Is there a rigorous definition of "quasiparticle"? Text

William R. Frensley

Jul14-08, 06:00 AM

chris.meyer123@googlemail.com wrote:
> Is there any (general accepted) conceptual precise and mathematically
> rigorous definition of the concepts "particle" and "quasiparticle" in
> the framework of non-relativistic quantum mechanics?
>
> I am mainly interested in quasiparticles, but I think if there is a
> rigorous definition of quasi-particle then it would be desirable to
> compare it with a rigorous definition of "particle", especially if the
> definition of quasiparticle is based on the concept of a particle (as
> I suppose).
>
> For me especially the mathematical rigor is important, but also to see
> how such a definition from mathematical physics is related to more
> imprecise treatments for example in solid-state or statistical physics
> textbooks.
>
> References to books or papers where (parts of) my questions were
> answered would be nice.
>
I have no idea what you expect in the way of a "rigorous definition."
Euclidean deductive systems have to be based upon a set of undefined
terms, and in physics "particle" is the first candidate one would
think of for this category.

One can define "particle" operationally as an appropriate degree of
freedom (e.g. a gauge group representation) of whatever quantum field
theory one believes in (here particle in the sense of electron versus
muon, not individuals of either type).

In this sense we would define "quasiparticle" as the quantized
model of any physical phenomenon which can be characterized
by a dispersion relation: \omega(k). (And, to be properly
"quasi", is not a simple elementary particle.) Thus, most
named quasiparticles are simply wave-like collective excitations
of a many-particle system: phonons, plasmons, magnons, or photons
(if in a dielectric medium rather than vacuum). The specification
that there be a meaningful dispersion relation implicitly requires
that the excitations be sufficiently long-lived as to be experimentally
accessible.

Quasiparticles are features of some less-detailed levels of abstraction
in the description of physical systems. What makes the concept useful
is that many of the dynamical details of the underlying system are
excluded from the analysis, leaving the parts of the system that are
presumably interesting. Thus quasiparticles follow from an
approximation to some underlying "exact" theory. Approximations are
never "rigorous" because their validity depends upon information
that is outside of the deductive system, in this case such things
as the characteristic energy scales of terrestrial or cryogenic systems,
and the length or time scales that are experimentally accessible.

Conceptually, any quasiparticle model is obtained from the exact,
microscopic, fully many-body theory by a sequence of unitary (or
cannonical) transformations and projections. The standard procedure
for the massless bosons listed above (and "massless boson" should
always be understood as "collective mode") is to derive the normal
plane-wave modes of the system classically and then apply the
quantization conditions to each mode separately. This certainly looks
like a rather ad hoc procedure, and indeed it is. The justification for
this is that, for such massless bosons, the classical and quantum
interference effects are essentially identical, and the quantum effects
show up more subtly, in the interaction with other particles.

A perhaps more interesting case is the conceptual development of massive
fermion quasiparticles, such as those in which the present message is
embodied (the electrons and holes in semiconductor devices). These are
clearly quasiparticles because they have "effective masses" which are
not equal to the free electron mass; quantum interference effects which
directly demonstrate the effective mass have been routinely observed
over the past 35 years. Now, with adequate computational effort, one
can in principle (and nearly in practice) adequately solve the many-body
problem of an ideal semiconductor crystal. The electron and hole
quasiparticles appear in the solution as the poles (or, for an infinite
crystal the branch cuts) in the Green's function. This is normally
considered proof of the quasiparticle model. But, really, one needs to
take a few more steps to demonstrate these properties. Ideal crystals,
by definition, do not contain macroscopic electric fields, so we need to
conceptually incorporate such fields to be able to demonstrate that the
excitations really do respond as if they were free particles. We do
this by applying a suitable unitary transformation whose result is an
"effective mass" Schrodinger equation in which the effect of the
microscopic periodic crystal potential is absorbed into the kinetic
energy term, leaving only the macroscopic potential as the apparent
driving force. In reality, there are an infinite number of these
equations, describing the infinite number of energy bands in the
crystal, but we simply focus on the band of interest. This last is
the application of the projection operator, that discards the degrees
of freedom that are not relevant to the problem. We can now invoke
the correspondence principle to explain the behavior of what appear to
be nearly-free classical quasiparticles.

I hope this sheds at least some light on the structure of the theory.

- Bill Frensley