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For a free complex Klein Gordon field find the unequal time commutator as \[ \big[\phi(x), \phi(y)\big] = i\Delta(x-y)\] and express your answer for \(\Delta(x)\) as an integral of the form \[\int dq e^{-iqx} \delta(q^2-\mu^2) \epsilon(q_0) \] You need not compute the integral.

• Argue that the function \(\Delta(x)\) is odd under change of sign of \(x\) and that it is Lorentz invariant.
• For spacelike \(x\) show that there exists a Lorentz frame such that \(x^\prime =-x\). Hence prove that the function \(\Delta(x)\) vanishes for space like \(x\).

Express the field momentum \begin{equation*} P^k = \int d^3x \big( \pi(x) \partial^k \phi(x) + \pi^*(x)\partial^k \phi(x)^*\big) \end{equation*} in terms of creation and annihilation operators.

• Starting from the Lagrangian for a complex scalar field obtain the Hamiltonian for a free complex Klein Gordon field.
• Write ETCR and for the quantized field prove that \begin{equation}\label{EQ01} \big[H, \pi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) . \end{equation}
• Does relation,\eqRef{EQ01}, hold only as equal time commutator or for \(H\) and \(\pi(x)\) at arbitrary different times ? Explain your answer.
• Use \EqRef{EQ01} to derive the usual Euler Lagrange equation of motion for the complex scalar field.

The Feynman propagator for Klein Gordon field corresponds to \(i\epsilon\) prescription as given below $$ \Delta_F(x-y) = \int d^4k {\exp(ikx)\over k^2-m^2+2i\epsilon k_0}~~,~~\epsilon>0 $$

• [(a)] Show the poles of the integrand in complex $k_0$ plane for $\epsilon>0$.
• [(b)] Evaluate the $k_0$ integral and write your answer for the Feynman propagator as integrals overs space components of momentum $k=(k_0\vec{k})$

$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}${}$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\ket}[1]{|#1\rangle}$ 
$\newcommand{\bra}[1]{\langle #1|}$

For a quantized real scalar field \(\phi(x)\), define functions \(i\Delta^{(\pm)}(x-y)\) defined by \begin{eqnarray} i\Delta^{(+)}(x-y) &=& \matrixelement{0}{\phi(x)\,\phi(y)}{0} \\ i\Delta^{(-)}(x-y) &=& - \matrixelement{0}{\phi(y)\,\phi(x)}{0}. \end{eqnarray} Use \(\phi^{(\pm)}(x)\) to denote positive and negative frequency parts of the scalar field \(\phi(x)\). Show that

• \(\Delta^{(\pm)}(x-y)\) are real.
• \(\Delta^{(+)}(x-y) = \matrixelement{0}{\big[\phi^{(+)}(x)\,,\,\phi^{(-)}(y)\big]}{0}\) \\ \(\Delta^{(-)}(x-y) = - \matrixelement{0}{\big[\phi^{(-)}(x)\,,\,\phi^{(+)}(y)\big]}{0}\).
• The Feynman propagator \(\Delta_F\) is defined as the vacuum expectation value of time ordered product of fields: \[ \Delta_F(x-y) = \matrixelement{0}{T\big(\phi(x)\phi(y)\big)}{0}.\] Show that the Feynman propagator satisfies the equation \[ (\Box + M^2) \Delta_F(x-y)= - \delta^{(4)}(x-y).\]

Hint Write \(\Delta_F(x-y)\) as \[\Delta_F(x-y)= \Delta^{(+)}(x-y)\theta(x_0-y_0) - \Delta^{(-)}(x-y)\theta(y_0-x_0).\]

$\newcommand{\Msc}{\mathscr M}$
For a decay process \(A (q) \to B(p) + C(\ell)\), where \(A\) and \(B\) are spin \(1/2\) fermions and \(C\) is a spin zero particle, the transition amplitude involves matrix element of the form \begin{equation} \Msc = \bar{u}^{(r)}_B(p)\big(g - g{'}\gamma_5\big) u^{(s)}_A(q) \end{equation} where \(u_B, u_A\) etc are Dirac spinors for the particles \(B\) and \(A\). We need to take absolute square of the matrix element and sum over all spins. Taking the adjoint, \(\dagger\), carefully gives the complex conjugate of \(\Msc\) \begin{eqnarray} \Msc^* &=& \{\bar{u}^{(r)}_B(p)\big(g - g{'}\gamma_5\big) u^{(s)}_A(q)\}^\dagger \\ &=& \{{u}^{(r)\dagger}_B(p)\gamma_0\big(g - g{'}\gamma_5\big) u^{(s)}_A(q)\}^\dagger\\ &=& \{{u}^{(s)\dagger}_A(q)\big(g^* - g^{{'}*}\gamma_5\big) \gamma_0 u^{(r)}_B(p)\} \\ &=& \{\bar{u}^{(s)}_A(q)\gamma_0\big(g^* - g^{{'}*}\gamma_5\big) \gamma_0 u^{(r)}_B(q)\}\\ &=& \{\bar{u}^{(s)}_A(q)\big(g^* + g^{{'}*}\gamma_5\big) u^{(r)}_B(p)\} \end{eqnarray} 

Therefore, the absolute square is given by \begin{eqnarray} \sum_{r,s} |\Msc|^2 &=& \Msc^* \Msc\\ &=& \sum_{r,s} \{\bar{u}^{(s)}_A(q)\big(g^* + g^{{'} *}\gamma_5\big) u^{(r)}_B(p) \} \times \big\{\bar{u}^{(r)}_B(p)\big(g - g{'}\gamma_5\big) u^{(s)}_A(q) \big\} \end{eqnarray}

Using projection operators, \begin{equation} \sum_r u^{(r)}(k) \bar{u}^{(r)}(k) = \frac{n\!\!\!{k}+ M }{2M} \end{equation} this expression can be written as trace of gamma matrices. To see this we write \(\Msc\) and \(\Msc^*\) terms explicitly in terms of Dirac indices as \begin{eqnarray} \Msc &=&\sum_{\alpha\beta} [\bar{u}^{(r)}_B(p)]_\alpha\,[\big(g -g{'}\gamma_5]_{\alpha\beta}\, [u^{(s)}_A(q)]_\beta\\ \Msc^*&=& \sum_{\sigma\rho}[\bar{u}^{(s)}_A(q)]_\sigma\,[\big(g^* + g^{{'}^*}\gamma_5\big)]_{\sigma\rho}\, [u^{(r)}_B(p)]_\rho \end{eqnarray} 

Therefore \(|\Msc|^2\), summed over all spins becomes
\begin{eqnarray} \sum_{r,s} |\Msc|^2 &=& \sum_{r,s}\sum_{\alpha\beta\sigma \rho} [\bar{u}^{(r)}_B(p)]_\alpha\,[\big(g-g{'}\gamma_5]_{\alpha\beta}\, [u^{(s)}_A(q)]_\beta [\bar{u}^{(s)}_A(q)]_\sigma\,[\big(g^* + g^{{'}^*}\gamma_5\big)]_{\sigma\rho}\, [u^{(r)}_B(p)]_\rho \end{eqnarray}

Now put \(u^{(r)}_A(p) \bar{u}^{(r)}_A(p)\) terms next to each other to get and use projection operators and complete spin sums over \(r,s\)
\begin{eqnarray} {\sum_{rs}|\Msc^2|} &=& \sum_{r,s}\sum_{\alpha\beta\sigma \rho} \Big(u^{(r)}_{B\rho}(p) \bar{u}^{(r)}_{B\alpha}(p)\Big)\,[\big(g -g{'}\gamma_5)]_{\alpha\beta}  \times \Big( u^{(s)}_{A\beta}(q) \bar{u}^{(s)}_{A\sigma} (Q)\Big) \,[\big(g^* + g^{{'}*}\gamma_5\big)]_{\sigma\rho}\\ &=& \sum_{\alpha\beta\sigma\rho} \left(\frac{n\!\!\!{p}+ M_B}{2M_B}\right)_{\rho\alpha} \Big(g -g{'}\gamma_5\Big)_{\alpha\beta}\, \left(\frac{n\!\!\!{q}+M_A}{2M_A}\right)_{\beta\sigma}\,\Big(g^* + g^{{'}*}\gamma_5\Big)_{\sigma\rho}\\ &=& \text{tr}\left\{\Big(\frac{n\!\!\!{p}+ M_B}{2M_B}\Big) \Big(g -g{'}\gamma_5\Big)\, \Big(\frac{n\!\!\!{q}+M_A}{2M_A}\Big)\Big(g^* + g^{{'}*}\gamma_5\Big)\right\} \end{eqnarray}
Thus problem of squaring the matrix element and summing over all spins has been reduced to computing trace of products of gamma matrices. \paragraph*{Question for You} Derive an general formula for spin sums taking \(|\Msc|^2\) to be of the form \begin{equation} \Msc= \bar{u}^{(s)}_B(p) \, \Gamma \, u^{(r)}_A(q) \end{equation} where \(\Gamma\) is an expression involving Dirac \(\gamma\) matrices. \paragraph*{Note} With this preparation you are now ready to compute life times and cross sections for processes involving fermions. \noindent Please go through the attached scan of computation of lifetime of \\ \(\Lambda^0 \to p + \pi^-\). \noindent Check all steps and factors carefully. Let me know if I have missed anything.

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Use Wick's theorem to evaluate the following free field time ordered operator product \[T\big(:\phi^4(x):\ :\phi^4(y):\big)\] and write it as a sum of normal ordered products and in terms of \(\Delta(x-y)=\matrixelement{0}{T(\phi(x)\phi(y))}{0}\). Which term(s), if any, will contribute to \(\pi-\pi\) scattering, if the field \(\phi(x)\) represents neutral pion field.

For \(\phi^4\)theory draw all one loop Feynman diagrams for four point function.