Abstract
We analyze vacuum tunneling in quantum field theory in a general formalism by using the Wigner representation. In the standard instanton formalism, one usually approximates the initial false vacuum state by an eigenstate of the field operator, imposes Dirichlet boundary conditions on the initial field value, and evolves in imaginary time. This approach does not have an obvious physical interpretation. However, an alternative approach does have a physical interpretation: in quantum field theory, tunneling can happen via classical dynamics, seeded by initial quantum fluctuations in both the field and its momentum conjugate, which was recently implemented in Ref. [1]. We show that the Wigner representation is a useful framework to calculate and understand the relationship between these two approaches. We find there are two, related, saddle point approximations for the path integral of the tunneling process: one corresponds to the instanton solution in imaginary time and the other one corresponds to classical dynamics from initial quantum fluctuations in real time. The classical approximation for the dynamics of the latter process is justified only in a system with many degrees of freedom, as can appear in field theory due to high occupancy of nucleated bubbles, while it is not justified in single particle quantum mechanics, as we explain. We mention possible applications of the real time formalism, including tunneling when the instanton vanishes, or when the imaginary time contour deformation is not possible, which may occur in cosmological settings.
Highlights
An alternative approach does have a physical interpretation: in quantum field theory, tunneling can happen via classical dynamics, seeded by initial quantum fluctuations in both the field and its momentum conjugate, which was recently implemented in Braden et al [arXiv:1806.06069]
We find there are two, related, saddle point approximations for the path integral of the tunneling process: one corresponds to the instanton solution in imaginary time and the other one corresponds to classical dynamics from initial quantum fluctuations in real time
The subject of quantum mechanical tunneling is an essential topic in modern physics, with a range of applications, including nuclear fusion [1], diodes [2], atomic physics [3], quantum field theory [4], and cosmological inflation [5]
Summary
The subject of quantum mechanical tunneling is an essential topic in modern physics, with a range of applications, including nuclear fusion [1], diodes [2], atomic physics [3], quantum field theory [4], and cosmological inflation [5]. In the context of a possible landscape of classically stable vacua in field theory, motivated by considerations in string theory [6], it is essential to determine the quantum tunneling rate from one vacuum to the next. The most famous approximation method, which is analogous to the WentzelKramers-Brillouin approximation in nonrelativistic quantum mechanics, involves the computation of the Euclidean instanton solution from one vacuum to another [10,11]. This leads to the well-known estimate for the decay rate per unit volume Γ ∝ e−SE, where SE is the bounce action of a solution of the classical equations of motion in imaginary time.
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