Weyl ordering of the hamiltonian

[Jack2013]

Hi , I can't understand the general formula for weyl ordering of the hamiltonian . It is written in Srednicki field theory book in page 68 . Can someone explain how to derive this formula ?

 

[tom.stoer]

You can't derive a specific ordering of the Hamiltonian; the ordering of operators is a quantization ambiguity which has no classical counterpart. Usually you have p²/2m for the kinetic energy of a particle but there is no qm principle which tells you that (px)(p/x)/2m is wrong. Different ordering schemes may result in different physics and you have to use an additional, independent physical principle in orer to select the "correct" one.

In case of a curved manifold with a free particle moving on that manifold one reasonable idea is to use the Laplace-Beltrami operator as kinetic energy; this results in a unique operator ordering.

Perhaps Srednicki explains something like that ...

 

[Bill_K]

The Weyl ordering tries to define a general prescription for operator ordering: complete symmetrization.

O(qⁿ p^m) ≡ 2^-n Σ qⁿ⁻ⁱ p^m qⁱ where the sum runs i=0 to n.

But difficult questions like operator ordering are a reason that people turned away from Langrangian formulation and were led instead to path integrals.