The point of departure is the classical
Hamiltonian formulation for space-times with a type I WIH
as an internal boundary, with fixed area
and charges
, where
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
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 [32
].
As in Figure 6 let us begin with a
partial Cauchy surface
whose internal boundary in
is a 2-sphere cross-section
of
and whose asymptotic boundary is a
2-sphere
at spatial infinity. The configuration
variable is an
connection
on
, where
takes values in the 3-dimensional
Lie-algebra
of
. Just as the
standard derivative operator acts on tensor fields and enables one
to parallel transport vectors, the derivative operator constructed
from
acts on fields with internal indices
and enables one to parallel transport spinors. The conjugate
momentum is represented by a vector field
with density weight 1 which also takes values in
; it is the analog of the Yang-Mills electric field.
(In absence of a background metric, momenta always carry a density
weight 1.)
can be regarded as a (density weighted)
triad or a ‘square-root’ of the intrinsic metric
on
:
, where
is the Cartan Killing metric
on
,
is the determinant of
and
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
. The presence of
represents an ambiguity in
quantization of geometry, analogous to the
-ambiguity in QCD. Just as the classical Yang-Mills
theory is insensitive to the value of
but the quantum
Yang-Mills theory has inequivalent
-sectors, classical
relativity is insensitive to the value of
but the quantum geometries based on different values
of
are (unitarily) inequivalent [94] (for details, see, e.g., [32
]).
Thus, the gravitational part of the phase space
consists of pairs
of fields
on
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:
In absence of internal boundaries, the quantum
theory has been well-understood since the mid-nineties (for recent
reviews, see, [162, 176, 32]). 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 on
, there is a self-adjoint operator
all of whose
eigenvalues are known to be discrete. The simplest eigenvectors are
represented by a single flux line, carrying a half-integer
as a label, which intersects the surface
exactly once, and the corresponding eigenvalue
of
is given by
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 where
is the well-understood bulk or volume Hilbert space
with ‘polymer-like excitations’, and
is the surface
Hilbert space of the
-Chern-Simons theory. As
depicted in Figure 12
, the polymer
excitations puncture the horizon. An excitation carrying a quantum
number
‘deposits’ on
an area equal to
. These contributions add up
to endow
a total area
. The surface Chern-Simons theory is therefore
defined on the punctured 2-sphere
. To incorporate the
fact that the internal boundary
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
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.
We will conclude by summarizing the nature of
geometry of the quantum horizon that results. Given any state
satisfying Equation (93), the curvature
of
vanishes everywhere except at the points at which the polymer
excitations in the bulk puncture
. 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
around that puncture. Each deficit angle is quantized and these angles add up to
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.