Editing Feynman diagram (section)

== Particle-path representation ==
Feynman diagrams were originally discovered by Feynman, by trial and error, as a way to represent the contribution to the S-matrix from different classes of particle trajectories.

=== Schwinger representation<!--'Schwinger representation' redirects here--> ===
The Euclidean scalar propagator has a suggestive representation:

:<math> \frac{1}{p^2+m^2} = \int_0^\infty e^{-\tau\left(p^2 + m^2\right)}\, d\tau </math>

The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.

:<math> \Delta(x) = \int_0^\infty d\tau e^{-m^2\tau} \frac{1}{ ({4\pi\tau})^{d/2}}e^\frac{-x^2}{ 4\tau}</math>

The contribution at any one value of {{mvar|τ}} to the propagator is a Gaussian of width {{math|{{sqrt|''τ''}}}}. The total propagation function from 0 to {{mvar|x}} is a weighted sum over all proper times {{mvar|τ}} of a normalized Gaussian, the probability of ending up at {{mvar|x}} after a random walk of time {{mvar|τ}}.

The path-integral representation for the propagator is then:

:<math> \Delta(x) = \int_0^\infty d\tau \int DX\, e^{- \int\limits_0^{\tau} \left(\frac{\dot{x}^2}{2} + m^2\right) d\tau'} </math>

which is a path-integral rewrite of the '''Schwinger representation'''<!--boldface per WP:R#PLA-->.

The Schwinger representation is both useful for making manifest the particle aspect of the propagator, and for symmetrizing denominators of loop diagrams.

=== Combining denominators ===
The Schwinger representation has an immediate practical application to loop diagrams. For example, for the diagram in the {{math|''φ''<sup>4</sup>}} theory formed by joining two {{mvar|x}}s together in two half-lines, and making the remaining lines external, the integral over the internal propagators in the loop is:

: <math> \int_k \frac{1}{k^2 + m^2} \frac{1}{ (k+p)^2 + m^2} \,.</math>

Here one line carries momentum {{mvar|k}} and the other {{math|''k'' + ''p''}}. The asymmetry can be fixed by putting everything in the Schwinger representation.

:<math> \int_{t,t'} e^{-t(k^2+m^2) - t'\left((k+p)^2 +m^2\right) }\, dt\, dt'\,. </math>

Now the exponent mostly depends on {{math|''t'' + ''t''′}},

:<math> \int_{t,t'} e^{-(t+t')(k^2+m^2) - t' 2p\cdot k -t' p^2}\,, </math>

except for the asymmetrical little bit. Defining the variable {{math|''u'' {{=}} ''t'' + ''t''′}} and {{math|''v'' {{=}} {{sfrac|''t''′|''u''}}}}, the variable {{mvar|u}} goes from 0 to {{math|∞}}, while {{mvar|v}} goes from 0 to 1. The variable {{mvar|u}} is the total proper time for the loop, while {{mvar|v}} parametrizes the fraction of the proper time on the top of the loop versus the bottom.

The [[Jacobian matrix and determinant#Jacobian determinant|Jacobian]] for this transformation of variables is easy to work out from the identities:

:<math> d(uv)= dt'\quad du = dt+dt'\,,</math>

and "[[exterior product|wedging]]" gives

:<math> u\, du \wedge dv = dt \wedge dt'\,</math>.

This allows the {{mvar|u}} integral to be evaluated explicitly:

:<math> \int_{u,v} u e^{-u \left( k^2+m^2 + v 2p\cdot k + v p^2\right)} = \int \frac{1}{\left(k^2 + m^2 + v 2p\cdot k - v p^2\right)^2}\, dv </math>

leaving only the {{mvar|v}}-integral. This method, invented by Schwinger but usually attributed to Feynman, is called ''combining denominator''. Abstractly, it is the elementary identity:

:<math> \frac{1}{AB}= \int_0^1 \frac{1}{\big( vA+ (1-v)B\big)^2}\, dv </math>

But this form does not provide the physical motivation for introducing {{mvar|v}}; {{mvar|v}} is the proportion of proper time on one of the legs of the loop.

Once the denominators are combined, a shift in {{mvar|k}} to {{math|''k''′ {{=}} ''k'' + ''vp''}} symmetrizes everything:

:<math> \int_0^1 \int\frac{1}{\left(k^2 + m^2 + 2vp \cdot k + v p^2\right)^2}\, dk\, dv = \int_0^1 \int \frac{1}{\left(k'^2 + m^2 + v(1-v)p^2\right)^2}\, dk'\, dv</math>

This form shows that the moment that {{math|''p''<sup>2</sup>}} is more negative than four times the mass of the particle in the loop, which happens in a physical region of [[Lorentz space]], the integral has a cut. This is exactly when the external momentum can create physical particles.

When the loop has more vertices, there are more denominators to combine:

:<math> \int dk\, \frac{1}{k^2 + m^2} \frac{1}{(k+p_1)^2 + m^2} \cdots \frac{1}{(k+p_n)^2 + m^2}</math>

The general rule follows from the Schwinger prescription for {{mvar|''n'' + 1}} denominators:

:<math> \frac{1}{D_0 D_1 \cdots D_n} = \int_0^\infty \cdots\int_0^\infty e^{-u_0 D_0 \cdots -u_n D_n}\, du_0 \cdots du_n \,.</math>

The integral over the Schwinger parameters {{mvar|u<sub>i</sub>}} can be split up as before into an integral over the total proper time {{math|''u'' {{=}} ''u''<sub>0</sub> + ''u''<sub>1</sub> ... + ''u<sub>n</sub>''}} and an integral over the fraction of the proper time in all but the first segment of the loop {{math|''v<sub>i</sub>'' {{=}} {{sfrac|''u<sub>i</sub>''|''u''}}}} for {{math|''i'' ∈ {{mset|1,2,...,''n''}}}}. The {{mvar|v<sub>i</sub>}} are positive and add up to less than 1, so that the {{mvar|v}} integral is over an {{mvar|n}}-dimensional simplex.

The Jacobian for the coordinate transformation can be worked out as before:

:<math> du = du_0 + du_1 \cdots + du_n \,</math>
:<math> d(uv_i) = d u_i \,.</math>

Wedging all these equations together, one obtains

:<math> u^n\, du \wedge dv_1 \wedge dv_2 \cdots \wedge dv_n = du_0 \wedge du_1 \cdots \wedge du_n \,.</math>

This gives the integral:

:<math> \int_0^\infty \int_{\mathrm{simplex}} u^n e^{-u\left(v_0 D_0 + v_1 D_1 + v_2 D_2 \cdots + v_n D_n\right)}\, dv_1\cdots dv_n\, du\,, </math>

where the simplex is the region defined by the conditions
:<math>v_i>0 \quad \mbox{and} \quad \sum_{i=1}^n v_i < 1 </math>
as well as
:<math>v_0 = 1-\sum_{i=1}^n v_i\,.</math>
Performing the {{mvar|u}} integral gives the general prescription for combining denominators:

:<math> \frac{1}{ D_0 \cdots D_n } = n! \int_{\mathrm{simplex}} \frac{1}{ \left(v_0 D_0 +v_1 D_1 \cdots + v_n D_n\right)^{n+1}}\, dv_1\, dv_2 \cdots dv_n </math>

Since the numerator of the integrand is not involved, the same prescription works for any loop, no matter what the spins are carried by the legs. The interpretation of the parameters {{mvar|v<sub>i</sub>}} is that they are the fraction of the total proper time spent on each leg.

=== 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.