Propagator
The kernel of the above Schrödinger differential operator in the big parentheses is denoted by K(x, t ;x′, t′) and called the propagator.The propagator may also be found by using a path integral: where L denotes the Lagrangian and the boundary conditions are given by q(t) = x, q(t′) = x′.For a time-translationally invariant system, the propagator only depends on the time difference t − t′, so it may be rewritten asThe latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,[7]where (As typical in relativistic quantum field theory calculations, we use units where the speed of light c and the reduced Planck constant ħ are set to unity.)The different choices for how to deform the integration contour in the above expression lead to various forms for the propagator., i.e., y causally precedes x, which, for Minkowski spacetime, means This expression can be related to the vacuum expectation value of the commutator of the free scalar field operator,This expression is Lorentz invariant, as long as the field operators commute with one another when the points x and y 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 Θ 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.They are often written with an explicit ε term although this is understood to be a reminder about which integration contour is appropriate (see above).This ε term is included to incorporate boundary conditions and causality (see below).In particular, unlike the commutator, the propagator is nonzero outside of the light cone, though it falls off rapidly for spacelike intervals.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.There is a nonzero probability amplitude to find a significant fluctuation in the vacuum value of the field Φ(x) 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.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.We see that the parts outside the light cone usually are zero in the limit and only are important in Feynman diagrams.The most common use of the propagator is in calculating probability amplitudes for particle interactions using Feynman diagrams.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.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.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.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).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.The differential equation satisfied by the propagator for a spin 1⁄2 particle is given by[13] where I4 is the unit matrix in four dimensions, and employing the Feynman slash notation.the equation becomes where on the right-hand side an integral representation of the four-dimensional delta function is used.the momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics is found to have form The iε downstairs is a prescription for how to handle the poles in the complex p0-plane.The name of the propagator, however, refers to its final form and not necessarily to the value of the gauge parameter.These functions are most simply defined in terms of the vacuum expectation value of products of field operators.