Dialog with the Bogdanovs (Part 4)

John Baez

From: baez@galaxy.ucr.edu (John Baez)
Newsgroups: sci.physics.research,sci.physics
Subject: Re: Physics bitten by reverse Alan Sokal hoax?
Date: Wed, 6 Nov 2002 20:33:35 +0000 (UTC)
Organization: UCR
Message-ID: (aqa32m$od5$1@glue.ucr.edu)


In article ,
I/G.Bogdanoff  wrote:

> John Baez wrote 

>> igor.bogdanov wrote:

>>> John Baez wrote:

>>>>For example, here's the beginning of [Igor's] 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. 

Note that I am still waiting for an explanation of this sentence.  

In particular, what does it mean to say the plane of oscillation 
of a pendulum is "aligned with the initial singularity"?   This 
was your previous attempt at explanation:

>>>in conjecture 4.9 (nothing more
>>>that an conjecture, by the way) we have considered that the
>>>2-dimensional plane of oscillation of the pendulum conserves the initial
>>singularity S for inertial reference, whatever the orientation of this
>>plane in physical space R3."

This attempt did not help at all.

I still don't what it means for a plane to "conserve the initial
singularity S for inertial reference".  What does it mean?

Here is another thing you said in attempting to explain the
above remark about Foucault's pendulum:

>>>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. 

I found this very unclear and proceeded to ask some questions.
Unfortunately, many of your answers to these questions were
equally unclear.   Thus, I will need to ask another set of questions!
It would be very good if this recursive process could terminate
at some point with me receiving a clear answer to my original question.  
The sooner this happens, the happier we both will be.

But let's continue with my questions and your answers:

>> 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. 

Okay, that makes sense.  We are studying the usual Friedman-LeMaitre-
Robertson-Walker "big bang" solution of Einstein's equations, and
you're referring to the big bang itself, where the scale factor R 
equals zero.  Strictly speaking only the limit R -> 0 makes sense,
not R = 0.  But I'll treat you as physicists and accept a little
sloppiness.

The trouble starts here:

>As far as our own model is concerned

Which model is that???

>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 ++++).

Ah!  You say you have *established* this.  That means you have
some sort of definite proof or calculation, at least at the physical 
level of rigor.  I would like to understand what you actually did.
So:

Firstly, tell me what model you are talking about.  

Secondly, tell me what mean by saying the 4-dimensional metric
"must be considered as" positive definite.  Are you saying it
*is* positive definite?  That would be very odd if you were 
talking about the FRLW cosmology, because here the metric is
never positive definite, not even in any vicinity (neighborhood)
of the initial singularity.  So, what's going on?

Thirdly, tell me how you *established* that the metric must be
considered as positive definite.

>This topic is described by topological field theory.

That might be interesting if true, but until you've explained
what you're talking about it's probably not worthwhile going
into this further.

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

>Answer : We consider here the "Heisenberg picture" and supergravity
>N=2.  

Okay, so you're studying observables in N = 2 supergravity.
That's a clear answer. 

>> 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). 

I have a lot of problems with these 2 sentences.

First, as I said before, it's not true that in the R -> 0 limit 
the Lorentzian metric *becomes* Riemannian.  You write "is replaced
by", which is much vaguer, but I don't know a sensible interpretation
of this phrase.

Also, I don't believe that in the R -> 0 limit "any dynamical
content of the theory is suppressed".  No matter how close we
get to the big bang, a lot of interesting dynamics is going on.
That's why people use the term "BANG"!  

Thirdly, I don't know what it means to say "the observables
are suppressed".  Are you saying some particular observables
are approaching zero, or something?  Which ones?

Fourth, I don't see why you say that "dynamical content is
suppressed" implies "observables are also suppressed".  

In short, I would say this doesn't make sense.

>Therefore observables must be replaced by what we call
>"pseudo observables". 

Must be?  Why?  You didn't really answer my question.

>In this case, the quantum field theory (real
>time) is replaced by topological field theory (imaginary time). 

Switching from real to imaginary time does *not*, by itself, 
turn a quantum field theory into a topological quantum field theory.

>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.

This sounds like a complicated way of saying that you're
switching to imaginary time, a common procedure in quantum
field theory.  You're doing this in N = 2 supergravity near
the big bang?  So you have worked out how the quantized version
of N = 2 supergravity evolves near the big bang in some sort
of supersymmetric version of the FRLW cosmology?  Let's see
some calculations!!!  Doing something like this would require 
a lot of work - and so far I haven't seen any of it.  

>> 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"... 

Which observables?  Calling them O1,... etcetera does not
say which ones they are.  We need to know exactly which ones
they are, if we're going to see this correspondence!

>Let's also consider homology cycles in the moduli space of 
>gravitational instantons. 

Which homology cycles?  Again, we need to know exactly which
observables get mapped to which homology cycles, if we are
going to see this correspondence.

>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. 

Whoa!  This sounds very impressive, but it doesn't give me
any of the details I need to see the correspondence.   I need specifics.  

>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). 

I would have to see these "topological arguments" to believe this.
But it's premature to worry about this until you answer many of
my more basic questions.  

>> 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.  

Again you seem to be making the mistake of saying switching
to imaginary time is enough to obtain a topological quantum 
field theory.  That's just not true.  If you're trying to say
something else, please clarify.

Okay.  Don't forget, you still owe me an answer to the most important
question of all:

>> 5) What does any of this have to do with Foucault's 
>> pendulum or the origin of inertia?  

Remember, my original question was: what does it mean to say 
the plane of oscillation of a pendulum is "aligned with the 
initial singularity"?   All this stuff about N = 2 supergravity,
imaginary time, "pseudo-observables" and the like came up as
part of your attempt to answer this question.  I don't see
what any of this stuff has to do with the original question.

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