Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

Abstract

We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schrödinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.-1 -There are a number of ways to try and sidestep this problem. One is to solve for the time evolution of the fields, or correlators of the fields, through some effective, systematically truncated, evolution equations. Good examples include the classical statistical approximation, where an ensemble mimicking the quantum initial density matrix are evolved using straightforward classical equations of motion. In cases where the occupation numbers are large, and the system is far from thermal equilibrium, this is a powerful and easily implementable approximation (see for instance [3][4][5][6][7]). Truncated Schwinger-Dyson equations for two-or higher-point correlators have been used to good effects for the approach to equilibrium, as well as in cosmological applications involving scalar and fermion fields (see [8][9][10][11]). Stochastic equations have also proven a valuable tool for concrete cases, that allow for a mapping of the full dynamics to such a description in a systematic way (see for instance [12,13]).In some cases, however, one must return to the direct computation of the path integral (1.2), but standard Monte-Carlo techniques, routinely and effectively applied to euclidean systems in QCD, fall short. A number of ways to try and resolve this sign problem have been proposed. Stochastic quantization is particularly promising, providing a Langevin type equation to sample the field space [14-16]. For a real-time path integral, the procedure leads to complex Langevin equations, driving the field φ away from the real axis sampling the entire complex plane. This effective doubling of the degrees of freedom helps to significantly improve convergence and one may show that under sensible assumptions, the correct physical result is achieved .Another remedy for the sign problem is to again complexify the real integration variables φ, but rather than doubling the dimensionality of field space R n → C n , we constrain the field variables to live on a...

A novel method to evaluate real-time path integral for scalar $ϕ^4$ theory

“…The approach, however, suffers from the notorious sign problem. To overcome the problem, the complex Langevin method [8,9] and the Lefshetz thimble method [10][11][12] are proposed.…”

Quantum cosmology of the flat universe via closed real-time path integral

“…In addition, for the Euclidean path integral quantum gravity, there is also the divergence problem as the Euclidean Einstein Hilbert action has no the lower bound [18]. Therefore, the Lorentz path integral attracted more and more attentions, see [19][20][21][22][23][24][25][26][27][28][29][30][31] and the references there in. For the case of the nonequilibrium quantum dynamics, the "trajectory" of the time variable forms a closed contour [32][33][34][35].…”

“…In fact, the WKB approximation and the steepest descent contour approximation in the path integral can be viewed as a simplified version of the Picard-Lefschetz theory, which we just consider the classical evolution trajectory of the system as the saddle point [14]. Recently, the Picard-Lefschetz theory attracts interests in theoretical physics [19][20][21][22][23][24][25][26][27][28][29][30][31][50][51][52][53][54][55]. In quantum cosmology, some researchers use this theory to study the tunneling of the universe, but some results remain controversial [50][51][52][53][54][55].…”

Parametrized path approach to vacuum decay

“…More recently, attempts have been made at developing a formalism using only real time coordinates in analogy with what can be done in non-relativistic Quantum Mechanics [24,25] or with numerical techniques used in cold atom like the Truncated Wigner Approximation [26,27]. It was also proposed that instantons may be generalized using Picard-Lefschetz thimbles [28][29][30][31] or other functional techniques [32,33].…”