Re: Dirac Gamma matrices including gamma^5, and the Spacetime Metric g_uv

Jay R. Yablon wrote:
When I asked "I am looking for any flaws you can identify in this line
of thought," I was hoping for a serious response.

I'm sorry that you thought that my response was not serious. I assure
you that it was.

Yes, the gamma^5 are staring us in the face and someone naiive could
just say, "gee, that ought to be a fifth spacetime dimension" without
more than superficial analysis, and I would then agree with the comments
made. However, let me elaborate my questions, please, because the
mathematics hangs together quite well and to not seriously consider this
and give a dismissive answer to me reflects a prejudice in thinking.

What does it mean for the mathematics to "hang together quite well"?
Yes, you've found an algebraic property of the algebra of Dirac gamma
matrices. It is an interesting property and it holds quite generally.
If C' is the complex Clifford algebra constructed over an n-dimentional
(n being an odd number) complex vector space and C is the Clifford
algebra constructed over an (n-1)-dimensional complex vector space.
Then C' is isomorphic to a direct sum of two copies of C. This means
that in any faithful matrix representation of C', we can find a basis
in wich every element of C' is represented by a matrix in block
diagonal form. Each of the blocks gives a matrix representation of C.
See for example:
http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras

This decomposition implies that there exist (several) homomorphisms
(linear multiplication-preserving maps) from C' to C. If P: C' -> C is
such a homomorphism, then {P(e_i),P(e_j)} = P{e_i,e_j} = delta_i,j, as
P(1) = 1, where the e_i are the rank-1 generators of C'. You've found
one such homomorphism from C' to C in the case n=5.

In other words, can we think of the Dirac gammas as
the "structure matrices of spacetime" which, via (1), give us an
alternative way to define a classical spacetime metric?

By construction, the generators of the Clifford algebra correspond to
an orthonormal basis in every tangent space. Knowing what "orthogonal"
means in every tangent space is equivalent to knowing the metric
tensor. So, yes, you can reconstruct the metric tensor from what's
called a Clifford bundle, just like you can from something called the
orthonormal frame bundle.

Can we equally think, that spacetime is alternatively
defined by its gamma matrices gamma^u, from which the g^uv may in turn
be deduced by (1)?

Now, here's the blind step: a space-time is not just the metric. A
space-time is a manifold with a defined on it. A manifold has a fixed
dimension. If the dimension is 4, try as you might, you'll never find 5
linearly independent vectors in a tangent space that are mutually
orthogonal, no matter how you construct the metric tensor.

10) Does this lead, at least roughly, to a "many-fingered" time sort of
notion which I recall Feynman once entertained? What is the
modern"conventional wisdom" and what other viewpoints are there on such
things as having more than one timelike dimension, e.g., two timelike
dimensions, including references which address this point? Has anyone
ever examined what quantum field theory would look like with a second
time dimension, that is, has anyone ever explored d = D + 2 QFT, for D=3
and otherwise? If so, where might I find such examination?

The question of wether theories with more than one time dimension have
been studied is completely separate from all the other questions about
gamma matrices. Yes, such theories have been considered, but apparently
no-one takes them seriously. This question has come up in this group
before. Here's what John Baez had to say on the topic:
news:b45r31$li7$1@xxxxxxxxxxxx
http://groups.google.ca/group/sci.physics.research/msg/c09890a5f78129ac

Igor

.

Relevant Pages
Loading