Canonical quantization with constraints

[mhill]

let be the Lagrangian [tex] (1/2)m( \dot x ^{2} + \dot y^{2}) - \lambda (x^{2}+y^{2}-R^{2}) [/tex]

with 'lambda' a Lagrange multiplier , and 'R' is the radius of an sphere.

basically , this would be the movement of a particle in 2-d with the constraint that the particle must move on an sphere of radius 'R' , my doubt is that i do not know how to quantizy it since [tex] p_{\lambda}=0 [/tex]

 

[strangerep]

let be the Lagrangian [tex] (1/2)m( \dot x ^{2} + \dot y^{2}) - \lambda (x^{2}+y^{2}-R^{2}) [/tex]

with 'lambda' a Lagrange multiplier , and 'R' is the radius of an sphere.

basically , this would be the movement of a particle in 2-d with the constraint that the particle must move on an sphere of radius 'R' , my doubt is that i do not know how to quantize it since [tex] p_{\lambda}=0 [/tex]

Have a look at the Wiki page for Dirac brackets and Dirac-Bergman quantization:
http://en.wikipedia.org/wiki/Dirac_bracket
It explains the essence of constrained quantization.

See also this paper:

R. Jackiw, "(Constrained) Quantization Without Tears", hep-th/9306075

 

[denian]

Canonical quantization with constraints is a powerful tool used in quantum mechanics to study systems with certain constraints. In this case, we have a Lagrangian with a constraint that the particle must move on a sphere of radius R. The Lagrange multiplier \lambda is introduced to enforce this constraint.

In order to quantize this system, we need to first determine the canonical momenta for the coordinates x and y. From the Lagrangian, we can see that the canonical momenta are given by p_x = m\dot x and p_y = m\dot y. However, we also have the constraint p_\lambda = 0, which means that the canonical momentum for \lambda is zero.

To quantize this system, we can use the canonical commutation relations [x,p_x] = i\hbar and [y,p_y] = i\hbar. However, since p_\lambda = 0, there is no corresponding coordinate for \lambda and thus no corresponding canonical commutation relation.

This means that we cannot directly quantize the Lagrange multiplier \lambda. Instead, we can use the constraint to eliminate it from the Hamiltonian and obtain a reduced Hamiltonian that only involves the coordinates x and y. We can then quantize this reduced Hamiltonian using the usual methods.

In summary, canonical quantization with constraints allows us to study systems with constraints in quantum mechanics. In this case, the Lagrange multiplier \lambda is used to enforce the constraint that the particle must move on a sphere of radius R. While we cannot directly quantize \lambda, we can use the constraint to obtain a reduced Hamiltonian and continue with the quantization process.