Welcome Guest!
For persistent fullscreen mode, use F11 function key.

# Documents

For page specific messages
For page specific messages
1382 records found.
Title & Summary Author Updated

### 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 3 days 3 hours ago

### QFT-14001

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

kapoor 3 days 3 hours ago

### QFT-06003

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 3 days 3 hours ago

### QFT-05005


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 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 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 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 $$$$\big[H, \phi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x)$$.$$

kapoor 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 \label{EQ01}
\big[H, \pi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) .
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 3 days 3 hours ago

### From Masters : Do not remain nameless

Summary (or abstract) not available for this node.
kapoor 3 days 13 hours ago

### Links for Cross Over Between Different Content Types

Summary (or abstract) not available for this node.
kapoor 3 days 15 hours ago