- #1

haushofer

Science Advisor

- 2,454

- 835

Hi,

I have a fairly simple question, in particular for the Nambu-Goto string,

[tex]

S = - T \int d^2 \sigma \sqrt{-\gamma}

[/tex]

where gamma is the induced metric on the worldsheet. The canonical momenta are

[tex]

p_{\mu} = - T\sqrt{-\gamma}\gamma^{a0}\partial_a x_{\mu}

[/tex]

From this it is quite straightforward to see that these momenta obey the two primary constraints

[tex]

p_{\mu}x'^{\mu} = 0, \ \ \ p_{\mu}p^{\mu} + T^2 x'_{\mu}x'^{\mu} = 0

[/tex]

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

[tex]

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

[/tex]

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 [itex]a\dot{x}^{\mu} + bx'^{\mu}[/itex] (what else could it be?), but is this the right approach?

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

I have a fairly simple question, in particular for the Nambu-Goto string,

[tex]

S = - T \int d^2 \sigma \sqrt{-\gamma}

[/tex]

where gamma is the induced metric on the worldsheet. The canonical momenta are

[tex]

p_{\mu} = - T\sqrt{-\gamma}\gamma^{a0}\partial_a x_{\mu}

[/tex]

From this it is quite straightforward to see that these momenta obey the two primary constraints

[tex]

p_{\mu}x'^{\mu} = 0, \ \ \ p_{\mu}p^{\mu} + T^2 x'_{\mu}x'^{\mu} = 0

[/tex]

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

[tex]

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

[/tex]

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 [itex]a\dot{x}^{\mu} + bx'^{\mu}[/itex] (what else could it be?), but is this the right approach?

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

Last edited: