canonical quantization of gravity [Archive]

alistair

Aug31-04, 04:55 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>\n\nWhat does canonical quantization of gravity mean?\nAnd how successful has it been compared to other methods\nfor quantizing gravity?\nWhat are the criteria for determining if quantization of gravity\nhas been successful?\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>What does canonical quantization of gravity mean?
And how successful has it been compared to other methods
for quantizing gravity?
What are the criteria for determining if quantization of gravity
has been successful?

Arnold Neumaier

Aug31-04, 03:41 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>alistair wrote:\n> What does canonical quantization of gravity mean?\n\nIt means: Write down the classical action, look at the corresponding\nfunctional integral as for QED or QCD, and try to compute S-matrix\nelements using the usual renormalization prescriptions for the integral\nfor the various Feynman diagrams.\n\n\n> And how successful has it been compared to other methods\n> for quantizing gravity?\n\nIt only works in the traditional way up to 1 loop; at higher loops one\nneeds more and more counterterms to make the resulting combination of\nintegrals finite. This is called \'nonrenormalizability\', and is the only\nblemish of canonical quantum gravity.\n\n\n> What are the criteria for determining if quantization of gravity\n> has been successful?\n\nIt depends on whom you ask. Many want a renormalizable theory;\nthen canonical quantum gravity is out.\n\nOthers treat canonical quantum gravity just as any other nonrenormalizable\neffective field theory, and fare well with it. See, for example,\nC.P. Burgess,\nQuantum Gravity in Everyday Life:\nGeneral Relativity as an Effective Field Theory\ngr-qc/0311082\n\nSee also several items in my theoretical physics FAQ at\nhttp://www.mat.univie.ac.at/~neum/physics-faq.txt\n\n\nArnold Neumaier\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>alistair wrote:
> What does canonical quantization of gravity mean?

It means: Write down the classical action, look at the corresponding
functional integral as for QED or QCD, and try to compute S-matrix
elements using the usual renormalization prescriptions for the integral
for the various Feynman diagrams.

> And how successful has it been compared to other methods
> for quantizing gravity?

It only works in the traditional way up to 1 loop; at higher loops one
needs more and more counterterms to make the resulting combination of
integrals finite. This is called 'nonrenormalizability', and is the only
blemish of canonical quantum gravity.

> What are the criteria for determining if quantization of gravity
> has been successful?

It depends on whom you ask. Many want a renormalizable theory;
then canonical quantum gravity is out.

Others treat canonical quantum gravity just as any other nonrenormalizable
effective field theory, and fare well with it. See, for example,
C.P. Burgess,
Quantum Gravity in Everyday Life:
General Relativity as an Effective Field Theory
http://www.arxiv.org/abs/gr-qc/0311082

See also several items in my theoretical physics FAQ at
http://www.mat.univie.ac.at/~neum/physics-faq.txt

Arnold Neumaier

Charles Torre

Sep1-04, 11:09 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>\nalistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408301330.338d7b35@posting.google. com>...\n> What does canonical quantization of gravity mean?\n> And how successful has it been compared to other methods\n> for quantizing gravity?\n> What are the criteria for determining if quantization of gravity\n> has been successful?\n\nI\'ll answer the first question, at least.\n\n"Canonical quantization" of a classical physical system\nusually means the following STRATEGY for making a quantum\ntheory that has the classical system as, well, its\nclassical limit.\n\n(1) Get your hands on the Hamiltonian formulation of the\ntheory (a.k.a. the "canonical" formulation). Then you will\nhave at the very least a phase space and a Hamiltonian\nfunction. (2) Represent the canonical coordinates and\nmomenta and the Hamiltonian and any other observables of\ninterest as self-adjoint operators on some Hilbert space.\nOne normally tries to represent the coordinates and momenta\nso that their commutator is the same as the classical\nPoisson bracket with a factor of i times Planck\'s constant.\n\nYou can find "canonical quantization" described in certain\ntexts, e.g., Merzbacher\'s text. The above is the basic\nidea, handed down by Dirac, I think. Of course, there are\nmany ambiguities and subtleties in this strategy and many\ninteresting and sophisticated generalizations. For example,\none sometimes tries to pick some preferred Lie algebra of\nobservables (functions on phase space - not necessarily\ncoordinates and momenta) and tries to find unitary\nrepresentations of the group generated by these\nobservables. If there are constraints - as there are in\nmost ways of doing canonical gravity - then one has to\ncontend with them as well. See, e.g., Dirac\'s nice little\nbook "Lectures on Quantum Mechanics" or Salisbury and\nSundermeyer\'s book "Constrained Dynamics".\n\nIn gravitation, the constraints are one of the principal\ndifficulties. There is a lot of work on the issues and\nproblems associated with trying to apply the canonical\nquantization ideas to gravitation. A seminal contribution\nis Bryce DeWitt\'s article, Phys. Rev. 160, 1113-1148\n(1967). Some more up to date references include the book\n"Lectures on Non-Perturbative Canonical Gravity", by Abhay\nAshtekar (1991) and the book "Quantum Gravity" by Claus\nKiefer (2004).\n\ncharlie\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>alistair@goforit64.fsnet.co.uk (alistair) wrote in message news:<861c1b21.0408301330.338d7b35@posting.google.com>...
> What does canonical quantization of gravity mean?
> And how successful has it been compared to other methods
> for quantizing gravity?
> What are the criteria for determining if quantization of gravity
> has been successful?

I'll answer the first question, at least.

"Canonical quantization" of a classical physical system
usually means the following STRATEGY for making a quantum
theory that has the classical system as, well, its
classical limit.

(1) Get your hands on the Hamiltonian formulation of the
theory (a.k.a. the "canonical" formulation). Then you will
have at the very least a phase space and a Hamiltonian
function. (2) Represent the canonical coordinates and
momenta and the Hamiltonian and any other observables of
interest as self-adjoint operators on some Hilbert space.
One normally tries to represent the coordinates and momenta
so that their commutator is the same as the classical
Poisson bracket with a factor of i times Planck's constant.

You can find "canonical quantization" described in certain
texts, e.g., Merzbacher's text. The above is the basic
idea, handed down by Dirac, I think. Of course, there are
many ambiguities and subtleties in this strategy and many
interesting and sophisticated generalizations. For example,
one sometimes tries to pick some preferred Lie algebra of
observables (functions on phase space - not necessarily
coordinates and momenta) and tries to find unitary
representations of the group generated by these
observables. If there are constraints - as there are in
most ways of doing canonical gravity - then one has to
contend with them as well. See, e.g., Dirac's nice little
book "Lectures on Quantum Mechanics" or Salisbury and
Sundermeyer's book "Constrained Dynamics".

In gravitation, the constraints are one of the principal
difficulties. There is a lot of work on the issues and
problems associated with trying to apply the canonical
quantization ideas to gravitation. A seminal contribution
is Bryce DeWitt's article, Phys. Rev. 160, 1113-1148
(1967). Some more up to date references include the book
"Lectures on Non-Perturbative Canonical Gravity", by Abhay
Ashtekar (1991) and the book "Quantum Gravity" by Claus
Kiefer (2004).

charlie