Dialog with the Bogdanovs (Part 3)

John Baez

From: igor.bogdanov@free.fr (I/G.Bogdanoff)
Newsgroups: sci.physics.research,sci.physics
Subject: Re: Physics bitten by reverse Alan Sokal hoax?
Date: Tue, 5 Nov 2002 18:34:20 +0000 (UTC)
Organization: http://groups.google.com/
Message-ID: (e8e077d9.0211041841.bba684e@posting.google.com)



baez@galaxy.ucr.edu (John Baez) wrote in message news:(aq1uoo$sr2$1@glue.ucr.edu)...

> In article ,
> igor.bogdanov  wrote:

> >On his webpage, Dr John Baez relates some aspects of the "Bogdanov
> >affair". We are very greatful regarding his effort to keep his page up
> >to date.

> You're welcome!
> 
> In what follows I will focus on the physics of your papers.
> The other issues surrounding this case are also fascinating,
> but I'd like to treat them separately, and perhaps in some 
> other forum, since sci.physics.research is mainly about physics.

Very good.

> >John Baez's text: 

> >>For example, here's the beginning of their paper "Topological Origin
> >>of Inertia:

> >>>We draw from the above that whatever the orientation, the plane of 
> >>>oscillation of Foucault's pendulum is necessarily aligned with the 
> >>>initial singularity marking the origin of physical space S^3, 
> >>>that of Euclidean space E^4 (described by the family of instantons
> >>>I_beta of whatever radius beta), and, finally, that of Lorentzian 
> >>>space-time M^4. 

> >Comment : It is not "their" paper but Igor's paper.

> Thanks; I've fixed this on my webpage:
> 
> http://math.ucr.edu/home/baez/bogdanov.html

> >We simply suggest that at 0 scale, the
> >observables must be replaced by the homology cycles in the moduli space
> >of gravitational instantons.  We then get a deep correspondence  -a
> >symmetry of duality- between physical theory and topological field
> >theory. 

> Yes, you say this in your paper.  However, you need to be 
> much more specific for there to be any substance to such a claim.
> Now is a good chance for you to do this.  For example:
> 
> 1) Could you please define "at 0 scale"?  

Answer : Let's begin first with a heuristic definition. In the FLRW
cosmology the scale factor R of the Universe can be reduced to R = 0.
On this singular region, the scale = 0 (O scale).  This point is the
"initial singularity" in the cosmological standard model. As far as
our own model is concerned, we have established that at the vincinity
of this "singular point" (0 scale) the 4 dimensional metric must be
considered as positive definite (euclidean signature ++++). This topic
is described by topological field theory.

> 2) You mention "the observables".  Observables in which 
> theory?

Answer : We consider here the "Heisenberg picture" and supergravity
N=2.   Here the observables are simply the Lorentzian metric of the
"spacetime system".   The dynamics of these observables is given by
the one parameter automorphisms group of the  algebra A  (Heisenberg
algebra) :

exp iht x A x exp -iht. 


> 3) You say "must be replaced".  Why must they be replaced?  
> And how?  

Answer : They must be replaced because at 0 scale any dynamical
content of the theory is suppressed. Consequently the observables are
also suppressed (ie. the lorentzian metric is replaced by an euclidean
metric). Therefore observables must be replaced by what we call
"pseudo observables". In this case, the quantum field theory (real
time) is replaced by topological field theory (imaginary time). Then
-as t goes to it-, the usual algebra of observales is replaced by an
algebra of  "pseudo observables" of the general form  :

exp -beta h x A x exp +beta h

In this case the dynamics in real time is replaced by a "pseudo
dynamic" in imaginary time  (or what we call an "euclidean evolution")
given by the one parameter semi-group of automorphisms of the ideal of
the algebra A.

> Presumably you are hinting at some correspondence between 
> observables in some theory and homology cycles in the moduli 
> space of gravitational instantons.  

Answer : Exactly.  

>Please describe this correspondence as precisely as possible.  Merely
>stating
>that it exists is not enough to convince us that it does.

Answer : Let's consider some physical observables  (lorentzian metric)
      01..,O2..,O"n"... Let's also consider homology cycles in the
moduli space of gravitational instantons. Now,  the correspondance
between observables and homology cycles is defined by a cohomological
field such that a correlation function of n physical observables   can
be interpreted as the  number of  intersections of  n cycles of
homology in moduli space of configurations of the instanton type  on
the fields f of the  theory. The point here is that observables O are
dependent of the lorentzian metric. But now, using topological
arguments, we can see that the correlation function 01..,O2..,O"n"..,
of those observables is topological (independent of the metric). More
precisely, the correlation function of the observables is a function
of the homology cycles H i defined on the moduli space of the 4 dim.
riemannian manifold. We can then construct a topological invariant out
of these homology cycles : the intersection of these cycles is a 
topological invariant.


> 4) You speak of a "deep correspondence" between some
> unspecified physical theory and some unspecified topological
> field theory.  Which theories are you talking about here?
> How does the correspondence go?

Answer : The "deep correspondance" between physical theory and
topological theory can be seen (on heuristic basis), as a simple
analytic continuation between lorentzian and euclidean metrics.  The
physical content of the theory is given by quantum field theory (whose
underlying metric is lorentzian) whereas the topological content of
the theory is described by topologcial field theory. The
correspondance can be seen as a duality (i-duality) between lorentzian
and euclidean metrics.

To be continued tomorrow...

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