Quantum General Relativity without Quantum Gravity -- a Field Law

Rock Brentwood

When coupling fermions with gravity, the first issues that stand out
are (1) there is no transformation behavior for fermions with respect
to coordinate transformations (i.e., they transform as scalars); (2)
instead, one has a transformation behavior under a localized version
of the Lorentz group; (3) the fermions see gravity as nothing more
than yet another gauge field -- one for the Lorentz group, and this is
how it is incorporated into the Dirac equation; and (4) a PREREQUISITE
for the definition of spinor field is that one must first know the
signature of the underlying spacetime , since the nature of the spinor
is critically dependent on it.

The last point, already, is strongly suggestive of the notion that the
metric g_{mu nu} or, equivalently, the frame h^a_{mu} (out of which
one defines g_{mu nu} = eta_{a b} h^a_{mu} h^b_{nu}, where eta_{ab} is
the Minkowski metric) have to already be in place before one can even
begin to construct the fermion field (quantized or classical).

This is a point of view that has been advanced by Sardanashvily,
Mangiarotti and others in the same general clique -- hence the notion
of the "fermion complex". A key argument here is that the
representation classes for Dirac matrices (gamma^mu) are actually tied
to INEQUIVALENT bundle structures, for inequivalent frame fields; that
cannot be brought together to form the basis of a single coherent
space in quantum theory. Hence, one sees the outline of a second major
point: the frame field is not so much a field, per se, as it is an
index of a family of coherent subspaces.

This ties in with an older idea (also indirectly advanced by Penrose
et. al.) that as a concomitant of the geometric formulation of the
Equivalence Principle (i.e. that the tangent space bundle TM and its
associated GL(4) frame bundle LM reduce to a SO(3,1) orthonormal frame
bundle FM) is that the frame is associated with a symmetry breaking
reducing GL(4) -> SO(3,1). The quotient space, in fact, GL(4)/
(SO(3,1)_{connected}) is topologically R^7 x S_3 -- the 10-dimensional
space where (frames modulo Lorentz invariance) reside. The frame is a
section over an (R^7 x S_3) bundle.

An intuitive argument for this was advanced by Penrose, in fact.
Pointing out that the very definition of "inertial" depends on
definition of the frame field, then one would expect that in two
states corresponding to different frame fields, the same
considerations that apply to mutually accelerating vacuua would apply
here as well: the respective state spaces are separated into distinct
coherent subspaces and cannot coherent superpose with one another.

Thus, in this point of view, h is no longer a field, per se, but the
index of the vacuum phase (or, equivalently, the coherent subspace).
Hence, in place of the single vacuum state |0> of the Wightman
formalism, one has a state |h>, associated with a frame h^a_mu.

If we take this idea seriously, then the natural formulation of
Einstein's equations in the quantum setting would just be
G^{mu nu}(h) = -k <h| T-hat^{mu nu} |h>
where G^{mu nu}(h) is constructed from the frame h, by (1) defining
the metric g from h, (2) defining the Levi-Civita connection Gamma
from g, (3) defining the Riemann tensor, Ricci scalar and Einstein
tensor from Gamma, g and g^{-1}.

On the right-hand side, the stress tensor is the (quantized) symmetric
stress tensor derived from the matter fields. In place of the "vacuum
expectation", one takes the h-expectation.

Had one been working solely in a Riemannian geometry, this would make
it essentially impossible to go the other way and write down equations
coupling gravity to matter (the "forward reaction equations"). But, in
a Riemannian-Cartan geometry, the connection is not just the Levi-
Civita connection. It splits into Gamma = Levi-Civita + K, where K is
the contorsion. In turn, the contorsion may be uniquely associated
with the torsion T. Each of these remains fully quantized. This is
despite the fact that the torsion T is the gauge-covariant derivative
of the frame h (for the covariant derivative here involves the
connection Gamma, which is half-quantized).

At this point, it's natural to ask where such an equation would come
from. The left hand side looks like a "defect" that emerges as a
result of quantizing the right-hand side (that is: with another round
of quantization *beyond* second quantization). The focus, thus, is
placed on the coefficient k. In a space-time of (n-1)+1 dimensions,
assuming the metric has units of [g_{mu nu}] = L^2/([x^{mu}]
[x^{nu}]), the coefficient has units [k] = L^{n-2}/[h-bar].

The first place we look is the Bekenstein Bound. This associates an
upper bound to the entropy of a spacetime region (more precisely, a 1-
parameter family of space-like (n-1) surfaces) with the area of its
boundary (the (n-2) surfaces bounding each). The upper bound is given
by a proportionality constant z with dimensions [z] = L^{n-2}/bit,
where "bit" = [Boltzmann's constant] is the unit of entropy.

The coefficient associated with the Bekenstein bound is related to k,
up to numeric factor, by k = z/h-bar. Moving it over to the left-hand
side, and switching sides, we have
<h| T^{mu nu} |h> = -h-bar/z G^{mu nu}(h).
In the limit k, z -> infinity, one has a 0 defect. Thus, whatever
appears on the (new) right-hand side must be arising from the finitude
of the Bekenstein bound, and may thus be regarded as a "defect"
associated with the finiteness of entropy associated with the (n-2)
boundaries. This is the above-mentioned "another round of
quantization".

The importance of indexing the states by the classical field h is that
(1) h, being classical, is in the background; (2) the very definition
of "area" requires a metric! (hence, h must already be in place).

How might this bound emerge? We need (n-2)-dimentional surfaces --
these play the role of horizons.

The usual formulation of the evolution law in the development of the
Noether Theorem and classical (and quantum) equations of motion
divides space-time up into a layering of global Cauchy surfaces;
reducing the manifold M topologically to M = S_{x,y,z} x [-infinity,
infinity]_t.

In fact, these formulations are not physically well-motivated, since
they make appeal to going off to infinity, which is not empirically
meaningful. Closely related to this are the technical conditions
required to actually make integrals consistent in the limit to
infinity.

Moreover, by taking the limit, one by-passes subtle nuances which are
present in the finite version of this picture that might, in fact, NOT
go away when one passes to the limit (i.e., non-trivial "horizon
effects" that linger in the limit to infinity).

So, in all cases, the requirement of physical realism points to a
construction whereby M is considered to be covered by COMPACT regions
M_0, each one layered (topologically) as M_0 = S_0 x [a,b], where S_0
is the compact spacelike region. We also assume that the covering of M
is refined enough that the S_0's are all simply connected.

Associated with this foliation is a 1-parameter family of 3-surfaces
(S_t: t = a to b), each topologically equivalent to S_0, and a 2-
parameter family of diffeomorphisms phi^t_u: S_u -> S_t, such that
phi^t_u phi^u_v = phi^t_v; (phi^t_u)^{-1} = phi^u_t; phi^t_t =
I_{S_t}. Fixing u = a, this leads to a time evolution S_t =
phi^t_a(S_a) as t ranges from a to b.

The boundaries of all the S_t are d(S_t) = H, tied to a single 2-
surface H, which plays the role of a "horizon". As one approaches the
horizon, the vector field d/dt -> 0.

The horizon actually comes into play in somewhat unexpected ways.
First, in writing down the Noether theorem, take a dynamic law given
by an action principle
integral L
where L is a Lagrangian 4-form. Write the variational of the
Lagrangian in terms of the field components (which we'll call q^a(x),
and field velocities v^a_{mu}(x) = dq^a/dx^mu) by
delta L = delta q^a F_a + delta v^a_mu P^{mu}_a.
Imposing the field kinematics (v^a_{mu} = dq^a/dx^{mu}) on the
variational, one gets
delta v^a_{mu} = d/dx^{mu} (delta q^a),
so that the variational of the Lagrangian reduces to (after
integrating by parts)
delta L = delta q^a (F^a - d/dx^{mu} P^{mu}_a) + d/dx^{mu} (delta
q^a P^{mu}_a).

The action principle asserts that for the actual field dynamics, the
variation of the Lagrangian over a region M_0 reduces to a variational
over its boundary. Thus, we may write
delta integral_{M_0} L = integral_{M_0} (delta q^a E_a)
+ integral_{dM_0} (delta q^a P^{mu}_a) (d^3x)_{mu}
where E_a is the "Euler-Lagrange" difference F_a - dP^{mu}_a/dx^{mu}
and
(d^3x)_0 = dx^1 ^ dx^2 ^ dx^3, (d^3x)_1 = -dx^0 ^ dx^2 ^ dx^3
(d^3x)_2 = dx^0 ^ dx^1 ^ dx^3, (d^3x)_3 = -dx^0 ^ dx^1 ^ dx^2.

This leads both to the field law
F_a = dP^{mu}_a/dx^{mu}
and the reduction of the variational to the boundary dM_0 = S_b - S_a
of the region M_0:
delta integral_{M_0} L = Q(b) - Q(a)
where
Q(t) = integral_{S_t} (delta q^a P^{mu}_a) (d^3x)_{mu}.

By generalizing the argument to the regions associated with all the
subintervals
M_{t0,t1} = S_0 x [t0, t1], where a < t0 < t1 < b
one finds
delta integral_{M_{t0,t1}} L = Q(t1) - Q(t0).

The backdrop to the Noether theorem is that if, after imposing the
field equations, the variational delta q^a is such that delta integral
L = 0, then one has the equality of all the Q's -- hence conservation.
The operator Q(t) is then the (time independent) "charge" associated
with the degree of freedom that led to the variational of q^a.

But, in this setting with the horizon present, the conservation law is
now LOCAL and tied directly to the horizon H, itself. This is seen as
follows.

In fact, this is a subtlety rarely seen in the literature -- (delta
q . P) is, itself, derivable from a "potential"!

Since the spatial regions have been assumed to be simply connected in
each region M_0, then from the conservation law
d(delta q^a P^{mu}_a)/dx^{mu} = 0,
one obtains a reduction to the divergence of a "potential"
delta q^a P^{mu}_a = dp^{mu nu}/dx^{nu}.
Thus, we may write the conserved "charge" operator Q as
integral_{S_t} = delta q^a P^{mu}_a) (d^3x)_{mu}
= integral_{S_t} dp^{mu nu}_a/dx^{nu} (d^3x)_{mu}
= integral_{dS_t} 1/2 p^{mu nu}_a (d^2x)_{mu nu}
= integral_H 1/2 p^{mu nu}_a (d^2x)_{mu nu}
= Q(H),
where the 2-forms are defined by
(d^2x)_{0 1} = dx^2 ^ dx^3; (d^2x)_{0 2} = -dx^1 ^ dx^3; etc.

Thus, the charge Q is seen not as a constant of motion, but as a
function of the horizon!

All the conserved charges become functions of the horizon H associated
with the region M_0. Out of this, one might expect to ultimately be
able to construct a generalized thermodynamics that relativizes all
the thermodynamics variables to H.

The second subtlety has to do with the effect of "cutting off" the
space-time on H. One normally thinks of a state as "evolving in time
t" and associated with an entire global Cauchy surface. Here, however,
the regions M_0 are compact, as are the spatial layers S_0. Thus, the
states reduce to improper mixtures that arise from cutting everything
off at the boundary H = dS_t, for each t.

Associated with this improper mixture is an entropy. And it's here
that one might impose the Bekenstein bound.

However, for regular spacetimes (those that are asymptotically flat),
as H is pushed off to infinity, the entropy also goes to infinity, if
we adopt the Bekenstein bound. That is, the "horizon effect" of the
cut-off actually INCREASES as we push the horizon out to infinity.

So, the cut-offs, the prevalence of mixed states (that is, improper
mixtures associated with the cut-off at H), and the essential
dependence of conserved quantities on H are the places where the
"horizon effect" enters non-trivially ... and in such a way that these
elements persist even in the limit as H is pushed out to infinity.

The view adopted here treats both the gravitational field and the
quantum fields as part of an open system. In effect, the Universe is
treated as an open system, with some sort of horizon H always present,
or possibly the lingering effects present even when the horizon falls
back out of view.

Most important of all: all of these constructions require that h be
first put into place. The evolution of the h-field, itself, is tied
directly to the construction of the local region M_0 and its foliation
(the S_t's), via a localized form of ADM. The evolution of the fields
is tied onto this background, within the S_t's already constructed.
The "forward-reaction" is of the connection Gamma (or torsion or
contorsion) with the fields; both are quantized. The "back-reaction"
of the classical part of the gravity field, h, is tied to the
effective Einstein law which, in turn, may be derivable as a
"Bekenstein bound defect" from the actual expression of <h| T^{mu nu} |
h>, itself.

In this way, the gravity field and Einstein equation emerge not as a
fundamental quantum field, but merely as a signal of the deviation
associated with the Bekenstein bound from the k -> infinity limit.

The one argument that's already led in this direction, of course, was
a 1995 paper published by Jacobson, which derived the Einstein
equation from the combination of the following assumptions:
* the first law (energy conservation)
* the second law dQ = T dS, where T is the Hawking-Unruh
temperature
* the "third law" -- a fixed proportionality of entropy with area,
when taken with respect to a distinguished family of "horizons".
From this, one derived both the Einstein equations and the constant of
proportionality.

What I'd be interested to see is if the Jacobson argument can be
adapted to the construction of horizons H, described above, in such a
way as to obtain the desired defect when the quantized T^{mu nu} are
relativized to a fixed classical background frame field h.