From: Arkadiusz Jadczyk (ark@cassiopaea.com)
Subject: Re: Bohr with no collapse
View: Complete Thread (2 articles)
Original Format
Newsgroups: sci.physics.research
Date: 2002-12-06 15:20:02 PST

On Fri, 6 Dec 2002 06:10:24 +0000 (UTC), "David K. Davis"
<dave@panix.com> wrote:

>Below is fragment from my web page (www.panix.com/~dave/quantum). I discuss
>a no-collapse interpretation of QM that I believe Bohr should have advanced.
>I believe that Bohr was a realist in the sense that he wanted to salvage as
>much of realism as possible. But Copenhagen has some vulnerabilities as a
>realist interpretation. I'm an opponent of all the quantum mind stuff. I
>believe it's gotten as much of hold as it has because of weaknesses in the
>way Copenhagen has been formulated.

What I can't find at this URL is how you describe a quantum system under
continuous observation, when a time series of data is being produced -
where the data consit of pairs (time,event)
as in the cloud chamber or on a photographic plate?

Do you have anything that could reproduce, or get even more powerful
results than :

``Particle Tracks, Events and Quantum
Theory", Progr. Theor.~Phys. {\bf 93} (1995), 631--646

http://xxx.lanl.gov/abs/hep-th/9407157

and

``On Quantum Jumps, Events and Spontaneous
Localization Models", {\em Found. Phys.} {\bf 25} (1995), 743--762

http://xxx.lanl.gov/abs/hep-th/9408021

Or how you model time of arrival, to compare, for instance, with

http://xxx.lanl.gov/abs/quant-ph/9602010

ark

On 3 Dec 2002 01:00:12 GMT, fiis5d@yahoo.com (Squark) wrote:

>> Are not the real difficulties in promoting any operator to the status of a
>> (quantum) observable linked to possible problems like, say, the set of
>> eigenvectors not forming a basis, rather than to the fact that the
>> associated eigenvalues can be complex?
>
>Not forming a basis _or_ not forming an orthogonal basis. The later thing
>is what happens to the annihilation / creation operators, for instance.

There is another generalization of an observable that uses a concept
similar to the above.

According to this generalization observable is given by a positive
operator valued measure dE(x) on some "space" X. Given a bounded real
function f on X we can always built a hermitian operator
f^= integral f(x) dE(x)
but this integral allows us only to calculate the mean value of f, not
of their moments. In fact, f^ can have infinitely many different
decomposition of the type as above.

A typical example is by taking the overcomplete basis phi(z)
of eigenstates of the annihilation operator, and to define
positive operator valued measure

E(Delta) = integral over Delta |phi(z) ><phi(z)|

(One needs to normalize it by using the overcopleteness relations)

This defines generalized observable on the phase space z=x+ip
and can be even used for "quantization" - that is to associate to
every classical function on the phase space a quantum operator
(coherent states quantization).

The problem that differs this quantization from Weyl's
quantization is that quantized (f square) is not the same
as (quantized f) square.

One can even generalize the prescription further, by assuming that F
can be a signed measure (not necessarily positive definite).

Again we can associate observables with real (or complex) functions.

Weyl quantization is of this type, and in case of Weyl's F, quantization
of square of a function of position or momentum alone is square of
the quantized function.

ark

dgoncz@aol.comp.mil ( Doug Goncz ) wrote:

> Well, on each iteration, a function system may provide a real answer
> or an integer answer. Screen pixels are integers. So what is the
> difference between those functions that round off iterations to a
> screen pixel and the functions iterating to a quantum (integer)
> parameter?

Quantum Iterated Function Systems are called quantum because their
dynamics comes from quantum dynamics. The arena for this dynamics is
either a complex projective space or some twistor space or the space
positive trace class oparators in a complex Hilbert space. "Parameters"
are not integers here.

> Keep in mind that a quantum wave function is usually normalized with
> a _real number_ multiplier to a unit area probability distribution.

In our case normalization is not necessary. Normalization is needed for
unitary quantum dynamics. The quantum dynamics discussed in QIFS is that
of an "open quantum system" (or of a quantum system under monitoring)
thus non-unitary. It conserves positivity but not the norm. Sometimes it
will conserve the trace, sometimes not.

ark

From: Arkadiusz Jadczyk (ark@cassiopaea.com)
Subject: Re: Magnetic monopole
View: Complete Thread (41 articles)
Original Format
Newsgroups: sci.physics.research
Date: 2002-12-10 22:53:18 PST

On Mon, 9 Dec 2002 15:12:41 +0000 (UTC), paradoxer@mailcity.com (Neil)
wrote:

>Isn't the vector potential (the A field) the whole problem with
>magnetic monopoles?

Indeed.

> If B = Curl A, then we can't get a reasonable
>field from the monopole without freaky, inelegant (and violating the
>principle of sufficient reason per which direction they attach from)
>strings of condensed A field attached to them.

Yes, but that is only due to lack of understanding of the nontrivial
topology that is involved.

(See the explanation of
>this vulgarity in, I think, one of the editions of Classical
>Electrodynamics by Jackson.)

Jackson doe not know about nontrivial fiber bundles, and he also
is not discussing quantum theory in a field of a magnetic monopole.
In classical theory it is easy to get rid of a vector potential. In
quantum mechanics in enters as a connection in a U(1) bundle.

> We shouldn't just blow off the A field as
>a fiction (aren't they all fictions at some level anyway?) because it
>has real consequences in quantum mechanics.
>Neil Bates

See the above. The really interesting problems start with quantum
theory.

I quantum theory you have canonical momentum and position
operators p and q. But when you want to write a Hamiltonian
you need (p-eA)^2 - that is you need "velocity" operator
(p-eA)/m not the canonical momentum.

But then, you may ask, why do not use velocity operators
form the start?

Well, components of the velocity do not commute when there
is magnetic filed present. So, you CAN write infinitesimal operators,
they will not commute, and then you want to see if you can "integrate"
the infinitesimal commutation relations by taking exponents of velocity.

And then you find that these exponents exist ONLY when magnetic
charge (if there is such) is quantized.

But the you want even more, you want an explicite form of your
velocity operators as differential operators. And here is where
topology comes, and when you get problems like when you want draw
the map of the globe on a flat paper. Yoo need to CUT it.

But this cut comes only from your attempt to represent globablly the
topologically non-trivial object on a topologically trivial paper!

ark

On Sun, 8 Dec 2002 09:15:32 +0000 (UTC), remove.haberg@matematik.su.se
(Hans Aberg) wrote:

>Thus, SU(2) will, via that integration, be enough to describe the QM field
>symmetries, but SO(3) will not, as one cannot capture the up/down spin
>distinction. One might word this by saying that one is not in actuality
>looking at space symmetries but at "infinitesimal intrinsic differential"
>symmetries.

Another way to say the same: 2Pi rotation of a spinor s gives not the
same spinor s, but -s. 4Pi rotation is needed to restore s. If we use
representations of SO(3) - we will always get the same object after 2Pi
rotation.

The fact that spinors can get negative sign after 2Pi rotation is
justified by "gauge invariance". In quantum theory phase of the wave
function is not observable (under normal circumstances, or following
"normal" textbooks) . Thus -s and s represent the same phsyical object.
In fact we could even have a more weird behavior: that a wave
function after 2Pi rotation can be any complex number of the form
exp(i phi), real phi. But it can be shown that such (projective)
representation of the rotation group is always equaivalent to one
where phi=0 or phi=Pi. Every projective representation of SO(3)
can be obtained from true (no phase factors) representation of
SU(2). Similar arguments act in the case of parallel transport of
spinors.

ark

On Thu, 19 Dec 2002 02:49:10 GMT, whopkins@csd.uwm.edu (Alfred Einstead)
wrote:

>> The basic idea is this: there's no god-given way to just take a smooth
>> manifold and define spinor fields on this manifold! To define spinor
>> fields you first need to pick a "spin structure".
>
>... which gets to a related question I've been meaning to ask for
>a while: what's the Lie derivative L_X psi of a spinor field psi
>by a vector X?

The problem with spinor Lie derivative comes from the fact that
if X is not a Killing vector field, then transport of an orthonormal
frame at x(t) back to x(0) along the flow of X is not an orthonormal
frame. The transported frame et(t) relates to e(0) by a general
linear transformation A(t). But, when t is small, this A(t) is close to
identity, and we can take the canonical polar decomposition
A(t)=L(t)P(t) and use L(t) which is then an orthogonal transformation
(far from identity L(t) could be a partial isometry only - determinant
0)
This way we get the formula. Of course some functorial
properties (nice commutators etc) are lost this way, but the formula
makes sense and has a clear geometrical sense.

ark

torre@cc.usu.edu (Charles Torre) wrote in message news:<aaec8787.0212210557.32d006fb@posting.google.com>...
> Arkadiusz Jadczyk <ark@cassiopaea.com> wrote:
>
> ...
>
> > The least confusing picture is, I believe,
> > one when you have different Hilbert space for each time (this picture
> > belongs originally to C. Piron, I think).
>
>
> If convenient, would you please provide a reference or two on this approach? Thanks.
>
> charlie

I posted the references in another reply - they should appear. But it
is very easy to understand the idea without any reference, so let me
explain it.

If we try to think of Galilei-Newton space-time, its characteristic
feature is absolute simultaneouity - so we that we have absolute time.
Space time events are grouped into equivalence classes of simultaneous
events. If E is the space of all events, then we have a fibration E->T
where T is time. Fibers are 3-dimensional. Fiber E_t over t is called
"space at time t". There is no natural identification of E_t with E_s
for t and s different. Each E_t carries its own
3-dimensional Riemannian metric, but there is no natural metric on E
(in fact, there is a contravariant degenerate metric, but that is a
different issue.)
So, we can build Hilbert space L^2(E_t) for each t, but there is no
natural
identification. Each "observer" (inertial or not) provides
identification, but change the observer and the identification will
change.
That is why a general relativistic (in Galei-Newton sense) quantum
mechanics
MUST have different Hilbert spaces for different times. So we have a
Hilbert
bundle over T. It is then only natural to look at the Schrodinger
equations as
a parallel transport in this bundle, with Hamiltonian H(t) (possibly
time-dependet) as a connection 1-form - and it is clear also why H(t)
will change with the observer - the same way connection coefficients
transform when we change trivialization in a vector bundle.
In this picture also the Feynman amplitude (that enter path inegral)
gets clear geometrical interpretation in terms of parallel transport.

ark

(Reposting as it was probably lost in the crash)

On Mon, 23 Dec 2002 05:49:29 +0000 (UTC), baez@galaxy.ucr.edu (John
Baez) wrote:

>In fact, Michael Weiss and I once had a nice long discussion of
>how to think of various discrete spectra in quantum mechanics
>as coming from "boxes" of different sorts. Here it is:
>
>http://math.ucr.edu/home/baez/harmonic.html
>
>Even for the harmonic oscillator there is a compact "box"
>lurking around, if you look hard enough....
>
>But of course, you don't *have* to think of it this way.

I agree that it is nice to know these things - magic in mathematics
always provides us with pleasure.

Yet a physicist thinks also in terms of dynamics, and asks
strange questions of the type: why should the wave function
care about being single valued? And how does it know whether
it is continuous or not? And "what is this S^3 (or B^2) that psi
is a function on? Or: what is the oscillator becomes anharmonic?
What if SU(2) symmetry breaks? Does quantum mechanics
breaks too?

Group theory, especially in quantum mechnics is a powerful tool.
It often enables us to solve difficul problems by clever manipulations:
for instance spectrum for the Kepler problem can be obtained from
group theory (cf. "Dynamical Groups" or "Spectrum Generating Algebras"
). But, it seems to me, the use of group theory lures us into sleep.
Another example: Kaluza-Klein theories where we assume that
the fibres are group manifolds or homogeneous spaces. Or: charge
quantization comes from cylindricity of S^1.... But does it really
answer our question about the nature of electric charge? And what if
the world is all discrete and if there are only rational numbers, no
continuum?

Good questions for the Holidays!

ark

On Fri, 3 Jan 2003 02:16:08 GMT, uchchwhash@hotmail.com (Pirated Dreams)
wrote:

>Let me explain. He says an electron is in P1(x1, t1), and afterwards
>in P2(x2, t2). The total probability amplitude is the sum over the
>various paths it can take. Suppose I believe him. But where does the
>momentum of the particle enter in this picture?

There are many way in which one can (or should) answer this question.
Since other knowledgable men will certainly address the momentum
and Fourier transform (as discussed in Feynman and Hibbs) and also
path integral in phase space, which is somewhat tricky, let me
just point out that momentum on a typical Feynman trajectory
is undefined. That is because a typical Feynman trajectory is
nondifferentiable. That is the mesure that enters path integral
is concentrated on nondifferentiable curves.

The exact properties depend on the Hamiltonian. It is important to
notice that there is evident assymmetry between the position and
the momentum in the Hamiltonian. In nonrelativistic case the dependence
on the momentum is at most quadratic. It is anly for harmonic oscillator
that position and momentum enter path integral in a symmetrical way.

ark

From: ark (ark@cassiopaea.com)
Subject: Re: Most general Liouvillian (dissipative Quantum Mechanics)
View: Complete Thread (2 articles)
Original Format
Newsgroups: sci.physics.research
Date: 2003-01-10 20:56:03 PST

Student <yssual@no-spam.yahoo.com> wrote in message
news:<3e1aad73$0$27688$626a54ce@news.free.fr>...

> Can you please confirm this is the most general Liouvillian for a density
> matrix (I found contradicting versions, this one seems the most likely,
> others in terms of two operators, sum_n,m h_n,m(v_n v_m rho + rho v_n v_m -
> 2 v_m rho v_n) are too vague on the conditions on h_n,m to be useful; if
> you know these conditions please let me know):
>
> L(rho) = -i[H,rho] - sum_n { rho v_n§ v_n + v_n§ v_n rho -2 v_n rho v_n§ }
>
> where I write v§ for the hermitean conjuguate of v (§ stands for dag), rho
> is the density matrix, [H, rho] the unitary dynamics and the sum the
> so-called Lindblad form. This is under assumption that each v_n is just
> *any* (2nd quantized) operator.
>
> Do you know of some (accessible) derivation of this formula, if exact?
>
> Regards.

See for instance formula (27) in

http://www.quantumfuture.net/quantum_future/topics.htm

and references there (mainly Lindblad and Gorini-Kossakowski)

This is the most general form assuming complete positivity. More
general form is possibly with positivity requirement alon (no
"complete positivity", which is somewhat stronger, as "completeness"
is equivalent to some kind of "stable positivity")

ark

On 30 Apr 2003 20:48:50 GMT, Jason <pristy@excite.com> wrote:

>It struck me that the more general representation is nothing more than
>the unitary rep of a group G for which there exists a surjective
>homomorphism f:G->P (P is the double cover of the proper Poincare
>group) with
>
>K = ker(f) = U(1).

I think you would benefit, if only a little, from reading the chapter
VII.3: Multipliers and group extensions from "Geometry of Quantum
Therory" by V.S. Varadarajan, Springer Verlag, 1985
(the first edition was published in two volumes, the particular
chapter is then in the second volume - which can be read independently
of the first volume). It is also very useful to consult the original
papers by V. Bargman on unitary ray representations.

ark

On 11 May 2003 07:16:36 GMT, Jason <pristy@excite.com> wrote:

>I just did some thinking about fields which are covariant under both
>gauge and Poincar=E9 transformations.

When talking about "gauge and Poincar=E9 transformations" one needs pay
special attention as to whether we talk about a "constant"
transformation, or about point-dependent transformation. To speak about
point dependent Poincare transformations (as one usually does with gauge
transformations) one has to enlarge the picture and do the
transformations in the tangent "affine bundle" or some other bundle
that needs to be soldered to the tangent bundle one way or another.

ark

On 8 May 2003 15:30:18 -0400, daryl@atc-nycorp.com (Daryl McCullough)
wrote:

>Godel's theorem doesn't say anything about the impossibility of
>developing a complete theory of physics, although it might imply
>limitations on what we can *prove* about such a theory.

Indeed. Godel's theorem may provide us with clues about completeness of
any "theory" of reality. It may indicate that any model of the
"universe" should always be "open". This idea may go well with the
philosphy of the late Karl Popper.

In his intellectual autobiography "Unended Quest", Popper writes:

"This is why the evolution of physics is likely to be an endless process
of correction and better approximation. And even if one dy we should
reach a stage where our theories were no longer open to correction,
since they were simply true, they would still not be complete - and we
would know it. For Godel's famous incompleteness theorem would come into
play: in view of the mathematical bacground of physics, at best an
infinite sequence of such true theories would be needed in order to
answer the problems which in any given (formalized) theory would be
undecidable" (Karl Popper, Unended Quest, Routledge, London 1993, p.
131)

Incompleteness theorem may also go well with the idea of intrinsic
irreversibility at some fundamental level (open quantum systems are
usually described by irreversible dynamics, semigroups rather than
groups. Incomplete knowledge suggests that pure states always evolve
into mixed states - that by using pure states and unitary dynamics we
cling to an utopia that will not let us build a "really unified
theory").

Frankly speaking I am not able to point at some DIRECT link between
Godel's incompleteess theorem and the philosophy of open systems
and irreversible dynamics. I am just sharing my "feelings" and
"intuitions" - those that I am using in chosing my own research
subjects, methods and tools. (Other physicists may have orthogonal
feelings and intuitions, and they may benefit from them. )

Some of these things are discussed in "Godel meets Einstein" by P.
Yourgrau, Open Court Publuishing Company, Chicago 1999, in particular
Chapter 7: "Being and Time".

ark

On 12 May 2003 22:05:20 -0400, Zig <jhannon@wisc.removeme.edu> wrote:

>next, if phase space is a group, and i know the commutation relations of
>the generators, then why does one say that this is the axiom of quantum
>mechanics? this is not an axiom at all, but is derived by purely
>mathematical considerations about the nature of the configuration space.
> the only thing i could come up with was the presence of the hbar. so
>my second question is this: in this context, what is the starting axiom
>of quatization? is it only that i should add a factor of hbar to the
>lie algebra?

Check

http://www.arxiv.org/abs/math-ph/0210006

Revisited gauge principle: towards a unification of space-time and
internal gauge interactions
by

V. Aldaya, J.L. Jaramillo, J. Guerrero

and references there.

From the abstract:

"The minimal coupling principle is revisited under the quantum
perspectives of the space-time symmetry. This revision is better
realized on a Group Approach to Quantization (GAQ) where group
cohomology and extensions of groups play a preponderant role."

Canonical quantization can be formulated in terms of CCR - which is
essentially a group extension problem. For many degrees of freedom
we deal with product of groups as momenta and positions for different
degrees of freedom commute - so nothing new comes this way - until we
go infinite number of degrees of freedom, where we can have
inequivalent irreducuble representations.

With some pain the same method applies to fremions (CCR), though the
group structure is completely different in this case.

ark

On 11 May 2003 07:17:24 GMT, Zig <jhannon@wisc.removeme.edu> wrote:

>what exactly is the connection between the physical field and
>irreducible representations of the symmetry group?

Roughly speaking, if phi(x) is a physical field, then phi(0) is a vector
in a vector space carrying an irreducible representation of the Lorentz
group.

Of course one has to be careful when psi(x) is a distribution rather
than a smooth function.

Also, often, we have to deal

a) with more than just a connected component of the Lorentz group - we
need inversions P and T as well
b) we need projective representations of The Lorentz group (or
representations of SL(2,C)) to deal with spin 1/2
c) we need to include gauge group actions (though normally it is argued
that these are not needed for "physical" (that is "observable") fields.
d) Once in a while the vector space of the representation will carry
indefinite scalar product - then the very concept of "irreducibility"
must be taken with care.

ark

On Thu, 15 May 2003 03:21:04 +0000 (UTC), baez@galaxy.ucr.edu (John
Baez) wrote:

>It's valid to ask - but unless you give me some formulas to compute
>probabilities that sum to 1 appropriately, my best guess is that the
>answer is no.

The problem is to get nonnegative numbers. Adding them to 1 is just a
question of normalization.

>The great thing about Hilbert spaces is that for any
>unit vector phi and any orthonormal basis psi_i, the numbers
>|<psi_i,phi>|^2 sum to 1. So, unit vectors represent "states"
>and orthonormal bases represent "complete sets of mutually exclusive
>states".
>
>I don't know how to get something this nice in a complex vector space
>with an indefinite sesquilinear form - except, as mentioned, by first
>throwing out the vectors of negative and zero norm.

Well, in EEQT the only thing we need are "transition probabilities".
(as they tell us what happens to the recording device). They are
computed from a formula of the form:

<psi, g^\star g psi>

and normalized. Thus we need to have transition operators g to have the
property that they map "physical" states into physical states. This
imposes restrictions on possible couplings, but that's all. The model
generating quantum fractals in a hyperbolic space works like that.
We have indefinite metric, yet the algorithm of dynamics of ierated
function system, with random selection of maps according to computed
probabilities works.

The point is that vectors of positive norm do not form a vector space,
so we can't just to throw negative and zero vectors out. We really need
all of them if we want to work with vector space and with linear
operators (which is convenient).

It is similar as with use of complex numbers. The imaginary part is
useful for computation even if physical quantities take real values.

ark

On Fri, 4 Apr 2003 23:50:13 +0000 (UTC), Arnold Neumaier
<Arnold.Neumaier@univie.ac.at> wrote:

>I repeatedly read that physical systems cannot be in a superposition
>of states with different charge. (This is a so-called superselection rule.)
>But I haven't seen a convincing explanation for the reason why,
>and have seen hints that the statement might be wrong.

This is connected to unobservability of the phase (global) of the wave
quantum mechanical functionm, and also to "unobservability" of the
electromagnetic potential. There is no proof of either, there are just
"arguments for". It may well happen that the phase is not so
"unimportant" and that charge is not a a superselection operator. This
will lead us beyond the present paradigm, even if just a little bit.

ark

On Wed, 23 Apr 2003 20:24:06 +0000 (UTC), Uncle Theodore
<uncletheodore@hotmail.com> wrote:

>But... Where does that "measurement" come from?

Here is my answer - which need not be the same as what others will say
on the subject:

A "measurement" can (and should) be described by a special kind of
coupling, that is NOT included in the Hamiltonian. This special coupling
changes the evolution of the quantum system (it becomes then the
evolution of the quantum system AND the measurement device).
The result of every measurement can be r epresented as a number (not an
operator). When this number is somehow recorded, this is an "event".
Events are classical by its very nature (yes-no, Boolean logic). Thus,
mathematically, the algebra of observables must have a central (Abelian)
part. The evolution of statistical states of a system COUPLED to an
events-producing device is described by a semigroup of positive definite
transformations (not *-automorphisms) - Lioville type, Lindblad's form.
When it comes to the description of an individual system, the evolution
is piecewise deterministic. There are random "jumps", ionce in a while.
RThese happen when events (motions of the pointer) take place.
Bertween jumps the evolution of the quantum state is non-unitary (the
norm of the state vector is decreasing, the rate of decrease dpends on
how "strong" the quantum system couples to the monitoring device.

For a review you may like to check

http://xxx.lanl.gov/abs/quant-ph/9812081

And if you have further questions - I will try to answer.

ark

From: Arkadiusz Jadczyk (ark@cassiopaea.com)
Subject: Re: Poincare lemma - generalization
View: Complete Thread (2 articles)
Original Format
Newsgroups: sci.physics.research
Date: 2003-05-20 23:09:04 PST

On 14 May 2003 16:04:30 -0400, deepa chavan <deepalichavan@yahoo.com>
wrote:

>Hello,
>
>The proof for the Poincare Lemma (which essentially
>says that - given a closed form, one can locally show
>that it is an exact form) is well known in the
>"continuous" case, i.e. differentiable manifolds. It
>is also known for cohomologies (simplices, etc).
>
>I have been wondering if there exists a proof for more
>general cases - for example does there exist such a
>lemma in, say, the discrete case?. And what about more
>general cases?.

There is one simple but interesting variation of the proof of the
Poincare Lemma.

Normally tp prove the Lemma we contract a region to a point along
straight lines in a coordinate space. In this way we can, for instance,
construct a vector potential A from the field strength F. We may call it
a "gauge". But we can choose contraction arbitrarily, for instance,
first to a hyperpalne, and then to a point, along the flow of some
vector field. In this way we can get "contraction gauges" - to each
contraction there corresponds a "gauge".

I never seen this published, but it is an easy and interesting exercise!

ark

On Tue, 20 May 2003 21:16:36 +0000 (UTC), Arnold Neumaier
<Arnold.Neumaier@univie.ac.at> wrote:

>We can measure at a given moment only observables from a commuting
>family, but given enough time we can make many measurements of
>noncommuting quantities.

There are two different things:

a) measuring a physical quantity of an individual system

b) measuring a quantity on an ensemble of systems "prepared in a
similar way."

We would like to apply quantum theory to individual systems (for
instance to an atom in a trap, to an electron (or a couple of them)
hitting a screen.

>The state of a coherent laser beam is not affected by the measurements made
>on it; hence one can obtain all the information its state contains by being
>patient enough. And this is routinely done in quantum optics.

There is a subtle point here. You say "laser beam". And when you say
that you really mean: an abstract object, that is abstracted in such a
way that all the parameters that CAN change due to measurement are
conveniently forgotten. or measurement that COULD change the state are
conveniently refused to be made.

Such convenient abstractions are routinely done not only in quantum
optics - true. So it is better always test our ideas on more fundmental
objects, where forgetting about spin, position or momentum is not that
easy, such as electrons, protons etc.

ark

On 14 May 2003 16:04:32 -0400, helbig@astro.multiCLOTHESvax.de (Phillip
Helbig (remove clothes to reply)) wrote:

>> It would be a great advance to cosmology to find a big bang model that
>> was stable given only cosmological parameters made of ordinary matter.
>
>NO non-empty classical cosmological model is stable in that sense.

Isn't it so that the answer depends on the particular distance function
that is being used for studying stability? In one metric the particular
configuration can be unstable, but it will be stable in another metric?

So the question is: how stable or unstable is the distance function that
we are using to decide about stability/instability of a given
configuration?

I know the answer may be difficult, nevertheless I think it is important
to realize that a model that is "microscopically unstable", may still be
stable if we disregard microscopical differences and "average"?

ark

On Wed, 21 May 2003 21:05:13 +0000 (UTC), minmo@eecs.berkeley.edu
(Domenico Campolo) wrote:

>Back to your answer to my posting, my understanding is that in the first
>of the two interpretations you mean vector bundle valued forms as opposed
>to R^n valued forms, is it correct?

Right

>As for Flanders, and also others like Do Carmo, it seems to me they might
>more or less implicitly refer to R^n valued forms.
>In fact, in section 4.1 "Moving Frames in E^3", Flander starts with:
>
> "... in dealing with vectors in Euclidean space, no matter where
> we draw them for picturesque purposes, when we deal with them
> analytically, they always start at the origin."

This is probably to distinguish between affine space and vector space.
Euclidean space is an affine space. But vectors are "tangent vectors at
a point". It starts at the point where it is tangent to the curve that
defines it. This point is the origin of the local tangent space.

>After reading your postings, it seems to me that the quoted sentence
>is equivalent to stating that R^n valued forms will be used, where R^n in
>this case is the tangent space at the origin o, i.e. T_oE^3.

E^3 oe E^n, in general. But in our particular case we do not need to
complicate things. R^3, or R^n is enough.

>(now I'm talking about the 0-forms e_i forming the moving frame and
>the 1-forms de_i)
>Is it correct?

I believe. The R^3 valued form on the bundle frame is then as follows:

take a path e_i(s)=e_i (x(s)) in the space of frames. Take the tangent
vector to this path, call it ksi:,

project it onto the manifold. Call the projection dzeta:

dzeta is tangent to x(s) at s=0.

decompose dzeta with respect to the frame e_i(0)

dzeta = dzeta^i e_i(x(0))

Our form associates ksi n numbers: the vector {dzeta^i} in R^n

>As for the implicit connection, the path independent Euclidean parallel
>transport is used which makes T_pE^3 (naturally?) isomorphic to T_oE^3
>for any p in E^3 and leads to
>d^2 e_i = 0 (==> zero 2-form curvature)
>
>Right?

I am not sure about the last - the devil is in the details. But yes,
E^3 is endowed with distant parallelism!

>Thanks for your attention,

Welcome,

ark

From: Arkadiusz Jadczyk (ark@cassiopaea.com)
Subject: Re: Quaternion Quantum Mechanics Question
View: Complete Thread (6 articles)
Original Format
Newsgroups: sci.physics.research
Date: 2003-06-05 20:06:53 PST

On Wed, 4 Jun 2003 22:58:11 +0000 (UTC), baez@galaxy.ucr.edu (John Baez)
wrote:

>Toby and I fixed this by defining a quaternionic vector space
>to be a special sort of *bimodule* over the quaternions - meaning
>that you can multiply vectors by quaternions on both the left
>and the right. The tensor product of bimodules is again a bimodule.

That is an interesting. I did not realize that this can be a way out.

Another problem with quaternionic quantum mechanics that I was
worrying about is missing the mysterious connection between
generators of symmetry groups (which are anti-hermitian operators)
and "observables" - which are hermitian. In the complex case we have
the"i" that does the job of associating conserved quantities with
abelian groups of symmetries (momentum observable, multiplied by "i"
is the generator of translations etc.). In quanternionic QM this
trick will not work. Or, perhaps, with the bimodules approach
we can have something similar also in the quaternionic case?

ark

On Wed, 4 Jun 2003 22:47:23 +0000 (UTC), frithjof.tillinger@gmx.de
(Frithjof Tillinger) wrote:

>There is a theorem that all two-dimensional manifolds are conformally
>flat. Does this theorem only hold for orientable manifolds or also for
>non-orientable manifolds?

The property of being conformally flat is a local property. Therefore it
does not distinguish between orientable and non-orientable manifolds.

See, for instance, chapter 3.9 "Conformally flat manifolds" in
"Curvature and homology" by S.I. Goldberg.

ark

On 8 Jun 2003 06:43:03 GMT, Blumschein <blumschein@et.uni-magdeburg.de>
wrote:

> I wonder if common physics already obeys the
>asymmetry of real time, being absolutely bound to the observed process
>of concern.

I am not sure if this will help in your search, but there is a part of
physics dealing with irreversible dynamics. The mathematical
structure that is used there is "dynamical semigroup" for "time
evolution". Semi-group means that the inverse of an evolution may not
exist. Thus going backward in time is kinda risky in this case.

Studying examples you will see that in many cases you can formally
compute "inverse", that is going in time backward, but this inverse
has "bad" properties. For instance you would get negative probabilitiese
for the past transitions etc.

You may enjoy reding the classic: "From being to becoming. Time and
complexity in the physical sciences" by Ilya Prigogine. There is also a
beautiful little book by David Ruelle, "Chance and Chaos", and there is
a chapter about irreversibility there. You will note the following
paragraph on page 113:

"I have described the interpretation of irreversibility that is
generally accepted by physicists. There are some dissenting voices, such
as that of Ilya Prigogine, but the disgreement is based on philosophical
prejudice rather than physical evidence. There is nothing wrong with
philosophical prejudice, it is invaluable in making discoveries in
physics. But in due time things have to be settled by careful comparison
of mathematical theories and physical experiments."

BTW: I have my own philosophical prejudice, and it is somewhat parallel
to that of Prigogine. And things are not "settled" yet.

Related terms "non-unitary dynamics", "irreversible dynamics (or
processes)", "dissipative dynamics", "Liouville equation", "Lindblad".

ark

On Fri, 13 Jun 2003 01:11:27 +0000 (UTC), galathaea@excite.com
(galathaea) wrote:

>I am intrigued by your question, but unfortunately do not think I
>could save you too much work. However, I thought I'd give my hand if
>only to see where you were going with things. I'm still weaker than I
>wish I was at these things, so please excuse any errors if there be
>any.
>
>Since the objects of the principle series are uniquely detmined by an
>character of Hom(A, U(1)) = A* based on the Cartan subgroup A of the
>Borel subgroup B of G on which the principle series is defined, the
>principle series representations are parametrized by objects of A*.
>And these correspond to the Weyl chambers of the group, and so it
>would appear that you are looking at a bundle of Weyl chambers. These
>have correspondences to bundles of volume chambers in the general
>works on line bundles, so perhaps this is the "geometric"
>interpretation you are after?

Thanks for the above. It helps, but now I see that I was not clear
enough.

When I wrote that I am interested in "say, principal series", a better
way would be to say "any infinite dimensional irreducible unitary
representation." In my particular case I am interested in principal
series, but my question, my curiosity, goes wider.

Let me elaborate. Think of a 4d manifold with Lorentzian structure, and
equipped with a spin structure as well. That is the standard starting
point of all theories of general relativity with spinors and with Dirac
equation.

We have several bundles there.

1) Frame bundle - that does not require
metric. With frame bundle we have associated bundles. We have tensors
and tensor densities.
2) Orthonormal frame bundle. As Lorentz transformations have determinant
1 (or -1), this bundle can not be used to define densities, but can be
used to define tensors.

Here I have a question that goes beyond the present subject, but if
somebody has an answer or a hint, I will appreciate it. It seems that
every finite dimensional representation of SO(3,1) can be extented to a
representation of GL(4,R). Is that true? Why? In other words, by
restricting ourselves to orthonormal frames instead of all linear
frames, we gain nothing as regards possible, finite-dimensional,,
geometrical objects. The only associated bundles are tensor bundles.

3) Spin frame bundle.

That is a principal bundle with SL(2,C) as the structure group that is
soldered to the bundle of orthonormal frames by a 2->1 homomorphism
that commutes with SL(2,C)->L, where L is the connected component of the
identity of SO(3,1).

4) Associated bundles, in particular S2,S2bar,S4, namely those
corersponding to the natural (and its complex conjugate) representation
of SL(2,C) on C^2, and also on C^4 - as needed for the 4-component
spinors and Dirac equation. And their finite tensor products.

**********
Now, we can also think of infinite-dimesnional representations of
SL(2,C). They define other associate bundles, with fibers of infinite
dimensions. Infinite diomensional spaces are usually function spaces.
My question is: can these fibers be interpreted in a natural way as
function or maps between fibers listed under 1,2,3,4 above?

I guess there will be only finite number of readers that will understand
my question, but if the set of answers is, at least, non-empty, or
"fuzzily" non-emprty, I will appreaciate.

Of course, if anything in what I wrote above is not 100% clear, please
let me know it, and I will try to do it better.

ark

On 19 Jun 2003 01:15:52 -0400, Oz <acoohdb@btopenworld.com> wrote:

>But I don't see why people think this process needs to be magic
>('collapse'), and not just a standard quantum mechanical evolution. I
>think that, allowing for the rather vast differences in knowledge, we
>agree on this.

Stabdard quantum mechanical evolution will not produce even one "event".
It will continuously change the wave function (or observable). Nothing
will EVER happen. No data EVER will be recorded. While in reality we are
collecting data and we are recording times at which they are recorded.
Then we try to understand how these data appeared.

Standard evolution will not produce one dot on a photographic plate.
(though it can be used to predict probability distribution if we collect
"sufficiently many data"). But what if we have just three points? How
did they come? And why this rather then other time?

A piecewise deterministic random process can model these, but not a
deterministic QM evolution (and the standard evolution IS
deterministic).

Or so I think ....

ark

On Fri, 20 Jun 2003 04:12:35 +0000 (UTC), waterballon@hotmail.com (Hari
Seldon) wrote:

>Hi,
>
>I'm confused again. I have been looking in nakahara, choquet & al and
>Frankel to figure out the meaning of the phrase "a connection on a
>vector bundle".
>First this.
>
>---
>
>1) Suppose we have a principal G-bundle P over a manifold M, then we
>can define a connection 1-form with values in g on P. Now when we take
>a local trivialisation, we can define a local connection 1-form on M.

Trivialization induces a (local) map M ->P and so we can pullback vector
valued forms from P to M

>2) According to Choquet we can define an exerior covariant derivative
>on P using the connection form on P. In Nakahara the same thing
>appears, although
>it is called the covariant derivative instead of exterior covariant
>derivative (In Frankel the concept of a covariant derivative on P does
>not appear i think). It is defined as: D phi(X_1, .., X_{r+1}) = d_P
>phi(X_1^h, ..., X_{r+1}^H)..
>3) Also in choquet a covariant derivative on associated VECTOR bundles
>is defined. The same happens in Nakahara. They have it defined as a
>map V:eps(M,E) ->eps(M,E) (x) /\^1(M) (= section of E -> section of E
>(x) 1-form on M).

Covariant derivative can be thought of as a special case of exterior
covariant derivative. Covariant derivative acts on bundle valued 0-forms
and produces 1-form.

>4) Frankel is the only one who defines an exterior covariant
>derivative on VECTOR bundles (Nakahara mentions the operator that does
>this, but doesnt give it a name).

Remember that there is one-to-one correspondence between equivariant
vector valued forms on P and associated vector bundle valued forms on M.
Once you manage this equivalence some of your puzzles will disappear.

ark

>Ok those were the things i noticed. Now my problem. Only Frankel
>speaks about a connection on associated VECTOR bundles. Both Nakahara
>and Choquet never mention anything about the concept of connections on
>(associated) VECTOR bundles. The problem with Frankel is that he does
>not explain how that connection relates to the connection of the
>principal bundle.
>
>So I'm hoping anyone can explain to me exactly how it works. Moreover
>it can very well be that I have missed something in Nakahara or
>Choquet and I would
>appreciate it if you let me know that too. If anyone knows a single
>book that explains all of this, let me know too!
>
>Thanks in advance,
>
>Seldon

On 21 Jun 2003 22:04:50 -0400, Helmut Moritz <moritz@phgg.tu-graz.ac.at>
wrote:

>Why do people not do this in a clean way by means of operators projecting
>onto our 3+1 space-time?

To project, you need to project "along something". So, question is how
do we get this something, or how this "something" arises "naturally". Of
course standard "spontanous compactification" does not answer this
question, but at least tries to indicate the answer by using the term
"spontaneous" compactification. That means that we believe that there
is, perhaps, some underlying dynamics that "causes" curling up of some
of the dimension. Then, with such a picture in mind, we are building
simple toy models, with symmetry groups, we select projection operators
etc.

ark

On Mon, 30 Jun 2003 23:07:15 +0000 (UTC), whopkins@csd.uwm.edu (Alfred
Einstead) wrote:

>None of this has anything to with the issue above, which is far more
>substantial, arising from the diffeomorphism invariance of General
>Relativity, and has also been discussed at length by Davies in his
>1995 monograph on time.

It may have something to do. What if general relativity is incomplete?
What if there are more than four space-dimensions? What if time is more
than one-dimensional? What if gravity needs to be only partly quantized?

Uf you think in narrow terms and if assumprtions of Isham and Davis
are your assumptions - then you are right. But what if you, following
other physicists, you try to be more flexible? Collapse of our 4D
univesrse may happen in external time that is NOT quantized.

And, btw, frankly speaking, do we really understand where is quantum
theory coming from? Some thing they know, some others admit thay don't.

Of course you may say "I don't like unquantized external time" , and I
can even understand it. On the other hand, I see it as a viable
hypothesis.

ark
P.S. For unstance Laurent Nottale argues that quantization can be
derived from fractal staructure of space-time, while Tifft speculates
on 3D time.

On Thu, 3 Jul 2003 20:03:33 +0000 (UTC), rthompson10@new.rr.com (DickT)
wrote:

>> In particular, the terms eigenvalue/vector probably comes from the
>> notion that they are special values/vectors belonging to a matrix
>> in the same way as a person's nose is her own nose.
>>
>>
>> Arnold Neumaier
>
>
>One English translation used to be proper, using the old sense you see
>in "proper noun". But both proper and characteristic risked
>misinterpretation, which I guess is why the untranslated eigen swept
>the field.

In Russian for instance the terms is "sobstvennyi vector"
which means "proper" in the sense "one who belons to" -
where it is left open whom it belongs to. And Arnold is right:
it belongs to the matrix, it is its characteristic property.
--

Arkadiusz Jadczyk

On 2 Jul 2003 06:05:40 -0400, Charles Francis <charles@clef.demon.co.uk>
wrote:

>>But there is no such system; real systems (except the whole universe
>>containing us) are never isolated.
>
>You contradict yourself. You agree that if the empirical statements
>above are accepted then we can interpret quantum systems as isolated
>between measurement, and then say there is no such thing. You need some
>sort of empirical basis if your claims are physics rather than
>speculation.

Perhaps the issue is that what is more general than the Schroedinger (or
even Dirac) equation, is Liouville equation. That is to say that pure
states (they are on the boundary of mixed states) are an utopia and
should be forgotten. The evolution, whether the system IS observed, or
is "going to be observed in the future", always adds an additional
"mix".

ark

On 4 Jul 2003 02:33:33 -0400, Igor Khavkine <k_igor_k@lycos.com> wrote:

>After this point it gets a little tough for me, I know that these forms are
>supposed to satisfy some relations involving the exterior differential,
>wedge product, and the structure constants of G's Lie algebra. Which seems
>rather natural, since all of these things are related to left invariant
>vector fields and their Lie brackets. After all, that is where the
>structure constants come from. But I haven't gone through the explicit
>calculations yet, so I don't have a definite feeling for how they are
>related.

The crucial and useful formula here is:

< d omega , X \wedge Y > = L_X <omega , Y> - L_Y <omega, X> - < omega ,
[X, Y] )

where L_X stands for Lie derative along X

valid for any differential form and any two vector fields.

In our case we take for omega the Maurer-Cartan form
on G, and for X and Y two elements of the basis
of left-invariant vector fields.

The first two terms in the above formula then vanish,
as we have Lie derivatives (here, just a differential of a function)
of constant functions. (exercise: why constant?)

The rest follows by comparing the third term of the above
formula with the other side of Maurer-Cartan structure equation.

ark

On Fri, 11 Jul 2003 05:39:40 +0000 (UTC), stargene@earthlink.net (Gene
Partlow) wrote:

>Is there a not too technical reference which I can check to see
>how see how such conservation falls out of the equations.

Hi,

When you say Klein-Gordon or Dirac equation, you should make it clear
whether you have in mind "free" eqations, or equations with an "external
potential". For free equations it is easy to answer your question: both
are invariant under the Lorentz group. All henerators of the Poincare
group commute with the Dirac and Klein-Gordon operators. Solutions
of these equations carry a unitary representation of the Poincare group,
and thus, in particular, of the rotation subgroup. (For the massless
case also dilations). If you add external potential, then things get
more tricky, especially when you consider adding a magnetic field.
Rotational invariance in a "spherically symmetric magnetic field"
is more tricky, because magnetic filed enters the equation
via electromagnetic potential rather than the field itself.

ark

On Sun, 13 Jul 2003 06:12:21 +0000 (UTC), Joris Vankerschaver
<Joris.Vankerschaver@UGent.be> wrote:

>> If e is an isomorphism, then the induced metric is non-degenerate,
>> and we can define orthonormal frames with respect to the induced
>> metric: we get the reduction of the tangent frame bundle
>> GL(n) --> O(n) or GL(n) --> O(p,q) - whatever is the signature of
>> the scalar product in T.
>
>Now I see, thanks! I really begin to grasp the philosophy behind these
>alternative formulations of GR now.
>
>In my original post, on the other hand, I was referring to the "standard"
>formulation of GR (i.e. with the Einstein-Hilbert Lagrangian on the frame
>bundle).

Perhaps I should add here that it is not only "philosophy". Certain
geometric configurations that cause troubles and look as "singular"
within "standard" formulation, can make mathematical sense within
one of the alternative formulations.

Whether this is "good" or "bad" should always be discussed though.
On one hand it may be "good", because we have a mathematical
formalism that can handle more situations (for instance Einstein-Rosen
bridge
is singular within the standard formalism, but makes perfect sense
within the expanded framework, where a connection in the
abstract bundle and possibly degenerate soldering form are
independent variables. The connection is smooth across the bridge, even
if the soldering form becomes an "into" rather than an "onto" map.).
On the other hand one may try to argue that more general solutions
are "unphysical". Sometimes lot of research needs to be done, and lot
of time have to pass before we find what is "physical" and what not!

Think of magnetic monople as an example? Physical or not?
We do not know. If it will be found one day that something
similar exists in Nature - we will decide it was physical all the time.
For a while we tend to think of it as being non-physical.

ark

On Sun, 20 Jul 2003 06:47:58 +0000 (UTC), "Daniel E. Platt"
<DanP57@ispwest.com> wrote:

>This implies the von Neumann argument was
>not sufficient to guarantee the production of a bubble chamber track.

But particle tracks can be easily produced, including the timing of each
"dot", from a slightly enhanced interpretation and formalism:

PARTICLE TRACKS, EVENTS AND QUANTUM THEORY.
Progr.Theor.Phys. 93 (1995), 631-646

http://xxx.lanl.gov/abs/hep-th/9407157

and

ON QUANTUM JUMPS, EVENTS AND SPONTANEOUS LOCALIZATION MODELS.
Found. Phys. 25(1995) 743-762

http://xxx.lanl.gov/abs/hep-th/9408021

The timing information allows us to find (from the model) the
approximate velocity, as in real bubble chamber experiments when the
forming of the track is monitored, for instance by a high speed camera.
So the theory and the experiment can be compared also in this respect.

As far as I know this kind of information could not be deduced from
previous models: Mott, Shiff or others. If anybody has some
information about any previous papers that could predict not only
linearity, but also the distribution of the timing, I will appreciate
such an information.

ark

On Wed, 23 Jul 2003 06:29:19 +0000 (UTC), Frithjof Tillinger
<frithjof.tillinger@gmx.de> wrote:

>What are the prerequisites for the theorem that two-dimensional manifolds
>are conformally flat? Under which conditions does this theorem hold?

Locally all two-dimensional Riemannian manifolds are conformally
equivalent.

ark

On Fri, 25 Jul 2003 14:52:44 +0000 (UTC), baez@math.ucr.edu wrote:

>In article <beq281$1ig$1@glue.ucr.edu>,
>Miguel Carrion <miguel@math-cl-n02.math.ucr.edu> wrote:

(snip)

>>If f : A -> A' is a *-algebra isomorphism, can one find unitary
>>(uu^*=1=u^*u) elements u in A and u' in A' such that u'f(a) = f(au)?

(snip)

>Hey! Wait a minute. There's something funny about your concept!

Indeed there is something funny! Take u=I and u'=I and the answer is
"yes"!!!

ark

On Thu, 24 Jul 2003 21:31:45 +0000 (UTC), Student
<yssual@no-spam.yahoo.com> wrote:

>Hi,
>
>Suppose you have two quantum systems in interaction. One is labelled 1 and
>the other is labelled 2. Can you work out the conditions which allow to say
>that the evolution of system 1 caused by system 2 is unitary and of the
>type:
>
>(d/dt)_{1->2} rho_1 = U rho_2 U^{\dag}
>
>where rho_i is the density matrix of state i, i=1,2, and U is some
>(unspecified) unitary matrix (\dag means h.c.). It is understood the time
>variation on rho_1 of the above type is due to rho_2 only (whence the
>notation 1->2 in time derivative) and that one probably has to add time
>variation due to rho_1's own dynamics.
>
>Regards.

When two systems interact, they form one system - the total system. And
the evolution of the total system, in general, when there is a real
interaction (that is mixed terms in the Hamiltonian), does not generate
a Markovian evolution of "subsystems."

ark

On Thu, 31 Jul 2003 21:46:14 +0000 (UTC), "greywolf42"
<mingstb@sim-ss.com> wrote:

>> They are consistent with QM because moving into the quantum domain we
>> move from the category of continuous or even smooth fiber bundles
>> to the category of measurable functions and operator algebras.
>
>I was under the impression we were discussing the physical universe, not
>arbitrary mathematics.. Maxwell's equations are continuous approximations in
>Maxwell's original derivation.

The mathematics here is not "arbitrary" - as you call it. The
mathematics here is the one that helps us to describe and predict
physical phenomena of interest.

>> In QM
>> dF=0 needs to hold only "almost everywhere", and this leads to the
>> possibility of having an infinite number of monopoles in the universe.
>
>But there is no physical basis for dF<>0.

There may be one. We do not know. Quantum theory is not yet completely
understood. Let me quote from J. A. Wheeler: " No prediction lends
itself to a more critical test than this, that every law of physics,
pushed to the extreme, will be found to be statistical and approximate,
not mathematically perfect, precise."

> Again, (as per the prior
>statements and history that you snipped) your view was popular in the 60's
>and 70's.

Popularity of some view is one thing, discussion of unpopular views can
also be revealing. Alpher and Herman predicted background radiation
already in 1948. As Herman wrote: " There was no doubt in our mind that
we had a very interesting result, but the reaction of the astronomical
community ranged from skeptical to hostile." So, perhaps, it is good to
be open-minded?

> And a great deal of effort was spent looking for such things.
>And nothing was found. To orders of magnitude lower incidences than
>theorists expected.

Maybe people were looking at wrong places? Maybe the theory is not
ready? Maybe the theory needs a major re-thinking and re-writing? Or
maybe we will find something else - which will also be interesting. And,
we do not know that "nothing has been found". Perhaps something has
been found but the phenomenon is "elusive".

> Certainly, absence of evidence is not evidence of
>absence. However, when theorists have to keep "moving the goalposts" after
>each "not found," one can infer a major flaw in the theory.

And yet theorists predicted antimatter. P.A.M. Dirac wrote:

"The pure mathematician who wants to set up all his work with absolute
accuracy is not likely to get very far in physics".

But he also wrote:

"There are, at present, fundamental problems in theoretical physics the
solution of which will presumably require a more drastic revision of our
fundamental concepts than any that have gone before. Quite likely, these
changes will be so great that it will be beyond the power of human
intelligence to get the necessary new ideas by direct attempts to
formulate the experimental data in mathematical terms. The theoretical
worker in the future will, therefore, have to proceed in a more direct
way. The more powerful method of advance that can be suggested at
present is to employ all resources of pure mathematics in attempts to
perfect and generalize the mathematical formalism that forms the
existing basis of theoretical physics, and after each success in this
direction, to try to interpret the new mathematical features in terms of
physical entities."

Of course we do not have to agree with all that Dirac wrote but, at
least, we should acknowledge that he may have had a point....

ark

On 31 Jul 2003 07:21:11 -0400, duncan_borkowski@yahoo.com (Duncan
Borkowski) wrote:

>I've been reading on Bohm's hidden variables (HV) theory and have a
>question about the stated inability to choose between that and the
>Copenhagen Interpretation (CI).
>
>My understanding of the proposed quantum computer is that it relies on
>the pre-wave-collapse indeterminacy posited by CI. So, if they get a
>QC to work, wouldn't that be proof that CI is proven over HV?
>
>Duncan Borkowski
>
>[Moderator's note: a sufficiently intelligent proponent of any
>interpretation of quantum theory can usually get it to explain
>everything explained by all other interpretations, since the
>difference is not in their experimental predictions, but in their
>account of what is "really happening". - jb]

In fact HV gives different experimental predictions than orthodox CI,
for non-standard experimental questions like for instance for the
tunnlling times. See Fig 2 and Fig 3 in

http://www.quantumfuture.net/quantum_future/papers/garda.htm

also

http://xxx.lanl.gov/abs/quant-ph/9911113

So, also for "typical" questions "intelligent proponents" should
certainly take care about "agreement" with well established
facts, a really intelligent ones would not refrain from adding
new predictions for experimental questions that are not within
the scope of CI-colored QM textbooks - like timing of events.

Quantum computers, which interface micro and macro, and which
work in real time, giving unique answers to unique questions
may well belong to this "non-standard" category.

ark

On 31 Jul 2003 08:04:40 -0400, waterballon@hotmail.com (Hari Seldon)
wrote:

>Hi,
>
>I wonder about the following. How does a lorentz transformation affect
>a spinor? I would like to know this in terms of spin structure,
>lorentz frame bundle, spinor bundle etc; so in mathematical language

Probably you are thinking of "spinor at a point." For that you need just
algebra.

Take 2-dimensional complex vector space V. Let W be its complex
conjugate. Let M be the hermitian part of the tensor product
V tensor W.

M is real four-dimensional, endowed with a natural (through the
construction) Minkowski metric (unique up to scale).

Think of elements of V as spinors and of elements of M as vectors.

Use Pauli matrices to associate to a complex basis in V a real basis in
M. Then when you apply GL(2,C) transformation to a basis in V,
the induced basis in M undergoes Lorentz transformation (or a dilation).

ark

On 31 Jul 2003 08:04:42 -0400, ysauvg@yahoo.com (Yvon Sauvageau) wrote:

>Although these effects are minute, it seems obvious to me that we end
>up with a set of stellar matter that is more energized than at the
>starting time. And by conservation of energy, the photon is bound to
>lose energy at a more or less constant rate as it travels through
>populated space. And consequently, we should observe a linear
>distance/redshift relationship.
>
>Would some of you care to comment on my conception?
>
>Thank you,
>
>Yvon Sauvageau
>Mountain View, California
>
>[Moderator's note: This effect is real but negligible. If it were the
>main cause of redshifts, distant galaxies would be substantially more
>redshifted when they were directly behind a bunch of other galaxies,
>since then photons from them would have interacted gravitationally
>with more matter before reaching us. This effect is not seen.
>Instead, we see "gravitational lensing" in this situation. - jb]

Two comments came to my mind:

The reason may be not so much in the "stellar" matter, but in "dark
matter" , provided this dark matter fills the space rather uniformly and
there is enough of it to dominate the effect. In fact, with such a
hypothesis, it should be possible to estimate the amount of such a dark
matter in space, provided we make some hypothesis on the form it is
distributed.

On the other hand it is not clear to me that such an effect should give
a redshift that is uniform (in the multiplicative sense) through all the
frequency spectrum. So a careful studying of (non)uniformity of the
redshift may give us a clue here.

ark

On 31 Jul 2003 08:04:43 -0400, Student <yssual@no-spam.yahoo.com> wrote:

>
>Dear all,
>
>I would like to know about your thinking of the phase of a particle, by
>which is meant (most probably) the phase of its wavefunction. It is often
>said to be unphysical, but far from being so when it is not constant and
>varies along with the "real" part of the wavefunction. Also I think it
>becomes very important when dealing with many particles, since it is the
>principal reason why interference behaves the strange way it does. Could
>you comment please and say in which cases it is known this phase plays a
>major role?
>
>Best regards.

Usually, when running programs visualizing wave function propagation
only the amplitude is plotted. But phase is also very interesting. You
can see bot amplitude AND the phase on our Java Wave Lab, when you can
see what happens to the phase when the wave scatters from a potential,
or when the wave meets a "detector" (normal, standard, quantum mechanics
does not know how to model detectors and events, but that is another
story ...)

Phase of the wave function is represented by an angle. So, you see
rotating wave. I think this representation is pretty unique!

http://www.quantumfuture.net/quantum_future/applets/eeqt.html

There is not much of a description yet, but you can play with
parameters, set potential and detectors etc.

NB. The program itself is an open source program.

ark

On Sun, 10 Aug 2003 22:23:38 +0000 (UTC), delphi29@excite.com (Delphi
Twentine) wrote:

>There is a paper by Peter Lynds that has apparently caught
>the attention of the media. It is to appear in Foundations
>of Physics Letters this month:

Foundations of Physics Letters is losing completely any credibility
it ever used to have. Check also

http://www.phy.cuhk.edu.hk/course/phy2002/forum/messages/339.html

http://www.phy.cuhk.edu.hk/course/phy2002/forum/messages/427.html
http://www.phy.cuhk.edu.hk/course/phy2002/forum/messages/430.html

ark