Re: boson calculus and half inter spin? huh?

Arnold Neumaier wrote:
> cyberkatru wrote:
> > I always thought bosons were integer spin. I also heard that you can't
> > get half integer spin out of combining integer spin components. OK, fine
> > but now I come accross a book that uses what it calls the Boson calculus
> > to get integer and also what seems to be half integer spin reps of of
> > su(2). What is going on here?
> >
> > The book is "Lie Groups and Algebras with applications to Physics,
> > Geometry and Mechanics" by Sattinger and Weaver. It is on page 52
> > (middle to bottom of the page).
>
>
> There is significant literature on bosonization. You might look the term
> up in scholar.google.com.

While bosonization does convert some fermionic theories into bosonic
ones (mostly in 1+1 dimensions), I don't think that's what the original
poster was interested in.

It looks like a question purely concerned with the mathematics of Lie
algebras. While I don't have the book by Sattinger and Weaver, I
consulted the massive monograph "Theory of group representations and
applications" by Barut and Raczka. Section 10.4 of the first volume
covers this topic briefly.

The main idea is to consider the algebra of ladder operators for
several harmonic oscillators. The only nontrivial commutation relations
are [a_i,a_j*] = delta_ij. Using them, it can be shown that the
operators of the form X_ij = a_i* a_j have the same commutation
relations as the standard generators of the Lie algebra gl(n), where n
is the number of oscillators:

[X_ij, X_kl] = delta_jk X_il - delta_il X_jk .

The rotation group and many other Lie groups can be realized as
subgroups of the matrix group GL(n), same goes for their Lie algebras
and gl(n). In other words, we can choose linear combinations of the
X_ij to generate the Lie algebra of su(2) or many other Lie groups. So,
once we have a representation of the algebra of these oscillators, by
restriction, we obtain a representation of the desired Lie algebra for
free.

A representation of the oscillator algebra is fairly easy to construct,
it can be summed up as the quantization of n independent harmonic
oscillators, and we already know how to do that. Here, I think, the
Stone-von Neumann theorem comes in and tells us that this
representation is irreducible and unique up to unitary transformation.
However, the resulting representation of any sub algebra will in
general be reducible. It seems this trick was first due to Schwinger
who used it to construct representations of su(2) by embedding it as a
subalgebra into the algebra of two harmonic oscillators.

More mathematically, this trick can be expressed as follows. For any
matrix Lie algebra, we can find a subalgebra of the algebra of several
harmonic oscillators which acts as an enveloping algebra of the desired
Lie algebra.

Hope this helps.

Igor

.

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