Primary constraints and Nambu-Goto action

[haushofer]

Hi,

I have a fairly simple question, in particular for the Nambu-Goto string,
$$S = - T \int d^2 \sigma \sqrt{-\gamma}$$
where gamma is the induced metric on the worldsheet. The canonical momenta are
$$p_{\mu} = - T\sqrt{-\gamma}\gamma^{a0}\partial_a x_{\mu}$$
From this it is quite straightforward to see that these momenta obey the two primary constraints
$$p_{\mu}x'^{\mu} = 0, \ \ \ p_{\mu}p^{\mu} + T^2 x'_{\mu}x'^{\mu} = 0$$
My question is: how do you systematically derive these constraints (not only for the string, but in particular)?

These primary constraints are due to the fact that the Jacobian of the transformation
$$p_{\mu} = \frac{\partial L}{\partial \dot{x}^{\mu}}$$
is not invertible, so it has to do something with this. The number of eigenvectors with eigenvalue zero of this Jacobian then, as I understand, gives the number of primary constraints. So the only thing I can think of is to calculate the Jacobian, and see if this anihilates the linear combination $a\dot{x}^{\mu} + bx'^{\mu}$ (what else could it be?), but is this the right approach?

Does anyone have a clear answer, or a good reference to this? Thanks! :)

 

[haushofer]

BTW, if this topic is more appropriate in another subforum, I don't mind to have it replaced.

 

[haushofer]

No-one?

 

[MathematicalPhysicist]

Have you tried at Physics Stackexchange?

 

[tom.stoer]

The general approach is called "constraint quantization". There are several different approaches (Gupta-Bleuler in QED, Dirac described a rather general concept, BRST, ...). I would start with Dirac's original paper.

 

[negru]

I don't think he needs quantization, this is classical stuff. But Dirac does have a whole book on constraints. You should find info in more advanced classical mechanics books too.

 

[tom.stoer]

A basic example is the canonical formalism for the free relativistic particle

 

[haushofer]

Hi,

indeed, I don't have to quantize, this is all classical. I've read Dirac's "lectures on QM" and his treatment of Hamiltonian analysis, but I can't really find in that text how one systematically finds the primary constraints.

For the free relativistic particle, one gets that the Jacobian $J_{\mu\nu} \equiv \frac{\partial p_{\mu}}{\partial \dot{x}^{\nu}}$ of the transformation

$$\dot{x}^{\mu} \rightarrow p_{\mu} = \frac{\partial L}{\partial \dot{x}^{\mu}}$$

annihilates the vector $\dot{x}^{\mu}$, but I only see that this is an indication that the momenta are dependent (the Jacobian has determinant zero and hence is not invertible); I don't see how one actually derives the primary constraint from that, but probably I'm missing something very basic.

 

[haushofer]

Have you tried at Physics Stackexchange?

No, I didn't know that site, but I will take a look :)

 

[Haelfix]

The canonical reference is Henneaux and Tetelboim "Quantization of gauge systems". They go over how to derive the first and second class constraints in great detail including all the subtleties in the first few chapters.

For a slightly easier read, there are likely many classical mechanics texts as well, but the real juice comes from the above.

 

[haushofer]

Thanks Haelfix, I'll look that one up! :)