Post by Urs Schreiber Post by Jack Tremarco Post by Kasper J. Larsen
Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold?
This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.
Jack, could you be a little bit more specific about what you mean here?
Post by Urs Schreiber
The only global condition that you have is the obvious one, that your
manifold admits spinors at all, globally, which means that it admits a spin
bundle which means that it is spin. This same condition is there for
Riemannian signature. So the answer is indeed Yes.
I am sorry, Urs, but this statement looks like word-juggling to me. -
If you don't specify how you define these terms,
how can we believe you that the whole construction is so "obvious"?
I would agree with Jack, that there are certain difficulties
with the definitions of spin structures, spinors, and Dirac operators
in pseudo-Riemannian spaces.
I had to look this up in the book:
"Spin-Strukturen und Dirac-Operatoren
ueber pseudoriemannschen Mannigfaltigkeiten"
by Helga Baum (Teubner-Verlag, Leipzig, 1981),
The results of this book have also appeared in English
without proofs in:
* Dlubek, Helga
Spinor-structures and Dirac-operators on pseudo-Riemannian manifolds.
Proceedings of the Conference on Differential Geometry and
its Applications (Nove Mesto na Morave, 1980), pp. 17--23,
Univ. Karlova, Prague, 1982.
* Baum, Helga
Spinor structures and Dirac operators on pseudo-Riemannian
manifolds. Bull. Polish Acad. Sci. Math. 33 (1985), no. 3-4, 165--171
Here are some facts from the book above:
First of all, the existence of pseudo-Riemannian structures
on manifolds is topologically restricting. -
For the existence of a Lorentz structure on a manifold M,
there has to exist a nowhere-vanishing line field on M.
Similar for pseudo-Riemannian metrics
of index (k,n-k) on an n-dimensional manifold,
there has to exist a splitting of the tangent bundle TM
into the direct sum
of a k-dimensional subbundle V and an (n-k)-dimensional subbundle W.
The first Stiefel-Whitney classes w_1 of these bundles,
which describe their orientability,
are invariants of the underlying pseudo-Riemannian structure.
Now, a metric of index (k,n-k) on M
gives us the bundle of orthonormal frames over M,
a principal bundle with structure group O(k,n-k).
The construction of Clifford algebra and Pin group Pin(k,n-k),
which is a natural double cover of O(k,n-k),
is very much like in the Riemannian case,
and there is plenty of literature about it, like, e.g.:
* Lawson, Michelsohn
Spin geometry. Princeton University Press, 1989.
* Harvey, Spinors and calibrations. Academic Press, Boston, MA, 1990.
A spin structure over the manifold M can be defined as reduction
of the principal O(k,n-k)-bundle of orthonormal frames
to a principal Pin(k,n-k)-bundle
which is compatible with the natural projection from Pin to O.
Already here appears the first slight complication:
While in Riemannian geometry the condition on a manifold
to carry a spin structure is that the second Stiefel-Whitney class
w_2(TM) be zero, in non-Riemannian geometry
the condition is connected to the splitting of M in space- and time-like
directions, TM = V + W, and it becomes:
w_2(TM) = w_1(V) u w_1(W).
This looks like a first reason to restrict oneself to
manifolds that carry space- and time-orientations,
and we have not yet started to define the spin group...
O.K., given a spin structure,
we can define the associated spinor bundle,
lift the Levi-Civita connection
to a compatible connection on the spinor bundle,
and define the Dirac operator.
Problem is, that all we can do in this setting is "geometric"
(we can e.g. define parallel spinors and harmonic spinors,
spinors annihilated by the Dirac operator)
but nothing "analytic", because for analytical considerations
we need the Dirac operator to act on a (complex!) *Hilbert space*. ...
Post by Urs Schreiber
It would be strange if otherwise, given that we do live in a Lorentzian
spacetime and we do observe spinors.
Note that the question "Can one define a Dirac operator D on a spin bundle
over M" is different from the question "Can we make sense of quantum field
theory of fermions on M." QFT on curved spaces is hard.
... But the problem does not start with QFT:
Even to do simple spectral theory for the Dirac operator,
we need a Hilbert space for it to act on.
Now Harvey goes in his book a great length
to prove that there exists a natural inner product on (real) spinors
that is compatible with the action of the Clifford algebra,
so that the Dirac operator would be *formally* self-adjoint
with respect to this product.
*BUT* this product may take values in R, C, or H,
and it may be symmetric or anti-symmetric,
and after complexification of the spinor bundle
(which we *must* inevitably do
to get a *complex* Hilbert space of spinor sections),
the inner product we obtain will - in the non-Riemannian case -
not be positive definite.
Now, in the later chapters of Helga Baum's book
there is a description for the definition of a "natural"
positive-definite hermitean product on the complex spinor bundle,
but to construct this product,
one has to fix a time- and a space-orientation on M,
i.e. one has to break the non-compact symmetry group
to the compact group SO(k)xSO(n-k).
But even then, the Dirac operator will *not* be essentially self-adjoint
in the Hilbert space defined by this scalar product,
only its "real and imaginary part" will be essentially self-adjoint;
and this makes that the spectrum of the Dirac operator
will not only consist of pure eigenvalues. -
There will also be essential spectrum and even rest spectrum.
Well, a Dirac operator with rest spectrum might
from the point of view of physics sound like nonsense,
but who can say, where we went wrong??