From: ark (arkatcassiopaea.com)
 Subject: Re: W*-Category 
 Newsgroups: sci.physics.research
 Date: 2001-10-08 15:10:06 PST 

On Wed, 26 Sep 2001 06:19:10 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>This is also an excellent book in itself! - but if it's too hard,
>just read the introduction and then jump to the bibliography.
>The introduction will help answer your question about why this
>stuff is important in physics. The one-line answer is: applications
>of categories to quantum theory often use C*-categories.

I would add to thi that W* category became especially important with
the discovery of Tomita-Takesaki theorem - it deals with von Neumann
algebras with a cyclic and separating vector.
In physics it was first related to the Kubo-martin-Schwinge condition
for thermal equilibrium, then it has found more potential
applications. BTW it also relates to the geometry of quantum states,
including Berry phase.

There are many textbooks available, some are less known than other,
and yet very good.

Among those less known I would recommend S. Stratila "Lectures on von
Neumann algebras" and Li Bing-Ren "Introduction to operator algebras"

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Clifford's geometry algebra
Newsgroups: sci.physics.research
Date: 2001-10-08 15:16:12 PST

On Sun, 7 Oct 2001 15:58:24 GMT, "David Corfield"
<david.corfieldatkcl.ac.uk> wrote:

>One of my interests in this is that a PhD student I'm supervising is working
>on the relationship between Clifford's math and his philosophy. It would
>make his thesis all the more striking the greater the latent power in
>Clifford's ideas. In any case, interesting philosophies deserve good ideas.

It seems to me that, as in many other areas, we are dealing here with
a particular lobby.
It is tru that Clifford algebras are useful, also in physics. It is
also true that there are amny results that you can get using
Clifford algebras, which can be also obtained, at a much
smaller cost, by using other algebraic and geometric techniques.

Let me give an example: W. Rodrigues writes his papers using
Clifford algebras - so that most of the volume of the paper deals
with the formal structures that are needed to present the result
in this particular language.
But, he often adds a sentence telling the reader that the particular
result can be as well obtained and expressed in a standard
language of differential forms (with co-differential, Laplace-Beltarmi
etc.).

In theories of gravity, especially those where gravity results from
some more fundamental structures, we need to tacle the problem
of "degenerate metric" or "degenerate soldering form." These theories
can hardly be based on a Clifford algebra. Although there are
generalization of Clifford algebra techniques to the case where
inner product can be degenerate, these do not belong to
the standard courses.

Clifford algebra is a particular structure and it offers a particular
computational technique.

What is perhaps interesting from a philosophical point of view is
the famous modulo 8 periodicity - which may somehow mirror something
very deep and fundamental in the structure of the universe....

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Clifford's geometry algebra
Newsgroups: sci.physics.research
Date: 2001-10-08 18:33:31 PST

On Sun, 7 Oct 2001 15:58:24 GMT, "David Corfield"
<david.corfieldatkcl.ac.uk> wrote:

>Let's see if this one works.
>
>Does anyone have any thoughts on the claims of the group at
>www.mrao.cam.ac.uk/~clifford that a huge opportunity has been overlooked
>for decades.

(snip)

Adding to comments from other people: Clifford algebras are also of
importance in quantum theory. Algebra of canonical anticommutation
relations (CAR) generated by fermionic creation-annihilation operators
is a Clifford algebra. In quantum theory we hace to deal also
with fermionic systems with infinitely many degrees of freedom - thus
we need infinite dimensional Clifford algebras. To deal with this
subject we have to invoke the theory of C* and their representations,
thus also W* algebras - which connects to another thread on this
group.

When dealing with systems that need indefinite metric (usually we
try to avoid it in quantum theory) - we have Clifford algebras with
indefinite scalar product. We move to more general category
or B* algebras (B for Banach).

ark

From: ark (arkatcassiopaea.com)
Subject: Re: string theory & Berry phase; quantum bundles
Newsgroups: sci.physics.research
Date: 2001-10-09 17:57:22 PST

On 8 Oct 2001 06:38:17 GMT, abergmanatPrinceton.EDU (Aaron J. Bergman)
wrote:

>>A quaternionic geometric phase could e.g. be associated with a rotation
>>subgroup of the Lorentz group (as opposed to the complex phase on
>>projective space), but I am not sure what this would mean.
>
>I'm not sure why it would be useful.

The same with me. Read Adler's book on QQM and also old papers
by Jauch, Emch, ....

Quaternionic Quantum Mechanics serves a different purpose than complex
one. We are not sure which purpose exactly. It is a good scheme from
the point of view of quantum logic and noncommutative probability.
It is not so good from the point of view of dynamics.
In complex QM there is one "i", and it stands in front of one Planck
constant. It serves as a crucial connection between group generators
(which are anti-Hermitian) and observables (which are Hermitian).
A generator of time translation, for instance, is anti-Hermitian, and
it "happens to be" iH/hbar, where H is energey observable. Thus
"energy conservation" etc. In Quaternionic QM we do not really know
what to do in similar cases. Should time be thre-dimensional (see
Tifft and Lehto) so as to connect to three imaginary units? If so,
then what?
There is also another problem in QQM - which stems from the fact that
quaternions do not commute: we ca not easily build composed systems
out of simple ones - because of the problem with tensor product of
quaternionic vector spaces.

I am not saying the there is no future for QQM. I am saying that I do
not know yet what this future can possibly be - because of the
problems sketched above.

One more comment: it is possible to take another approach to
Berry phaes (originally due to Armin Uhlmann). In this other
approach you consider parallel transport along density matrices
rather than along pure states. In this approach Berry phase has
values in unitary operators, not just a complex phase.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: People in the Group/Backgrounds
Newsgroups: sci.physics.research
Date: 2001-10-11 10:45:52 PST

On 10 Oct 2001 01:04:00 GMT, baezatgalaxy.ucr.edu (John Baez) wrote:

>On the other hand, studying at a university also has its problems!
>Besides costing a lot, the main problem is that you can easily fall
>into the trap of believing that your teachers know everything you
>need to learn, and that all you need to do is go to class and pass
>tests, and everything will be fine.
>
>In short: there are problems with getting degrees from universities
>and problems with studying on your own, so the only solution is to
>do BOTH.

I would add to the above one more thing: as soon as possible, while
studying, do all you can so as to be able to go to as many conferences
and seminars with guest scientists as possible. Do not be shy!!!
You must overcome your shyness. Very important. When you have some
idea or some question - approach speakers during
a break, or a lunch, or a party, and ask your questions. Do you
homework during the night, and next day ask the followup..

This makes a real differerence. Sometimes much of your future
work depends on a single exchange with just one bright scientist!

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Kaluza-Klein theories
Newsgroups: sci.physics.research
Date: 2001-10-12 14:33:41 PST

On 10 Oct 2001 07:17:24 -0700, mcartoajatmat.ulaval.ca (Mihai Cartoaje)
wrote:

>I would like to know what the current state of non-quantum
>Kaluza-Klein theories are. I know that using them one can predict the
>four forces. Have there been other fields predicted like the
>Klein-Gordon or Dirac fields?

Part of the mathematical formalism related to this question is
in Chapter seven of "Riemannian geometry, fiber bundles and
Kaluza-Klein theories." (World Scientific, I am one of the authors).
This chapter deals with "Harmonic analysis and dimensional reduction."

However, the starting assumptions in this book are much too
restrictive (and thus not realistic as far as physics is concerned)
For an alternative approach, not quite in the line
of mainstream phsyics, and yet, in my opinion, interesting, search
the net for papers by P.S. Wesson.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Kaluza-Klein theories
Newsgroups: sci.physics.research
Date: 2001-10-15 21:20:04 PST

On 14 Oct 2001 20:51:41 GMT, "alon" <newsatbezeqint.net> wrote:

>why isnt P.S Wesson approach in the mainstream of physics?

Perhaps I will withdraw this statement. The definition of "mainstream
physics" will always be debatable.

Instead let me quote from P.S. Wesson paper: "String theory is
'promising ...,' one worker said, '... and promising, and promising'
[116]. M-theory which (unlike superstring theory) is not
perturbative, is even more opaque (...)".

Another quote: "(...) we do not assume compactification of the extra
dimension or assume it space-like."

ark

From: Arkadiusz Jadczyk (arkatcassiopaea.com)
Subject: Re: Kaluza-Klein theories
Newsgroups: sci.physics.research
Date: 2001-11-30 16:57:57 PST

On Thu, 29 Nov 2001 05:15:22 +0000 (UTC), "Danny Ross Lunsford"
<antimatter33atworldnet.att.net> wrote:

>> 2) Through each point in 5d space passes only one geodesic
>
>this should read
>
>> 2) Through each point in 5d space passes only one geodesic
>> that returns to the same point in the same direction as it left it
>
>This is part of the "projective" assumption.
>
>-ross

Sorry, I posted my message before reading your correction.

In 1938 Einstein and Bermann generalized KK assumptions:
"we ascribe physical reality to the fifth dimension whereas in Kaluza's
theory this fifth dimension was introduced only in order to obtain new
components of the metric tensor representing the electromagnetic field."

"We have, therefore to take the fifth dimension seriously although we
are not encouraged to do so by plain experience."

"The most essential point of our theory is the replacing of the
hypothesis 2, of rigorous cylindricity by the assumption that space is
closed (or periodic) in the x0- direction ."

x0 is the fifth dimension in their notation.

See Ref. 43 in
http://xxx.lanl.gov/abs/gr-qc/9805018

ark

From: Arkadiusz Jadczyk (arkatcassiopaea.com)
Subject: Re: Kaluza-Klein theories
Newsgroups: sci.physics.research
Date: 2001-11-30 16:57:28 PST

On Thu, 29 Nov 2001 01:55:31 GMT, Danny Ross Lunsford
<antimatter33atworldnet.att.net> wrote:

>For this reason Klein sought to abandon the cylinder condition. Because
>the correct transformation properties of gm5 ~ Am (they have to look
>like 4d gauge transformations) depends to some extent on the cylinder
>condition, it turns out that the best you can do to preserve this
>essential feature is to make the assumptions
>
>1) The dependence on x5 is angular (i.e. x5 is compactified)
>2) Through each point in 5d space passes only one geodesic

Are you sure 2), as written above, experesses what you have in mind?

Through each point in 5d space passes infinitely many geodesics.

Perhaps what you tried to say is that 2) the fibers are closed
geodesics???

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac theory on the KG kernel (?)
Newsgroups: sci.physics.research
Date: 2001-10-09 21:24:22 PST

On Mon, 8 Oct 2001 20:59:48 +0200, "Urs Schreiber"
<Urs.Schreiberatuni-essen.de> wrote:

>Given an n-dimensional pseudo-Riemannian manifold (M,g).
>
>Let
>
> K |psi> = 0
>
>be a Klein-Gordon-type equation for a particle propagating on (M,g)
>and let
>
> D |psi> = 0
>
>be the corresponding Dirac equation, where the generalized Dirac
>operator D satisfies
>
> D o D = K

I would be careful here. On a curved space square of the Dirac
operator is the "spinor Laplacian", not the ordinary one. You get an
extra 1(1/4) scalar curvature term. It makes real difference when your
manifold is not of constant curvature.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac theory on the KG kernel (?)
Newsgroups: sci.physics.research
Date: 2001-10-10 19:54:12 PST

On Wed, 10 Oct 2001 09:46:43 +0200, "Urs Schreiber"
<Urs.Schreiberatuni-essen.de> wrote:

>Not if one uses e.g. an "N=2" Dirac operator where the curvature
>terms of both "handednesses" cancel mutually. The simplest example is
>the Dirac operator on the exterior bundle given by
>
> D = d + d*
>
>where d and d* are the exterior and coexterior derivative,
>respectively.

When trying to compare kernels of two operators, both operators
must act on the same space.

Which space? 0-forms? On zero-forms d+d* acts as d alone. Thus its
kernel consists of constant function - whatever the metric is.

Not very interesting.

If not zero forms, then what else? Square of d+d* gives
Laplace-Beltrami operator, not the Laplace operator - the two differ
by curvature terms on forms of higher order.

Therefore, I would insist that one should try, first, to specify the
question: which exactly operators? Which exactly space they are
supposed act upon.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac theory on the KG kernel (?)
Newsgroups: sci.physics.research
Date: 2001-10-12 14:31:35 PST

On Thu, 11 Oct 2001 16:34:49 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>>Which space?
>
>The space of all differential forms. That's what Schreiber
>meant by "on the exterior bundle".

All right. My fault. When he wrote "operator of Klein-Gordon type",
I did not get that what he really means is the Laplace-Beltrami
operator. (If that is what he really meant?)

Now, space of all differential form, when the metric is
pseudo-Riemannian, is an indefinite metric space, not a Hilbert space.
One needs to be careful with spectral decomposition here indeed.
Years ago I wrote a paper dealing with the peculiarities of spectral
decomposition of selfadjoint operators on Krein spaces -
but it was about bounded operators. So it will not be useful here.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: General covariance, background
independence
Newsgroups: sci.physics.research
Date: 2001-10-15 21:24:19 PST

On 12 Oct 2001 21:38:56 GMT, baezatgalaxy.ucr.edu (John Baez) wrote:

>In article <%Esw7.53277$3d2.2474513atbgtnsc06-news.ops.worldnet.att.net>,
>AG <solgravitasatatt.net> wrote:
>>Does "general covariance" mean "expressible as tensor equations?"
>Did I just hear someone open a can of worms?
>
>The term "general covariance" has a long and convoluted history,
>since it goes back at least to Einstein's early work, where he
>had strong intuitions about what made general relativity different
>from all previous theories, but said some misleading stuff, because
>he hadn't yet figured everything out.
>
>Taken literally, "general covariance" means covariance under all
>coordinate transformations. A modern translation would be "diffeomorphism
>covariance".

I would like to add to the above just few comments. First is that it
can also mean: "use only NATURAL operations." Of course the question
is which are natural operations? For this I recommend the book by
Kolar, Michor and Slovak "Natural operations in differntial geometry."
Second, when you peruse the book, you will notice that there is no
chapter on spinors. Are not spinors natural objects too? Is not Dirac
operator a natural operator? And here one needs to be careful.
Diffeomorphism invariance of spinors is trcky - to say the least.

But I am not going to open the can of worms even more....

ark

From: ark (arkatcassiopaea.com)
Subject: Re: D vs. E in vacuum
Newsgroups: sci.physics.research
Date: 2001-10-19 13:32:49 PST

On 18 Oct 2001 00:56:34 GMT, "Eric Forgy" <forgyatuiuc.edu> wrote:

>After all that mess, the short answer to the question "Is D a pseudo
>vector?" is apparent. Although D is a 2-form, that does not mean that D is a
>pseudo vector because D is the Hodge dual of a 1-form E and the HODGE STAR
>PICKS UP AN ADDITIONAL SIGN UNDER A PARITY TRANSFORMATION (in this case, but
>not in general).
>
>Thanks for a nice question! It made me think. I hope my answer makes sense.

One observation that is not strictly related to the original equation,
but, somehow, is relevant:

The relation between D and E, as well as the relation between H and B,
in vacuum, involves Hodge operator on 2-forms. Hodge operators on
2-forms in 4d (as well as hodge operators on n/w forms in nd) is
conformally invariant. Thsi is an easy exercise. But it less known
that Hodge * operator on 2-forms in (-+++) signature , with the
properies *^2=-1 and Hermitian (with respect to the natural inner
product) is sufficient to define uniquely the light cone (=
conformal) structure of space time.

ark

From: Charles Torre (TORREatcc.usu.edu)
Subject: Re: D vs. E in vacuum
Newsgroups: sci.physics.research
Date: 2001-10-21 21:58:44 PST

ark <arkatcassiopaea.com> writes:
> Also earlier: "Electromagnetic permeability of the vacuum and the
> light cone structure"
> A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94.
> Online at http://www.quantumfuture.net/quantum_future/emp.htm with extra
> info.
>

Nice. I will keep your result showing how Hodge * on 2-forms 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 Kaluza-Klein 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

From: ark (arkatcassiopaea.com)
Subject: Re: position operator
Newsgroups: sci.physics.research
Date: 2001-10-23 22:46:28 PST

On Tue, 23 Oct 2001 18:54:05 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>In article <vu_A7.38581$ev2.45505atwww.newsranger.com>,
>pollux <pollux_000atyahoo.com> wrote:
>>In page 24 of Peskin/Schroeder, they say that the quantized Klein-
>>Gordon field \phi(x) (here x and p denote the spatial part of the
>>corresponding 4-vectors in a fixed frame) acting on the vacuum
>>state creates a particle at position x.
>Really? That's a sort of odd thing to say - if this were true,
>it would be very hard to understand why the inner product of phi(x)|0>
>and phi(y)|0> is nonzero [....]

(snip)

>People have studied other candidates as well. The whole exercise
>is of very limited use in practical particle physics, but it's sort
>of fun to figure this stuff out.

I would add to the above one comment. Operationally, there is a
difference between "creating a particle" or "preparing a particle
state" at (x,t) and "detecting a particle at (x,t)". The "measurement"
issue is a complex one. If one studies the issues, one finds that
standard prescriptions (or axioms) of QM (Born's interpretation,
transition probabilities") are valid only as approximations.

As for Newton-Wigner analysis - yes, it is really fun to follow their
argument, however their assumptions are too strong. They
assume projection valued measure, while there is no reason
for it. It could be POV (positive-operator-valued) as well,
and probably it is, and then their reasoning does not apply.

But in old days POV measure were not that popular as they
are today.

P.S. There is a rather popular myth that all "observables" in quantum
theory are necessarily represented by Hermitian linear operators. To
see that it isn't necessarily so check the discussion of "time of
arrival" in Helv. Phys. Acta 69 (1996) 613-636, available also online:

http://www.quantumfuture.net/quantum_future/papers/time/time.html

That is how I learned my lesson....

ark

From: ark (arkatcassiopaea.com)
Subject: Re: EF theory
Newsgroups: sci.physics.research
Date: 2001-11-01 19:43:12 PST

On Thu, 1 Nov 2001 20:31:28 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>Consider EF theory on a G-bundle P -> M over an orientable
>smooth manifold M of dimension n, without cosmological constant
>term. The Lagrangian is
>
>tr(E ^ F)
>
>where E is an ad(P)-valued (n-2)-form and F is the curvature
>of the connection A on P.
>
>1) What are the classical equations of motion?

Perhaps you guys know it already, but this question can be answered
using Souriau's method. I did similar calculations for point particles
and strings with charges valued in Lie(G)* (Higgs charge), I could
try it with EF setting. It should be specified whether we are looking
for point particles or for strings. For strings, of course, classical
equations of motion can not be specified until we postulate the
necessary internal structure ("tension" tensor).

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Time travel
Newsgroups: sci.physics.research
Date: 2001-11-06 14:42:09 PST

On 6 Nov 2001 02:55:26 GMT, Moataz Emam <emamatphysics.umass.edu>
wrote:

>Physics looks at both space and time as one inseparable structure, a geometric
>manifold. So, in that sense, the past and the future do exist, because the
>manifold exists as a whole.

Probably you meant "Einstein theory of relativity" rather than
"physics". Einstein theory of relativity is just one particular
theory, and will probablu be superceeded by another theory.
Quantum theory, for instance, looks at space and time as
separate structures. Some physicists keep it in mind that Lorentz
group symmetry may be violated. Only experiment can tell.

ark

ubject: Re: Unification of Electromagnetism and Gravity
Newsgroups: sci.physics.research
Date: 2001-11-07 19:04:02 PST

On Tue, 6 Nov 2001 20:35:34 +0000 (UTC), Grace Shellac
<furlong.physics.holidayatsingtech.com> wrote:

>I've been under the impression that gravity and electromagnetism
>actually hasn't been unified. I asked Jerrold Marsden down at CalTech
>if this was true.
>
>He responded: "...there is a well defined set of equations called the
>Einstein-Maxwell equations that already unifies E and M with
>gravity. They are discussed in standard books on general relativity."
>
>"Thus, I would say that most researchers believe that these theories
>are already unified. The big question is how to unify quantum theory
>with gravity."
>
>So, what is the deal here. Am I misinformed?

No, you are not misinformed. You got a direct information about one
particular point of view. To have a full picture you need to read
what other physicists think of this problem. There are at least three
other ways.

> Or is Prof. Marsden
>mistaken?

Prof Marsden represents here a conservative trend, I would say,
and a short description of one vesrsion of "already unified"
theory (originated by Rainich) you can find in "Gravitation" by Misner
et al.

> Doing a google.com search didn't seem to help as there
>doesn't seem to be a consensus that gravity and electromagnetism are or
>have been satisfactorily unified. I see web pages by people like
>Sweetster and others but no definitive comment that indicates that
>anyone actually thinks gravity and electromagnetism have been unified.

See above. Alternatives are:

a) try to describe gravitation as a function of electromagnetic
phenomena
b) try to describe electromagnetism as a function of gravitational
phenomena
c) try to derive both from more fundamental fields
d) try to derive both by making use of alternative geometries
(Finsler, Einstein-Cartan, gauge theories, more than four space-time
dimensions, noncommutative geometry etc
e) try to derive Theory of Everything, and EM+G will come
in a big bag of all kinds of goodies.

>Help! What's the latest opinion?

Different physicists will have different opinions. There is no
consensus. Marsden is certainly right that we have also
problems with adding quantum theory - and here you will find not so
many physicists that will tell you that there are no problems there,

The above is simply understanding of the present situation, based on
my own research experience, not an autoritative answer.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Time operator
Newsgroups: sci.physics.research
Date: 2001-11-07 23:59:46 PST

On Wed, 7 Nov 2001 22:32:39 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>>The loop quantum
>>gravity approach seems to take both views into account also, although I
>>understand that they have a problem with how to define a time quantum
>>operator.
>
>What's a "time quantum operator"? You mean an operator that
>corresponds to the observable we hope we're measuring when we
>look at a clock? More precisely, an operator T canonically
>conjugate to the energy H:
>
>[H, T] = i hbar ?
>
>The lack of such an operator is a quite general feature of
>quantum theory, nothing special about loop quantum gravity.
>String theory doesn't have such an operator either. It's not
>a problem.

When discussing quantum-mechanical observables it is useful
to make a distinction between Scroedinger and Heisenberg
picture. Relativistic local quantum filed theory teaches us that it
usually less confusing to use Heisenberg picture. Having this
in mind, there is no such thing as a "position operator x of a
particle". But, for each time t, there is a position operator x(t) for
a particle at time t. Usually operators x(t1) and x(t2) do not
commute!
Similarly, even if only conceptually, it makes no sense to speak
about "time operator in general". But we can think of "time operator
t(x) for a particle to arrive at x". It makes no sense to think of
"time in itself, or "time of a clock" as observable, because each
observable needs a prescription of how to "measure it". Ther is
no prescription as to how to measure time. Time in the morning,
or time in the evening? Time at 6AM or time at 5PM? It makes
no even sense.
Nevertheless, without much thinking, just for curiosity, people
were looking for an operator "T" satisfying [H, T] = i hbar and
indeed soon they found that it can't exist as a regular operator,
mainly because energy is usually bounded from one side, and
also because energy spectrum is not continuously homogeneus.
Clever formal substitutes for [H, T] = i hbar have been invented
in the past - just to find a solution
From the physical point of view a well defined observable for a single
particle is "time of arrival at a point" or "time of crossing of a
surface".
The subject and different versions of these observables, and their
relation to measurements of "tunnelling time" are discussed in
the current literature. The complication here is that many particles
will never arrive at x, thus we need to "normalize the probability",
the result being that such a time of arrival is described by
a nonlinear operator.

Additional complications arise when quantum mechanics
is treated relativistically or on a curved background.

Indeed, as John explained, all that has nothing to do with
loop quantum gravity - rather it has to do with clearly defining
meaning of observables.

One more comment is important. Quantum mechanics is usually
about "particles". But it need not be so. Suppose instead of
thinking about quantum mechanics of particles, we are conteplating
quantum mechanics of extended objects - for instance wave fronts.
In THAT case it makes sense to ask: "at which time a given wave front,
will arrive at x. We will get answer to this experimental question
almost always. Thus, in this case, we expect that for each each
time-like geodesic gamma there will be a linear time operator t(gamma)
describing time at which observer described by gamma will register
arrival of the wave front.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Lorentz invariace of inner products
Newsgroups: sci.physics.research
Date: 2001-10-29 17:36:06 PST

[Moderator's note: I screwed up, so it seemed that this post was
written by me, when in fact it was by "ark" - jb]

On 21 Oct 2001 02:38:57 GMT, Iuval clejan <clejanatmindspring.com>
wrote:

>The thing that bothers me is
>that under a passive boost (change of reference frames) this spinor
>does not transform in a unitary manner, for example it does not
>preserve its norm

One thing I would like to add to what John already wrote:
"who sets the norm?"

You write about "norm" - but this norm is an artifficial
construct in the framework of finite dimensional representations
of the Lorentz group. On the other hand, you will usually find that
there IS a "norm" that conserved - which comes from indefinite
metric (i.e. scalar product that is invariant but not positive
definite). Perhaps this "norm" has some physical meaning?
Like for instance in Minkowski space.....

Normally we think that there is a little if any relation between
the fact that Minkowski space scalar product is indefinite, and
the fact that quantum theory is noncommutative. But, note,
that because of the indefinite scalar product, the double cones
in Minkowski space (with natural intersection and union operations)
form a complete orthomodular lattice, a "quantum logic" that is
non-Boolean, it is not even modular.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Lorentz invariace of inner products
Newsgroups: sci.physics.research
Date: 2001-11-08 11:34:20 PST

On Tue, 06 Nov 2001 20:32:56 -0500, Iuval clejan
<clejanatmindspring.com> wrote:

>Now here we have a finite dimensional irreducible unitary
>representation of a non compact group (Lorentz group plus parity)????
>Can someone please explain this?
>
>-Iuval

This has already been discussed. John Baez addressed this problem
already. You can find answers in previous
posts on the subject. But let me say it again: standard probabilistic
intepretation of quantum mechanics needs positive definite scalar
product, not just "inner product". What you call "unitary
representation" is indeed unitary but with respect to "indefinite
metric" scalar product : there are vectors with <v|v>=-1. When your
space can be decomposed into a direct sum of positive and negative
definite subspaces, it is called a Krein space. Studying of operators,
group representations, spectral theory on Krein spaces is a separate
branch of mathematics. Probabilistic interpretation of quantum
mechanics needs Hilbert space, not a Krein space.

This being said, I need to add a note: the so called Gupta-Bleuler
quantization of EM field uses indefinite metric. Then a "physical
subspace" is selected, which is semi-definite, by imposing
gauge condition. Then a Hilbert space is cosnstructed by taking
a quotient with respect to the "zero-norm" subspace. But this use
of indefinite metric, although having the same source as in your
example (Minkowski space signature) is not the same what
you are talking about when discussing finite dimensional
"unitary" representations. Second note: as the number of papers
is growing on the subject of "negative probability" - one day
physicists may be back to indefinite metric as in your example,
and quantum mechanical interpretation may be given to
finite dimensional "unitary" representations of the Lorentz
group (possibly in terms of pre-geometry, ur-geometry or whatever),
but this is a phantasy right now.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Gauss and Euclidean Space
Newsgroups: sci.physics.research
Date: 2001-11-07 23:19:52 PST

On Mon, 5 Nov 2001 14:49:30 GMT, "AG" <solgravitasatatt.net> wrote:

>As Henri Poincare said, any attempt to discover by experiment the geometry
>of space is futile: "Measurement is never of space itself, but always of
>empirically given physical objects in space, whether rigid rods or light
>rays. Regarding the structure of space as such, experiment can tell us
>nothing; it can tell us only of the relations that hold among material
>objects." [quoted in M. Jammer/Concepts of space]

Interpretation of every experiment whatsoever is always through some
underlying theory. The very fact that what we are doing constitutes
and experiment is already based on prior assumptions. If the
underlying theory of space and its geometry is such that it allows
experiments to determine some aspects of space geoemetry - then
it is so.

In my opinion Poincare, in the above statement, was exhibiting
the way of thinking of a philosopher-mathematician. Physicists
and engineers have somewhat different mentality.
For many years I was collaborating with Rudolf Haag (one of
the pioneers of algebraic quantum field theory). His first
education was of an engineer. If you read his recent
papers, for instance "Objects, events and localization",
you will realize that he proposes to use experiments to
detect "a granular structure of space."

Reference: R. Haag, "Objects, events and localization",
in "Quantum Future. From Wolta and Como to the Present and Beyond."
Springer, 1999

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Gauss and Euclidean Space
Newsgroups: sci.physics.research
Date: 2001-11-12 13:49:51 PST

On Sat, 10 Nov 2001 22:37:55 GMT, "AG" <agorgunatatt.net> wrote:

>I was unable to interpret this as a proposal to detect the granular
>structure of "space." Is yours a literal quote from the paper? Because he
>only writes "there is some granular structure," he does not say "...granular
>structure of space."

You are right. He does not say "granular structure of space". A short
summary of his position is contained in his other paper "On the
lessons of quantum physics" [Helv. Phys. Acta. 66(1996), 679-681]

"An event is a fact and irreversibility is inherent in this concept.
According to widespread belief the fundamental laws are time-reversal
invariant and irreversibility appears only on the macroscopic level
dues to coarse grained description. This should be seriously
reconsidered in Quantum Physics.(...) we can assert that as soon as we
accept the existence of discrete facts independent of the cognition
by an observer we are forced to an evolutionary picture where in the
progress of time new events are realized and the events are connected
by directed causal ties which in the simplest case, are particles and
when ultimately space-time is related to the pattern of events."

Thus, it seems, he considers space-time as emerging in a limit, not as
a priori given with its geometry.

Perhaps I should also add that in "objects, Events and Localization"
Haag is quoting papers by Ph. an myself on EEQT as "In a series of
papers (Blanchard and Jadczyk 1995) Ph. Blanchard and A. Jadczyk have
described a formalism which generates real event in the interaction of
an atomic object with a macroscopic measuring device.(...) The scheme,
called 'event enhanced quantum theory' introduces irreversible
decisions into the interaction process and yields a good
phenomenological description of the quantum measurement process. It is
related in spirit to points 1 and 5 of the strategy outlined above but
allocates events only to the interaction between small and a large
system retaining the ad hoc distinction between a classical and a
quantum level of the world."

I should explain here that I consider classical-quantum and
small-large as two different divisions. Large can be quantum, and
small can be classical. In our formalism we never use limit of "large
quantum systems." We simply admit a part that is classical, we allow
for the algebra to have a nontrivial center.

Back to Haag and Poincare and whether measurements tell us something
about geometry of space and time and other structures of interest:
While discussing algebraic quantum field theory in 1964 or so, Haag
stressed the importance of weak topology where neighborhoods are
determined by finite sets of measurements.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Phase space of all possible universes
Newsgroups: sci.physics.research
Date: 2001-11-05 18:19:26 PST

On Fri, 26 Oct 2001 17:55:39 +0000 (UTC), p.kinsleratic.ac.uk wrote:

>> Yes, exactly: because the *process* of observation forces the system
>> into an eigenstate. Nobody talks about probabilities in the context of
>> QM when the system under consideration is in an eigenstate. After it is
>> observed, there's no "probability" anywhere. It's in that state and
>> that is all.
>
>An observation forces the system into an eigenstate of the measurement
>operator. A subsequent measurement using a _different_ measurement
>will push the system into an eigenstate of that different operator.

Although this is one of the standard axioms of quantum mechanics,
there is no compelling reason that it must be so. In fact there are
good reasons to believe that this statement is an oversimplification.

First of all there is no reason to associate every observation with a
self-adjoint operator. Second, to even discuss this question within
physics we need a framework within which the very concept of
observation is part of the dynamics. Moreover, since we are talking
here about early stage of the universe, it is clear that "observation"
should not imply an "observer". Wheeler's concept of a Big Eye
observing the early cosmos from the Future is a nice metaphor, but is
not yet part of any formalism that I know about.

So, how can we define an "observation", so that it works also when
there is no "observer?" Following Bohr and Bell (and early Stapp) we
can define it as a coupling between the quantum and the classical
part of the system (the Universe in our case). If we apply now
Lindblad's theory, generalized by Christensen to arbitrary C*
algebras, and if we apply the theory of piecewise deterministic
processes (a typical process governing stock market and its crashes),
we find out that an observation results in "events", and these
events involve "quantum jump". Typically a quantum jump
maps |psi> into A|psi>, but the operator A need not be Hermitean and,
moreover, the resulting state needs not be an eigenstate of some
particular "observable". This happens only when A is a projection
operator on a one-dimensional eigenvector of some "observable"
that defines the observation. It is a very particular, very special
case.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Phase space of all possible universes
Newsgroups: sci.physics.research
Date: 2001-11-10 18:41:05 PST

On Thu, 8 Nov 2001 03:00:04 GMT, "Urs Schreiber"
<Urs.Schreiberatuni-essen.de> wrote:

>Following Zeh et al., however, we can define it as the coupling
>between a quantum system with few degrees of freedom and a "bath" of
>a large number of (also quantum) degrees of freedom, which leads to
>decoherence of the "small" system, mimicking "state reduction". This
>is more than a metaphor and indeed part of a formalism (actually of
>*the* formalism).

Yes, I know. But I disagree. If you couple a quantum system with a few
degrees of freedom with a large number of degrees of freedom - you
still have a quantum system and unitary linear evolution. Nothing will
ever "happen", there will be no wave packet reduction. Things will
happen "in the limit of infinite time", but nothing will ever happen
in finite time. Of course you can try to argue that time large enough
is infinite FAPP. But what is FAPP for one person, in one
circumstances, will not necessarily be FAPP for another person,
in different circumstances. FAPP is subjective, and we are striving
at making physics objective - as much as it can be. Hepp and Araki,
also Namichi, were discussing thse questions long ago.
My point of view here is that the problem above is telling
us something important.

Decoherence formalism is interesting, sure .... FAPP.

Quting from Kiefer: "The most importante feature of decoherence
besides its ubiquity is its irreversibility". [in "Quantum Future",
Ph. Blanchard and A. Jadczyk ed]

You will not get irreversibility in a quantum system, however large,
from a unitary dynamics! You can get it by replacing your system
with another system, obtained in some clever limit. But thsi will
not change the fact that YOUR original system, the one that
obeys unitary evolution, is reversible.

So, what to do? One way is to accept FAPP methods.
Another way is to accept the facts and relax axioms
of QM that are due to Bohr, von Neumann and others.

1) Istead of assuming that the algebra of observables
is L(H) (all bounded operators on H), let us assume that
it is a general C* algbenra, not necessarily simple, possibly
with a nontrivial center (superselection observables)
2) Instead of assumint that time evolution is given by a
one parameter group of aotomorphism, let us assume
that it is given by a semigroup of positive-definite
maps (positivity has a physical meaning, multiplication
of quantum operators does not have such a meaning)

So, we have these two options. Most physicists
follow the first options. But Prigogine, Haag and
some other propose to examine the possibility
of irreversibility on the fundamental level.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Phase space of all possible universes
Newsgroups: sci.physics.research
Date: 2001-11-14 09:52:10 PST

On 13 Nov 2001 04:38:29 GMT, Urs Schreiber
<Urs.Schreiberatuni-essen.de> wrote:

>> But I disagree. If you couple a quantum system with a few
>> degrees of freedom with a large number of degrees of freedom - you
>> still have a quantum system and unitary linear evolution. Nothing will
>> ever "happen", there will be no wave packet reduction. Things will
>> happen "in the limit of infinite time", but nothing will ever happen
>> in finite time.
>
>Since we were talking about cosmology: Do we even need a
>notion of time evolution for decoherence to be of interest and
>useful?
> Doesn't it all boil down to the statement that: "When
>looking at a generic, highly entangled system and ignoring
>most of its degrees of freedom, the remaining degrees of
>freedom will look like a mixture, not like a superposition."?

You see, I disagree here. Once we are talking about cosmology,
it does not matter whether. Nobody is "looking", and it does not
matter whether we ignore something or not. Once it is there - it is
there. They may "look" like a mixture, but this is subjective
(objectively they are not a mixture), and we aim at an objective
description.

>In my current understanding the role of any time evolution in
>decoherence is merely to evolve a special state into a generic
>state to which the above sentence applies, much like time
>evolution in statistical mechanics is merely there to
>thermalize everything and bring the theory into position.

Yes, and it is subjective. There is no objective decoherence,
except after taking some kind of a limit, or after taking some
kind of subjective coarse-graining.

>
>> 1) Istead of assuming that the algebra of observables
>> is L(H) (all bounded operators on H), let us assume that
>> it is a general C* algbenra, not necessarily simple, possibly
>> with a nontrivial center (superselection observables)
>> 2) Instead of assumint that time evolution is given by a
>> one parameter group of aotomorphism, let us assume
>> that it is given by a semigroup of positive-definite
>> maps (positivity has a physical meaning, multiplication
>> of quantum operators does not have such a meaning)
>
>What would replace the usual path-integral formalism in this
>setting?

Mensky wrote a series of papers on path inegral formalism for systems
under continuous observation. He was, in fact, deriving a form of
disiipation from a modification of path inegral. I do not think that
he has the final answer, but he has at least a good starting point.
Check his "Continuously Measured Systems,Path Integrals and
Information", quant-ph/0007089, as well as other papers.
The path inegral formalism comes from Trotter's formula, and it
applies to an arbitrary evolution kernel, but in QM the tricky thing
is using space, or phase space variables in the propagator.
Every Abelian C* algebra is essentially algebra of functions on some
set. There is also "central decomposition" - probably it should be
applied here, but how exactly? I do not know.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac equation under Lorentz transformations
Newsgroups: sci.physics.research
Date: 2001-11-10 18:39:27 PST

On Fri, 9 Nov 2001 00:01:22 +0000 (UTC), mcartoajatmat.ulaval.ca (Mihai
Cartoaje) wrote:

>I believe I am beginning to understand how the Dirac equation
>transforms under Lorentz transformations. The Dirac equation is,
>
>(i gamma^mu del_mu - m) Psi(x) = 0.
>
>The gamma matrices transform as,
>
>gamma^mu ' = S gamma^mu S^(-1) = Lambda^mu_nu gamma^nu,

I would be careful with wording here, because it may lead to
a wrong understanding of what is going on.
I wouldn't say "gamma matrices transform". I would say
gamm matrices have the property. What transforms is
\Psi and \del. Then, because gammas have this property,
we find that the equation is invariant. Here is a schematic
sequence (I will skip the mass)

gamma del Psi = 0

gamma del S^{-1}S Psi =0

gamma S^{-1} del S Psi =0

S^{-1} S gamma S^{-1} del S Psi =0

thus (multiply with S from the left)

S gamma S^{-1} del S Psi =0

thus (properties of gammas)

gamma L del S Psi = 0

now

L del = del^prime
S Psi = Psi^prime

thus

gamma del^prime Psi^prime = 0

The deeper understanding being that we change the spinorial
frame by a constant Lorentz transformation, and that del_mu
are derivations along vectors of this frame. Gamma matrices
are constants here (they keep their standard form).
What is not constant are gamma^mu del_mu.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac equation under Lorentz transformations
Newsgroups: sci.physics.research
Date: 2001-11-14 09:52:49 PST

On Tue, 13 Nov 2001 21:45:53 +0000 (UTC), ultraman2002athotmail.com
(mandro) wrote:

>the electron is a section q of a vector bundle B on a manifold
>M, the two "lorenz coordinate systems on M" in your case minkowsky space
>are f, g : M --> R^4
>since--and only since, they determine orthonormal basis fields
>E_i , E'_i i = 1,,,4 on M
>we can say they determine trivializations t, t': B --> M x C^4
> (t, t' are like coordinates on B)

Not exactly. You need more than an orthonormal basic field, you need a
spinorial frame, and there are two spinorial frames over each
orthonormal frames. You need a spinorial frame to trivialize the
particular B. But you may call it a nuisance here.

>
>now f o g^{-1} = L is a lorenz transformation
>
> t o t'^{-1} = p(L) is a SL(2, C) transformation
>
>
>then the precise content of that statement is that
>
> t o q o f^{-1} "is the wave fn in one coord system"
> and t' o q o g^{-1} " is the wave fn in the other coord system"
>
>and, the dirac equation would have work in both (rouhly speaking)

Yes, but the above says nothing about the Dirac operator. The fact
that the Dirac operator is invariant follows from its definition, and
need to be checked. Of course, in a sense, it follows from the fact
the Dirac operator is a "natural" operator, but to see how natural
it is, you need to contemplate its definition.

>By the way did you know that you dot really need two coordinate systems
>to show this all you really need is two vector basis fields
>just consider
> t o q and t' o q
> enjoy

Yes, but still, there is a work that need to be done to show that the
Dirac operator is a well defined operator defined on sections of B.
This work is equivalent to showing the invariance of the Dirac
equation.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Dirac equation under Lorentz transformations
Newsgroups: sci.physics.research
Date: 2001-11-19 19:23:54 PST

On Fri, 16 Nov 2001 14:46:45 GMT, Mihai Cartoaje
<mcartoajatmat.ulaval.ca> wrote:

>> Psi itself does not know how to tranform to a frame that
>> is not an orthonormal frame.
>
>Have you used the assumption that the gamma matrices are invariant to
>reach such a conclusion? On sci.physics, I have posted a LaTeX document
>showing how to transform the wavefunction so as to keep the Dirac
>equation form invariant assuming the gamma matrices form a contravariant
>system. It is only one page of easy reading. Shall you tell me where
>I have made a mistake or made the assumption that the coordinate systems
>are orthonormal? If you want to read it it is under the thread "Dirac
>equation under coordinate tranformations". You can use groups.google.com
>to find it if it has been removed from the server.

Hi,

It seem you propose to use Psi transforming as a scalar, rather
than as a spinor. So you want to have a multiplet (as your Psi has
values in C^4) of spin zero particles, instead of a doublet of spin
1/2 particles (as it is with the standard Dirac equation).

Now, the good thing about the standard Dirac equation is that it has a
conserved probablity current, with positive definite probability
density.

You should also attempt to propose your definition of the probability
current with your method. You should be cateful though, because
in the standard approach gamma_0 is being used. But in your
approach there is no one gamma_0, you have different
gammas at each point and for each coordinate system.

So, what would be your formula for the probability current
that a) defines a conserved vector field b) probability density
is non-negative?

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Lorentz invariance of inner products
Newsgroups: sci.physics.research
Date: 2001-10-31 09:27:29 PST

On Tue, 30 Oct 2001 01:37:05 +0000 (UTC), baezatgalaxy.ucr.edu (John
Baez) wrote:

>In particular, this fact just means that you cannot make up a
>Lorentz-invariant quantum theory where the Hilbert space of
>states is finite-dimensional.
>
>(Unless you use a trivial representation, which would be boring.)
>
>This is one reason why people needed to invent quantum field theory!
>In quantum field theory, the space of states is an *infinite-dimensional*
>*unitary* representation of the Lorentz group (or its double cover).

A little correction: You do not need quantum field theory for that.
The Hilbert space of solutions of Dirac equation carries an
infinite-dimensional projective unitary representation of the Lorentz
group. Thus the above can not be thought of as a reason for
inventing quantum field theory. People invented quantum field
theory because of Klein paradox and other problems of relativistic
quantum mechanics in external fields.

I am making this note just in case some kid is reading this
newsgroup ....

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Lorentz invariance of inner products
Newsgroups: sci.physics.research
Date: 2001-11-02 19:19:35 PST

On 2 Nov 2001 07:33:16 GMT, Iuval clejan <clejanatmindspring.com>
wrote:

>ohn Baez wrote:
>
>>
>>
>> We all agree that it's good to have a norm, or more precisely an inner
>> product, to interpret a vector space as a space of "state vectors"
>> in quantum theory. However, the spin-1/2 representation does not
>> have a Lorentz-invariant norm. What this means is that in the context
>> of special relativity, it is quite artificial to pick any particular
>> norm on this space - whatever you pick, a Lorentz boost will change it!
>>
>>
>> And this is consistent with the fact that we should not attempt to
>> use this space as a space of state vectors in the context of special
>> relativity + quantum theory.
>
>So what space should we use? The infinite dimensional unitary reps of the
>Lorentz group have all possible spin values, so they are not what we want when
>we talk about spin 1/2 particles.

Let's concentrated on these things that can be handled mathematically,
that is free fields. A free relativisttic quantum field has a particle
number operator which splits the Fock space into n-particle subspaces.
For n=0 we have vacuum which carries a trivial representation of
SL(2,C). For n=1 we have usually a direct sum of few irreducible
representations of the same spin. (why "few"? because we have
particles and antitiparticles, and possibly some other "internal
symmetries"). Apart of possible subtleties the n=1 (one-particle)
space is isomorphic to the space of solutions of the appropriate
wave equation. The subtleties involve the fact that solutions
of classical wave quation split into positive and negative energies.
Also, the classical wave equation can be real, while in quantum field
theory we need complex Hilbert space.

> The infinite dimensional (time and light-like)
>unitary reps of the Poincare group are probably OK though (I am looking at my
>group theory book), but since the Lorentz group is just the Poincare group
>without translations, these should be equivalent, so I am still confused.
>
>> >There is absolutely no relation between the double cones in Minkowski
>> >and quantum logic except a mathematical one. There is no physics there
>> >at all.

And why so? Here is an example, When we look for a relativistic
position operator, we need into account causal structure of the
Minkowski space. The natural regions for localization are double
cones (diamonds). We want states to be orthogonal when localized in
spacelike diamonds. These diamonds form a non-Boolean lattice.
We want to represent this non-Boolean lattice, together with the
symmetry group. The natural lattice to represent a non-Boolean
lattice is the lattice of subspaces of a Hilbert space. This way
you can rediscover Dirac equation.

>> This relation probably *is* irrelevant to your question, so I don't
>> know why ark mentioned it. However, it's *not* irrelevant to physics!

I mention it, becuase there are finite dimensional representation of
SO(3,1), but with indefinite metric. Thus they can not have a standard
quantum mechanical probabilistic interpretation. However more and more
people are looking into a possibility of using signed measures, thus
"negative probabilities" (Feynman wrote a paper on this subject,
recently Noyes wrote a paper too). This is NOT a standard theory
to be told kids, but this is a valid research avenue.

>> The point is this: Minkowski spacetime and the space of 2x2 hermitian
>> matrices are equivalent representations of the Lorentz group. The
>> causal ordering in Minkowski spacetime corresponds to the usual
>> ordering on the space of 2x2 hermitian matrices, where A < B if B - A
>> has positive eigenvalues. And if we restrict attention to
>> projection operators, which correspond to "propositions" in quantum
>> logic, this ordering corresponds to "implication": we obtain a
>> complete orthomodular lattice of propositions.

I am not talking about representing points. I am talking about
generalization of Mackey imprimitivity theorem, when spectral
measure is replaced by the lattice of diamond sets in the Minkowski
space, with its natural orthogonal complement operation.
There is no such a theory in fact. But there are partial results.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: Lorentz invariance of inner products
Newsgroups: sci.physics.research
Date: 2001-11-15 19:28:52 PST

On Thu, 15 Nov 2001 05:16:25 GMT, George Jones
<george_llew_jonesatyahoo.com> wrote:

>Bottom line: unitarity in quantum theory is defined with respect to a
>positive definite inner product.
>
>Regards,
>George

This would be true if you formulate is as "usually unitarity in
quantum theory is defined with respect to a positive definite inner
product."

The word "usually" is important here. If you do not use this word -
than what you said is incorrect. There are books about "indefinite
metric." For a recent paper on indefinite metric in C* algebraic
formulation see e.g.

http://www.lqp.uni-goettingen.de/lqp/papers/98/12/98122300.html

It is important to undertand that there is no reason for the scalar
product to be positive definite. The only thing that needs to be
positive definite are probabilities of events coming from physical
measurements.

To see how can it be achieved see for intance
Ph. Blanchard and A, Jadczyk
Found. Phys. 26 (1996) 1669-1681:
quant-ph/9610028

We write there:

"We extend the ideas of L.P. Horwitz and C. Piron and we propose a
relativistic version of Event Enhanced Quantum Theory, with an event
generating algorithm for spin one-half particle detectors. The
algorithm is based on proper time formulation of the relativistic
quantum theory. Although we use indefinite metric, all the
probabilities controlling the random process of the detector clicks
are non--negative. "

Thus it is only ""usually unitarity in quantum theory is defined with
respect to a positive definite inner product." Usually, not always.

ark

From: ark (arkatcassiopaea.com)
Subject: Re: unitary reps of Lorentz group
Newsgroups: sci.physics.research
Date: 2001-11-16 18:48:43 PST

On Tue, 13 Nov 2001 07:56:10 +0000 (UTC), Toby Bartels
<tobyatmath.ucr.edu> wrote:

>>George Jones wrote:
>>>As Ark, John Baez, and Pertti Lounesto have pointed out, it is
>>>possible to define a Lorentz-invariant scalar product on the space of
>>>Dirac spinors, but this scalar product is not positive definite, and
>>>so is not a true inner product.
>>You mean the norm has to be positive definite (the inner product of
>>course can be negative or zero)
>What else could "positive definite" mean when speaking of a scalar product?
>(The way that you've interpreted is useless, as you essentially note.)
>By definition, a bilinear form is _positive_ or _definite_
>(the 2 properties are logically independent, so I separate them)
>iff its associated quadratic form is respectively positive or definite.
>I can't say with certainty that I've ever seen this before,
>but it's how I would naturally use and interpret the terms.

One needs to be careful with terminology. Inner product means often
the same as "nondegenerate sesquilinear form".
We can then specialize: "definite inner product" which means positive
or negative definite
or
"indefinite inner product" which means <x|x> is positive
for some vectors and negative for some other vectors.

For instance in "On Noncommutative and semi-Riemannian Geometry"
http://xxx.lanl.gov/abs/math-ph/0110001
the Author writes uses explicitly the term "indefinite inner product"
(in relation to the Dirac operator). He is also introducing the
concept of a Krein space - worth reading.

ark