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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 6}\\
%{\Large DRAFT/not a final version}

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%\item
%
%Beginning from eq. (4.2.17) of Chaichian and Nelipa derive
%eq. (4.2.19). 


\item 

The Majorana spinor operator is defined as
\begin{displaymath}
\chi(x)=\sum_{s=\pm \frac 12} {\int{ \frac{d^3k}{(2 \pi)^3} \frac 1{2
k^0} \left[ b(k,s) u(k,s) e^{-i k \cdot x} +\eta b^\dagger (k,s)
v(k,s) e^{i k \cdot x} \right] } },
\end{displaymath}
where $|\eta|=1$. Show that $\chi^c=\eta^* \chi$. Dirac spinor
operator $\psi$ can be defined as a sum of two Majorana spinor
operators having opposite phase factor $\eta$:
\begin{displaymath}
\psi= \frac 1{\sqrt{2}} \left( \chi_1 + \chi_2 \right) .
\end{displaymath}
What are the creation and annihilation operators $c$ and $d$ of the
Dirac spinor expressed in terms of Majorana operators $b_{1,2}$? Show
that $c$ and $d$ obey canonical anticommutation relations. \\ Hint:
$v=u^c$ and $u=v^c$ ($\psi^c=C {\overline{\psi}}^T$).

\item

Derive the commutation relations for the creation and annihilation
operators of real scalar fields $a^\dagger(p)$, $a(p)$ from the
assumption that the particle states are symmetric under change of
particles. From this derive the commutation relation for the field
operator $\hat \psi (x)$ and the canonical impulse $\hat \Pi (x)$.

\item 
Assume that you have a theory containing a heavy real scalar field
$\phi$ and a light fermion field $\psi$ having interaction Hamiltonian
$\hat H_I=\lambda \hat {\overline{\psi}} \hat \psi \hat \phi$. Write
down the transition amplitude for the decay of the boson (momentum
$p_0$) to two fermions (momenta $p_1$, $p_2$) in leading order in
terms of the spinor eigenvectors $u$ and $v$, that is, calculate the
matrix element
\begin{displaymath}
S_{fi}=-i  \int { d^4 x \langle p_1 p_2 | \hat H_I | p_0 \rangle } .
\end{displaymath}
Would it make any difference if the fermion field would be Dirac or
Majorana field? 

%\item In addition to the residual $U(1)_{em}$ gauge symmetry
%associated with the electromagnetic interaction, the standard model
%has some global symmetries (phase transitions related to lepton and
%baryon number) surviving the spontaneous breaking of $SU(2)_L \times
%U(1)_Y$. Find them, and interpret them. Is it possible to break them?


\end{list}


Return the solved problems to the downstairs box in Laurikainen
building on Monday October 26th 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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