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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 5}\\
%{\Large DRAFT/not a final version}

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\item
Conjugation operator $C$ is usually defined in group theory to be the
operator that leaves $\psi^T C \psi$ invariant, where $\psi$ is a
representation of some group. If the group in question is homogenous
Lorentz group and $\psi$ are spin-$\frac 12$ representations of it,
then $C$ should satisfy $C \gamma^\mu C^{-1}=-\gamma^{\mu T}$,
$C^\dagger C=1$ (convince yourself that you can define such an
object). {\em (a)} Replace the Dirac spinor $\psi$ in the Lagrangian
of QED (C\&N section 2.3) with $\psi \rightarrow \psi^c=C \gamma^0
\psi^*$ and show that everything remains the same --- except the
electric charge changes in sign. ${\em (b)}$ Show that $\psi^T C \psi$
is really a Lorentz singlet (at least in infinitesimal
transformations).  \\ Hint: For literature see e.g. Peskin and
Schröder p. 64, Halzen and Martin, ... \\ {\small (This way you could
create a {\em Majorana} mass term $m_M \psi^T C \psi$ for spin-$\frac
12$ particles carrying no unbroken $U(1)$ quantum numbers, like
electric charge. Neutrinos would be an obvious candidate. Obviously(?)
lepton number conservation would however be violated.)}

\item 
Show that
\begin{displaymath}
\int^\infty_{-\infty}{ \left( \prod_i{ \frac {dy_i}{ \sqrt{2 \pi}} }
\right) \exp \left(- \frac 12 y_i A^{ij} y_j \right) } =
\left(\det A \right) ^{-\frac 12} .
\end{displaymath}

\item 
What is
\begin{displaymath}
\int { \prod_i{d\eta_i^*} \prod_i{d\eta_i} \exp \left( - \eta_i^*
A^{ij} \eta_j \right) }
\end{displaymath}
for anticommuting Grassman variables?

\item
Study the $\lambda \phi ^4$ model specified by the Lagrangian
\begin{eqnarray}
{\cal{L}} = \frac 12 \partial^\mu \phi \partial_\mu \phi - \frac 12 m^2 \phi^2- \frac
1{4!} \lambda \phi ^4  .
\end{eqnarray}
where $\phi$ is a real scalar field. Beginning from this Lagrangian
derive the path-integral form of the vacuum-vacuum transition
amplitude following the lead of Chaichian and Nelipa
eqs. (4.1.18--22). Write the equivalent of eq. (4.1.22) explicitly in
discrete form (like in eqs. (4.1.14,17,...)) to familiarize yourself
with the notation. Could you in principle calculate the amplitude (in
discretized form) numerically?

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Return the solved problems to the downstairs box in Laurikainen
building on Monday October 19th at 12 o'clock at latest. There are two
problems solving sessions: Tuesdays 16-18 in the Exercise room
(harjoitussali) in the basement of the Laurikainen bldg [English] and
Fridays 10-12 in Hall 5, Tallqvist bldg [Finnish]. \\ The www-page of
this course is: http://www.physics.helsinki.fi/{\%}7Epuolamak/hitu/

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