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arXiv:2507.11608

Coherent State Path Integral Reveals Unexpected Vacuum Structure in Thermal Field Theory

Rens Roosenstein, Maximilian Attems, William Alexander Horowitz

Published Jul 15, 2025• 0 citations
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Abstract

We construct the path integral formulation of the partition function for a free scalar thermal field theory using coherent states, first in the ladder operator basis and then in the field operator basis. In so doing, we provide for the first time a mapping in quantum field theory between the field-basis coherent states and ladder-basis coherent states. Using either basis, one finds terms missed in the usual path integral derivation, which we identify as the vacuum energy contribution to the partition function. We then extend the field-basis coherent state method to the interacting $ϕ^4$ theory. In addition to the vacuum energy contribution, one finds a coupling of a vacuum expectation value to the mass term that is absent in the existing literature.

DOI
10.48550/arxiv.2507.11608

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We construct the path integral formulation of the partition function for a free scalar thermal field theory using coherent states, first in the ladder operator basis and then in the field operator basis. In so doing, we provide for the first time a mapping in quantum field theory between the field-basis coherent states and ladder-basis coherent states. Using either basis, one finds terms missed in the usual path integral derivation, which we identify as the vacuum energy contribution to the partition function. We then extend the field-basis coherent state method to the interacting ϕ4ϕ^4ϕ4 theory. In addition to the vacuum energy contribution, one finds a coupling of a vacuum expectation value to the mass term that is absent in the existing literature.
The path-integral quantization of thermal scalar, vector, and spinor fields is performed newly in the coherent-state representation. In doing this, we choose the thermal electrodynamics and φ 4 theory as examples. By this quantization, correct expressions of the partition functions and the generating functionals for the quantum thermal electrodynamics and φ 4 theory are obtained in the coherent-state representation. These expressions allow us to perform analytical calculations of the partition functions and generating functionals and therefore are useful in practical applications. Especially, the perturbative expansions of the generating functionals are derived specifically by virtue of the stationary-phase method. The generating functionals formulated in the position space are re-derived from the ones given in the coherent-state representation.
The path-integral quantization of thermal scalar, vector, and spinor fields is performed newly in the coherent-state representation. In doing this, we choose the thermal electrodynamics and ψ4 theory as examples. By this quantization, correct expressions of the partition functions and the generating functionals for the quantum thermal electrodynamics and ψ4 theory are obtained in the coherent-state representation. These expressions allow us to perform analytical calculations of the partition functions and generating functionals and therefore are useful in practical applications. Especially, the perturbative expansions of the generating functionals are derived specifically by virtue of the stationary-phase method. The generating functionals formulated in the position space are re-derived from the ones given in the coherent-state representation.
The boundary conditions corresponding to the Matsubara formalism for the T>0T > 0T>0 partition function may be introduced as {\em constraints} in the path integral for the vacuum amplitude. We implement those constraints with time-independent Lagrange multipliers and, by integrating out the original fields, we obtain an alternative representation for the partition function, in terms of the Lagrange multipliers as dynamical fields. The resulting functional integral has the appealing property of involving only ddd-dimensional, {\em time independent} fields, and looks like a nonlocal version of the classical partition function. We develop this formalism within the context of the scalar and Dirac fields.
This chapter shows how to construct path integral representation using a closure relation based on the eigenstates of the creation or the annihilation operators. It examines how coherent states are used to obtain a path integral representation of the partition function as well as calculating directly the trace defining the partition function. The chapter explores quantum field theory using the Feynman functional and diagrammatic techniques as foundations to apply Quantum Field Theory to a broad range of topics in physics. The minimum uncertainty wave packets for the harmonic oscillator were published as coherent states by Schrodinger. Coherent states have played an important role in many branches of physics and, in particular, quantum field theory and quantum optics. The most important property of coherent states is their overcompleteness in the Fock space, which implies that any vector of the Fock space can be expanded in terms of coherent states.
We make a rigorous computation of the relative entropy between the vacuum state and a coherent state for a free scalar in the framework of algebraic description of quantum field theory (AQFT). We study the case of the Rindler wedge. Previous calculations including path integral methods and computations from the lattice give a result for such relative entropy which involves integrals of expectation values of the energy-momentum stress tensor along the considered region. However, the stress tensor is in general nonunique. That means that if we start with some stress tensor, then we can ``improve'' it adding a conserved term without modifying the Poincar\'e charges. On the other hand, the presence of such an improving term affects the naive expectation for the relative entropy by a nonvanishing boundary contribution along the entangling surface. In other words, this means that there is an ambiguity in the usual formula for the relative entropy coming from the nonuniqueness of the stress tensor. The main motivation of this work is to solve this puzzle. We first show that all choices of stress tensor except the canonical one are not allowed by positivity and monotonicity of the relative entropy. Then we fully compute the relative entropy between the vacuum and a coherent state in the framework of AQFT using the Araki formula and the techniques of modular theory. After all, both results coincide and give the usual expression for the relative entropy calculated with the canonical stress tensor.