Re: Quantization....
From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 08/27/04
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Date: Fri, 27 Aug 2004 07:40:25 +0000 (UTC)
Frank Hellmann wrote:
>>Quantization just means constructing a quantum theory for a theory
>>of which we already know what is the classical limit
>
> Implicitly assuming that the process of taking the limit is exactly
> the one used in NRQM. An assumption that fails in almost everything
> beyond NRQM,
Haven't you heard of NRQED, NRQCD, etc? All relativistic field theories
have reasonable nonrelativistic limits, though, at present, the limit
cannot be taken in a mathematical sense since there is no mathematical
formulation of QED etc.
> already in QED it's not straightforward to recover the
> field picture, the dirac field that serves to quantize the electron
> has no classical meaning
It only needs to have a qunatum meaning.
> and in non abelian gauge field theories the
> classical theories we construct them from are not a good description
> of their classical limit (asymptotic freedom and all).
They are. The usual 1-loop results are low order expansions around the
classical limit.
> the fact that straightforward path integral quantization of the
> Einstein-Hilbert action is apparently very very ill defined.
Straightforward path integral quantization of the anharmonic oscillator
is already ill-defined. What counts are the final results with which one
works numerically, not the intermediate formal detours.
To 1-loop order (which is all one needs for all applications of
quantum general relativity in the near future), there is no problem
with quantum gravity. At higher loops, one simply needs to take account
of a few more coupling constants for each order. There is nothing
intrinsically wrong here, as long as one doesn't want to have a full
theory.
The common problem to all relativistic QFTs is that they lack a
firm mathematical basis; but this has nothing to do with the problem
of quantization.
> Consider the various generalizations of QM considered and how they
> appear naturally in a different approach:
> gr-qc/9706069
This looks like an interesting paper. The symplectic formulation of QM
might well be the one that generalizes to a full theory of quantum
gravity. But even thatr would involve 'quantizing gravity'!
>>>And now it seems that "quantizing gravity" and "constructing a quantum
>>>theory of gravity" are used as (almost) equivalent!
>>
>>Two names for exactly the same problem.
>
> Yet implying different approaches! "Quantizing gravity" implies
> starting with GR whereas "constructing a quantum theory of gravity"
> means ending up with GR in some limit! That this is considered
> equivalent is precisely what I question!
To construct the theory for the first time (and to motivate it later
for each new generation of physics) one _must_ go the first way,
since how can we guess what the right theory should look like if
we didn't. To verify that the theory is indeed quantum gravity one needs
to take the second path. Thus both paths are complementary, and give
the same result in case of success.
That none of the current approaches to quantum gravity is close to
any sort of success is a different matter.
Arnold Neumaier
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