On 12/18/2013 4:27 PM, Stephen Paul King wrote:
Ever attempt to do a particular calculation with an actual infinite dimensional Hilbert space?


Why not? Sure, you can mod out (using symmetries and other tricks) all of the infinite dimensions except some finite subset,

You can calculate all the eigenfunctions of a finite square well.

but that is the act that introduces the bias that I am pointing at! The actual Hilbert spaces used to do calculi are finite dimensional.

Even if you only find a finite subset of eigenfunctions, the calculation is still done in an infinite dimensional space. If you create a wave packet it consists of infinitely many momentum eigenfunctions. I don't see that cutting the

No, Choosing a basis and choosing a coordinate system is NOT a convenience. You must do it.

Try reading Robert Wald's "Quantum Field Theory in Curved Spacetime". He seldom chooses a coordinate system.

Especially in GR, where one cannot define the manifold unless there is a choice of coordinate system on the patches of local space-time used to define the manifold - which is then run through the diffeomorphism mill... AFAIK, there is no global manifold that can be defined that does not involve the requirement of stitching together of local patches of space-time (defined per individual events) into manifolds of arbitrary size.

But that doesn't require choosing a specific coordinate system, and in fact for most manifolds it is impossible to choose a single coordinate system.

What must be remembered is that the "stitching operation" is very restrictive, one cannot connect patches that have events with different (other than an infinitesimal) values of momenta and position associated with each. The math of GR is amazing once one is familiar with it...

?? Coordinate patches have momenta??  That's amazing all right.


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