correlation function (quantum field theory)
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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 ...
, correlation functions, often referred to as correlators or
Green's functions In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...
, are vacuum expectation values of
time-ordered In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). H ...
products of field operators. They are a key object of study in quantum field theory where they can be used to calculate various observables such as
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 ...
elements.


Definition

For a scalar field theory with a single field \phi(x) and a
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, \Omega\rangle at every event (x) in spacetime, the n-point correlation function is the vacuum expectation value of the time-ordered products of n field operators in the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
G_n(x_1,\dots, x_n) = \langle \Omega, T\, \Omega\rangle. Here T\ is the time-ordering operator for which orders the field operators so that earlier time field operators appear to the right of later time field operators. By transforming the fields and states into the interaction picture, this is rewritten as G_n(x_1, \dots, x_n) = \frac, where , 0\rangle is the ground state of the free theory and S
phi Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet. In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voicele ...
/math> is the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
. Expanding e^ using its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
, the n-point correlation function becomes a sum of interaction picture correlation functions which can be evaluated using
Wick's theorem Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihila ...
. A diagrammatic way to represent the resulting sum is via
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduce ...
, where each term can be evaluated using the position space Feynman rules. The series of diagrams arising from \langle 0, e^, 0\rangle is the set of all vacuum bubble diagrams, which are diagrams with no external legs. Meanwhile, \langle 0, \phi(x_1)\dots \phi(x_n)e^, 0\rangle is given by the set of all possible diagrams with exactly n external legs. Since this also includes disconnected diagrams with vacuum bubbles, the sum factorizes into (sum over all bubble diagrams)\times(sum of all diagrams with no bubbles). The first term then cancels with the normalization factor in the denominator meaning that the n-point correlation function is the sum of all Feynman diagrams excluding vacuum bubbles G_n(x_1, \dots, x_n) = \langle 0, T\, 0\rangle_. While not including any vacuum bubbles, the sum does include disconnected diagrams, which are diagrams where at least one external leg is not connected to all other external legs through some connected path. Excluding these disconnected diagrams instead defines connected n-point correlation functions G_n^c(x_1, \dots, x_n) = \langle 0, T\, 0\rangle_ It is often preferable to work directly with these as they contain all the information that the full correlation functions contain since any disconnected diagram is merely a product of connected diagrams. By excluding other sets of diagrams one can define other correlation functions such as one-particle irreducible correlation functions. In the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
, n-point correlation functions are written as a functional average G_n(x_1, \dots, x_n) = \frac. They can be evaluated using the partition functional Z /math> which acts as a
generating functional In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series ...
, with J being a source-term, for the correlation functions G_n(x_1, \dots, x_n) = (-i)^n \frac \left.\frac\_. Similarly, connected correlation functions can be generated using W = -i \ln Z /math> as G_n^c(x_1, \dots, x_n) = (-i)^ \left.\frac\_.


Relation to the S-matrix

Scattering amplitudes can be calculated using correlation functions by relating them to the S-matrix through the
LSZ reduction formula In quantum field theory, the LSZ reduction formula is a method to calculate ''S''-matrix elements (the scattering amplitudes) from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the L ...
\langle f, S, i\rangle = \left \int d^4 x_1 e^ \left(\partial^2_ + m^2\right)\rightcdots \left \int d^4 x_n e^ \left(\partial_^2 + m^2\right)\right\langle \Omega , T\, \Omega\rangle. Here the particles in the initial state , i\rangle have a -i sign in the exponential, while the particles in the final state , f\rangle have a +i. All terms in the Feynman diagram expansion of the correlation function will have one propagator for each external leg, that is a propagators with one end at x_i and the other at some internal vertex x. The significance of this formula becomes clear after the application of the Klein–Gordon operators to these external legs using \left(\partial^2_ + m^2\right)\Delta_F(x_i,x) = -i\delta^4(x_i-x). This is said to amputate the diagrams by removing the external leg propagators and putting the external states
on-shell In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", ...
. All other off-shell contributions from the correlation function vanish. After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a
Fourier transformation 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 ...
operation where the integration is over the internal point positions x that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the amputated correlation functions with on-shell external states. It is common to directly deal with the momentum space correlation function \tilde G(q_1, \dots, q_n), defined through the Fourier transformation of the correlation function (2\pi)^4 \delta^(q_1+\cdots + q_n) \tilde G_n(q_1, \dots, q_n) = \int d^4 x_1 \dots d^4 x_n \left(\prod^n_ e^\right) G_n(x_1, \dots, x_n), where by convention the momenta are directed inwards into the diagram. A useful quantity to calculate when calculating scattering amplitudes is the matrix element \mathcal M which is defined from the S-matrix via \langle f, S - 1 , i\rangle = i(2\pi)^4 \delta^4 \mathcal M where p_i are the external momenta. From the LSZ reduction formula it then follows that the matrix element is equivalent to the amputated connected momentum space correlation function with properly orientated external momenta i \mathcal M = \tilde G_n^c(p_1, \dots, -p_n)_. For non-scalar theories the reduction formula also introduces external state terms such as polarization vectors for photons or spinor states for fermions. The requirement of using the connected correlation functions arises from the
cluster decomposition In physics, the cluster decomposition property states that experiments carried out far from each other cannot influence each other. Usually applied to quantum field theory, it requires that vacuum expectation values of operators localized in bou ...
because scattering processes that occur at large separations do not interfere with each other so can be treated separately.


See also

*
Effective action In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective ac ...
*
Green's function (many-body theory) In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators. The name comes from ...
*
Partition function (mathematics) The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing ...


References


Further reading

* Altland, A.; Simons, B. (2006). ''Condensed Matter Field Theory''.
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
. * Schroeder, D.V.; Peskin, M., ''An Introduction to Quantum Field Theory''.
Addison-Wesley Addison-Wesley is an American publisher of textbooks and computer literature. It is an imprint of Pearson PLC, a global publishing and education company. In addition to publishing books, Addison-Wesley also distributes its technical titles through ...
. {{DEFAULTSORT:Correlation Function (Quantum Field Theory) Quantum field theory Covariance and correlation