spin
pusch
> Would some kind and knowledgeable person explian spin to me.
>
> Is it just another "energy state" or a quantum number to do the
> book-keeping, or is it a real quantity like the angular momentum of a
> spining knuckel ball?
It is not an energy-state; it is a vector-valued quantum-mechanical
operator that obeys the same commutation-relations as the ``orbital''
angular momentum, and is interpreted as representing the ``intrinsic''
angular momentum of a quantum particle. It is not itself a quantum number,
but it is _associated_ with two quantum numbers: the spin-squared S^2,
and the component of the projection of spin along any fixed direction
(conventionally taken to be along the Z-axis, S_z). If a state is a
simultaneous eigenstate of S^2, and S_z, the eigenvalue of S^2 is
\hbar \sqrt[s(s+1)], where the spin quantum-number `s' is an integer
or half an odd integer, i.e., one of {0, 1/2, 1, 3/2, 2, 5/2, ... },
and the eigenvalues of S_z are members of the set {-s, -(s-1), -(s-2), ... ,
(s-2), (s-1), s} times \hbar.
The spin is definitely ``real'' in the sense that it really =DOES= carry
angular momentum: For example, the photons in a beam of circularly polarized
light have their spins either parallel or antiparallel to its direction,
depending on whether the polarization is right-handed or left-handed,
and it has been observed experimentally that if such a polarized beam
is directed at an absorbing target on a low-friction support, the target
will begin to rotate in the same direction as the spins of the photons.
So a circularly polarized beam of light _does_ carry angular momentum.
(I do not recall if this experiment has ever been carried out using
any particles other than photons; it would be difficult to perform
using protons or electrons because they are charged, but in principle
it should certainly be possible to repeat it using a polarized beam
of neutrons, which are spin-1/2 particles likje protons and electrons.)
Most physicists believe that it is inappropriate to visualize the spin as
representing anything ``spinning;'' Pauli, for example, referred to the spin
of the spin-1/2 electron as being a ``non-classical two-valued-ness.''
However, a minority of workers, such as Assim Barut and David Hestenes
believe that the spin is related to the ``zitterbewegung'' motion of the
electron, which is a periodic oscillation of the expected velocity of the
electron at twice its de Broglie frequency. (In Dirac's theory of the
electron, the ``momentum'' and ``velocity'' operators are no longer
proportional to each other, and while the momentum is a constant of the
motion of a free electron, its velocity is _NOT_ constant, but precesses
about the momentum. Hestenes has shown that this velocity-precession
implies that the streamlines of the ``probability current'' for a
Dirac-particle wave-packet circulate in the plane perpendicular to the
expected spin, so that in some sense it is the _wave-packet_ that is
``rotating.'' Many of Hestenes' papers on this subject are online at:
http://modelingnts.la.asu.edu/html/GAinQM.html
However, I again caution your that Barut's and Hestenes' interpretation
that zitterbewegung and spin are two aspects of the same thing is =NOT=
generally accepted by most physicists (even though I personally happen
to be one of the few that do...).
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Kenneth Simpson
if anyone has formulated electroweak interactions or the
standard model in Hestene's geometric calculus?
-- Ken
pusch wrote:
>
> Many of Hestenes' papers on this subject are online at:
>
> http://modelingnts.la.asu.edu/html/GAinQM.html
>
--
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Kenneth Simpson Well Connected Computing, Inc.
Email: k...@wellconnected.com 1001 Bridgeway
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pusch
> Hi - on a slightly different note, would you happen to know
> if anyone has formulated electroweak interactions or the
> standard model in Hestene's geometric calculus?
Yes --- Hestenes himself, in:
``Space-Time Structure of the Weak and Electromagnetic Interactions,''
D. Hestenes, Foundations of Physics, v.12, pp.153--168 (1982)
Hestenes does not appear to have this paper online, but if you cannot find
it at your local University Library, you might try writing him directly;
it's possible he still has reprints left.
Hestenes also _very_ briefly sketches the natural SU(2)xU(1) structure of
his reformulation of the Dirac equation on p.63 of the online draft of his
book, ``Space-Time Calculus.'' The basic idea is,
1.) Rotors acting on a Hestenes-spinor from the right instead of the left
induce an ``isospin'' transformation instead of a spin transformation.
2.) Reducing the most general Space-Time calculus form of the Dirac equation
to ``Hestenes standard form'' involves singling out a pair of arbitrary
timelike and spacelike ``isospin-basis vectors;'' this does not ``break''
the Lorentz-symmetry of the equations, however, since these vectors are
arbitrary, any more than choosing an orthonormal basis tetrad ``breaks''
Lorentz symmetry. Moreover, a different choice of ``isospin basis vectors''
can always be transformed to the ``standard isospin gauge'' by a global
rotor acting from the right --- i.e., by performing a global ``isospin
gauge transformation.''
3.) Since these ``isospin basis vectors'' are arbitrary but fixed, in the
spirit of Yang-Mills theory, one can imagine ``gauging'' this global
transformation (or at least part of it). Holding the timelike vector
fixed, the arbitrary direction of the spacelike vector and the ``rotor''
nature of the transformation implies an SU(2) covariance; there is also
a second, more subtle covariance involving local rotations _about_ the
spacelike vector, which brings in the U(1) covariance already discussed
in Hestenes' early papers on the Dirac equation. To compensate for these
local ``isospin gauge transformations,'' one must of course also introduce
corresponding ``gauge-field'' vector-potentials; by adding corresponding
``kinetic terms'' to the Lagrangian, these gauge-potentials then become
dynamical fields in their own right.
The one loose end in all this is, of course, the Higgs fields; these *can*
be introduced into the Hestenes formalism, but they are somewhat unnatural.
(However, many feel that elementary Higgses are ``unnatural'' even in the
``Standard Model'' except possibly as auxiliary degrees of freedom of an
``effective theory,'' and have sought various ways of eliminating them from
the theory --- e.g., ``technicolor'' theories --- so some might feel that
Higgses being ``unnatural'' in the Hestenes approach is simply one more sign
that they are indeed ``auxiliary'' fields that will not appear in a ``Final
Theory.'')
One possible (albeit speculative) approach would be to to look to the
remaining timelike ``isospin basis vector,'' which so far has still been
held ``rigid.'' By allowing the timelike ``isospin basis-vector'' to vary
as well (modulo still remaining timelike), it turns out that this introduces
_exactly_ the correct set of effective degrees of freedom to produce a
``non-compact Higgs model;'' the problem with this approach is that the
resulting theory is non-renormalizable, so it would imply that the
``Standard Model'' can only be regarded as an ``effective'' field-theory.
(Again, some would not consider this a disadvantage, but a sign that one
should look ``deeper'' for a more elegant ``Final Theory.'')
Finally, there is one point that appeared in Hestenes' first book,
``Space-Time Algebra,'' and in a couple of his early papers, but which for
some reason he has not chosen to pursue in recent years: By dropping the
requirement that the Dirac field be a ``Hestenes-spinor'' and allowing it
to be a general Clifford-number, one instead arrives at a very elegant
equation
\clifford_deriv \Psi = m \Psi \gamma_5
which is equivalent to the ``Dirac-Gursey equations.'' The Dirac-Gursey
equations describe a coupled pair of Dirac-spinors with the same mass,
which have very a natural interpretation as an ``isospin doublet.''
IMO, it is this equation, not the ``Dirac-Hestenes equation'' that is
the most natural starting-point for a theory, since it automatically
implies that all particles occur in ``isospin doublets'' with an
SU(2)xU(1) structure, and the only remaining mysteries are:
A.) Where does color SU(3) come from ?, and
B.) Why do the blasted things come in ``Xerox(tm) copies''
(i.e., what is the solution to the ``generation problem'') ?
IIRC, Hestenes provided a partial solution to (A.) in the Found. Phys.
paper cited above --- but NOT one that most physicists would find palatable:
One *can* construct a ``natural'' SU(3) action on a Hestenes-spinor, but only
if one is willing to pay the horrible price of breaking Lorentz invariance!
Hestenes proposes that one possible ``out'' to this impasse is to again
regard QCD as being only an ``effective'' theory --- and moreover, one that
only applies within the _interior_ of a hadron in its proper ``rest'' frame.
This approach solves the ``confinement'' problem (quarks are not ``real''
particles, but rather ``effective'' particles that are only defined inside
a hadron); however, it clearly introduces a host of others, such as:
What ``rest frame'' should one use to compute QCD effects in a hadronic
collision? IMO, the only way this approach will be ``natural'' is if it
proves possible to construct a _NONLINEAR_ Clifford-algebraic theory of
hadrons along somewhat the same lines as the Skyrme model, and if this model
allows identification of the quarks and gluons as ``effective'' low-energy
excitations of the Skyrme-like soliton via the Hestenes SU(3) construction.
However, both Hestenes' approach and my take on it clearly have serious
problems --- in particular, how is one to explain the success of the
``asymptotic freedom of QCD'' interpretation of hadronic cross-section
scaling, even to the highest energies so far observed ??? One would
instead have expected strong deviations from QCD would occur once the
energies of the collisions exceed the natural energy-scale associated
with the ``rigidity'' of the effective quark and gluon excitations about
the nonlinear skyrme-like soliton, which is unlikely to be much more than
the order of magnitude of the effective quark-masses, which I would expect
to be the ``current-quarkk'' masses, not the ``constituent-quark'' masses.
However, in the case of the ``up/down'' doublet, the ``current-quark'' masses
are only a few MeV, so such deviations *should* certainly have been observed
by now !!! The only ``outs'' I've ever been able to come up with are:
1.) The ``effective QCD'' inside the skyrme-like solitons must be a
_topological_ field theory instead of just a ``weak-field excitation;'' or
2.) The ``effective QCD'' inside the skyrme-like solitons is the result of
``extra dimensions,'' as in the recently renascent Kaluza-Klein theories,
or their offspring, the ``brane'' theories.
A variation on theme (2.) is an old proposal by J.P. Vigier, in which he
derives the Gell-Mann/Okubo mass-formula by assuming a modified Kaluza-Klein
model in which the interior of the hadron has an ``Einstein/de Sitter''
effective geometry, while the exterior has a Lorentzian effective geometry;
however, one then needs a ``natural'' explanation for how the scale of the
effective geometry of spacetime can change from the (presumably) Planck-scale
radii of the exterior, to the ~1 fermi or so of the hadronic interior.
This seems more than a little implausible on the face of it --- but may not
be *quite* so outrageous in some of the recent ``brane'' models, in which
the ``domain wall'' upon which the Universe lives and which is embedded
in ``extra dimensions,'' has a ``thickness'' that may perhaps be as high as
a FRACTION OF A MILLIMETER --- =MUCH= closer to the ~1 fermi or so Vigier
postulates for the hadronic interior than the Planck scale of KK theory !!! :-/
BUT, I believe I've rampantly speculated enough for one day... ;-I
pusch
> Hi - on a slightly different note, would you happen to know
> if anyone has formulated electroweak interactions or the
> standard model in Hestene's geometric calculus?
While it doesn't *directly* relate to electroweak theory, you might also be
interested in the paper ``A Multivector Derivative Approach to Lagrangian
Field Theory,'' by A. N. Lasenby, C. J. L. Doran and S. F. Gull,
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/lag_field.html
In it, the authors show how to apply Geometric Calculus techniques to
derive the Euler-Lagrange equations of motion, and also give a generalized
form of Noether's Theorem. There are also a number of other interesting
papers on the Geometric Calculus and the Dirac Equation at this website.