QF, action, EoM, mass of particle, on-shell

[binbagsss]

Homework Statement

Question attached.

I am stuck on part d, but give my workings to all parts of the question below.

Homework Equations

The Attempt at a Solution

[/B]
a) EoM given by E-L equations:
Gives $-\partial_u\partial^u \phi + m^2 \phi - \frac{\lambda \phi^3}{3!} =0$
b) $L=T- V$, $T$ the kinetic energy, $V$ the potential energy

So , assuming a signature of $(-,+,+,..)$ , $L=\frac{1}{2} (\partial_t \phi)^2-\frac{1}{2} (\partial_i \phi )^2+\frac{1}{2}m^2 \phi^2 - \frac{\lambda}{4!}\phi^4-\frac{3m^4}{2\lambda}$

So $-V= -\frac{1}{2} (\partial_i \phi )^2+\frac{1}{2}m^2 \phi^2 - \frac{\lambda}{4!}\phi^4-\frac{3m^4}{2\lambda}$

c) for a constant field the first deriviative term of the EoM vanishes. So we have :

$m^2 \phi - \lambda \frac{\phi^3}{3!} = 0$
$\phi (m^2-\lambda \frac{\phi^2}{3!}) =0$
So $\phi_c = \pm m \sqrt{\frac{3!}{\lambda}}$

To check which have the lowest energy evaluate $T+ V$

d)

I get: (I can't do tilda in LaTex so I've changed it to $\phi=\phi_c+\phi'(x)$ )

$S= \int d^2 x (\frac{-1}{2} \partial_u \phi' \partial^u \phi' + \frac{1}{2}m^2 (\phi_c + \phi')^2 - \frac{\lambda}{4!} (\phi_c + \phi')^4 -\frac{3m^4}{2\lambda})$

Should I expand only to first order in $\phi'(x)$?

Since $\phi_c$ satisfies the EoM , the EoM becomes:

$-\partial_u\partial^u \phi' + m^2 \phi' -\frac{\lambda}{3!}\phi^3 =0$

I have no idea how get the corresponding mass. I know that if the EoM is obeyed, the particle is on-shell and $p^up_u = -m^2$ ? I think my answer lies in this but I'm not sure what to do

Many thanks in advance.

 

[mark726]

Thank you for sharing your attempt at solving the given problem. It seems like you have a good understanding of the concepts involved and have made some progress towards finding the solution.

To answer your question about part d, yes, you should expand only to first order in $\phi'(x)$. This is because we are interested in small fluctuations around the classical solution $\phi_c$, and expanding to higher orders would involve higher powers of these fluctuations which are negligible in comparison to the classical solution.

To find the corresponding mass, you can use the relation you mentioned: $p^up_u = -m^2$. This is known as the dispersion relation and relates the energy and momentum of a particle to its mass. In this case, since we are considering a scalar field, the momentum is given by $p^u = \partial^u \phi$ and the energy is given by $p_u = \partial_u \phi$. Substituting these into the dispersion relation, you can solve for $m^2$.

I hope this helps. Good luck with your solution!