Propagator
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In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal lines of Feynman diagrams. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions.
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Momentum space propagator
For example, a free scalar field, which has spinless quanta, classically obeys the Klein-Gordon equation
- <math>(\square+m^2)\Phi=0<math>
where <math>\square<math> is another way of writing <math>\partial^2 = \partial^\mu \partial_\mu = {\partial^2 \over \partial t^2} - {\partial^2 \over \partial x^2} - {\partial^2 \over \partial y^2} - {\partial^2 \over \partial z^2}<math>.
(As typical in relativistic quantum field theory calculations, we use units where the speed of light is 1.)
Its momentum-space propagator is obtained by making the identification <math>i\partial^\mu \Rightarrow p^\mu<math> and formally taking the reciprocal of the operator acting on the field:
- <math>G = 1 / (\square+m^2) \quad \Rightarrow \quad G(p) = -1/(p^2-m^2+i\epsilon)<math>
where <math>m<math> is the particle's mass and <math>p^2<math> is short for <math>p^\mu p_\mu<math>, the inner product of the four-momentum with itself.
An infinitesimal imaginary number is usually added to the denominator to incorporate causality (see below).
For purposes of Feynman diagram calculations it is usually convenient to write this with an additional overall factor of <math>-i<math> (conventions vary).
Position space propagator
In order to get the propagator in position space (i.e. the first meaning mentioned above), one has to perform a Fourier transformation on energy-momentum space:
- <math>G(x-x') = -\int{{d^4 p\over (2\pi)^4} \, e^{-i p \cdot (x-x')}}\, {1\over(p^2-m^2+i\epsilon)}<math>
Here <math>x<math> and <math>x'<math> are two points in spacetime, and the dot in the exponent is a four-vector inner product.
This expression can be derived directly from the field theory as the vacuum expectation value of the time-ordered product of two quantum fields, that is, the product always taken such that the time ordering of the spacetime points is the same:
| <math> G(x-x') <math> | <math> = i<0|T(\Phi(x) \Phi(x'))|0> <math> |
| <math> = i<0| [ \Theta(x^0 - x'^0) \Phi(x)\Phi(x') + \Theta(x'^0 - x^0) \Phi(x')\Phi(x) ] |0> <math> |
where <math>\Theta (x) = \left \{ \begin{matrix} 1 & \mbox{for} & x \ge 0 \\ 0 & \mbox{for} & x < 0 \end{matrix} \right. <math>
This expression is Lorentz invariant as long as the field operators commute with one another when the points <math>x<math> and <math>x'<math> 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, then show that the <math>\Theta<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.
Faster than light?
This function 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 traveling 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 <math>\Phi(x)<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.
In terms of 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 disappear 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 causality violation is involved.
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
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 momentum-space propagator used in Feynman diagrams for a Dirac field representing the electron in quantum electrodynamics has the form
- <math>{i (\gamma^\mu p_\mu + m) \over p^2 - m^2 + i \epsilon}<math>
where the <math>\gamma^\mu<math> are the same matrices appearing in the covariant formulation of the Dirac equation. It is sometimes written
- <math>{i \over \gamma^\mu p_\mu - m + i\epsilon} <math> or <math>{i \over p\!\!\!/ - m + i\epsilon} <math>
for short.
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\epsilon }<math>
where <math>g^{\mu \nu}<math> is the metric tensor.
References
- Griffiths, David J., Introduction to Elementary Particles, New York: John Wiley & Sons, 1987. ISBN 0-471-60386-4
- Itzykson, Claude, and Zuber, Jean-Bernard, Quantum Field Theory, New York: McGraw-Hill, 1980. ISBN 0-07-032071-3
- Pokorski, Stefan, Gauge Field Theories, Cambridge: Cambridge University Press, 1987. ISBN 0-521-36846-4 (Has useful appendices of Feynman diagram rules, including propagators, in the back.)