currently investigated.

DOI
10.48550/arxiv.hep-th/0001058

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This paper presents the operator form of the effective potential V governing the time evolution in 2 and 3 and n dimensional subspace of states. The general formula for the n dimensional case is considered the starting point for the calculation of the explicit formulae for 2 and 3 dimensional degenerate and non-degenerate cases. We relate the 2 and 3 dimensional cases to some physical systems which are currently investigated.
The Lee, Oehme and Yang (LOY) theory of time evolution in two state subspace of states of the complete system is discussed. Some inconsistencies in assumptions and approximations used in the standard derivation of the LOY effective Hamiltonian, H(LOY), governig this time evolution are found. Eliminating these inconsistecies and using the LOY method, approximate formulae for the effective Hamiltonian, H(eff), governing the time evolution in this subspace (improving those obtained by LOY) are derived. It is found, in contradistinction to the standard LOY result, that in the case of neutral kaons ( - ), cannot take the zero value if the total system the preserves CPT--symmetry. Within the use of the method mentioned above formulae for H(eff) acting in the three state (three dimensional) subspace of states are also found.
Approximate formulas are given for the effective Hamiltonian ${\mathit{H}}_{\mathrm{\ensuremath{\parallel}}}$(t) governing the time evolution in a subspace ${\mathit{scrH}}_{\mathrm{\ensuremath{\parallel}}}$ of the state space scrH. It is proved that this approximation is correct for any Hamiltonian H of the system under consideration at the early-time period. The approximate form of the survival amplitude for a given state improving Fleming's estimation in the short-time region is found and the properties of a decay rate for small, intermediate, and long times are discussed.
The relations are discussed among a number of different formulations of perturbation theory for an effective Hamiltonian of a general quantum system. Although the definitions and the formulations are apparently different, it is shown that the effective Hamiltonians, which have appeared in the history of time-independent approaches, can be written simply in a unified form. Expansion formulae are given explicitly for both the non-Hermitian and Hermitian effective Hamiltonians, and a discussion is given on a method for summing up the expansion series and obtaining a convergent result.
Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.
Already in the case of finite dimensional Hilbert spaces [Formula: see text] the general form of density matrices ρ is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces in terms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices ρ(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of [Formula: see text] is determined. As a special case we recover, for instance, the Lindblad form.
We present a general method for calculating Rényi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends. Technically, we consider the case when the boundary operator implementing the change of boundary conditions is degenerate under the Virasoro algebra. By exploiting the null-vectors conditions, we derive the ordinary differential equations that govern the scaling functions associated to the entropies. In particular, we provide an explicit formula for the second Rényi entropy. Additionally, we identify and compute the leading finite-size corrections to compare our theoretical results with numerical data for the Ising and three-state Potts critical chains.