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Consider scalar field theory $$ \mathcal L = \frac{1}{2} (\partial \phi)^2 -V(\phi).$$ I want to understand Hamiltonian formalism in light-cone coordinates. I choose convention $$x^{\pm}=\frac{1}{\sqrt{2}}(x^0 \pm x^3)$$ and obtain $$\pi \equiv \frac{\partial \mathcal L}{\partial (\partial_+ \phi)}=\partial_- \phi$$ and the following Hamiltonian: $$ \mathcal H= \frac{1}{2}(\nabla_{\perp}\phi)^2+V(\phi).$$ It doesn't depend on $\pi$ and I get Hamilton's equation $\partial_+ \phi=0$, which is incorrect. This leads to following question: what did I do wrong and how to proceed correctly in this case. Of course my final goal is quantization of the theory.

Remark: I suspect that my problem is related to the fact that there are characteristic curves of this theory tangent to hypersurface $x^+$. Nevertheless, I hope there is a way around.

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  • $\begingroup$ I computed $\pi \partial_+ \phi - \mathcal L$. I checked that this is equivalent to computing component $T^+_+$ of energy-momentum tensor. $\endgroup$ – Blazej Apr 26 '17 at 17:43
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There are at least two approaches, which yield the same result:

  1. Dirac-Bergmann analysis: There is a second-class constraint $$\chi~:=~\pi-\partial_-\phi~\approx~0, \tag{1}$$ which leads to the Dirac-bracket (5).

  2. Jackiw-Faddeev method: Recall that $x^+$ is light-cone time, so that the Lagrangian density $${\cal L}~=~\partial_-\phi ~\partial_+\phi -{\cal H}, \qquad {\cal H}~:=~\frac{1}{2}(\partial_{\perp}\phi)^2+{\cal V}(\phi), \tag{2}$$ is already on first-order form. The symplectic one-form potential can be transcribed from the kinetic term in eq. (2): $$ \vartheta(x^+) ~=~\int\! dx^- d^2x^{\perp} ~\partial_-\phi(x)~\mathrm{d}\phi(x), \tag{3} $$ where $\mathrm{d}$ denotes the exterior derivative in infinitely many dimensions. The symplectic two-form is then $$ \omega(x^+)~=~\mathrm{d}\vartheta(x^+)$$ $$~=~\frac{1}{2}\int\! dx^- d^2x^{\perp}\int\!dy^- d^2y^{\perp}~(-2)\delta^{\prime}(x^-\!-\!y^-)~\delta^2(x^{\perp}\!-\!y^{\perp})~\mathrm{d}\phi(x)\wedge\mathrm{d}\phi(y).\qquad \tag{4}$$ The equal-time Dirac bracket on fundamental fields is the inverse matrix of the matrix for the symplectic two-form (4): $$ \{\phi(x^+,x^-,x^{\perp}),\phi(x^+,y^-,y^{\perp})\}_{DB} ~=~-\frac{1}{4} {\rm sgn}(x^-\!-\!y^-) \delta^2(x^{\perp}\!-\!y^{\perp}) .\tag{5} $$ One may check that Hamilton's equation $$ \partial_+\phi(x)~\approx~\{\phi(x),H(x^+)\}_{DB}, \qquad H(x^+)~:=~\int\! dx^- d^2x^{\perp}~{\cal H}(x), \tag{6}$$ reproduces Euler-Lagrange (EL) equation.

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  • $\begingroup$ I missed the "obvious" fact that my momentum is not independent of position and the system is constrained. Thanks. $\endgroup$ – Blazej Apr 26 '17 at 22:00

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