Old Quantum Theory & Quantization of Action

[tom.stoer]

"Old quantum theory" was derived using "quantization of action" in phase space

$\oint p\,dp = nh$

Does "quantization of action" still make sense using canonical quantization?

 

[atyy]

http://physics.njnu.edu.cn/users/papers/20120331092408.pdf

"Berry and Tabor investigated the relation of the Gutzwiller periodic orbit formalism to the EBK torus quantization. They found that the algebraic sum of the contributions from all the periodic orbits orbits gives the density of states of the integrable system."

Predrag Cvitanovic's textbook has quite a few chapters on quantum chaos. The chapter on semiclassical quantization talks about the Gutzwiller trace formula. http://chaosbook.org/

 

[tom.stoer]

I know (knew) some of the results of Gutzwiller et al. But I think that in the PI "quantization of action" does no longer exist, b/c all paths do contribute. So "quantization of action" is a classical concept to derive QM, it applies to some special solutions, but I don't see it anywhere in the final theory, at least not at the fundamental level.

 

[tom.stoer]

My summary is that - except for semiclassical approximations - there is nothing like "quantization of action" in quantum mechanics.

Any ideas?

 

[neerajareen]

There is. In many cases, path integrals are too difficult to solve exactly. Therefore physics tend use lattice grid calculations and in order to sum over different trajectories. This invoked divided space time into little 4D volumes and considering the action over each of these 4D cubes. This method first shows up in QCD calculations. See
http://theory.physics.helsinki.fi/~qftgroup/paco/Panero.pdf

 

[tom.stoer]

I worked with lattice QCD and other Monte Carlo simulations; there is no fundamental quantization of action, it is introduced via the discretization only.

 

[atyy]

My summary is that - except for semiclassical approximations - there is nothing like "quantization of action" in quantum mechanics.

Any ideas?

I agree. The canonical commutation relations are primary. Then "quantization of action" is useful in the semiclassical regime, using EBK or Gutzwiller like formulae. The "action" in the path integral is not semiclassical, but as you said, isn't quantized, and must be related to the canonical formulation using things like Osterwalder-Schrader conditions.

I have to confess I'm intrigued by Gutzwiller trace formula because of the Berry-Keating conjecture about quantum mechanics and the Riemann hypothesis.