- #1
ismaili
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Dear all,
I'm reading Polchinski's text of string theory. In section 1.3, he demonstrates how to quantize the free point particle in the light-cone gauge. I'm confused with a step in the follows.
Begin with the action,
[tex]S = \frac{1}{2}\int d\tau\left(\eta^{-1}\dot{X}^\mu\dot{X}_\mu - \eta m^2\right)[/tex]
Choose the light-cone gauge
[tex] X^+(\tau) = \tau [/tex]
Then the action becomes,
[tex] S' = \frac{1}{2} \int d\tau \left(-2\eta^{-1}\dot{X}^- + \eta^{-1}\dot{X}^i\dot{X}^i - \eta m^2 \right)[/tex]
Thus, the Hamiltonian is
[tex] H = p_-\dot{X}^- + p_i\dot{X}^i - L
= \frac{p^ip^i+m^2}{2p^+} [/tex]
I can follow these till now, but he says later which I don't understand how he does that:
"The remaining momentum component [tex]p_+[/tex] is determined in terms of the others as follows. The gauge choice relates [tex]\tau[/tex] and [tex]X^+[/tex] translations, so [tex]H=-p_+ = p^-[/tex]. The relative sign between [tex]H[/tex] and [tex]p_+[/tex] arises because the former is active, and the later passive."
Q1: How does he get this relation [tex]H=-p_+ = p^-[/tex]?
Q2: Why [tex]H[/tex] is active and [tex]p_+[/tex] is passive? What does he mean by active and passive?
Thanks very much for any instructions!
I'm reading Polchinski's text of string theory. In section 1.3, he demonstrates how to quantize the free point particle in the light-cone gauge. I'm confused with a step in the follows.
Begin with the action,
[tex]S = \frac{1}{2}\int d\tau\left(\eta^{-1}\dot{X}^\mu\dot{X}_\mu - \eta m^2\right)[/tex]
Choose the light-cone gauge
[tex] X^+(\tau) = \tau [/tex]
Then the action becomes,
[tex] S' = \frac{1}{2} \int d\tau \left(-2\eta^{-1}\dot{X}^- + \eta^{-1}\dot{X}^i\dot{X}^i - \eta m^2 \right)[/tex]
Thus, the Hamiltonian is
[tex] H = p_-\dot{X}^- + p_i\dot{X}^i - L
= \frac{p^ip^i+m^2}{2p^+} [/tex]
I can follow these till now, but he says later which I don't understand how he does that:
"The remaining momentum component [tex]p_+[/tex] is determined in terms of the others as follows. The gauge choice relates [tex]\tau[/tex] and [tex]X^+[/tex] translations, so [tex]H=-p_+ = p^-[/tex]. The relative sign between [tex]H[/tex] and [tex]p_+[/tex] arises because the former is active, and the later passive."
Q1: How does he get this relation [tex]H=-p_+ = p^-[/tex]?
Q2: Why [tex]H[/tex] is active and [tex]p_+[/tex] is passive? What does he mean by active and passive?
Thanks very much for any instructions!