- #1

- 649

- 3

## Homework Statement

Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.

## Homework Equations

[tex]

\{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta A}{\delta \pi}\frac{\delta B}{\delta \phi}

[/tex]

[tex]

j_{a}^{\mu}=\frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}A_{a}^{\nu}\partial_\nu \phi - A_{a}^{\mu}\mathcal{L}

[/tex]

[tex]

\pi = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi)}

[/tex]

[tex]

\Box \phi - m^2 \phi=0

[/tex]

[tex]

Q_a = \int d^3 x j^{0}_{a}

[/tex]

## The Attempt at a Solution

Starting with the action

[tex]

\mathcal{L}=\frac{1}{2}(\partial_\mu \phi \partial^\mu \phi -m^2 \phi^2)

[/tex]

for the conserved currents for both translations and lorentz transformations I obtain:

[tex]

p^\mu = \int d^3 x (\partial^0 \phi \partial^\mu \phi - g^{0\mu}\mathcal{L})

[/tex]

[tex]

m^{\alpha\beta}=\int d^3 x (\theta^{0\alpha}x^\beta - \theta^{0\beta}x^\alpha)

[/tex]

with

[tex]

\theta^{\alpha\beta}=\partial^\alpha \phi \partial^\beta \phi - g^{\alpha\beta}\mathcal{L}

[/tex]

The for [itex]\{ p^\mu ,p^\nu \}[/itex] I get

[tex]

\frac{\delta p^\mu}{\delta \phi}=g^{0\mu}m^2 \phi \quad\text{and}\quad \frac{\delta p^\mu}{\delta \pi} = \partial^\mu \phi - g^{0\mu}\partial^0 \phi

[/tex]

which gives

[tex]

\begin{align}

\{ p^\mu ,p^\nu \} &= \int d^4 x (g^{0\mu}m^2 \phi)(\partial^\nu \phi - g^{0\nu}\partial^0 \phi)-(\partial^\mu \phi - g^{0\mu}\partial^0 \phi)(g^{0\nu}m^2 \phi) \\

&= \int d^4 x m^2(g^{0\mu} \phi \partial^\nu \phi - \partial^\mu \phi g^{0\nu}\phi) \\

&= \int d^4 x m^2 g^{0\mu}\frac{1}{2} \partial^\nu (\phi^2) - \int d^4 x m^2 g^{0\nu}\frac{1}{2} \partial^\mu (\phi^2) =0

\end{align}

[/tex]

since the field vanishes on the boundary. Now if I have done this right... to compute [itex]\{ p^\mu , m^{\alpha\beta} \}[/itex] I can borrow the momentum stuff, and for the Ms I get

[tex]

\frac{\delta m^{\alpha\beta}}{\delta \pi}=(\partial^\alpha \phi - g^{0\alpha}\partial^0 \phi)x^\beta - (\partial^\beta \phi - g^{0\beta}\partial^0 \phi)x^\alpha

[/tex]

and

[tex]

\frac{\delta m^{\alpha}}{\delta \phi}=g^{0\alpha}m^2 \phi x^\beta - g^{0\beta}m^2 \phi x^\alpha

[/tex]

Now at this point I have tried to compute the poisson bracket of those above guys, however I cannot seem to retrieve the appropriate form... I know I need to get a superpostion of momenta but I can't seem to get the momentum to come back out. The things I have tried range from integration by parts of various terms, to substituting in the the EOM for the [itex]m^2[/itex] term and trying to integrate by parts from that. But no luck as of yet. Could someone give me a solid push in how to manipulate these guys into the correct end form, thanks.

Last edited: