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[QUE/QFT-05003]Node id: 2301pageProve 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 \( \int d^3x \big[f_q(x) (f_q(x) \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p(x))\big]\)
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22-04-14 07:04:26 |
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[QUE/QFT-05002]Node id: 2300pageStarting from the Lagrangian for a complex scalar field obtain the
Hamiltonian for a free complex Klein Gordon field.
and show that quantized field obeys \(\begin{equation}\big[H, \phi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) \end{equation}.\) |
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22-04-14 07:04:49 |
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[QUE/QFT-05001]Node id: 2299page
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 does it hold 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 Klein Gordon field.
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22-04-14 07:04:32 |
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[QUE/QFT-05005]Node id: 2303page\(\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\)
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 \(\vec{k}, \vec{q}\). |
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22-04-13 22:04:15 |
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[QUE/QFT-05004]Node id: 2302page 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. |
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22-04-13 22:04:54 |
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[QUE/QFT-05007] QFT-PROBLEMNode id: 4035pageLet \(\displaystyle N=-\Big(\frac{i\beta\vec{\alpha}\cdot\vec{p} }{2mc}\Big)\, f\Big(\frac{|\vec{p}|}{mc}\Big)\)
- Prove that \begin{equation} \exp(iN) = \cos\big(\frac{|\vec{p}|f}{2mc} \big) + \frac{\beta\vec{\alpha}\cdot\vec{p}}{|\vec{p}|} \sin \big(\frac{|\vec{p}|f}{2mc} \big), \end{equation} where \(H\) is Dirac Hamiltonian \(H=c\vec{\alpha}.\cdot\vec{p} + \beta mc^2\)
- Find the real function \(f\) such that \begin{equation} H{'} = e^{iN} H e^{-iN} \end{equation} is free of operators odd operators. For this choice of \(f\) \[ H{'} = \beta c \sqrt{|\vec{p}|^2 + m^2c^2}.\]
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22-02-04 08:02:19 |
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[QUE/QFT-05006] QFT-PROBLEMNode id: 4034pageFor 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 \(x\).
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22-02-04 08:02:38 |
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[QUE/QFT-05005] QFT-PROBLEMNode id: 4350page$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}${}$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}${}$\newcommand{\ket}[1]{|#1\rangle}$ {} $\newcommand{\bra}[1]{\langle #1|}$
For a 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 \(\vec{k}, \vec{q}\). |
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22-02-02 20:02:12 |
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[QUE/QFT-05004] QFT-PROBLEMNode id: 4349pageExpress 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. |
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22-02-02 20:02:34 |
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[QUE/QFT-05003] QFT-PROBLEMNode id: 4348pageProve that the free particle solutions \(f_q(x)\) obey the orthononality relations \begin{eqnarray} i \int d^3x f_q^*(x) \overleftrightarrow{\partial}_0 f_p(x) = 2\omega_q\delta(\vec{q}-\vec{p}).\nonumber \end{eqnarray} and find the value of \( \int d^3x f_q(x) \overleftrightarrow{\partial}_0 f_p(x)\). |
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22-02-02 20:02:18 |
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[QUE/QFT-05001] QFT-PROBLEMNode id: 4346page
- 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.
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22-02-02 19:02:29 |
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[QUE/QFT-05002] QFT-PROBLEMNode id: 4347pageStarting 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}\label{eq01} \big[H, \pi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) . \end{equation} |
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22-02-02 19:02:39 |
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