Quantum field theory in curved spacetime
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In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, quantum field theory in curved spacetime (QFTCS) is an extension of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
from
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
to a general
curved spacetime Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
. This theory treats spacetime as a fixed, classical background, while giving a quantum-mechanical description of the matter and energy propagating through that spacetime. A general prediction of this theory is that particles can be created by time-dependent
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
s (multi
graviton In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
pair production Pair production is the creation of a subatomic particle and its antiparticle from a neutral boson. Examples include creating an electron and a positron, a muon and an antimuon, or a proton and an antiproton. Pair production often refers specific ...
), or by time-independent gravitational fields that contain horizons. The most famous example of the latter is the phenomenon of
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
emitted by
black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
.


Overview

Ordinary
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
, which form the basis of
standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
, are defined in flat
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. For non-zero
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
s, on curved spacetimes quantum fields lose their interpretation as asymptotic
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s. Only in certain situations, such as in asymptotically flat spacetimes (zero
cosmological Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an
S-matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
. Even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given spacetime). Another observation is that unless the background
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
has a global timelike
Killing vector In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...
, there is no way to define a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
or ground state canonically. The concept of a vacuum is not invariant under
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
s. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If ''t''′(''t'') is a diffeomorphism, in general, the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of exp 'ikt''′(''t'')will contain negative frequencies even if ''k'' > 0.
Creation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
correspond to positive frequencies, while annihilation operators correspond to negative frequencies. This is why a state which looks like a vacuum to one observer cannot look like a vacuum state to another observer; it could even appear as a
heat bath In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
under suitable hypotheses. Since the end of the 1980s, the
local quantum field theory The Haag–Kastler axiomatic framework for quantum field theory, introduced by , is an application to local quantum physics of C*-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT). The axioms are stated in ...
approach due to
Rudolf Haag Rudolf Haag (17 August 1922 – 5 January 2016) was a German theoretical physicist, who mainly dealt with fundamental questions of quantum field theory. He was one of the founders of the modern formulation of quantum field theory and he identifi ...
and
Daniel Kastler Daniel Kastler (; 4 March 1926 – 4 July 2015) was a French theoretical physicist, working on the foundations of quantum field theory and on non-commutative geometry. Biography Daniel Kastler was born on March 4, 1926, in Colmar, a city of no ...
has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems mentioned above arising from the absence of a preferred reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables.


Applications

Using
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
in quantum field theory in curved spacetime geometry is known as the semiclassical approach to
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. This approach studies the interaction of quantum fields in a fixed classical spacetime and among other thing predicts the creation of particles by time-varying spacetimes and
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
. The latter can be understood as a manifestation of the
Unruh effect The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
where an accelerating observer observes black body radiation. Other prediction of quantum fields in curved spaces include, for example, the radiation emitted by a particle moving along a geodesic and the interaction of
Hawking radiation Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
with particles outside black holes. This formalism is also used to predict the primordial density
perturbation Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbat ...
spectrum arising in different models of
cosmic inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from  seconds after the conjectured Big Bang singularity ...
. These predictions are calculated using the Bunch–Davies vacuum or modifications thereto.


Approximation to quantum gravity

The theory of quantum field theory in curved spacetime may be considered as an intermediate step towards
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. QFT in curved spacetime is expected to be a viable approximation to the theory of quantum gravity when spacetime curvature is not significant on the Planck scale. However, the fact that the true theory of quantum gravity remains unknown means that the precise criteria for when QFT on curved spacetime is a good approximation are also unknown. Gravity is not
renormalizable Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similarity, self-similar geometric structures, that are used to treat infinity, infinities arising in calculated ...
in QFT, so merely formulating QFT in curved spacetime is not a true theory of quantum gravity.


See also

*
General relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
* History of quantum field theory *
Local quantum field theory The Haag–Kastler axiomatic framework for quantum field theory, introduced by , is an application to local quantum physics of C*-algebra theory. Because of this it is also known as algebraic quantum field theory (AQFT). The axioms are stated in ...
*
Statistical field theory Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industri ...
*
Topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
*
Quantum geometry In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to the Planck length. At these d ...
*
Quantum spacetime In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebr ...


References


Further reading

* * * *


External links


Summary Chart of Intro Steps to Quantum Fields in Curved Spacetime
A two-page chart outline of the basic principles governing the behavior of quantum fields in general relativity. {{quantum field theories, state=expanded Quantum field theory Quantum gravity