From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: The Dirac sea
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Mon, 11 Jan 93 22:30:06 GMT

Michael Weiss writes:

You've already explained what you mean by "complex structure".  Namely,
let H be a Hilbert space over C; the complex structure is the mapping v -->
iv, where v is in H and i=sqrt(-1), as usual.

The next obvious question is what is H for Dirac's electrons?  Normally
I'd suppose the answer is 4-spinor fields, or (confusing the representation
with the abstraction) L^2 functions from spacetime to C^4.

----
Your description of what a complex structure is makes it a bit unclear
as to what "picking the right one" means.  The point is that the
classical phase space is a real vector space V and to make it into a
complex vector space one needs to pick a linear operator J:V -> V with
J^2 = -1; then we can think of V as a complex vector space with iv = Jv.
In the case of bosons, the classical phase space is a symplectic vector
space (i.e., equipped with a nondegenerate antisymmetric bilinear
pairing), while in the case of fermions, it's an orthogonal vector space
(i.e., equipped with a positive definite symmetric bilinear pairing).  When
we have picked a complex structure, the antisymmetric (resp. symmetric) pairing
is supposed to become the imaginary (resp. real) part of a (complex) inner
product on V, which then can be completed to obtain a complex Hilbert
space.  Note that given the antisymmetric (resp. symmetric) pairing, the
requirement that it be the imaginary (resp. real) part of a complex
inner product places a constraint on any candidate complex structure.
This constraint is far from enough to determine the complex structure,
though; often one can pin down the right one by the constraint that the
Hamiltonian of the theory be nonnegative.  There is much more to say
about this (like why one needs a complex structure to define creation
and annihilation operators, and how changing the complex structure can
amount to switching creation and annihilation operators), most of which
is in my book with Segal and Zhou, but I won't go into it now.

Instead, I'll just answer your question!  :-)  You have almost described the
right "classical phase space" for charged spin-1/2 particles.  (The notion of a
"classical phase space" sounds funny in this situation but that's
because we are taught bosonic classical physics, not fermionic!)  In
other words, as a vector space OVER THE REALS one uses L^2 functions
from space (R^3 - not spacetime!) to C^4.  This is an orthogonal vector
space where one uses the real part of the usual complex inner product as
the positive definite symmetric bilinear pairing.  Dirac's error (in
retrospect) was to use ordinary multiplication by i as the complex
structure!  One should instead solve the Dirac equation and decompose
the vector space as the direct sum of two parts, the
"positive-frequency" and "negative-frequency" parts, and let the complex
structure J be multiplication by i on the first part but -i on the
second part!  (One can do these projections by taking the Fourier
transform of the solution; the positive frequency part is supported on
the forwards mass hyperboloid, while the negative frequency part is
supported on the backwards mass hyperboloid.)  

There is, in fact, a unique way to make the solutions of the Dirac
equation into a complex Hilbert space such that 1) the inner product is
invariant under the action of the Poincare group and 2) the self-adjoint
generator of time evolution is nonnegative.  I have just described it;
proving it's unique is done in our book.  Using the "obvious but wrong"
complex structure (plain old multiplication by i) and the orthogonal
structure described above gives a complex Hilbert space satisfying 1)
but not 2) -- that is, one gets holes!
Xref: sparky sci.physics:22372 news.groups:25243 sci.misc:1984