Editing Path integral formulation (section)

=== Propagator ===

In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.

The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point {{mvar|x}} to point {{mvar|y}} in time {{mvar|T}}:
: <math>K(x, y; T) = \langle y; T \mid x; 0 \rangle = \int_{x(0)=x}^{x(T)=y} e^{i S[x]} \,Dx.</math>

This is called the [[propagator]]. To obtain the final state at {{math|''y''}} we simply apply {{math|''K''(''x'',''y''; ''T'')}} to the initial state and integrate over {{math|''x''}} resulting in:
: <math>\psi_T(y) = \int_x \psi_0(x) K(x, y; T) \,dx = \int^{x(T)=y} \psi_0(x(0)) e^{i S[x]} \,Dx.</math>

For a spatially homogeneous system, where {{math|''K''(''x'', ''y'')}} is only a function of {{math|(''x'' − ''y'')}}, the integral is a [[convolution]], the final state is the initial state convolved with the propagator:
: <math>\psi_T = \psi_0 * K(;T).</math>

For a free particle of mass {{mvar|m}}, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian:
: <math>K(x, y; T) \propto e^\frac{i m(x - y)^2}{2T}.</math>

Taking the Fourier transform in {{math|(''x'' − ''y'')}} produces another Gaussian:
: <math>K(p; T) = e^\frac{i T p^2}{2m},</math>
and in {{mvar|p}}-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending {{math|''K''(''p''; ''T'')}} to be zero for negative times, gives Green's function, or the frequency-space propagator:
: <math>G_\text{F}(p, E) = \frac{-i}{E - \frac{\vec{p}^2}{2m} + i\varepsilon},</math>
which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the {{mvar|p}}-space representation.

The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in {{mvar|E}} will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of {{mvar|E}} where there is no singularity. This guarantees that {{mvar|K}} propagates the particle into the future and is the reason for the subscript "F" on {{mvar|G}}. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.

It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the Gaussian {{mvar|t}} is replaced by {{math|−''t''}}. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:
: <math>G_\text{B}(p, E) = \frac{-i}{-E - \frac{i\vec{p}^2}{2m} + i\varepsilon}.</math>
Given the nearly identical only change is the sign of {{mvar|E}} and {{mvar|ε}}, the parameter {{mvar|E}} in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.

For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time):
: <math>K(x - y, \Tau) = \int_{x(0)=x}^{x(\Tau)=y} e^{i \int_0^\Tau \sqrt{\dot{x}^2 - \alpha} \,d\tau}.</math>

The integral above is not trivial to interpret because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function {{math|''K''(''x'' − ''y'', ''τ'')}} can be evaluated when the sum is over paths in Euclidean space:
: <math>K(x - y, \Tau) = e^{-\alpha \Tau} \int_{x(0)=x}^{x(\Tau)=y} e^{-L}.</math>

This describes a sum over all paths of length {{math|Τ}} of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to {{math|Τ}}, and each step is less likely the longer it is. By the [[central limit theorem]], the result of many independent steps is a Gaussian of variance proportional to {{math|Τ}}:
: <math>K(x - y,\Tau) = e^{-\alpha \Tau} e^{-\frac{(x - y)^2}{\Tau}}.</math>

The usual definition of the relativistic propagator only asks for the amplitude to travel from {{mvar|x}} to {{mvar|y}}, after summing over all the possible proper times it could take:
: <math>K(x - y) = \int_0^\infty K(x - y, \Tau) W(\Tau) \,d\Tau,</math>
where {{math|''W''(Τ)}} is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant {{mvar|α}}:
: <math>K(x - y) = \int_0^\infty e^{-\frac{(x - y)^2}{\Tau} -\alpha \Tau} \,d\Tau.</math>

This is the [[Feynman diagram#Schwinger representation|Schwinger representation]]. Taking a Fourier transform over the variable {{math|(''x'' − ''y'')}} can be done for each value of {{math|Τ}} separately, and because each separate {{math|Τ}} contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in {{mvar|p}}-space, the propagator can be reexpressed simply:
: <math>K(p) = \int_0^\infty e^{-\Tau p^2 - \Tau \alpha} \,d\Tau = \frac{1}{p^2 + \alpha},</math>
which is the Euclidean propagator for a scalar particle. Rotating {{math|''p''<sub>0</sub>}} to be imaginary gives the usual relativistic propagator, up to a factor of {{math|−''i''}} and an ambiguity, which will be clarified below:
: <math>K(p) = \frac{i}{p_0^2 - \vec{p}^2 - m^2}.</math>

This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by [[partial fractions]]:
: <math>2 p_0 K(p) = \frac{i}{p_0 - \sqrt{\vec{p}^2 + m^2}} + \frac{i}{p_0 + \sqrt{\vec{p}^2 + m^2}}.</math>

For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near {{math|''p''<sub>0</sub> {{=}} ''m''}}. When convolving with the propagator, which in {{mvar|p}} space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near {{math|''p''<sub>0</sub> {{=}} ''m''}}, the dominant first term has the form
: <math>2m K_\text{NR}(p) = \frac{i}{(p_0 - m) - \frac{\vec{p}^2}{2m}}.</math>

This is the expression for the nonrelativistic [[Green's function]] of a free Schrödinger particle.

The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.

The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where {{math|''t'' → −∞}} of the first term must vanish, while the {{math|''t'' → +∞}} limit of the second term must vanish. In the Fourier transform, this means shifting the pole in {{math|''p''<sub>0</sub>}} slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:
: <math>K(p) = \frac{i}{p_0 - \sqrt{\vec{p}^2 + m^2} + i\varepsilon} + \frac{i}{p_0 - \sqrt{\vec{p}^2+m^2} - i\varepsilon}.</math>

Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of {{math|''p''<sub>0</sub>}}. The terms can be recombined:
: <math>K(p) = \frac{i}{p^2 - m^2 + i\varepsilon},</math>
which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The {{mvar|ε}} term introduces a small imaginary part to the {{math|''α'' {{=}} ''m''<sup>2</sup>}}, which in the Minkowski version is a small exponential suppression of long paths.

So in the relativistic case, the Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the [[Feynman–Stueckelberg interpretation|interpretation]] of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.

Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Green's function that is only nonzero in the future in a relativistically invariant theory.