All Questions
26
questions
3
votes
1
answer
287
views
Time-evolution operator in QFT
I am self studying QFT on the book "A modern introduction to quantum field theory" by Maggiore and I am reading the chapter about the Dyson series (chapter 5.3).
It states the following ...
- 613
1
vote
0
answers
134
views
Clarification on interaction picture in QFT
Say we want to calculate $\langle f(t_2)|O|i(t_1)\rangle$. Where $O$ is an arbitrary operator. We can treat the states as stationary and then evolve the operator
$$\langle f(0)|O(t)|i(0)\rangle\\O(t) =...
user310742
2
votes
0
answers
143
views
LSZ reduction formula for scalar field
I am using Schwartz QFT and the LSZ reduction formula at pp 70. The scalar field was written as
$$
\phi(x)=\phi(\vec{x}, t)=\int \frac{d^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_p}}\left[a_p(t) e^{-i p ...
- 534
-1
votes
1
answer
92
views
Commutation $[H_0, \phi_0(\vec{x},t)]$ in the Heisenberg picture [closed]
Studying from Schwartz "Quantum Field Theory and the Standard Model" p. 23, I got to the part where he discusses time dependence of the field operator $\phi$ and the annihilation/creation ...
- 99
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
8
votes
3
answers
2k
views
How to avoid paradoxes about time-ordering operation?
(Original title: is time-odering operator a linear operator?)
I'm confused with two formulas, one of which is
$$
\mathcal{T} \exp \left [-\frac{\mathrm{i}}{\hbar} \int_{t_0}^t \mathrm{d} t' \hat{H}_I(...
- 695
1
vote
0
answers
63
views
Interacting field operator at diffrent time in Peskin and Schroeder'sbook [duplicate]
Related questions
Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder)
About an expression of Peskin and Schroeder
I saw this question has been asked a couple of times in different ...
- 213
2
votes
0
answers
83
views
Why do we say $\exp(-iHt)$ is holomorphic for $t$ in the lower half plane when $H>0$?
I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane ...
- 81
1
vote
0
answers
104
views
Expanding field operators at a fixed time $t_0$ (from Peskin/Schroeder)
This relates to the bottom of page 83 in Peskin and Schroeder.
The following claim is made:
At any fixed time $t_0$ we can of course expand $\phi$ in terms of ladder operators
$$\phi(\textbf{x},t_0)=\...
- 6,963
2
votes
1
answer
547
views
Dyson's formula $\phi^4$ theory
I have some difficulties when calculating the amplitude of a S-matrix using Wick's theorem. The evolution of the $U$-matrix is
\begin{align}
U(t, t_0) = T \exp(-i\int_{t_0}^{t}H_I(t') dt')=1 - i\int_{...
- 146
1
vote
1
answer
348
views
A step in the derivation of the unitary time-evolution operator (time-ordered exponential)
I'm reading introductory quantum field theory, from the book Relativistic Quantum Physics by T. Ohlsson. In the derivation of the unitary time-evolution operator, chapter 11.2, there is an equation ...
- 1,215
4
votes
1
answer
363
views
Time ordering in correlation function in QFT
In Peskin & Schroeder, "An Introduction to Quantum Field theory", chapter 4, the author derives the 2 point correlation function:
$$\langle \Omega|P{\phi(x)\phi(y)}|\Omega \rangle = \...
- 1,023
5
votes
1
answer
217
views
Do photonic creation (anihilation) operators $a^\dagger$ ($a$) taken at different times commute?
This question was spawned from a discussion with my supervisor on the nature of the second-order correlation function
\begin{align}
G^{(2)} (t_1,t_2) &= <\Psi| E^-(t_1) E^-(t_2) E^+(t_2) E^+(...
- 53
0
votes
2
answers
626
views
Commutation of field operators with differents times
We know that field operators in the Heisemberg (or interacting) picture satisfy the commutation relations
$$\{ \hat{\psi}(\textbf{r},t), \hat{\psi}^{\dagger}(\textbf{r}',t) \} = \delta( \textbf{r} - \...
- 587
1
vote
3
answers
386
views
Initial values of creation/annihilation operators
I have a question about creation/annihilation operators. For example, if I have an evolution equation for annihilation operator of photon
$$ \frac{da_k}{dt} = -i \omega_k a_k$$
I obviously obtain
$$...
- 31