Current Sources and Generating Functionals

An external source turns a current into a controlled probe. Functional derivatives with respect to that source generate current insertions, the logarithm of the vacuum functional selects connected correlators, and a simultaneous transformation of all sources packages the Ward identities into one functional equation. This language also keeps two facts visible: coincident derivatives can contain local contact terms, and coupling to a nondynamical source is not the same operation as gauging a symmetry.

Here $a$ is a prescribed nondynamical source and is never integrated over. The page treats time-ordered source functionals and their connected and 1PI Ward constraints, leaving global bundle data and transport to later pages.

Required background. Quantum Currents, Improvements, and Conservation supplies the normalized composite-current insertion and its improvement freedom. The Generating Functional supplies the normalized source-dependent path integral and functional-differentiation rules used below.

Helpful background. The 1PI Effective Action and Mean-Field Equations supplies the ordinary Legendre transform used to distinguish connected current kernels from 1PI vertices.

The source-coupled vacuum functional

Let $a_\mu^A(x)$ couple to currents labelled by $A$. Near zero source, write

$$\begin{aligned} S[\Phi;a] ={}&S_0[\Phi] \\ &+\int\mathrm d^d x\, a_\mu^A[j_A^\mu]_R \\ &+O(a^2). \end{aligned}$$

The $O(a^2)$ terms are not optional when a finite background-source symmetry requires them. They do not affect the first derivative at $a=0$, but they do contribute local terms to higher derivatives.

We divide by the source-free vacuum functional and define

$$\begin{aligned} Z[a] &=\int\mathcal D\Phi\,e^{iS[\Phi;a]}, \\ \mathcal Z[a] &=\frac{Z[a]}{Z[0]}, \\ W[a] &=-i\log\mathcal Z[a], \\ \mathcal Z[a] &=e^{iW[a]}. \end{aligned}$$

Vacuum boundary conditions, the state, and the regulator are part of this definition. Source transformations will be taken compactly supported unless a boundary term is displayed explicitly.

Source derivatives generate current insertions

Define the source-dependent current by

$$\mathcal J_A^\mu(x;a) \equiv \frac{\delta S[\Phi;a]} {\delta a_\mu^A(x)}.$$

Direct differentiation of the Lorentzian exponential gives

$$\begin{aligned} \frac{\delta Z}{\delta a_\mu^A(x)} &=iZ[a] \left\langle\mathcal J_A^\mu(x;a)\right\rangle_a, \\ \frac{\delta W}{\delta a_\mu^A(x)} &=\left\langle\mathcal J_A^\mu(x;a)\right\rangle_a. \end{aligned}$$

For a strictly linear, source-independent coupling, repeated derivatives at zero source give the full time-ordered correlators,

$$\left. \frac{1}{i^n} \frac{\delta^n\mathcal Z} {\delta a_1\cdots\delta a_n} \right|_{a=0} =\left\langle \mathrm T\{j_1\cdots j_n\} \right\rangle_0.$$

Here each abbreviated $a_k$ carries its spacetime, Lorentz, and current labels. The factors of $i$ follow from the declared $e^{iS}$ convention. This source-differentiation construction is developed at Schwartz 2014, § 14.3, pp. 261–264.

Connected kernels and local pieces

Taking one more derivative of $W$ yields

$$\begin{aligned} &\frac{\delta^2W} {\delta a_\mu^A(x)\,\delta a_\nu^B(y)} \\ ={}&i\left\langle \mathrm T\{\mathcal J_A^\mu(x;a) \mathcal J_B^\nu(y;a)\} \right\rangle_{c,a} \\ &+\left\langle \frac{\delta\mathcal J_A^\mu(x;a)} {\delta a_\nu^B(y)} \right\rangle_a. \end{aligned}$$

The first term is connected because differentiating the logarithm subtracts the product of one-point functions. The second term is a local seagull or contact contribution generated by nonlinear source dependence. Thus, for a strictly linear coupling,

$$\begin{aligned} &\left\langle \mathrm T\{j_A^\mu(x)j_B^\nu(y)\} \right\rangle_c \\ ={}&-i\left. \frac{\delta^2W} {\delta a_\mu^A(x)\,\delta a_\nu^B(y)} \right|_{a=0}. \end{aligned}$$

Weinberg constructs $W$ as the connected functional and fixes the same Lorentzian sign conventions at Weinberg 1996, Vol. II, § 16.1, pp. 63–68. At coincident points, renormalized composite products and local functionals of $a$ can shift the second derivative by delta functions and their derivatives. These scheme-dependent local pieces do not change a separated-point correlator, but they are indispensable in a complete Ward identity.

The kernel above is time ordered. A retarded response function requires a causal prescription rather than merely relabelling this derivative; real-time response is handed off to the thermal and many-body volumes.

Why the scalar source needs a seagull

The threaded complex scalar makes the nonlinear term explicit. In a topologically trivial patch define

$$D_\mu(a)=\partial_\mu-ia_\mu.$$

Then

$$\begin{aligned} (D_\mu\phi)^\dagger D^\mu\phi ={}&\partial_\mu\phi^\dagger\partial^\mu\phi \\ &+a_\mu j^\mu +a_\mu a^\mu\phi^\dagger\phi, \end{aligned}$$

with

$$j^\mu =i\left( \phi^\dagger\partial^\mu\phi -(\partial^\mu\phi^\dagger)\phi \right).$$

Consequently,

$$\begin{aligned} \left.\frac{\delta S}{\delta a_\mu(x)}\right|_{a=0} &=j^\mu(x), \\ \left. \frac{\delta^2S} {\delta a_\mu(x)\,\delta a_\nu(y)} \right|_{a=0} &=2\eta^{\mu\nu}\phi^\dagger\phi(x) \\ &\quad\times\delta^{(d)}(x-y). \end{aligned}$$

In a symmetry-compatible renormalized prescription, define $\mathcal K^{\mu\nu}(x,y) \equiv\left.\delta^2W/ (\delta a_\mu(x)\delta a_\nu(y))\right|_{a=0}$. It has the schematic form

$$\begin{aligned} \mathcal K^{\mu\nu}(x,y) ={}&i\Bigl\langle \mathrm T\{[j^\mu]_R(x) \\ &\qquad\times[j^\nu]_R(y)\} \Bigr\rangle_c \\ &+2\eta^{\mu\nu} \left\langle[\phi^\dagger\phi]_R(x)\right\rangle \\ &\qquad\times\delta^{(d)}(x-y) \\ &+\mathcal K_{\mathrm{ct}}^{\mu\nu}(x,y). \end{aligned}$$

The last line allows finite local counterterm choices. A bare term $\int a_\mu j^\mu$ correctly generates the first insertion at $a=0$, but by itself it misses the scalar seagull and is not invariant under finite local source transformations. This is the functional version of the warning about current-current contacts on the preceding page.

A source-space Ward identity

Let $\lambda^r(x)$ denote sources for any other operators whose insertions are needed. Suppose the regulated action, measure, state, and boundary data are invariant under a simultaneous localized transformation of the fields and sources. In the absence of an anomaly,

$$\begin{aligned} 0=\delta_\alpha W ={}&\int\mathrm d^d x\Bigl[ \frac{\delta W}{\delta a_\mu^A}\, \delta_\alpha a_\mu^A \\ &\qquad +\frac{\delta W}{\delta\lambda^r}\, \delta_\alpha\lambda^r \Bigr]. \end{aligned}$$

Differentiating this single equation with respect to the $\lambda^r$ produces the contact terms associated with every transformed insertion. Schwartz derives the equivalent localized change-of-variables and insertion identities at Schwartz 2014, §§ 14.8.1–14.8.2, pp. 278–280. The displayed source-space equation is the present page’s reformulation; the cited section does not construct a general background-current functional.

Three qualifications are essential. A measure Jacobian adds an anomaly functional, explicit breaking adds transforming coupling sources or a breaking insertion, and a noncompact transformation can add boundary flux. None of these effects is erased by writing the identity in source notation.

Connected and 1PI functionals are different objects

To define the ordinary one-particle-irreducible functional, introduce sources $K_I$ for the dynamical fields while keeping $a$ external:

$$\begin{aligned} Z[K,a] ={}&\int\mathcal D\Phi\, \exp\!\Bigl\{ iS[\Phi;a] \\ &\qquad +i\int\mathrm d^d x\,K_I\Phi^I \Bigr\}. \end{aligned}$$

With $W[K,a]=-i\log(Z[K,a]/Z[0,0])$, set

$$\begin{aligned} \bar\Phi^I &=\frac{\delta W}{\delta K_I}, \\ \Gamma[\bar\Phi,a] &=W[K,a] -\int\mathrm d^d x\,K_I\bar\Phi^I. \end{aligned}$$

The Legendre transform gives

$$\frac{\delta\Gamma}{\delta\bar\Phi^I}=-K_I, \qquad \left.\frac{\delta\Gamma}{\delta a}\right|_{\bar\Phi} =\left.\frac{\delta W}{\delta a}\right|_K.$$

Thus $W$ generates connected current correlators, whereas $\Gamma$ is 1PI with respect to propagators of the dynamical fields transformed through $K_I$. Derivatives of $\Gamma$ with respect to $a$ describe current insertions in vertices that are 1PI in those dynamical lines. Merely Taylor-expanding $W[a]$ does not produce ordinary 1PI current vertices. Legendre transforming with respect to the composite-current source $a$ would define a different current-density effective action and requires a separate invertibility analysis. Weinberg states both the usual construction and the composite-source qualification at Weinberg 1996, Vol. II, § 16.1, pp. 63–68; The One-Particle-Irreducible Effective Action develops the field Legendre transform.

For a transformation linear or affine in the dynamical fields, with $K_I$ transformed contragrediently, the mean field transforms in the same representation and the symmetry equation passes through the Legendre transform:

$$\begin{aligned} 0={}&\int\mathrm d^d x\Bigl[ \frac{\delta\Gamma}{\delta a_\mu^A}\, \delta_\alpha a_\mu^A \\ &\quad +\frac{\delta\Gamma}{\delta\bar\Phi^I}\, \delta_\alpha\bar\Phi^I \\ &\quad +\frac{\delta\Gamma}{\delta\lambda^r}\, \delta_\alpha\lambda^r \Bigr]. \end{aligned}$$

On a mean-field solution, the middle term vanishes. Off shell it is part of the 1PI Ward identity and must not be discarded. For a nonlinear field transformation, this simple formula need not follow from the Legendre transform alone; sources for the composite variations or a more general functional identity may be required. The threaded scalar $U(1)$ below is linear and obeys the displayed equation.

Threaded scalar: exact, spurionic, and discrete symmetry

For the scalar convention $\phi\mapsto e^{i\alpha(x)}\phi$, the source completion above is invariant under

$$a_\mu\mapsto a_\mu+\partial_\mu\alpha.$$

Now include the controlled interaction

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2,$$

and temporarily regard $h(x)$ as a source. The family is covariant if

$$h\mapsto e^{-iN\alpha}h, \qquad h^*\mapsto e^{+iN\alpha}h^*.$$

Assuming an invariant regulator and measure, compact support, and no anomaly, source covariance says

$$\begin{aligned} W[&a+\mathrm d\alpha, e^{-iN\alpha}h, e^{+iN\alpha}h^*] \\ &=W[a,h,h^*]. \end{aligned}$$

Its infinitesimal local form is

$$\partial_\mu\frac{\delta W}{\delta a_\mu} +iNh\frac{\delta W}{\delta h} -iNh^*\frac{\delta W}{\delta h^*} =0.$$

Define the breaking insertion in the same prescription by $[\mathcal B]_R =iN\bigl(h[\phi^N]_R-h^*[(\phi^\dagger)^N]_R\bigr)$. Since $\delta W/\delta h$ inserts $[\phi^N]_R$, setting the sources to fixed values gives

$$\begin{aligned} \partial_\mu\langle[\mathcal J^\mu]_R\rangle ={}&-iN\Bigl[ h\langle[\phi^N]_R\rangle \\ &\qquad -h^*\langle[(\phi^\dagger)^N]_R\rangle \Bigr] \\ ={}&-\langle[\mathcal B]_R\rangle. \end{aligned}$$

Differentiating before setting charged operator sources to zero restores the insertion contact terms of the Ward–Takahashi identity. If instead a nonzero constant $h$ is held fixed and not transformed, the continuous $U(1)$ is explicitly broken. Its exact stabilizer is the constant subgroup $\mathbb Z_N$, whose finite Ward identity cannot be obtained by differentiating with respect to an infinitesimal parameter.

Common pitfalls

Forgetting the Lorentzian factor. With $W=-i\log\mathcal Z$, the second derivative is $i$ times the connected time-ordered correlator, not the correlator itself.

Dropping nonlinear source terms. A linear coupling fixes the first insertion at zero source. Local source covariance and higher response can require seagulls such as $a^2\phi^\dagger\phi$.

Erasing contact-term freedom. Local counterterms shift coincident response kernels. Separated-point data alone do not fix those terms.

Calling connected current kernels 1PI. The ordinary $\Gamma$ is obtained by transforming the elementary-field sources $K_I$, not automatically by transforming the current source $a$.

Reading a time-ordered kernel as causal response. Retarded, advanced, Euclidean, and time-ordered functions obey different prescriptions even when analytic continuation relates them.

Confusing a source with a dynamical gauge field. No integral over $a$ has been introduced, so no new gauge constraint or gauge-boson Hilbert space has appeared.

Check your understanding

Starting from the scalar source transformations of $a_\mu$, $h$, and $h^*$, derive the local identity for $W$. Then explain why it yields explicit continuous breaking but still permits an exact $\mathbb Z_N$ symmetry when a nonzero constant $h$ is held fixed.

Check

The infinitesimal variation is

$$\begin{aligned} 0=\int\mathrm d^d x\,\alpha\Bigl[ &-\partial_\mu\frac{\delta W}{\delta a_\mu} -iNh\frac{\delta W}{\delta h} \\ &+iNh^*\frac{\delta W}{\delta h^*} \Bigr], \end{aligned}$$

where the $a_\mu$ term was integrated by parts. Arbitrariness of $\alpha$ gives

$$\partial_\mu\frac{\delta W}{\delta a_\mu} +iNh\frac{\delta W}{\delta h} -iNh^*\frac{\delta W}{\delta h^*}=0.$$

Holding $h$ fixed removes its compensating transformation and leaves the nonzero breaking insertion on the right-hand side of the current identity. A constant transformation still preserves $h\phi^N$ whenever $e^{iN\alpha}=1$. Those transformations form $\mathbb Z_N$, but they have no nontrivial infinitesimal neighborhood.

What to carry forward

Source differentiation organizes current insertions, connected kernels, local contacts, and 1PI identities without turning the probe into a dynamical field. Coupling to Background Gauge Fields and Bundles supplies the global geometric meaning of $a$ and its non-Abelian transformation law. Spurions, Local Counterterms, and Symmetry Response classifies the local ambiguities, while Background Fields versus Dynamical Gauging separates probing a symmetry from summing over gauge configurations.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996. DOI