Entropy and area.

The ground state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed. ∗ This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098, and in part by the National Science Foundation under Grant Nos. AST91-20005 and PHY91-16964. † E-mail: [email protected]. On leave from Department of Physics, University of California, Santa Barbara, CA 93106. This document was prepared as an account of work sponsored by the United States Government. 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Lawrence Berkeley Laboratory is an equal opportunity employer. A free, massless, scalar, quantum field (which could just as well represent, say, the acoustic modes of a crystal, or any other three-dimensional system with dispersion relation ω = c|~k| ) is in its nondegenerate ground state, |0〉. We form the ground state density matrix, ρ0 = |0〉〈0|, and trace over the field degrees of freedom located inside an imaginary sphere of radius R. The resulting density matrix, ρout, depends only on the degrees of freedom outside the sphere. We now compute the associated entropy, S = −Tr ρout log ρout. How does S depend on R ? Entropy is usually an extensive quantity, so we might expect that S ∼ R3. However, this is not likely to be correct, as can be seen from the following argument. Consider tracing over the outside degrees of freedom instead, to produce a density matrix ρin which depends only on the inside degrees of freedom. If we now compute S′ = −Tr ρin log ρin, we would expect that S′ scales like the volume outside the sphere. However, it is straightforward to show that ρin and ρout have the same eigenvalues (with extra zeroes for the larger, if they have different rank), so that in fact S′ = S [1]. This indicates that S should depend only on properties which are shared by the two regions (inside and outside the sphere). The one feature they have in common is their shared boundary, so it is reasonable to expect that S depends only on the area of this boundary, A = 4πR2. S is dimensionless, so to get a nontrivial dependence of S on A requires another dimensionful parameter. We have two at hand: the ultraviolet cutoff M and the infrared cutoff μ, both of which are necessary to give a precise definition of the theory. (For a crystal, M would be the inverse atomic spacing, and μ the inverse linear size, in units with h̄ = c = 1.) Physics in the interior region should be independent of μ, which indicates that perhaps S will be as well. We therefore expect that S is some function of M2A. In fact, as will be shown below, S = κM2A, where κ is a numerical constant which depends only on the precise definition of M that we adopt. This result bears a striking similarity to the formula for the intrinsic entropy of a black hole, SBH = 1 4M 2 Pl A, where MPl is the Planck mass and A is the surface area of the horizon of the black hole [2]. The links in the chain of reasoning establishing this formula are remarkably diverse, involving, in turn, classical geometry, thermodynamic analogies, and quantum field theory in curved space. The result is thus rather mysterious. In particular, we would like to know whether or not SBH has anything to do with the number of quantum states accessible to the black hole. As a black hole evaporates and shrinks, it produces Hawking radiation whose entropy, SHR, can be computed by standard methods of statistical mechanics. One finds, after the 1 black hole has shrunk to negligible size, that SHR is a number of order one (depending on the masses and spins of the elementary particles) times the original black hole entropy [3]. This calculation of SHR is done by counting quantum states, and the fact that SBH ≃ SHR lends support to the idea that SBH should also be related to a counting of quantum states. It is then tempting think of the horizon as a kind of membrane [4], with approximately one degree of freedom per Planck area. However, in classical general relativity, the horizon does not appear to be a special place to a nearby observer, so it is hard to see why it should behave as an object with local dynamics. The new result quoted above indicates that S ∼ A is a much more general formula than has heretofore been realized. It shows that the amount of missing information represented by SBH is about right, in the sense that we would get the same answer in the vacuum of flat space if we did not permit ourselves access to the interior of a sphere with surface area A, and set the ultraviolet cutoff to be of order MPl (perfectly reasonable for comparison with a quantum theory that includes gravity). Furthermore, getting S ∼ A clearly does not require the boundary of the inaccessible region to be dynamical, since in our case it is entirely imaginary. To establish that S = κM2A for the problem at hand, let us begin with the simplest possible version of it: two coupled harmonic oscillators, with hamiltonian H = 1 2 [ p1 + p 2 2 + k0(x 2 1 + x 2 2) + k1(x1 − x2) ] . (1) The normalized ground state wave function is ψ0(x1, x2) = π −1/2(ω+ω−)1/4 exp [ −(ω+x+ + ω−x−)/2 ] , (2) where x± = (x1 ± x2)/ √ 2, ω+ = k 1/2 0 , and ω− = (k0 + 2k1) 1/2. We now form the ground state density matrix, and trace over the first (“inside”) oscillator, resulting in a density matrix for the second (“outside”) oscillator alone: ρout(x2, x ′ 2) = ∫ +∞ −∞ dx1 ψ0(x1, x2)ψ ∗ 0(x1, x ′ 2) = π−1/2(γ − β)1/2 exp [ −γ(x2 + x′2 2 )/2 + βx2x2 ] , (3) where β = 4(ω+ − ω−)/(ω+ + ω−) and γ − β = 2ω+ω−/(ω+ + ω−). We would like to find the eigenvalues pn of ρout(x, x ′): ∫ +∞ −∞ dx′ ρout(x, x′)fn(x′) = pnfn(x) , (4)