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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.

**Many-body theory** is an area of physics which provides the framework for understanding the collective behavior of large numbers of interacting particles, often on the order of Avogadro's number. In general terms, many-body theory deals with effects that manifest themselves only in systems containing large numbers of constituents. While the underlying physical laws that govern the motion of each individual particle may be simple, the study of the collection of particles can be extremely complex. In some cases emergent phenomena may arise which bear little resemblance to the underlying elementary laws.

In quantum field theory, the *n*-point correlation function is defined as the functional average of a product of field operators at different positions

**Creation 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 lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.

- Spatially uniform case
- Basic definitions
- Two-point functions
- Spectral representation
- General case
- Basic definitions 2
- Two-point functions 2
- Spectral representation 2
- See also
- References
- Books
- Papers
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The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point 'Green's functions' in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)

In quantum mechanics, a **Hamiltonian** is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

We consider a many-body theory with field operator (annihilation operator written in the position basis) .

The Heisenberg operators can be written in terms of Schrödinger operators as

In physics, the **Heisenberg picture** is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.

In physics, the **Schrödinger picture** is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant with respect to time. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.

and the creation operator is , where is the grand-canonical Hamiltonian.

In statistical mechanics, a **grand canonical ensemble** is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

Similarly, for the imaginary-time operators,

[Note that the imaginary-time creation operator is not the Hermitian conjugate of the annihilation operator .]

In real time, the -point Green function is defined by

where we have used a condensed notation in which signifies and signifies . The operator denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left.

In imaginary time, the corresponding definition is

where signifies . (The imaginary-time variables are restricted to the range from to the inverse temperature .)

**Note** regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point () thermal Green function for a free particle is

and the retarded Green function is

where

is the Matsubara frequency.

Throughout, is for bosons and for fermions and denotes either a commutator or anticommutator as appropriate.

(See below for details.)

The Green function with a single pair of arguments () is referred to as the two-point function, or propagator. In the presence of both spatial and temporal translational symmetry, it depends only on the difference of its arguments. Taking the Fourier transform with respect to both space and time gives

where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of , as usual).

In real time, we will explicitly indicate the time-ordered function with a superscript T:

The real-time two-point Green function can be written in terms of 'retarded' and 'advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by

and

respectively.

They are related to the time-ordered Green function by

where

is the Bose–Einstein or Fermi–Dirac distribution function.

The thermal Green functions are defined only when both imaginary-time arguments are within the range to . The two-point Green function has the following properties. (The position or momentum arguments are suppressed in this section.)

Firstly, it depends only on the difference of the imaginary times:

The argument is allowed to run from to .

Secondly, is (anti)periodic under shifts of . Because of the small domain within which the function is defined, this means just

for . Time ordering is crucial for this property, which can be proved straightforwardly, using the cyclicity of the trace operation.

These two properties allow for the Fourier transform representation and its inverse,

Finally, note that has a discontinuity at ; this is consistent with a long-distance behaviour of .

The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by

where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian *H* − *μN*, with eigenvalue *E _{α}*.

The imaginary-time propagator is then given by

and the retarded propagator by

where the limit as is implied.

The advanced propagator is given by the same expression, but with in the denominator.

The time-ordered function can be found in terms of and . As claimed above, and have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane.

The thermal propagator has all its poles and discontinuities on the imaginary axis.

The spectral density can be found very straightforwardly from , using the Sokhatsky–Weierstrass theorem

where P denotes the Cauchy principal part. This gives

This furthermore implies that obeys the following relationship between its real and imaginary parts:

where denotes the principal value of the integral.

The spectral density obeys a sum rule,

which gives

as .

The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function

which is related to and by

and

A similar expression obviously holds for .

The relation between and is referred to as a Hilbert transform.

We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as

Due to translational symmetry, it is only necessary to consider for , given by

Inserting a complete set of eigenstates gives

Since and are eigenstates of , the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving

Performing the Fourier transform then gives

Momentum conservation allows the final term to be written as (up to possible factors of the volume)

which confirms the expressions for the Green functions in the spectral representation.

The sum rule can be proved by considering the expectation value of the commutator,

and then inserting a complete set of eigenstates into both terms of the commutator:

Swapping the labels in the first term then gives

which is exactly the result of the integration of ρ.

In the non-interacting case, is an eigenstate with (grand-canonical) energy , where is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes

From the commutation relations,

with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply , leaving

The imaginary-time propagator is thus

and the retarded propagator is

As β→∞, the spectral density becomes

where α = 0 corresponds to the ground state. Note that only the first (second) term contributes when ω is positive (negative).

We can use 'field operators' as above, or creation and annihilation operators associated with other single-particle states, perhaps eigenstates of the (noninteracting) kinetic energy. We then use

where is the annihilation operator for the single-particle state and is that state's wavefunction in the position basis. This gives

with a similar expression for .

These depend only on the difference of their time arguments, so that

and

We can again define retarded and advanced functions in the obvious way; these are related to the time-ordered function in the same way as above.

The same periodicity properties as described in above apply to . Specifically,

and

for .

In this case,

where and are many-body states.

The expressions for the Green functions are modified in the obvious ways:

and

Their analyticity properties are identical. The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.

If the particular single-particle states that are chosen are `single-particle energy eigenstates', i.e.

then for an eigenstate:

so is :

and so is :

We therefore have

We then rewrite

therefore

use

and the fact that the thermal average of the number operator gives the Bose–Einstein or Fermi–Dirac distribution function.

Finally, the spectral density simplifies to give

so that the thermal Green function is

and the retarded Green function is

Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.

In electrodynamics, **elliptical polarization** is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.

In quantum chemistry and molecular physics, the **Born–Oppenheimer** (**BO**) **approximation** is the assumption that the motion of atomic nuclei and electrons in a molecule can be treated separately. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics. The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity, but is then often used as a starting point for more refined methods.

In physics, specifically in quantum mechanics, a **coherent state** is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent states arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well. The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

**Second quantization**, also referred to as **occupation number representation**, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields are thought of as field operators, in a manner similar to how the physical quantities are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were developed, most notably, by Vladimir Fock and Pascual Jordan later.

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).

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In quantum field theory, the **LSZ reduction formula** is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.

The **Havriliak–Negami relaxation** is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In mathematics, the **Fubini–Study metric** is a Kähler metric on projective Hilbert space, that is, on a complex projective space **CP**^{n} endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

In mathematics, specifically in symplectic geometry, the **momentum map** is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including **symplectic** (**Marsden–Weinstein**) **quotients**, discussed below, and symplectic cuts and sums.

A **quasiprobability distribution** is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Although quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, *the ability to yield expectation values with respect to the weights of the distribution*, they all violate the *σ*-additivity axiom, because regions integrated under them do not represent probabilities of mutually exclusive states. To compensate, some quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis, and elsewhere.

**Sinusoidal plane-wave solutions** are particular solutions to the electromagnetic wave equation.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In a field of mathematics known as differential geometry, a **Courant geometry** was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on , called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.

In mathematics — specifically, in stochastic analysis — an **Itô diffusion** is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion of a particle subjected to a potential in a viscous fluid. Itô diffusions are named after the Japanese mathematician Kiyosi Itô.

A **symmetric, informationally complete, positive operator-valued measure** (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem.

In thermal quantum field theory, the **Matsubara frequency** summation is the summation over discrete imaginary frequencies. It takes the following form

This article summarizes important identities in exterior calculus.

- Bonch-Bruevich V. L., Tyablikov S. V. (1962):
*The Green Function Method in Statistical Mechanics.*North Holland Publishing Co. - Abrikosov, A. A., Gorkov, L. P. and Dzyaloshinski, I. E. (1963):
*Methods of Quantum Field Theory in Statistical Physics*Englewood Cliffs: Prentice-Hall. - Negele, J. W. and Orland, H. (1988):
*Quantum Many-Particle Systems*AddisonWesley. - Zubarev D. N., Morozov V., Ropke G. (1996):
*Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory*(Vol. 1). John Wiley & Sons. ISBN 3-05-501708-0. - Mattuck Richard D. (1992),
*A Guide to Feynman Diagrams in the Many-Body Problem*, Dover Publications, ISBN 0-486-67047-3.

- Bogolyubov N. N., Tyablikov S. V. Retarded and advanced Green functions in statistical physics, Soviet Physics Doklady, Vol.
**4**, p. 589 (1959). - Zubarev D. N., Double-time Green functions in statistical physics, Soviet Physics Uspekhi
**3**(3), 320–345 (1960).

- Linear Response Functions in Eva Pavarini, Erik Koch, Dieter Vollhardt, and Alexander Lichtenstein (eds.): DMFT at 25: Infinite Dimensions, Verlag des Forschungszentrum Jülich, 2014 ISBN 978-3-89336-953-9

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