Sources, Operators, and Generating Functionals

The main idea

The AdS/CFT dictionary begins with a simple piece of quantum field theory. To measure a local operator $\mathcal O(x)$, couple it to a classical source $J(x)$ and study how the partition function changes. In the cleanest convention,

$$Z[J] = \left\langle \exp\!\left(\int d^d x\,J(x)\mathcal O(x)\right) \right\rangle, \qquad W[J]=\log Z[J].$$

Then $W[J]$ is the generating functional of connected correlation functions:

$$\frac{\delta W}{\delta J(x)}=\langle \mathcal O(x)\rangle_J, \qquad \frac{\delta^2 W}{\delta J(x)\delta J(y)} = \langle\mathcal O(x)\mathcal O(y)\rangle_{J,\mathrm{conn}},$$

and similarly for higher derivatives.

Holography turns this source calculus into a gravitational boundary-value problem. The source $J(x)$ becomes the leading boundary datum of a bulk field $\phi(z,x)$, and the generating functional is computed from the renormalized on-shell bulk action:

$$W_{\mathrm{CFT}}[J] = \log Z_{\mathrm{CFT}}[J] = \log Z_{\mathrm{string}}[\phi\to J] \simeq -S_{\mathrm{bulk}}^{\mathrm{ren}}[\phi_{\mathrm{cl}};J].$$

This is the core of the Gubser-Klebanov-Polyakov/Witten prescription. The later pages will derive the bulk side. This page builds the QFT side carefully.

A source $J^I(x)$ couples to a CFT operator $\mathcal O_I(x)$ and defines a generating functional $W[J]$. Functional derivatives of $W[J]$ produce connected correlators. In AdS/CFT, the same source is the boundary value of a bulk field $\phi^I$, while the renormalized on-shell action computes $W[J]$ in the semiclassical regime.

The important lesson is not merely that $J$ is “dual to” $\phi$. The useful statement is operational:

$$\boxed{ \text{source dependence of the CFT partition function} = \text{boundary-condition dependence of the bulk partition function}. }$$

Once this is understood, local correlators, one-point functions, Ward identities, transport coefficients, and response functions are all variations of the same idea.

Sources are background fields, not quantum operators

A source is a classical function or background field that appears in the definition of the path integral. For a scalar operator, the deformation is schematically

$$S \longrightarrow S_J = S - \int d^d x\,J(x)\mathcal O(x)$$

in the convention where $Z[J]=\langle e^{\int J\mathcal O}\rangle$. The source is not integrated over. It is an external knob.

This distinction matters. The operator $\mathcal O(x)$ fluctuates quantum mechanically. The source $J(x)$ labels which QFT problem one is solving. Differentiating with respect to $J$ inserts $\mathcal O$ into correlation functions.

More generally, a QFT can be coupled to a collection of sources:

$$Z[J^I,A_\mu^a,g_{\mu\nu},\ldots] = \left\langle \exp\!\left[ \int d^d x\sqrt g\, \left( J^I\mathcal O_I +A_\mu^a J_a^\mu +\frac{1}{2}h_{\mu\nu}T^{\mu\nu} +\cdots \right) \right] \right\rangle.$$

Here $J^I$ sources scalar or spinning operators $\mathcal O_I$, $A_\mu^a$ is a background gauge field for a conserved current $J_a^\mu$, and the metric $g_{\mu\nu}$ sources the stress tensor. More invariantly, the stress tensor is defined by varying the theory with respect to the background metric, not by adding an arbitrary $h_{\mu\nu}$ in flat space.

In holography, these sources become boundary data:

CFT source	CFT operator	Bulk field
scalar source $J(x)$	scalar operator $\mathcal O(x)$	scalar field $\phi(z,x)$
background gauge field $A_\mu^{(0)}(x)$	conserved current $J^\mu(x)$	bulk gauge field $A_a(z,x)$
boundary metric $g_{\mu\nu}^{(0)}(x)$	stress tensor $T^{\mu\nu}(x)$	bulk metric $g_{ab}(z,x)$
spinor source $\eta(x)$	fermionic operator $\Psi(x)$	bulk spinor $\psi(z,x)$
source for a line or defect	nonlocal observable	string, brane, or defect geometry

The superscript $(0)$ on $A_\mu^{(0)}$ and $g_{\mu\nu}^{(0)}$ means “leading boundary value,” not necessarily “small perturbation.” The boundary metric is part of the definition of the CFT background.

Dimensions of sources

The source has a scaling dimension fixed by the operator it couples to. In units where $\hbar=1$, the deformation of the action must be dimensionless:

$$\int d^d x\,J(x)\mathcal O(x) \quad \text{dimensionless}.$$

If $\mathcal O$ has scaling dimension $\Delta$, then

$$[J]=d-\Delta.$$

Thus:

Operator dimension	Source dimension	Name of deformation
$\Delta<d$	$[J]>0$	relevant
$\Delta=d$	$[J]=0$	marginal
$\Delta>d$	$[J]<0$	irrelevant

This is the CFT version of a radial statement in AdS. A source for a relevant operator grows toward the infrared, so its bulk field tends to backreact strongly in the interior. A source for an irrelevant operator is small in the infrared but dangerous near the ultraviolet boundary; it usually requires specifying UV data beyond the low-energy theory.

For a conserved current $J^\mu$, the dimension is fixed by conservation:

$$\Delta[J^\mu]=d-1.$$

The source $A_\mu$ therefore has dimension one, as expected for a gauge field. For the stress tensor,

$$\Delta[T_{\mu\nu}]=d,$$

and its source, the metric $g_{\mu\nu}$, is dimensionless.

Full correlators from $Z[J]$, connected correlators from $W[J]$

Assume $Z[0]=1$ for simplicity. Expanding the exponential gives

$$Z[J] = 1 +\int d^d x\,J(x)\langle\mathcal O(x)\rangle +\frac{1}{2}\int d^d x d^d y\, J(x)J(y)\langle\mathcal O(x)\mathcal O(y)\rangle +\cdots.$$

Therefore derivatives of $Z[J]$ generate ordinary correlation functions:

$$\left. \frac{\delta^n Z[J]}{\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0} = \langle \mathcal O(x_1)\cdots\mathcal O(x_n) \rangle.$$

The logarithm $W[J]=\log Z[J]$ removes disconnected pieces. Its expansion is the cumulant expansion:

$$W[J] = \sum_{n=1}^{\infty}\frac{1}{n!} \int d^d x_1\cdots d^d x_n\, J(x_1)\cdots J(x_n) \langle \mathcal O(x_1)\cdots \mathcal O(x_n) \rangle_{\mathrm{conn}}.$$

Hence

$$\boxed{ \left. \frac{\delta^n W[J]}{\delta J(x_1)\cdots\delta J(x_n)} \right|_{J=0} = \langle \mathcal O(x_1)\cdots\mathcal O(x_n) \rangle_{\mathrm{conn}}. }$$

This distinction is indispensable in large-$N$ holography. The classical bulk action computes $W[J]$, not $Z[J]$ directly. Exponentiating the classical answer gives $Z[J]$:

$$Z[J]\simeq e^{-S_{\mathrm{bulk}}^{\mathrm{ren}}[J]}.$$

At leading large $N$, $S_{\mathrm{bulk}}^{\mathrm{ren}}$ is of order $N_{\mathrm{eff}}^2$. Its functional derivatives generate connected correlators with the expected large-$N$ scaling.

The one-point function in a source background

The first derivative of $W[J]$ gives the expectation value in the presence of the source:

$$\langle\mathcal O(x)\rangle_J = \frac{\delta W[J]}{\delta J(x)}.$$

This formula is more important than it looks. It tells us that the source-dependent one-point function contains all connected correlators. Expanding around $J=0$,

$$\langle\mathcal O(x)\rangle_J = \langle\mathcal O(x)\rangle +\int d^d y\,J(y) \langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}}$$ $$+\frac{1}{2}\int d^d y d^d z\,J(y)J(z) \langle\mathcal O(x)\mathcal O(y)\mathcal O(z)\rangle_{\mathrm{conn}} +\cdots.$$

Thus the response of a one-point function to a small source is a two-point function. The nonlinear response is controlled by higher connected correlators.

This is the field-theory origin of the holographic rule:

$$\text{boundary source} \quad\longrightarrow\quad \text{solve bulk equations} \quad\longrightarrow\quad \text{differentiate on-shell action to obtain }\langle\mathcal O\rangle_J.$$

Later, for a scalar field in AdS$_{d+1}$, the near-boundary expansion will have the form

$$\phi(z,x) = z^{d-\Delta} \left(J(x)+\cdots\right) + z^{\Delta} \left(A(x)+\cdots\right),$$

for standard quantization away from special integer cases. The coefficient $J(x)$ is the source. The coefficient $A(x)$ is related to the expectation value, after counterterms and possible local terms are included:

$$\langle\mathcal O(x)\rangle_J \sim (2\Delta-d)A(x)+\text{local terms in sources}.$$

The exact normalization depends on the bulk action convention. The conceptual point is invariant: the source is the boundary condition, while the vev is obtained from the renormalized canonical momentum.

Euclidean sign conventions

There are two common conventions. They differ by signs, not by physics.

Correlator convention

This page mostly uses

$$Z[J] = \left\langle e^{\int J\mathcal O}\right\rangle, \qquad W[J]=\log Z[J].$$

Then connected correlators are direct derivatives of $W[J]$:

$$G_n^{\mathrm{conn}} = \left.\frac{\delta^n W}{\delta J^n}\right|_{J=0}.$$

In this convention, the semiclassical holographic relation is

$$W_{\mathrm{CFT}}[J] \simeq -S_{\mathrm{bulk}}^{\mathrm{ren}}[J].$$

Euclidean free-energy convention

In holographic renormalization it is often cleaner to insert the source into the Euclidean action as

$$S_E[J]=S_E[0]+\int d^d x\sqrt g\,J\mathcal O$$

and define

$$\mathcal W_E[J] = -\log Z_E[J].$$

Then

$$\langle\mathcal O(x)\rangle_J = \frac{1}{\sqrt g}\frac{\delta \mathcal W_E}{\delta J(x)},$$

while connected two-point functions carry an extra sign:

$$\langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}} = - \frac{1}{\sqrt{g(x)}\sqrt{g(y)}} \left. \frac{\delta^2\mathcal W_E}{\delta J(x)\delta J(y)} \right|_{J=0}.$$

In this convention, the classical gravity statement is often written simply as

$$\mathcal W_E[J] = S_{\mathrm{bulk}}^{\mathrm{ren}}[J].$$

Both conventions are common. A trustworthy holographic calculation must state which one is being used, especially for Euclidean correlators and stress-tensor variations. Lorentzian retarded correlators require further $i\epsilon$ and boundary-condition prescriptions, which appear later in the course.

Functional derivatives with several sources

Real CFTs have many operators. Let $J^I(x)$ denote a collection of sources for operators $\mathcal O_I(x)$:

$$Z[J] = \left\langle \exp\!\left(\int d^d x\,J^I(x)\mathcal O_I(x)\right) \right\rangle.$$

Repeated derivatives give matrix-valued correlators:

$$\frac{\delta W}{\delta J^I(x)} = \langle\mathcal O_I(x)\rangle_J,$$ $$\frac{\delta^2 W}{\delta J^I(x)\delta J^J(y)} = \langle\mathcal O_I(x)\mathcal O_J(y)\rangle_{J,\mathrm{conn}}.$$

If operators mix under renormalization, the sources mix oppositely. This is why the source vector $J^I$ should be thought of as a coordinate system on the space of deformations. Changing renormalization scheme can redefine both operators and sources by local terms.

In holography, this mixing appears as coupled bulk fields. A scalar can mix with the trace of the metric near the boundary; a vector can mix with metric perturbations at finite density; metric perturbations in different channels can mix depending on momentum and symmetry. The source/operator formalism remains the organizing principle even when the equations are coupled.

Metric and current sources

The metric and background gauge fields deserve special attention because their source dependence encodes Ward identities.

For a theory on a background metric, define the generating functional $W[g]$ by

$$Z[g]=e^{W[g]}.$$

In the correlator convention, a common definition is

$$\langle T^{\mu\nu}(x)\rangle = \frac{2}{\sqrt g}\frac{\delta W}{\delta g_{\mu\nu}(x)},$$

up to sign conventions associated with whether one varies $g_{\mu\nu}$ or $g^{\mu\nu}$. In the Euclidean free-energy convention $\mathcal W_E=-W$, one often writes instead

$$\langle T_{\mu\nu}(x)\rangle = \frac{2}{\sqrt g}\frac{\delta \mathcal W_E}{\delta g^{\mu\nu}(x)},$$

again with convention-dependent signs. The invariant statement is this: the stress tensor is the response of the theory to a change of background geometry.

Similarly, if $A_\mu$ is a background gauge field for a conserved current, then

$$\langle J^\mu(x)\rangle_A = \frac{1}{\sqrt g}\frac{\delta W}{\delta A_\mu(x)}$$

in the correlator convention. Holographically, $A_\mu^{(0)}$ is the boundary value of a bulk gauge field. The expectation value $\langle J^\mu\rangle$ is obtained from the renormalized radial electric flux.

These definitions are not just formal. They are how one computes:

Quantity	Source derivative
current two-point function	$\delta\langle J^\mu\rangle/\delta A_\nu$
stress-tensor two-point function	$\delta\langle T^{\mu\nu}\rangle/\delta g_{\rho\sigma}$
conductivity	real-time current response to $A_i$
viscosity	real-time stress response to metric shear $h_{xy}$
charge density	one-point function from $A_t^{(0)}=\mu$
energy density and pressure	one-point function from boundary metric variations

This is why transport theory, thermodynamics, and holographic renormalization all start from the same object: a source-dependent generating functional.

Ward identities from source invariance

Ward identities are consequences of redundancies or symmetries of the source-dependent generating functional. This is the most economical way to derive them.

Global symmetry and background gauge invariance

Suppose $\mathcal O$ has charge $q$ under a global $U(1)$ symmetry, and we turn on a background gauge field $A_\mu$ and source $J$:

$$W[A_\mu,J,J^*].$$

Background gauge invariance says that $W$ is invariant under

$$\delta_\alpha A_\mu=\partial_\mu\alpha, \qquad \delta_\alpha J=iq\alpha J, \qquad \delta_\alpha J^*=-iq\alpha J^*.$$

Varying $W$ and integrating by parts gives the Ward identity

$$\nabla_\mu\langle J^\mu\rangle = -iqJ\langle\mathcal O\rangle +iqJ^*\langle\mathcal O^*\rangle,$$

up to anomaly terms if the symmetry is anomalous. When $J=0$ and the symmetry is non-anomalous, this reduces to current conservation:

$$\nabla_\mu\langle J^\mu\rangle=0.$$

In the bulk, this Ward identity follows from bulk gauge invariance and the radial constraint equation.

Diffeomorphism Ward identity

If sources vary in spacetime, energy-momentum is exchanged with the external backgrounds. For scalar sources and background gauge fields, the diffeomorphism Ward identity takes the schematic form

$$\nabla_\mu\langle T^{\mu}{}_{\nu}\rangle = F_{\nu\mu}\langle J^\mu\rangle +\langle\mathcal O_I\rangle\nabla_\nu J^I +\text{anomaly terms}.$$

The first term is the Lorentz force density from the background gauge field. The second says that a spacetime-dependent coupling can inject momentum into the system. In holography, this identity is the boundary form of the bulk momentum constraint.

Weyl Ward identity

For a CFT deformed by scalar sources, Weyl invariance implies

$$\langle T^\mu{}_{\mu}\rangle = \sum_I (\Delta_I-d)J^I\langle\mathcal O_I\rangle +\mathcal A[g,J]$$

for sources with no beta functions, where $\mathcal A[g,J]$ is the conformal anomaly and possible local source-dependent anomaly. More generally, quantum beta functions replace $(\Delta_I-d)J^I$ by $\beta^I(J)$.

This formula is one of the cleanest ways to remember relevant deformations: a source with $\Delta_I\neq d$ explicitly breaks scale invariance.

Contact terms and scheme dependence

Functional derivatives are distributions. When two insertion points coincide, contact terms appear. For example, adding a local functional of the source,

$$W[J]\longrightarrow W[J] +\frac{a}{2}\int d^d x\,J(x)^2,$$

changes the two-point function by

$$\langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}} \longrightarrow \langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}} +a\,\delta^{(d)}(x-y).$$

Separated-point correlators are unchanged. Coincident-point data are scheme-dependent unless protected by anomalies or Ward identities.

This is not a nuisance specific to holography. It is ordinary QFT. Holographic renormalization makes it visible because the on-shell action diverges near the AdS boundary and must be supplemented by local counterterms:

$$S_{\mathrm{bulk}}^{\mathrm{ren}} = \lim_{\epsilon\to0} \left( S_{\mathrm{bulk}}^{z\geq\epsilon} +S_{\mathrm{GHY}}^{z=\epsilon} +S_{\mathrm{ct}}^{z=\epsilon} \right).$$

The finite local part of $S_{\mathrm{ct}}$ is a scheme choice. In the CFT, it is the freedom to add local functionals of background sources.

A practical rule:

$$\boxed{ \text{Separated-point correlators are usually less scheme-dependent; contact terms require conventions.} }$$

Many apparent disagreements in holographic two-point functions are disagreements about contact terms, normalization, or Euclidean sign conventions.

Sources versus expectation values

A central trap in AdS/CFT is confusing a source with a vacuum expectation value. In a CFT, the difference is sharp:

$$J(x) \quad\text{is chosen in the definition of the path integral},$$

whereas

$$\langle\mathcal O(x)\rangle_J \quad\text{is computed from the path integral}.$$

In the bulk, both appear as coefficients in the near-boundary expansion of the same field. That is why the confusion is tempting.

For a scalar in standard quantization,

$$\phi(z,x) = z^{d-\Delta}\left(J(x)+\cdots\right) +z^\Delta\left(A(x)+\cdots\right).$$

The leading coefficient $J(x)$ is the source. The subleading coefficient $A(x)$ is related to the vev. But local terms can mix into the precise relation:

$$\langle\mathcal O(x)\rangle_J = \mathcal N_\Delta A(x)+\mathcal F_{\mathrm{local}}[J,g_{\mu\nu}^{(0)}](x).$$

When the scalar mass lies in the alternate-quantization window, the roles of the two coefficients can be exchanged by a Legendre transform. That is not a contradiction; it means the same bulk boundary condition can define different CFTs or different choices of operator dimension. This will be discussed after the mass-dimension relation.

Legendre transforms and effective actions

The generating functional $W[J]$ is source-based. Sometimes one wants a functional of expectation values instead. Define

$$\sigma(x)=\frac{\delta W[J]}{\delta J(x)}=\langle\mathcal O(x)\rangle_J.$$

The Legendre transform is

$$\Gamma[\sigma] = \int d^d x\,J(x)\sigma(x)-W[J],$$

where $J$ is eliminated in favor of $\sigma$. Varying gives

$$\frac{\delta\Gamma}{\delta\sigma(x)}=J(x).$$

Thus stationary points of $\Gamma$ at $J=0$ describe possible expectation values in the undeformed theory.

In ordinary QFT, $\Gamma$ is the one-particle-irreducible effective action. In holography, Legendre transforms appear in at least three important places:

1. alternate quantization for scalars near the Breitenlohner-Freedman window;
2. multi-trace deformations, especially double-trace flows;
3. changing ensembles, for example between fixed chemical potential and fixed charge density.

The moral is that sources and vevs are conjugate variables. Holographic boundary conditions choose which variable is fixed.

Multi-trace sources

A source for a single-trace operator is not the same as a multi-trace deformation. Compare

$$\delta S=\int d^d x\,J(x)\mathcal O(x)$$

with

$$\delta S=\frac{f}{2}\int d^d x\,\mathcal O(x)^2.$$

The first changes the one-point source. The second changes the dynamics of the theory. At large $N$, double-trace deformations are especially tractable because factorization makes $\mathcal O^2$ behave semiclassically. Holographically, they often correspond to mixed boundary conditions relating the source-like and vev-like coefficients:

$$A(x)\sim f J(x)$$

or, depending on quantization, the inverse relation.

This is a preview. The main point for this page is that the generating functional can be enlarged to include sources for composite operators, but each source corresponds to a distinct deformation of the QFT.

Euclidean sources, Lorentzian sources, and states

A source is not always merely a deformation of the Hamiltonian. Its interpretation depends on where it is turned on.

Euclidean sources

A Euclidean source over the full Euclidean spacetime computes ordinary Euclidean correlators. A Euclidean source on part of a Euclidean manifold can prepare a state. For example, a source on a Euclidean cap can prepare an excited state on the boundary time slice where the cap ends.

Lorentzian sources

A Lorentzian time-dependent source changes the Hamiltonian:

$$H(t)=H_0-\int d^{d-1}x\,J(t,\mathbf x)\mathcal O(t,\mathbf x)$$

in the correlator convention. It injects energy and creates real-time response. Retarded functions are obtained by differentiating the causal response of one-point functions:

$$\delta\langle\mathcal O(t,\mathbf x)\rangle = \int dt' d^{d-1}y\, G_R(t,\mathbf x;t',\mathbf y)J(t',\mathbf y)+O(J^2),$$

where

$$G_R(x,y) = -i\theta(t_x-t_y)\langle[\mathcal O(x),\mathcal O(y)]\rangle.$$

The Euclidean generating functional alone does not automatically specify the retarded prescription in a black-hole background. In Lorentzian holography, one must impose causal boundary conditions, such as infalling conditions at horizons. This returns later in the pages on real-time correlators and quasinormal modes.

How this page becomes the GKPW prescription

The QFT source calculus says:

$$\langle\mathcal O_1\cdots\mathcal O_n\rangle_{\mathrm{conn}} = \left. \frac{\delta^n W_{\mathrm{CFT}}[J]}{\delta J_1\cdots\delta J_n} \right|_{J=0}.$$

AdS/CFT says:

$$W_{\mathrm{CFT}}[J] = \log Z_{\mathrm{string}}[\phi\to J].$$

In the classical supergravity limit,

$$\log Z_{\mathrm{string}}[\phi\to J] \simeq -S_{\mathrm{bulk}}^{\mathrm{ren}}[\phi_{\mathrm{cl}};J].$$

Therefore

$$\boxed{ \langle\mathcal O_1\cdots\mathcal O_n\rangle_{\mathrm{conn}} = - \left. \frac{\delta^n S_{\mathrm{bulk}}^{\mathrm{ren}}[J]}{\delta J_1\cdots\delta J_n} \right|_{J=0} }$$

in the correlator convention used above. If one uses the Euclidean free-energy convention, the signs are reshuffled but the source dependence is the same.

The computational algorithm is:

1. choose a boundary source $J(x)$;
2. solve the bulk equations with boundary behavior $\phi\to J$;
3. evaluate and renormalize the on-shell action;
4. differentiate with respect to $J$;
5. set $J=0$ unless computing in a deformed theory or background source.

This is the seed of Witten diagrams. Expanding the classical solution and on-shell action perturbatively in $J$ generates tree-level diagrams. Including bulk quantum loops gives $1/N$ corrections.

Common mistakes

Mistake 1: Confusing the source with the operator

The source $J$ is a classical background. The operator $\mathcal O$ is a quantum observable. The statement $J\leftrightarrow\phi$ really means that $J$ is the boundary condition for the bulk field $\phi$, not that $J$ is a fluctuating CFT operator.

Mistake 2: Confusing the source with the vev

Both the source and the vev appear in the asymptotic expansion of a bulk field. The source is fixed; the vev is computed. In standard quantization, the source is the leading coefficient for $\Delta>d/2$ scalar operators, while the vev is related to the subleading coefficient after renormalization.

Mistake 3: Forgetting connected versus disconnected correlators

Derivatives of $Z[J]$ generate full correlators. Derivatives of $W[J]=\log Z[J]$ generate connected correlators. The classical bulk action computes the connected generating functional.

Mistake 4: Treating contact terms as universal

Adding finite local counterterms changes coincident-point terms. Momentum-space correlators can shift by polynomials in momentum. These terms are often scheme-dependent, although anomalies and Ward identities can fix special local pieces.

Mistake 5: Ignoring sign conventions

A formula for $\langle\mathcal O\mathcal O\rangle$ can differ by a minus sign depending on whether the source is written as $+\int J\mathcal O$ in the exponent or $+\int J\mathcal O$ in the Euclidean action. Always track whether the generating functional is $W=\log Z$ or $\mathcal W_E=-\log Z_E$.

Exercises

Exercise 1: Connected correlators from $W[J]$

Let

$$Z[J] = \left\langle e^{\int J\mathcal O}\right\rangle, \qquad W[J]=\log Z[J].$$

Show that

$$\left.\frac{\delta^2 W}{\delta J(x)\delta J(y)}\right|_{J=0} = \langle\mathcal O(x)\mathcal O(y)\rangle - \langle\mathcal O(x)\rangle\langle\mathcal O(y)\rangle.$$

Solution

First,

$$\frac{\delta W}{\delta J(x)} = \frac{1}{Z[J]}\frac{\delta Z[J]}{\delta J(x)}.$$

Differentiating again gives

$$\frac{\delta^2 W}{\delta J(x)\delta J(y)} = \frac{1}{Z}\frac{\delta^2 Z}{\delta J(x)\delta J(y)} - \frac{1}{Z^2} \frac{\delta Z}{\delta J(x)} \frac{\delta Z}{\delta J(y)}.$$

At $J=0$ and $Z[0]=1$,

$$\left.\frac{\delta Z}{\delta J(x)}\right|_{J=0} = \langle\mathcal O(x)\rangle,$$

and

$$\left.\frac{\delta^2 Z}{\delta J(x)\delta J(y)}\right|_{J=0} = \langle\mathcal O(x)\mathcal O(y)\rangle.$$

Therefore

$$\left.\frac{\delta^2 W}{\delta J(x)\delta J(y)}\right|_{J=0} = \langle\mathcal O(x)\mathcal O(y)\rangle - \langle\mathcal O(x)\rangle\langle\mathcal O(y)\rangle.$$

This is the connected two-point function.

Exercise 2: Dimension of a source

A scalar primary operator has scaling dimension $\Delta$. Use dimensional analysis of

$$\delta S=-\int d^d x\,J(x)\mathcal O(x)$$

to determine the scaling dimension of $J$. Which sources are relevant, marginal, and irrelevant?

Solution

The action is dimensionless. Since $d^d x$ has dimension $-d$ and $\mathcal O$ has dimension $\Delta$, we need

$$[J]-d+\Delta=0.$$

Thus

$$[J]=d-\Delta.$$

If $\Delta<d$, the source has positive mass dimension and the deformation is relevant. If $\Delta=d$, the source is dimensionless and the deformation is marginal. If $\Delta>d$, the source has negative mass dimension and the deformation is irrelevant.

Exercise 3: A Ward identity from background gauge invariance

Let $W[A_\mu,J,J^*]$ be invariant under

$$\delta A_\mu=\partial_\mu\alpha, \qquad \delta J=iq\alpha J, \qquad \delta J^*=-iq\alpha J^*.$$

Using

$$\frac{1}{\sqrt g}\frac{\delta W}{\delta A_\mu}=\langle J^\mu\rangle, \qquad \frac{1}{\sqrt g}\frac{\delta W}{\delta J}=\langle\mathcal O\rangle,$$

derive the corresponding Ward identity.

Solution

The variation of $W$ is

$$\delta W = \int d^d x\sqrt g\, \left( \langle J^\mu\rangle\partial_\mu\alpha +iq\alpha J\langle\mathcal O\rangle -iq\alpha J^*\langle\mathcal O^*\rangle \right).$$

Integrating the first term by parts gives

$$\delta W = \int d^d x\sqrt g\,\alpha \left( -\nabla_\mu\langle J^\mu\rangle +iqJ\langle\mathcal O\rangle -iqJ^*\langle\mathcal O^*\rangle \right),$$

ignoring boundary terms. Since $\alpha(x)$ is arbitrary and $\delta W=0$, we obtain

$$\nabla_\mu\langle J^\mu\rangle = iqJ\langle\mathcal O\rangle -iqJ^*\langle\mathcal O^*\rangle.$$

This is current conservation modified by explicit charged sources. For $J=0$ and no anomaly, it becomes $\nabla_\mu\langle J^\mu\rangle=0$.

Exercise 4: Contact terms from finite counterterms

Suppose the connected generating functional is shifted by a finite local term,

$$W[J]\longrightarrow W[J]+\frac{a}{2}\int d^d x\,J(x)^2.$$

How does the two-point function change? Does this affect separated points?

Solution

The added term contributes

$$\frac{\delta}{\delta J(x)} \left(\frac{a}{2}\int d^d u\,J(u)^2\right) = aJ(x),$$

and therefore

$$\frac{\delta^2}{\delta J(x)\delta J(y)} \left(\frac{a}{2}\int d^d u\,J(u)^2\right) = a\delta^{(d)}(x-y).$$

Thus

$$\langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}} \longrightarrow \langle\mathcal O(x)\mathcal O(y)\rangle_{\mathrm{conn}} +a\delta^{(d)}(x-y).$$

This affects only coincident points. For $x\neq y$, the separated-point correlator is unchanged. In momentum space, this contact term appears as a momentum-independent polynomial contribution.

Exercise 5: Legendre transform

Let

$$\sigma(x)=\frac{\delta W[J]}{\delta J(x)}.$$

Define

$$\Gamma[\sigma]=\int d^d x\,J(x)\sigma(x)-W[J],$$

where $J$ is treated as a functional of $\sigma$. Show that

$$\frac{\delta\Gamma}{\delta\sigma(x)}=J(x).$$

Solution

Vary $\Gamma$:

$$\delta\Gamma = \int d^d x\, \left( \delta J(x)\sigma(x)+J(x)\delta\sigma(x) \right) - \int d^d x\, \frac{\delta W}{\delta J(x)}\delta J(x).$$

Using

$$\frac{\delta W}{\delta J(x)}=\sigma(x),$$

the terms proportional to $\delta J$ cancel. Therefore

$$\delta\Gamma = \int d^d x\,J(x)\delta\sigma(x),$$

which implies

$$\frac{\delta\Gamma}{\delta\sigma(x)}=J(x).$$

At zero source, stationary points of $\Gamma$ describe possible expectation values in the undeformed theory.

Further reading

For the original source/boundary-value formulation of AdS/CFT, see Gubser, Klebanov, and Polyakov, Gauge Theory Correlators from Non-Critical String Theory, and Witten, Anti de Sitter Space and Holography. For the systematic treatment of one-point functions, counterterms, Ward identities, and holographic renormalization, see Skenderis, Lecture Notes on Holographic Renormalization. For the role of Legendre transforms and alternate quantization, see Klebanov and Witten, AdS/CFT Correspondence and Symmetry Breaking.