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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