Quantum field theory in curved spacetime (QFTCS) is a field of study that focuses on problems that arise when considering a quantum field on a fixed, curved spacetime. It allows the study of quantum effects in strong gravitational fields, and has led to many interesting conclusions, such as the Unruh effect and the Hawking effect.
Brief Summary
Relativistic quantum-field-theory (QFT) is most usually studied in inertial reference frames in Minkowski spacetime, which means it implements the principles of special-relativity. While this approach is excellent for the study of, e.g., particle-physics, it does not consider the influence of strong gravitational fields nor allows one to consider what happens in accelerated frames of reference. Quantum Field Theory in Curved Spacetimes deals with these situations.
The general idea is to weaken the assumptions of Minkowski spacetime in usual QFT and allow for a more general background metric. For example, one might want to study what happens to quantum fields in the proximity of black-holes or how they behave once one considers the usual spacetime solutions of cosmology. In this sense, QFTCS merges QFT and general-relativity (GR). However, it should be clear it is not a theory of quantum-gravity, since one assumes the background spacetime to be classical.
Many researchers in QFTCS do not believe it to be a fundamental theory, in the sense that they still believe gravity is inherently quantized and, as a consequence, QFTCS is not a complete description of our Universe. However, the framework does allow for a deeper understanding of the interplay between quantum-mechanics and GR and has led to many interesting conclusions across the years, some of which include the unruh-effect and the prediction of hawking-radiation.
Relativity of "Particles"
One of the main results of QFTCS is the fact that the notion of a "particle" is not fundamental, but rather an observer-dependent concept. This is not an issue at all, since quantum field theory is a theory of fields, not of particles.
In summary, particles in QFT are understood as excitations of the fields. To understand excitations as particles, one decomposes the field in Fourier modes and assigns positive-frequency solutions to particles and negative-frequency solutions to anti-particles. While this is a straightforward procedure in usual QFT, QFTCS considers more general spacetimes which not always have enough structure for these sorts of decompositions to be made.
In stationary spacetimes, one can use the notion of frequency associated to the stationary Killing field to decompose the quantum field. In many situations, the need for a choice of a Killing can lead to interesting predictions.
Unruh Effect
As a first example, one might consider the unruh-effect. In Minkowski spacetime, one usually chooses to work with the timelike Killing field corresponding to inertial time. However, on the so-called right Rindler wedge, one can instead choose to work with the timelike Killing field associated with Lorentz boosts. This Killing field is parallel to the orbits of accelerating observers, and hence can be interpreted as the notion of time of these observers.
Suppose then the quantum field is in the inertial vacuum, meaning an inertial observer sees no particles. In this situation, one can also decompose the field in terms of Rindler time, and arrive at the conclusion that the accelerated observers will see a thermal distribution of particles at the Unruh temperature $T_U = \frac{a}{2\pi}$ (where $a$ is the acceleration and we use units with $\hbar = c = 1$) where the inertial observers see none. This prediction exhibits the fact that the notion of particle is observer-dependent.
Hawking Radiation
Another example is the prediction of hawking-radiation. Suppose there is a star that collapses to a black hole. If the quantum field on the spacetime was originally on its vacuum state before the collapse happens, a distant, stationary observer will see the black hole emitting particles in a thermal distribution after the collapse happens. The temperature of the thermal distribution is the Hawking temperature, given by $T_H = \frac{\kappa}{2 \pi}$, where $\kappa$ is the surface gravity of the black hole and we employ units with $\hbar = c = G = 1$.
Resources
Some references for QFTCS are
Books
- R. M. Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics;
- N. D. Birrell & P. C. W. Davies' Quantum Fields in Curved Space.
- L. Parker & D. Toms' Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity
- V. F. Mukhanov & S. Winitzki's Introduction to Quantum Effects in Gravity
- S. A. Fulling's Aspects of Quantum Field Theory in Curved Spacetime
Review Papers
- S. Hollands & R. M. Wald. "Quantum fields in curved spacetime". Physics Reports 574, pp. 1–35, 2015. arXiv: 1401.2026 [gr-qc].
Phys.SE Posts
- Suggested reading for quantum field theory in curved spacetime (introductory references)
- Modern treatment of effective QFT in curved spacetime (advanced references)
- How to implement a Hilbert space on a manifold? (review on how to construct QFTCS)
