All Questions
57
questions
0
votes
0
answers
63
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Questions about computing the commutator of the Lorentz generator
I am computing the commutator of the Lorentz generators, from the Eqn (3.16) to Eqn (3.17) in Peskin & Schroeder.
$$
\begin{aligned}
J^{\mu\nu} &= i(x^\mu \partial^\nu - x^\nu \partial^\mu ) &...
- 41
2
votes
2
answers
132
views
Commutator of conjugate momentum and field for complex field QFT
In Peskin & Schroeder's Introduction to QFT problem 2.2a), we are asked to find the equations of motion of the complex scalar field starting from the Lagrangian density. I want to show that:
$$i\...
- 538
0
votes
0
answers
63
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Action of Conjugate momentum $\hat{\pi}$ on $\hat{\phi}$ eigenstate [duplicate]
So I am trying to solve ex. 14.3 in Schwartz textbook "Quantum Field Theory and the Standard Model"
and in the second requirement, he wanted me to show that the action of the conjugate ...
3
votes
1
answer
117
views
Show that $i/2m\int d^3\vec x\hat\pi(\vec x)\partial^2_i\hat\phi(\vec x)=1/(2\pi)^3\int d^3\vec p E(\vec p)\hat a(\vec p)^\dagger\hat a(\vec p)$ [closed]
Show that the quantum field for the Hamiltonian, $$\hat H=\frac{i}{2m}\int d^3 \vec x\hat{\pi}(\vec x)\partial^2_i\hat{\phi}(\vec x)\tag{1}$$
can be written as $$\int \frac{d^3\vec p}{(2\pi)^3}E(\vec ...
- 295
0
votes
1
answer
66
views
Clarification Needed for The Klein-Gordon Field Acting on the Vacuum State (Peskin and Schroeder)
In Peskin and Schroesder's Introduction to Quantum Field Theory, section 2.3, the Klein Gordon Field has the expression
$$
\phi(x,t) := \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2\omega_{p}}} [a_{...
- 23
1
vote
1
answer
214
views
Simplify a product of annihilation/creation operators
Take $\psi^\dagger(r) = \sum_i c^{\dagger}_i φ∗_i (r)$, $\psi(r) = \sum_i c_iφ_i(r)$, $c^\dagger_i$ and $c_i$ are fermionic creation/annihilation operators.
I want to calculate:
$\langle{0}|\psi(r2)\...
- 325
0
votes
1
answer
90
views
Greiner´s Field Quantization question [closed]
I upload a screenshot of Greiner´s book on QFT. I don´t understand one step. I need help understanding equation (3), what are the mathematical steps in between?
Greiner, Field Quantization, page 245 (...
- 15
1
vote
1
answer
52
views
Momentum commutation relations in QFT [closed]
Given the following relations for scalar fields at equal time $t$:
$$ [\phi_{\alpha}(x) , \pi_{\beta} (x')] = i \delta_{\alpha \beta} \delta( x - x') $$
$$ [\phi_{\alpha}(x) , \phi_{\beta} (x')] = [\...
- 1,124
-2
votes
1
answer
96
views
Why $\langle k,n|\varphi(x)|0\rangle = e^{-ikx} \langle k,n|\varphi(0)|0\rangle$? (Srednicki's Quantum field theory)
I'm reading Srednicki's Quantum field theory, p.94 and trying to understand some statement :
Why is the underlined statement true?
Q.1. What is the energy-momentum operator $P^{\mu}$?
Q.2. Why can we ...
- 323
2
votes
0
answers
359
views
Momentum of the Dirac field in terms of creation/annhilation operators [closed]
In Peskin & Schroeder, the Momentum operator of the Dirac field is given as:
$$
{\bf P}=\int d^3x \psi^\dagger \left(-i\nabla\right) \psi=\int \frac{d^3p}{(2\pi)^3}\sum_s {\bf p}(a_p^{s\dagger}a_p^...
- 129
4
votes
1
answer
310
views
How can I find an operator originally expressed in terms of raising and lowering operators in terms of the field operators?
I'm following this book on QFT called "Quantum Field Theory of Point Particles and Strings" by Brian Hatfield. After the end of the scalar field theory section on Exercise 3.6, it asks us to ...
- 775
0
votes
1
answer
260
views
Tong QFT Problem set 2, question 6: Normal ordering of angular momentum operator
I've been studying Tong's QFT notes and am trying to do problem sheet 2, question 6. here.
We are asked to take the classical angular momentum of the field,
$\begin{align}
Q_i &= \epsilon_{ijk}\...
- 213
0
votes
0
answers
41
views
Polar decomposition of a complex scalar field theory
In the text I am referring to, the field was substituted in terms of a number density and phase:
$$\psi(x) = \sqrt(ρ(x))e^{iθ(x)}.$$
While quantizing the field, a commutation relation was imposed:
$$[\...
4
votes
2
answers
612
views
Heisenberg's picture on complex field operators
I've been reading David Tong's lecture notes on QFT, and specifically on Lecture 2, he writes (section 2.6, eq. 2.8.3)
$$e^{i\hat{H}t}\,\hat{a}_{\vec{p}}\,e^{-i\hat{H}t}\,=\,e^{-iE_{\vec{p}}t}\,\hat{a}...
0
votes
1
answer
255
views
How are the creation and annihilation operators constructed for a single mode?
I'm reading Weinberg's QFT, and he defines the creation and annihilation operators as
\begin{align}
(a_k)_{n_1',n_2',\dots,n_1,n_2,\dots}&=\sqrt{n_k}\delta_{n_k',n_k-1}\prod_{j\ne k}\delta_{n_j',...
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