Lagrangian visualisation and Uncertainty Principle

[exponent137]

Probably, the essence of quantum theory (QT) is principle of uncertainty (HUP).

The essence of QT is also the fact that Fourier transformation of wave function in phase(?) space gives wave function in momentum space. If one wave function is Gaussian (and so both ones) this gives HUP.

Very useful function in quantum mechanics are also Lagrangian and Action principle. Feynman used them in visualization of QED in his book "QED: The Strange Theory of Light and Matter".

But Lagrangian cannot be so easily visualized as Hamiltonian, for instance.

Wharton also find Lagrangian useful in QT:

Indeed, for classical particles and fields, there's a perfect match between the initial data one
uses to constrain the Lagrangian and the amount of classical data one is permitted under the HUP. In Fermat's principle, if you know the initial light ray position, the HUP says you can't know the initial angle.

http://fqxi.org/data/essay-contest-files/Wharton_FQX4.pdf

Above Fourier analysis is used in derivation of HUP. Another aspect in derivation of HUP is use of commutator [x,p_x]. Can Lagrangian be another aspect?

 

[exponent137]

Now I succeded to find something:
\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}.
The canonical commutation relations then amount to
[x_i,\pi_j] = i\hbar\delta_{ij}, \,
http://en.wikipedia.org/wiki/Canonical_commutation_relation

If \pi_i=p_i this is the most simple version, what I searched. But I please for visualization and explanation of this?