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In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
event described by the respective diagram. These may also be viewed as the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
of the wave operator appropriate to the particle, and are, therefore, often called ''(causal) Green's functions'' (called "''causal''" to distinguish it from the elliptic Laplacian Green's function).


Non-relativistic propagators

In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a
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 ...
to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t). Consider a system with Hamiltonian . The Green's function ( fundamental solution) for the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is a function : G(x, t; x', t') = \frac \Theta(t - t') K(x, t; x', t') satisfying : \left( i\hbar \frac - H_x \right) G(x, t; x', t') = \delta(x - x') \delta(t - t'), where denotes the Hamiltonian written in terms of the coordinates, denotes the Dirac delta-function, is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
and is the kernel of the above Schrödinger differential operator in the big parentheses. The term ''propagator'' is sometimes used in this context to refer to , and sometimes to . This article will use the term to refer to (see Duhamel's principle). This propagator may also be written as the transition amplitude : K(x, t; x', t') = \big\langle x \big, \hat(t, t') \big, x' \big\rangle, where is the
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation In mathematics, a unitary representation of a grou ...
time-evolution operator for the system taking states at time to states at time . Note the initial condition enforced by \lim_ K(x, t; x', t') = \delta(x - x'). The quantum-mechanical propagator may also be found by using a path integral: : K(x, t; x', t') = \int \exp \left frac \int_t^ L(\dot, q, t) \, dt\rightD (t) where the boundary conditions of the path integral include . Here denotes the Lagrangian of the system. The paths that are summed over move only forwards in time and are integrated with the differential D (t)/math> following the path in time. In non-relativistic
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is specified by the equation : \psi(x, t) = \int_^\infty \psi(x', t') K(x, t; x', t') \, dx'. If only depends on the difference , this is a
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the initial wave function and the propagator.


Basic examples: propagator of free particle and harmonic oscillator

For a time-translationally invariant system, the propagator only depends on the time difference , so it may be rewritten as : K(x, t; x', t') = K(x, x'; t - t'). The propagator of a one-dimensional free particle, obtainable from, e.g., the path integral, is then Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel, The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity, : \exp \left( -\frac \left( \frac \mathsf^2 + \frac m\omega^2 \mathsf^2 \right) \right) :: = \exp \left( -\frac \mathsf^2\tan\frac \right) \exp \left( -\frac\mathsf^2 \sin(\omega t) \right) \exp \left( -\frac \mathsf^2 \tan\frac \right), valid for operators \mathsf and \mathsf satisfying the Heisenberg relation mathsf,\mathsf= i\hbar. For the -dimensional case, the propagator can be simply obtained by the product : K(\vec, \vec'; t) = \prod_^N K(x_q, x_q'; t).


Relativistic propagators

In relativistic quantum mechanics and quantum field theory the propagators are
Lorentz-invariant In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
. They give the amplitude for a
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 ...
to travel between two
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
points.


Scalar propagator

In quantum field theory, the theory of a free (or non-interacting) scalar field is a useful and simple example which serves to illustrate the concepts needed for more complicated theories. It describes spin-zero particles. There are a number of possible propagators for free scalar field theory. We now describe the most common ones.


Position space

The position space propagators are Green's functions for the Klein–Gordon equation. This means that they are functions satisfying : (\square_x + m^2) G(x, y) = -\delta(x - y), where : are two points in
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 ...
, : \square_x = \tfrac - \nabla^2 is the d'Alembertian operator acting on the coordinates, : is the Dirac delta-function. (As typical in relativistic quantum field theory calculations, we use units where the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
and
Planck's reduced constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equiv ...
are set to unity.) We shall restrict attention to 4-dimensional
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 ...
. We can perform a
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 the equation for the propagator, obtaining : (-p^2 + m^2) G(p) = -1. This equation can be inverted in the sense of distributions, noting that the equation has the solution (see Sokhotski–Plemelj theorem) : f(x) = \frac = \frac \mp i\pi\delta(x), with implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements. The solution is where : p(x - y) := p_0(x^0 - y^0) - \vec \cdot (\vec - \vec) is the 4-vector inner product. The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator. The choice of contour is usually phrased in terms of the p_0 integral. The integrand then has two poles at : p_0 = \pm \sqrt, so different choices of how to avoid these lead to different propagators.


Causal propagators


=Retarded propagator

= A contour going clockwise over both poles gives the causal retarded propagator. This is zero if is spacelike or if (i.e. if is to the future of ). This choice of contour is equivalent to calculating the
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, :G_\text(x,y) = \lim_ \frac \int d^4p \, \frac = -\frac \delta(\tau_^2) + \Theta(x-y)\Theta(\tau_^2)\frac Here :\Theta (x) := \begin 1 & x \ge 0 \\ 0 & x < 0 \end is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
and :\tau_:= \sqrt is the proper time from to and J_1 is a Bessel function of the first kind. The expression y \prec x means causally precedes which, for Minkowski spacetime, means :y^0 < x^0 and \tau_^2 \geq 0 ~. This expression can be related to the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
of the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of the free scalar field operator, :G_\text(x,y) = i \langle 0, \left \Phi(x), \Phi(y) \right, 0\rangle \Theta(x^0 - y^0) where :\left Phi(x),\Phi(y) \right= \Phi(x) \Phi(y) - \Phi(y) \Phi(x) is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
.


=Advanced propagator

= A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if is spacelike or if (i.e. if is to the past of ). This choice of contour is equivalent to calculating the limit : G_\text(x,y) = \lim_ \frac \int d^4p \, \frac = -\frac\delta(\tau_^2) + \Theta(y-x)\Theta(\tau_^2)\frac This expression can also be expressed in terms of the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
of the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of the free scalar field. In this case, :G_\text(x,y) = -i \langle 0, \left \Phi(x), \Phi(y) \right0\rangle \Theta(y^0 - x^0)~.


Feynman propagator

A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948. This choice of contour is equivalent to calculating the limit :G_F(x,y) = \lim_ \frac \int d^4p \, \frac = \begin -\frac \delta(s) + \frac H_1^(m \sqrt) & s \geq 0 \\ -\frac K_1(m \sqrt) & s < 0.\end Here :s:= (x^0 - y^0)^2 - (\vec - \vec)^2, where and are two points in
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 ...
, and the dot in the exponent is a four-vector
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. is a
Hankel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
and is a
modified Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...
. This expression can be derived directly from the field theory as the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
of the '' time-ordered product'' of the free scalar field, that is, the product always taken such that the time ordering of the spacetime points is the same, : \begin G_F(x-y) & = -i \lang 0, T(\Phi(x) \Phi(y)), 0 \rang \\ pt& = -i \left \lang 0, \left Theta(x^0 - y^0) \Phi(x)\Phi(y) + \Theta(y^0 - x^0) \Phi(y)\Phi(x) \right, 0 \right \rang. \end This expression is Lorentz invariant, as long as the field operators commute with one another when the points and are separated by a spacelike interval. The usual derivation is to insert a complete set of single-particle momentum states between the fields with Lorentz covariant normalization, and then to show that the functions providing the causal time ordering may be obtained by a contour integral along the energy axis, if the integrand is as above (hence the infinitesimal imaginary part), to move the pole off the real line. The propagator may also be derived using the path integral formulation of quantum theory.


Momentum space propagator

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 the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators. They are often written with an explicit term although this is understood to be a reminder about which integration contour is appropriate (see above). This term is included to incorporate boundary conditions and
causality Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
(see below). For a 4-momentum the causal and Feynman propagators in momentum space are: :\tilde_\text(p) = \frac :\tilde_\text(p) = \frac :\tilde_F(p) = \frac. For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of (conventions vary).


Faster than light?

The Feynman propagator has some properties that seem baffling at first. In particular, unlike the commutator, the propagator is ''nonzero'' outside of the light cone, though it falls off rapidly for spacelike intervals. Interpreted as an amplitude for particle motion, this translates to the virtual particle travelling faster than light. It is not immediately obvious how this can be reconciled with causality: can we use faster-than-light virtual particles to send faster-than-light messages? The answer is no: while in
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
the intervals along which particles and causal effects can travel are the same, this is no longer true in quantum field theory, where it is
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
s that determine which operators can affect one another. So what ''does'' the spacelike part of the propagator represent? In QFT the
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 ...
is an active participant, and
particle number The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which i ...
s and field values are related by an uncertainty principle; field values are uncertain even for particle number ''zero''. There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field if one measures it locally (or, to be more precise, if one measures an operator obtained by averaging the field over a small region). Furthermore, the dynamics of the fields tend to favor spatially correlated fluctuations to some extent. The nonzero time-ordered product for spacelike-separated fields then just measures the amplitude for a nonlocal correlation in these vacuum fluctuations, analogous to an EPR correlation. Indeed, the propagator is often called a ''two-point correlation function'' for the free field. Since, by the postulates of quantum field theory, all observable operators commute with each other at spacelike separation, messages can no more be sent through these correlations than they can through any other EPR correlations; the correlations are in random variables. Regarding virtual particles, the propagator at spacelike separation can be thought of as a means of calculating the amplitude for creating a virtual particle- antiparticle pair that eventually disappears into the vacuum, or for detecting a virtual pair emerging from the vacuum. In Feynman's language, such creation and annihilation processes are equivalent to a virtual particle wandering backward and forward through time, which can take it outside of the light cone. However, no signaling back in time is allowed.


Explanation using limits

This can be made clearer by writing the propagator in the following form for a massless photon: : G^\varepsilon_F(x, y) = \frac. This is the usual definition but normalised by a factor of \varepsilon. Then the rule is that one only takes the limit \varepsilon \to 0 at the end of a calculation. One sees that : G^\varepsilon_F(x, y) = \frac if (x - y)^2 = 0, and : \lim_ G^\varepsilon_F(x, y) = 0 if (x - y)^2 \neq 0. Hence this means that a single photon will always stay on the light cone. It is also shown that the total probability for a photon at any time must be normalised by the reciprocal of the following factor: : \lim_ \int , G^\varepsilon_F(0, x), ^2 \, dx^3 = \lim_ \int \frac \, dx^3 = 2 \pi^2 , t, . We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.


Propagators in Feynman diagrams

The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams. These calculations are usually carried out in momentum space. In general, the amplitude gets a factor of the propagator for every ''internal line'', that is, every line that does not represent an incoming or outgoing particle in the initial or final state. It will also get a factor proportional to, and similar in form to, an interaction term in the theory's Lagrangian for every internal vertex where lines meet. These prescriptions are known as ''Feynman rules''. Internal lines correspond to virtual particles. Since the propagator does not vanish for combinations of energy and momentum disallowed by the classical equations of motion, we say that the virtual particles are allowed to be
off 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 ...
. In fact, since the propagator is obtained by inverting the wave equation, in general, it will have singularities on shell. The energy carried by the particle in the propagator can even be ''negative''. This can be interpreted simply as the case in which, instead of a particle going one way, its antiparticle is going the ''other'' way, and therefore carrying an opposing flow of positive energy. The propagator encompasses both possibilities. It does mean that one has to be careful about minus signs for the case of fermions, whose propagators are not
even function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
s in the energy and momentum (see below). Virtual particles conserve energy and momentum. However, since they can be off shell, wherever the diagram contains a closed ''loop'', the energies and momenta of the virtual particles participating in the loop will be partly unconstrained, since a change in a quantity for one particle in the loop can be balanced by an equal and opposite change in another. Therefore, every loop in a Feynman diagram requires an integral over a continuum of possible energies and momenta. In general, these integrals of products of propagators can diverge, a situation that must be handled by the process of renormalization.


Other theories


Spin

If the particle possesses spin then its propagator is in general somewhat more complicated, as it will involve the particle's spin or polarization indices. The differential equation satisfied by the propagator for a spin particle is given by :(i\not\nabla' - m)S_F(x', x) = I_4\delta^4(x'-x), where is the unit matrix in four dimensions, and employing the
Feynman slash notation In the study of Dirac fields in quantum field theory, Richard Feynman invented the convenient Feynman slash notation (less commonly known as the Dirac slash notation). If ''A'' is a covariant vector (i.e., a 1-form), : \ \stackrel\ \gamma^1 A_1 + ...
. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation, :S_F(x', x) = \int\frac\exp\tilde S_F(p), the equation becomes : \begin & (i \not \nabla' - m)\int\frac\tilde S_F(p)\exp \\ pt= & \int\frac(\not p - m)\tilde S_F(p)\exp \\ pt= & \int\fracI_4\exp \\ pt= & I_4\delta^4(x'-x), \end where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus :(\not p - m I_4)\tilde S_F(p) = I_4. By multiplying from the left with :(\not p + m) (dropping unit matrices from the notation) and using properties of the gamma matrices, : \begin \not p \not p & = \frac(\not p \not p + \not p \not p) \\ pt& = \frac(\gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) \\ pt& = \frac(\gamma_\mu \gamma_\nu + \gamma_\nu\gamma_\mu)p^\mu p^\nu \\ pt& = g_p^\mu p^\nu = p_\nu p^\nu = p^2, \end the momentum-space propagator used in Feynman diagrams for a Dirac field representing the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
in quantum electrodynamics is found to have form : \tilde_F(p) = \frac = \frac. The downstairs is a prescription for how to handle the poles in the complex -plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written :\tilde_F(p) = = for short. It should be remembered that this expression is just shorthand notation for . "One over matrix" is otherwise nonsensical. In position space one has :S_F(x-y) = \int \frac \, e^ \frac = \left( \frac + \frac \right) J_1(m , x-y, ). This is related to the Feynman propagator by :S_F(x-y) = (i \not \partial + m) G_F(x-y) where \not \partial := \gamma^\mu \partial_\mu.


Spin 1

The propagator for a
gauge boson In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of ga ...
in a
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
is :. The general form with gauge parameter , up to overall sign and the factor of i, reads : -i\frac. The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter , up to overall sign and the factor of i, reads : \frac+\frac. With these general forms one obtains the propagators in unitary gauge for , the propagator in Feynman or 't Hooft gauge for and in Landau or Lorenz gauge for . There are also other notations where the gauge parameter is the inverse of , usually denoted (see gauges). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter. Unitary gauge: :\frac. Feynman ('t Hooft) gauge: :\frac. Landau (Lorenz) gauge: :\frac.


Graviton propagator

The graviton propagator for Minkowski space in
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 ...
is :G_ = \frac - \frac = \frac, where D is the number of spacetime dimensions, \mathcal^2 is the transverse and traceless spin-2 projection operator and \mathcal^0_s is a spin-0 scalar multiplet. The graviton propagator for (Anti) de Sitter space is :G = \frac + \frac, where H is the
Hubble constant Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
. Note that upon taking the limit H \to 0 and \Box \to -k^2, the AdS propagator reduces to the Minkowski propagator.


Related singular functions

The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. We follow the notation in Bjorken and Drell. See also Bogolyubov and Shirkov (Appendix A). These functions are most simply defined in terms of the
vacuum expectation value In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle ...
of products of field operators.


Solutions to the Klein–Gordon equation


Pauli–Jordan function

The commutator of two scalar field operators defines the
Pauli Pauli is a surname and also a Finnish male given name (variant of Paul) and may refer to: * Arthur Pauli (born 1989), Austrian ski jumper * Barbara Pauli (1752 or 1753 - fl. 1781), Swedish fashion trader *Gabriele Pauli (born 1957), German politi ...
Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...
function \Delta(x-y) by :\langle 0 , \left \Phi(x),\Phi(y) \right, 0 \rangle = i \, \Delta(x-y) with :\,\Delta(x-y) = G_\text (x-y) - G_\text(x-y) This satisfies :\Delta(x-y) = -\Delta(y-x) and is zero if (x-y)^2 < 0.


Positive and negative frequency parts (cut propagators)

We can define the positive and negative frequency parts of \Delta(x-y), sometimes called cut propagators, in a relativistically invariant way. This allows us to define the positive frequency part: :\Delta_+(x-y) = \langle 0 , \Phi(x) \Phi(y) , 0 \rangle, and the negative frequency part: :\Delta_-(x-y) = \langle 0 , \Phi(y) \Phi(x) , 0 \rangle. These satisfy :\,i \Delta = \Delta_+ - \Delta_- and :(\Box_x + m^2) \Delta_(x-y) = 0.


Auxiliary function

The anti-commutator of two scalar field operators defines \Delta_1(x-y) function by :\langle 0 , \left\ , 0 \rangle = \Delta_1(x-y) with :\,\Delta_1(x-y) = \Delta_+ (x-y) + \Delta_-(x-y). This satisfies \,\Delta_1(x-y) = \Delta_1(y-x).


Green's functions for the Klein–Gordon equation

The retarded, advanced and Feynman propagators defined above are all Green's functions for the Klein–Gordon equation. They are related to the singular functions by :G_\text(x-y) = -\Delta(x-y) \Theta(x_0-y_0) :G_\text(x-y) = \Delta(x-y) \Theta(y_0-x_0) :2 G_F(x-y) = -i \,\Delta_1(x-y) + \varepsilon(x_0 - y_0) \,\Delta(x-y) where \varepsilon(x_0-y_0) is the sign of x_0-y_0.


Notes


References

* (Appendix C.) * (Especially pp. 136–156 and Appendix A) * (section Dynamical Theory of Groups & Fields, Especially pp. 615–624) * * * * * * * * ''(Has useful appendices of Feynman diagram rules, including propagators, in the back.)'' * *Scharf, G. (1995). ''Finite Quantum Electrodynamics, The Causal Approach.'' Springer. {{ISBN, 978-3-642-63345-4.


External links


Three Methods for Computing the Feynman Propagator
Quantum mechanics Quantum field theory Theoretical physics Mathematical physics