From: phfrom@nyx.uni-konstanz.de (Hartmut Frommert)
Subject: Re: BAEZIAN Physics and astronomy [was: Space-time IS a semi-Riemannian manifold (?)
Organization: Dept. of Physics, University of Constance
Date: Thu, 8 Oct 1992 15:05:45 GMT

jbaez@riesz.mit.edu (John C. Baez) writes:

>Let me add in response to Hartmut Frommert's advocacy of modifications
>of GR... it is certainly worth studying these and people have.  No
>matter what one does along these lines one will probably want to
>quantize the theory eventually, and quantizing old-fashioned Einstein
>gravity will probably be good practice.  (Unless these other theories
>turn out to be significantly EASIER to quantize.  Occaisionally one
                              ^^^^^^^^^^^^^^^^^^
    Unfortunately there are too many different opinions on what this means.
>hears this claim but not for torsionful or nonmetric connection
>theories.)  

At least for one issue of quantization, the RENORMALIZABILITY, there is an
alternative that was shown to be renormalizable, in contrast to standard
Einstein GR (that with the Einstein-Hilbert action): The R^2 theory. See

  K.S. Stelle. Renormalization of higher-derivative quantum gravity.
   Phys Rev D, 16(4): 953--969, August 1977

for the proof (which is VERY technical). 

The same issue is still open for the scalar-tensor theories: It is up to
now NOT clear (at least I found no reference) if the substitution of the 
Newtonian grav'l constant G by some scalar field (square), i.e.

  G^{-1} --> a * |Phi|^2                                        (1)

gives a renormalizable theory without R^2 terms (it was shown, I think by
B.S. DeWitt, that the scalar-tensor theory WITH R^2 remains renormalizable,
this is also rather easy if you take Stelle's proof as granted). At least
the dimensionality of G as coupling constant can be abandoned that way.
Also if you embed the particle physics' standard model in curved spacetime, 
I don't see any preference of the Einstein-Hilbert action over that with
G substituted by (1).

You may take (but must of course not) the Phi cited above to co-incide with 
one Higgs field of elementary particle physics. To my knowledge this was 
first proposed (but only partially outlined) by

  A. Zee. Broken-symmetric theory of gravity. Phys Rev Lett, 42(7): 417--421,
   February 1979. 

Then if in the ground-state Phi_0 the relation

  | Phi_0 |^2 = v^2

holds, the gravitational constant is generated by the Higgs mechanism:

  G = 1 / (a*v^2)

As on the other hand v^2 is given by the gauge boson masses, there arise some
interesting relations, which cannot be presented appropriately in this 
article, but has some surprising properties due to the non-minimal coupling
of the Higgs field to geometry.

For torsion and nonmetricity, if you want to forbid them strictly, you MUST
explain why they don't appear, if you take curvature into account. Reason:
If you do not forbid them *explicitely*, they are contained in the 
Einstein-Hilbert Lagrangian (and thereby action) hidden as terms in the 
curvature scalar R (also in all covariant derivatives, of course). 
So you must BY HAND add the Riemannian constraints of torsion-freeness and 
metricity, which IMHO is not very satisfactory. If you do NOT forbid 
nonmetricity and torsion this way, you may ask how you can modify the 
theory -- from different aspects, i.e. in the simplest way, or looking for 
the most general version of the theory that is still renormalizable (to beat 
this issue here).

To mention your essential point, John: While the theories presented above 
are as consistent with the present experimental situation (and in some 
theoretical points superior and more general) than standard GR (Riemannian
+ Einstein-Hilbert), they are AS-ARE not quantum theories, of course. 
(BTW, IMHO many theories on cosmic loopings also are no QTs, however :-). 
But there is some evidence that they have properties which make them closer 
related to that. The central problem of QG stays (IMHO) the question how to 
give geometry a quantum interpretation (quantum geometry ?), i.e. questions 
like
* how can metric and connection (or curvature, nonmetricity, and torsion)
  be interpreted (or derived) quantum theoretically ?
* is there a "set" of "classical" spacetime"s" (i.e. 4-manifolds) 
  which form a Hilbert space ?
* is the "quantum state" of ST described by some wave function, i.e. 
  probability to find ST in this or that state ?
* does ST show geometrical quantum fluctuations (i.e. of metric, connection,
  etc.) and what are their effects ?
All these appear to be still open (:-))

[stuff deleted]

>I don't go around saying "The
>spacetime manifold MUST DIE!!"  

Do you think it would care ? :-))
--
 Hartmut Frommert                 <phfrom@nyx.uni-konstanz.de>
 Dept of Physics, Univ of Constance, P.O.Box 55 60, D-W-7750 Konstanz, Germany
                                           -- Eat whale killers, not whales --
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