Mathematical Physics and Quantum Field Theory, Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 159–164 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

The Bisognano-Wichmann theorem for charged states and the conformal boundary

of a black hole ^{∗}

Roberto Longo

Dedicated to Eyvind Wichmann on his 70th birthday

Abstract This note concerns the study of the incremental entropy of a quantum black hole, based on Operator Algebra methods. Our results are based on the results presented in the references [6, 11, 12, 13].

1 Case of Rindler spacetime

Let M = ℝ^{4} be the Minkowski spacetime and O ⊂ M → A(O) be a net of von Neumann algebras generated by a local Wightman field on M. Let W ⊂ M be a wedge region; I may assume W = {x : x_{1} > x_{0}} as any other wedge is a Poincaré translate of this special one. The Bisognano-Wichmann theorem [2] describes the modular structure associated with (A(W), Ω), with Ω the vacuum vector (which is cyclic and separating for A(W) by the Reeh-Schlieder theorem):

Δ^{it} = U(Λ_{W} (−2πt)),  (1)
J = U(R)Θ.  (2)

Here Δ and J are the Tomita-Takesaki modular operator and conjugation associated with (A(W), Ω), Λ_{W} is the one-parameter group of pure Lorentz transformations in the x_{1}-direction, U is the unitary representation of the Poincaré group, R is the rotation of π around the x_{1}-axis, and Θ is the anti-unitary PCT transformation.

In particular, for any fixed a > 0, the rescaled pure Lorentz transformations Λ_{W} give rise to a one-parameter automorphism group α_{t} = Ad U (Λ_{W} (at)) of A(W) satisfying the KMS thermal equilibrium condition with respect to ω|_{A(W)} at inverse temperature β = \frac{2π}{a}, with ω = (Ω, ·Ω).

As is known, the Rindler spacetime may be identified with W. The pure vacuum state ω is thus a (mixed) thermal state when restricted to the algebra of W. Now, if K = −i\frac{d}{dt}U(Λ_{W} (t))|_{t=0}, then H = aK is the proper Hamiltonian for a uniformly accelerated observer with acceleration a in the x_{1}-direction. As noticed by Sewell [16], the Bisognano-Wichmann theorem is thus essentially equivalent to the Hawking-Unruh effect in the case of a Rindler black hole.

∗Mathematics Subject Classifications: 81T05, 46L60, 47L90, 83C57.

Key words: Bisognano-Wichmann Theorem, black hole, modular theory.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University and University of North Texas.

Published July 12, 2000.

Suppose now one adds a short range charge (superselection sector) localized in the exterior of the black hole: according to DHR [3], this amounts to considering the net A in a different irreducible localized representation. I may assume that Haag duality holds and that this representation is realized by a localized endomorphism ρ of the C^{∗}-algebra 𝔄(M), where, for an unbounded region R, 𝔄(R) denotes the C^{∗}-algebra generated by the A(O)’s with O ⊂ R bounded. I may further choose the charge ρ to be localized in W and to have finite statistics.

Now, as ρ is transportable, it turns out that ρ|_{𝔄(W)} is normal, and I denote by ρ|_{A(W)} its weakly continuous extension to A(W). With Φ_{ρ} the left inverse of ρ (the unique completely positive map Φ_{ρ} : 𝔄(M) → 𝔄(M) with Φ_{ρ} · ρ = id), the state φ_{ρ} = ω · Φ_{ρ}|_{A(W)} is clearly invariant under the automorphism group α_{t}^{ρ} = Ad U_{ρ} (Λ_{W} (at)) of A(W), where U_{ρ} is the unitary representation of P_{+}^{↑} in the representation ρ.

The modular structure associated with φ_{ρ} is given by the following theorem [11].

Theorem 1.1

(i) α^{ρ} satisfies the KMS condition with respect to φ_{ρ} at the same inverse temperature β = \frac{2π}{a};

(ii) log Δ_{Ω,ξ_{ρ}} = −2πK_{ρ} + log d(ρ) = −βH_{ρ} + log d(ρ) .

Here K_{ρ} is the infinitesimal generator of U_{ρ}(Λ_{W} (·)), ξ_{ρ} is the vector representative of φ_{ρ} in the natural cone of (A(W), Ω), d(ρ) is the DHR statistical dimension of ρ [3], and Δ_{Ω,ξ_{ρ}} is the relative modular operator.

Now H_{ρ} = aK_{ρ} is the proper Hamiltonian, in the representation ρ, for the uniformly accelerated observer, so in particular one sees by (i) that the Bisognano-Wichmann theorem extends with its Hawking-Unruh effect interpretation in the charged representation ρ. This is to be expected, inasmuch as the addition of a single charge should not alter the temperature of a thermodynamical system.

Point (ii) shows, however, a new feature: log(e^{−βH_{ρ}}Ω, Ω) may be interpreted as the incremental partition function between the states ω and φ_{ρ} on A(W). The incremental free energy is then given by

F (ω|φ_{ρ}) = −β^{−1} log(e^{−βH_{ρ}}Ω, Ω) ,

and indeed it turns out that the thermodynamical relation dF = dE − T dS holds, namely

F (ω|φ_{ρ}) = φ_{ρ}(H_{ρ}) − β^{−1}S(ω|φ_{ρ}) ,

where S is Araki’s relative entropy.

As the statistical dimension takes only integer values by the DHR theorem [3], one has as a corollary that the values of the incremental free energy are quantized:

Corollary 1.2 The possible values of the incremental free energy are

F (ω|φ_{ρ}) = −β^{−1} log n, n = 1, 2, 3, …

The index-statistics theorem [10] relates d(ρ) to the square root of a Jones index [9]; this and other, see [11], leads to the formula

F (ω|φ_{ρ}) = −\frac{1}{2}β^{−1}S_{c}(ρ),

where S_{c}(ρ) = d(ρ)^{2} is the conditional entropy of ρ.

Notice now that the above discussion is subject to several restrictions. I have focused on a particular spacetime, the Rindler wedge W, and considered endomorphisms of 𝔄(W) that are restrictions of endomorphisms of 𝔄(M). Moreover, I have analyzed thermodynamical variables associated with the exterior of the black hole, without a direct relation to, say, the entropy of the black hole.

In the following I shall generalize this discussion, leading to a certain clarification of the above aspects.

2 Case of a globally hyperbolic spacetime

I shall now consider more realistic black holes spacetimes, namely globally hyperbolic spacetimes a with bifurcate Killing horizon. By considering charges localizable on the horizon, we shall get quantum numbers for the increment of the entropy of the black hole itself, rather than of the outside region.

A main point here is that the restriction of the net A of local observable von Neumann algebras to each of the two horizon components 𝔥_{+} and 𝔥_{−} gives rise a conformal net on S^{1}, a general fact that is obtained by applying Wiesbrock’s characterization of conformal nets [18], as discussed in [12, 6]. This is analogous to the holography on the anti-de Sitter spacetime that independently appeared in the Maldacena-Witten conjecture [14, 19], proved by Rehren [15]. Yet their context differs inasmuch as the anti-de Sitter spacetime is not globally hyperbolic, and the holography there is a peculiarity of that spacetime, rather than a general phenomenon.

Finally, I shall deal with general KMS states, besides the Hartle-Hawking temperature state. In this context, a non-zero chemical potential [1] can appear (see [11]).

The extension of the DHR analysis of superselection sectors [3] to a quantum field theory on a curved globally hyperbolic spacetime has been pursued in [6], and I refer to this paper for the necessary background material. I recall here, however, the construction of conformal symmetries for the observable algebras on the horizon of the black hole.

Let V be a (d+1)-dimensional globally hyperbolic spacetime with a bifurcate Killing horizon. A typical example is given by the Schwarzschild-Kruskal manifold that, by the Birkoff theorem, is the only spherically symmetric solution of the Einstein-Hilbert equation; one might first focus on this specific example, as the more general case is treated similarly. I denote by 𝔥_{+} and 𝔥_{−} the two codimension-1 submanifolds that constitute the horizon 𝔥 = 𝔥_{+} ∪ 𝔥_{−}. I assume that the horizon splits V into four connected components, the future, the past, and the “left and right wedges” that I denote by R and L (in the Minkowski spacetime R = W and L = W′).

Let κ = κ(V) be the surface gravity, namely, denoting by χ the Killing vector field, the equation on 𝔥

∇g(χ, χ) = −2κχ ,  (3)

with g the metric tensor, defines a function κ on 𝔥, that is actually constant on 𝔥 [17]. If V is the Schwarzschild-Kruskal manifold, then

κ(V) = \frac{1}{4M} ,

where M is the mass of the black hole. In this case R is the exterior of the Schwarzschild black hole.

Our spacetime is R, and V is to be regarded as a completion of R.

Let A(O) be the von Neumann algebra on a Hilbert space H of the observables localized in the bounded diamond O ⊂ R. I assume Haag duality, properly infinite A(O), and the Borchers property B (see [7]). The Killing flow Λ_{t} of V gives rise to a one parameter group of automorphisms α of the quasi-local C^{∗}-algebra 𝔄(R), since R ⊂ V is a Λ-invariant region.

I now consider a locally normal α-invariant state φ on 𝔄(R), that restricts to a KMS state at inverse temperature β on the horizon algebra, as I will explain. It is clearly the case that φ is KMS on all 𝔄(R).

For convenience, I shall assume that the net 𝔄 is already in the GNS representation of φ, hence φ is represented by a cyclic vector ξ. Let’s denote by R_{a} the wedge R “shifted by” a ∈ ℝ along, say, 𝔥_{+} (see [6]). If I = (a, b) is a bounded interval of ℝ_{+}, I set

C(I) = 𝔄(R_{a})′′ ∩ 𝔄(R_{b})′, 0 < a < b .

One obtains in this way a net of von Neumann algebras on the intervals of (0, ∞), where the Killing automorphism group α acts covariantly by rescaled dilations.

I can now state my assumption: φ|_{ℭ(0,∞)} is a KMS state with respect to α at inverse temperature β > 0. Here ℭ(0, ∞) is the C^{∗}-algebra generated by all C(a, b), b > a > 0 (for the Schwarzschild spacetime, cf. [16]).

It follows that the restriction of the net to the black hole horizon 𝔥_{+} has many more symmetries than on global spacetime.

Theorem 2.1 ([6]). The Hilbert space H_{0} = \overline{C(I)ξ} is independent of the bounded open interval I.

The net C extends to a conformal net ℝ of von Neumann algebras acting on H_{0}, where the Killing flow corresponds to the rescaled dilations.

This theorem says in particular that one may compactify ℝ to the circle S^{1}, extend the definition of C(I) for all proper intervals I ⊂ S^{1}, find a unitary positive energy representation of the Möbius group PSL(2, ℝ) acting covariantly on C, so that the rescaled dilation subgroup is the Killing automorphism group.

If ξ is cyclic for ℭ on H, namely H_{0} = H (as is true for the free field in Rindler case, see [6]), then C automatically satisfies Haag duality on ℝ. Otherwise one would pass to the dual net C^{d} of C, which is is automatically conformal and strongly additive [5]. In the following I assume that C is strongly additive.

2.1 Charges localizable on the horizon

I now consider an irreducible endomorphism ρ of 𝔄(R) with finite dimension that is localizable in an interval (a, b), b > a > 0, of 𝔥_{+}, namely ρ acts trivially on 𝔄(R_{b}) and on C(I) if Ī ⊂ (0, a), therefore ρ restricts to a localized endomorphism of ℭ(0, ∞).

This last requirement is necessary to extend ρ to a normal endomorphism of π_{φ}(ℭ(0, ∞))′′, a result obtained by conformal methods [13].

Theorem 2.2 With the above assumptions, if ρ is localized in an interval of 𝔥_{+}, then d(ρ|_{ℭ(0,∞)}) has a normal extension to the weak closure ℭ(0, ∞)′′ with dimension d(ρ).

Let σ be another irreducible endomorphism of 𝔄 localized in (a, b) ⊂ 𝔥_{+} and denote by φ_{ρ} and φ_{σ} the thermal states for the Killing automorphism group in the representation ρ. As shown in [11], φ_{ρ} = φ · Φ_{ρ}, where Φ_{ρ} is the left inverse of ρ, and similarly for σ.

The increment of the free energy between the thermal equilibrium states φ_{ρ} and φ_{σ} is expressed as in [11] by

F (φ_{ρ}|φ_{σ}) = φ_{ρ}(H_{ρσ̄}) − β^{−1}S(φ_{ρ}|φ_{σ}).

Here S is the Araki relative entropy, and H_{ρσ̄} is the Hamiltonian on H_{0} corresponding to the charge ρ and the charge conjugate to σ̄ localized in (−∞, 0) as in [11] (by [4], a transportable localized endomorphism ρ of C with finite dimension is Möbius covariant). In particular, if σ is the identity representation, then H_{ρσ̄} = H_{ρ} is the Killing Hamiltonian in the representation ρ.

By the analysis in [13], one can write formulas in the case of two different KMS states φ_{ρ} and φ_{σ}. In particular

F (φ_{σ}|φ_{ρ}) = \frac{1}{2}β^{−1}(S_{c}(σ) − S_{c}(ρ)) + μ(φ_{σ}|φ_{ρ}),  (4)

where μ(φ_{σ}|φ_{ρ}) is the chemical potential, cf. [1].

The above formula gives a canonical splitting for the incremental free energy. The first term is symmetric under charge conjugation, the second term is antisymmetric. The quantization of the possible values of the symmetric part follows as in Corollary 1.2.

164 The Bisognano-Wichmann theorem for charged states

3 Quantum index theorem

The above discussion is part of a general aim concerning a quantum index theorem. This point of view is discussed in [13].

References

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[2] J. Bisognano, E. Wichmann: On the duality condition for a Hermitean scalar field, J. Math. Phys. 16, 985 (1975).

[3] S. Doplicher, R. Haag, J.E. Roberts: Local observables and particle statistics I, Commun. Math. Phys. 23, 199-230 (1971), II Commun. Math. Phys. 35, 49-85 (1974).

[4] D. Guido, R. Longo: Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148, 521 (1992).

[5] D. Guido, R. Longo, H.-W. Wiesbrock: Extension of conformal nets and superselection structures, Commun. Math. Phys. 192, 217-244 (1998).

[6] D. Guido, R. Longo, J.E. Roberts, R. Verch: Charged sectors, spin and statistics in quantum field theory on curved spacetimes, Rev. Math. Phys. (to appear).

[7] R. Haag “Local Quantum Physics”, Springer-Verlag (1996).

[8] P.D. Hislop, R. Longo: Modular structure of the local algebras associated with the free massless scalar field theory, Commun. Math. Phys. 84, 71 (1982).

[9] V. F. R. Jones: Index for subfactors, Invent. Math. 72 (1983) 1-25.

[10] R. Longo: Index of subfactors and statistics of quantum fields. I & II, Commun. Math. Phys. 126, 217–247 (1989) & Commun. Math. Phys. 130, 285-309 (1990).

[11] R. Longo: An analogue of the Kac-Wakimoto formula and black hole conditional entropy, Commun. Math. Phys. 186 (1997), 451-479.

[12] R. Longo: Abstracts for the talks at the Oberwolfach meetings on “C^{∗}-algebras”, February 1998, and “Noncommutative geometry”, August 1998.

[13] R. Longo: Notes for a quantum index theorem, Preprint, March 2000.

[14] J. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252.

[15] H.K. Rehren: Algebraic holography, Preprint, May 1999.

[16] G.L. Sewell: Quantum fields on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. 141 (1982), 201.

[17] R.M. Wald, “Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics”, University of Chicago Press, 1994.

[18] H.-W. Wiesbrock: Conformal quantum field theory and half-sided modular inclusions of von Neumann algebras, Commun. Math. Phys. 158 (1993), 537.

[19] E. Witten: Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998), 253-291.

Roberto Longo Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, I–00133 Roma, Italy. e-mail: longo@mat.uniroma2.it