Broken symmetry and particle number conservation
Don
its guises in solid state physics than in HEP, so I shall elaborate on
the points that are unclear to me in examples from the former class.
I can understand how a "real" symmetry like rotational symmetry is
broken, e.g. in an (infinitely large) ferromagnet. After all, I can
directly observe the direction of the magnetic moment.
However, I still do not understand the breaking of gauge symmetry in a
superconductor, which has served as an important model in the
development of the Higgs model etc.
The different possible states with broken gauge symmetry are
characterized by a continous angle alpha. They are quite peculiar
objects in as far as they do not correspond to a fixed number of
electrons. As in a (even very large) superconductor particle number
will always be fixed, I conclude that symmetry is never really
broken.
Some authors try to justify the use of states with unsharp particle
number simply as a convenient mathematical trick (...in the
thermodynamical limit, it doesn't matter anyhow, which ensemble is
used), which allows to use an independent particle picture after
Bogoliubov transformation.
I tried to clarify this by reading an article by Rudolph Haag, "The
Mathematical Structure of the Bardeen-Cooper-Schrieffer Model. Nuovo
Cimento, vol 25, pp. 287, (1962).
Haag finds that the states of broken symmetry form an irreducible
representation of the algebra spanned by the creation and anihilation
operators, while the states with exact particle numbers are reducible
representations.
Although I do not know too much about representation theory of
operator algebras, I thought that irreducible representations are some
way equivalent to pure states while a reducible representation
corresponds to a mixed state. Is this correct or what significance do
have irreps over reducible representations if not? If true this would
indicate that the broken symmetry states have a much more profound
meaning beyond being a mathematical convenience.
Ulf Klein
Hi Don,
the BCS state of superconductivity is not an eigenstate of the
particle
number operator. Thus there is a uncertainty DeltaN in particle
number.
In the thermodynamic limit (BCS theory for infinite number of
electrons)
DeltaN -> \infty but DeltaN/N -> 0. As a consequence of number-phase
uncertainty relation the phase \phi becomes sharply defined in the
N-> \infty limit (This leads to a macrocopic quantum state).
It is probably true that a different statistical framework (other
than
grand-canonical in standard BCS theory) will - in the thermodynamical
limit - lead to identical results. I have never seen a concrete work
using
this alternative formulation (but it might exist).
The symmetry that is broken (typically) in the superconducting state
is
gauge symmetry, as you wrote. It is not that intuitively accessible
as
broken rotational symmetry in the ferromagnet but it is analogous.
The simplest formulation to see this is Ginzburg-Landau theory. Above
T_c the order parameter vanishes, the phase is undefined and
consequently one has perfect gauge symmetry. Below Tc a finite
order parameter exists and consequently the phase must take a
definite
value (even if it is arbitrary). Gauge symmetry is broken.
Best
Ulf