Making GR & QM compatible without QG (was: GR & QM incompatibility issues)

markwh04@yahoo.com

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Bilge wrote:\n> Then, let me restate it in a more obvious way. The (apparent)\n> incompatibility between general relativity and quantum mechanics\n> refers to the attempts to develop a quantum theory of gravity\n> that reduces to general relativity in the classical limit.\n\nThat\'s not a problem at all, and (in fact) doesn\'t even require quantum\ngravity. As Jacobson showed in 1995, the mere imposition of the\nBekenstein bound in conjunction with the laws of thermodynamics,\nimplies general relativity. To quote his conclusion: gravity need not\nbe quantized as a fundamental field any more than phonons do.\n\nThe way he showed this is to employ the Raychaudhuri equation for\nvolume contraction at a local "Rindler horizon". This gives you an\nequation expressing the horizon area in terms of the energy\ninflux/outflux.\n\nThe imposition of the Bekenstein bound equates the horizon area, up to\nproportion, to the entropy associated with the phase averaging over the\nhorizon. Using this expression, a formulation of the 2nd law: d(heat)\n= (Temperature) d(Entropy) is formed, the temperature being that\nassociated with the thermal state arising from the relative state phase\naveraging over the horizon -- thus being expressible in terms of the\nRicci tensor.\n\nThis gives you the right hand side of the Einstein equation. The left\nhand side (d(Heat)), is equated to the flux of matter/energy. This\ngives you the left hand side of the Einstein equation -- up to a term\nproportional to the metric. The extra term is fixed (up to a\ncosmological constant) by invoking the conservation of energy -- as is\nusually done in the derivation of the field equations.\n\nAny quantum theory which admits a formulation of thermodynamics + a\nBekenstein bound -- regardless of whether it explicitly includes a\ntheory of gravity or not -- will therefore yield Einstein\'s equations\nof gravity as a consequence of the compatibility requirements for local\nRindler horizons.\n\n> It has little to do with the question of doing quantum theory in\n> curved spacetime. Hawking radiation, for example, is predicted\n> from doing quantum field theory in curved spacetime.\n\nIn a way, Jacobson\'s result is a major generalization of the earlier\nHawking result, going the extra step further to FULLY bring the field\nequations under the scope of thermodynamics.\n\n> Precisely. General relativity is generally considered to be the\n> classical limit of some quantum theory of gravity.\n\nThe end result is that GR is shown to be the classical limit of\n*GENERIC* quantum theory, with no special need for any quantum theory\nof gravity.\n\nThe two missing elements from Jacobson are\n(1) extension of the foregoing to globally non hyperbolic spacetimes\n(2) an implementation of the Bekenstein bound.\n\nItem (2) is where string theory may come into play, for instance. As\nfor item (1), in fact, is something Jacobson is ideally suited for!\nSince you generally don\'t have a global "t" parameter in the absence of\nglobal hyperbolicity (nor a universal state space), you\'re forced to\ntreat everything locally.\n\nYou can still, then, talk about locally hyperbolic regions. Such a\nregion R is characterized as follows. It has associated with it a\ntime-like vector field X of compact support. The region is bounded by\n2 spacelike hypersurfaces: Boundary(R) = R(1) - R(0). Associated with\nthis is a flow exp(tX), which maps the initial hypersurface R(0) ->\nR(t) for t between 0 and 1. On each hypersurface is a (n-1) form,\nn(t), such that n(t) ^ X gives you a volume form.\n\nThis is, in fact, the standard prescription (e.g. Lecture Notes in\nPhysics, 107) for setting up a symplectic structure.\n\nThe Hamiltonian is given by an expression of the form\nH = Lie_X(phi) dL/d(phi) - X L\nwhere Lie_X is the Lie derivative associated with X. On each\nhypersurface R(t) this gives you an integral (integrated using n(t)),\nH(t). The commutator [F, H(t)] = i h-bar F\'(t) gives you the\nHeisenberg equations.\n\nSince the boundary of the region is R(1) - R(0), then all the layers\nR(t) share a common frontier, Boundary(R(t)) = H, for all t. This "H"\nplays the role of Jacobson\'s local Rindler Horizon.\n\nConsidering all the possible ways of setting up regions, you get a\nrequirement for compatibility from imposing the Bekenstein bound on\neach such H. This translates into Jacobson\'s arguments.\n\nThe novel element not in Jacobson is how compatibility is enforced\nbetween the overlapping parts of two locally hyperbolic regions. Since\nthe spacetime in general need not be globally hyperbolic, then what\npasses for timelike within one region may be spacelike or even timelike\nwith the opposite orientation, seen from the perspective of another\nregion.\n\nThis is, in fact, a form of the "time-traveller paradox".\n\nTaking two points x, y; which are timelike in region R1, and spacelike\nin region R2; the commutators [A(x),A(y)]_{R1} != 0, while\n[A(x),A(y)]_{R2} = 0!\n\nInvariably, this means that the state space must be thermal! What\npasses for quantum noise from the perspective of region R1 in virtue of\nthe non-commutatitivy of the operators, will be seen as thermal noise\nform the perspective of R2.\n\nThe best way to see how this works is as follows.\n\nIf you think of a time-like path traversing from x to y wholly\ncontained in R1, at some point it has to exit R2 and reenter it. Those\ntwo points cross one or both of the boundary hypersurfaces, R2(1) or\nR2(0). The quantum noise seen from the perspective of R1, shows up as\nthermal noise associated with the cut-off that took place on the\nboundary, and shows up as a boundary fluctuation. In other words, the\nextra information corresponding to the correlation between A(x), A(y)\nis encoded into the boundary R2(1), R2(0).\n\nThe usual, Wightmann, formalism of QFT is the first casualty. In place\nof the axiom that the vacuum be a pure state that is Poincare\'\ninvariant, you have the axiom that it is a thermal state. If you also\nimpose the condition that it be a thermal state at positive\ntemperature, then this may be enough to also subsum the Wightmann\n"spectral axiom", thereby rendering its ad hoc formulation unnecessary.\n\nThe first two laws of thermodynamics, applied locally to the region,\ngive you the Einstein equations.\n\nThe third law may (via a general argument posted in s.p.r. not too long\nago, concerning the nature of negative temperature and negative energy)\ngive you a spectral gap away from 0 energy -- a prerequisite to\nestablishing a particle interpretation and a scattering theory\nformalism. In Wightmann this needs to be separately postulated. Here,\nit may arise as a consequence.\n\nSo, with the extension and with a way of implementing the Bekenstein\nbound, you have a general approach to quantum theory that automatically\nsubsumes general relativity, thereby achieving the desired goal through\nthe back door without the need to "quantize gravity" -- exactly as\nJacobson foretold.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Bilge wrote:
> Then, let me restate it in a more obvious way. The (apparent)
> incompatibility between general relativity and quantum mechanics
> refers to the attempts to develop a quantum theory of gravity
> that reduces to general relativity in the classical limit.

That's not a problem at all, and (in fact) doesn't even require quantum
gravity. As Jacobson showed in 1995, the mere imposition of the
Bekenstein bound in conjunction with the laws of thermodynamics,
implies general relativity. To quote his conclusion: gravity need not
be quantized as a fundamental field any more than phonons do.

The way he showed this is to employ the Raychaudhuri equation for
volume contraction at a local "Rindler horizon". This gives you an
equation expressing the horizon area in terms of the energy
influx/outflux.

The imposition of the Bekenstein bound equates the horizon area, up to
proportion, to the entropy associated with the phase averaging over the
horizon. Using this expression, a formulation of the 2nd law: d(heat)
= (Temperature) d(Entropy) is formed, the temperature being that
associated with the thermal state arising from the relative state phase
averaging over the horizon -- thus being expressible in terms of the
Ricci tensor.

This gives you the right hand side of the Einstein equation. The left
hand side (d(Heat)), is equated to the flux of matter/energy. This
gives you the left hand side of the Einstein equation -- up to a term
proportional to the metric. The extra term is fixed (up to a
cosmological constant) by invoking the conservation of energy -- as is
usually done in the derivation of the field equations.

Any quantum theory which admits a formulation of thermodynamics + a
Bekenstein bound -- regardless of whether it explicitly includes a
theory of gravity or not -- will therefore yield Einstein's equations
of gravity as a consequence of the compatibility requirements for local
Rindler horizons.

> It has little to do with the question of doing quantum theory in
> curved spacetime. Hawking radiation, for example, is predicted
> from doing quantum field theory in curved spacetime.

In a way, Jacobson's result is a major generalization of the earlier
Hawking result, going the extra step further to FULLY bring the field
equations under the scope of thermodynamics.

> Precisely. General relativity is generally considered to be the
> classical limit of some quantum theory of gravity.

The end result is that GR is shown to be the classical limit of
*GENERIC* quantum theory, with no special need for any quantum theory
of gravity.

The two missing elements from Jacobson are
(1) extension of the foregoing to globally non hyperbolic spacetimes
(2) an implementation of the Bekenstein bound.

Item (2) is where string theory may come into play, for instance. As
for item (1), in fact, is something Jacobson is ideally suited for!
Since you generally don't have a global "t" parameter in the absence of
global hyperbolicity (nor a universal state space), you're forced to
treat everything locally.

You can still, then, talk about locally hyperbolic regions. Such a
region R is characterized as follows. It has associated with it a
time-like vector field X of compact support. The region is bounded by
2 spacelike hypersurfaces: Boundary(R) [itex]= R(1) - R(0)[/itex]. Associated with
this is a flow [itex]\exp(tX),[/itex] which maps the initial hypersurface R(0) ->
[itex]R(t)[/itex] for t between and 1. On each hypersurface is [itex]a (n-1)[/itex] form,
n(t), such that n(t) [itex]^ X[/itex] gives you a volume form.

This is, in fact, the standard prescription (e.g. Lecture Notes in
Physics, 107) for setting up a symplectic structure.

The Hamiltonian is given by an expression of the form
[itex]H = Lie_X(\phi) dL/d(\phi) - X L[/itex]
where [itex]Lie_X[/itex] is the Lie derivative associated with X. On each
hypersurface R(t) this gives you an integral (integrated using n(t)),
H(t). The commutator [F, H(t)] = i h-bar F'(t) gives you the
Heisenberg equations.

Since the boundary of the region [itex]is R(1) - R(0),[/itex] then all the layers
R(t) share a common frontier, Boundary(R(t)) [itex]= H,[/itex] for all t. This "H"
plays the role of Jacobson's local Rindler Horizon.

Considering all the possible ways of setting up regions, you get a
requirement for compatibility from imposing the Bekenstein bound on
each such H. This translates into Jacobson's arguments.

The novel element not in Jacobson is how compatibility is enforced
between the overlapping parts of two locally hyperbolic regions. Since
the spacetime in general need not be globally hyperbolic, then what
passes for timelike within one region may be spacelike or even timelike
with the opposite orientation, seen from the perspective of another
region.

This is, in fact, a form of the "time-traveller paradox".

Taking two points x, y; which are timelike in region R1, and spacelike
in region R2; the commutators [itex][A(x),A(y)]_{R1} != 0,[/itex] while
[itex][A(x),A(y)]_{R2} = 0![/itex]

Invariably, this means that the state space must be thermal! What
passes for quantum noise from the perspective of region R1 in virtue of
the non-commutatitivy of the operators, will be seen as thermal noise
form the perspective of R2.

The best way to see how this works is as follows.

If you think of a time-like path traversing from x to y wholly
contained in R1, at some point it has to exit R2 and reenter it. Those
two points cross one or both of the boundary hypersurfaces, R2(1) or
R2(0). The quantum noise seen from the perspective of R1, shows up as
thermal noise associated with the cut-off that took place on the
boundary, and shows up as a boundary fluctuation. In other words, the
extra information corresponding to the correlation between A(x), A(y)
is encoded into the boundary R2(1), R2(0).

The usual, Wightmann, formalism of QFT is the first casualty. In place
of the axiom that the vacuum be a pure state that is Poincare'
invariant, you have the axiom that it is a thermal state. If you also
impose the condition that it be a thermal state at positive
temperature, then this may be enough to also subsum the Wightmann
"spectral axiom", thereby rendering its ad hoc formulation unnecessary.

The first two laws of thermodynamics, applied locally to the region,
give you the Einstein equations.

The third law may (via a general argument posted in s.p.r. not too long
ago, concerning the nature of negative temperature and negative energy)
give you a spectral gap away from energy -- a prerequisite to
establishing a particle interpretation and a scattering theory
formalism. In Wightmann this needs to be separately postulated. Here,
it may arise as a consequence.

So, with the extension and with a way of implementing the Bekenstein
bound, you have a general approach to quantum theory that automatically
subsumes general relativity, thereby achieving the desired goal through
the back door without the need to "quantize gravity" -- exactly as
Jacobson foretold.

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