Holographic Entropy Bound and Local Quantum Field Theory

Abstract

The maximum entropy that can be stored in a bounded region of space is in dispute: it goes as volume, implies (nongravitational) microphysics; it goes as the surface area, asserts the "holographic principle." Here I show how the holographic bound can be derived from elementary flat-spacetime quantum field theory when the total energy of Fock states is constrained gravitationally. This energy constraint makes the Fock space dimension (whose logarithm is the maximum entropy) finite for both Bosons and Fermions. Despite the elementary nature of my analysis, it results in an upper limit on entropy in remarkable agreement with the holographic bound.PACS numbers: 04.70.-s, 04.62.+v, 04.60.-m, 03.67.-a An outstanding recent puzzle in gravitational physics is to find a local, microscopic explanation for the "holographic principle" [1], which asserts that the maximum entropy that can be stored inside a bounded region R in 3-space must be proportional to the surface area A(R) [as opposed to the volume V (R)] of the region:where k B is Boltzmann's constant, and l p = G/c 3 is the Planck length. The most compelling conceptual evidence for the holographic bound comes from black-hole physics and thermodynamics. If there were a physical system enclosed in R whose entropy exceeded S max , it would be possible to violate the second law in the following way: First, one could dump as much energy into R as necessary to bring it to the threshold of gravitational collapse. This process can only increase the entropy contained in R, making it exceed S max even further. One could then tip the system into full gravitational collapse, leaving nothing but a black hole inside R. The resulting event horizon, being contained in R, necessarily has surface area no larger than A(R). But according to the Bekenstein formula [2], the entropy of this black hole, given by the right hand side of Eq. (1) with A(R) replaced by the horizon area, cannot exceed S max . Thus gravitational collapse would appear to cause a sudden decrease in entropy, violating the second law of thermodynamics.The holographic principle presents a puzzle since derivations based on standard (non-gravitational) microphysics yield an entropy bound proportional to the volume V (R) instead of the surface area. To discuss this in the simplest microscopic model, let me choose R to be a standard three-dimensional spacelike cube of size L in Minkowski space, and consider a real, massless (linear) scalar field φ confined in R. The Fock space is built out of the modes of the field φ, which are the positive frequency solutions of the scalar wave equation φ = 0 that vanish on ∂R. These modes are given (up to normalization) by the solutions sin( k · x − ω k t), where ω k = c| k|, and the admissible wave vectors k are labelled by non-negative integers m x , m y , m z : (k x , k y , k z ) = (π/L)(m x , m y , m z ). I will often use single-letter labels i, j etc. to denote a composite multi-index like (m x , m y , m z ). Mode counting and summing various quantities over the modes (and all my...

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