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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 1}\\

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\item
Find the solutions to the Klein-Gordon equation $$(\partial^\mu
\partial_\mu+m^2) \psi(x)=0 .$$ 

\item
Find the solutions to the Dirac equation
$$( i \partial_\mu \gamma^\mu - m ) \psi(x)=0,$$ and interpret them
physically (see e.g. Bj\"orken-Drell, Merzbacher, Itzykson-Zuber,
Halzen-Martin or any other textbook in advanced quantum physics). {\em
Total} angular momentum should be conserved, that is, it should
commute with the Hamiltonian for Dirac spinor. What is the total
angular momentum operator for Dirac spinor (see e.g. Halzen and
Martin, especially exercise 5.4)? 

\item A representation for a linear algebra, called Clifford
algebra, can be constructed from the matrices 
$\gamma_\mu$ in the following way:
$$
\Gamma^S = 1,\: \Gamma^V_\mu =\gamma_\mu, \: \Gamma^T_{\mu\nu} = 
\frac{\dot{\imath}}{2} [\gamma_\mu,\gamma_\nu],\: \Gamma^A_\mu = \gamma_5
\gamma_\mu,\: \Gamma^P = \gamma_5 .
$$
Show that this set
 $\{\Gamma^a\}^{16}_{a=1}$ 
is complete, i.e. it  generates the 16 dimensional vector space of the
linear algebra. Hint: You should show
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\item $\;\;$  $(\Gamma^a)^2 = \pm 1$,
\item  $\;\;$ $\forall \Gamma^a \neq 1\; \exists \Gamma^b$ s.e. 
$\Gamma^a\Gamma^b = -\Gamma^b\Gamma^a$,
\item $\;\;$ so that $ {\rm Tr}(\Gamma^b) =0$, 
$\Gamma^b \neq 1, a \neq b$,
\item $\;\;$ $\Gamma^a\Gamma^b = d_{abc}\Gamma^c$, $d^4_{abc}=1$, 
$\Gamma^c \neq 1$, $a \neq b$
\item $\;\;$so that  $\lambda_a \Gamma^a = 0 \Rightarrow \lambda_a = 
0$. 
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Show the above for as many a,b that it is sufficiently
believable.(Itzykson-Zuber, Bj\"orken-Drell)

\item 
Show in detail that the bilinear covariants
$$
\bar{\psi}(x)\psi(x), \;\; \bar{\psi}(x)\gamma_5 \psi(x),\;\;{\rm 
and}\;\;\: \bar{\psi}(x)\gamma_\mu \psi(x)
$$
of the spinor $\psi(x)$ transform in the 
Lorentz transformations $\Lambda$ as a scalar, pseudoscalar and 
vector.\\
Hint:  You should show  ${\Lambda^\nu}_\alpha{\Lambda_\mu}^\alpha = 
{g^\nu}_\mu$, $\gamma^0S^\dagger
(\Lambda)\gamma^0 = S^{-1}(\Lambda)$, $S^{-1}(\Lambda)\gamma_5 S(\Lambda)=
\det(\Lambda) \gamma_5$. Also note that the matrix $S(\Lambda)$ can be 
defined requiring the covariance of the Dirac equation.
(See e.g. Bj{\"o}rken-Drell I or Schweber for 
Lorentz transformations and inversions of space)

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General advice: Avoid writing the gamma matrices and spinors in
component form as much as possible - use the general properties of the
gamma matrices as far as you can! You might find Juha's appendix for
the last year's course quite useful in dealing with gamma matrices
(see the web page).

Return the solved exercises to the downstairs box in Laurikainen
building on Monday September 21st at 12 o'clock at latest. The first
problem session is on Tuesday 16-18 [English] (as yet I have no idea
where, you can read it from the web page or ask me later) and the
second is on Friday 10-12 in Hall 5, Tallqvist building [Finnish].

The
www-page of this course is:
http://www.physics.helsinki.fi/{\%}7Epuolamak/hitu/
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