exponential map of operators

[F. B.]

Hi there!

The exponential map of the differential operator can be regarded as a translation operator, eg:

Exp( a * diff_x ) f( x ) = f( x + a )

This can be seen by Taylor expanding both sides.

Other kinds of operators generate different transformations.

What about adding this feature to the Operator object in the quantum module? It could be very useful to physics.

[Stefan Krastanov]

It would certainly be nice. If you have the time go on. If you need
any help just ask here or on a pull request.

[F. B.]

I know this problem is related to the exponential map between lie algebras and lie groups, do you have any reference on this issue?

By the way, I'm still pondering how to integrate quantum module's Operator with the vtensor in currently working on. Specifically, should qapply be applied automatically upon contraction?

Is it better to create a new module to deal with operators on tensors or integrate quantum's operators?

Partial derivative (and on differential manifolds also covariant derivative) should be represented as operators with a tensor index or just as class methods?

[Stefan Krastanov]

The `quantum` module reimplements a lot of stuff and it is in no way
connected to the tensor module, the diffgeom module, combinatorics,
matrix or any other module that is related to it in theory. If you
want them to work together you will have a lot of refactoring to do on
both sides.

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[F. B.]

You're right, better keep things separate.

By the way, I added two new methods to my last commit to MultiArray in the valued tensor PR: applyfunc and diff.

Besides, I'm currently pondering how to add "diff" capabilities to valued tensors, see https://github.com/sympy/sympy/pull/2041#issuecomment-17679576

[Aaron Meurer]

I'm not sure he was suggesting to keep things separate. There is a lot of stuff in the quantum module that is purely mathematical that *should* be moved out into some separate submodule, and made to work with the other parts of SymPy. The quantum module could then just subclass these things and add their own physics conventions, printers, and so on.

Aaron Meurer

--

[Stefan Krastanov]

Sorry for being unclear. I do share Aaron's opinion.

[F. B.]

OK, well, the physics module indeed should not implement abstract mathematical stuff in its inside, I agree.

The point is, modern physics is highly dependent on indexed data, that's why I started the PR to add valued tensors.

I think that creating some mathematical modules could be very useful for the physics module, by name:

• Lie groups and algebra, with both matrix representations and differential representations.
  
• complete the valued tensor PR.
• Generalize "diff" to act on abstract objects, create a "differential operator" object independent of the physics module.

Representations of Lie groups both as matrices and as differential operators are both used in physics, that's why I suggested the exponential map of an operator.

Regarding the "diff", I am currently considering to create a differentiation for indexed data, specifically, it would be very useful to derive a quantity with respect to an indexed variable (which would result in an indexed object).

I'm still not sure on the notation to implement such a "diff", but I'm pretty sure that such an operator is of paramount importance to the physics module.