Renormalized One-Point Functions and the Variational Problem

Renormalized one-point functions are finite canonical responses obtained by varying the complete renormalized action with respect to independent boundary sources. The response coefficient alone is not the answer: local contact terms, source mixing, field normalization, constraints, and the chosen Dirichlet, Neumann, or mixed variational problem all contribute. A correct formula must make the boundary variation finite and integrable.

Required background. Scalar counterterms construct a finite scalar action. Metric counterterms and the boundary stress tensor fix gravitational variation. Helpful background. Renormalized composite-operator insertions explains local source mixing on the QFT side.

First application. Vary the renormalized scalar action at fixed source and identify the response coefficient and local contact contribution to the one-point function.

Renormalized canonical momenta

Let $q^A$ denote all induced cutoff fields: the metric, scalar sources, gauge potentials, and any additional form data. Define

$$\Pi_A^{\mathrm{ren}}(\epsilon,x) =\frac{1}{\sqrt\gamma} \frac{\delta(S_{\mathrm{reg}}+S_{\mathrm{ct}})}{\delta q^A(x)}.$$

After rescaling each field by its leading radial weight, the finite limit gives the expectation value conjugate to its boundary source. For a scalar in standard quantization,

$$\langle\mathcal O\rangle =\mathcal N(2\Delta-d)\phi_{(2\Delta-d)} +\mathcal F_{\mathrm{local}}[\phi_{(0)},g_{(0)},a],$$

while

$$\langle J^i\rangle =\frac{1}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta a_i}, \qquad \langle T^{ij}\rangle =\frac{2}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta g_{(0)ij}}.$$

These are definitions in the selected source convention. Some literature varies with respect to $g^{ij}$ rather than $g_{ij}$ or uses the Lorentzian generating functional $W=-i\log Z$; the resulting signs must be translated before comparing coefficients.

Scalar source-response derivation

For the Euclidean scalar normalization

$$S=\frac{\mathcal N}{2} \int\sqrt G\left[(\nabla\phi)^2+m^2\phi^2\right],$$

the regulated canonical momentum is $\pi_\phi=\mathcal N\,n\cdot\partial\phi$. Counterterm variation cancels all terms determined locally by $\phi_{(0)}$ that diverge as $\epsilon\to0$. The first undetermined term contributes

$$\delta S_{\mathrm{ren}} =\mathcal N\int\mathrm d^d x\sqrt{g_{(0)}} \left[(2\Delta-d)\phi_{(2\Delta-d)} +\mathcal F_{\mathrm{local}}\right]\delta\phi_{(0)}.$$

The factor $2\Delta-d$ comes from the difference of the two radial exponents. It is independently recovered from the renormalized Klein–Gordon symplectic flux, which makes this a useful normalization check.

If the regular interior solution is linear in the source,

$$\phi_{(2\Delta-d)}(x) =\int\mathrm d^d y\,\mathcal K(x,y)\phi_{(0)}(y),$$

then differentiating once more gives the connected two-point function. Only the nonlocal part of $\mathcal K$ is invariant under finite local counterterms.

Mixing and contact terms

When several bulk fields share quantum numbers or interact, the asymptotic coefficients mix. Write sources $J^a$ and renormalized operators $\mathcal O_a$. A finite local source redefinition $J'^a=J'^a(J,\partial J)$ changes the one-point vector by the functional Jacobian and adds contacts. The invariant comparison is a separated-point correlator, a Ward identity, an integrated charge, or another quantity insensitive to the chosen local basis.

Metric dependence creates further mixing. For a scalar source,

$$\nabla_i\langle T^i{}_j\rangle =\langle\mathcal O\rangle\partial_j\phi_{(0)}+\cdots,$$

so changing the scalar counterterm also changes consistent contact terms in $T_{ij}$. One cannot adjust the scalar one-point function while leaving all stress-tensor contacts untouched.

Boundary conditions define the source

Dirichlet variation holds $\delta\phi_{(0)}=0$ when solving the bulk problem and treats $\phi_{(0)}$ as the source when differentiating the generating functional. In the alternate-quantization window, a Legendre transform exchanges the source and response. A mixed condition adds a multi-trace functional and changes the conjugate pair. The correct one-point function is always the coefficient of the independent variation in the final action.

This gives an adversarial test. Add a finite scalar counterterm and separately add a double-trace boundary term. The first changes $\mathcal F_{\mathrm{local}}$ but not the separated-point nonanalytic kernel. The second changes the boundary condition and hence the full response kernel. Treating both as “scheme” fails this test.

Completion checks

• the on-shell variation is finite for every allowed independent source variation;
• scalar, gauge, and momentum constraints reproduce the Ward identities;
• mixed second derivatives of $S_{\mathrm{ren}}$ agree, up to understood anomalies;
• the Euclidean kernel has the expected reflection and reality properties;
• analytic continuation states which Lorentzian correlator and $i0$ prescription is obtained;
• finite local changes affect only the contact structures they are allowed to affect.

The variational method supplies expectation values in a specified state and scheme. It does not determine the state without an interior condition, nor does it turn a bottom-up bulk action into a complete dual theory.

The variational extraction of renormalized one-point functions and its constraint checks follow the systematic holographic-renormalization construction of Bianchi, Freedman, and Skenderis 2002; the link supports the finite-source-variation claim, not a choice of finite scheme.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bianchi, M., Freedman, D. Z., and Skenderis, K. “Holographic Renormalization.” Nuclear Physics B 631 (2002): 159–194. DOI. arXiv.
• de Haro, S., Solodukhin, S. N., and Skenderis, K. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. DOI. arXiv.
• Papadimitriou, I., and Skenderis, K. “Correlation Functions in Holographic RG Flows.” Journal of High Energy Physics 2004, 075 (2004). DOI. arXiv.