Quantum General Relativity without Quantum Gravity - a Field Law

  1. Quantum General Relativity without Quantum Gravity -- a Field Law

    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.
     
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