Lectures on Holographic Renormalization

Abstract

We provide a pedagogical introduction to the method of holographic renormalization, in its Hamiltonian incarnation. We begin by reviewing the description of local observables, global symmetries, and ultraviolet divergences in local quantum field theories, in a language that does not require a weak coupling Lagrangian description. In particular, we review the formulation of the Renormalization Group as a Hamiltonian flow, which allows us to present the holographic dictionary in a precise and suggestive language. The method of holographic renormalization is then introduced by first computing the renormalized two-point function of a scalar operator in conformal field theory and comparing with the holographic computation. We then proceed with the general method, formulating the bulk theory in a radial Hamiltonian language and deriving the Hamilton–Jacobi equation. Two methods for solving recursively the Hamilton–Jacobi equation are then presented, based on covariant expansions in eigenfunctions of certain functional operators on the space of field theory couplings. These algorithms constitute the core of the method of holographic renormalization and allow us to obtain the holographic Ward identities and the asymptotic expansions of the bulk fields.

Appendices

Appendix

ADM Identities

A few identities relating to the ADM decomposition (84) of the metric are collected in this appendix. In particular, in matrix form, the metric (84) and its inverse are

$$\begin{aligned} g=\left( \begin{matrix} N^2+N_kN^k &{} N_i \\ N_i &{} \gamma _{ij}\end{matrix}\right) , \quad g^{-1}=\left( \begin{matrix} 1/N^2 &{} -N^i/N^2 \\ -N^i/N^2 &{} \gamma ^{ij}+N^iN^j/N^2\end{matrix}\right) , \end{aligned}$$

(190)

where the indices $i=1,\ldots ,d$ are raised and lowered respectively with $\gamma ^{ij}$ and $\gamma _{ij}$. Moreover, the Christoffel symbols $\varGamma ^\rho _{\mu \nu }[g]$ can be decomposed into the following components in terms of N, $N_i$ and $\gamma _{ij}$:

$$\begin{aligned} \begin{aligned} \varGamma ^r_{rr}&=N^{-1}\left( \dot{N}+N^i\partial _i N-N^iN^jK_{ij}\right) ,\\ \varGamma ^r_{ri}&=N^{-1}\left( \partial _iN-N^jK_{ij}\right) ,\\ \varGamma ^r_{ij}&=-N^{-1}K_{ij},\\ \varGamma ^i_{rr}&=-N^{-1}N^i\dot{N}-ND^iN-N^{-1}N^iN^j\partial _jN+\dot{N}^i+N^jD_jN^i+2NN^jK^i_j\\ {}&\quad +N^{-1}N^iN^kN^lK_{kl},\\ \varGamma ^i_{rj}&=-N^{-1}N^i\partial _jN+D_jN^i+N^{-1}N^iN^kK_{kj}+NK^i_j,\\ \varGamma ^k_{ij}&=\varGamma ^k_{ij}[\gamma ]+N^{-1}N^kK_{ij}. \end{aligned} \end{aligned}$$

(191)

Hamilton–Jacobi Primer

In this appendix we collect a few essential facts about HJ theory in classical mechanics. For an in-depth account of HJ theory we refer the interested reader to [24, 40]. A more abstract exposition can be found in [41].

Let $\mathcal{Q}$ be the configuration space of a point particle described by the action¹⁴

$$\begin{aligned} S=\int ^t dt' L(q,\dot{q};t), \end{aligned}$$

(192)

where $q^\alpha$ are coordinates on $\mathcal{Q}$. In the Hamiltonian formalism the generalized coordinates $q^\alpha$ and the canonical momenta

$$\begin{aligned} p_\alpha =\frac{\partial L}{\partial \dot{q}^\alpha }, \end{aligned}$$

(193)

are independent variables parameterizing the phase space, which is isomorphic to the cotangent bundle $T^*\mathcal{Q}$ of the configuration space $\mathcal{Q}$. The cotangent bundle is a symplectic manifold with a canonical closed 2-form (symplectic form)

$$\begin{aligned} {\Omega }=dp_\alpha \wedge dq^\alpha . \end{aligned}$$

(194)

Since is closed, it can be locally expressed as

(195)

where

$$\begin{aligned} \varTheta =p_\alpha dq^\alpha , \end{aligned}$$

(196)

is known as the canonical 1-form, or pre-symplectic form. The Hamiltonian, given by the Legendre transform of the Lagrangian,

$$\begin{aligned} H(p,q;t)=p_\alpha \dot{q}^\alpha -L, \end{aligned}$$

(197)

is a map $H:T^*\mathcal{Q}\longrightarrow \mathbb {R}$ and governs the time evolution of the dynamical system through Hamilton’s equations

$$\begin{aligned} \dot{q}^\alpha = \frac{\partial H}{\partial p_\alpha },\quad \dot{p}_\alpha = -\frac{\partial H}{\partial q^\alpha }. \end{aligned}$$

(198)

At this point it is instructive to distinguish two cases, depending on whether the Hamiltonian depends explicitly on time t or not.

• Time-independent systems

  A section, s, of the cotangent bundle is a map $s:\mathcal{Q}\longrightarrow T ^*\mathcal{Q}$, providing a 1-form over each point $q\in \mathcal{Q}$. A closed section of $T^*\mathcal{Q}$ is locally exact and so it can be written as $s=d\mathcal{W}$ for some function $\mathcal{W}(q)$ on $\mathcal{Q}$. Under the isomorphism between phase space and the cotangent bundle this means that locally

  $$\begin{aligned} p_\alpha =\frac{\partial \mathcal{W}(q)}{\partial q^\alpha }. \end{aligned}$$

  (199)

  Moreover,

  

  (200)

  These results hold for any closed section s of $T^*\mathcal{Q}$. The HJ theorem relates certain closed sections, s, of the cotangent bundle to solutions of Hamilton’s equations (198). In particular,

  $$\begin{aligned} d\left( H\circ s\right)&=\left( \frac{\partial H}{\partial q^\alpha }+\frac{\partial ^2\mathcal{W}}{\partial q^\beta \partial q^\alpha }\frac{\partial H}{\partial p_\beta }\right) dq^\alpha \nonumber \\&=\left( \frac{\partial H}{\partial q^\alpha }+\dot{p}^\alpha \right) dq^\alpha +\frac{\partial ^2\mathcal{W}}{\partial q^\beta \partial q^\alpha }\left( \frac{\partial H}{\partial p_\beta }-\dot{q}^\beta \right) dq^\alpha , \end{aligned}$$

  (201)

  which implies that the following two statements are equivalent (see Theorem 2.1 in [42]):

  • (i) If $\sigma :\mathbb {R}\rightarrow \mathcal{Q}$ satisfies the first of Hamilton’s equations in (198), then $s\circ \sigma$ satisfies the second Hamilton equation.

  • (ii) $d\left( H\circ s\right) =0$.

  Hence, a closed section $s=d\mathcal{W}$ of the cotangent bundle that satisfies the (time-independent) HJ equation

  $$\begin{aligned} H\circ s=H\left( \frac{\partial \mathcal{W}}{\partial q^\alpha },q^\beta \right) =E, \end{aligned}$$

  (202)

  where E is some constant, provides a solution of Hamilton’s equations.

• Time-dependent systems

  In order to accommodate systems with a Hamiltonian that explicitly depends on time we extend the configuration space by including time as a generalized coordinate so that $\mathcal{Q}_\mathrm{ext}=\mathcal{Q}\times \mathbb {R}$ is now the extended configuration space. Phase space is accordingly extended by including $-H$ as the canonical momentum conjugate to t. This extended phase space is isomorphic to the cotangent bundle $T^*\mathcal{Q}_\mathrm{ext}$, which carries the canonical symplectic form

  $$\begin{aligned} {\Omega }_\mathrm{ext}=d\varTheta _\mathrm{ext}=dp_\alpha \wedge dq^\alpha -dH\wedge dt. \end{aligned}$$

  (203)

  Moreover, to Hamilton’s equations we can now append the equation

  $$\begin{aligned} \dot{H}=\frac{\partial H}{\partial t}. \end{aligned}$$

  (204)

  A closed section of $T^*\mathcal{Q}_\mathrm{ext}$ can be locally written as $s=d\mathcal{S}$ for some function on $\mathcal{Q}_\mathrm{ext}$, and consequently

  $$\begin{aligned} p_\alpha =\frac{\partial \mathcal{S}(q;t)}{\partial q^\alpha },\quad -H=\frac{\partial \mathcal{S}(q;t)}{\partial t}, \end{aligned}$$

  (205)

  which imply that

  $$\begin{aligned} \varTheta _\mathrm{ext}\circ s=d\mathcal{S},\quad {\Omega }_\mathrm{ext}\circ s=0. \end{aligned}$$

  (206)

  It follows that

  $$\begin{aligned} 0&=d\left( H\circ s+\frac{\partial \mathcal{S}}{\partial t}\right) \nonumber \\&=\left[ \frac{\partial H}{\partial q^\alpha }+\dot{q}^\beta \frac{\partial ^2\mathcal{S}}{\partial q^\beta \partial q^\alpha }+\frac{\partial ^2\mathcal{S}}{\partial t\partial q^\alpha }+\frac{\partial ^2\mathcal{S}}{\partial q^\beta \partial q^\alpha }\left( \frac{\partial H}{\partial p_\beta }-\dot{q}^\beta \right) \right] dq^\alpha \nonumber \\&\quad +\left[ \frac{\partial H}{\partial t}+\dot{q}^\alpha \frac{\partial ^2\mathcal{S}}{\partial t\partial q^\alpha }+\frac{\partial ^2\mathcal{S}}{\partial t^2}+\frac{\partial ^2\mathcal{S}}{\partial q^\beta \partial t}\left( \frac{\partial H}{\partial p_\beta }-\dot{q}^\beta \right) \right] dt\nonumber \\&=\left[ \frac{\partial H}{\partial q^\alpha }+\dot{p}^\alpha +\frac{\partial ^2\mathcal{S}}{\partial q^\beta \partial q^\alpha }\left( \frac{\partial H}{\partial p_\beta }-\dot{q}^\beta \right) \right] dq^\alpha \nonumber \\&\quad +\left[ \frac{\partial H}{\partial t}-\dot{H}+\frac{\partial ^2\mathcal{S}}{\partial q^\beta \partial t}\left( \frac{\partial H}{\partial p_\beta }-\dot{q}^\beta \right) \right] dt, \end{aligned}$$

  (207)

  which allows us to generalize the HJ theorem to time-dependent Hamiltonians. Namely, a closed section $s=d\mathcal{S}$ of $T^*\mathcal{Q}_\mathrm{ext}$ that satisfies the HJ equation

  $$\begin{aligned} H\circ s+\frac{\partial \mathcal{S}}{\partial t}=H\left( \frac{\partial \mathcal{S}}{\partial q^\alpha },q^\beta ;t\right) +\frac{\partial \mathcal{S}}{\partial t}=0, \end{aligned}$$

  (208)

  provides a solution to Hamilton’s equations.

A few comments are in order at this point. Firstly, note that the HJ formalism for time-dependent Hamiltonians reduces to that for time-independent Hamiltonians upon setting

$$\begin{aligned} \mathcal{S}(q;t)=\mathcal{W}(q)-E t. \end{aligned}$$

(209)

The function $\mathcal{S}(q;t)$ is known as Hamilton’s principal function, while $\mathcal{W}(q)$ is called the characteristic function. Secondly, the expressions (94) for the canonical momenta and the Hamiltonian should be familiar from quantum mechanics. Indeed, Hamilton’s principal function $\mathcal{S}(q;t)$ is related to the WKB wavefunction by

(210)

and so the expressions (94) are respectively the coordinate representation of the momentum operator and the identification of the Hamiltonian with the time evolution operator.

Finally, Hamilton’s principal function $\mathcal{S}(q;t)$, defined as a solution of the HJ equation (208), is closely related to the on-shell action. To elucidate the relation, consider the action (192) on the semi-infinite line $(-\infty ,t]$. A general variation of the action (192) gives

$$\begin{aligned} \delta S=\int ^t dt' \left( \frac{\partial L}{\partial q^\alpha }\delta q^\alpha +\frac{\partial L}{\partial \dot{q}^\alpha }\delta \dot{q}^\alpha \right) = \int ^t dt' \left( \frac{\partial L}{\partial q^\alpha }-\frac{d}{dt'}\left( \frac{\partial L}{\partial \dot{q}^\alpha }\right) \right) \delta q +\left. \frac{\partial L}{\partial \dot{q}^\alpha }\delta q^\alpha \right| _t. \end{aligned}$$

(211)

To ensure that the variational principle implies the equations of motion we need to impose the boundary condition $\delta q^\alpha =0$ at $t'=t$. The on-shell action therefore becomes a function of the fixed but arbitrary boundary condition $q^\alpha (t)$, namely $S_{\text {on}\text {-}\text {shell}}(q;t)$, while

$$\begin{aligned} \left. p_\alpha \right| _t=\left. \frac{\partial L}{\partial \dot{q}^\alpha }\right| _t=\frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial q^\alpha }. \end{aligned}$$

(212)

Moreover,

$$\begin{aligned} \dot{S}_{\text {on}\text {-}\text {shell}}=L=\frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial t}+\frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial q^\alpha }\dot{q}^\alpha , \end{aligned}$$

(213)

and so $S_{\text {on}\text {-}\text {shell}}$ satisfies the HJ equation (208):

$$\begin{aligned} 0=p_\alpha \dot{q}^\alpha -L+\frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial t}=H\left( \frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial q^\alpha },q^\beta ;t\right) +\frac{\partial S_{\text {on}\text {-}\text {shell}}}{\partial t}. \end{aligned}$$

(214)

We therefore conclude that the on-shell action as a function of the arbitrary but fixed boundary condition q(t), $S_{\text {on}\text {-}\text {shell}}(q;t)$, can be identified with Hamilton’s principal function $\mathcal{S}(q;t)$. The fact that the on-shell action is a solution of the HJ equation is the fundamental reason for the critical role that HJ theory has in holographic renormalization.