Aspects of Quantum Field Theory in Curved Spacetime

London Mathematical Society Student Texts

by Stephen A. Fulling

This introduction to the theory of quantum fields in curved spacetime, intended for mathematicians, arose from a course taught to graduate students and is designed for self-study or advanced courses in relativity and quantum field theory. The style is informal and some knowledge of general relativity and differential geometry is assumed, yet the author does supply background material on function analysis and quantum field theory as required. Physicists should also gain a sound grasp of various aspects of the theory, some of which have not been particularly emphasized in the existing review literature.

Genres: Physics

328 pages, Hardcover

First published January 1, 1989

Reviews

[Medhat ullah · Rating 5 out of 5]

Quantum Field Theory (QFT) is the theoretical framework wherein quantum mechanics and special relativity coalesce to formulate a paradigmatic description of fundamental particles and their interactions. It operates within the paradigm of fields as operator-valued distributions, effectively mapping points in spacetime to operators on an infinite-dimensional Hilbert space. This formalism facilitates the quantization of classical fields, yielding a sophisticated interplay between commutation relations and causal structure dictated by Lorentz invariance.
Key aspects of QFT include:
Path Integral Formalism: A profound reformulation of quantum mechanics, leveraging functional integrals over field configurations to compute transition amplitudes, encoded via the generating functional Renormalization Group Dynamics: The theoretical scaffold addressing divergences through the reparametrization of coupling constants, with beta functions dictating the flow of these parameters under energy scale transformations.Spontaneous Symmetry Breaking and the Higgs Mechanism: Crucial phenomena where the vacuum expectation value of scalar fields induces mass for gauge bosons, elucidating the origin of mass in the Standard Model via Goldstone's theorem.
Operator Product Expansion (OPE): A technique unraveling the short-distance behavior of operator products, pivotal in defining scaling dimensions and anomalous dimensions within conformal field theory (CFT).Gauge Invariance and Yang-Mills Theories: A cornerstone ensuring internal symmetries are local, underpinning non-abelian gauge theories, and leading to phenomena such as asymptotic freedom and confinement in Quantum Chromodynamics (QCD).Perturbative and Non-Perturbative Techniques: While Feynman diagrams provide a perturbative visualization of scattering amplitudes via Dyson series, non-perturbative methods like lattice QFT and instantons probe strong coupling regimesAnomalies and Topological Effects: Chiral anomalies and the Atiyah-Singer index theorem reveal deep connections between quantum fields, topology, and conserved currents.Supersymmetry and Beyond: Extensions incorporating superalgebra to unify bosons and fermions, offering potential resolutions to the hierarchy problem and leading to insights within string theory and M-theory.QFT's multi-layered abstractions, from the quantization of classical Lagrangians to the algebraic structures underpinning operator algebras, make it an indispensable scaffold for modern physics, blending mathematical rigor with empirical robustness. Its ultimate aim? To reconcile the quantum microcosm with the cosmological macrodynamics, paving pathways to quantum gravity.
From the vantage point of a theoretical physicist deeply entrenched in the labyrinthine abstractions of quantum formalism, the edifice of Quantum Field Theory (QFT) represents not merely a calculational apparatus but a profound ontological framework. It encapsulates the synthesis of Lorentz-covariant dynamics and quantized excitations over a background manifold, thereby engendering a fertile interplay between locality, causality, and gauge symmetries.The path integral formalism, with its functional overmeasure on an infinite-dimensional configuration space, serves as an elegant instantiation of Feynman’s sum-over-histories paradigm. It simultaneously encodes perturbative expansions and paves the way for semi-classical analyses, including instantonic contributions. This duality between operatorial and path-integral formulations hints at the deeper categorical underpinnings that may unify disparate QFT domains under a topos-theoretic umbrella.Renormalization, far from being a mere computational sleight of hand, embodies a profound insight into the scale-dependent nature of physical laws. Wilsonian renormalization elucidates how effective theories emerge via coarse-graining, while the Callan-Symanzik equations delineate the flow of coupling constants in the ultraviolet (UV) and infrared (IR) regimes. This multifaceted tapestry underscores the fractal-like structure of quantum vacua and its role in shaping phase transitions.Gauge symmetries, rendered local via covariant derivatives, are not merely invariances but constraints that define the quantum dynamics. The Yang-Mills framework, with its fiber bundle topology, inherently connects gauge fields to curvature tensors, where non-Abelian structures give rise to intricate phenomena like quark confinement and mass gaps—a non-perturbative enigma that beckons deeper elucidation via lattice computations or AdS/CFT dualities.The quantum vacuum, far from being a barren state, is a seething plenum of zero-point fluctuations. Spontaneous symmetry breaking, through vacuum expectation values, exemplifies the non-trivial topology of field space. The Higgs mechanism, meanwhile, seamlessly reconciles gauge invariance with mass acquisition, reflecting a deeper unitarity-preserving interplay between Goldstone bosons and longitudinal gauge modes.Anomalies, often perceived as pathological, reveal subtle consistency conditions linking symmetries to topology. The axial anomaly, for instance, manifests as a chiral current non-conservation and connects seamlessly with the index theorem, thereby illuminating the interplay between spacetime geometry and quantum fields.Supersymmetry and higher-dimensional extensions, such as string theories, underscore the incompleteness of the QFT paradigm in accommodating quantum gravity. Yet, they also point toward tantalizing unifications, wherein spacetime geometry, topology, and quantum entanglement are interwoven in ways that challenge conventional intuitions.

[albin james · Rating 5 out of 5]

The Hissing Of Summer Lawns

by Joni Mitchell

He bought her a diamond for her throat
He put her in a ranch house on a hill
She could see the valley barbecues
From her window sill
See the blue pools in the squinting sun
Hear the hissing of summer lawns

He put up a barbed wire fence
To keep out the unknown
And on every metal thorn
Just a little blood of his own
She patrols that fence of his
To a latin drum
And the hissing of summer lawns
Darkness
Wonder makes it easy
Darkness
With a joyful mask
Darkness
Tube's gone darkness darkness darkness
No color no contrast

A diamond dog
Carrying a cup and a cane
Looking through a double glass
Looking at too much pride and too much shame
There's a black fly buzzing
There's a heat wave burning in her master's voice
The hissing of summer lawns

He gave her his darkness to regret
And good reason to quit him
He gave her a roomful of Chippendale
That nobody sits in
Still she stays with a love of some kind
It's the lady's choice
The hissing of summer lawns

© 1975; Crazy Crow Music

via http://jonimitchell.com/music/song.cf...

[Brian Powell · Rating 4 out of 5]

An interesting collection of topics in the field, though far from a comprehensive or even pedagogical treatment. Very heavy on the operator theory perspective, which sets this book apart from other popular treatments like Birrell & Davies. Though a useful perspective, this approach really only provides the garnish to the meat and potatoes of more mainstream textbooks. Section 7, on particle production in the expanding universe, is however perhaps the cleanest take on this topic in the literature.