Editing Path integral formulation (section)

== Quantum field theory ==
{{Quantum field theory}}
Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
: <math>[\varphi(x), \partial_t \varphi(y)] = i \delta^3(x - y)</math>
for two simultaneous spatial positions {{mvar|x}} and {{mvar|y}}, and this is not a relativistically invariant concept. The results of a calculation ''are'' covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the [[continuum limit]], this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a [[renormalization|careful limiting procedure]].

The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to [[operator product expansion|operator product rules]], which are new relations difficult to see in the old formalism.

Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete.

The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the [[Grassmann variable]] – which also allowed changes of variables to be done naturally, as well as allowing [[Faddeev–Popov ghost|constrained quantization]].

The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities [[Anomaly (physics)|fail]].

=== 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.

=== Functionals of fields ===

However, the path integral formulation is also extremely important in ''direct'' application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a [[field (physics)|field]] over all space.  The action is referred to technically as a [[functional (mathematics)|functional]] of the field: {{math|''S''[''ϕ'']}}, where the field {{math|''ϕ''(''x<sup>μ</sup>'')}} is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. ''One'' such given function {{math|''ϕ''(''x<sup>μ</sup>'')}} of [[spacetime]] is called a ''field configuration''. In principle, one integrates Feynman's amplitude over the class of all possible field configurations.

Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these [[functional integral]]s mathematically precise.

Such a functional integral is extremely similar to the [[partition function (statistical mechanics)|partition function]] in [[statistical mechanics]].  Indeed, it is sometimes ''called'' a [[partition function (quantum field theory)|partition function]], and the two are essentially mathematically identical except for the factor of {{mvar|i}} in the exponent in Feynman's postulate 3. [[Analytic continuation|Analytically continuing]] the integral to an imaginary time variable (called a [[Wick rotation]]) makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals.

=== Expectation values ===

In [[quantum field theory]], if the [[action (physics)|action]] is given by the [[functional (mathematics)|functional]] {{mathcal|S}} of field configurations (which only depends locally on the fields), then the [[time-ordered]] [[vacuum expectation value]] of [[polynomially bounded]] functional {{mvar|F}}, {{math|{{angbr|''F''}}}}, is given by
: <math>\langle F \rangle = \frac{\int\mathcal{D}\varphi F[\varphi]e^{i\mathcal{S}[\varphi]}}{\int\mathcal{D}\varphi e^{i\mathcal{S}[\varphi]}}.</math>

The symbol {{math|∫{{mathcal|D}}''ϕ''}} here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization.

=== As a probability ===
Strictly speaking, the only question that can be asked in physics is: ''What fraction of states satisfying condition {{mvar|A}} also satisfy condition {{mvar|B}}?'' The answer to this is a number between 0 and 1, which can be interpreted as a [[conditional probability]], written as {{math|P(''B''{{!}}''A'')}}. In terms of path integration, since {{math|P(''B''{{!}}''A'') {{=}} {{sfrac|P(''A''∩''B'')&nbsp;|&nbsp;P(''A'')}}}}, this means
: <math>\operatorname{P}(B\mid A) = \frac
{\sum_{F \subset A \cap B}\left| \int\mathcal{D}\varphi O_\text{in}[\varphi]e^{i\mathcal{S}[\varphi]}  F[\varphi]\right|^2}
{\sum_{F \subset A} \left|\int\mathcal{D}\varphi O_\text{in}[\varphi] e^{i\mathcal{S}[\varphi]} F[\varphi]\right|^2},</math>
where the functional {{math|''O''<sub>in</sub>[''ϕ'']}} is the superposition of all incoming states that could lead to the states we are interested in. In particular, this could be a state corresponding to the state of the Universe just after the [[Big Bang]], although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals, it is naturally normalised.

=== Schwinger–Dyson equations ===
{{Main|Schwinger–Dyson equation}}

Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.

In the language of functional analysis, we can write the [[Euler–Lagrange equation]]s as
: <math>\frac{\delta \mathcal{S}[\varphi]}{\delta \varphi} = 0</math>
(the left-hand side is a [[functional derivative]]; the equation means that the action is stationary under small changes in the field configuration).  The quantum analogues of these equations are called the [[Schwinger–Dyson equation]]s.

If the [[functional measure]] {{math|{{mathcal|D}}''ϕ''}} turns out to be [[Translational symmetry|translationally invariant]] (we'll assume this for the rest of this article, although this does not hold for, let's say [[nonlinear sigma model]]s), and if we assume that after a [[Wick rotation]]
: <math>e^{i\mathcal{S}[\varphi]},</math>
which now becomes
: <math>e^{-H[\varphi]}</math>
for some {{mvar|H}}, it goes to zero faster than a [[Multiplicative inverse|reciprocal]] of any [[polynomial]] for large values of {{mvar|φ}}, then we can [[integration by parts|integrate by parts]] (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:
: <math>\left\langle \frac{\delta F[\varphi]}{\delta \varphi} \right\rangle = -i \left\langle F[\varphi]\frac{\delta \mathcal{S}[\varphi]}{\delta\varphi} \right\rangle</math>
for any polynomially-bounded functional {{mvar|F}}. In the [[deWitt notation]] this looks like<ref>{{cite journal |first=Jean |last=Zinn-Justin |date=2009 |title=Path integral |journal=Scholarpedia |volume=4 |issue=2 |doi=10.4249/scholarpedia.8674 |bibcode=2009SchpJ...4.8674Z |at=8674|doi-access=free }}</ref>
: <math>\left\langle F_{,i} \right\rangle = -i \left\langle F \mathcal{S}_{,i} \right\rangle.</math>

These equations are the analog of the [[on-shell]] EL equations. The time ordering is taken before the time derivatives inside the {{math|{{mathcal|S}}<sub>,''i''</sub>}}.

If {{mvar|J}} (called the [[source field]]) is an element of the [[dual space]] of the field configurations (which has at least an [[affine structure]] because of the assumption of the [[translational invariance]] for the functional measure), then the [[generating functional]] {{mvar|Z}} of the source fields is '''defined''' to be
: <math>Z[J] = \int \mathcal{D}\varphi e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}.</math>

Note that
: <math>\frac{\delta^n Z}{\delta J(x_1) \cdots \delta J(x_n)}[J] = i^n \, Z[J] \, \left\langle \varphi(x_1)\cdots \varphi(x_n)\right\rangle_J,</math>
or
: <math>Z^{,i_1\cdots i_n}[J] = i^n Z[J] \left \langle \varphi^{i_1}\cdots \varphi^{i_n}\right\rangle_J,</math>
where
: <math>\langle F \rangle_J = \frac{\int \mathcal{D}\varphi F[\varphi]e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}}{\int\mathcal{D}\varphi e^{i\left(\mathcal{S}[\varphi] + \langle J,\varphi \rangle\right)}}.</math>

Basically, if {{math|{{mathcal|D}}''φ'' ''e''<sup>''i''{{mathcal|S}}[''φ'']</sup>}} is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of [[Quantum field theory|QFT]], unlike its Wick-rotated [[statistical mechanics]] analogue, because we have [[time ordering]] complications here!), then {{math|{{angbr|''φ''(''x''<sub>1</sub>) ... ''φ''(''x<sub>n</sub>'')}}}} are its [[moment (mathematics)|moments]], and {{mvar|Z}} is its [[Fourier transform]].

If {{mvar|F}} is a functional of {{mvar|φ}}, then for an [[Operator (mathematics)|operator]] {{mvar|K}}, {{math|''F''[''K'']}} is defined to be the operator that substitutes {{mvar|K}} for {{mvar|φ}}. For example, if
: <math>F[\varphi] = \frac{\partial^{k_1}}{\partial x_1^{k_1}}\varphi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\varphi(x_n),</math>
and {{mvar|G}} is a functional of {{mvar|J}}, then
: <math>F\left[-i\frac{\delta}{\delta J}\right] G[J] = (-i)^n \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\delta}{\delta J(x_1)} \cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\frac{\delta}{\delta J(x_n)} G[J].</math>

Then, from the properties of the [[functional integral]]s
: <math>\left \langle \frac{\delta \mathcal{S}}{\delta \varphi(x)} [\varphi] + J(x)\right\rangle_J = 0</math>
we get the "master" Schwinger–Dyson equation:
: <math>\frac{\delta \mathcal{S}}{\delta \varphi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J] + J(x)Z[J] = 0,</math>
or
: <math>\mathcal{S}_{,i}[-i\partial]Z + J_i Z = 0.</math>

If the functional measure is not translationally invariant, it might be possible to express it as the product {{math|''M''[''φ''] {{mathcal|D}}''φ''}}, where {{mvar|M}} is a functional and {{math|{{mathcal|D}}''φ''}} is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the [[target space]] is diffeomorphic to {{math|'''R'''<sup>''n''</sup>}}. However, if the [[target manifold]] is some topologically nontrivial space, the concept of a translation does not even make any sense.

In that case, we would have to replace the {{mathcal|S}} in this equation by another functional
: <math>\hat{\mathcal{S}} = \mathcal{S} - i\ln M.</math>

If we expand this equation as a [[Taylor series]] about ''J'' {{=}} 0, we get the entire set of Schwinger–Dyson equations.