No Intrinsic Decoherence of Inflationary Cosmological Perturbations †
Abstract
:1. Introduction
1.1. Gravitational and Quantum Field Perturbations—Some Clarifying Remarks
1.1.1. Which Quantities in Cosmological Perturbations Are Intrinsically Quantum?
1.1.2. Tensor Perturbations: Gravitational Waves. Quantized Tensor Perturbations: Gravitons
1.1.3. Perturbations: Deterministic Variables. Fluctuations: Stochastic Variables
1.2. Model Studies, Key Findings and Organization
- Such a Wigner function does not correspond to a physical state
- (a)
- it violates the bound of the Wigner function when ,
- (b)
- if the system started in a pure state, the final state is no longer pure even though the evolution is unitary,
- (c)
- the purity of the state is greater than unity,
- (d)
- this implies that the corresponding density matrix has negative eigenvalues,
- The Robertson-Schrödinger uncertainty relation for a pure state is violated,
- The commutator of the canonical operator, like , at different times becomes commutative,
- The equal-time commutation relation of the canonical variables vanishes so the canonical pair commutes.
2. Decoherence in Cosmology: Highlights of Past Four Decades
Background (5 Stages) and Methodology (4 Veins)
3. Quantum States in a Closed System Do Not Turn Classical
3.1. Heisenberg Equation
3.2. Gaussian Pure State
3.3. Wigner Function and Density Matrix Elements
4. Quantum Mechanical Examples
4.1. Harmonic Oscillator: Semiclassical Limit
4.2. Free Particle
4.3. Inverted Linear Oscillator
5. Inflation Field
5.1. Canonical Variables Remain Noncommutating
5.2. Particle Creation: Numbers and Coherence
5.3. Entanglement: An Indelible Signifier of Quantumness
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Preservation of the Uncertainty Function under Squeeze Transformation
Appendix B. Semiclassical Limit of the Harmonic Oscillator
1 | Here, the center of attention is the quantum perturbations of the inflation field, that is, (see next subsection); the mean field is governed by a potential which engineers the inflationary dynamics, such as ‘slow-roll’, etc. |
2 | There is plenty of truth in the idiom,“truth is always simple, but simplicity is not always the truth”. |
3 | The background field is often assumed to be classical, but this needs to be proven rather than assumed to be automatically valid. The regarded as a mean field keeps its quantum nature. (The difference between a mean field and a classical field is explained in, e.g., [38].) One can show how readily the mean field is decohered by its quantum fluctuations, such as treated in [25]. |
4 | Note gravitational waves are weak metric perturbations. Gravitons are quantized short wavelength linear perturbations off of a smooth spacetime manifold, in the nature of collective excitations. Gravitons are governed by general relativity, a low energy theory for the macroscopic structure of spacetime, a far cry from quantum gravity, defined as theories for the microscopic structures of spacetime functional at the Planck scale [42] |
5 | Despite the similarity in form with the bilinear type of coupling between a quantum Brownian oscillator interacting with a bath of many modes, when two fields are bilinearly coupling, only one mode of the system field interacts with one mode of the bath field, the physics is totally different. It is like two equal subsystems interacting. One would not see dissipation or decoherence; the energy and phase information only pass from one to another back and forth. A large number of modes in the bath is needed to see dissipation and decoherence in the system. |
6 | For a pure state , the density matrix operator is . We thus have . In addition, any pure Gaussian state can be reached from the vacuum state by a suitable unitary transformation. Since the vacuum has minimal uncertainty, that is, and since the unitary transformation preserve the Robertson-Schrödinger uncertainty principle, the resulting pure Gaussian state then has . |
7 | Although in [32], the authors did not explicitly write the Wigner function into a form proportional to a delta function for the free particle case, their Equation (63) and Figure 1 served the same end. Besides, the Wigner function of the cosmological perturbations in their Equation (29) takes the delta-function form. They obtained their Equations (15) and (16) by keeping only the dominant contributions. According to the analysis in our Section 3 these results in [32] are thus problematic. |
8 | |
9 | Or instead of representation, we may use the representation,
|
10 | |
11 | For example, in [1], the Wigner funciton of the cosmological perturbations in its Equation (46) is written into a delta function form, and when only the leading term of what is resulting in its Equation (51) is kept, the conclusion of ‘decoherence without decoherence’ is conveniently yet haphazardly drawn. |
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Hsiang, J.-T.; Hu, B.-L. No Intrinsic Decoherence of Inflationary Cosmological Perturbations. Universe 2022, 8, 27. https://doi.org/10.3390/universe8010027
Hsiang J-T, Hu B-L. No Intrinsic Decoherence of Inflationary Cosmological Perturbations. Universe. 2022; 8(1):27. https://doi.org/10.3390/universe8010027
Chicago/Turabian StyleHsiang, Jen-Tsung, and Bei-Lok Hu. 2022. "No Intrinsic Decoherence of Inflationary Cosmological Perturbations" Universe 8, no. 1: 27. https://doi.org/10.3390/universe8010027