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 from
Minkowski spacetime to a general
curved spacetime. 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 fields (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), 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 a ...
emitted by
black holes.
Overview
Ordinary
quantum field theories, which form the basis of
standard model, 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 iner ...
, 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 eq ...
s, on curved spacetimes quantum fields lose their interpretation as asymptotic
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from ...
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), 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 has a global timelike
Killing vector, there is no way to define a
vacuum 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 ...
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 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 d ...
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 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 va ...
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 middl ...
in quantum field theory in curved spacetime geometry is known as the
semiclassical approach to
quantum gravity. 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 a ...
. The latter can be understood as a manifestation of the
Unruh effect 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 a ...
with particles outside black holes.
This formalism is also used to predict the primordial density
perturbation spectrum arising in different models of
cosmic inflation. 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. 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-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
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
*
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