\input phyzzx

\def\bx{{\bf x}}
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\def\a{\hat a^{\vphantom\dagger}}
\def\ad{\hat a^\dagger}
%\def\bd{\hat b^\dagger}
%\def\bp{{\bf p}}
%\def\kl{{\bk,\lambda}}
%\def\klp{{\bk',\lambda'}}
\def\dummy{{\phantom{0\mu}}}
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	\copy0\kern-\wd0\kern 0.1\wd2\box0 }
\def\pmbold#1{\mathpalette\PMBOLD{#1}}

\hsize=6.8in	\hoffset=-0.2in	%\vsize=8.5in	\overfullrule=0pt
\noindent PHY--396~K.\quad
Problem set \#7.\quad
Due October 21, 1998.
\par\smallskip\hrule\kern1pt\hrule\vbox{}\medskip

\pointbegin
Consider Lorentz-invariant parity-invariant interactions between
4 Dirac fields $\overline\Psi_1$, $\Psi_2$, $\overline\Psi_3$
and $\Psi_4$.
Generally, such terms have form of products of two bilinear currents
with similar Lorentz properties, namely
$$
S_{12}S_{34}\,,\quad
V_{12}^\mu V^\dummy_{34\,\mu}\,,\quad
T_{12}^{\mu\nu} T^\dummy_{34\,\mu\nu}\,,\quad
A_{12}^\mu A^\dummy_{34\,\mu}\quad {\rm and}\quad
P_{12}P_{34}\,,
\eqn\JJottf
$$
where $S_{12}\overline\Psi_1\Psi_2$, 
$V_{12}^\mu=\overline\Psi_1\gamma^\mu\Psi_2$, \etc, \etc,
and ditto for the $S_{34},\ldots,P_{34}$.
Naturally, in addition to interactions~\JJottf\ of the form
$\overline\Psi_1\Gamma^A\Psi_2\,\overline\Psi_3\Gamma_A\Psi_4$
one should also consider the interactions of the form
$\overline\Psi_1\Gamma^A\Psi_4\,\overline\Psi_3\Gamma_A\Psi_2$,
namely
$$
S_{14}S_{32}\,,\quad
V_{14}^\mu V^\dummy_{32\,\mu}\,,\quad
T_{14}^{\mu\nu} T^\dummy_{32\,\mu\nu}\,,\quad
A_{14}^\mu A^\dummy_{32\,\mu}\quad {\rm and}\quad
P_{14}P_{32}\,.
\eqn\JJoftt
$$
However, it turns out that all the bilinear products~\JJoftt\
are nothing but linear combinations of the bilinear products~\JJottf;
this is known as {\sl Fierz rearrangement} of the Dirac
bilinear products.

\pointcon
Algebraically, Fierz rearrangement amounts to an identity
$$
\sum_{A\in R}\,(\Gamma^A)_{ab}\,(\Gamma_A)_{cd}\
= \sum_{R'}\,C_{R,R'} \sum_{B\in R'}\,
	(\Gamma^B)_{ad}\,(\Gamma_B)_{cb}
\eqn\Fierz
$$
where $R$ and $R'$ denote Lorentz-related sets of
$\Gamma^A$ or $\Gamma^B$ matrices;
specifically, $R$ and $R'$ run over sets $\{1\}$ (scalar),
$\{\gamma^\mu\}$ (vector), $\{\gamma^{\mu\nu}\}$ (tensor),
$\{\gamma^5\gamma^\mu\}$ (axial vector) and $\{\gamma^5\}$ (pseudoscalar).
Your task is to prove this identity and to calculate the coefficients
$C_{R,R'}$ of the Fiertz rearrangement.

\spointbegin
Show that \Fierz\ is equivalent to
$$
\sum_{A\in R}\,\Gamma^A\Gamma^C\Gamma_A\
= \sum_{R'}\,C_{R,R'} \sum_{B\in R'}\,\tr(\Gamma^C\Gamma_B)\,\Gamma^B .
\eqn\Lemma
$$

\spoint
For our choice of the $\Gamma^B$ matrices,
$\tr(\Gamma^C\Gamma_B)=\pm4\delta^C_B$.
Use this fact to prove~\Lemma\ and to calculate the coefficients
$C_{R,R'}$.

\pointcon
In addition to the Lorentz-and-parity-invariant four-fermion
interactions \JJottf\ there also Lorentz-invariant but not
parity-invariant interactions
$$
S_{12}P_{34}\,,\quad
P_{12}S_{34}\,,\quad
V_{12}^\mu A^\dummy_{34\,\mu}\,,\quad
A_{12}^\mu V^\dummy_{34\,\mu}\quad {\rm and}\quad
\epsilon_{\kappa\lambda\mu\nu}\,T^{\kappa\lambda}_{12}\,T_{34}^{\mu\nu}.
\eqn\NoParity
$$
Naturally, there are Fierz rearrangements of these bilinear products too.

\spoint
Working out the details of Fierz rearrangements of the parity-breaking
four-fermion interactions is too much work.
However, one particular rearrangement
is of great importence for the theory of low-energy weak interactions:
$$
\overline\Psi_1(1-\gamma^5)\gamma^\mu\Psi_2\,
\overline\Psi_3(1-\gamma^5)\gamma_\mu\Psi_4\
=\ \overline\Psi_1(1-\gamma^5)\gamma^\mu\Psi_4\,
\overline\Psi_3(1-\gamma^5)\gamma_\mu\Psi_2\,.
\eqno\eq
$$
Prove this.

\medskip
\point %2
In theories involving both bosons and fermions,
one often has to combine commutation and
anti-commutation relations of various operators, depending on the overall
statistics of the operators involved.
For that purpose, it is useful to define a `mixed' commutator bracket
$$
[\hat A,\hat B\}\ \buildrel \scriptscriptstyle{\rm def}\over= \
\hat A\hat B\ -\ (-1)^{AB}\hat B\hat A
\eqn\MixedDef
$$
where $(-1)^{AB}$ is $-1$ if both $\hat A$ and $\hat B$ have overall Fermi
statistics (\ie, each comprises an odd number of {\sl fermionic}
creation/annihilation operators --- the number of bosonic creation/annihilation
operators does not matter) and $+1$ in all other cases.

\spointbegin
Verify the Leibniz rule for the mixed brackets:
$[\hat A,\hat B\hat C\} = [\hat A,\hat B\}\hat C
+ (-1)^{AB}\hat B[\hat A,\hat C\}$
and write down a similar rule for the
$[\hat A\hat B,\hat C\}$.

\spoint
Similarly, express $[\hat A\hat B,\hat C\hat D\}$ in terms of appropriate
mixed brackets of $\hat A$ or $\hat B$ with $\hat C$ or $\hat D$.

\spoint
Prove the `mixed' Jacobi identity
$$
(-1)^{CA}[\hat A,[\hat B,\hat C\}\}\ +\ (-1)^{AB}[\hat B,[\hat C,\hat A\}\}\
+\ (-1)^{BC}[\hat C,[\hat A,\hat B\}\}\ =\ 0.
\eqn\MixedJacobi
$$
In other words (and notations),
$$
\vcenter{\openup 1\jot \ialign{&$\displaystyle{{}#}$\enspace\hfil\cr
	   [\hat B_1,[\hat B_2,\hat B_3]] &
	+\ [\hat B_2,[\hat B_3,\hat B_1]] &
	+\ [\hat B_3,[\hat B_1,\hat B_2]] &
	=\ 0,\cr
	   [\hat B_1,[\hat B_2,\hat F]] &
	+\ [\hat B_2,[\hat F,\hat B_1]] &
	+\ [\hat F,[\hat B_1,\hat B_2]] &
	=\ 0,\cr
	   \{\hat F_1,[\hat F_2,\hat B]\} &
	-\ \{\hat F_2,[\hat B,\hat F_1]\} &
	+\ [\hat B,\{\hat F_1,\hat F_2\}] &
	=\ 0,\cr
	   [\hat F_1,\{\hat F_2,\hat F_3\}] &
	+\ [\hat F_2,\{\hat F_3,\hat F_1\}] &
	+\ [\hat F_3,\{\hat F_1,\hat F_2\}] &
	=\ 0,\cr
}}\eqn\MixedJacobiSpelledOut
$$
where `$B$' and `$F$' indicate the overall statistics of the operator involved.

\medskip
\point %3
Finally, an exercise in fermionic creation and annihilation operators and their
anticommutation relations.
$$
\{\a_\alpha,\a_\beta\}\ =\ \{\ad_\alpha,\ad_\beta\}\ =\ 0,\quad
\{\a_\alpha,\ad_\beta\}\ =\ \delta_{\alpha,\beta}\,.
\eqn\CCR
$$

\spointbegin
Calculate the commutators $[\ad_\alpha\a_\beta,\ad_\gamma]$, $[\ad_\alpha\a_\beta,\a_\delta]$
and $[\ad_\alpha\a_\beta,\ad_\gamma\a_\delta]$.

\spoint %b
Calculate the commutators of Dirac Hamiltonian with all the creation and annihilation
operators and verify that the time dependence of the Dirac field $\hat\Psi(\bx,t)$
is correct.

\spoint %c
Consider two one-body operators $\hat A_1$ and $\hat B_1$
and let $\hat C_1$ be their commutator, $\hat C_1=[\hat A_1,\hat B_1]$.
Let $\hat A$ be the second-quantized forms of $\hat A_{\rm tot}$,
$$
\hat A\ =\sum_{\alpha,\beta}\bra\alpha\hat A_1\ket\beta\,\ad_\alpha\a_\beta\, ,
\eqn\OneParticle
$$
and ditto for the second-quantized $\hat B$ and $\hat C$.
\subpar
Verify that $[\hat A,\hat B]=\hat C$.

\pointcon
The second quantized form of a two-body additive operator
$$
\hat B_{\rm tot}\ =\ \half\sum_{i\neq j}\,
\hat B_2( i{\rm th\over}\ {\rm and}\ j{\rm th\over}\ {\rm particles})
$$
acting on identical fermions is
$$
\hat B\ =\ \half\sum_{\alpha,\beta,\gamma,\delta}
\bra{\alpha,\beta}\hat B_2\ket{\gamma,\delta}\,
	\ad_\alpha\ad_\beta\a_\delta\a_\gamma\,.
\eqn\TwoParticles
$$
This expression is similar to its bosonic counterpart, but note the reversed order
of the annigilation operators $\a_\delta$ and $\a_\gamma$.

\spoint %d
Calculate the commutator $[\ad_\mu\a_\nu,\ad_\alpha\ad_\beta\a_\gamma\a_\delta]$.

\spoint %e
Consider a one-body operator $\hat A_1$ and a two-body operator $\hat B_2$
and let\brk $\hat C_2=\left[\bigl(\hat A_1(1\coeff{\rm st}{})
 +\hat A_1(2\coeff{\rm nd}{})\bigr),\hat B_2\right]$.
Show that the respective second-quantized operators in the fermionic Fock space
satisfy $\hat C=[\hat A,\hat B]$.

\pointcon
Finally, consider a Hamiltonian involving both one-body and two-body forces,
$$
\hat H\ =\sum_{\alpha,\beta}\bra\alpha\hat T_1\ket\beta\,\ad_\alpha\a_\beta\
+\ \half\sum_{\alpha,\beta,\gamma,\delta}
    \bra{\alpha,\beta}\hat V_2\ket{\gamma,\delta}\,
	\ad_\alpha\ad_\beta\a_\delta\a_\gamma
\eqno\eq
$$
and a quantum state of the form $\ket{[1,2,\ldots,N]}$.

\spoint %f
Calculate the energy of that state and show that
$$\eqalign{
\bra{[1,2,\ldots,N]}\hat H\ket{[1,2,\ldots,N]}\
    = \sum_{\alpha=1}^N \bra\alpha\hat T_1\ket\alpha\ &
+\ \half\sum_{\alpha,\beta=1}^N \bra{\alpha,\beta}\hat V_2\ket{\alpha,\beta}\cr
&-\ \half\sum_{\alpha,\beta=1}^N \bra{\alpha,\beta}\hat V_2\ket{\beta,\alpha} .\cr
}\eqn\Energy
$$
The last term here is known as the {\sl exchange energy}.
\subpar
Hint: Use
$$
\eqalign{
\a_\alpha\ket{[1,2,\ldots,N]}\ =\ \bra{[1,2,\ldots,N]}\ad_\alpha\ &
=\ 0\quad{\rm for}\ \alpha> N,\cr
\ad_\alpha\ket{[1,2,\ldots,N]}\ =\ \bra{[1,2,\ldots,N]}\a_\alpha\ &
=\ 0\quad{\rm for}\ \alpha\le N.\cr
}\eqno\eq
$$

\bye
