Quantum General Relativity without Quantum Gravity  a Field Lawby Rock Brentwood Tags: field, gravity, quantum, relativity 
#1
Jun2708, 05:00 AM

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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 10dimensional 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 That^{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 LeviCivita connection Gamma from g, (3) defining the Riemann tensor, Ricci scalar and Einstein tensor from Gamma, g and g^{1}. On the righthand side, the stress tensor is the (quantized) symmetric stress tensor derived from the matter fields. In place of the "vacuum expectation", one takes the hexpectation. 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 RiemannianCartan geometry, the connection is not just the Levi Civita connection. It splits into Gamma = LeviCivita + 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 gaugecovariant derivative of the frame h (for the covariant derivative here involves the connection Gamma, which is halfquantized). 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 righthand side (that is: with another round of quantization *beyond* second quantization). The focus, thus, is placed on the coefficient k. In a spacetime of (n1)+1 dimensions, assuming the metric has units of [g_{mu nu}] = L^2/([x^{mu}] [x^{nu}]), the coefficient has units [k] = L^{n2}/[hbar]. 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 spacelike (n1) surfaces) with the area of its boundary (the (n2) surfaces bounding each). The upper bound is given by a proportionality constant z with dimensions [z] = L^{n2}/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/hbar. Moving it over to the lefthand side, and switching sides, we have <h T^{mu nu} h> = hbar/z G^{mu nu}(h). In the limit k, z > infinity, one has a 0 defect. Thus, whatever appears on the (new) righthand 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 (n2) boundaries. This is the abovementioned "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 (n2)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 spacetime 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 wellmotivated, 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 bypasses 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., nontrivial "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 1parameter family of 3surfaces (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 4form. 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 "EulerLagrange" 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 2forms 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 spacetime 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 cutoff actually INCREASES as we push the horizon out to infinity. So, the cutoffs, the prevalence of mixed states (that is, improper mixtures associated with the cutoff at H), and the essential dependence of conserved quantities on H are the places where the "horizon effect" enters nontrivially ... 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 hfield, 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 "forwardreaction" is of the connection Gamma (or torsion or contorsion) with the fields; both are quantized. The "backreaction" 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 HawkingUnruh 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. 


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