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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! :)
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