Light-cone gauge quantisation of point particle

[ismaili]

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,

$$S = \frac{1}{2}\int d\tau\left(\eta^{-1}\dot{X}^\mu\dot{X}_\mu - \eta m^2\right)$$

Choose the light-cone gauge

$$X^+(\tau) = \tau$$

Then the action becomes,

$$S' = \frac{1}{2} \int d\tau \left(-2\eta^{-1}\dot{X}^- + \eta^{-1}\dot{X}^i\dot{X}^i - \eta m^2 \right)$$

Thus, the Hamiltonian is

$$H = p_-\dot{X}^- + p_i\dot{X}^i - L = \frac{p^ip^i+m^2}{2p^+}$$

I can follow these till now, but he says later which I don't understand how he does that:
"The remaining momentum component $$p_+$$ is determined in terms of the others as follows. The gauge choice relates $$\tau$$ and $$X^+$$ translations, so $$H=-p_+ = p^-$$. The relative sign between $$H$$ and $$p_+$$ arises because the former is active, and the later passive."

Q1: How does he get this relation $$H=-p_+ = p^-$$?

Q2: Why $$H$$ is active and $$p_+$$ is passive? What does he mean by active and passive?

Thanks very much for any instructions!

 

[eljose79]

Thank you for your question. The relation H = -p_+ = p^- can be understood by looking at the Hamiltonian and momentum components in the light-cone gauge. In this gauge, the Hamiltonian is given by H = p_-\dot{X}^- + p_i\dot{X}^i - L, where p_-\dot{X}^- represents the kinetic energy and p_i\dot{X}^i represents the potential energy. Since the gauge choice X^+(\tau) = \tau relates translations in time (\tau) and X^+, we can write H = -p_+ = p^-. This means that the Hamiltonian is equivalent to the minus sign of the momentum in the X^+ direction.

In terms of active and passive, this refers to the way the Hamiltonian and momentum are acting on the system. The Hamiltonian is considered active because it is the generator of time translations, while the momentum is considered passive because it is a conserved quantity that describes the motion of the system.

I hope this helps clarify the confusion. If you have any further questions, please don't hesitate to ask. Good luck with your studies!

 

[Entropia]

I would first like to commend you for delving into the complex world of string theory and attempting to understand the light-cone gauge quantization of a point particle. It is a challenging subject, but one that holds great potential for understanding the fundamental nature of our universe.

To answer your first question, the relation H=-p_+ = p^- can be obtained by using the equations of motion for the momenta, which can be derived from the Hamiltonian. In the light-cone gauge, the equations of motion for the momenta are given as p^+=0 and p^i=\dot{X}^i. Using these equations, we can solve for p^+ in terms of p^- and p^i, which leads to the relation H=-p_+ = p^-. This relation is important because it shows that the Hamiltonian, which represents the energy of the system, is equal to the negative of the momentum in the light-cone direction.

To answer your second question, Polchinski is referring to the concept of active and passive transformations in symplectic geometry. In this context, active transformations are those that change the coordinates of the system, while passive transformations leave the coordinates unchanged but change the momenta. In the case of the light-cone gauge, the gauge choice X^+(\tau)=\tau is an active transformation that relates translations in \tau to translations in X^+. This is why the Hamiltonian, which represents the active transformation, is related to the passive momentum component p_+.

I hope this helps clarify the steps in the quantization process and the terminology used. Keep exploring and questioning, as that is the essence of science. Good luck in your studies!