Stern Gerlach and spiining ball electrons
Pete
Hello
In his notes on Lie Groups and Quantum Mechanics, Michael Weiss gives the
results of Stern and Gerlach 's experiment as evidence that the spinning
ball model of electron spin is wrong - see
http://math.ucr.edu/home/baez/lie/node11.html
I'm a little confused about this and would appreciate it if someone could
explain why a classical model of the electron as a spinning ball is
incompatible with the results.
One of the reasons given seems to be that we would expect a continuous range
of spin angles rather than the observed two states. But given the relative
sizes of the electron field and that of the apparatus, wouldn't we expect
the apparatus to induce a polarisation no matter what the original spin
direction. Iow, wouldn't the apparatus put a torque on a spinning field
which would force it to align with the field. Passing through a subsequent
orthogonal field would again force the spin axis into one of two stable
states with (in this case) a 50/50 probability. So why do we have to abandon
our spinning ball model?
Thanks
Pete
Michael Weiss
: I'm a little confused about this and would appreciate it if someone could
: explain why a classical model of the electron as a spinning ball is
: incompatible with the results.
:
: One of the reasons given seems to be that we would expect a continuous range
: of spin angles rather than the observed two states. But given the relative
: sizes of the electron field and that of the apparatus, wouldn't we expect
: the apparatus to induce a polarisation no matter what the original spin
: direction. Iow, wouldn't the apparatus put a torque on a spinning field
: which would force it to align with the field. Passing through a subsequent
: orthogonal field would again force the spin axis into one of two stable
: states with (in this case) a 50/50 probability. So why do we have to abandon
: our spinning ball model?
You didn't read footnote 5! Here it is:
[begin footnote]
Why wouldn't the electron simply snap into alignment with the magnetic
field? Answer: the spinning electron would act like a gyroscope, and precess
in response to the torque exerted by the field. Thus it would maintain its
angle of inclination to the field.
[end footnote]
I should add that history is a good deal more complicated than this footnote
suggests--- spectroscopy had a lot to do with discovery of electron spin,
probably more than the S-G experiment. I've sketched what I know of the
history in:
http://math.ucr.edu/home/baez/spin/spin.html
I was using the S-G experiment partly as a pedagogical hook. I stole the
idea from Feynman's Lectures on Physics, volume III --- you might want to
take a look at that for a lot more on spin.
I guess the biggest heap of evidence for the QM description of spin comes
from the success of the Pauli exclusion principle.
Pete
"Michael Weiss" <mic...@spamfree.net> wrote in message
news:jI2X9.12423$e34....@nwrddc04.gnilink.net...
> Pete wondered about the Stern-Gerlach experiment:
>> ..... wouldn't the apparatus put a torque on a spinning field which
>> would force it to align with the field. Passing through a
>> subsequent >>orthogonal field would again force the spin axis into
>> one of two stable states with (in this case) a 50/50
>> probability. So why do we have to abandon our spinning ball model?
> You didn't read footnote 5! Here it is:
>
> [begin footnote]
> Why wouldn't the electron simply snap into alignment with the magnetic
> field? Answer: the spinning electron would act like a gyroscope, and precess
> in response to the torque exerted by the field. Thus it would maintain its
> angle of inclination to the field.
> [end footnote]
Yes you're right I didn't see that - thanks for putting me straight.
In fact, I did turn to Feynman vol III where he also discusses the S-G
experiment, and it became obvious that the idealised apparatus acts as a
filter and not as an aligner. This is emphasised by the results presented on
page 5-9, especially (5.14) & (5.17) where we see that the effect of
splitting and recombining is entirely null.
But, can't I still picture a spinning ball with the proviso that it doesn't
interact with a magnetic field in a classical/continuous way, but in a
manner in which the force is quantised parallel or anti-parallel wrt to the
field? After all, the behaviour of the electron in a series of S-G
experiments seems to suggest that spin axis has some form of intrinsic angle
or phase even if an individual measurement only ever gives an up/down
result.
Or should I abandon any attempt to geometricize spin and try to think just
in terms of some abstract Hilbert/state space?
> I guess the biggest heap of evidence for the QM description of spin comes
> from the success of the Pauli exclusion principle.
I've been wanting to get some form of understanding of Fermi-Dirac
statistics for many years but it still seems like magic. Even Feynman
doesn't seem to clarify this - at least for me.
Pete
Michael Weiss
: But, can't I still picture a spinning ball with the proviso that it
doesn't
: interact with a magnetic field in a classical/continuous way, but in a
: manner in which the force is quantised parallel or anti-parallel wrt to
the
: field? ....
: Or should I abandon any attempt to geometricize spin and try to think just
: in terms of some abstract Hilbert/state space?
Sure you can picture the electron as a spinning ball! I usually think of it
as silvery, with just a faint hint of mist clinging to it.
Nothing wrong with these pictures, as long as you remember that they *are*
just pictures. Pedagogical or mnemonic crutches, useful for those (like me)
whose like their math served with a generous dollop of imagery on top.
(Once upon a time I wrote a post, "Anschaulichkeit vs. Abscheulishkeit",
about the differing visual preferences of Heisenberg and Schroedinger.)
This geometrical picture will carry you pretty far. Pauli used it to
explain several features of some spectrum or other (I forget the exact
historical details) a few years before "real" QM came along.
In some ways, it's a little like the "expanding balloon" picture in GR. It
works pretty well, so long as you keep tucked away in the back of your mind
a list of things wrong with it.
The list for the "spinning ball electron" would include three main points:
a) Quantization.
b) A 360 degree rotation reverses the quantum mechanical phase, and does not
return the electron to its initial state (unlike a 720 degree rotation).
c) Don't ask for the radius of the electron. Classical physicists (Lorentz,
for example) tried to build literal models of the electron as an actual ball
of charge; these models always ran into difficulties.
Point (b) is the juciest aspect of the whole business. Here's how I put it
in http://math.ucr.edu/home/baez/spin/node15.html:
[quote]
The groups SU(2) and SO(3) have the same Lie algebra, and as we've seen,
the Lie algebra begets the angular momentum operators. This above all is why
classical pictures can carry us so far, even for half-integer spin. Push it
far enough though, and any classical picture will finally break down, for
SU(2) and SO(3) are different groups.
[end quote]
: I've been wanting to get some form of understanding of Fermi-Dirac
: statistics for many years but it still seems like magic. Even Feynman
: doesn't seem to clarify this - at least for me.
I think it *is* magic. I understand some bits and pieces, but the
spin-statistics theorem still eludes me. (However, John Baez's
http://math.ucr.edu/home/baez/spin.stat.html is probably a good place to
start.)
Squark
> Or should I abandon any attempt to geometricize spin and try to think just
> in terms of some abstract Hilbert/state space?
I just want to add you _can_ geometricize the state space of spin,
i.e. the projective space corresponding to Hilbert state space.
That later is just a sphere for a spin 1/2 quantum rigid rotator,
but it's somewhat more tricky for a full fledged particle. There
are several ways to think of the state space of the later, one of
them as follows:
Each state consists of two parts:
1) A point in the quaternion projective space of quaternion
wavefunctions q(x) over space. "Quaternions" here is the ring
formed by taking the direct sum of the pseudo-vectors with the
scalars, and defining multiplication by
(a, v)(b, w) = (ab - <v,w>, aw + bv + v x w)
2) A function k from E \ Z to our lovely sphere from before, where
E (isomorphic to R^3) is physical space and Z = {x | q(x) = 0} is
the set of zeroes of q.
Those two must satisfy the compatibility condition
q(x) k(x) q(x)^-1 = q(y) k(y) q(y)^-1
For any two points x, y in E. This makes sense due to the fact our
sphere should really be the unit sphere in the space of
pseudovectors.
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and change the
extension in the obvious way)
Pete
>
> Sure you can picture the electron as a spinning ball! I usually think of
it
> as silvery, with just a faint hint of mist clinging to it.
I was rather hoping to disperse that mist ;-)
Seriously, by "spinning ball" we're really talking about some intrinsic
geometry - an axis/plane for example.
The real question is whether the electron's properties can be explained by
some structure within standard space-time or whether it's behaviour
necessitates something beyond or instead of that. The web page you direct me
to states that "half-integer spin is fundamentally non-classical" so I guess
you are saying that electron spin does not "live" within normal Euclidean
space? This seems to be different from the orbital distribution of an
electron in a hydrogenic atom frx - or am I misunderstanding you?
> In some ways, it's a little like the "expanding balloon" picture in GR.
It
> works pretty well, so long as you keep tucked away in the back of your
mind
> a list of things wrong with it.
>
> The list for the "spinning ball electron" would include three main points:
>
> a) Quantization.
Is spin quantisation fundamentally different from orbital/radial
quantisation for which we have a fairly clear geometric picture?
>
> b) A 360 degree rotation reverses the quantum mechanical phase, and does
not
> return the electron to its initial state (unlike a 720 degree rotation).
Ok, I guess this is the big one and it's probably the area where I most need
to improve my understanding. So let me see if I have this right.
Spin angular momentum is similar to orbital momentum (the two are jointly
conserved), but measurements on lone electrons and the orbital degeneracy of
atomic electrons require that it be quantised in half the units of orbital
momentum. Orital and spin angular momentum seem to inhabit the same space
since they interact and intermingle.
I'm confused about the origin of the 720 degree rotation and by the proof in
your web page that electron spin cannot be charecterised by standard
Euclidean groups. The proof presented seems to rest on the need to represent
quantum states using a complex Hilbert Space. However if instead of an
algebraic sqrt(-1) we use a geometric sqrt(-1) such as a bivector then
everything would seem to proceed in exactly the same way except that the
complex component has a clear geometric interpretation as a circle within
some arbitrary Euclidian plane and SU(2) reduces to a more geometric
quaternion which lives entirely within E^3.
Now I may be totally wrong here, and there may be some fundamental reason
why we can't interpret spin in terms of some hidden axis but if this is the
case then I'd really like to understand that reason. It seems to me that the
wave/particle duality is enough of a mystery without assigning a
mysteriously abstract interpretation to the "complex" nature of the wave
function when a clear geometric interpretation is equally viable.
>
> c) Don't ask for the radius of the electron. Classical physicists
(Lorentz,
> for example) tried to build literal models of the electron as an actual
ball
> of charge; these models always ran into difficulties.
>
Again, are these difficulties fundamentally different from those facing the
models for orbital/radial oribtal distribution. Aren't there different ways
of defining both atomic and electron radii.
> Point (b) is the juciest aspect of the whole business. Here's how I put
it
> in http://math.ucr.edu/home/baez/spin/node15.html:
>
> [quote]
> The groups SU(2) and SO(3) have the same Lie algebra, and as we've seen,
> the Lie algebra begets the angular momentum operators. This above all is
why
> classical pictures can carry us so far, even for half-integer spin. Push
it
> far enough though, and any classical picture will finally break down, for
> SU(2) and SO(3) are different groups.
> [end quote]
I'm stepping outside my area of expertise, but can't we construct SU(2) from
SO(3) and U(1) (the circle) ?
>
> : I've been wanting to get some form of understanding of Fermi-Dirac
> : statistics for many years but it still seems like magic. Even Feynman
> : doesn't seem to clarify this - at least for me.
>
> I think it *is* magic. I understand some bits and pieces, but the
> spin-statistics theorem still eludes me. (However, John Baez's
> http://math.ucr.edu/home/baez/spin.stat.html is probably a good place to
> start.)
Thanks for that pointer. Just a quick confirmation - the Pauli exclusion
principle has nothing to do with electron charge does it? I mean even
non-charged fermions (neutrinos) would be forbidden from occupying the same
state - right?
Thanks again for being kind enough to respond, it really does help and it is
certainly appreciated.
Pete
Arnold Neumaier
>
> "Pete" <Pe...@removebtclick.com> wrote in message news:<b0m97k$2av$1...@sparta.btinternet.com>...
> > Or should I abandon any attempt to geometricize spin and try to think just
> > in terms of some abstract Hilbert/state space?
>
> I just want to add you _can_ geometricize the state space of spin,
> i.e. the projective space corresponding to Hilbert state space.
> That later is just a sphere for a spin 1/2 quantum rigid rotator,
> but it's somewhat more tricky for a full fledged particle. There
> are several ways to think of the state space of the later, one of
> them as follows:
It may be interesting to note that there are also classical geometric
models for particles with spin; they have an 8-dimensional phase space
instead of the 6-dimensional one for spin 0, and carry a nontrivial
representation of the Poincare group by a Poisson bracket in this
larger phase space. Again the sphere is involved.
The Poisson representations of the Poincare group were classified by
R. Arens in two papers:
Comm Math Phys 21 (1971), 139-149
J. Math. Phys 12 (1971), 2415-2422.
Arnold Neumaier
Michael Weiss
: ...I guess
: you are saying that electron spin does not "live" within
: normal Euclidean space?
: This seems to be different from the orbital distribution of an
: electron in a hydrogenic atom frx - or am I misunderstanding you?
You understand me just fine --- though bear in mind that we haven't given a
precise definition for "living in normal Euclidean space". From what you
write below, I believe we are interpreting that phrase differently.
: Is spin quantisation fundamentally different from orbital/radial
: quantisation for which we have a fairly clear geometric picture?
I'd say no, with a similar caveat about the phrase "fundamentally
different".
: Spin angular momentum is similar to orbital momentum (the two are jointly
: conserved), but measurements on lone electrons and the orbital degeneracy
: of
: atomic electrons require that it be quantised in half the units of orbital
: momentum.
Correct.
: Orital and spin angular momentum seem to inhabit the same space
: since they interact and intermingle.
That's a legitimate way to interpret the phrase "inhabit the same space",
but it's not the interpretation I had in mind when I wrote those notes on
spin.
: I'm confused about the origin of the 720 degree rotation and by the proof
: in
: your web page that electron spin cannot be characterised by standard
: Euclidean groups. The proof presented seems to rest on the need to
: represent
: quantum states using a complex Hilbert Space.
"Proof" is a bit strong. For a proof, we'd need a precise mathematical
statement of the hypotheses and the conclusion. I do appeal to a precise
mathematical result: irreducible finite-dimensional unitary representations
of SO(3) are carried only by odd-dimensional C^n's. And n=2j+1; that's just
the definition of j. So for j=1/2, n=2 and so no irrep of SO(3).
But to get from "X inhabits 3-space" to "the spin degrees of freedom of X
carry an irreducible finite-dimensional unitary representation of SO(3)" ---
well, it takes a bit of hand-waving. However, I wasn't trying to give a
formal proof; just explain how things look in the usual standard formalism
of QM.
There's a bit more I could have said. For one thing, I could have discussed
spin-vectors. A spin-vector is a geometric (or maybe quasi-geometric)
object; once you've set up the right kind of coordinate system, you can
represent spin-vectors by pairs of complex numbers.
What does a spin-vector look like? Well, for starters you have a regular
3-d vector; to this you attach a "flag", that is a half-plane whose edge
lies along the vector; to reinforce the imagery, we'll call the vector the
flagpole of the spin-vector.
However, if (u,v) represents a spin-vector s, then
(exp(i theta) u, exp(i theta) v) represents a spin-vector with the same
flagpole, but with the flag rotated through an angle of 2 theta. So minus s
has the same flagpole and the same flag as s. In general, rotating a
spin-vector through 360 degrees results in minus the original spin-vector.
That's pretty strange behavior for a geometric object.
You can learn more about spin-vectors in Chapter 1 of _Spinors and
Space-time_, by Penrose and Rindler.
: However if instead of an
: algebraic sqrt(-1) we use a geometric sqrt(-1) such as a bivector then
: everything would seem to proceed in exactly the same way except that the
: complex component has a clear geometric interpretation as a circle within
: some arbitrary Euclidian plane and SU(2) reduces to a more geometric
: quaternion which lives entirely within E^3.
I'm not sure what you mean here. By a bivector, I assume you mean a
rank-two antisymmetric contravariant tensor, or in other terminology, the
wedge product of two vectors. These have spin 2, not spin 1/2.
: I'm stepping outside my area of expertise, but can't we construct SU(2)
: from
: SO(3) and U(1) (the circle) ?
Well, "construct" could mean lots of things, but I'm not aware of any such
construction. SU(2) is the double-cover of SO(3); in other words, there's a
really nice 2-to-1 map from SU(2) onto SO(3). (By "really nice", I mean
it's both a group homomorphism, and a covering map in the topological
sense.) I don't know of any map from SO(3) x U(1) to SU(2) with such nice
properties.
: Thanks for that pointer. Just a quick confirmation - the Pauli exclusion
: principle has nothing to do with electron charge does it? I mean even
: non-charged fermions (neutrinos) would be forbidden from occupying the
: same
: state - right?
Right.
Squark
> It may be interesting to note that there are also classical geometric
> models for particles with spin; they have an 8-dimensional phase space
> instead of the 6-dimensional one for spin 0, and carry a nontrivial
> representation of the Poincare group by a Poisson bracket in this
> larger phase space. Again the sphere is involved.
Well, those models exactly conform with the "spinning ball picture",
the one difference being we only consider the angular momentum of the
spinning ball as a degree of freedom, ignoring the orientation of its
axes.
Arnold Neumaier
Squark wrote:
> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message
>news:<3e478890$0$8780$3b21...@news.univie.ac.at>...
> > It may be interesting to note that there are also classical geometric
> > models for particles with spin; they have an 8-dimensional phase space
> > instead of the 6-dimensional one for spin 0, and carry a nontrivial
> > of the Poincare group by a Poisson bracket in this
> > larger phase space. Again the sphere is involved.
> Well, those models exactly conform with the "spinning ball picture",
> the one difference being we only consider the angular momentum of the
> spinning ball as a degree of freedom, ignoring the orientation of its
> axes.
No, it is rather a spinning point.
There is a significant mathematical difference:
The spinning ball has a 10-dimensional phase space, and carries
*no* associated Poincare representation - the radius introduces
non-covariant features.
Arnold Neumaier
Pete
> : However if instead of an
> : algebraic sqrt(-1) we use a geometric sqrt(-1) such as a bivector then
> : everything would seem to proceed in exactly the same way except that the
> : complex component has a clear geometric interpretation as a circle within
> : some arbitrary Euclidian plane and SU(2) reduces to a more geometric
> : quaternion which lives entirely within E^3.
> I'm not sure what you mean here. By a bivector, I assume you mean a
> rank-two antisymmetric contravariant tensor, or in other terminology, the
> wedge product of two vectors. These have spin 2, not spin 1/2.
Almost. Recall that I was also wanting to work within a geometric algebra
framework where we can define the geometric product of two vectors as v1*v2
= v1.v2 + v1^v2. I was wanting to use a pure bivector s3 =e1^e2 which is
equivalent to e1*e2. In this case we have s3*s3 = -1 so we can write a
solution for a free electron in the form phi=exp[s3*h(kx-wt)] where phi can
now be interpreted as a unit quaternion which defines a rotation by an angle
2*h(kx-wt) in the plane e1^e2. So we see that if we plant your flagpole in
the plane defined by s3 then we find that -phi has the same flag and
flagpole as phi. Iow, our "spinning ball" has to rotate twice to get phi
back to the same phase.
Is this not exactly the behaviour you were describing, but now with a clear
geometric interpretation? Or have I got it the wrong way round - am I in
fact describing a particle with spin 2 as you suggest?
This argument is presented more fully (and no doubt more correctly) by David
Hestenes in section VII of
http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf. where he appears
to argue for a much more classical interpretation of electrodynamics.
Pete
Squark
> Squark wrote:
> > Well, those models exactly conform with the "spinning ball picture",
> > the one difference being we only consider the angular momentum of the
> > spinning ball as a degree of freedom, ignoring the orientation of its
> > axes.
>
> No, it is rather a spinning point.
>
> There is a significant mathematical difference:
> The spinning ball has a 10-dimensional phase space,
The full fledged phase space is 6 + 3 + 3 = 12 dimensional, the two
3's coming from the orientation of the ball's axes and the angular
momentum (which is the canonically conjugate thing). If you ignore
the orientation (as I said you should before) you get a 9 dimensional
phase space, and if you fix the total angular momentum you get an 8
dimensional one.
> and carries
> *no* associated Poincare representation - the radius introduces
> non-covariant features.
Okay, relativistically you might have problems with a finite radius.
Basically, the difference between the "spinning point" and the
spinning ball is the moment of inertia, but that difference
disappears once you "forget the orientation" as I suggested. Now,
I'm not sure whether you can easily incorporate finite moment of
inertia into a relativistic model if you really want to, but I
priori I'd guess you can - just think of the radius as an
infinitesimal "proper radius".
John Devers
> I usually think of it
> as silvery, with just a faint hint of mist clinging to it.
>
> Nothing wrong with these pictures, as long as you remember that they *are*
> just pictures.
Let's see if I can get rid of that silver lining for you;-)
http://www.physics.purdue.edu/topaz/research/emagcoupl.html
In this artist's rendering, a "bare" electron is denoted by the bright
spot at the center of the figure. The wispy white lines represent
electric field lines radiating out from the electron. Virtual
particle-antiparticle pairs popping into and out of the vacuum are
represented by blue-gold ellipses; the blue side, corresponding to a
positively-charged particle, is nearer to the electron. This
polarization effect reduces the effective charge of the electron that
we observe at a large distance.
Dennis Harp/Purdue University
http://www.physics.purdue.edu/topaz/research/electron.gif
And this one is kewl too.
http://link.aps.org/abstract/PRL/v89/e135506
Visualization of an electronic 'tubular image state' around a
nanotube. The electron's attraction to its image charge combines with
the centrifugal-barrier repulsion to localize it far from the
nanotube, while defects in the tube could localize it as well in the
longitudinal direction, as depicted.
http://ojps.aip.org/prl/covers/images/lg89-13.jpg
Michael Weiss
: Almost. Recall that I was also wanting to work within a geometric algebra
: framework ...
Ah, now that changes things a bit! I remember you talking about a geometric
interpretation, but I don't think you uttered the magic phrase "geometric
algebra" before in this thread.
I've heard of Hestenes' GA, but I'm not familiar with it--- I haven't read
any articles about it.
: This argument is presented more fully (and no doubt more correctly) by
David
: Hestenes in section VII of
: http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf. where he
appears
: to argue for a much more classical interpretation of electrodynamics.
Thanks for the reference. I'll take a look (when I find the time!); maybe
we can pick up the discussion after that.
Michael Weiss
Pete wrote:
: Almost. Recall that I was also wanting to work within a geometric algebra
: framework ...
: Is this not exactly the behaviour you were describing, but now with
: a clear geometric interpretation? ...
: This argument is presented more fully (and no doubt more correctly) by
: David Hestenes in section VII of
: http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf. where he
: appears to argue for a much more classical interpretation of
electrodynamics.
OK, I've had a chance to glance through the Hestenes article. As I
mentioned in another follow-up, I wasn't familiar with geometric algebra.
(I'm still not, except for perusing the article, and doing a search on this
newsgroup for "geometric algebra", which turned up a bunch of interesting
threads.)
Anyway, yes, it looks like the algebra Hestenes calls G3 in his paper does
contain an isomorphic copy of the quaternions, so you can do all
the usual spin 1/2 stuff from that viewpoint.
So, is it fair to say (as I did in
http://math.ucr.edu/home/baez/spin/node15.html)
that "half-integer spin is fundamentally non-classical", or that a spin 1/2
object doesn't "inhabit" ordinary physical 3-space?
I guess it all depends on what you mean by "fundamentally non-classical", or
by "inhabiting 3-space". When I picture a regular old spinning ball, I
don't think of something that becomes minus itself (whatever that means)
when you rotate it 360 degrees. If you can find a toy store that carries
tops like that, let me know about it--- I'll be all set for Christmas!
Starting from this naive sort of visualization, I was attempting to draw the
connections with the elementary representation theory of SU(2) and SO(3).
From another standpoint, it's quite legitimate to emphasize the continuity
between quantum and classical notions--- a huge topic, upon which dozens of
books have been written.
You asked a while back:
: Or should I abandon any attempt to geometricize spin and try to think just
: in terms of some abstract Hilbert/state space?
Some textbooks do say things like that; there's a passage in Feynman's
Lectures where he grumbles about grad students who don't want to learn
anything new, and so insist on thinking about spin as if it were just the
same as classical angular momentum.
I think that's a bit extreme; I find the geometric connections and analogies
very illuminating, but with the caveats and provisos we've already
discussed.
In another branch of this thread, Arnold Neumaier mentions some intriguing
facts I wasn't aware of:
: It may be interesting to note that there are also classical geometric
: models for particles with spin; they have an 8-dimensional phase space
: instead of the 6-dimensional one for spin 0, and carry a nontrivial
: representation of the Poincare group by a Poisson bracket in this
: larger phase space. Again the sphere is involved.
Details, please!
Oz
[NB No, I haven't abandoned the other thread, but a reply takes serious
thought and I've become sidetracked on kets, something I think I also
need to grok; a bit anyway.]
>Sure you can picture the electron as a spinning ball! I usually
>think of it as silvery, with just a faint hint of mist clinging to
>it.
Hmm. Mine is golden and entirely mist. Sort of graded, denser in the
middle and extending faintly off to infinity. It also has a wavy density
structure that sort of turns inside out (gradually) if you rotate it 360
deg. The waves may be sort of 'flowing outwards' (attenuated by
distance) if I am being particularly precise.
>Nothing wrong with these pictures, as long as you remember that they
>*are* just pictures. Pedagogical or mnemonic crutches, useful for
>those (like me) whose like their math served with a generous dollop
>of imagery on top. (Once upon a time I wrote a post,
>"Anschaulichkeit vs. Abscheulichkeit", about the differing visual
>preferences of Heisenberg and Schroedinger.)
Hmm. What I try and do is incorporate what I know about the object into
the visual analogy. That is, for *each* characteristic I try and put a
modification to the image. That way I can imagine some interactions. I
am led to believe my image of a free photon is less than ideal, though.
>In some ways, it's a little like the "expanding balloon" picture in
>GR. It works pretty well, so long as you keep tucked away in the
>back of your mind a list of things wrong with it.
Or at least include them in your image somehow. Of course to be able to
do this you have to know about the 'wrong things about it'. Getting this
info is often like getting blood out of stone.
>The list for the "spinning ball electron" would include three main
>points: a) Quantization.
I cheat. A particle is obviously quantised, this is easy. I do the
quantisation bit by enforcing it during a measurement. That is it is the
measurement that forces (say) spin up or down. It works in much the same
way that you (effortlessly) imagine either finding or not finding a
particle. The latter is not strange (who ever found half an electron) so
with a bit of worldview twisting it's not so hard to imagine that you
can't find an electron with a quarter spin, because that isn't an
electron.
>b) A 360 degree rotation reverses the quantum mechanical phase, and
>does not return the electron to its initial state (unlike a 720
>degree rotation).
This one is real hard. The above is the best I can do right now.
"Turn the wavy bit inside out". A bit like when you turn a gyroscope,
sort of.
>c) Don't ask for the radius of the electron. Classical physicists
>(Lorentz, for example) tried to build literal models of the electron
>as an actual ball of charge; these models always ran into
>difficulties.
I don't like point particles at all. I consider them a mathematical
convenience. A hard ball is just as bad.
Entanglement, now that's really fun!
Mind you, I am a total amateur.
You would be wise to disregard what I say.
Oz
This post is worth absolutely nothing and is probably fallacious.
Note: soon (maybe already) only posts via despammed.com will be accepted.
Arnold Neumaier
> Arnold Neumaier wrote:
> > Squark wrote:
> > > Well, those models exactly conform with the "spinning ball picture",
> > > the one difference being we only consider the angular momentum of the
> > > spinning ball as a degree of freedom, ignoring the orientation of its
> > > axes.
> > No, it is rather a spinning point.
> >
> > There is a significant mathematical difference:
> > The spinning ball has a 10-dimensional phase space,
> The full fledged phase space is 6 + 3 + 3 = 12 dimensional, the two
> 3's coming from the orientation of the ball's axes and the angular
> momentum (which is the canonically conjugate thing).
Please explain how you mean that. In my understanding, the angular
momentum cannot be canonically conjugate to anything, since its
components do not commute under any Poisson bracket.
> If you ignore the orientation (as I said you should before) you get
> a 9 dimensional phase space, and if you fix the total angular momentum
> you get an 8 dimensional one.
Sorry, I cannot translate this into mathematics. Please give generators
of the 12D space and the 9D subspace (or quotient space?) you think of,
together with their Poisson brackets, so that one can check that
functions on each of the three spaces you mention are closed under
the bracket. Only then can one legitimately talk about phase spaces.
Arnold Neumaier
Arnold Neumaier
Well, we got it both wrong; the situation is more complicated
than intelligent guessing based on general knowledge would suggest.
I spent the weekend to figure out some details.
In the nonrelativistic case, there is no 12D phase space at all
but _every_ rigid body (not only the ball) is described by a
9D phase space generated by p_i, q_i, J_i with the standard
commutation relation realized as Poisson bracket. p and q are
canonically conjugate but J has no conjugate partner.
S=J-q x p is the spin of the rigid body (= angular momentum in
the rest frame of the center of mass). S^2 is a Casimir operator
(= has vanishing brackets with everything). Thus the set of
points (p,q,J) with constant S^2 =s^2 is a symplectic manifold
of dimension 8. Free motion of a rigid body has constant S^2
iff the inertia tensor is a multiple of the identity.
This holds, e.g., for the ball, but also for the cube or another
regular solid (and in general for what is called a eutactic star).
Moreover, seen from far away (without being able to probe the
intrinsic structure), the dynamics remains the same even if the
body is not rigid but the dynamics preserves its inertia.
In particular, the cube and the ball have the same dynamics;
therefore, if a particle is a spinning ball, it could as well
be a spinning cube or dodecahedron, or something that oscillates
between a cube and a sphere!
In the relativistic case, there is - assuming the known force
laws - no such thing as a rigid spinning body; see, e.g.,
the discussion by Michael Weiss in
http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html
But at least in special relativity there are 'rigid bodies in
the Born sense', which also carry a 9D phase space with a natural
action of the Poincare group.
(Michael Weiss writes: 'Rigid motion can occur in SR only
through a conspiracy of forces', which I take to mean: not induced
by the known classical forces.)
This phase space is a 9D hyperboloid in a 10D space spanned by
the generators of the Poincare group, given by the mass shell
p^2=m^2. For m>0, there is another Casimir, W^2, which
taken constant produces an 8D symplectic manifold.
The remarkable thing is that this happens to be isomorphic to
the nonrelativistic one, as exhibited by Arens.
Thus I renounce my statement that we have a spinning point;
instead the free classical electron (having the Poincare symmetry)
is an extended object with the inner structure of a eutactic
star, which can be viewed as a rigid - or pulsating - ball
(or cube, or tetrahedron, etc.) in the Born sense. Its size,
shape, or pulsation frequencies cannot be deduced from
these general considerations.
An interacting electron loses the Poincare invariance;
this is consistent with this picture since the ball (or cube)
is most likely deformed by the impending forces.
Whether the electron is indivisible is another matter, but lacking
evdence we may currently take it well to be so.
Since Michael Weiss asked for details:
The connections mentioned above are nontrivial, and I haven't
yet worked out a _nice_ scheme.
Arens' paper (Comm. Math. Phys. 21 (1971), 139-149) is not easy to
read. He also has a second paper (I have it at home - J.Math.Phys
1972??) classifying all transitive Poisson representations of
the Poincare group, but this is even more sketchy. However, one
sees that the massless case is essentially different -
classical photons sit in a noncanonical 6-dimensional
symplectic space. I hope to understand this soon somewhat better.
There appear to be (unquoted) relations to a paper by Pryce
(Proc. Roy. Soc. Ser. A 195, 62-81 (1949)) on the relativistic
center of mass, which should give a neat description
(after polishing the contents).
Arnold Neumaier
Arnold Neumaier
> In the relativistic case, there is - assuming the known force
> laws - no such thing as a rigid spinning body; see, e.g.,
> the discussion by Michael Weiss in
> http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html
> But at least in special relativity there are 'rigid bodies in
> the Born sense', which also carry a 9D phase space with a natural
> action of the Poincare group.
> (Michael Weiss writes: 'Rigid motion can occur in SR only
> through a conspiracy of forces', which I take to mean: not induced
> by the known classical forces.)
After more thinking, I no longer understand this remark.
The strange thing is that also in the nonrelativistic case,
rigid motion can occur only through a conspiracy of forces.
And the nuclear forces ensure in both cases that the forces
conspire to a sufficiently high degree to make approximate
rigid bodies an everyday phenomenon.
> This phase space is a 9D hyperboloid in a 10D space spanned by
> the generators of the Poincare group, given by the mass shell
> p^2=m^2. For m>0, there is another Casimir, W^2, which
> taken constant produces an 8D symplectic manifold.
>
> The remarkable thing is that this happens to be isomorphic to
> the nonrelativistic one, as exhibited by Arens.
>
> Since Michael Weiss asked for details:
> The connections mentioned above are nontrivial, and I haven't
> yet worked out a _nice_ scheme.
> Arens' paper (Comm. Math. Phys. 21 (1971), 139-149) is not easy to
> read. He also has a second paper (I have it at home - J.Math.Phys
> 1972??)
No, it was: Math. Phys 12 (1971), 2415-2422.
> classifying all transitive Poisson representations of
> the Poincare group, but this is even more sketchy. However, one
> sees that the massless case is essentially different -
> classical photons sit in a noncanonical 6-dimensional
> symplectic space. I hope to understand this soon somewhat better.
There is a readable summary account in pp. 25-30 of the book
Henry Bacry,
Localizability and space in quantum physics,
Springer, Berlin 1988.
The terminology there is more modern than that of Arens;
it boils down to studying the coadjoint orbits of the
Poincare group. But there are not enough details to get an
explicit representation with which one can do calculations
easily. For example, I'd like to have it explicit enough to
the the nonrelativisitc limit, relating the 8D spinning
massive relativistic particle to its nonrelativistic cousin.
Another place where one might get some information is
N.P. Landsman
Mathematical Topics between Classical and Quantum Mechanics
Springer, New York 1998.
A table of contents is at
http://remote.science.uva.nl/~npl/book.html
suggesting that Chapter IV.3.1-5 describes what is needed.
But I haven't yet seen the book and cannot say what it
contributes to our problem.
> There appear to be (unquoted) relations to a paper by Pryce
> (Proc. Roy. Soc. Ser. A 195, 62-81 (1949)) on the relativistic
> center of mass, which should give a neat description
> (after polishing the contents).
The construction (e) of Pryce exhibits in the Lie-Poisson algebra
of the Poincare group a representation of the Lie algebra
h(3) + so(3), where h(3) is the Heisenberg algebra over R^3,
and shows that the Poincare generators can be expressed in
terms of this h(3)+so(3). The construction only works for
massive particles (because of a division by the mass),
and gives the desired representation of the Poincare group on
R^6 x S^2.
Arnold Neumaier
Squark
> > The full fledged phase space is 6 + 3 + 3 = 12 dimensional, the two
> > 3's coming from the orientation of the ball's axes and the angular
> > momentum (which is the canonically conjugate thing).
>
> Please explain how you mean that. In my understanding, the angular
> momentum cannot be canonically conjugate to anything, since its
> components do not commute under any Poisson bracket.
Maybe I did use the term too loosely here, I'm not quite sure it
applies. Anyway, I'm too tired right now to dive into the rest of
the (some it, apparently quite interesting) stuff you wrote on the
manner, so I'll only explain the 12D phase-space approach _is_
viable. It goes like this:
1) We may define a "location-less rigid rotator". The
configuration space of this thingie is the SO(3)-torsor of frames
in physical space (they represent the axes of the rotator). The
Lagrangian is the usual "Newtonian particle" Lagrangian for an
SO(3)-invariant metric and zero potential. Of course, there are a
lot of such metrics, though the choice of it at a given point
fixes it. In fact, the space of metrics is the space of possible
moments of inertia. The 6D phase space follows.
2) The 12D phase space is obtained by taking the product of the
above with the usual 6D phase-space of a point-particle.
Arnold Neumaier
>
> I'll only explain the 12D phase-space approach _is_ viable.
> It goes like this:
> 1) We may define a "location-less rigid rotator". The
> configuration space of this thingie is the SO(3)-torsor of frames
> in physical space (they represent the axes of the rotator). The
> Lagrangian is the usual "Newtonian particle" Lagrangian for an
> SO(3)-invariant metric and zero potential. Of course, there are a
> lot of such metrics, though the choice of it at a given point
> fixes it. In fact, the space of metrics is the space of possible
> moments of inertia. The 6D phase space follows.
> 2) The 12D phase space is obtained by taking the product of the
> above with the usual 6D phase-space of a point-particle.
OK; I see what you mean. The configuration space of a moving frame
is R^3 x O(3) of dimension 6, given by translations and rotations
in 3-space. Its phase space is a 12-dimensional symplectic manifold.
Taking equivalence classes of frames with the same origin (which
amounts to looking at the frame from so far away that one cannot
recognize its orientation) factors this space down to a 9-dimensional
space R^6 x R^3.
But fixing the total angular momentum |J| in this space (as you
suggested in your earlier mail) gives a manifold of dimension 7
only, since J^2 is not a Casimir. Instead, one has to separate
the motion of the center of mass and fix the remaining intrinsic
angular momentum S^2, which is classically spin^2 with real-valued
spin and quantum mechanically spin*(spin+1) with half integer spin.
Thus a nonrelativistic spinning massive particle should be
considered as a moving frame corresponding to a rigid body
too small to care about its orientation.
(It cannot be just a point, since it is difficult to imagine
how a point should have spin.)
Arnold Neumaier
Squark
> But fixing the total angular momentum |J| in this space (as you
> suggested in your earlier mail) gives a manifold of dimension 7
> only, since J^2 is not a Casimir. Instead, one has to separate
> the motion of the center of mass and fix the remaining intrinsic
> angular momentum S^2, which is classically spin^2 with real-valued
> spin and quantum mechanically spin*(spin+1) with half integer spin.
Sorry, I might have formulated myself inaccurately. Of course I
meant fixing the "spin" (i.e. total angular momentum with respect
to the center-of-mass).
> Thus a nonrelativistic spinning massive particle should be
> considered as a moving frame corresponding to a rigid body
> too small to care about its orientation.
More generally, it could be an ideally spherically symmetric body.
Arnold Neumaier
>
> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<3e5f7cca$0$13160$3b21...@news.univie.ac.at>...
> > Thus a nonrelativistic spinning massive particle should be
> > considered as a moving frame corresponding to a rigid body
> > too small to care about its orientation.
>
> More generally, it could be an ideally spherically symmetric body.
In my opinion, this is less generally. It could be any body whose
inertia is spherically symmetric. But I think the inertia cannot be
detected from far away; hence it can be any body within the accurcay
of current experiments.
Arnold Neumaier
Squark
This is not always true. If the particle has magnetic moment, for
instance, torque could be exerted on it through a magnetic field
and thus the moment of inertia necessarily comes into play.
(Electric dipole moment would do just as well, of course, and
basically anything which adds a spin-dependant term to the
Hamiltonian)
Arnold Neumaier
Squark wrote:
>
> Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<3e6499ed$0$14448$3b21...@news.univie.ac.at>...
> > Squark wrote:
> > >
> > > Arnold Neumaier <Arnold....@univie.ac.at> wrote in message news:<3e5f7cca$0$13160$3b21...@news.univie.ac.at>...
> >
> > > > Thus a nonrelativistic spinning massive particle should be
> > > > considered as a moving frame corresponding to a rigid body
> > > > too small to care about its orientation.
> > >
> > > More generally, it could be an ideally spherically symmetric body.
> >
> > In my opinion, this is less generally. It could be any body whose
> > inertia is spherically symmetric. But I think the inertia cannot be
> > detected from far away; hence it can be any body within the accurcay
> > of current experiments.
>
> This is not always true. If the particle has magnetic moment, for
> instance, torque could be exerted on it through a magnetic field
> and thus the moment of inertia necessarily comes into play.
> (Electric dipole moment would do just as well, of course, and
> Hamiltonian)
Agreed. So it could be any body whose inertia ellipsoid is a sphere.
Arnold Neumaier
Squark
> Agreed. So it could be any body whose inertia ellipsoid is a sphere.
Yeah, in principle, but it seems problematic to quotient by the
"configuration variables", i.e. identify the various orientations
of the body, if it isn't spherically symmetric. In other words,
if it's not spherically symmetric one cannot avoid it having the
additional orientation degrees of freedom, besides the spin.