LQG commutation relations and causality [Archive]

mad2physicist@gmail.com

Mar18-05, 12:41 PM

<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>I\'ve been trying to understand how loop quantum gravity gets its\ncommutation relations. Specifically, as Wald points out in his General\nRelativity book, no one knew what sort of commutation relations to give\nquantum GR, since ordinarily one sets [phi(x),phi(y)]=0 for x,y\nspacelike separated, but in quantum GR we don\'t know what it means to\nbe spacelike separated.\n\nNow, as far as I can figure out, LQG takes its commutation relations\nfrom the poisson brackets of the ADM formalism. Since the observables\nare (I think?) taken from the traces of holonomies of the connection A\nand their canonically conjugate momenta E. In the ADM formalism their\npoisson brackets are proportional to a delta function,\n{A(x),E(y}=kdelta(x-y). This allows one to get similar poisson\nbrackets for the loop observables, which vanish if the two loops are\ncompletely seperated.\n\nSo, my question is, doesn\'t this method of getting the poisson brackets\nin the ADM formalism assume that the slice on which we are working is\ncompletely spacelike (after all, if it weren\'t we wouldn\'t get the\npoisson brackets proportional to a delta function, I don\'t think)? If\nso, then how can we assume that those same commutation relations will\ncarry over into the quantum domain, where its possible that the slice\ncould be in a superposition of being spacelike and being timelike (in\nplaces)? I don\'t see how this fixes the question of giving good\ncommutation relations, seeing how it now seems to just assume that the\nslice we are working on is a completely spacelike slice, and that\ndoesn\'t strike me as a valid assumption (although perhaps it is, and I\njust don\'t understand why).\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>I've been trying to understand how loop quantum gravity gets its
commutation relations. Specifically, as Wald points out in his General
Relativity book, no one knew what sort of commutation relations to give
quantum GR, since ordinarily one sets [\phi(x),\phi(y)]=0 for x,y
spacelike separated, but in quantum GR we don't know what it means to
be spacelike separated.

Now, as far as I can figure out, LQG takes its commutation relations
from the poisson brackets of the ADM formalism. Since the observables
are (I think?) taken from the traces of holonomies of the connection A
and their canonically conjugate momenta E. In the ADM formalism their
poisson brackets are proportional to a \delta function,
{A(x),E(y}=kdelta(x-y). This allows one to get similar poisson
brackets for the loop observables, which vanish if the two loops are
completely seperated.

So, my question is, doesn't this method of getting the poisson brackets
in the ADM formalism assume that the slice on which we are working is
completely spacelike (after all, if it weren't we wouldn't get the
poisson brackets proportional to a \delta function, I don't think)? If
so, then how can we assume that those same commutation relations will
carry over into the quantum domain, where its possible that the slice
could be in a superposition of being spacelike and being timelike (in
places)? I don't see how this fixes the question of giving good
commutation relations, seeing how it now seems to just assume that the
slice we are working on is a completely spacelike slice, and that
doesn't strike me as a valid assumption (although perhaps it is, and I
just don't understand why).

whopkins@csd.uwm.edu

Mar19-05, 02:04 AM

<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>mad2physicist@gmail.com wrote:\n> I\'ve been trying to understand how loop quantum gravity gets its\n> commutation relations. Specifically, as Wald points out in his General\n> Relativity book, no one knew what sort of commutation relations to give\n> quantum GR, since ordinarily one sets [phi(x),phi(y)]=0 for x,y\n> spacelike separated, but in quantum GR we don\'t know what it means to\n> be spacelike separated.\n\nThat\'s the same argument made elsewhere, such as the 1990 World\nScientific book on Gauge Theory and Quantum Gravity by Nakanishi and\nOjima.\n\nOn the general question: that\'s the problem. LQG doesn\'t yet live in\nthe 4-dimensional world. That\'s partly what prompted things like Spin\nFoam by Baez et. al. to try and handle the problem of dynamics.\n\nBut, overall, the issue of dynamics is mostly unresolved.\n\nIt really boils down to this: is the light cone in the background or\nnot?\n\nIf not, then it lives not at the operator level, but at the state level\nand the light cone is smeared -- which means you have a novel feature\n(heretofore unmentioned by name except by me) called Light Cone\nTunnelling.\n\nIn that setting, the microcausality axiom is false. Commutators have\nto come from elsewhere or simply not be there at all with the whole\nquantum world itself pulled out of the hat entirely from the classical\nworld (the Einstein\'s Revenge Scenario). That is, indeed, possible in\nthe presence of the very causal anomalies characteristic of the very\nglobal non-hyperbolicity that would block the construction of a "t"\ncoordinate for quantizing with.\n\nquant-ph/9706018\nThe Logic of Quantum Mechanics Derived From Classical General\nRelativity\nMark Hadley\nAbstract:\nFor the first time it is shown that the logic of quantum mechanics can\nbe derived from Classical Physics. An orthomodular lattice of\npropositions, characteristic of quantum logic, is constructed for\nmanifolds in Einstein\'s theory of general relativity. A particle is\nmodelled by a topologically non-trivial 4-manifold with closed timelike\ncurves - a 4-geon, rather than as an evolving 3-manifold.\n[... etc. ...]\n\nThe only other alternative is to work entirely from self-consistency.\nFor example, Hojman & Shepley\'s "No Lagrangian? No Quantization!"\n(1990, JMP 32) pulls out of the hat the structure of a system with a\nHamiltonian in the classical limit. I showed, not too long ago, here\nhow this can be extended to get you a canonically quantized system\n(with a Hamiltonian quadratic in the momenta) WITHOUT the classical\nlimit.\n\nBut both approaches start out with an assumption concerning\n*equal-time* commutators (i.e., in effect, the microcausality axiom).\n\nThe general approach is to work out what results by requiring that the\noperator algebra for the field quantities be compatible with the field\nequations and propagate consistently under them. This is NOT a trivial\nrequirement and leads to constraints on the possible forms of the\ncommutators, but the constraints may not be tight enough.\n\nThe simplifying assumption I made is that the commutators are c-numbers\nwhen expressed in terms of a suitable set of coordinates (or field\nquantities) subsequently deemed "fundamental". Maybe, one can assume\nhere as well that a linear subspace of quantities which have c-number\ncommutators and generates the entire operator algebra exists and then\nwork out the compatibility with the equations of motion to yield\nsomething substantial.\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>mad2physicist@gmail.com wrote:
> I've been trying to understand how loop quantum gravity gets its
> commutation relations. Specifically, as Wald points out in his General
> Relativity book, no one knew what sort of commutation relations to give
> quantum GR, since ordinarily one sets [\phi(x),\phi(y)]=0 for x,y
> spacelike separated, but in quantum GR we don't know what it means to
> be spacelike separated.

That's the same argument made elsewhere, such as the 1990 World
Scientific book on Gauge Theory and Quantum Gravity by Nakanishi and
Ojima.

On the general question: that's the problem. LQG doesn't yet live in
the 4-dimensional world. That's partly what prompted things like Spin
Foam by Baez et. al. to try and handle the problem of dynamics.

But, overall, the issue of dynamics is mostly unresolved.

It really boils down to this: is the light cone in the background or
not?

If not, then it lives not at the operator level, but at the state level
and the light cone is smeared -- which means you have a novel feature
(heretofore unmentioned by name except by me) called Light Cone
Tunnelling.

In that setting, the microcausality axiom is false. Commutators have
to come from elsewhere or simply not be there at all with the whole
quantum world itself pulled out of the hat entirely from the classical
world (the Einstein's Revenge Scenario). That is, indeed, possible in
the presence of the very causal anomalies characteristic of the very
global non-hyperbolicity that would block the construction of a "t"
coordinate for quantizing with.

http://www.arxiv.org/abs/quant-ph/9706018
The Logic of Quantum Mechanics Derived From Classical General
Relativity
Mark Hadley
Abstract:
For the first time it is shown that the logic of quantum mechanics can
be derived from Classical Physics. An orthomodular lattice of
propositions, characteristic of quantum logic, is constructed for
manifolds in Einstein's theory of general relativity. A particle is
modelled by a topologically non-trivial 4-manifold with closed timelike
curves - a 4-geon, rather than as an evolving 3-manifold.
[... etc. ...]

The only other alternative is to work entirely from self-consistency.
For example, Hojman & Shepley's "No Lagrangian? No Quantization!"
(1990, JMP 32) pulls out of the hat the structure of a system with a
Hamiltonian in the classical limit. I showed, not too long ago, here
how this can be extended to get you a canonically quantized system
(with a Hamiltonian quadratic in the momenta) WITHOUT the classical
limit.

But both approaches start out with an assumption concerning
*equal-time* commutators (i.e., in effect, the microcausality axiom).

The general approach is to work out what results by requiring that the
operator algebra for the field quantities be compatible with the field
equations and propagate consistently under them. This is NOT a trivial
requirement and leads to constraints on the possible forms of the
commutators, but the constraints may not be tight enough.

The simplifying assumption I made is that the commutators are c-numbers
when expressed in terms of a suitable set of coordinates (or field
quantities) subsequently deemed "fundamental". Maybe, one can assume
here as well that a linear subspace of quantities which have c-number
commutators and generates the entire operator algebra exists and then
work out the compatibility with the equations of motion to yield
something substantial.

Thomas Larsson

Mar20-05, 01:39 AM

<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>mad2physicist@gmail.com wrote in message news:<1111075242.057432.114640@g14g2000cwa.googleg roups.com>...\n> I\'ve been trying to understand how loop quantum gravity gets its\n> commutation relations. Specifically, as Wald points out in his General\n> Relativity book, no one knew what sort of commutation relations to give\n> quantum GR, since ordinarily one sets [phi(x),phi(y)]=0 for x,y\n> spacelike separated, but in quantum GR we don\'t know what it means to\n> be spacelike separated.\n>\n> So, my question is, doesn\'t this method of getting the poisson brackets\n> in the ADM formalism assume that the slice on which we are working is\n> completely spacelike (\n\nYes, this is a major concern. The problems of reconciliating general\ncovariance with a priori folitations are discussed e.g. in\n\nhttp://www.arxiv.org/abs/gr-qc/0412059\nGeneral Relativity Histories Theory\nAuthors: Ntina Savvidou\n\nAbstract: The canonical description is based on the prior choice of a\nspacelike foliation, hence making a reference to a spacetime metric.\nHowever, the metric is expected to be a dynamical, fluctuating quantity\nin quantum gravity. After presenting the developments in the History\nProjection Operator histories theory in the last seven years--giving\nspecial emphasis on the novel temporal structure of the formalism--we\nshow how this problem can be solved in the histories formulation of\ngeneral relativity. We implement the 3+1 decomposition using\nmetric-dependent foliations which remain spacelike with respect to all\npossible Lorentzian metrics. This allows us to find an explicit relation\nof covariant and canonical quantities which preserves the spacetime\ncharacter of the canonical description. In this new construction we have\na coexistence of the spacetime diffeomorphisms group Diff(M) and the\nDirac algebra of constraints.\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>mad2physicist@gmail.com wrote in message news:<1111075242.057432.114640@g14g2000cwa.googlegroups. com>...
> I've been trying to understand how loop quantum gravity gets its
> commutation relations. Specifically, as Wald points out in his General
> Relativity book, no one knew what sort of commutation relations to give
> quantum GR, since ordinarily one sets [\phi(x),\phi(y)]=0 for x,y
> spacelike separated, but in quantum GR we don't know what it means to
> be spacelike separated.
>
> So, my question is, doesn't this method of getting the poisson brackets
> in the ADM formalism assume that the slice on which we are working is
> completely spacelike (

Yes, this is a major concern. The problems of reconciliating general
covariance with a priori folitations are discussed e.g. in

http://www.arxiv.org/abs/http://www.arxiv.org/abs/gr-qc/0412059
General Relativity Histories Theory
Authors: Ntina Savvidou

Abstract: The canonical description is based on the prior choice of a
spacelike foliation, hence making a reference to a spacetime metric.
However, the metric is expected to be a dynamical, fluctuating quantity
in quantum gravity. After presenting the developments in the History
Projection Operator histories theory in the last seven years--giving
special emphasis on the novel temporal structure of the formalism--we
show how this problem can be solved in the histories formulation of
general relativity. We implement the 3+1 decomposition using
metric-dependent foliations which remain spacelike with respect to all
possible Lorentzian metrics. This allows us to find an explicit relation
of covariant and canonical quantities which preserves the spacetime
character of the canonical description. In this new construction we have
a coexistence of the spacetime diffeomorphisms group Diff(M) and the
Dirac algebra of constraints.