Quantum horizon geometry

7.2 Quantum horizon geometry

For simplicity of presentation, in this section we will restrict ourselves to type I WIHs, i.e., the ones for which the only non-zero multipole moment is the mass monopole. The extension to include type II horizons with rotations and distortion will be briefly summarized in the next Section 7.3. Details can be found in [9

, 10 , 19

, 8

, 23

The point of departure is the classical Hamiltonian formulation for space-times $ℳ$ with a type I WIH $Δ$ as an internal boundary, with fixed area $a 0$ and charges $Q α 0$ , where $α$ runs over the number of distinct charges (Maxwell, Yang–Mills, dilaton, …) allowed in the theory. As we noted in Section 4.1, the phase space $Γ$ can be constructed in a number of ways, which lead to equivalent Hamiltonian frameworks and first laws. However, so far, the only known way to carry out a background independent, non-perturbative quantization is through connection variables [31 ].

As in Figure 6 let us begin with a partial Cauchy surface $M$ whose internal boundary in $ℳ$ is a 2-sphere cross-section $S$ of $Δ$ and whose asymptotic boundary is a 2-sphere $S ∞$ at spatial infinity. The configuration variable is an $SU (2)$ connection $Aia$ on $M$ , where $i$ takes values in the 3-dimensional Lie-algebra $su(2)$ of $SU (2)$ . Just as the standard derivative operator acts on tensor fields and enables one to parallel transport vectors, the derivative operator constructed from $i A a$ acts on fields with internal indices and enables one to parallel transport spinors. The conjugate momentum is represented by a vector field $Pia$ with density weight 1 which also takes values in $su(2)$ ; it is the analog of the Yang–Mills electric field. (In absence of a background metric, momenta always carry a density weight 1.) $a P i$ can be regarded as a (density weighted) triad or a ‘square-root’ of the intrinsic metric $&tidle;qab$ on $S$ : $2 a b ab (8πG γ) Pi P jδij = &tidle;q &tidle;q$ , where $δij$ is the Cartan Killing metric on $su(2)$ , $&tidle;q$ is the determinant of $q&tidle;ab$ and $γ$ is a positive real number, called the Barbero–Immirzi parameter. This parameter arises because there is a freedom in adding to Palatini action a multiple of the term which is ‘dual’ to the standard one, which does not affect the equations of motion but changes the definition of momenta. This multiple is $γ$ . The presence of $γ$ represents an ambiguity in quantization of geometry, analogous to the $θ$ -ambiguity in QCD. Just as the classical Yang–Mills theory is insensitive to the value of $θ$ but the quantum Yang–Mills theory has inequivalent $θ$ -sectors, classical relativity is insensitive to the value of $γ$ but the quantum geometries based on different values of $γ$ are (unitarily) inequivalent [93 ] (for details, see, e.g., [31 ]).

Thus, the gravitational part of the phase space $Γ$ consists of pairs $(Aia,Pai )$ of fields on $M$ satisfying the boundary conditions discussed above. Had there been no internal boundary, the gravitational part of the symplectic structure would have had just the expected volume term:

where $Σabi := ηabcPci$ is the 2-form dual to the momentum $Pic$ and $δ1$ , and $δ2$ denote any two tangent vectors to the phase space. However, the presence of the internal boundary changes the phase space-structure. The type I WIH conditions imply that the non-trivial information in the pull-back $i A←a$ of $i A a$ to $S$ is contained in a $U (1 )$ connection $i Wa := ←A ari$ on $S$ , where $i r$ is the unit, internal, radial vector field on $S$ . Its curvature 2-form $Fab$ is related to the pull-back $←Σiab$ of the 2-form $Σiab$ to $S$ via

The restriction is called the horizon boundary condition. In the Schwarzschild space-time, all the 2-spheres on which it is satisfied lie on the horizon. Finally, the presence of the internal boundary modifies the symplectic structure: It now acquires an additional boundary term

with

The new surface term is precisely the symplectic structure of a well-known topological field theory – the $U (1)$ -Chern–Simons theory. The symplectic structures of the Maxwell, Yang–Mills, scalar, and dilatonic fields do not acquire surface terms. Conceptually, this is an important point: This, in essence, is the reason why (for minimally coupled matter) the black hole entropy depends just on the area and not, in addition, on the matter charges.

In absence of internal boundaries, the quantum theory has been well-understood since the mid-nineties (for recent reviews, see, [164 , 177 , 31 ]). The fundamental quantum excitations are represented by Wilson lines (i.e., holonomies) defined by the connection and are thus 1-dimensional, whence the resulting quantum geometry is polymer-like. These excitations can be regarded as flux lines of area for the following reason. Given any 2-surface $𝕊$ on $M$ , there is a self-adjoint operator $ˆA𝕊$ all of whose eigenvalues are known to be discrete. The simplest eigenvectors are represented by a single flux line, carrying a half-integer $j$ as a label, which intersects the surface $𝕊$ exactly once, and the corresponding eigenvalue $a𝕊$ of $ˆ A 𝕊$ is given by

Thus, while the general form of the eigenvalues is the same in all $γ$ -sectors of the quantum theory, their numerical values do depend on $γ$ . Since the eigenvalues are distinct in different $γ$ -sectors, it immediately follows that these sectors provide unitarily inequivalent representations of the algebra of geometric operators; there is ‘super-selection’. Put differently, there is a quantization ambiguity, and which $γ$ -sector is actually realized in Nature is an experimental question. One appropriate experiment, for example a measurement of the smallest non-zero area eigenvalue, would fix the value of $γ$ and hence the quantum theory. Every further experiment – e.g., the measurement of higher eigenvalues or eigenvalues of other operators such as those corresponding to the volume of a region – would provide tests of the theory. While such direct measurements are not feasible today we will see that, somewhat surprisingly, the Hawking–Bekenstein formula (87

) for the entropy of large black holes provides a thought experiment to fix the value of $γ$ .

Recall next that, because of the horizon internal boundary, the symplectic structure now has an additional surface term. In the classical theory, since all fields are smooth, values of fields on the horizon are completely determined by their values in the bulk. However, a key point about field theories is that their quantum states depend on fields which are arbitrarily discontinuous. Therefore, in quantum theory, a decoupling occurs between fields in the surface and those in the bulk, and independent surface degrees of freedom emerge. These describe the geometry of the quantum horizon and are responsible for entropy.

In quantum theory, then, it is natural to begin with a total Hilbert space $ℋ = ℋV ⊗ ℋS$ where $ℋV$ is the well-understood bulk or volume Hilbert space with ‘polymer-like excitations’, and $ℋ S$ is the surface Hilbert space of the $U (1)$ -Chern–Simons theory. As depicted in Figure 12 , the polymer excitations puncture the horizon. An excitation carrying a quantum number $j$ ‘deposits’ on $S$ an area equal to $∘ -------- 8π ℓ2Pl j(j + 1)$ . These contributions add up to endow $S$ a total area $a0$ . The surface Chern–Simons theory is therefore defined on the punctured 2-sphere $S$ . To incorporate the fact that the internal boundary $S$ is not arbitrary but comes from a WIH, we still need to incorporate the residual boundary condition (89 ). This key condition is taken over as an operator equation. Thus, in the quantum theory, neither the triad nor the curvature of $W$ are frozen at the horizon; neither is a classical field. Each is allowed to undergo quantum fluctuations, but the quantum horizon boundary condition requires that they have to fluctuate in tandem.

Figure 12:

Quantum horizon. Polymer excitations in the bulk puncture the horizon, endowing it with quantized area. Intrinsically, the horizon is flat except at punctures where it acquires a quantized deficit angle. These angles add up to endow the horizon with a 2-sphere topology.

An important subtlety arises because the operators corresponding to the two sides of Equation (89 ) act on different Hilbert spaces: While $Fˆ$ is defined on $ℋS$ , $ˆ←Σ ⋅ r$ is defined on $ℋV$ . Therefore, the quantum horizon boundary condition introduces a precise intertwining between the bulk and the surface states: Only those states $Ψ = ∑ Ψi ⊗ Ψi i V S$ in $ℋ$ which satisfy

can describe quantum geometries with WIH as inner boundaries. This is a stringent restriction: Since the operator on the left side acts only on surface states and the one on the right side acts only on bulk states, the equation can have solutions only if the two operators have the same eigenvalues (in which case we can take $Ψ$ to be the tensor product of the two eigenstates). Thus, for solutions to Equation (93

) to exist, there has to be a very delicate matching between eigenvalues of the triad operators $ˆ←Σ ⋅ r$ calculated from bulk quantum geometry, and eigenvalues of $Fˆ$ , calculated from Chern–Simons theory. The precise numerical coefficients in the surface calculation depend on the numerical factor in front of the surface term in the symplectic structure, which is itself determined by our type I WIH boundary conditions. Thus, the existence of a coherent quantum theory of WIHs requires that the three cornerstones – classical isolated horizon framework, quantum mechanics of bulk geometry, and quantum Chern–Simons theory – be united harmoniously. Not only should the three frameworks be mutually compatible at a conceptual level, but certain numerical coefficients, calculated independently within each framework, have to match delicately. Remarkably, these delicate constraints are met, whence the quantum boundary conditions do admit a sufficient number of solutions.

We will conclude by summarizing the nature of geometry of the quantum horizon that results. Given any state satisfying Equation (93 ), the curvature $F$ of $W$ vanishes everywhere except at the points at which the polymer excitations in the bulk puncture $S$ . The holonomy around each puncture is non-trivial. Consequently, the intrinsic geometry of the quantum horizon is flat except at the punctures. At each puncture, there is a deficit angle, whose value is determined by the holonomy of $W$ around that puncture. Each deficit angle is quantized and these angles add up to $4π$ as in a discretized model of a 2-sphere geometry. Thus, the quantum geometry of a WIH is quite different from its smooth classical geometry.