# Primary constraints and Nambu-Goto action

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! :)

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

No-one?

MathematicalPhysicist
Gold Member
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.

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

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.

Have you tried at Physics Stackexchange?
No, I didn't know that site, but I will take a look :)

Haelfix