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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 3}\\

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\item 
Show that the energy-momentum tensor
$${\cal T}_{\nu\mu} = \partial_\mu u_i \frac{\delta{\cal
L}}{\delta(\partial^\nu u_i)} - {\cal L}g_{\mu\nu}$$ satisfies the
continuity equation $\partial^\mu{\cal T}_{\mu\nu} =0$ when the
Euler-Lagrange equation is in force. The continuity equation means
that something is conserved; what is it? What is the Hamiltonian
density ${\cal H} = {\cal T}_{00}$ for a complex scalar field having
Lagrangian density
\begin{displaymath}
{\cal{L}}=(\partial_\mu \phi)^* (\partial^\mu \phi)-m^2 \phi^*
\phi-\frac 12 \lambda (\phi^* \phi)^2 {\rm{ ?}}
\end{displaymath} \\
Hint: You should consider transformation properties of fields in space
translations and apply Noether theorem remembering that in
translations the Lagrangian does not remain invariant. For literature
see e.g. Bailin and Love, chapter 3. (This should be quite a short
calculation...)

\item 
Study the model with $N$ real scalar fields $\phi_i$, $i=1,...,N$,
having global $O(N)$ symmetry
\begin{displaymath}
{\cal{L}} = \frac 12 \partial_\mu \phi^i \partial^\mu \phi_i + \frac
12 m^2 \phi^i \phi_i- \frac 14 \lambda (\phi^i \phi_i)^2 .
\end{displaymath}
Study the spontaneous symmetry breaking. Show that the Goldstone
theorem is correct, that is, that there is a massless Goldstone boson
for each generator breaking the symmetry of the vacuum. What is the
residual symmetry group? Show that the generators of the residual
symmetry group leave the vacuum invariant.

Hint: See Abers, Lee, Phys. Rep. 9C (1973) 14--19 (a copy can be found
from the exercise box). $\frac 12 (N-1)N$ generators $T_{ij}$ of the
orthogonal rotation group $O(N)$ can be written as
$(T_{ij})_{kl}=-i(\delta_{ik} \delta_{jl}-\delta_{il}
\delta_{jk})$. You can start by choosing a VEV $\left< \phi \right>$
for the field. Then you can expand the fields around the VEV as
usually. There should be one massive field and $N-1$ massless
fields. You should see the residual symmetry group quite easily. You
should identify the generators of $O(N)$ that do not belong to the
residual symmetry group, denoted by $K_i$. When you operate with these
generators to the vacuum state you should get $K_i \left< \phi \right>
\ne 0$, i.e. they do not leave the vacuum invariant.

%\item 
%Study in detail spontaneous symmetry breaking in locally $U(1)$
%invariant theory having Lagrangian
%\begin{displaymath}
%{\cal{L}} = {\overline{\psi}} i \gamma^\mu D_\mu \psi + (D^\mu
%\phi)^* (D_\mu \phi) + m^2 \phi^* \phi-\frac 14 \lambda (\phi^*
%\phi)^4 -\frac 14 F^{\mu \nu} F_{\mu \nu} ,
%\end{displaymath}
%where $D_\mu=\partial_\mu+igA_\mu$, $F_{\mu \nu}=\partial_\mu
%A_\nu-\partial_\nu A_\mu$ and $\phi$, $\psi$ and $A_\mu$ are scalar,
%spinor and vector fields. What happens to the Goldstone boson? Can
%$A_\mu$ be a electromagnetic field? How could you make the fermion
%massive?


\item 
Dirac spinors are not irreducible representations of the homogeneous
Lorentz group. Show this by finding an operator ($=\gamma_5$) that
commutes with the generators of the homogeneous Lorentz group and the
Hamiltonian for the massless Dirac spinor. Show that for massless
Dirac spinor $\gamma_5 \psi_D=\pm \psi_D$. Define
$\psi_{DL(R)}=P_{L(R)} \psi_D$. Write the Lagrangian
\begin{displaymath}
{\cal{L}}=\frac i2 \left[ {\overline{\gamma}}_D \gamma^\mu
\partial_\mu \psi_D- (\partial_\mu {\overline{\gamma}}_D) \gamma^\mu
\psi_D \right]-m_D {\overline{\gamma}}_D \psi_D
\end{displaymath}
in terms of its chiral components $\psi_{DL(R)}$ ($m_D\ne 0$) and show
that at massless limit the chiral components decouple and that there
are separate equations of motion for both of them. \\ Hint: $P_L=\frac
12 (1-\gamma_5)$, $P_R=\frac 12 (1+\gamma_5)$. For literature see any
textbook on advanced relativistic quantum
mechanics. (E.g. Itzykson-Zuber, Halzen-Martin, ...)



\end{list}


Return the solved problems to the downstairs box in Laurikainen
building on Monday October 5th at 12 o'clock at latest. The www-page
of this course is: \\ http://www.physics.helsinki.fi/{\%}7Epuolamak/hitu/
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