Propagator

Template:Short description Template:About Template:Use American EnglishTemplate:Quantum field theory In quantum mechanics 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 event described by the respective diagram. Propagators may also be viewed as the inverse 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).<ref>The mathematics of PDEs and the wave equation, p 32., Michael P. Lamoureux, University of Calgary, Seismic Imaging Summer School, August 7–11, 2006, Calgary.</ref><ref>Ch.: 9 Green's functions, p 6., J Peacock, FOURIER ANALYSIS LECTURE COURSE: LECTURE 15.</ref>

Non-relativistic propagators

In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a particle to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).

The Green's function G for the Schrödinger equation is a function <math display="block">G(x, t; x', t') = \frac{1}{i\hbar} \Theta(t - t') K(x, t; x', t')</math> satisfying <math display="block">\left( i\hbar \frac{\partial}{\partial t} - H_x \right) G(x, t; x', t') = \delta(x - x') \delta(t - t'),</math> where Template:Math denotes the Hamiltonian, Template:Math denotes the Dirac delta-function and Template:Math is the Heaviside step function. The kernel of the above Schrödinger differential operator in the big parentheses is denoted by Template:Math and called the propagator.<ref group=nb> While the term propagator sometimes refers to Template:Mvar as well, this article will use the term to refer to Template:Mvar.</ref>

This propagator may also be written as the transition amplitude <math display="block">K(x, t; x', t') = \big\langle x \big| U(t, t') \big| x' \big\rangle,</math> where Template:Math is the unitary time-evolution operator for the system taking states at time Template:Mvar to states at time Template:Mvar.Template:Sfn Note the initial condition enforced by <math display="block">\lim_{t \to t'} K(x, t; x', t') = \delta(x - x').</math> The propagator may also be found by using a path integral:

<math>K(x, t; x', t') = \int \exp \left[\frac{i}{\hbar} \int_{t'}^{t} L(\dot{q}, q, t) \, dt\right] D[q(t)],</math>

where Template:Mvar denotes the Lagrangian and the boundary conditions are given by Template:Math. The paths that are summed over move only forwards in time and are integrated with the differential <math>D[q(t)]</math> following the path in time.Template:Sfn

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 given by

<math>\psi(x, t) = \int_{-\infty}^\infty \psi(x', t') K(x, t; x', t') \, dx'.</math>

If Template:Math only depends on the difference Template:Math, this is a convolution of the initial wave function and the propagator.

Examples

Template:See also For a time-translationally invariant system, the propagator only depends on the time difference Template:Math, so it may be rewritten as <math display="block">K(x, t; x', t') = K(x, x'; t - t').</math>

The propagator of a one-dimensional free particle, obtainable from, e.g., the path integral, is then Template:Equation box 1 = \left(\frac{m}{2\pi i\hbar t}\right)^{\frac{1}{2}} e^{-\frac{m(x-x')^2}{2i\hbar t}}.</math> |border colour = #0073CF |bgcolor = #F9FFF7}}

Similarly, the propagator of a one-dimensional quantum harmonic oscillator is the Mehler kernel,<ref>E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, (1937) 158–164.</ref><ref>Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) Template:ISBN. Section 44.</ref> Template:Equation box 1 \exp\left(-\frac{m\omega\big((x^2 + x'^2) \cos\omega t - 2xx'\big)}{2i\hbar \sin\omega t}\right).</math> |border colour = #0073CF |bgcolor = #F9FFF7}} The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,<ref>Kolsrud, M. (1956). Exact quantum dynamical solutions for oscillator-like systems, Physical Review 104(4), 1186.</ref> <math display="block">\begin{align} &\exp \left( -\frac{it}{\hbar} \left( \frac{1}{2m} \mathsf{p}^2 + \frac{1}{2} m\omega^2 \mathsf{x}^2 \right) \right) \\ &= \exp \left( -\frac{im\omega}{2\hbar} \mathsf{x}^2\tan\frac{\omega t}{2} \right) \exp \left( -\frac{i}{2m\omega \hbar}\mathsf{p}^2 \sin(\omega t) \right) \exp \left( -\frac{im\omega }{2\hbar} \mathsf{x}^2 \tan\frac{\omega t}{2} \right), \end{align}</math> valid for operators <math>\mathsf{x}</math> and <math>\mathsf{p}</math> satisfying the Heisenberg relation <math>[\mathsf{x},\mathsf{p}] = i\hbar</math>.

For the Template:Mvar-dimensional case, the propagator can be simply obtained by the product <math display="block">K(\vec{x}, \vec{x}'; t) = \prod_{q=1}^N K(x_q, x_q'; t).</math>

Relativistic propagators

In relativistic quantum mechanics and quantum field theory the propagators are Lorentz-invariant. They give the amplitude for a particle to travel between two spacetime events.

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 Template:Math satisfying <math display="block">\left(\square_x + m^2\right) G(x, y) = -\delta(x - y),</math> where

• Template:Mvar are two points in Minkowski spacetime,
• <math>\square_x = \tfrac{\partial^2}{\partial t^2} - \nabla^2</math> is the d'Alembertian operator acting on the Template:Mvar coordinates,
• Template:Math is the Dirac delta function.

(As typical in relativistic quantum field theory calculations, we use units where the speed of light Template:Mvar and the reduced Planck constant Template:Mvar are set to unity.)

We shall restrict attention to 4-dimensional Minkowski spacetime. We can perform a Fourier transform of the equation for the propagator, obtaining <math display="block">\left(-p^2 + m^2\right) G(p) = -1.</math>

This equation can be inverted in the sense of distributions, noting that the equation Template:Math has the solution (see Sokhotski–Plemelj theorem) <math display="block">f(x) = \frac{1}{x \pm i\varepsilon} = \frac{1}{x} \mp i\pi\delta(x),</math> with Template:Mvar implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.

The solution is Template:Equation box 1{p^2 - m^2 \pm i\varepsilon},</math> |border colour = #0073CF |bgcolor=#F9FFF7}} where <math display="block">p(x - y) := p_0(x^0 - y^0) - \vec{p} \cdot (\vec{x} - \vec{y})</math> 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 <math>p_0</math> integral.

The integrand then has two poles at <math display="block">p_0 = \pm \sqrt{\vec{p}^2 + m^2},</math> 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 Template:Mvar is spacelike or Template:Mvar is to the future of Template:Mvar, so it is zero if Template:Math.

This choice of contour is equivalent to calculating the limit, <math display="block">G_\text{ret}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(x^0 - y^0)}{2\pi} \delta(\tau_{xy}^2) + \Theta(x^0 - y^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}.</math>

Here <math display="block">\Theta (x) := \begin{cases} 1 & x \ge 0 \\ 0 & x < 0 \end{cases}</math> is the Heaviside step function, <math display="block">\tau_{xy}:= \sqrt{ (x^0 - y^0)^2 - (\vec{x} - \vec{y})^2}</math> is the proper time from Template:Mvar to Template:Mvar, and <math>J_1</math> is a Bessel function of the first kind. The propagator is non-zero only if <math>y \prec x</math>, i.e., Template:Mvar causally precedes Template:Mvar, which, for Minkowski spacetime, means

<math>y^0 \leq x^0</math> and <math>\tau_{xy}^2 \geq 0 ~.</math>

This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator, <math display="block">G_\text{ret}(x,y) = -i \langle 0| \left[ \Phi(x), \Phi(y) \right] |0\rangle \Theta(x^0 - y^0),</math> where <math display="block">\left[\Phi(x), \Phi(y) \right] := \Phi(x) \Phi(y) - \Phi(y) \Phi(x).</math>

Advanced propagator

A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if Template:Mvar is spacelike or if Template:Mvar is to the past of Template:Mvar, so it is zero if Template:Math.

This choice of contour is equivalent to calculating the limit<ref>Template:Cite book</ref> <math display="block"> G_\text{adv}(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2\pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{(p_0 - i\varepsilon)^2 - \vec{p}^2 - m^2} = -\frac{\Theta(y^0-x^0)}{2\pi}\delta(\tau_{xy}^2) + \Theta(y^0-x^0)\Theta(\tau_{xy}^2)\frac{m J_1(m \tau_{xy})}{4 \pi \tau_{xy}}. </math>

This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case, <math display="block">G_\text{adv}(x,y) = i \langle 0|\left[ \Phi(x), \Phi(y) \right]|0\rangle \Theta(y^0 - x^0)~.</math>

Feynman propagator

A contour going under the left pole and over the right pole gives the Feynman propagator, introduced by Richard Feynman in 1948.<ref>Template:Citation</ref>

This choice of contour is equivalent to calculating the limit<ref>Template:Cite book</ref> <math display="block">G_F(x,y) = \lim_{\varepsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p \, \frac{e^{-ip(x-y)}}{p^2 - m^2 + i\varepsilon} = \begin{cases} -\frac{1}{4 \pi} \delta(\tau_{xy}^2) + \frac{m}{8 \pi \tau_{xy}} H_1^{(1)}(m \tau_{xy}) & \tau_{xy}^2 \geq 0 \\ -\frac{i m}{ 4 \pi^2 \sqrt{-\tau_{xy}^2}} K_1(m \sqrt{-\tau_{xy}^2}) & \tau_{xy}^2 < 0. \end{cases} </math>

Here, Template:Math is a Hankel function and Template:Math is a modified Bessel function.

This expression can be derived directly from the field theory as the vacuum expectation value 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, <math display="block"> \begin{align} G_F(x-y) & = -i \lang 0|T(\Phi(x) \Phi(y))|0 \rang \\[4pt] & = -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{align}</math>

This expression is Lorentz invariant, as long as the field operators commute with one another when the points Template:Mvar and Template:Mvar 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 Template:Math 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.

Dirac propagator

Introduced by Paul Dirac in 1938.<ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Momentum space propagator

The Fourier transform 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 Template:Mvar term although this is understood to be a reminder about which integration contour is appropriate (see above). This Template:Mvar term is included to incorporate boundary conditions and causality (see below).

For a 4-momentum Template:Mvar the causal and Feynman propagators in momentum space are:

<math>\tilde{G}_\text{ret}(p) = \frac{1}{(p_0+i\varepsilon)^2 - \vec{p}^2 - m^2}</math>

<math>\tilde{G}_\text{adv}(p) = \frac{1}{(p_0-i\varepsilon)^2 - \vec{p}^2 - m^2}</math>

<math>\tilde{G}_F(p) = \frac{1}{p^2 - m^2 + i\varepsilon}. </math>

For purposes of Feynman diagram calculations, it is usually convenient to write these with an additional overall factor of Template:Mvar (conventions vary).

Faster than light?

Template:More citations needed section 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 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 commutators that determine which operators can affect one another.

So what does the spacelike part of the propagator represent? In QFT the vacuum is an active participant, and particle numbers 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 Template:Math 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 particle: <math display="block">G^\varepsilon_F(x, y) = \frac{\varepsilon}{(x - y)^2 + i \varepsilon^2}.</math>

This is the usual definition but normalised by a factor of <math>\varepsilon</math>. Then the rule is that one only takes the limit <math>\varepsilon \to 0</math> at the end of a calculation.

One sees that <math display="block">G^\varepsilon_F(x, y) = \frac{1}{\varepsilon} \quad\text{if}~~~ (x - y)^2 = 0,</math> and <math display="block">\lim_{\varepsilon \to 0} G^\varepsilon_F(x, y) = 0 \quad\text{if}~~~ (x - y)^2 \neq 0.</math> Hence this means that a single massless particle 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: <math display="block"> \lim_{\varepsilon \to 0} \int |G^\varepsilon_F(0, x)|^2 \, dx^3

= \lim_{\varepsilon \to 0} \int \frac{\varepsilon^2}{(\mathbf{x}^2 - t^2)^2 + \varepsilon^4} \, dx^3
= 2 \pi^2 |t|.

</math> 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 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 functions 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 Template:Frac

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 Template:Frac particle is given by<ref>Template:Harvnb</ref>

<math>(i\not\nabla' - m)S_F(x', x) = I_4\delta^4(x'-x),</math>

where Template:Math is the unit matrix in four dimensions, and employing the Feynman slash notation. This is the Dirac equation for a delta function source in spacetime. Using the momentum representation, <math display="block">S_F(x', x) = \int\frac{d^4p}{(2\pi)^4}\exp{\left[-ip \cdot(x'-x)\right]}\tilde S_F(p),</math> the equation becomes

<math>

\begin{align} & (i \not \nabla' - m)\int\frac{d^4p}{(2\pi)^4}\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & \int\frac{d^4p}{(2\pi)^4}(\not p - m)\tilde S_F(p)\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & \int\frac{d^4p}{(2\pi)^4}I_4\exp{\left[-ip \cdot(x'-x)\right]} \\[6pt] = {} & I_4\delta^4(x'-x), \end{align} </math>

where on the right-hand side an integral representation of the four-dimensional delta function is used. Thus

<math>(\not p - m I_4)\tilde S_F(p) = I_4.</math>

By multiplying from the left with <math display="block">(\not p + m)</math> (dropping unit matrices from the notation) and using properties of the gamma matrices, <math display="block">\begin{align} \not p \not p & = \tfrac{1}{2}(\not p \not p + \not p \not p) \\[6pt] & = \tfrac{1}{2}(\gamma_\mu p^\mu \gamma_\nu p^\nu + \gamma_\nu p^\nu \gamma_\mu p^\mu) \\[6pt] & = \tfrac{1}{2}(\gamma_\mu \gamma_\nu + \gamma_\nu\gamma_\mu)p^\mu p^\nu \\[6pt] & = g_{\mu\nu}p^\mu p^\nu = p_\nu p^\nu = p^2, \end{align}</math>

the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form

<math> \tilde{S}_F(p) = \frac{(\not p + m)}{p^2 - m^2 + i \varepsilon} = \frac{(\gamma^\mu p_\mu + m)}{p^2 - m^2 + i \varepsilon}.</math>

The Template:Math downstairs is a prescription for how to handle the poles in the complex Template:Math-plane. It automatically yields the Feynman contour of integration by shifting the poles appropriately. It is sometimes written

<math>\tilde{S}_F(p) = {1 \over \gamma^\mu p_\mu - m + i\varepsilon} = {1 \over \not p - m + i\varepsilon} </math>

for short. It should be remembered that this expression is just shorthand notation for Template:Math. "One over matrix" is otherwise nonsensical. In position space one has <math display="block">S_F(x-y) = \int \frac{d^4 p}{(2\pi)^4} \, e^{-i p \cdot (x-y)} \frac{\gamma^\mu p_\mu + m}{p^2 - m^2 + i \varepsilon} = \left( \frac{\gamma^\mu (x-y)_\mu}{|x-y|^5} + \frac{m}{|x-y|^3} \right) J_1(m |x-y|).</math>

This is related to the Feynman propagator by

<math>S_F(x-y) = (i \not \partial + m) G_F(x-y)</math>

where <math>\not \partial := \gamma^\mu \partial_\mu</math>.

Spin 1

The propagator for a gauge boson in a gauge theory depends on the choice of convention to fix the gauge. For the gauge used by Feynman and Stueckelberg, the propagator for a photon is

<math>{-i g^{\mu\nu} \over p^2 + i\varepsilon }.</math>

The general form with gauge parameter Template:Math, up to overall sign and the factor of <math>i</math>, reads

<math> -i\frac{g^{\mu\nu} + \left(1-\frac{1}{\lambda}\right)\frac{p^\mu p^\nu}{p^2}}{p^2+i\varepsilon}.</math>

The propagator for a massive vector field can be derived from the Stueckelberg Lagrangian. The general form with gauge parameter Template:Math, up to overall sign and the factor of <math>i</math>, reads

<math> \frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}+\frac{\frac{k_\mu k_\nu}{m^2}}{k^2-\frac{m^2}{\lambda}+i\varepsilon}.</math>

With these general forms one obtains the propagators in unitary gauge for Template:Math, the propagator in Feynman or 't Hooft gauge for Template:Math and in Landau or Lorenz gauge for Template:Math. There are also other notations where the gauge parameter is the inverse of Template:Mvar, usually denoted Template:Mvar (see [[Gauge fixing#Rξ gauges|Template:Math gauges]]). The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.

Unitary gauge:

<math>\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{m^2}}{k^2-m^2+i\varepsilon}.</math>

Feynman ('t Hooft) gauge:

<math>\frac{g_{\mu\nu}}{k^2-m^2+i\varepsilon}.</math>

Landau (Lorenz) gauge:

<math>\frac{g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}}{k^2-m^2+i\varepsilon}.</math>

Graviton propagator

The graviton propagator for Minkowski space in general relativity is <ref>Quantum theory of gravitation library.uu.nl</ref> <math display="block">G_{\alpha\beta~\mu\nu} = \frac{\mathcal{P}^2_{\alpha\beta~\mu\nu}}{k^2} - \frac{\mathcal{P}^0_s{}_{\alpha\beta~\mu\nu}}{2k^2} = \frac{g_{\alpha\mu} g_{\beta\nu}+ g_{\beta\mu}g_{\alpha\nu}- \frac{2}{D-2} g_{\mu\nu}g_{\alpha\beta}}{k^2},</math> where <math>D</math> is the number of spacetime dimensions, <math>\mathcal{P}^2</math> is the transverse and traceless spin-2 projection operator and <math>\mathcal{P}^0_s</math> is a spin-0 scalar multiplet. The graviton propagator for (Anti) de Sitter space is <math display="block">G = \frac{\mathcal{P}^2}{2H^2-\Box} + \frac{\mathcal{P}^0_s}{2(\Box+4H^2)},</math> where <math>H</math> is the Hubble constant. Note that upon taking the limit <math>H \to 0</math> and <math>\Box \to -k^2</math>, the AdS propagator reduces to the Minkowski propagator.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Related singular functions

Template:Further The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. These functions are most simply defined in terms of the vacuum expectation value of products of field operators.

Solutions to the Klein–Gordon equation

Pauli–Jordan function

The commutator of two scalar field operators defines the Pauli–Jordan function <math>\Delta(x-y)</math> by<ref>Template:Cite journal</ref><ref name="BD">Template:Cite book</ref>

<math>\langle 0 | \left[ \Phi(x),\Phi(y) \right] | 0 \rangle = i \, \Delta(x-y)</math>

with

<math>\,\Delta(x-y) = G_\text{ret} (x-y) - G_\text{adv}(x-y)</math>

This satisfies

<math>\Delta(x-y) = -\Delta(y-x)</math>

and is zero if <math>(x-y)^2 < 0</math>.

Positive and negative frequency parts (cut propagators)

We can define the positive and negative frequency parts of <math>\Delta(x-y)</math>, sometimes called cut propagators, in a relativistically invariant way.

This allows us to define the positive frequency part:

<math>\Delta_+(x-y) = \langle 0 | \Phi(x) \Phi(y) |0 \rangle, </math>

and the negative frequency part:

<math>\Delta_-(x-y) = \langle 0 | \Phi(y) \Phi(x) |0 \rangle. </math>

These satisfy<ref name="BD"/>

<math>\,i \Delta = \Delta_+ - \Delta_-</math>

and

<math>(\Box_x + m^2) \Delta_{\pm}(x-y) = 0.</math>

Auxiliary function

The anti-commutator of two scalar field operators defines <math>\Delta_1(x-y)</math> function by

<math>\langle 0 | \left\{ \Phi(x),\Phi(y) \right\} | 0 \rangle = \Delta_1(x-y)</math>

with

<math>\,\Delta_1(x-y) = \Delta_+ (x-y) + \Delta_-(x-y).</math>

This satisfies <math>\,\Delta_1(x-y) = \Delta_1(y-x).</math>

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<ref name="BD"/>

<math>G_\text{ret}(x-y) = \Delta(x-y) \Theta(x^0-y^0) </math>

<math>G_\text{adv}(x-y) = -\Delta(x-y) \Theta(y^0-x^0) </math>

<math>2 G_F(x-y) = -i \,\Delta_1(x-y) + \varepsilon(x^0 - y^0) \,\Delta(x-y) </math>

where <math>\varepsilon(x^0-y^0)</math> is the sign of <math>x^0-y^0</math>.

See also

• Source field
• LSZ reduction formula

References

• Template:Cite book (Appendix C.)
• Template:Cite book (Especially pp. 136–156 and Appendix A)
• Template:Cite book
• Template:Cite book (section Dynamical Theory of Groups & Fields, Especially pp. 615–624)
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Citation
• Template:Cite book
• Template:Cite book
• Template:Cite book (Has useful appendices of Feynman diagram rules, including propagators, in the back.)
• Template:Cite book
• Scharf, G. (1995). Finite Quantum Electrodynamics, The Causal Approach. Springer. Template:ISBN.

External links

• Three Methods for Computing the Feynman Propagator