Editing Path integral formulation (section)

=== 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.