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Principia
Physica... or
Caveat Emptor?
An unusal
paper appeared recently on the Los
Alamos eprint server: "Principia
Physica" by Mukul Patel (grqc/0002012)
This is an unusual paper in the sense that it has not a single reference
although it acknowledges Prof. Phil Parkers of TWSU. The abstract of the
paper states: "A comprehensive physical theory explains all aspects of
the physical universe, including quantum aspects, classical aspects, relativistic
apects, their relationships and unifications." With such bold claims,
the paper is strange indeed.
"Principia
Physica" was brought to my attention by a colleague, a fellow member
of an international physics discussion group that is very actively pursuing
avenues of "New Physics" on the WWW. Being not only a theoretical
physicist, but a mathematical physicist as well, I was asked to check
out the claims of this paper including the mathematical terms, but it
proved to be not an easy task, to say the least! The author and his "associates,"
such as they may be, were not only evasive, but aggressively so.
I approached
this request in the same manner as any other contact between colleagues
where certain courtesies are understood and extended in the neverending
pursuit of greater knowledge of the universe in which we live.
But, I must
have stumbled onto something else unawares. At this point in time I can
only conjecture that 1) the paper is a joke; or 2) the paper is NOT a
joke. If it is not a joke, then I will leave other conjectures to the
reader.
The following
is a series of emails exchanged in the process of trying to determine
the identity of the author, his academic affiliations/credentials, and
obtaining some straightforward answers to technical questions.
Dear Colleagues,
[This was addressed to the discussion group]
I normally
don't say much except that which relates strictly to physics or math.
But, I think it is time for an exception. Most physicists are aware of
the "rivalries" that exist between mathematicians and physicists, so I
don't need to explain it... But, because of my position as a mathematical
physicist, I am often asked by theoretical physicists to "get down and
dirty" with the math, which I am always happy to do.
The following
exchange of emails is rather lengthy but, it is certainly worth looking
at for some strange clues about the possible existence of a "secret group"
out there deliberately spreading lies and disinformation. In the above
case, Mukul Patel's "Principia Physica", I had a look at the paper...
First of
all,there seems to be another version of this paper, sent by the author
to a "group of physicists". This other version is, supposedly, written
in a more serious way.
Second, I
am in touch with the author, and he slowly withdraws from the formulations
of the official version, and starts to rephrase all ....
It is possible
that Mukul Patel indeed has something interesting to say, but makes "errors"
on purpose. What purpose? I do not know.
To be more
specific: I checked his "academic affiliations," by writing to the institution
in question:
From: Arkadiusz
Jadczyk
To: weinerts@sckans.edu
Subject: info/inquiry
Date sent: Tue, 29 Feb 2000 17:35:30 0400
Sirs,
Two weeks ago a research paper appeared on Los Alamos eprint server xxx.lanl.gov
No: grqc/0002012. The paper is signed by Mukul Patel from Department
of Mathematics, Southwestern College. As an expert in differential geometry
I was asked by a group of physicists to review and comment on this paper.
After reading it I became curious about the background of the author.
Physicists
publish their papers with references and their backgrounds are available
for all. If a mathematician writes a paper, physicists normally assume
the paper is mathematically correct. This is not the case with the aforementioned
new paper, therefore I am wondering about institutional affiliation of
the author.
Please, let
me know if Mukul Patel is indeed associated with the Department of Mathematics
of your College, and in what capacity.
Thank you,
Sincerely,
Arkadiusz
Jadczyk
I never received
a reply from the uni (added on March 23: just got the
answer), but did receive a reply from the email listed on the paper,
to which I had addressed a similar inquiry. The second inquiry and reply
are as follows:
From: Arkadiusz
Jadczyk
To: arfaei@jinx.sckans.edu
Subject: Principia Physica
Date sent: Tue, 29 Feb 2000 08:43:29 0400
Dear Mehri Arfaei,
I am somewhat puzzled by the paper "Principia Physica" (grqc/0002012)
apparently authored by Mukul Patel, Department of Mathematics, Sothwestern
College. I noticed that Mukul Patel is not listed as a faculty memeber.
What is his association with your Department?
Thank you,
ark
To which I received the reply:
Date sent:
Tue, 29 Feb 2000 13:03:23 0600
From: Mehri Arfaei
Subject: Re: Principia Physica
He is a friend of this department, and has been working here for a while.
I tried to send your emails to him, including this one, but for some
reason it doesn't go through. sorry about that. Of course I don't understand
why should it matter!
best regard
Mehri Arfaei
I then wrote
to the mathematician who was referred to in the paper, (who I had to search
to find as no e mail address was given) and received the following reply:
Date sent:
Wed, 1 Mar 00 01:15:46 CST
From: PPARKER@TWSUVM.UC.TWSU.EDU
Subject: Re: Principia Physica
Date: Tue, 29 Feb 2000 09:13:06 0400
From: "Arkadiusz Jadczyk"
I am reading, with some puzzlement, a paper "Principia Physica" submitted
to Los Alamos eprint server by Mukul Patel (grqc/0002012)
In the acknowledgement
section the Author is thanking you for "critical assesement". Did you
indeed study this paper? Who is the author? I am puzzled, because the
paper, on one hand, shows pretty good knowledge of certain aspects of
differential geometry, but also it contains unsupported claims. Very strange....
He's a
former student of mine (some 12 years or so ago). I did read through
it and offered some advice. Some was taken and the rest wasn't. So it
goes. I thought it needed more work and refinement, especially regarding
the provision of at least reasonable support for all claims, even if
not genuine mathematical proofs. I must admit I did not expect him to
"publish" it at this (what I thought of as an) early stage, but I assume
he has his own agenda.
If you
wish to conttact Mukul directly, I can provide an email address if one
was not included.
Phil Parker 
URL http://www.math.twsu.edu/Faculty/Parker/
As will be
noticed by these two foregoing responses, there seemed to be some discrepancy
between the "versions" of who and what Mukul Patel was, as well as the
"connections" he was trying to establish by his references to Professor
Parker, so I wanted to clear this up. I wrote the following:
From: Arkadiusz
Jadczyk
To: Mehri Arfaei Subject:
Re: Principia Physica
Copies to: weinerts@sckans.edu
Date sent: Wed, 1 Mar 2000 04:06:16 0400
On 29 Feb 00, at 13:03, Mehri Arfaei wrote:
"He is a friend of this department, and has been working here
for a while. I tried to send your emails to him, including this one,
but for some reason it doesn't go through. sorry about that. Of course
I don't understand why should it matter! best regard Mehri Arfaei
"
Dear Collegue,
I am including below an email from Prof. Parker. If Mukul Patel is
indeed your friend, I would suggest that you ask him to withdraw the
paper that involves your email address and Department of Mathematics
of Southwestern College  at least until errors and unsubstantiated
claims are removed. That is a normal procedure and will save the time
of many physicist who expect that a paper published in Los Alamos archive
satisfy certain standard criteria. Unfortunately I am not able to contact
Mukul Patel himself. Emails are bounced.
Sincerely,
Arkadiusz
Jadczyk
This was
followed by a standard inquiry:
To: PPARKER@TWSUVM.UC.TWSU.EDU
Subject: Re: Principia Physica
Date sent: Wed, 1 Mar 2000 04:24:12 0400
On 1 Mar 00, at 1:15, PPARKER@TWSUVM.UC.TWSU.EDU wrote: > If you wish
to conttact Mukul directly, I can provide an email > address if one
was not included. > > Phil Parker
Thank you for your kind information. I tried to contact him directly,
but he has stopped responding. I think the paper is wrong, but perhaps
you can convince me that it is not.... I have many problems with the
paper, but the first one deals with affine nature of his maps omega_{xy}.
I asked Mukul to explain, here is what he wrote:
Dear Colleague,
thanks
for your kind interest in the paper. if we had required that f(x)
= y, then the natural candidate would be the "ordinary derivative"....i.e.
a LINEAR map from tangent space at x to that at y. But as things stand
f(x) maybe a point differnt than y. so we have a linear map (the derivative)
from tangent space at x to that at f(x)...now we apply our principle
to the pair f(x) and y....again there is a derivative map form tangent
space at f(x) to that at f(f(x)), and we appply the prinicple to the
pair of paoints f(f(x)) and y....we repeat the process and get successive
linear maps from f^n(x) to f^(n+1)(x)...for each n we have a compisiton
from tangent space at x to that of f^n(x)...it can be show that the
sequence of points f^n(x) converges to y and the wbove compistion
converges to a linear map from tnagent space at x to that at y. But
we have an extra pice of information...the sequence of points f^n(x)....this
determines a vector at y. so what we have is a Linear map from x to
y AND vector at y. In short, we have an AFFINE map form x to y (tangent
spaces at these)....this map is what i calle "roughly speaking, an
'affine derivative of f'"....i hope this clarifies the vague allusion
to the term affine. ************
to which I replied:
Patel
wrote:
"But we have an extra pice of information...the sequence of points
> f^n(x)....this determines a vector at y."
Hi, I
am still trying to understand your paper. I do not see how a sequence
point y_n on a manifold, even converging to a point y, determines a
tangent vector at y. Will you please provide a proof/construction?
Did you
think about this problem when reading the paper? Although, I do not
understand how he extends affine map omega from vectors to tensors.
Is there a method that I am not aware of? I think you can't extend affine
map to tensors, because extension depends of decomposition of the tensor....?
Sincerely,
ark
This was
followed by a rather astonishing reply from Professor Parker who, apparently,
objected to my inquiries:
Date sent:
Wed, 1 Mar 00 12:42:09 CST
From: PPARKER@TWSUVM.UC.TWSU.EDU
Subject: you erred
You quoted without permission: that's not only bad manners, rude, and
thoughtless, its also a violation of federal US law. I suspect you are
merely a frustrated jerk who is trying to cause trouble because you
are so uncreative yourself. You will receive no further communications
from me, either directly or indirectly. You owe me an apology.
Phil Parker 
URL http://www.math.twsu.edu/Faculty/Parker/
So, the
project having taken an unusual turn, I responded:
To: PPARKER@TWSUVM.UC.TWSU.EDU
Subject: Re: you erred
Date sent: Wed, 1 Mar 2000 14:08:12 0400
When I
am asked by members of the international physics community to examine
a paper and pronounce on its usefulness, I act in the standard way which
includes being open with all, since it saves time and energy for all
concerned. For the record: the standard way is that ALL is quotable,
as long as proper attribution is made, and no specific request for confidentiality
is appended. Papers which are evidently WRONG, should not be published.
As a referee for journals, you should know it. I don't apologize for
doing my job.
ark
I did not
receive a reply from Parker, but DID receive info from Patel, indicating
clearly that he WAS in communication with Parker and Arfaei:
Date sent:
Wed, 01 Mar 2000 16:06:38 0800
To: "Arkadiusz Jadczyk"(by way of Mehri Arfaei )
From: mukulp
Subject: Re: Principia Physica
Dear Colleague,
i recieved the following communication from you addressed to arfaei..
I am glad to learn that you are 'reviewing' the paper for a 'group of
physicists'. If you had mentioned that earlier, i could have responded
in more elaborate and precise fashion. I must comment though that some
of your objections are rather hasty. For example, you kindly informed
me that one needs a measure or a volume form to integrate. Now this
point is already addressed to in a footnote on the same page. If you
had been reading more carefully, i am sure you could not have missed
it, as it is referenced exactly where that integral occurs.
I humbly
suggest that you reserve your opinion until AFTER you have read the
whole paper, and you direct these same to me personally, rather than
running around bothering my aquaintances and colleagues. Another thing,
since you have failed to tell me what your background is, i had to go
look up on the net. While this 'group of physicists' might consider
you an expert on differential geometry, judging from your papers, i
am inclined not to quite concur with their opinion.
More importantly,
i suggest that you also consider the physical significance of the IDEAS
presented in the paper. I assure you that all your technical objections
can be and will be answered in detail. Judging from your website, you
seem to be of the opinion that understanding "what actually happens"
should be the ultimate goal and the "ptolemian" quantum theory ought
to be replaced. Given this conviction of yours, which i wholeheartedly
agree with, i am sure that you will find it in your heart to consider
the concepts and how they successfully do the job, rather than being
foccussed completely on the technicalities...
I assure
you again, that i will answer all the questions you have, in one single
communication. (Indeed, i am in the process of writing down all the
"there exists" "for all" and "if and only iffs" of the technical side,
AS WELL AS, a detailed analysis of the whole conceptual framework. And
that detailed paper promises to be of a length not quite appropriate
for publication in a journal.)
Also,
you obviously seem to have rather too exalted a view of the "standards
of rigor" at los almos archive. There are hundreds of papers there which
don;t even come close your idea of rigor there. I sincerely believe
that all your comments are rather caused by "a mathematician writes
a physics paper". I hope you will be able to put this obvious prejudice
and the prejudice about "institutional affiliation" aside and review
the worth of the ideas in answering the questions you yourself have
deemed ncessary in your worthy project, "Quantum Future".
Sincerely,
Mukul Patel
p.s. i
also suggest that you review the paper as i submitted to "the group
of physicists" rather than an older version at the archive.
So, essentially,
he was saying that there was a version of the paper that was posted at
Los Alamos that was DIFFERENT from the one he had submitted to a "group
of physicists."
Now, apparently, he had the idea that I was affiliated with this "group
of physicists" to whom he had submitted the "other" version of his paper...
To answer
his inquiry about my background in differential geometry, I forwarded
the following:
From: Arkadiusz
Jadczyk
To: mukulp
Subject: Re: Principia Physica
Date sent: Wed, 1 Mar 2000 17:56:52 0400
On 1
Mar 00, at 16:06, mukulp wrote:
While this 'group of physicists' might consider you an expert on differential
geometry, judging from your papers, i am inclined not to quite concur
with their opinion.
From:
http://www.math.washington.edu/~hillman/Relativity/wrong.html Dr.
Arkadiusz Jadczyk (yup, he has a Ph.D. in physics!) is a curious case
indeed. On the one hand, he seems believe in lots of weird, weird stuff.
(He complains that his colleagues at the Institute of Theoretical Physics,
University of Wroclaw asked him to move this page off their website;
it is easy to see why they didn't want their institute associated with
this stuff.) On the other hand, he has coauthored a serious book on
Riemannian geometry with R. Coqueraux, who is known to me for his perfectly
sane papers on CayleyKlein geometries,, and he has coauthored numerous
papers with Phillipe Blanchard, who is known to me for his excellent
review paper on the classical theories of Julia and Fatou (as in the
MandelbrotHubbard set). So I figure Jadczyk must be sane on even numbered
days and channel with the Cassiopeians on odd numbered days.
Well, today
is an odd day....
ark
Apparently
Mr. Patel was quite impressed by SOMETHING in the above quote. Whether
it was dependent on the odd or even days, I am not quite sure. Nevertheless,
he responded:
Date sent:
Wed, 01 Mar 2000 19:04:00 0800
To: "Arkadiusz Jadczyk"
From: mukulp
Subject: Re: Principia Physica
:):) now we are speaking the same language. (not the mathemtically obscure
text on geometry, but the communicating with casseopians.)
Look, looks
like we got off on the wrong foot with one another. let us put this
thing behind us and concentrate on the problems at hand. You and i both
agree that physics has SEVERE problems, AND most physicsts are shy of
admitting those. I congratulate on your vision to see this, and the
courage to speak out.
Before
i attempt to answer your objections, with all due respect, i want to
record my objections to your tactics of reviewing: i find it extremely
offensive that you misguided me into believing that you are merely "reading"
my paper. On top of that you go and bother my exprofessor, and my current
academic abode, and then you crossrefer them to each other...and send
private communications to third parties without express permission of
their respective writers. You must agree that this is way outside of
your regular duty as judging the content of the paper.
That being
so, i am not holding it agaist you, as long as you don;t insist on lecturing
me on how one needs a measure to integrate.
About your
latest objection, this is what i have to say: not every converging seuqence
determine a vector. But this one does beuase of the way it arises. In
fact, even this "sequence" is just a first approximation that i thought
of on the spur of the moment, to answer a question by someone who i
thought must be a casual amateur reading my paper. In the actual formulation
of the principle, i don;t even need that sequence. And of course it
IS a technical matter (even if what you call an error IS an error indeed.)
The crux of the matter is that what we have is a (nonlocal) *affine*
connection. The upshot is that its infinitesimal aspect is a usual affine
connection. BUT this affine connection has two components, the linear
componet and a "rotational" component (corresponding to the semidirect
product of the affine group in to the linear gorup and R^4). The interesting
thig is that this rotational aspect turns out to be the "canonical form"
which is equivalent to a tensor field on the base space, which is given
by the identity transformation on each tangent space.
Now you
must be familiar with the fact that canonical form in itself does not
represent any extra sturcture on a manifold. So essentially what we
have is a linear connection. (plus the canonical form). Basically, the
"field" consistes of the exterior derivative of the linear connection
with respoect to itself AND the exterior covariant derivative of the
canonical form with respect to the former. So all our information IS
ineed contained in the linear part of the connection..so, even if your
hasty assertion is correct and certain sequence does not determine a
vector as advertised, it is not a serious fundamental problem of my
theory. I implore you again, to stipulate temporarily that we do have
what i call a "nonlocal [affine] connection".
Based on
that, see if other ideas have any significance at all in your opinion.
Believe me you will answers to many of the questions you have outlined
in your "Quantum Fuuture" project....and more. meanwhile i will try
to write up a precise formulation which would spin the heads of even
the most pedantic mathemticians. One professional courtsey you can do
to me is tell me exactly what "group of physicsists" has asked you to
review this paper? Because if this group is what i think it is, you
are reading the wrong version of the paper.
Again, he
talks about "two versions" of the paper. The fact that he likes my wife's
Cassiopaeans has nothing to do with the scientific approach that gives
structure and form to my work as a physicist.
I replied:
To: mukulp Subject:
Re: Principia Physica
Date sent: Wed, 1 Mar 2000 22:40:12 0400
Dear Mukul,
As you
certainly know certain requests for a "review" are confidential. We
are dealing with such a case. Perhaps you are right, perhaps I am reading
the wrong, that is the "official" version. If you like to send me the
"right" version  please do it. Otherwise I will have to follow blindly
my duty  which will consume my time and yours.
ark
He responded:
Date sent:
Thu, 02 Mar 2000 12:25:25 0800
To: "Arkadiusz Jadczyk"
From: mukulp
Subject: Re: Principia Physica
Dear Ark,
i am certianly
not aware of a protocol where the body requesting a review is "confidential".
Referees are often "confidential", of course. At any rate, this is the
case. I have submitted a version of the paper to a "group of physicists".
This version slightly differs from the one you seem to be looking at.
I am assuming your "group of physicists" asked you to review the version
on the xxx archive and not the one i submitted to my "group of physicists".
If that is the case, it is possible that your "group" and mine are different.
One question before i send you the other version: did this "group of
physicists" say or imply that I have formally submitted my paper to
them for publication in a print journal? I am sure you can tell me that
much. I think you owe me that information before i waste any more of
my time and yoursi mean what is the point in convincing a person
of the verity of my paper if I (or my journal publisher) have not asked
him to 'review' my paper?
mukul
So, we seem
to be getting into the realm of DELIBERATE DISINFORMATION. So, having
some personal objections to folks who post lies and obfuscations in the
public arena which seem to be designed to lead the majority of physicists
who don't have a good handle on the more complex math (not because they
are not GOOD physicists! Believe me, I KNOW it!) I wrote back to Patel:
From:
Arkadiusz Jadczyk
To: mukulp Subject: Re: Principia Physica
Date sent: Thu, 2 Mar 2000 15:14:57 0400
Hi Mukul,
Yes, probably our "groups of physicists" are nonoverlapping. So, let
me proceed with the only version that is official. I am back to my question:
how you define "affine derivative". Affine derivative of WHAT? And how
it defines affine map between tangent spaces. ark P.S. I am ready to
accept: "I do not really need all these constructions, let us start
with an arbitrary affine connection, and we will construct all our f's
and omegas from this connection. Then, if we want to reverse the precess,
we certainly, after sufficiently hard and, in principle, useless work,
can do it and find axioms that fs must satisfy ..." Or something to
thi s extent. Otherwise I will press you to fix all wrong or unsuported
claims (and you know you are making such claims, believing that they
will be undetected, like with "sequence defining a tangent vector".
Believe me, I will detect all such little sweet swindles.
Best,
ark
Well, needless
to say, Patel must have decided that physicists/mathematicians as a group,
those who are searching for answers, do not deserve to read the "right"
version of his paper... and he wrote back:
Date sent:
Thu, 02 Mar 2000 20:26:42 0800
To: "Arkadiusz Jadczyk"
From: mukulp Subject:
Re: Principia Physica
hi ark,
get lost.
How dare
you badger me as if you are doing me a favor by reading my paper???
i didn't ask you to "process" me. So you can do whatever you want. I
will take my "sweet swindles" directly to intelligent physicists. Actually,
i already have. It is exactly your type of bitter old bastions of mediocrity
that has helped keep physics in the quagmire that it has been for over
a centurey now. Look at you, instead of thinking about physics, you
are boasting about coauthoring obscure books on geometry that no respectable
geometer reads. I am not much impressed by anyone lesser than Gauss,
Riemann, Cartan, Ehressman etc. So it is hard to intimidate me using
your pathetic little book that nobody reads. And your work speaks for
itselfwhich isn't much i must add. I consider it beneath by dignity
to even hold a conversation with you. Do not attempt to write to this
address any more. Because if you do write any more unsolicited mail,
i am going public with all your infringements of protocols, personal
slanders, and other such strongarm tactics.
Note
added October 26, 2001
The statement above about the book "that no respectable geometer
reads" somewhat contradicts a recent post to the newsgroup sci.physics.research
in the "Re:
D vs. E in vacuum" thread (post no 25 in this thread".
Quoting from this post:
From:
Charles Torre (TORRE@cc.usu.edu)
Message 25 in thread
Subject: Re: D vs. E in vacuum
Newsgroups: sci.physics.research
View this article only
Date: 20011021 21:58:44 PST
ark <ark@cassiopaea.com> writes:
> Also earlier: "Electromagnetic permeability of the vacuum
> and the light cone structure"
> A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 9194.
> Online at http://www.quantumfuture.net/quantum_future/emp.htm
> with extra info.
Nice.
I will keep your result showing how Hodge * on 2forms determines
conformal structure in mind next time the issue comes up, which it
does from time to time.
As long
as I am writing, I might mention (changing the subject entirely) that
your series of papers with Coquereaux and the corresponding book (all
from long ago) on KaluzaKlein reductions have been very useful to
us here at USU since we are working on various aspects of symmetry
reduction in gravitational theories. I would highly recommend them
to anyone who is interested in such things.
charlie
It
is always useful to see that different people have different opinions,
but it is even more useful to try to understand a basis for each opinion.
So, what
to make of it all? I don't know. Throughout the exchange, whenever a math
question was asked, which was the main focus, (and in the exchange quoted
above I skipped much of the technical math email exchange between us),
answers between mathematicians are given as a courtesy  and this is standard
 the answer always seemed to twist around to "why do you want to know?"
or "who ARE you that you want to know?" etc. And then, when called on
these antics, both Patel and Parker jump into the highly defensive insult
mode. I just simply could not get a straight answer about the math questions,
nor could I get a reason for why the "wrong" paper was posted for the
public, and the "right" one was reserved for "a group of physicists."
His attitude
toward hardworking physicists who haven't sold their scientific principles
is that they are "bitter old bastions of mediocrity that has helped
keep physics in the quagmire that it has been for over a centurey now."
Everyone
knows that there are many, MANY physicists who are working day and night,
wading through mountains of papers, theories, and the quagmire of disinformation,
to which Mr. Mukul Patel has now contributed so willingly by posting a
version of a paper that he also freely admits is full of errors, but "so
what?" In point of fact, based on all of the above, how can we know
that the "right" version of the paper is even "right?"
Well, I have
long thought about the idea that there IS a group of physicists who DO
have some REAL answers... of course, that is mere speculation. Well, I
guess you can draw your own conclusions.
Some of
the technical exchange can be found here.
At my last
attempt I tried to find out about a missing term in an important formula.
Here what I got:
Date
sent: Thu, 02 Mar 2000 20:56:30 0800
To: "Arkadiusz Jadczyk"
From:
"Arkadiusz Jadczyk"
(by way of mukulp )
Subject: Re: Principia Physica
The following message is undelieverable due to a filter set on this
server.

 
Hi Mukulp, I got a problem with the following: On p. 10 you write
"Here, G is the Riemann curvature tensor field of the metric connection
\gamma" But what happened to \delta\wedge\delta term which is also
in Gothic g? ark
Mukul
Patel must have set up a "filter", and does not want to
discuss missing terms.... Perhaps someone else will also find this
missing term?

Date
sent: Thu, 23 Mar 2000 17:30:34 0600
From: Reza Sarhangi
Subject: info/inquiry
Sirs, Two weeks ago a research paper appeared on Los Alamos eprint
server xxx.lanl.gov No: grqc/0002012..................
Dr. Jadczyk, Mr. Mukul Patel was an adjunct faculty in our department.
Currently he does not have any contract with our department or with
the college. Regards, Reza Sarhangi, Ph.D. Chair, Mathematics and
Computer Science Southwestern College 100 College Street Winfield,
KS 67156
For
the Expert:
At
this place I am making some comments that address some the technicalities
of the paper.
Recently
there have been many discussion of torsion theories (EinsteinCartan,Shipov,
....), and we are dealing here with torsion, so some readers may like
to have some help in assessing some of the geometrical ideas presented
in the paper.
In
the email exchange quoted above Mukul states " this affine connection
has two components, the linear componet and a "rotational" component
(corresponding to the semidirect product of the affine group in to the
linear gorup and R^4). The interesting thing is that this rotational
aspect turns out to be the "canonical form" which is equivalent to a
tensor field on the base space, which is given by the identity transformation
on each tangent space."
There
is nothing unusual in the above. It is a standard knowledge from textbooks
on differential geometry (KobayashiNomizu
for example) that a theory with torsion can be equivalently presented
as "gauge theory of the affine group". It is also known that
the translational part of so obtained affine connection is then nothing
else than the canonical form (KobayashiNomizu,
Vol 1, Ch III.3, Proposition 3.1), and that translational part of the
curvature coincides with torsion Proposition 3.4). So presenting the
theory with torsion as as a gauge theory of he affine group living on
the bundle of affine frames adds nothing new. I believe that Mukul is
unaware of the vast literature on this subject (It is true that many
of the relevant papers were written by physicists, and Mukul, as he
frankly admits, is reading only Cartan, Riemann, Ehresmann and Gauss.
If that is true he does not even know KobayashiNomizu
standard monograph). In fact, KobayashiNomizu
(Vol 1, Ch. III.3, p. 127) discusses also "generalized affine connections",
when the translational part of the connection form does not necessarily
comes from the canonical form. WE do not know why Mukul neglects this
possibility....
This
being said let me add also the following: one CAN go beyond the above
scheme by "unsoldering" the principal bundle from the bundle
of affine frames. In such a case one gets indeed interesting new possibilities.
In particular the famous EinsteinRosen bridge is a nonsingular configuartion
in such a theory. Part of this program has been discussed in a classical
paper by K. Pilch ( "Geometrical meaning of the Poincare group
gauge theory", Lett. Math. Phys. 4 (1980) p. 4951), another part
in my
own paper "Vanishing Vierbein in gauge theories of gravitation",
available on this web site. This generalization allows for "phase
transitions" (changes of signature) of the metric. This interesting
idea belongs probably to A. Sakharov, and gets recently more and more
attention (Prof. R. Jackiv brought recently to my attention his own
paper "Gauge theories for gravity on a line", while F. W.
Hehl kindly informed me about his own ideas about "Avoiding
degenerate coframes in an affine gauge approach to quantum gravity").
Links to my own papers dealing with multidimensional gravitation can
be found on my KaluzaKlein
page.
Physicists
usually assume that there is a metric tensor. Mukul claims that metric
tensor is, by itself, nonmeasurable, that only curvature is measurable.
The standard wisdom of general relativity is that nine of ten components
of the metric tensor are measurable. Whether this is indeed the case
is disputable, but Mukul does not address this standard wisdom at all.
His theory is a theory with metric tensor, or with "auxiliary metric
tensor." Again, a pure affine theory is not a new invention. J.
Kijowski has published a couple of papers on pure affine unified field
theory. The main idea is to construct the Lagrangian density from the
determinant of the (nonsymmetric) Ricci tensor, and DEFINE metric as
canonically conjugate object to the Ricci tensor (therefore metric,
in this approach, is also nonsymmetric).
Mukul's
paper starts with assuming that, for every two points (x,y) and a sufficiently
small neighborhood U of x there is a map f that maps U onto a neighborhood
V of y. Unfortunately Mukul is not using a precise mathematical terminology,
so we can not really know what are his assumptions. But, I guess, he
is resoning as follows: suppose we have an affine connection, then if
things are not too weird we can connect x and y by a unique geodesic
gamma. Then a neighbourhood of x is mapped, via exponential map (see
KN, Ch. III.8, Proposition 8.2), onto a neighborhood of zero vectors
in the tangent space Tx at x. We can also use the exponential map to
identify V with a neighborgood of zero vector of the tangent space Ty
at y. But now, by parallel transport (KN, Ch.II.4, p. 130) we have
affine map between Tx and Ty. If our connection is complete (KN, Ch.
III.6), then composing: inverse exponentila map at x with parallel transport
affine map with exponential map at y, we get map f from U to V. This
is, I guess, what Mukul has in mind.
Now,
what I wrote below (in italics) is wrong .... So, do not read it. I
have to think for a little while how to change it, so that will be correct....
The
map such constructed maps geodesics through x into geodesics through
y. If our connection is more general than admitted by Mukul,
(that is when the translational part of the connection is a general
11 tensor, then this last property (that f maps geodesics through
x into geodesics through y) does not have to hold. This follows from
the fact exponential map maps straight lines through zero into geodesics
if we have affine connction, but not necessarily so if have generalized
affine connection.
I keep
it here, so that readers can see that it is a natural process in creating
math that one has a clear idea of how to make things right, but making
them really right takes some time, some thinking, some computation,
and sometimes a complete turn .... It is only natural that when working
with math one makes mistakes. What is important is to be willing to
DISCUSS all possible errors, and all possible improvements. My error
in the reasoning above is caused by the fact that for a while I have
forgotten that affine map sends zero vector into nonzero vector. Therefore
geodesics through x is mapped into straight line in Ty, but not through
zero. Therefore exponential map will not, in general, send it into a
geodesic through y. By the way, instead of using the term "geodesic
line" I should use the term "autoparallel", because the
term "geodesics" is usually reserved for SHORTEST lines. But
we do not have metric to measure the distance!
.....
After sleeping
over the problem, I give up. I do not know how to make this "affine
business" to work. It causes other problems too. Mukul's formula
(1.1) makes no
sense in affine category. I do not know how he extends his omega to
tensors. Integral involves addition, and addition is not defined in
affine category, only subtraction. So, I am able to make some sense
only when we take the linear part of the connection (the translational
part is anyhow given by the canonical form, so there is no additional
information there ....)
Before
going further on, let me explain what is this "affine" extra
translation that causes problems: suppose we have a connection (some
will call linear connection, some will call it affine connection). Suppose
we have two points, x and y and a path C(t) connecting x and and y:
C(t0)=x, C(t1)=y. Then, by parallel transport we have linear isomorphism,
say Fc: Tx > Ty. Take now a frame e at x, and transport it paralelly
along C. Let e(t) be paralelly transported frame. Let v(t) be the tangent
vector to C at t. Let v[i](t) be components of v(t) with respect to
e(t). Let a[i] (t) be the integral of a[i](s)ds from s=t0 to s=t1. Then,
using frame e(t1), a[i](t1) defines tangent vector in Ty. This is the
extra affine shift. Notice that if C is a geodesic then v[i](t) is a
constant function, and so the affine shift is equal to a multiple of
the tangent vector. For what KobayshiNomizu calls generalized affine
connection, calculation of the affine shift would involve application
of the 11 tensor (of the translational part of the connection) to the
tangent vector first.
Let me
move now to Mukul's formula (1.1):
I will
assume his omega is linear rather than affine. Question is: which volume
form to take? If the formula is valid for one volume form, it will not
be valid for another. He mentions "class of permissible fields",
but we do not know what he means. His condition (1.1) is a strong condition,
it is an integral equation. It imposes some kind of "harmonicity"
on "physical fields". One possibility is that his connection
admits a parallel volume form, then this form is unique up to a constant
scalar, but even then we would have a dependence on this scalar. There
is one possibility to make Mukul's equation invariant, namely to assume
that the connection defines parallel transport not of vectors, but of
vector densities.... But even then we have a problem with (1.1) ...
So I do not know.... It would take me a couple of hours to think of
a possible way of giving a meaning to Mukul's formula (1,1).
After having
a one hour stroll, the following two ideas came to my mind:
1) Perhaps
in his formula (1.1) Mukul wants to take "average" rather
than "sum". That is, he would extend his integral over some
open neighborhood V, divide it by the volume of V, and then take the
limit of V going to infinity, assuming the limit exists. In this way
the formula would be independent of a constant scalar factor, and it
can be considered as "balance formula" or "equilibrium
formula", when "external" sources are not present.
Or, possibly,
he assumes that spacetime is compact (R4 plus lightcone at infinity?),
and then he assumes that the volume form is normalized. But, even thenm
this will work only if the volume form is unique up to a constant. So
that we do not have to divide by the volume of V. Do we need to assume
that our affine connection has a parallel volume form? We will come
to this question later on when discussing curvature 2form. Is it assumed
to be traceless? Notice that once we have a connection with torsion,
we have a whole 1parameter family of connections (as we can add a scalar
multiple of the torsion)  thus we have some freedom of manipulation....
2) If we
have a map f: U>V, f(x)=y, then the first jet of f defines a map
from Tx to Ty. This is how Mukul proposes to get a connection. But we
can continue. We can take second jet of f. This would map second order
frames at x to second order frames at y. In particular this would map
connection at x to connection at y. If so defined connection at y happens
to be the same as the connection obtained from first order jets, then
it would impose strong selfconsistency condition on the family of maps
f. Whether Mukul takes it into account or not  I do not know....
You may
ask why I am so preoccupied with this paper, which is so "USER
UNFRIENDLY"? Well, first, it fits my field of professional interest
and it touches many points that I was thinking about through the years.
On the other hand it is a fun to be "filtered out"  so that
I am forbidden to talk directly to Mukul or Phil Parker. Strange, strange
are these mathematicians.... Fortunately we have Internet ... so I can
talk to everybody who is interested and willing to use search engines.
To
be continued... maybe.....
