Breakdown of the semi-classical approximation in the path integral and black hole thermodynamics

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Breakdown of the semi-classical approximation in the path integral and black hole thermodynamics

Based upon work with Bob McNees

Daniel Grumiller

Center for Theoretical Physics Massachusetts Institute of Technology

University of Washington,June 2007

hep-th/0703230

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Outline

Introduction

Euclidean Path Integral

Dilaton Gravity in 2D

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics 2/32

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Outline

Introduction

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Introduction 3/32

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Black Hole Thermodynamics - Why?

B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: dS ≥0

Classical General Relativity

I Four Laws(Bardeen, Carter, Hawking, 1973)

I Gedankenexperiments with entropy

(Bekenstein, 1973)

Black Hole Analogues

I Sonic Black Holes(Unruh, 1981)

I Hawking effect in condensed matter?

Black Hole Thermodynamics

Quantum Gravity

I Semiclassical approximation?

I Microstate counting(Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997)

Dual Formulations

I AdS/CFT(Maldacena 1997, Gubser, Klebanov, Polyakov 1998, Witten 1998)

I Hawking-Page transition

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B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: dS ≥0

Classical General Relativity

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

I Microstate counting(Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997)

Dual Formulations

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B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: dS ≥0 Classical General Relativity

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

I Microstate counting(Strominger, Vafa, 1996;

Ashtekar, Corichi, Baez, Krasnov, 1997)

Dual Formulations

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B-H: S= _4G^A

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

Dual Formulations

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B-H: S= _4G^A

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

Dual Formulations

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Black Hole Thermodynamics - How?

Many different approaches available...

Approach:

I Physical arguments

I QFT on fixed BG

I Conformal anomaly

I Gravitational anomaly

I Euclidean path integral

Advantage:

I Very simple

I Rigorous, plausible

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

I Not convincing “first time”-derivation of Hawking effect

I Convenient short-cut to obtain thermodynamical partition function

I Rather insensitive to matter coupling

I Useful insights about gravitational actions!

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Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

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Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

(12)

Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

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Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

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Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

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Approach:

I QFT on fixed BG

I Conformal anomaly

Advantage:

I Very simple

I Rigorous, simple

I Plausible, simple

I Very simple

Drawback:

I ad-hoc!

I lengthy

I too special?

I additional input?

I physical?

Employ Euclidean Path Integral Approach

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Outline

Introduction

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 6/32

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Main Idea

Consider Euclidean path integral (Gibbons, Hawking, 1977)

Z = Z

DgDX exp

−1

~I_E[g, X]

I g: metric,X: scalar field

I Semiclassical limit (~→0): dominated by classical solutions (?)

I Exploit relationship between Z and Euclidean partition function Z ∼e^−β^Ω

I Ω: thermodynamic potential for appropriate ensemble

I β: periodicity in Euclidean time

Requires periodicity in Euclidean time and accessibility of semi-classical approximation

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Main Idea

Z = Z

DgDX exp

−1

~I_E[g, X]

I g: metric, X: scalar field

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Main Idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Main Idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Main Idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Semiclassical Approximation

Consider small perturbation around classical solution

IE[gcl+δg, Xcl+δX] =IE[gcl, Xcl] +δIE[gcl, Xcl;δg, δX]

+1

2δ²I_E[g_cl, X_cl;δg, δX] +. . .

I The leading term is the ‘on-shell’ action.

I The linear term should vanish on solutions gcl and Xcl.

I The quadratic term represents the first corrections. If nothing goes wrong:

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

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IE[gcl+δg, Xcl+δX] =IE[gcl, Xcl]+δIE[gcl, Xcl;δg, δX]

+1

I Theleading term is the ‘on-shell’ action.

I The linear term should vanish on solutions gcl and Xcl.

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

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+1

I Thelinear term should vanish on solutionsgcl and Xcl.

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

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+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

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+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

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What could go Wrong?

...everything!

Accessibility of the semiclassical approximation requires

1. I_E[g_cl, X_cl]>−∞ 2. δI_E[g_cl, X_cl;δg, δX] = 0 3. δ²I_E[g_cl, X_cl;δg, δX]≥0

Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below

2. Violated: First variation of action not zero for all field configurations contributing to path integral due to boundary terms

δIE

EOM∼

Z

∂M

dx√ γ

h

π^abδγab+π_XδX i

6= 0

3. Frequently violated: Gaussian integral may diverge Focus in this talk on thesecond problem!

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...everything!

Accessibility of the semiclassical approximation requires 1. I_E[g_cl, X_cl]>−∞

2. δI_E[g_cl, X_cl;δg, δX] = 0 3. δ²I_E[g_cl, X_cl;δg, δX]≥0

Typical gravitational actions evaluated on black hole solutions:

1. Violated: Action unbounded from below

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

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...everything!

2. δI_E[g_cl, X_cl;δg, δX] = 0

3. δ²I_E[g_cl, X_cl;δg, δX]≥0

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

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...everything!

2. δI_E[g_cl, X_cl;δg, δX] = 0

3. δ²I_E[g_cl, X_cl;δg, δX]≥0

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

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...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

3. Frequently violated: Gaussian integral may diverge

Focus in this talk on thesecond problem!

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...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

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Outline

Introduction

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 10/32

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The Action

...for a review cf. e.g. DG, W. Kummer and D. Vassilevich,hep-th/0204253

Standard form of the action:

I_E =−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ γL(X)

I Dilaton X defined via coupling to Ricci scalar

I Model specified by kinetic and potential functions for dilaton

I Dilaton gravity analog of Gibbons-Hawking-York boundary term: coupling of X to extrinsic curvatureof (∂M, γ)

Variational principle: fix X and induced metricγ at∂M

Note: additional boundary term allowed consistent with classical solutions, variational principle and symmetries!

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The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ γL(X)

I Model specified by kinetic and potential functions for dilaton

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The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ γL(X)

I Model specified by kineticandpotential functions for dilaton

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The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√ γXK

− Z

∂M

dx√ γL(X)

I Dilaton gravity analog of Gibbons-Hawking-York boundary term:

coupling of X to extrinsic curvatureof(∂M, γ) Variational principle: fix X and induced metricγ at∂M

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The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√

γXK−

Z

∂M

dx√ γL(X)

I Dilaton gravity analog of Gibbons-Hawking-York boundary term:

coupling of X to extrinsic curvatureof(∂M, γ) Variational principle: fix X and induced metricγ at∂M

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Selected List of Models

Black holes in(A)dS, asymptotically flat or arbitrary spaces

Model U(X) V(X)

1. Schwarzschild (1916) −_2X¹ −λ²

2. Jackiw-Teitelboim (1984) 0 ΛX

3. Witten Black Hole (1991) −_X¹ −2b²X

4. CGHS (1992) 0 −2b²

5.(A)dS2 ground state (1994) −_X^a BX

6. Rindler ground state (1996) −_X^a BX^a

7. Black Hole attractor (2003) 0 BX⁻¹

8. Spherically reduced gravity (N >3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2)

9. All above: ab-family (1997) −_X^a BX^a+b

10. Liouville gravity a be^αX

11. Reissner-Nordstr¨om (1916) −_2X¹ −λ²+^Q_X²

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X

13. Katanaev-Volovich (1986) α βX²−Λ

14. BTZ/Achucarro-Ortiz (1993) 0 ^Q_X² −_4X^J₃−ΛX

15. KK reduced CS (2003) 0 ¹₂X(c−X²)

16. KK red. conf. flat (2006) −¹₂tanh (X/2) AsinhX 17. 2D type 0A string Black Hole −_X¹ −2b²X+^b²_8π^q²

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Selected List of Models

Black holes in(A)dS,asymptotically flatorarbitrary spaces

Model U(X) V(X)

1. Schwarzschild (1916) −_2X¹ −λ²

2. Jackiw-Teitelboim (1984) 0 ΛX

3. Witten Black Hole (1991) −_X¹ −2b²X

4. CGHS (1992) 0 −2b²

5.(A)dS2 ground state (1994) −_X^a BX

6. Rindler ground state (1996) −_X^a BX^a

7. Black Hole attractor (2003) 0 BX⁻¹

8. Spherically reduced gravity (N >3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2)

9. All above: ab-family (1997) −_X^a BX^a+b

10. Liouville gravity a be^αX

11. Reissner-Nordstr¨om (1916) −_2X¹ −λ²+^Q_X²

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X

13. Katanaev-Volovich (1986) α βX²−Λ

14. BTZ/Achucarro-Ortiz (1993) 0 ^Q_X² −_4X^J₃−ΛX

15. KK reduced CS (2003) 0 ¹₂X(c−X²)

16. KK red. conf. flat (2006) −¹₂tanh (X/2) AsinhX 17. 2D type 0A string Black Hole −_X¹ −2b²X+^b²_8π^q²

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Equations of Motion (EOM) Extremize the action: δI_E = 0

U(X)∇µX∇νX−1

2gµνU(X)(∇X)²−gµνV(X) +∇µ∇νX−gµν∇²X= 0 R+∂U(X)

∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]

I Generalized Birkhoff theorem: at least one Killing vector

I Orbits of this vector are isosurfaces of the dilaton L_kX=k^µ∂µX = 0

I Choose henceforth ∂Mas X= const.hypersurface

Adapted coordinate system (LapseandShift for radial evolution) X =X(r) ds² = N(r)²

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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∂X (∇X)²+ 2U(X)∇²X−2∂V(X)

∂X = 0

| {z }

:=ξ(r)⁻¹

dr²+ ξ(r)

|{z}

=k^µkµ

(dτ+ N^τ(r)

| {z }

:=0

dr)²

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Solutions

I Define two model-dependent functions Q(X) :=Q0+

Z X

dX U˜ ( ˜X) w(X) :=w₀−2

Z X

dX V˜ ( ˜X)e^{Q( ˜}^X)

I Q0 andw0 are arbitrary constants (essentially irrelevant)

I Construct all classical solutions

∂rX=e^−Q(X) ξ(X) =e^Q(X) w(X)−2M

Constant of motion M (“mass”) characterizes classical solutions

I Absorb Q₀ into rescaling of length units

I Shift w₀ such that M = 0 ground state solution

I Restrict to positive mass sector M ≥0

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Solutions

Z X

dX U˜ ( ˜X) w(X) :=w₀−2

Z X

dX V˜ ( ˜X)e^{Q( ˜}^X)

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Solutions

Z X

dX U˜ ( ˜X) w(X) :=w₀−2

Z X

dX V˜ ( ˜X)e^{Q( ˜}^X)

(50)

Solutions

Z X

dX U˜ ( ˜X) w(X) :=w₀−2

Z X

dX V˜ ( ˜X)e^{Q( ˜}^X)

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Black Holes

Horizons

Solutions with M ≥0exhibit (Killing) horizons for each solution of w(X_h) = 2M

Killing norm² ξ(X) =e^Q(X) w(X)−2M

≥0onX_h ≤X <∞ Assumption 1

If there are multiple horizons we take the outermost one Asymptotics

X → ∞: asymptotic region of spacetime; most models: w(X)→ ∞

Consider only models where w(X)→ ∞ as X→ ∞ Assumption 2

Consequence: ξ(X)∼e^Qw asX → ∞, i.e.,ξ asymptotes to ground state

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Black Holes

Horizons

Killing norm² ξ(X) =e^Q(X⁾ w(X)−2M

If there are multiple horizons we take the outermost one

Asymptotics

Consequence: ξ(X)∼e^Qw asX → ∞, i.e.,ξ asymptotes to ground state

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Black Holes

Horizons

Consequence: ξ(X)∼e^Qw asX→ ∞, i.e.,ξ asymptotes to ground state

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Black Holes

Horizons

Consequence: ξ(X)∼e^Qw asX→ ∞, i.e.,ξ asymptotes to ground state

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Black Hole Temperature

Standard argument: absence of conical singularity requires periodicity in Euclidean time

The gτ τ component of the metric vanishes at the horizon Xh

Regularity of the metric requires τ ∼τ +β with periodicity β = 4π

∂_rξ rh

= 4π w⁰(X)

Xh

I Ifξ →1 atX → ∞: β⁻¹ is temperature measured ’at infinity’

I Denote inverse periodicity by T :=β⁻¹= ^w⁰_4π^(X) Xh

I Proper local temperature related toβ⁻¹ by Tolman factor T(X) = 1

pξ(X)β⁻¹ So far no action required but only a line-element

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Outline

Introduction

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Free Energy 17/32

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Free Energy?

Not yet!

Given the black hole solution, can we calculate the free energy?

Z ∼ exp

−1

~I_E[g_cl, X_cl]

∼e^{−β F}

Need a limiting procedure to calculate the action. Implement this in a coordinate-independent way by putting a regulator on the dilaton.

X ≤X_reg

Evaluating the on-shell action leads to three problems

1. On-shell actionunbounded from below (cf. second assumption) I_E^reg =β 2M−w(X_reg)−2π X_hT

→ −∞ 2. First variation of actionnot zero for all field configurations

contributing to path integral due to boundary terms

3. Second variation of actionmay lead to divergent Gaussian integral

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Free Energy? Not yet!

Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞ 2. First variation of actionnot zero for all field configurations

contributing to path integral due to boundary terms

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Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞

2. First variation of actionnot zero for all field configurations contributing to path integral due to boundary terms

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Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞

2. First variation of actionnot zero for all field configurations contributing to path integral due to boundary terms