What is the most general mathematical framework for quantum

[lfqm]

1.- The hilbert space approach does not include distributions (free particle, for example) nor mixed states.
2.- The C* algebra approach does not account for unbounded operators.
3.- Rigged Hilbert space approach does not include mixed states.

I'm not sure about path integral formulation... so I ask... What is the most general mathematical framework for quantum mechanics

 

[thephystudent]

Maybe such mathematical framework cannot exist because it would imply the existence of number operators, hereby violating Gödels incompleteness theorem?

 

[George Jones]

Lfqm, why do you think 1. and 3. don't include mixed states?

 

[vanhees71]

I'd say all three approaches together make up a consistent rigorous mathematical framework for non-relativistic QT.

 

[thephystudent]

I just started reading a bit on operator algebra's and apparently you need c*-algebra's to deal with situations where an infinite amount of particles are present, as in the thermodynamic limit. (http://www.springer.com/us/book/9783540170938)

 

[bhobba]

Lfqm, why do you think 1. and 3. don't include mixed states?

Off course it does. That's because states are not elements of a vector space but are operators.

Study Ballentine.

This book gives our most powerful, complete and rigorous formulation:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

In practice most physicists use Diracs formulation which is made rigorous by the rigged hilbert space approch.

Thanks
Bill