Black hole entropy and the renormalization group

Alejandro Satz

University of Maryland

Based on work in collaboration with Ted Jacobson

13th Marcel Grossmann Meeting - Stockholm, July 2nd 2012

Outline

1. Introduction: Entanglement entropy and black hole entropy

2. Renormalization group and black hole entropy

3. Free fields

4. Interacting fields

5. Conclusions

1. Introduction

Entanglement entropy

Quantum (pure) state |ψ〉on a region A∪B, can be written in a basis of products of states for each

region. The reduced density matrix 𝜌A formed by tracing over the B states has nonzero entropy:

The entanglement entropy has ultraviolet divergences. With a UV short-distance cutoff 𝜀, the leading order scales as the area of the d-2 surface between A and B:

It can be proven that SA = SB.

1. Introduction

Entanglement entropy

Quantum (pure) state |ψ〉on a region A∪B, can be written in a basis of products of states for each

region. The reduced density matrix 𝜌A formed by tracing over the B states has nonzero entropy:

The entanglement entropy has ultraviolet divergences. With a UV short-distance cutoff 𝜀, the leading order scales as the area of the d-2 surface between A and B:

On a black hole background, the restriction of the global Hartle-Hawking vacuum to the exterior is a thermal state at the Hawking temperature

entanglement entropy equals thermal entropy of the fields in the exterior

It can be proven that SA = SB.

1. Introduction

Entanglement entropy

Quantum (pure) state |ψ〉on a region A∪B, can be written in a basis of products of states for each

region. The reduced density matrix 𝜌A formed by tracing over the B states has nonzero entropy:

The entanglement entropy has ultraviolet divergences. With a UV short-distance cutoff 𝜀, the leading order scales as the area of the d-2 surface between A and B:

On a black hole background, the restriction of the global Hartle-Hawking vacuum to the exterior is a thermal state at the Hawking temperature

entanglement entropy equals thermal entropy of the fields in the exterior

It can be proven that SA = SB.

Using a Euclidean partition function representation for the density matrix:

“Off-shell” computation, introduces a conical singularity

2. Introduction

Canonical partition function

Consider a partition function including both matter and metric fields. The matter fields integrate out to give the effective action for gravity, and gravity is treated “classically” with a saddle point evaluation:

ḡ = ḡ(β) solution of the effective equations of motion.

2. Introduction

Canonical partition function

Consider a partition function including both matter and metric fields. The matter fields integrate out to give the effective action for gravity, and gravity is treated “classically” with a saddle point evaluation:

ḡ = ḡ(β) solution of the effective equations of motion.

Then by thermodynamics:

The W-term is the contribution of the matter fields to the entropy. The on-shell procedure and the off-shell one agree since:

and the second term vanishes at the on-shell metric.

2. Introduction

Canonical partition function

Consider a partition function including both matter and metric fields. The matter fields integrate out to give the effective action for gravity, and gravity is treated “classically” with a saddle point evaluation:

ḡ = ḡ(β) solution of the effective equations of motion.

Then by thermodynamics:

The W-term is the contribution of the matter fields to the entropy. The on-shell procedure and the off-shell one agree since:

and the second term vanishes at the on-shell metric.

If 𝛤0 can be approximated as the Einstein-Hilbert action with Λ=0 and Gibbons-Hawking boundary term, then ḡ is Euclidean Schwarzschild, and SBH=A/4G0 comes only from the boundary term.

2. Introduction

Renormalization of the effective action

In a one-loop approximation, the matter contribution to the effective action is:

for a free, massless scalar field. Using the heat kernel expansion:

The divergences in W are absorbed into the bare coefficients of Sb to render a finite 𝛤0. The entropy gets automatically renormalized in the same way as the effective action:

(d=4)

2. Introduction

Non-minimal coupling

for a massless scalar; similar for gauge fields, gravitons (with gauge fixing and FP ghosts)

The non-minimal coupling gives a potentially negative contribution to the entropy.

This is not entanglement entropy, which is positive definite. It can be interpreted as Wald entropy:

For the Einstein-Hilbert action, this matches SBH = A/4G. When there is a nonminimally coupled quantum scalar, it subtracts to it:

so we have:

“contact term”

(Wald 1993, Jacobson 1994)

3. Renormalization group

RG conceptualization of BH entropy

Instead of the integrating out all the matter degrees of freedom at once, integrate out only the modes between the UV cutoff scale Λ ∼1/𝜀

and an intermediate scale k < Λ.

The contributions from k < p < Λ add up with Sb[g] into a flowing effective gravitational action 𝛤k[g] , which reduces to 𝛤0 as k→0.

The unintegrated “lower” modes are still “quantum” and contribute entanglement/thermal entropy.

As one moves the RG scale k, the total entropy remains constant while the balancing between the different contributions changes.

Λ

k

0

3. Renormalization group

Problems

• The separation between upper and lower modes is clearly definable in the Euclidean, through a cut-off function of the Laplacian -𝛻2. How do we interpret back each term of the entropy in the Lorentzian?

• The uncertainty principle makes it impossible to have simultaneosly a sharp spatial division of states (to define entanglement entropy accross the horizon) and a sharp separation in momentum space (to define the RG flow).

• But in the Euclidean there is no black hole interior!

• For an interacting theory, the Wilsonian effective action for the lower modes will be nonlocal. This impedes going back from Z to 𝜌 = exp(-βH) to have an entanglement entropy interpretation for their contribution to the entropy.

• If there are mass thresholds, we can argue that integrating past one of them “kills” the nonlocalities: there is a local expansion in inverse powers of the mass scale.

4. Free fields

Minimally coupled scalar

4. Free fields

Minimally coupled scalar

The effective action for gravity 𝛤k satisfies the Wetterich flow equation:

4. Free fields

Minimally coupled scalar

The effective action for gravity 𝛤k satisfies the Wetterich flow equation:

It encompasses the effects of the modes above k. Those of the lower modes are accounted for in a partition function with an inbuilt UV cutoff at scale k. The total entropy is distibuted as:

since for the cutoff function the running of G is given by

4. Free fields

Non-minimal coupling

Then “gravitational” contribution to the entropy, from 𝛤k[g], is:

and the entropy of the quantum modes below k is distributed as:

4. Free fields

Non-minimal coupling

Then “gravitational” contribution to the entropy, from 𝛤k[g], is:

and the entropy of the quantum modes below k is distributed as:

Note: Reuter and Becker have recently argued that the nonminimal term of the bare action does not contribute to the boundary term of the effective action, and that bulk and boundary G get different renormalizations. (Conflicting with previous results by Barvinsky and Solodukhin 1995)

If so, then the 𝜉 term should really be absent from both expressions!

5. Interacting fields

Polchinski equation in curved space

5. Interacting fields

Polchinski equation in curved space

flow equation

5. Interacting fields

Polchinski equation in curved space

flow equation

Decompose:

non-kinetic terms of matter action

gravitational effective action

Then we can trace their evolution separately. The first encodes the gravitational entropy, and the second, under a path integral UV-cutoffed at k, the entropy of the matter field. Neither is uniquely defined, since they depend on the choice of cutoff function.

5. Interacting fields

Momentum space entanglement?

Text

Balasubramanian, McDermott and Van Raamsdonk (2011) point out that in an interacting quantum field theory, there is mutual entanglement between modes above and below a certain scale.

A simple order of magnitude estimate shows that this entanglement entropy scales as the volume of spacetime and hence dominates over the spatial horizon entanglement entropy.

This entropy seems absent from our description.

5. Interacting fields

Momentum space entanglement?

Text

Balasubramanian, McDermott and Van Raamsdonk (2011) point out that in an interacting quantum field theory, there is mutual entanglement between modes above and below a certain scale.

A simple order of magnitude estimate shows that this entanglement entropy scales as the volume of spacetime and hence dominates over the spatial horizon entanglement entropy.

This entropy seems absent from our description.

Answer:

It is misleading to describe the entropy computed from a partition function over the lower modes using Sk as “the entropy of the lower modes”.

The partition function with a Wilsonian action is equivalent to the full theory, it does not involve tracing over the upper modes and losing information. We are just identifying an “effective gravitational” and an “effective quantum matter” term in this description.

6. Conclusions

Summary

• Black hole entropy, in a semiclassical approximation, comes from the full effective action for gravity evaluated on shell.

• We reconceptualize this, through the introduction of a sliding RG scale, as coming partly from a flowing gravitational effective action, partly from the Wilsonian effective action for the quantum theory.

• In an “induced gravity” scenario there is no bare gravitational contribution; in the deep UV regime only the matter fields are present.

• For free fields (with minimal or non-minimal coupling), there is a clean interpretation for both contributions to the entropy, as the entropy of the “lower” and “upper” modes.

• (However, it is problematic to trace back this distinction to the Lorentzian theory!)

• For interacting fields, we can still track both contributions with the Polchinski equation, but we can not in general make this interpretation, due to momentum entanglement and nonlocalities in the effective action.

6. Conclusions

Outlook

• Application to a theory with mass thresholds where the description at different RG scales is qualitatively different.

• Cf. Kabat, Shenker and Strassler (1995) on black hole entropy in the O(n) model.

• Can the framework for interacting theories be adapted for quantum gravity?

• If quantum gravity is asymptotically safe, there is no substantial need for an overall UV cutoff Λ. Cf. work of Reuter et al., especially Reuter and Becker (2012).

• Can we formulate the RG running of the entropy without leaving the Lorentzian picture?

• Clarify better under which conditions can “the entropy of the lower modes” be identified with entanglement entropy.