## What's a Fermionic Condensate?

#### John Baez

#### January 31, 2004

This week the
media have been talking in
very vague
terms about a new thing called
a "fermionic condensate" - they say it's a new state of matter,
but they're not saying much about what it is!
Here's an article that's a bit clearer:
Basically, there had been two known ways of getting fermions
to form pairs, which act approximately like bosons, and can thus
"condense" - meaning that a whole bunch of them get into the same
state.
One way was for the fermions to literally stick together: for
example, some protons, neutrons and electrons (all fermions)
can stick together and form helium-4, which is a boson... and
these bosons can then form a condensate known as "superfluid helium".

A fancy way of saying that the fermions stick together is to
say that they have strongly correlated *positions*: if you know
where one is, you've got a good idea where its mates are.

The other option was for the fermions to get strongly correlated
*momenta*. The classic example is a superconductor, where electrons
form "Cooper pairs", which are bosons. The two electrons in a
Cooper pair aren't close together in position - after all, they
repel each other due their electric charge. However, they
have almost the same momentum!

The new "fermionic condensate" allows physicists to interpolate
between these two extremes: they can now get some fermions to
correlate in ways that are "between" position correlation and
momentum correlation. Even better, they can adjust the type of
correlation by changing an external magnetic field.

In more mathematical terms, we used to be able to form pairs
of bosons in "position space" or in "momentum space".
Now we can do it more generally. Yay!

The position space
and momentum space representation of wavefunctions are important,
and nicely related by the Fourier transform - but there are infinitely
many other representations, and we can try to get correlations in
any one. News reports about this being useful for new high-temperature
superconductors are somewhat missing the point. It may ultimately be
useful - but what it is right now is *beautiful!*

It's not surprising that the media are having a tough time explaining
this, though!

© 2004 John Baez

baez@math.removethis.ucr.andthis.edu