All Questions
23
questions
2
votes
0
answers
65
views
Calculating LSZ reduction for higher order in fields terms
Consider a theory with only a single massless scalar field $\phi(x)$ and a current $J^\mu(x)$ which can be polynomially expanded as fields and their derivatives and spacetime
\begin{align}
J^\mu(x) = ...
- 78
1
vote
1
answer
74
views
Quantization of a massless scalar
Let $t$:time, $r$:distance, and $u=t-r$.
Since any massless particle should propagate along u=const. , we need to change the asymptotic infinity of a massless scalar from time infinity to null ...
- 11
2
votes
0
answers
77
views
LSZ theorem for trivial scattering
The $1\to1$ scattering amplitude is trivial and is given by (take massless scalars for simplicity)
$$
\tag{1}
\langle O(\vec{p}) O^\dagger(\vec{p}\,')\rangle = (2 | \vec{p}\,|) (2\pi)^{D-1} \delta^{(...
- 181
0
votes
1
answer
149
views
$S$-matrix from interacting picture
I’ve been reading a lot about the interaction picture, and I’m trying to string the ideas behind it together.
Essentially, the goal is to calculate something like $<f(\infty)|i(-\infty)>$. We ...
user310742
5
votes
1
answer
575
views
Vacuum, creation and annihilation operators in interacting QFT
I am reading the QFT book by M. Schwartz. More specifically, I have issues with the section about LSZ. I am puzzled with the way the creation and annihilation operators from the free theory act there.
...
- 705
2
votes
0
answers
143
views
LSZ reduction formula for scalar field
I am using Schwartz QFT and the LSZ reduction formula at pp 70. The scalar field was written as
$$
\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
2
votes
1
answer
84
views
Proof of boost generator $K_0$ commute with $S$-matix in Weinberg QFT 1
In Weinberg QFT Vol.1, Weinberg defines a boost operator K when there exists an interaction $V$ as
$$\textbf{K}=\textbf{K}_0+\textbf{W}, \tag{3.3.20}$$
where $\textbf{W}$ is expected as a correction ...
- 21
4
votes
2
answers
281
views
Derivation in LSZ Reduction Formula
In deriving the LSZ formula, a crucial step is to show
$$\langle|a_{p}^{\dagger}|\rangle=-i\int dx^0 \int \mathrm{d}^{3} x \partial_{0}\langle | e^{-i p\cdot x} \overleftrightarrow{\partial_{0}} \phi(...
- 381
7
votes
1
answer
747
views
Confusion about in and out states, interacting Hilbert space etc, referring to Weinberg QFT
There are many posts related to this issue on this site, but I have found none that answer my specific questions about this matter.
I review my understanding of Weinbergs approach. There are probably ...
- 1,129
1
vote
0
answers
99
views
In the derivation of LSZ formula, why do we need $\langle k| \phi(0)|0 \rangle =1$? (Srednicki's book)
In the section 5 of the book, it says
The LSZ formula is valid provided that the field obeys
$$\langle 0|\phi(x)|0\rangle=0, \langle k|\phi(x)|0\rangle=1.$$
The second one is needed to ensure one-...
- 995
2
votes
1
answer
547
views
Dyson's formula $\phi^4$ theory
I have some difficulties when calculating the amplitude of a S-matrix using Wick's theorem. The evolution of the $U$-matrix is
\begin{align}
U(t, t_0) = T \exp(-i\int_{t_0}^{t}H_I(t') dt')=1 - i\int_{...
- 146
1
vote
2
answers
165
views
Localised wavepacket (basic question from Srednicki chap. 5)
I'm currently working through Srednicki and I am confused by a couple of lines in chap. 5 (the LSZ reduction formula). He wants a wavepacket localised around $\mathbf{k = k}_1$ and $\mathbf{x} = 0$ at ...
- 143
2
votes
1
answer
1k
views
$\phi^4$-theory, S-matrix Feynman diagram to first order from Peskin and Schroeder
This relates to page 111 in Peskin and Schroeder.
We have the $\phi^4$ S-matrix for a 2-particle to 2-particle scattering reaction:
$$-i\frac{\lambda}{4!}\int d^4x \langle p_1p_2|\mathcal T\left(\phi(...
- 6,963
5
votes
1
answer
158
views
On the creation of wave packets with particular properties in quantum field theory
At the start of chapter 5 of Mark Srednicki's lecture notes on quantum field theory we define an operator that creates a particle that is "localised in momentum space near $\mathbf {k_1}$, and ...
- 6,963
2
votes
1
answer
213
views
Confusion on wave packet and creation operator in Mark Srednicki's book
In Mark Srednicki's QFT book, section $5$, he mentions following things:
$a^{\dagger}({\bf k})$ creates a particle with momentum $k$ and is given by
\begin{equation}
a^{\dagger}(k)=-i\int d^3x [e^{ikx}...
- 995