States of Low Energy are a class of exact Hadamard states for free quantum fields on cosmological spacetimes whose structure is fixed at all scales by a minimization principle. The original construction was for Friedmann–Lemaître geometries and is here generalized to anisotropic Bianchi I geometries relevant to primordial cosmology. In addition to proving the Hadamard property, systematic series expansions in the infrared and ultraviolet are developed. The infrared expansion is convergent and induces in the massless case a leading spatial long distance decay that is always Minkowski-like but anisotropy modulated. The ultraviolet expansion is shown to be equivalent to the Hadamard property, and a non-recursive formula for its coefficients is presented.
REFERENCES
1.
I.
Khavkine
and V.
Moretti
, “Algebraic QFT in curved spacetime and quasifree Hadamard states: An introduction
,” in Advances in Algebraic Quantum Field Theory
, edited by R.
Brunetti
, C.
Dappiaggi
, K.
Fredenhagen
, and J.
Yngvason
(Springer
, 2014
), pp. 191
–251
; arXiv:1412.5945 [math-ph].2.
S.
Hollands
and R. M.
Wald
, “Quantum fields in curved spacetime
,” Phys. Rep.
574
, 1
–35
(2015
); arXiv:1401.2026 [gr-qc].3.
C.
Gerard
, Microlocal Analysis of Quantum Fields on Curved Spacetimes
(European Mathematical Society
, 2019
).4.
S. A.
Fulling
, F. J.
Narcowich
, and R. M.
Wald
, “Singularity structure of the two-point function in quantum field theory in curved spacetime, II
,” Ann. Phys.
136
, 243
–272
(1981
).5.
C.
Dappiaggi
, V.
Moretti
, and N.
Pinamonti
, Hadamard States from Light-Like Hypersurfaces
(Springer
, 2017
); arXiv:1706.09666 [math-ph].6.
W.
Junker
and E.
Schrohe
, “Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties
,” Ann. Henri Poincare
3
, 1113
–1181
(2002
); arXiv:math-ph/0109010.7.
N.
Afshordi
, S.
Aslanbeigi
, and R. D.
Sorkin
, “A distinguished vacuum state for a quantum field in a curved spacetime: Formalism, features, and cosmology
,” J. High Energy Phys.
2012
(08
), 137
; arXiv:1205.1296 [hep-th].8.
M.
Brum
and K.
Fredenhagen
, “‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes
,” Classical Quantum Gravity
31
, 025024
(2014
); arXiv:1307.0482 [gr-qc].9.
H.
Olbermann
, “States of low energy on Robertson–Walker spacetimes
,” Classical Quantum Gravity
24
, 5011
–5030
(2007
); arXiv:0704.2986 [gr-qc].10.
S. A.
Fulling
, “Remarks on positive frequency and Hamiltonians in expanding universes
,” Gen. Relativ. Gravitation
10
, 807
–824
(1979
).11.
R.
Banerjee
and M.
Niedermaier
, “Bonus properties of states of low energy
,” J. Math. Phys.
61
, 103511
(2020
); arXiv:2006.08685 [math-ph].12.
M.
Martín-Benito
, R. B.
Neves
, and J.
Olmedo
, “Non-oscillatory power spectrum from states of low energy in kinetically dominated early universes
,” Front. Astron. Space Sci.
8
, 702543
(2021
); arXiv:2104.14850 [gr-qc].13.
M.
Martín-Benito
, R. B.
Neves
, and J.
Olmedo
, “States of low energy in bouncing inflationary scenarios in loop quantum cosmology
,” Phys. Rev. D
103
, 123524
(2021
); arXiv:2104.03035 [gr-qc].14.
S.
Nadal-Gisbert
, J.
Navarro-Salas
, and S.
Pla
, “Low-energy states and CPT invariance at the big bang
,” Phys. Rev. D
107
, 085018
(2023
); arXiv:2302.08812 [gr-qc].15.
A.
Álvarez-Domínguez
, L. J.
Garay
, M.
Martín-Benito
, and R. B.
Neves
, “States of low energy in the Schwinger effect
,” JHEP
2023
, 93.16.
K.
Them
and M.
Brum
, “States of low energy in homogeneous and inhomogeneous expanding spacetimes
,” Classical Quantum Gravity
30
, 235035
(2013
); arXiv:1302.3174 [gr-qc].17.
C. A.
Mantica
and L. G.
Molinari
, “Generalized Robertson–Walker spacetimes—A survey
,” Int. J. Geom. Methods Mod. Phys.
14
, 1730001
(2017
); arXiv:1612.07021 [math-ph].18.
B.
Cropp
and M.
Visser
, “Any spacetime has a Bianchi type I spacetime as a limit
,” Classical Quantum Gravity
28
, 055007
(2011
); arXiv:1008.4639 [gr-qc].19.
G.
Montani
, M. V.
Battisti
, R.
Benini
, and G.
Imponente
, Primordial Cosmology
(World Scientific
, Singapore
, 2009
).20.
G.
Maniccia
, M.
De Angelis
, and G.
Montani
, “WKB approaches to restore time in quantum cosmology: Predictions and shortcomings
,” Universe
8
, 556
(2022
); arXiv:2209.04403 [gr-qc].21.
J.
Ritchie
, “Bianchi I ‘asymptotically Kasner’ solutions of the Einstein scalar field equations
,” Classical Quantum Gravity
39
, 135007
(2022
); arXiv:2205.10930 [gr-qc].22.
E.
Nungesser
, “Isotropization of non-diagonal Bianchi I spacetimes with collisionless matter at late times assuming small data
,” Classical Quantum Gravity
27
, 235025
(2010
); arXiv:1007.0184 [gr-qc].23.
R.
Haag
, H.
Narnhofer
, and U.
Stein
, “On quantum field theory in gravitational background
,” Commun. Math. Phys.
94
, 219
(1984
).24.
C. J.
Fewster
, “A general worldline quantum inequality
,” Classical Quantum Gravity
17
, 1897
–1911
(2000
); arXiv:gr-qc/9910060.25.
R.
Banerjee
and M.
Niedermaier
, “The spatial functional renormalization group and Hadamard states on cosmological spacetimes
,” Nucl. Phys. B
980
, 115814
(2022
); arXiv:2201.02575 [hep-th].26.
C.
Fewster
, “Modified Green-hyperbolic operators
,” Sigma
19
, 057 (2023).27.
M. J.
Radzikowski
, “Micro-local approach to the Hadamard condition in quantum field theory on curved space-time
,” Commun. Math. Phys.
179
, 529
–553
(1996
).28.
B. S.
Kay
and R. M.
Wald
, “Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon
,” Phys. Rep.
207
, 49
–136
(1991
).29.
A.
Strohmaier
, “Microlocal analysis
,” edited by C. Bär, K. Fredenhagen Lect. Notes Phys.
786
, 85
(Springer, 2009
).30.
Here , the space of distributions, is the topological dual of .
31.
L.
Hörmander
, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis
(Springer
, 1983
).32.
This corrects a copying error in Eq. (71) of Ref. 11.
33.
J.
Swieca
, “Goldstone’s theorem and related topics
,” in Proceedings, Summer School on Theoretical Physics, Cargèse Lectures in Physics: Cargese, France, September 1969
, edited by D.
Kastler
(Gordon and Breach
, New York
, 1970
), Vol. 4
.34.
J.
Lopuszanski
, An Introduction to Symmetry and Supersymmetry in Quantum Field Theory
(World Scientific
, 1991
).35.
N. G. C.
Bär
and F.
Pfäffle
, Wave Equations on Lorentzian Manifolds and Quantization
(European Mathematical Society
, 2007
).36.
Y.
Decanini
and A.
Folacci
, “Off-diagonal coefficients of the Dewitt-Schwinger and Hadamard representations of the Feynman propagator
,” Phys. Rev. D
73
, 044027
(2006
); arXiv:gr-qc/0511115.37.
K.-T.
Pirk
, “Hadamard states and adiabatic vacua
,” Phys. Rev. D
48
, 3779
–3783
(1993
); arXiv:gr-qc/9211003.38.
F. D.
Mazzitelli
, J. P.
Paz
, and M. A.
Castagnino
, “Cauchy data and Hadamard singularities in time-dependent backgrounds
,” Phys. Rev. D
36
, 2994
–3001
(1987
).39.
D.
Bernard
, “Hadamard singularity and quantum states in Bianchi type-I space-time
,” Phys. Rev. D
33
, 3581
–3589
(1986
).40.
I.
Avramidi
, Heat Kernel Method and its Applications
(Birkhäuser
, 2015
).41.
I. G.
Avramidi
and R.
Schimming
, “A new explicit expression for the Korteweg–De Vries hierarchy
,” Math. Nachr.
219
, 45
–64
(2000
); arXiv:solv-int/9710009.42.
I.
Polterovich
, “Heat invariants of Riemannian manifolds
,” Isr. J. Math.
119
, 239
(2000
).43.
S.
Rida
, “Explicit formulae for the powers of a Schrödinger-like ordinary differential operator
,” J. Phys. A: Math. Gen.
31
, 5577
(1998
).44.
Z.
Avetisyan
and R.
Verch
, “Explicit harmonic and spectral analysis in Bianchi I–VII-type cosmologies
,” Classical Quantum Gravity
30
, 155006
(2013
); arXiv:1212.6180 [math-ph].45.
D.
Saadeh
, S. M.
Feeney
, A.
Pontzen
, H. V.
Peiris
, and J. D.
McEwen
, “How isotropic is the Universe?
,” Phys. Rev. Lett.
117
, 131302
(2016
); arXiv:1605.07178 [astro-ph.CO].46.
M.
Niedermaier
, “Nonstandard action of diffeomorphisms and gravity’s anti-Newtonian limit
,” Symmetry
12
, 752
(2020
).47.
L. E.
Parker
, “Quantized fields and particle creation in expanding universes. I
,” Phys. Rev.
183
, 1057
–1068
(1969
).48.
C.
Lüders
and J.
Roberts
, “Local quasiequivalence and adiabatic vacuum states
,” Commun. Math. Phys.
134
, 29
–63
(1990
).49.
J.
Matyjasek
, “Quantum fields in Bianchi type I spacetimes: The Kasner metric
,” Phys. Rev. D
98
, 104054
(2018
); arXiv:1811.00993 [gr-qc].50.
Here, as well as in the Proof of Theorem A.1 and Lemma A.1, the statement “f ◦ ı is an even function of ζ” for , , means that there is an open ball B ∋ 0 contained in such that (f◦ı)(τ, ℘, ζ) = (f◦ı)(τ, ℘, −ζ) for all .
51.
Although for notational brevity we omit the adiabatic order n, both Rp and Fp implicitly depend on it.
© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.