Currently I am rewriting the geometric algebra (Clifford algebra) and
calculus module for sympy again. Current work is being kept separately
at github.com/brombo/GA for now. You may want to look at tensor
sections in the "GA Notes" and "LaTeX docs" directories.
On 02/02/2014 07:28 AM, F. B. wrote:
Hi all!
I was considering that it would be great to have the diffgeom module
and the tensor module work together, as tensors are also part of
differential geometry arising on the tangent and cotangent spaces of
manifolds.
The main problem I face is that in the tensor module, indices of a
tensor can be declared as belonging to different types. For example,
gamma matrices can be declared as a (Lorentz, Spinor, Spinor) tensor.
The question is, how to characterize such a tensor from a differential
geometric perspective?
The Lorentz and Spinor indices are indices carrying two different
representation of the symmetry of the universe, they correspond to two
representation of a Lie algebra, and have their own transformation
laws. The point is, in SymPy there is no such advanced infrastructure
which is able to handle principal bundles, so I was wondering if there
can be an easier approach to this problem.
When I consider the Riemann tensors, for example, R(a, -b, -c, -d),
this is an element of the tensor product space (T, V, V, V), where T
is the tangent space, and V is the cotangent space, of the same base
manifold, i.e. the space time manifold.
Do you think that the gamma matrices, as their indices do not belong
to the same spaces, can be viewed as a tensor in some power of the
tangent space of the product space of two manifolds, say the spacetime
and something like a Clifford Algebra which represents the spinor space?
It would be useful to be able to declare a link to a manifold in the
object *TensorIndexType*, e.g.:
|
L =TensorIndexType('L')
M =Manifold('M')
L.manifold =M
|
in such a way, tensors depending only on *L* would be immediately
linked to manifold *M*, and it would be possible to use the already
implemented algorithms in the diffgeom module to perform covariant and
Lie derivative, as well as compute the Riemann tensor, Ricci tensor
from the metric tensor.
The problem remains in mixed indices tensors. Any ideas on how to
overcome this?
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