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QFT-15005


The interaction Hamiltonian for pion decay \(\pi^-\longrightarrow e^- + \bar{\nu} \) can be written as \[H_\text{int} = \frac{g}{\sqrt{2}}\bar{\psi}_e(x)\gamma_\mu(1-\gamma_5)\psi_\nu(x) \partial^\mu \phi_{\pi^-}(x) + h.c. \]

  1. Show that the decay rate is given by \[\Gamma = \frac{g^2}{8\pi} \frac{m_e^2(m_\pi^2-m_e^2)^2}{m_\pi^3}.\]
  2. Assuming that the coupling constant for \(\pi^-\longrightarrow \mu^- + \bar{\nu} \) is equal to that for the electron decay, calculate the branching ratio \[\frac{\Gamma(\pi^-\longrightarrow \mu^- + \bar{\nu})}{\Gamma((\pi^-\longrightarrow e^- + \bar{\nu})} \]
kapoor kapoor's picture 3 days 3 hours ago

QFT-14001

   Prove that
  \(\begin{equation}
    \int_{t_0}^ t \, dt_1 \int_{t_0}^ {t_1} dt_2\, H^\prime_I(t_1)
      H^\prime_I(t_2) = \frac{1}{2}      \int_{t_0}^ t \, dt_1 \int_{t_0}^ t \,
       dt_2  T\big( H^\prime_I(t_1) H^\prime_I(t_2)\big)
       \end{equation}\)

kapoor kapoor's picture 3 days 3 hours ago

QFT-06003

\(\newcommand{\ket}[1]{|#1\rangle} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\)
Compute the matrix element  \[ \matrixelement{s,\vec{q}}{J_\mu(x)}{r,\vec{p}}\] where \(J_\mu(x)\) is the current for a Dirac particle and \( \ket{s,\vec{q}}\) and \(\ket{r,\vec{p}} \) denote  one particle states of spin half particle with spin and momenta as specified.

kapoor kapoor's picture 3 days 3 hours ago

QFT-05005

\(\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\)

For a real free Klein Gordon field, mass \(m\),  compute  \[ \matrixelement{0}{\phi(x)\phi(y)}{\vec{k}, \vec{q}}\] and show that the result is  properly symmetrized wave function for two identical bosons with momenta \(\vec{q},\vec{p}\). Here \(\ket{\vec{k}, \vec{q}}\) is the state with two bosons with momenta \(\vec{k}, \vec{q}\).

kapoor kapoor's picture 3 days 3 hours ago

QFT-05004

 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.

kapoor kapoor's picture 3 days 3 hours ago

QFT-05003

Prove that the free particle solutions of Klein Gordon equation \(f_q(x)\), given by
\[ f_q(x) = \frac{1}{\sqrt{(2\pi)^3}} e^{-iqx},\]  obey the orthononality relations \[ \begin{eqnarray}   i \int d^3x f_q^*{x} \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p^*(x) = 2\omega_q\delta(\vec{q}-\vec{p}). \end{eqnarray} \] and find the value of \( \int d^3x \big[f_q(x) (f_q(x) \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p(x))\big]\)

                    

kapoor kapoor's picture 3 days 3 hours ago

QFT-05002

Starting from the Lagrangian for a complex scalar field obtain the
Hamiltonian for a free complex Klein Gordon field.
and show that quantized field obeys \(\begin{equation}\big[H, \phi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) \end{equation}.\)

kapoor kapoor's picture 3 days 3 hours ago

QFT-05001


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

kapoor kapoor's picture 3 days 3 hours ago

From Masters : Do not remain nameless

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kapoor kapoor's picture 3 days 13 hours ago

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kapoor kapoor's picture 3 days 15 hours ago

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