How particle spin follows from relativity and quantum mechanics
Dr Tim
So far I have grasped the Schroedinger equation, general relativity,
exterior differential forms, matrix algebra, Hamiltonian theory, and I
can see how electromagnetism follows from adding an extra rolled up
dimension to general relativity.
So now I come to spin. Lets see if I understand correctly.
The Dirac equation is a relativistic treatment of a free particle, for
example an electron. The equation can be divided into electronic and
positronic parts, but the parts do not separate completely because
there are cross terms. If we try to eliminate the positronic part,
we get some unexpected terms in the electron equation. These terms
are interepreted as "spin".
Thus "spin" can be seen as the SHADOW OF THE POSITRONIC PART of the
electron solution.
Is this correct, or am I wrong as usual?
Arkadiusz Jadczyk
wrote:
>So far I have grasped the Schroedinger equation, general relativity,
>exterior differential forms, matrix algebra, Hamiltonian theory, and I
>can see how electromagnetism follows from adding an extra rolled up
>dimension to general relativity.
If you are interested in spin, a good thing to add to your list would be
group theory and unitary representations. The simples way to derive spin
is by studying unitaru group representations and quantu mechanics via
imprimitivity systems.
The book by Jauch may be of help, laso the two volumes by
Varadarajan's "Geometry of Quantum Theory" and references there.
Spin comes out automatically from projective unitary representations
of SU(2). You do not need to embed it into Lorentz group as in the Dirac
equation. Pauli's Hamiltonian for free particle with spin comes from
(somewhat extended) analysis (Mackey theorem) of imprimitivity system:
Euclidean group of RxR^3 plus covariant (generalized) spectral measure,
along the line described in
"Logics Generated by Causality Structures.
Covariant Representations of the Galilean Logic"
available online at
http://www.cassiopaea.org/quantum_future/jadpub.htm#ceja75
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
Alejandro Rivero
news:7fed4c14.02061...@posting.google.com
>
> So now I come to spin. Lets see if I understand correctly.
> The Dirac equation is a relativistic treatment of a free particle, for
> example an electron. The equation can be divided into electronic and
> positronic parts, but the parts do not separate completely because
> there are cross terms. If we try to eliminate the positronic part,
> we get some unexpected terms in the electron equation. These terms
> are interepreted as "spin".
>
> Thus "spin" can be seen as the SHADOW OF THE POSITRONIC PART of the
> electron solution.
>
> Is this correct, or am I wrong as usual?
Here you have been wronged, I believe, because of the PCT
symmetry. You have read a Tempotal inversion (giving the antiparticle)
as a Parity inversion (giving the oppossite spin).
The point to Spin should be to meet the nonrelatistic Pauli equation,
should it? I am not sure because I remember something about wrong
interpretations of Stern Gerlach experiments and spins.
Then we have, of course, the spin-statistics theorem. That is
other good clue to the meaning of spin.
Last, by not least, we have the question of the connection
between spin and geometry.
Yours
Alejandro
--
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John Baez
Dr Tim <timro...@paradise.net.nz> wrote:
>So far I have grasped the Schroedinger equation, general relativity,
>exterior differential forms, matrix algebra, Hamiltonian theory, and I
>can see how electromagnetism follows from adding an extra rolled up
>dimension to general relativity.
Holy Kaluza!
>So now I come to spin.
>Thus "spin" can be seen as the SHADOW OF THE POSITRONIC PART of the
>electron solution.
>
>Is this correct, or am I wrong as usual?
Maybe you can see things this way if you want, but personally I think
it's better to say spin arises when you have a particle that transforms
in a nontrivial way when you rotate it. After all, spin makes sense
even before you get to special relativity, antiparticles, and all that.
Projective unitary representations of the rotation group - that's what
*I* think spin is about!
Dr Tim
Thanks everyone.
And yes, I do have some group theory and various algebrae
(Clifford, Grassman, Lie, Boole...)
I can see I have made two mistakes.
1) I have confused particle spin with magnetic moment.
And it seems you can't just multiply spin by charge
to get magnetic moment, either.
2) The Dirac equation reveals two effects
(i) the positron and
(ii) the magnetic moment of the electron.
But this doesn't mean that magnetic moment
has anything much to do with the existence of positrons.
My current understanding is that Dirac's equation shows
that *every* charged particle has a magnetic moment
proportional to its charge.
Does this mean that there are no spinless charged particles?
Please pardon my ignorance - I am looking for a cure!
Radu Grigore
ba...@galaxy.ucr.edu (John Baez) wrote in message news:<aegna1$l4o$1...@glue.ucr.edu>...
> >Thus "spin" can be seen as the SHADOW OF THE POSITRONIC PART of the
> >electron solution.
>
> Projective unitary representations of the rotation group - that's what
> *I* think spin is about!
I think this is a cool mathematical object that explains what a spin
is and what can it do. But form the subject line I would guess the
original question is more something like: "Where from did we get the
idea that there _is_ something named _spin_. Is there any mathematical
reason?".
Anyway this is what *I* would like to know... :-)
bye,
radu
Mark
timro...@paradise.net.nz (Dr Tim) writes:
>Does this mean that there are no spinless charged particles?
There are no fundamental spin 0 fields at all.
The only possible exceptions are the Higgs, which has no charge
(and has never been seen), and the Dark Energy field (= Higgs?),
which looks like a spin 0 field, but obviously has no charge.
Jeffery
whop...@alpha2.csd.uwm.edu (Mark) wrote in message news:<afctra$n48$1...@uwm.edu>...
There are lots of spin-0 particles. You have all of the Higgs
particles, not just one. In MSSM, there are five Higgs particles. You
have all the supersymmetric partners of the fermions. String theory
predicts the dilaton which is spin-0. You have the axion, which is the
goldstone boson resulting from Peccei-Quinn symmetry breaking.
Jeffery Winkler
John Baez
Radu Grigore <radug...@ieee.org> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<aegna1$l4o$1...@glue.ucr.edu>...
>> Projective unitary representations of the rotation group - that's what
>> *I* think spin is about!
>I think this is a cool mathematical object that explains what a spin
>is and what can it do. But form the subject line I would guess the
>original question is more something like: "Where from did we get the
>idea that there _is_ something named _spin_. Is there any mathematical
>reason?".
There are lots of reasons we got the idea there is something named
spin. One is that electrons have a magnetic moment. Another is the
periodic table: the observed shells are only consistent with Pauli
exclusion if the electron has an internal degree of freedom which
comes in two states. The history of quantum mechanics is convoluted,
and I forget most of it, but very crudely: experiment pushed us into
grappling with spin, and eventually people realized it's all about
projective representations of the rotation group. Certainly we owe
a lot to Pauli and his matrices!
Michael Weiss
: The history of quantum mechanics is convoluted,
: and I forget most of it, but very crudely: experiment pushed us into
: grappling with spin, and eventually people realized it's all about
: projective representations of the rotation group.
See http://math.ucr.edu/home/baez/spin/spin.html for some of this history.