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).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.Ch.: 9 Green's functions, p 6., J Peacock, FOURIER ANALYSIS LECTURE COURSE: LECTURE 15.
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
satisfying
where denotes the Hamiltonian, denotes the Dirac delta-function and is the Heaviside step function. The kernel of the above Schrödinger differential operator in the big parentheses is denoted by and called the propagator.While the term propagator sometimes refers to as well, this article will use the term to refer to .
This propagator may also be written as the transition amplitude
where is the unitary time-evolution operator for the system taking states at time to states at time . Note the initial condition enforced by
The propagator may also be found by using a path integral:
where denotes the Lagrangian and the boundary conditions are given by . The paths that are summed over move only forwards in time and are integrated with the differential following the path in time.
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
If only depends on the difference , this is a convolution of the initial wave function and the propagator.
Examples
For a time-translationally invariant system, the propagator only depends on the time difference , so it may be rewritten as
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,E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, (1937) 158–164.Wolfgang Pauli, Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) . Section 44.
The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,Kolsrud, M. (1956). Exact quantum dynamical solutions for oscillator-like systems, Physical Review 104(4), 1186.
valid for operators and satisfying the Heisenberg relation .
For the -dimensional case, the propagator can be simply obtained by the product
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 satisfying
where are two points in Minkowski spacetime with metric signature , 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 and the reduced Planck constant 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 This equation can be inverted in the sense of distributions, noting that the equation has the solution (see Sokhotski–Plemelj theorem)
with implying the limit to zero. Below, we discuss the right choice of the sign arising from causality requirements.
Using a particular expression for the Dirac delta function, the solution is:
where
is the 4-vector Minkowski 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 integral.
The integrand then has two poles at
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 is to the future of , so it is zero if .
This choice of contour is equivalent to calculating the limit,
Here
is the Heaviside step function,
is the proper time from to , and is a Bessel function of the first kind. The propagator is non-zero only if , i.e., causally precedes , which, for Minkowski spacetime, means and
This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,
where
Advanced propagator
A contour going anti-clockwise under both poles gives the causal advanced propagator. This is zero if is spacelike or if is to the past of , so it is zero if .
This choice of contour is equivalent to calculating the limit
This expression can also be expressed in terms of the vacuum expectation value of the commutator of the free scalar field. In this case,
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
Here, is a Hankel function and 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,
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 two functions, one for the particle and one for its anti-particle, 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.
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 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 (see below).
For a 4-momentum the causal and Feynman propagators in momentum space are:
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 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 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:
This is the usual definition but normalised by a factor of . Then the rule is that one only takes the limit at the end of a calculation.
One sees that
and
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:
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.