For larger values of p g the two tunneling events merge. Real and complex solutions in the quantum modified inverted sine-Gordon potential. The inverted potential on the real axis is shown in black, the real bounce and associated critical and turning points are shown in red, the pair of complex bions and associated turning as well as critical points are blue, and the real bion is shown in green.
Path Integrals in Quantum Mechanics : Jean Zinn-Justin :
The singular limit is shown as the dashed line. Note that the vacuum properties are governed by the real and complex bion solutions. Editors' Suggestion. Abstract Authors Article Text — Introduction. Abstract Authors Article Text Introduction.
Abstract In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semiclassical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex saddle points, even when the parameters in the action are real. Research Areas. Quantum field theory. Classical solutions in field theory Path-integral methods. Issue Vol. Authorization Required. Log In.
Gaussian path integrals, like finite dimensional Gaussian integrals, can be calculated explicitly. Generalizing to the interacting case the definition given for the free particle in the section "Discussion" , correlation functions are the moments of the measure associated to the integrand in the path integral. Each term in the series can then be evaluated using Wick's theorem and the explicit form of the Gaussian two-point function.
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For larger values of the expansion parameter, series summation methods are required. Following Feynman Feynman , quantum time-evolution here we refer to real physical time can be described in terms of oscillatory path integrals. The formulation of quantum mechanics in terms of path integrals actually explains why equations of motion in classical mechanics can be derived from a variational principle. The leading order contribution is then obtained by expanding the path around the classical path, keeping only the quadratic term in the deviation and performing the corresponding Gaussian integration.
From the mathematical point of view, it is much more difficult to define rigorously the real-time path integral than the imaginary-time statistical path integral. A possible strategy involves, when applicable, to calculate physical observables for imaginary time and then to proceed by analytic continuation. The purpose of this section is to illustrate with a simple example the evaluation of statistical or imaginary time path integrals in the semi-classical approximation.
Path integral formulation
It is more technical and can be omitted in a first reading. Indeed, it can be shown that the classically forbidden barrier penetration appears, in the semi-classical limit, as formally related to classical evolution in imaginary time. Since the potential 35 is not bounded from below, it is first necessary to define the quantum Hamiltonian.
One looks for non-trivial saddle points, here non-constant solutions of the classical equations of motion derived from the Euclidean action 34 , which correspond formally to evolution in imaginary time. These solutions are called instantons. We have presented only the simplest form of path integrals, which for the point of view of quantum mechanics involve only a classical Lagrangian with the general form For more general Lagrangians or Hamiltonians, one encounters new problems in the definition of path integrals.
Correspondingly, the naive continuum form of the path integral is not defined because the continuum limit depends explicitly on the time-discretized form of the path integral and leads to a one-parameter family of different theories. The underlying quantum Hamiltonian is then uniquely determined by demanding either its hermiticity or equivalently its gauge invariance.
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This regularization does not violate gauge invariance but violates hermiticity of the Hamiltonian. In the general case, the interpretation of this path integral reflects the problems of quantizing classical Hamiltonians and the order of operators in products. The Hamiltonian path integral has mainly a heuristic value except in the semi-classical limit.
Up to now, we have described the path integral formalism relevant for distinct quantum particles.
But quantum particles are either bosons , obeying the Bose-Einstein statistics or fermions , governed by Fermi-Dirac statistics. To describe the quantum evolution of several identical and thus indiscernible quantum particles, the path integral formulation has to be generalized. In the case of bosons, it is based on the coherent states holomorphic formalism and the Hilbert space of analytic entire functions. The path integral approach has proved extremely useful for the understanding of the most complex problems in quantum field theory, cosmology, and condensed matter physics.
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The book deals with systems that have an infinite number of degrees of freedom. It discusses the general physical background and concepts of the path integral approach used, followed by a detailed presentation of the most typical and important applications as well as problems with either their solutions or hints how to solve them. Each chapter is self-contained and can be considered as an independent textbook. It provides a comprehensive, detailed, and systematic account of the subject suitable for both students and experienced researchers.
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