All Questions
47
questions
2
votes
2
answers
148
views
Particle Creation by a Classical Source (on-shell mass momenta)
It is noted in Peskin and Schroeder's QFT text that the momenta used in the evaluation of the field operator $\phi(x)$ are "on mass-shell": $p^2=m^2$. Specifically, this is in relation to ...
- 4,399
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 $$\int \frac{d^3\vec p}{(2\pi)^3}E(\vec ...
- 295
0
votes
1
answer
80
views
Confusion on the signs in the complex scalar field [closed]
I saw there are different ways we can write down the complex scalar field. For example, in most textbooks I can find, this is defined as
$$\phi(x) =\int \dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_p}}\...
- 1,783
0
votes
0
answers
111
views
Total momentum operator for the KG field
This question pertains to Equation (2.33) in Peskin and Schroeder:
$$
\hat{\vec P}=-\int d^3\!x\,\hat\pi(\vec x)\vec\nabla\hat\phi(\vec x)=\int d^3\!p\,\vec p\,\hat a_{\vec p}^\dagger\,\hat a_{\vec p}
...
- 713
4
votes
2
answers
248
views
Quantization of non-relativistic complex scalar field
I found that the taking the non-relativistic limit of the Lagrgangian for complex scalar fields gives
$$\mathcal{L} = i\dot{\psi}\psi^* -\frac{1}{2m}\nabla\psi \nabla\psi^*.\tag{1}$$
Now, when we ...
- 193
-1
votes
1
answer
208
views
Field operators on vacuum
What do the field operators $\psi$ and $\pi$ produce when they act on vacuum $|0>$ state?
Here,
$$\psi(\vec{x}) = \int \frac{d^3p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_p}}\left(a_p e^{i\vec{p}\cdot \vec x} ...
- 21
3
votes
1
answer
119
views
Can we obtain an exact answer when we sum Feynman diagrams to all orders?
Consider a $\phi^4$ theory in QFT. Following Peskin & Schroeder's QFT chapter 4, we can do some calculations of correlation functions using perturbation expansion. On their book's page 83, they ...
- 1,421
1
vote
2
answers
265
views
What are Test Functions in QFT (physics context)?
In terms of mode function solutions of the KG equation, the field operator can be written as
$$ \hat{\phi}(x) = \int_{R^3} d^3\textbf{p}\space \left(u_\textbf{p} \hat{a}_\textbf{p}+ u^*_\textbf{p}\hat{...
- 136
2
votes
0
answers
78
views
QFT Formalism, Relation between different POVs
A Klein-Gordon field on a Minkowski background can be written in the following expansion
$$ \hat{\phi}(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}} (\hat{a}_p e^{-ip x} + \hat{a}^\dagger_p e^{...
- 136
0
votes
0
answers
123
views
Calculating the contraction of a field operator and a creation operator
In Folland's Quantum Field Theory (section 6.4) he considers a field:
$$\phi_\pi = \sum_\tau \int f(\textbf{q})\big[u(\textbf{q}, \tau, \pi)a(\textbf{q}, \tau, \pi) e^{-iq_\mu x^\mu} + v(\textbf{q}, \...
- 3,350
9
votes
3
answers
3k
views
How do we know that all quantum fields are Fourier transforms of creation and annihilation operators?
In Folland's book Quantum Field Theory, he says
...we start out with classical field equations and a relativistically invariant Lagrangian from which they are derived, then replace the classical ...
- 3,350
5
votes
1
answer
305
views
What does it mean to apply a creation or annihilation operator to a free field, e.g. $\langle 0|a(p)\varphi(x)| 0 \rangle$?
I am self studying Quantum Field Theory, and I am starting to get a little lost. So far, I have studied free fields and some basic computations involving them, such as creation and annihilation ...
- 3,350
1
vote
1
answer
54
views
Generalized conjugate momenta and the generalized Fourier transform
Let the Lagrangian be a functional of $\hat{\phi}$ and $\partial_{\mu}\hat{\phi}$, i.e. $\hat{L} = L(\hat{\phi},\partial_{\mu}\hat{\phi})$, where $\hat{\phi}$ is an operator.
The conjugate momenta is ...
- 1,023
0
votes
0
answers
153
views
Time dependent operators in QFT
In Quantum field theory, how does one define time-dependent operators? For example, let me generalize the operator fermion $\psi$:
$\psi(x) = \int \frac{d^3p}{2 (\pi)^3} \frac{1}{\sqrt{2E_p}} \sum_s \...
- 1,023
1
vote
1
answer
91
views
What's the rationale of replacing the Fourier coefficients in a field expansion by operators?
Let's take a look on the particular case of the Fourier expansion of the Klein-Gordon field:
$$\psi (x,t) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_0(p)}[a(p)e^{i(E_0(p)t-px)}+a^\star (p)e^ {-i(E_0(p)t-...
- 500