Quantum Field Theory in Curved Spacetime (2nd Edition) (2308.14517v5)

Abstract: The 2023 second edition of a 2006 encyclopedia article on mathematical aspects of quantum field theory in curved spacetimes (QFTCST). Section-titles (with new sections indicated with stars) are: Introduction and preliminaries, Construction of a $*$-algebra for a real linear scalar field on globally hyperbolic spacetimes and some general theorems, *More about (quasifree) Hadamard states, Particle creation and the limitations of the particle concept, Theory of the stress-energy tensor, *More about the intersection of QFTCST with AQFT and the Fewster-Verch No-Go Theorem, Hawking and Unruh effects, *More about (classical and) quantum fields on black hole backgrounds, Non-globally hyperbolic spacetimes and the time-machine question, *More about QFT on non-globally hyperbolic spacetimes, Other related topics and some warnings. The article contains many references. It also includes a review of, and also compares and contrasts, recent results on the implications of QFTCST for the question of the instability of three sorts of Cauchy horizon -- first those inside black holes such as especially Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter, second the compactly generated Cauchy horizons of spacetimes in which time-machines get manufactured, and third the Cauchy horizon of the spacetime which is believed to describe evaporating black holes and which underlies (one version of) the black hole information-loss puzzle.

• The paper establishes a rigorous *-algebra framework for real scalar fields in globally hyperbolic spacetimes, forming the basis for quasifree Hadamard states.
• It highlights the intrinsic limitations of the particle concept in curved spacetime and analyzes key results like the Fewster-Verch no-go theorem.
• The study examines astrophysical phenomena such as Hawking and Unruh effects while discussing implications for black hole thermodynamics and quantum gravity.

Quantum Field Theory in Curved Spacetime: An Examination

The paper "Quantum Field Theory in Curved Spacetime" provides an extensive review of the developments in the theoretical framework of quantum field theory within curved spacetime (QFTCST). Authored by Bernard S. Kay, the document embodies a comprehensive discourse on various mathematical aspects and significant field results pertaining to QFTCST, especially emphasizing their implications in astrophysical and cosmological contexts. Below is a concise analysis of the topics addressed and their implications for ongoing research in theoretical physics.

Core Topics and Theoretical Insights

The paper encompasses the establishment and analysis of a $*$-algebra for real linear scalar fields in globally hyperbolic spacetimes. This establishment serves as a foundational cornerstone for the discussion and construction of concepts like quasifree Hadamard states. The paper goes into detail about the intrinsic limitations of the particle concept in non-flat spacetimes, particularly highlighting scenarios such as particle creation and the stress-energy tensor theory.

The document then transitions into a nuanced discussion on the intersection of QFTCST with algebraic quantum field theory (AQFT). It introduces the Fewster-Verch no-go theorem, which states the impossibility of a locally covariant preferred state construction, a revelation with substantial implications for the ontology of quantum states in differing spacetime structures.

Implications of Black Hole Thermodynamics

Particularly noteworthy is the examination of Hawking and Unruh effects, which underscore the thermal properties of black holes. The analysis of quantum fields on black hole backgrounds reiterates the unique role and adaptability of quasifree states. There is an exploration of the Hartle-Hawking-Israel state and further discussions address the stability of Cauchy horizons, both in classical and quantum frameworks, especially in de Sitter and Kerr-de Sitter spacetimes. Recent conjectures regarding quantum effects on Cauchy horizons have sparked significant interest and point toward further investigations into cosmic censorship conjectures.

Non-Globally Hyperbolic Spacetimes

The article tackles the intriguing "time machine" question by considering non-globally hyperbolic spacetimes. Through microlocal methods, it is argued that time machines may not be constructible, supporting Hawking's theory of chronology protection. This limitation is closely examined through quantum field theory, emphasizing the implications on future research in spacetimes that potentially allow closed timelike curves.

Conclusion and Future Directions

The paper concludes with a discussion of the prospects for combining QFTCST with semiclassical and quantum gravity. It underscores the importance of continued research in understanding quantum field behaviors under various spacetime geometries and the potential for uncovering new insights into the fundamental structure of the universe. Research in effective quantum field theory as applied to background-independent scenarios also promises new avenues for exploring quantum gravity's effects.

In the proposal of future developments, the paper speculates on the necessity for further mathematical advancements to achieve a unified theory of quantum gravity, with potential insights from semiclassical solutions breaking down at the limits of their applicability. The paper's implications, ranging from black hole thermodynamics to time-machine impossibilities, lay essential groundwork for advancing the horizons of theoretical physics and understanding quantum phenomena in cosmological models.