Abstract
We present the calculation of the Feynman path integral in real time for tunneling in quantum mechanics and field theory, including the first quantum corrections. For this purpose, we use the well-known fact that Euclidean saddle points in terms of real fields can be analytically continued to complex saddles of the action in Minkowski space. We also use Picard-Lefschetz theory in order to determine the middle-dimensional steepest- descent surface in the complex field space, constructed from Lefschetz thimbles, on which the path integral is to be performed. As an alternative to extracting the decay rate from the imaginary part of the ground-state energy of the false vacuum, we use the optical theorem in order to derive it from the real-time amplitude for forward scattering. While this amplitude may in principle be obtained by analytic continuation of its Euclidean counterpart, we work out in detail how it can be computed to one-loop order at the level of the path integral, i.e. evaluating the Gaußian integrals of fluctuations about the relevant complex saddle points. To that effect, we show how the eigenvalues and eigenfunctions on a thimble can be obtained by analytic continuation of the Euclidean eigensystem, and we determine the path-integral measure on thimbles. This way, using real-time methods, we recover the one-loop result by Callan and Coleman for the decay rate. We finally demonstrate our real-time methods explicitly, including the construction of the eigensystem of the complex saddle, on the archetypical example of tunneling in a quasi-degenerate quartic potential.
Highlights
Tunneling is one of the signature phenomena of quantum theory
One motivation is the use of the real-time amplitude in an optical theorem for tunneling, that leads us to the decay rate through a route that is alternative to calculating the imaginary part of the ground-state energy at the false vacuum, or solving for wave functions in the WKB approximation
The interpretation of the real-time false-vacuum to false-vacuum amplitude in terms of the optical theorem provides a new relation between the functional techniques of calculating decay rates to real-time dynamics
Summary
Tunneling is one of the signature phenomena of quantum theory. The most prominent example realized in nuclear physics is alpha decay, but there are important technical applications such as the tunneling microscope. In appendix A, the Gaußian approximation to the Euclidean path integral using Picard-Lefschetz theory is reviewed, and in appendix B, we summarize various methods of calculating the one-loop functional determinant These results can be compared with the decay rate inferred in appendix C using the WKB approximation from the imaginary part of the zero-point energy of the false vacuum, or, more directly, from the probability current that flows toward the global ground state. This way, we provide a comprehensive survey of the computation of the first quantum corrections to tunneling, to which we can relate our results from the functional approach in real time.
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