All Questions
15
questions
2
votes
1
answer
159
views
Confusion regarding the $S$-matrix in Quantum Field Theory
In his Harvard lectures on QFT, Sidney Coleman defines the $S$-matrix as,
$$ S \equiv U_{I}(\infty, -\infty) $$
Where $U_{I}(-\infty, \infty)$ is the time evolution operator in the interaction picture....
- 322
1
vote
1
answer
112
views
$S$-matrix in Dirac picture
Let's define the interaction Hamiltonian as
$$\hat{H}(t) = \hat{H}_{\text{S}}+\hat{V}_{\text{S}}(t)\tag{1}$$
Where $\hat{V}_{\text{S}}\in \mathcal{L}(\mathcal{H})$ represents time-dependent ...
user375448
17
votes
2
answers
1k
views
What physical processes other than scattering are accounted for by QFT? How do they fit into the general formalism?
For background, I'm primarily a mathematics student, studying geometric Langlands and related areas. I've recently been trying to catch up on the vast amount of physics knowledge I'm lacking, but I've ...
- 171
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
$$
\phi(x)=\phi(\vec{x}, t)=\int \frac{d^3 p}{(2 \pi)^3} \frac{1}{\sqrt{2 \omega_p}}\left[a_p(t) e^{-i p ...
- 534
3
votes
0
answers
201
views
Vacuum matrix elements
On page 87, section 7.2.3 titled Vacuum matrix elements of Quantum Field Theory and the Standard Model by Matthew Schwartz, the author writes that the vacuum state $|\Omega>$ is annihilated by the ...
- 904
4
votes
1
answer
207
views
The reasoning of the definition of $S$-matrix
The definition of the $S$-matrix is given by
$$S=\lim_{t_{f}\rightarrow\infty}\lim_{t_{i}\rightarrow-\infty}U(t_{f},t_{i}).$$
Where $U(t_{f},t_{i})$ is the evolution operator, given by the $$U(t_{f},...
- 525
1
vote
0
answers
83
views
State time-evolution in the Interaction picture
What is the Schrödinger-like equation
$$i\frac{d}{dt}|\psi(t)\rangle_I=V_I|\psi(t)\rangle_I$$
telling us for the behavior of the interaction picture state vectors, $|\psi(t)\rangle_I$, at infinity/...
- 3,992
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
3
votes
1
answer
616
views
Why is there a negative sign in the time evolution operator when defining in/out states? (Peskin/Schroeder)
This relates to Peskin & Schroeder's QFT book, equation 4.70 on page 104.
To define in and out states we take our initial state and evolve it far into the past, and do the same for our final state....
- 6,963
2
votes
2
answers
419
views
Usefulness of $| {\rm in}\rangle$ and $| {\rm out}\rangle$ states in S-matrix description of QFT
I am currently reading Niklas Beisert's lecture notes on QFT, Chapter 10, on the scattering matrix $S$.$^1$ My main confusion lies in the construction of $\vert \rm in \rangle$ and $\vert \rm out \...
- 1,215
2
votes
1
answer
215
views
Energy Interpretation of Quantum Effective Action From Weinberg's "The Quantum Theory of Fields"
In section 16.3 of Weinberg, he attempts to prove that the effective potential energy $V(\phi)$ is equal to the minimum energy density of a state with field expectation value $\phi$. I am confused ...
- 3,514
1
vote
1
answer
233
views
What people mean by "state evolving with the interacting/free theory"?
This is a quite basic question but I confess it is something I didn't get up to this point.
When defining the Moller operators and hence the $\cal{S}$-matrix one usually considers "states $\Psi$ ...
- 36.4k
0
votes
0
answers
100
views
Why can we use time-dependent perturbations when evaluating the S-matrix?
Suppose we have Hamiltonian $H_0 + V$. When working in the interaction picture we may derive the evolution operator of $|\psi_I(0)\rangle$ which is given by $$S(t,t_0) = T\left[\exp \left( -i \int_{...
- 238
0
votes
1
answer
459
views
Time Evolution of Asymptotic Free States in QFT
In equation (4.70) of Peskin, he states that
$$_{out}\langle \mathbf{p_1, p_2, \cdots} | \mathbf{k_A,k_B}\rangle_{in} = \lim_{T\rightarrow \infty}\langle \mathbf{p_1, p_2, \cdots} | e^{-iH(2T)} |\...
- 4,780
9
votes
3
answers
960
views
Why is there a time dependence in the Heisenberg states of the Haag-Ruelle scattering theory?
I'm reading R. Haag's famous book "Local Quantum Physics: Fields, Particles, Algebras", 2nd edition, and I'm very puzzled by the way he treats the Heisenberg picture in the Haag-Ruelle ...
- 687