Editing Feynman diagram (section)

=== Scattering ===
The correlation functions of a quantum field theory describe the scattering of particles. The definition of "particle" in relativistic field theory is not self-evident, because if you try to determine the position so that the uncertainty is less than the [[compton wavelength]], the uncertainty in energy is large enough to produce more particles and antiparticles of the same type from the vacuum. This means that the notion of a single-particle state is to some extent incompatible with the notion of an object localized in space.

In the 1930s, [[Eugene Wigner|Wigner]] gave a mathematical definition for single-particle states: they are a collection of states that form an irreducible representation of the [[Poincaré group]]. Single particle states describe an object with a finite mass, a well defined momentum, and a spin. This definition is fine for protons and neutrons, electrons and photons, but it excludes quarks, which are permanently confined, so the modern point of view is more accommodating: a particle is anything whose interaction can be described in terms of Feynman diagrams, which have an interpretation as a sum over particle trajectories.

A field operator can act to produce a one-particle state from the vacuum, which means that the field operator {{mvar|''φ''(''x'')}} produces a superposition of Wigner particle states. In the free field theory, the field produces one particle states only. But when there are interactions, the field operator can also produce 3-particle, 5-particle (if there is no +/− symmetry also 2, 4, 6 particle) states too. To compute the scattering amplitude for single particle states only requires a careful limit, sending the fields to infinity and integrating over space to get rid of the higher-order corrections.

The relation between scattering and correlation functions is the LSZ-theorem: The scattering amplitude for {{mvar|n}} particles to go to {{mvar|m}} particles in a scattering event is the given by the sum of the Feynman diagrams that go into the correlation function for {{math|''n'' + ''m''}} field insertions, leaving out the propagators for the external legs.

For example, for the {{math|''λφ''<sup>4</sup>}} interaction of the previous section, the order {{mvar|λ}} contribution to the (Lorentz) correlation function is:

:<math> \left\langle \phi(k_1)\phi(k_2)\phi(k_3)\phi(k_4)\right\rangle = \frac{i}{k_1^2}\frac{i}{k_2^2} \frac{i}{k_3^2} \frac{i}{k_4^2} i\lambda \,</math>

Stripping off the external propagators, that is, removing the factors of {{math|{{sfrac|''i''|''k''<sup>2</sup>}}}}, gives the invariant scattering amplitude {{mvar|M}}:

:<math> M = i\lambda \,</math>

which is a constant, independent of the incoming and outgoing momentum. The interpretation of the scattering amplitude is that the sum of {{math|{{abs|''M''}}<sup>2</sup>}} over all possible final states is the probability for the scattering event. The normalization of the single-particle states must be chosen carefully, however, to ensure that {{mvar|M}} is a relativistic invariant.

Non-relativistic single particle states are labeled by the momentum {{mvar|k}}, and they are chosen to have the same norm at every value of {{mvar|k}}. This is because the nonrelativistic unit operator on single particle states is:

:<math> \int dk\, |k\rangle\langle k|\,. </math>

In relativity, the integral over the {{mvar|k}}-states for a particle of mass m integrates over a hyperbola in {{math|''E'',''k''}} space defined by the energy–momentum relation:

:<math> E^2 - k^2 = m^2 \,.</math>

If the integral weighs each {{mvar|k}} point equally, the measure is not Lorentz-invariant. The invariant measure integrates over all values of {{mvar|k}} and {{mvar|E}}, restricting to the hyperbola with a Lorentz-invariant delta function:

:<math> \int \delta(E^2-k^2 - m^2) |E,k\rangle\langle E,k|\, dE\, dk = \int {dk \over 2 E} |k\rangle\langle k|\,.</math>

So the normalized {{mvar|k}}-states are different from the relativistically normalized {{mvar|k}}-states by a factor of
:<math>\sqrt{E} = \left(k^2-m^2\right)^\frac14\,.</math>

The invariant amplitude {{mvar|M}} is then the probability amplitude for relativistically normalized incoming states to become relativistically normalized outgoing states.

For nonrelativistic values of {{mvar|k}}, the relativistic normalization is the same as the nonrelativistic normalization (up to a constant factor {{math|{{sqrt|''m''}}}}). In this limit, the {{math|''φ''<sup>4</sup>}} invariant scattering amplitude is still constant. The particles created by the field {{mvar|φ}} scatter in all directions with equal amplitude.

The nonrelativistic potential, which scatters in all directions with an equal amplitude (in the [[Born approximation]]), is one whose Fourier transform is constant—a delta-function potential. The lowest order scattering of the theory reveals the non-relativistic interpretation of this theory—it describes a collection of particles with a delta-function repulsion. Two such particles have an aversion to occupying the same point at the same time.