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arXiv:gr-qc/9212008 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Coherent state representation of quantum fluctuations in the early Universe

Andrew Matacz

Published Jan 15, 1994• 69 citations
arXivHTMLPDF

Abstract

Using the squeezed state formalism the coherent state representation of quantum fluctuations in an expanding universe is derived. It is shown that this provides an interesting alternative to the Wigner function as a phase space representation of quantum fluctuations. The quantum to classical transition of fluctuations is simply implemented by decohering the density matrix in this representation. The entropy of the decohered vacua is derived and is shown to agree with previous results in the high squeezing limit. It is shown that the decoherence process can significantly change the predictions derived using pure states. In particular, scale-invariant power spectra are not obtained for any decoherence process, though it can be obtained by decohering in the coherent state representation.

DOI
10.1103/physrevd.49.788

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•Feb 1, 1992•3 citations
Using the squeezed state formalism the coherent state representation of quantum fluctuations in an expanding universe is derived. It is shown that this provides an interesting alternative to the Wigner function as a phase space representation of quantum fluctuations. The quantum to classical transition of fluctuations is simply implemented by decohering the density matrix in this representation. The entropy of the decohered vacua is derived and is shown to agree with previous results in the high squeezing limit. It is shown that the decoherence process can significantly change the predictions derived using pure states. In particular, scale-invariant power spectra are not obtained for any decoherence process, though it can be obtained by decohering in the coherent state representation.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
Functional integrals provide a powerful and useful formalism for deriving both perturbative and nonperturbative results in quantum theory. Here we consider their application to the problem of formulating the dynamics of coherent and squeezed states. We derive the functional integral representation of the coherent state density matrix by combining the closed time path representation of the density matrix in terms of functional integrals in configuration space with the projection of coherent states onto configuration space states. In this formalism the atomic variables can, without loss of generality, be integrated out exactly yielding a single functional integral which describes the combined quantum dynamics of both the atomic and the coherent states.
Many quantum systems in various fields, a classical signal spontaneously appears without any specific operations/measurements. For example, the initiation of the spontaneous symmetry breaking, the appearance of the off-diagonal long-range order in the Bose-Einstein condensate, the transient process of the quantum fluctuations in the early Universe, etc.  We are interested in the basic principle of these transient processes in quantum systems which shows classical signal spontaneously. We start our observation that some instability often promotes the squeezing of the original quantum system. This is usually interpreted as the particle production or the Bogoliubov shift of the vacuum. At this stage, everything is Unitary and reversible. For example, Hawking radiation around the Black Hole, the thermal state associated with the accelerated observer (Unruh effect), and the particle production in the expanding universe, etc. are all in this category. By using the Schwinger-Keldysh in-in formalism of quantum field theory, we find that these squeezed states are associated with non-dissipative classical fluctuations. We call them dry-noise. Then applying the generalized effective action method and deriving the Langevin equation, we find that these dry-noise triggers, through nonlinear interactions, to develop the coherent state which can be classical if sufficiently developed. This is the origin of the classical signal in the full quantum systems. The appearance of the squeezed state through the tachyonic instability and the development of the coherent state through the nonlinearity seem to be a general transient process that a classical signal spontaneously develops. These states appear more generally not only in the quantum optics. We finally show that this formalism is widely applicable; initiation dynamics of the spontaneous symmetry breaking in quantum systems such as BEC, Black hole entropy, Unruh detectors, and quantum cosmology. These analyses may give some implications on the origin of the irreversibility associated with the transition from quantum to classical. The detail description is in arxiv.org/abs/1810.11293 and the references therein.
After a brief summary of the four main veins in the treatment of decoherence and quantum to classical transition in cosmology since the 1980s, we focus on one of these veins in the study of quantum decoherence of cosmological perturbations in inflationary universe, the case when it does not rely on any environment. This is what ‘intrinsic’ in the title refers to—a closed quantum system, consisting of a quantum field which drives inflation. The question is whether its quantum perturbations, which interact with the density contrast giving rise to structures in the universe, decohere with an inflationary expansion of the universe. A dominant view which had propagated for a quarter of a century asserts yes, based on the belief that the large squeezing of a quantum state after a duration of inflation renders the system effectively classical. This paper debunks this view by identifying the technical fault-lines in its derivations and revealing the pitfalls in its arguments which drew earlier authors to this wrong conclusion. We use a few simple quantum mechanical models to expound where the fallacy originated: The highly squeezed ellipse quadrature in phase space cannot be simplified to a line, and the Wigner function cannot be replaced by a delta function. These measures amount to taking only the leading order in the relevant parameters in seeking the semiclassical limit and ignoring the subdominant contributions where quantum features reside. Doing so violates the bounds of the Wigner function, and its wave functions possess negative eigenvalues. Moreover, the Robertson-Schrödinger uncertainty relation for a pure state is violated. For inflationary cosmological perturbations, in addition to these features, entanglement exists between the created pairs. This uniquely quantum feature cannot be easily argued away. Indeed, it could be our best hope to retroduce the quantum nature of cosmological perturbations and the trace of an inflation field. All this points to the invariant fact that a closed quantum system, even when highly squeezed, evolves unitarily without loss of coherence; quantum cosmological perturbations do not decohere by themselves.