[time 17] Re: Reformulating Weyl's Unified Field Theory

Hitoshi Kitada (hitoshi@kitada.com)
Mon, 15 Mar 1999 13:55:50 +0900

Dear Stephen,

I agree that Weyl's book "Space-Time-Matter" is a beautiful book. However, it
is concerned only with classical aspect, not with quantum mechanical aspect as
you quote:

-----Original Message-----
From: Stephen P. King <stephenk1@home.com>
To: Time List <time@kitada.com>
Date: Sunday, March 14, 1999 12:57 PM
Subject: [time 15] eformulating Weyl's Unified Field Theory

>Here we go!
>Michael J. Martin, University of Missouri
> Reformulating Weyl's Unified Field Theory of 1919
> In 1918-19, Hermann Weyl introduced the first classical unified
>field theory, a theory that united electromagnetism and general
>relativity into one geometrical framework. This theory, the forerunner
>of modern gauge theory, involved transforming the metric 2-tensor with a
>"scale factor" that, in turn, transformed the line element, causing
>lengths of parallel transported vectors, as well as their orientations,
>to change. This length dependence was later dismissed, and reformulated
>by London and Fock as wave function phase dependence. We attempt another
>reformulation, whereby the volume element varies on parallel transport
>(volume instead of length) unless it is transformed into a tensor
>density. This transformation introduces a gauge potential, the
>contracted Christoffel symbol, which maintains invariance under
>covariant differentiation.
> The electromagnetic potential, in pure gauge form, is shown to be
>equivalent to a constant multiple of the contracted Christoffel symbol.
>The gravitational field is introduced into wave mechanics via the
>contracted Christoffel symbol, which appears as an integrable phase
>factor. The contracted Christoffel symbol and the electromagnetic
>4-potential are related via a simple gauge transformation under this
> Density Transformations in a Gauge Unification of Gravity and
> The Lagrangian which describes an electron wavefunction in QED
>remains invariant under local phase transformations through a gauge
>transformation of the electromagnetic 4-potential, which is the gauge
>connection appearing in the QED covariant derivative. We postulate that
>similarly, the Lagrangian which describes the electron wavefunction in
>both an electromagnetic and gravitational field remains invariant under
>local phase transformations of the wave function density, via the
>introduction of the contracted Christoffel symbol as gauge connection in
>the covariant derivative. The resulting transformations, of the QED
>Lagrangian density, are explored. The quanta of the gravitational field
>would appear to be both massless and of infinite range, as the
>transformations are nearly identical to U(1). We explore the case of
>wave function --> wave function density, and wave function
>density-->wave function density, where a density is defi.....
>searching further...

I think there arises a problem when we consider quantum-mechanical aspect of
things as we discussed:

-----Original Message-----
From: Hitoshi Kitada <hitoshi@kitada.com>
To: stephenk1@home.com <stephenk1@home.com>
Date: Saturday, March 13, 1999 2:41 PM
Subject: RE: Quantum groups (was: Re: Poisson groups/actions)

>Dear Stephen,
>I seem to have underestimated John Baez.
>-----Original Message-----
>From: Stephen Paul King <stephenk1@home.com>
>To: hitoshi@kitada.com <hitoshi@kitada.com>
>Date: Saturday, March 13, 1999 12:24 PM
>Subject: Re: Quantum groups (was: Re: Poisson groups/actions)
>>Dear Hitoshi,
>> John Baez himself seems to be agreeing with your arguments! :)
>>On 12 Mar 1999 20:33:28 GMT, you wrote:
>>>In article <36E692F5.A8CDCB86@phys.lsu.edu>,
>>>Vesselin Gueorguiev <vesselin@baton.phys.lsu.edu> wrote more or less this:
>>>>I have had problems convincing myself that x,y,z should become
>>>Me too - at least now. Back when I was a grad student, I was really
>>>excited about this idea. This is how I thought: the hydrogen atom is
>>>saved from collapse by quantum mechanics. Because momentum and position
>>>don't commute, the electron can't have both a definite position and
>>>a definite momentum. This keeps it from falling into the proton.
>>>Now in general relativity we have a problem similar to electron collapse:
>>>the singularity that occurs when a large mass collapses. So maybe we
>>>can solve this problem too using quantum mechanics. How? Well, maybe
>>>we should go a bit further and say, not just that momentum and position
>>>don't commute, but that the diffferent components of position don't
>>>commute! Then nothing can even have a definite position. So there's
>>>no way for matter to collapse down to a single point!
>>>It sounded convincing to me back then. I still believe that quantum
>>>mechanics will eliminate the singularities in general relativity. But
>>>I no longer think the trick is to make different components of position
>>>commute. Here's why I think this idea is wrong. In general relativity,
>>>the components of position depend on the coordinate system you use. They
>>>are not really observable quantities. We don't just go out and measure
>>>the x coordinate of something. Fundamentally, what we do is measure the
>>>relation of one physical object to another. Since the components of
>>>position are not really physical observables, we shouldn't quantize them!
>His point agrees with my standpoint.
>>>More precisely, since they depend on an arbitrary choice of coordinates,
>>>we shouldn't quantize them unless we can do so in a way that doesn't
>>>depend on our choice of coordinates. But there appears to be no such
>>>way. I tried to find one and I couldn't. Recently lots more people have
>>>been playing this game, and none of them has succeeded in this either.
>>>Whenever people quantize spacetime by making the coordinates
>>>they break the diffeomorphism symmetry down to a much smaller group.
>Exactly! He seems to have noticed the defect of the present activities of
>>>Now of course this is not a definitive argument. The final test is
>>>experiment. But I've decided that the motivation for making the
>>>components of position noncommutative is too weak, and the problems
>>>with it too great.
>>>>It seems that they have an argument for this but it is not too appealing
>>>> "...the smaller the cube the bigger the energy... generates a black
>>>> ... Thus it is reasonable to assume that the operators corresponding to
>>>> coordinates x, y, and z do not commute..."
>>>Right, that's kind of how the argument goes. The problem with this
>>>argument is that there's a huge logical gap right before the "Thus".
>>>Why should we assume this? Why not something else? I've never seen
>>>a plausible answer.
>Best wishes,

How do you think of the problems which arise when we consider
quantum-mechanical aspects of things in the context of Weyl?

Best wishes,
Hitoshi Kitada

This archive was generated by hypermail 2.0b3 on Sat Oct 16 1999 - 00:29:45 JST