Subject: gauge transformations w/quaternions II
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/15
Message-Id: <5lfajd$1i9$1@agate.berkeley.edu>
Newsgroups: sci.physics.research
Hello:
In this post I will address Achille Hui's insightful criticism (dejanews
1997/04/22) of work I presented (1997/04/20) concerning gauge
transformations of the Maxwell equations with quaternions. I hope
to show that a quaternion field gauge transformation requires a
continuity equation for the vector portion of the quaternion as well
as the standard continuity equation for the scalar portion. Before I
get to the specific mathematical analysis, let me first make a few
general comments about gauge transformations.
The ability to choose a gauge for the Maxwell equations has been
discussed in this newsgroup before. A gauge corresponds to how one
measures the electrmagnetic 4-potential, phi and A. Within well-
defined limitations, one is free to choose a standard for the 4-
potential. For example, if one decides that the divergence of A
should always be zero, this is called choosing the Coulomb gauge.
Gauge transformations are a very powerful technique for solving
practical problems. Gauge symmetry also has profound implications
theoretically, particularly for quantum field theory. Spacetime-
dependent gauge symmetry leads to the gauge fields that form the
fabric of the standard model, U(1)xSU(2)xSU(3).
My initial work exploring gauge transformations with quaternions
suggested an _additional_ constraint, one that will be elaborated on
and extended later. This is potentially the most controversial aspect.
I would expect that particle physicists to ardently defend their
standard-model right to choose a gauge :-) In my proposal to be
outlined below, there still is the freedom to choose a gauge so long as
the additional constraint is insignificant, which is the case in the
classical electrodynamics region. Under relativistic conditions, a
continuity-like equation for the vector portion of the quaternion is
required.
The history of physics is a history of increasing constraints imposed
by new mathematical descriptions of nature. The fuzzy philosophy
of Aristotle was replaced by the precision of Newton's three laws.
Calculus provided the lens to see the workings of Nature's clock. The
instantaneous transmission of light was limited to c by special
relativity. The Minkowski metric and the Lorentz group created the
analytical tools required. The arbitrary frequencies of classical
atomic systems was constrained by quantum mechanics using
complex Hilbert spaces. A key component of quantum field theory is
the freedom to choose a gauge and its implied symmetry. I am
suggesting that this freedom be further constrained under relativistic
conditions. From a mathematical perspective, I want to replace the
notion that all terms in an equation be tensors of the same rank (all
scalars, all vectors, all anti-symmetric second rank tensors...) In
their place, I put quaternions, which can locally represent events,
fields, transformations, operators, or whatever, and still be subject to
the same mathematical rules. It's free market mathematical
democracy on a local level (makes me sound like a Republican :-)
Enough philosophy...
The E and B fields can be generated from the 4-potential phi and A
with the following quaternion equation:
(d/dt + Del)(phi&
nbsp;+ A)
?scalar eq?
|
-
E |
B
= d phi/dt -
div A + Grad phi + dA/dt
+ Curl A
The vector part is familiar. The scalar equation does not arise in the
typical analysis of electromagnetism, although it has the form of a
gauge relation. To try and define its role, I examined a gauge
transformation with a scalar field as is usually done:
L = dg/dt -
Grad g
Achille Hui correctly pointed out that for the sake of logical
consistency, I should be using a quaternion field. Therefore let me
repeat the exercise.
Consider the following quaternion field L, where small g is the scalar
and big G the 3-vector:
L = dg/dt -
Grad g + div G + dG/dt
The terms with the small g are the typical scalar field gauge
transformation. The terms with the big G are my guess at a
reasonable extension of the scalar field to a quaternion field. Big G is
handled in a manner that parallels small g, namely there is a time
derivative and del operation that transforms vector big G into the
scalar div G, just like scalar small g was transformed to vector Grad g.
Transform the quaternion potential with L.
phi -
> phi' = phi -
dg/dt + div G
A -
> A' = A
+ Grad g + dG/dt
Substitute these into the first equation:
(d/dt + Del)(phi - dg/dt + div G
+ A + Grad g + dG/dt)
= d phi/dt - d^2 g/dt^2 + d div G/dt /* time der. of scalars */
+ dA/dt + d Grad g/dt + d^2 G/dt^2 /* time der. of vectors */
+ Grad phi - Grad dg/dt + Grad div G /* Grads of scalars */
- div A - div Grad g - div dG/dt /* divs of vectors */
+ Curl A + Curl Grad g + Curl dG/dt /* Curls of vectors */
?scalar eq?
|
-
E |
B
= d phi/dt -
div A + Grad phi + dA/dt
+ Curl A
- d^2 g/dt^2 -
div Grad g
+ d^2 G/dt^2 +
Grad div G + Curl dG/dt
An equation should not be altered by a gauge transformation, but
this does not appear to be the case with L at first glance! Let me
write the final 5 terms in a suggestive way.
rho
J
|
|
d
dg
- -- --
- div Grad g
dt dt
d
dG
dG
+ -- --
+ Grad div G +
Curl --
dt dt
dt
|
|
|
A
phi
A
Since the first two terms must always be zero, the proposed mapping
forms the continuity equation:
d rho/dt +
div J = 0
As Achille Hui pointed out, the J must be irrotational (Curl J = Curl
Grad g = 0). (This may be due to my choice for big G which lacked a
curl, but I haven't analyzed that case yet). It is noteworthy that the
continuity equation arises naturally from an analysis of a quaternion
gauge transformation.
The final three terms with the proposed map reform the vector
potential, and thus B and -E. They will be zero only if B - E = 0. In
some sense, this is like a continuity equation for the vector field big
G. The time rate of change of A must balance the gradient phi and
the
curl of A, which all can be expressed in terms of the same
vector field big G. For those folks that have lasted this long, I must
admit I was struck by the way this new constraint neatly parallels
the scalar continuity equation.
On possible response: "Elegance, schmellegance, this is wrong
'cause
I've used gauge transformations before that don't make this demand
and they work!" Remember that any constraint has regions
where it
is applicable. Newtonian physics works until the speed of light is
approached. Classical atomic physics works until the number of
particles gets small. The vector continuity constraint is not relevant
for classical electrodynamics. The classical region of spacetime
happens for intervals where
|delta t| >>
> |delta t, x, y, z|
== classical
An analogous region for a gauge transformation is
|dg/dt - Grad g| >>> |div G
+ dG/dt| == classical
EM
In this region, the transformation then simplifies to
phi -
> phi' = phi - dg/dt
A -
> A' = A +
Grad g
This is precisely the type of transformation that has been used so
often before, and it is still part of the quaternion formulation of the
Maxwell equations, but now it is constrained to classical
electrodynamics.
The question now concerns the validity of gauge freedom of the
Maxwell equations under relativistic conditions, a cornerstone of
quantum field theory. I prefer to tie down the gauge
transformations under relativistic conditions with a continuity
equation for the vector field that is so nicely symmetric with the
scalar continuity equation so I can avoid the funny business of
regularization and renormalization. I enjoy playing with this new
way of looking at old math.
I apologize for the length of this post, but Achille Hui raised a solid
objection and it took me this long (both in time and space :-) to make
a solid answer.
Doug Sweetser
http://world.std.com/~sweetser
Surfing a wave of quaternion logic,
trying to catch the BIG one.
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