Editing Path integral formulation (section)

== Localization ==
The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in [[local quantum field theory]] we would restrict everything to lie within a finite ''causally complete'' region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.

=== Ward–Takahashi identities ===
{{main|Ward–Takahashi identity}}

Now how about the [[on shell]] [[Noether's theorem]] for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.

Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a [[gauge symmetry]], but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a [[Lagrangian (field theory)|Lagrangian]], and that
: <math>Q[\mathcal{L}(x)]=\partial_\mu f^\mu (x)</math>
for some function {{mvar|f}} where {{mvar|f}} only depends locally on {{mvar|φ}} (and possibly the spacetime position).

If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless {{math|''f'' {{=}} 0}} or something. Here, {{mvar|Q}} is a [[Derivation (abstract algebra)|derivation]] that generates the one parameter group in question. We could have [[antiderivation]]s as well, such as [[BRST quantization|BRST]] and [[supersymmetry]].

Let's also assume
: <math>\int \mathcal{D}\varphi\, Q[F][\varphi]=0</math>
for any polynomially-bounded functional {{mvar|F}}. This property is called the invariance of the measure, and this does not hold in general. (See ''[[anomaly (physics)]]'' for more details.)

Then,
: <math>\int \mathcal{D}\varphi\, Q\left[F e^{iS}\right][\varphi]=0,</math>
which implies
: <math>\langle Q[F]\rangle +i\left\langle F\int_{\partial V} f^\mu\, ds_\mu\right\rangle=0</math>
where the integral is over the boundary. This is the quantum analog of Noether's theorem.

Now, let's assume even further that {{mvar|Q}} is a local integral
: <math>Q=\int d^dx\, q(x)</math>
where
: <math>q(x)[\varphi(y)] = \delta^{(d)}(X-y)Q[\varphi(y)] \,</math>
so that\
: <math>q(x)[S]=\partial_\mu j^\mu (x) \,</math>
where
: <math>j^{\mu}(x)=f^\mu(x)-\frac{\partial}{\partial (\partial_\mu \varphi)}\mathcal{L}(x) Q[\varphi] \,</math>
(this is assuming the Lagrangian only depends on {{mvar|φ}} and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that {{math|''q''(''x'')}} is the generator of a symmetry (i.e. we are ''not'' insisting upon the [[gauge principle]]), but just that {{mvar|Q}} is. And we also assume the even stronger assumption that the functional measure is locally invariant:
: <math>\int \mathcal{D}\varphi\, q(x)[F][\varphi]=0.</math>

Then, we would have
: <math>\langle q(x)[F] \rangle +i\langle F q(x)[S]\rangle=\langle q(x)[F]\rangle +i\left\langle F\partial_\mu j^\mu(x)\right\rangle=0.</math>

Alternatively,
: <math>q(x)[S]\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=\partial_\mu j^\mu(x)\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>

The above two equations are the Ward–Takahashi identities.

Now for the case where {{math|''f'' {{=}} 0}}, we can forget about all the boundary conditions and locality assumptions. We'd simply have
: <math>\left\langle Q[F]\right\rangle =0.</math>

Alternatively,
: <math>\int d^dx\, J(x)Q[\varphi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>