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{\Large Theories of particle physics} \hfill 
{\large Fall 1998, Exercise 2}\\

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\item

Obtain the Euler-Lagrange equations of motion for theories
whose Lagrange densities are
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\item ${\cal L}_0(\varphi,\partial_\mu\varphi) =\frac{1}{2}[\partial^\mu
\varphi(x)\partial_\mu\varphi(x) - M^2 \varphi^2(x)]$
\item ${\cal L}_0(\varphi,\partial_\mu\varphi) = 
\partial^\mu\varphi^*(x)
\partial_\mu\varphi(x) - M^2 \varphi^*(x)\varphi(x)$
\item ${\cal L}_0(A_\mu,\partial_\mu A_\nu) = -\frac{1}{4}
 F^{\mu\nu}F_{\mu\nu}$
\item ${\cal L}_0(\psi,\partial_\mu\psi) =\frac{\dot{\imath}}{2}[
\bar{\psi}(x)\partial\!\!\! / \psi(x) - 
(\partial^\mu\bar{\psi}(x))\gamma_\mu \psi(x)] - m\bar{\psi}(x)\psi(x)$.
\end{list}

\item Show that the Lagrangians 
\begin{displaymath}
{\cal L}_0(\varphi,\partial_\mu\varphi) = \partial^\mu\varphi^*(x)
\partial_\mu\varphi(x) - m^2 \varphi^*(x)\varphi(x)
\end{displaymath}
and
\begin{displaymath}\label{LAGspi}
{\cal L}_0(\psi,\partial_\mu\psi) =\frac{\dot{\imath}}{2}[
\bar{\psi}(x)\partial\!\!\! / \psi(x) - (\partial^\mu\bar{\psi}(x))\gamma_\mu
\psi(x)] - M\bar{\psi}(x)\psi(x),
\end{displaymath}
where $\psi=(\psi_1,\psi_2)^T$,
are invariant in global infinitesimal SU(2)-transformations
$$
\varphi(x)\rightarrow\varphi'(x)=\varphi(x),
\psi^a(x)\rightarrow
\psi'^a(x)=(\delta^{ab}-\frac{\dot{\imath}}{2}\varepsilon^k{\tau_k}^{
a b }) \psi_b(x),
$$
i.e. $\varphi(x)$ is isosinglet and $\psi(x)$ isodoublet.\\
Hint: ${\bar{\psi}} = ({\bar{\psi_1}},{\bar{\psi_2}})$ and
$A\!\!\! / \equiv A_\mu \gamma^\mu$. 

%What would change if the SU(2) transformation would be local? Would
%the above Lagrangians remain invariant in local SU(2)
%transformations?\\ Hint: In global transformations the parameter
%$\varepsilon$ is constant everywhere. In local transformations the
%parameter $\varepsilon$ is, by definition, a function of place
%($\varepsilon^k=\varepsilon^k(x)$). Look at the derivatives.

\item What are the conserved currents in global U(1)-transformations for
the Lagrangian
\begin{eqnarray*}\label{LAGQED}
{\cal L} &=&\frac{\dot{\imath}}{2}[
\bar{\psi}(x)\partial\!\!\! / \psi(x) - \partial^\mu\bar{\psi}(x)\gamma_\mu
\psi(x)] - M\bar{\psi}(x)\psi(x)  \\&&
-e\bar{\psi}(x)\gamma_\mu\psi(x)A^\mu
- \frac{1}{4} F_{\mu\nu}F^{\mu\nu}.
\end{eqnarray*}
How would you interpret the result?

\item (Corrected version, clarified hints and notation.)
\begin{displaymath}
{\cal{L}}={\overline{\psi}}(i \gamma^\mu D_\mu-m) \psi-\frac 14
F^{\mu \nu}_a F_{\mu \nu}^a ,
\end{displaymath}
where
\begin{displaymath}
D^\mu=\partial^\mu+i g T^a A^\mu_a ,
\end{displaymath}
\begin{displaymath}
F^{\mu \nu}_a=\partial^\mu A^\nu_a-\partial^\nu A^\mu_a-g f_{abc}
A^\mu_b A^\nu_c .
\end{displaymath}
$T_a$ are generators of some simple Lie group obeying commutation
relation $[T_a,T_b]=i f_{abc} T_c$. 

Show that the Lagrangian is invariant in {\em finite} local (=gauge)
transformations
\begin{displaymath}
\psi(x) \rightarrow U(x) \psi(x), 
\end{displaymath}
\begin{displaymath}
A^\mu(x) \rightarrow U(x)
(A^\mu(x)-\frac ig \partial^\mu ) U^{-1}(x)=U(x) A^\mu(x)
U^{-1}(x)-\frac ig U(x) (\partial^\mu U^{-1}(x)) ,
\end{displaymath}
where $U(x)=e^{-i g T^a \Lambda_a(x)}$ and $A_\mu(x)= A^a_\mu(x) T_a$.

Hint: The first part of the Lagrangian and the Yang-Mills Lagrangian
are separately gauge invariant. You can begin by showing that the
covariant derivative transforms as $D^\mu(x) \psi(x) \rightarrow U(x)
D^\mu(x) \psi (x)$, as it should. You can use this to show that the
first part of the Lagrangian is gauge invariant. The Yang-Mills
Lagrangian can be written in matrix form using $F^{\mu \nu}=F^{\mu
\nu}_a T^a(=- \frac ig [D^\mu,D^\nu])$ and the normalization property
of the simple Lie groups ${\rm{Tr}} T_k T_l=2 \delta_{kl}$. (If you
want, you can take the simple Lie group to be SU(2); then
$T_a=\tau_a$, $a=1,2,3$, and $f_{abc}=2 \epsilon_{abc}$. This is not
necessary, though.) For helpful literature on the topic see e.g. Bailin
and Love, chapter 9 ``Gauge field theories''.

\end{list}


Return the solved problems to the downstairs box in Laurikainen
building on Monday September 28th at 12 o'clock at latest. The first
problem session is on Tuesday 16-18 in the Exercise room
(harjoitussali), Laurikainen building, [English] 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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