All Questions
28
questions
1
vote
0
answers
101
views
Discontinuity of the scattering amplitude and optical theorem
The generalized optical theorem is given by:
\begin{equation}\label{eq:optical_theorem}
M(i\to f) - M^*(f\to i) = i \sum_X \int d\Pi_X (2\pi)^4 \delta^4(p_i-p_X)M(i\to X)M^*(f\to X).\tag{Box 24.1}
...
- 53
2
votes
1
answer
106
views
Scattering Amplitude & Unitarity
In Srednicki's Quantum Field Theory chapter 11, the probability of a $2 \to n$ scattering process is calculated to be
$$
P = \frac{|\left<f|i\right>|^2}{\left<f|f\right>\left<i|i\right&...
- 148
2
votes
1
answer
321
views
Weinberg, Effective Field Theories
Weinberg in his QFT Volume 1 points out in Chapter 12, section 12.3, near Fig. 12.4 (Is Renormalizability necessary?) that for expansions in EFTs in powers of $k/M$, where $k$ is the energy scale of ...
- 433
2
votes
2
answers
326
views
Proof of unitarity of gauge-invariant $S$-matrix in Peskin and Schroeder
I'm reading chapter 9.4 "Quantization of the electromagnetic field" of Peskin's and Schroeder's book.
When proving the unitarity of the gauge-invariant $S$-matrix, a trick is used.
$$
SS^\...
- 546
7
votes
1
answer
780
views
What does it mean for QFT to be unitary?
I understand the statement that 'X QFT is unitary' is shorthand for saying 'the S-matrix of X QFT is unitary', cf. e.g. this Phys.SE post.
Is there some definition of unitarity that is stronger than ...
- 1,022
1
vote
0
answers
382
views
Why does S-matrix have poles?
Why does unitarity require amplitudes to have singularities? I can see this by constructing amplitudes for specific processes, but what is the rationale behind the general rule? For concreteness, ...
- 2,758
6
votes
1
answer
2k
views
QFT: relation between Cutkosky's cutting rules and the optical theorem
I'm self-studying QFT and am trying to see the relation between Cutkosky rules and the optical theorem, which are presented together as consequences of unitarity in almost every elementary ...
- 61
7
votes
1
answer
601
views
What does QFT say about non-linear processes?
In QFT one can use the S matrix theory. We have a IN free system in the far past. It interacts in a black box in the present and there is a free OUT system in the far future.
We have OUT = S IN with ...
- 708
1
vote
1
answer
734
views
Unitarity and amplitudes
In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes.
I want to understand statement:
In a local theory of ...
- 5,707
4
votes
1
answer
220
views
For which values of lambda the euclidean two-point function $(p^2 +m^2)^{-\lambda}$ is reflection positive
The case $\lambda=1$ is well known free field kernel. What about $\lambda$ in between 0 and 1 ??
... for $\lambda>1$ I have a proof that the kernel is not reflection positive ,
...
- 41
5
votes
1
answer
184
views
Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?
The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
- 103k
6
votes
1
answer
443
views
Perturbative proof of unitarity of $S$-matrix in QED
In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
7
votes
1
answer
3k
views
Optical theorem in QFT
I've been working with the Optical theorem in the case in which final and initial states are equals and I have the following doubt. Let's write the scattering matrix $S$ as:
$$S = 1 + i·T \tag1$$
...
- 1,597
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
10
votes
1
answer
2k
views
Allowed Field Re-definitions in QFT
I am trying to understand which field redefinitions are allowed in a QFT. The textbooks I have read appear to treat this topic flippantly. I assume that one cannot arbitrarily manipulate the ...
- 223