Aspekty kvantové teorie pole v zakřivených prostoročasech

• Thesis title in Czech: Aspekty kvantové teorie pole v zakřivených prostoročasech
• Thesis title in English: Aspects of quantum field theory in curved spacetimes
• Key words: kvantové teorie pole|zakřivené prostoročasy
• English key words: quantum field theory|curved spacetimes
• Academic year of topic announcement: 2024/2025
• Thesis type: diploma thesis
• Department: Institute of Theoretical Physics (32-UTF)
• Supervisor: Dr. Dražen Glavan, Ph.D.
• Advisors: Alexander Vikman

Guidelines

The first part of the project will be devoted to introducing the student to the basic concepts of quantum field theory (QFT) in curved spacetimes. The emphasis will be on cosmological spacetimes and the phenomenon of gravitational particle production (Ref. [1,2]). This mostly concerns free (non-interacting) QFT on expanding backgrounds. The
second part of the project deals with more advanced concepts of interacting QFT in curved spacetimes. This will introduce the student to some of the current problems researchers face when computing quantum loop corrections to cosmological observables in the early Universe. Depending on the progress, proclivity, and interest of the student, one of the following projects will be chosen:

(1) Different vacua in cosmological QFT. (Ref. [1,2])
The dynamic nature of the cosmological expansion prevents the existence of a unique vacuum for QFTs. Nevertheless, different physically motivated choices exist. The student will critically assess the feasibility of different choices of vacua in various cosmological eras and will investigate whether the renormalizability of observables is allowed for such states. This will help understand how reasonable are choices of quantum states at the beginnings of various cosmological eras, such as reheating.

(2) The freedom in adiabatic regularization. (Ref. [2,3]).
QFT observables in curved spacetimes exhibit very similar ultraviolet divergence issues known from the Minkowski spacetime. Multiple methods of regularizing these divergences exist. In this project, the student will examine the freedom and ambiguities in the "adiabatic regularization" scheme that is well adapted to cosmology. This regularization scheme involved subtracting the contributions of ultraviolet modes from the mode functions of the quantum field. The freedom in adiabatic regularization will be compared to the known renormalization scheme ambiguities in dimensional regularization. This will help understand the apparent disagreements between different computations.

(3) Linearization instability of quantum gauge fields. (Ref. [4,5,6]).
When electromagnetism is considered in a spacetime with spatially closed manifolds, such as a sphere, it is curious that the zero mode contains three physical polarizations instead of only two transverse ones known from the textbooks. However, when the electromagnetic field is coupled to charged matter this extra degree of freedom disappears and is substituted by the constraint that the total electric charge on a closed manifold must vanish (electric field lines must start and end somewhere!). This is a mathematical phenomenon of linearization instability where there is no smooth limit from the interacting theory to the free one. This has consequences for interacting QFT in curved spacetimes with compact spatial sections, such as some slicings of the de Sitter spacetime. The student will examine the quantization on a flat torus of an EM field in a family of covariant/average/multiplier gauges, and will construct the photon propagator. This will give an insight into related problems in quantum gravity appearing in the de Sitter space.

(4) Construction of non-linear gauge-invariant gravitational perturbations in the de Sitter spacetime. (Ref. [7,8])
The question of proper observables in perturbative quantum gravity during inflation is still open. Here we shall construct gauge-invariant perturbations of the metric around exponentially expanding de Sitter spacetime, in the pure gravity theory with a cosmological constant. The Hamiltonian formalism for theories with first-class constraints will be used. Such perturbations are appropriate for quantization and their correlators will be gauge-independent one-loop observables. This will allow for proper implementation and examination of the recently proposed de Sitter quantum break time [8].

References

[1] V. Mukhanov, S. Winitzki, "Introduction to quantum effects in gravity", Cambridge University Press, 2007, DOI:https://doi.org/10.1017/CBO9780511809149
[2] L. Parker, D. Toms, "Quantum Field Theory in Curved Spacetime", Cambridge University Press, 2009, DOI:10.1017/CBO9780511813924
[3] A. Ferreiro, F. Torrenti, "Ultraviolet-regularized power spectrum without infrared distortions in cosmological spacetimes", Phys.Lett.B 840 (2023) 137868, arXiv:2212.01078 [gr-qc]
[4] S. P. Miao, R.P. Woodard, "Transforming to Lorentz Gauge on de Sitter", J.Math.Phys. 50 (2009) 122502, arXiv:0907.4930 [gr-qc]
[5] D. Glavan, "Photon quantization in cosmological spaces", arXiv:2212.13975 [hep-th]
[6] D. Glavan, T. Prokopec: "Photon propagator in de Sitter space in the general covariant gauge", JHEP 05 (2023) 126, arXiv:2212.13982 [gr-qc]
[7] D. Langlois, "Hamiltonian formalism and gauge invariance for linear perturbations in inflation", Class.Quant.Grav. 11 (1994) 389-407
[8] G. Dvali, C. Gomez, S. Zell, "Quantum Break-Time of de Sitter", JCAP 06 (2017) 028, arXiv:1701.08776 [hep-th]

Preliminary scope of work in English

This project will introduce the student to the field of nonequilibrium QFT in cosmological spacetimes that is rapidly becoming indispensable in studies of the early Universe. In fact, the quantum fluctuations during the period of primordial inflation are believed to be the source of the famed temperature fluctuations in the cosmic microwave background. Further rapid advances of modern cosmology are opening up the possibility to probe not only particle physics lying beyond the well-established standard model but also in perturbative quantum gravity by studying quantum effects in the extreme conditions of the early expanding Universe. During the first stage of this project, the student will be introduced to the basic concepts of QFT in expanding cosmological spacetimes and to the effect of gravitational particle production. During the second stage, the student will apply the acquired knowledge to study one of the problems of interacting QFT in curved spacetimes. The student will work closely with the supervisor, and in addition will have the opportunity to interact with the members of the Central European Institute for Cosmology and Fundamental Physics (CEICO) of FZU, and to attend regular seminars and journal club meetings there.