Advise needed
michael kagalenko
physics (with intro to said invariance) would you wizards recommend ?
Thank you.
John C. Baez
Assuming I count as a wizard.... First you should decide whether you
are primarily interested in 4-dimensional physics or 2-dimensional
physics. The conformal group is finite-dimensional in the former case,
infinite-dimensional in the latter, so the two topics are quite
different in flavor. As for 4-dimensional physics, you might try
Penrose and Rinder's Spinors and Spacetime (2 volumes). As for
2-dimensional physics, you might try Kaku's two books on string theory.
(Especially for the latter, you will need to consult the references to
get a handle on the vast literature.) There are also lots of
applications of 2-dimensional conformal invariance to condensed matter
theory, but I don't know an especially good starting-point.
U16...@uicvm.uic.edu
Baez) says:
>Penrose and Rinder's Spinors and Spacetime (2 volumes). As for
>2-dimensional physics, you might try Kaku's two books on string theory.
>(Especially for the latter, you will need to consult the references to
>get a handle on the vast literature.) There are also lots of
Kaku's books will be helpfull for the references, but you will be much
better off reading the original references. A better review is
Ginsparg's "Applied Conformal Theory" appearing in Les Houches 1988
Fields,Strings and Critical Phenomena.
Alison Chaiken
spin-statistics theorem. That is: why are spin-(n + 1/2) particles
fermions, and spin-n particles bosons? I have seen textbook
demonstrations that this is the case according to what type of Lie
group the particle transforms under, but I must admit I don't get it.
In particular, is there an intuitive explanation as to why spin-1/2
particles can't have a zero S_z state? After all, zero is a possible
geometrical projection of 1/2, just as it is a possible geometrical
projection of 1.
Sorry to dredge up "1920's murk." Guess I should have asked Vicki
Weisskopf when I was at MIT.
--
Alison Chaiken ali...@wsrcc.com
(510) 422-7129 [daytime] or cha...@cmsgee.llnl.gov
Look if you like, but you will have to leap.
Matt Austern
> Hmm, sounds like someone out there should be able to explain the
> spin-statistics theorem. That is: why are spin-(n + 1/2) particles
> fermions, and spin-n particles bosons? I have seen textbook
> demonstrations that this is the case according to what type of Lie
> group the particle transforms under, but I must admit I don't get it.
> In particular, is there an intuitive explanation as to why spin-1/2
> particles can't have a zero S_z state? After all, zero is a possible
> geometrical projection of 1/2, just as it is a possible geometrical
> projection of 1.
Unfortunately, I don't know of any proof that doesn't require a fair
amount of heavy mathematics. Also, I should add: I don't think that
you can show this except in the context of quantum field theory.
The standard reference for this subject is Streater and Wightman's
book, _PCT, Spin and Statistics, and All That_, which is a book on
axiomatic field theory---that is, a book on the attempts to put
quantum field theory on a rigorous footing. It does include a proof
of the spin-statistics theorem. Unfortunately, this proof is on a
rather formal level: you write down the Green's functions, do some
analytic continuations, turn the crank, and the result pops out. Each
step in the process is easy enough to follow, but the proof still
seems like magic.
Now, when it comes to the CPT theorem, I'm a bit happier, because I
know how to prove it rigorously, but I also know a nonrigorous
"proof": you can't write down a Lagrangian that violates CPT. Does
anyone know of a "proof" of the spin-statistics theorem that starts
with Lagrangian field theory? For a sloppy physicist like me, that
would be good enough.
--
Matthew Austern Just keep yelling until you attract a
(510) 644-2618 crowd, then a constituency, a movement, a
aus...@lbl.bitnet faction, an army! If you don't have any
ma...@physics.berkeley.edu solutions, become a part of the problem!
John C. Baez
>spin-statistics theorem.
Once I posted a sketch of the proof of (part of) this theorem, but
I no longer have it. Does anyone out there still have it??? If so,
please post it or email it. This question comes up now and then and now I
wish I had kept my little explanation.
For now, let me just say that the proof depends crucially on the
machinery of quantum field theory - you are right that it doesn't follow
from quantum mechanical considerations of the sort you referred to.
The key ingredients of the proof are (it seems to me) Poincare
invariance (i.e. special relativistic symmetry) and energy positivity.
Daniel E. Platt
The way I remember it is that commutators are used to construct Green functions
for Bose (integral spin) fields and anti-commutators are used to construct
Green functions (propogators) for Fermi (half-integral) spin systems. If
you try to construct a field whose internal symmetry acts half-integral
but which is built with a commutator algebra, the Green functions have
space-like behavior (acausal).
The connection with statistics is that the number operator looks like Psi* Psi
(where * is now the adjoint operator on a state vector that looks like the
number of excitations at each point or momentum; multiple excitations look like
more than one particle with each excitation being a particle; you can either
postulate a field theory and many-partical theories pop out or you can take
a simple field theory, and artificially build a multiple-particle 'Fock space'
that acts the same way; either way its '2nd quantization'; from that, interactions
can look like things that destroy photons and create electron-positron pairs, etc).
Raising and lowering operators (looks sort of like the harmonic oscillator) are used
to represent creation and destruction of particles (say in electron/positron
anihilation coupled with photon creation) can be constructed. Since Fermi fields
(anticommutation) can only create one particle before destroying it again, they
obey Fermi statistics. Since Bose fields (commutation) support any number of
excitations, they obey Bose statistics. Quantum indistinguishability is built in.
There's no way in the machinery to count any more information than how many
electrons have this polarization and momentum or that polarization and momentum.
The bottom line is that if you try to construct a multiparticle theory of quantum
particles, you get a quantum field theory. If you make Fermions, you will use
an anticommutating system. If you make bosons, you will use commutators. If
you try to construct a Green function (propogator for moving the system through
interactions), for integral spin fermions, you get an acausal Green function.
If you try to construct a Green function with half-integral spin for Bose
systems, you get an acausal Green function.
First place I saw this stuff was Bjorken and Drell. I think that Itzykson
and Zuber does it too. I also think that Bugoliubov does it, but I don't
think of him as being that readable.
Wish I could say it shorter.
Dan
--
-------------------------------------------------------------------------------
Daniel E. Platt pl...@watson.ibm.com
The views expressed here do not necessarily reflect those of my employer.
-------------------------------------------------------------------------------
Michael Weiss
Feynman that attempts to explain the spin-statistics theorem "with all the
gears showing" (approximate quote). I couldn't follow it (except for the
Balinese dancer trick at the end). Still the reference may be useful to some:
Elementary Particles and the Laws of Physics: The 1986 Dirac
Memorial Lectures, by Feynman, Richard P., and Weinberg, Steve.
Cambridge University Press, 1987, ISBN: 0-521-34000-4
In fact, I'd be interested to know what someone who does understand this
stuff thinks of it.
SCOTT I CHASE
>Hmm, sounds like someone out there should be able to explain the
>spin-statistics theorem. That is: why are spin-(n + 1/2) particles
>fermions, and spin-n particles bosons? I have seen textbook
>demonstrations that this is the case according to what type of Lie
>group the particle transforms under, but I must admit I don't get it.
>In particular, is there an intuitive explanation as to why spin-1/2
>particles can't have a zero S_z state? After all, zero is a possible
>geometrical projection of 1/2, just as it is a possible geometrical
>projection of 1.
This is not a proof, but just a few words of explanation. Have you
ever plowed through one of the basic books on quantum field theory, such
as Bjorken and Drell? In such a book you will see the way one
second-quantizes the Klein-Gordon and Dirac equations, the basic equations
which govern spin-0 and spin-1/2 fields. It is straightforward to
*try* to quantize the K-G equation using anticommutators - you find that
you can't do it. If I remember correctly, you get an unstable ground state,
i.e., an infinite number of states with ever decreasing energy. Similarly,
if you *try* to quantize the Dirac equation with commutators, the same
sort of thing happens.
This does not, of course, prove the general theorem, but if you work
through the commutator algebra, trying to build up the eigenspectrum
using creation and annihilation operators along with the basic commutators
or anticommutators (as the case may be), you get a good feeling for
what goes wrong, without all the fancy group theory.
-Scott
--------------------
Scott I. Chase "It is not a simple life to be a single cell,
SIC...@CSA2.LBL.GOV although I have no right to say so, having
been a single cell so long ago myself that I
have no memory at all of that stage of my
life." - Lewis Thomas
SCOTT I CHASE
>
>Now, when it comes to the CPT theorem, I'm a bit happier, because I
>know how to prove it rigorously, but I also know a nonrigorous
>"proof": you can't write down a Lagrangian that violates CPT. Does
Sure I can. Oh... you mean a *local* Lagrangian. Never mind.
Matt McIrvin
>>particles can't have a zero S_z state? After all, zero is a possible
>>geometrical projection of 1/2, just as it is a possible geometrical
>>projection of 1.
This is a quite separate issue from the spin-statistics theorem, and
it's much simpler. I suggest looking in a book like Sakurai's
_Modern Quantum Mechanics_; the approach taken, I believe, is to
build up the possible spin states with raising and lowering operators
(which can only change a component of angular momentum by hbar, so
you don't get zero if you start with hbar/2), then prove that there
aren't any more.
--
Matt McIrvin I read Usenet just for the tab damage!
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John C. Baez
In article <C0s1...@wsrcc.com> ali...@wsrcc.com (Alison Chaiken) writes:
>Hmm, sounds like someone out there should be able to explain the
>spin-statistics theorem.
Here's a brief sketch of some ideas that appear in the proof
of the spin-statistics theorem... it is crucial to realize that this
proof
relies on the formalism of relativistic quantum field theory, not just
quantum mechanics. The key axioms for quantum fields that one must
assume
in order to deduce this theorem are that the fields are invariant under
the Poincare group (the special relativistic symmetry group), that there
is a vacuum state that is invariant under this group, that all states
can
be built up from the vacuum by applying field operators, that the
Hamiltonian is bounded below, and locality, in the sense that the fields
either commute or anticommute at spacelike separations. The theorem
then says that integer-spin fields commute at spacelike separations,
while
half-integer-spin fields anticommute. To really make the theorem
precise
one must make ones axioms precise. One can use, for example, the
Garding-Wightman axioms (which is what my sketch implicitly does), or
else one can work with the Haag-Ruelle axioms, which use C*-algebras
and are arguably more fundamental, though further from the language
of "practical" quantum field theory.
I will note that the proof I am outlining comes from:
Introduction to axiomatic quantum field theory / by N. N. Bogolubov, A.
A.
Logunov, I. T. Todorov ; authorized translation from the Russian ms. by
Stephen A. Fulling and Ludmila G. Popova ; edited by Stephen A. Fulling.
1st
English ed. Reading, Mass. : W. A. Benjamin, Advanced Book Program,
1975.
All the mistakes are mine; some are deliberate attempts to make the
proof seem simpler than it is, but there may well be some accidental mistakes
too. I will skip all sorts of steps and try to present the logical outline.
Let F(x) be a field transforming as one of spin s (here s either integer
or half-integer; I'm not worrying about left-handed and right-handed spinors).
Let v be the vacuum state and let
f(x - y) = <v, F(x) F*(y) v>
g(x - y) = <v, F*(x) F(y) v>
Here F* is the adjoint of F, and of course the expectation values
really do depend only on x - y by translation invariance. Now suppose
the wrong commutation relations hold, i.e., for x and y spacelike
separated
we have
F(x) F*(y) = -(-1)^{2s} F*(y) F(x)
instead of what we "should have" (no extra minus sign). This implies
f(x) + (-1)^{2s} g(-x) = 0 . 1)
g(x) is called a
"Wightman function" and these are known to transform in a nice way
under the Poincare group, because of the Poincare symmetry of the
field theory. In fact, one can use the fact that the Hamiltonian is
bounded below to analytically continue the Wightman functions to
part of the complexification of Minkowski space (C^4 instead of R^4),
and
the Wightman functions then transform in a nice way under the
complexification of the Poincare group. This is rather technical,
and it requires care to do correctly, but heuristically one can
just let all your coordinates (t,x,y,z) become complex and hope
that your formulas extend in the "obvious way".
The parity/time-reversal operator PT lives in the connected component
of
the complexified Lorentz group (or more precisely, its double cover).
This group, by the way, is just SL(2,C) + SL(2,C) (direct sum of 2
copies of
the double cover of the Lorentz group). PT corresponds to the map
(t,x,y,z) -> -(t,x,y,z)
which we may think of as being a 180 degree rotation in the
(x,y) plane followed by a 180 degree rotation in the complexified (t,z)
plane.
(One needs to complexify precisely to get away with the second rotation.)
Thus we should be able to derive how a spin-s field should transform
under PT assuming we know how it transforms under the complexified
Lorentz group. And by analytic continuation, this should be calculable
from knowing how the field transforms under the plain old Lorentz
group - i.e., its spin!
But this is a calculation I don't want to do here, since I haven't figured
out how to do it simply, even cheating a little here and there. Maybe
sometime I'll explain this part and finish writing up a nice little
spin-statistics FAQ... but I will offer a raincheck on this one.
Let me simply assert, then, that PT transforms the field in such a way
that the covariance of the Wightman functions implies
g(x) = (-1)^{2s} g(-x) .
It follows from 1), then, that
f(x) + g(x) = 0 ,
or recalling what f and g are,
<F*(x)v, F*(y)v> + <F(x)v, F(y)v> = 0
for all spacelike separated x, y . (Here one should really smear the fields
by test functions!) Now letting y approach x along a suitable spacelike
direction one gets in the limit
||F*(x)v||^2 + ||F(x)v||^2 = 0.
Thus F(x)v = 0 for all x . A big fat theorem called the
Wightman reconstruction theorem says one can reconstruct a quantum field
from its Wightman functions, and using this we can show F(x) = 0 for all x.
Thus no (nonzero) field can have commutation relations that don't match it's
spin correctly.