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Questions tagged [analyticity]

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7
votes
3answers
753 views

Is holomorphicity the real reason for non-renormalization in supersymmetry?

Seiberg traced the nonrenormalization of supersymmetric theories to holomorphicity of the superpotential in chiral superspace. However, this overlooks the fact that with a different number of ...
3
votes
2answers
219 views

Using Wick Rotation to calculate Generating Function in Minkowski Space

The question arises when I'm reading over the section "3.3.1 Minkowski Space" in page 16-17 in the following link: https://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf It is ...
2
votes
0answers
33 views

Method of pole shifting in Scattering theory [closed]

I asked a question Related to scattering theorey in Mathematics stalk exchange but no one answer to it. I read in a QFT book that this trick of shifting the poles is called Feynman's trick. https://...
2
votes
1answer
46 views

Method of pole shifting (feyman's trick) in Scattering theory vs contour deformation trick

I am studying Scattering theory but I am stuck at this point on evaluating this integral $G(R)={1\over {4\pi^2 i R }}{\int_0^{\infty} } {q\over{k^2-q^2}}\Biggr(e^{iqR}-e^{-iqR} \Biggl)dq$ Where $ ...
4
votes
2answers
86 views

Is there any “real” use of complex analysis in quantum mechanics? [closed]

After learning some quantum mechanics, I see a lot of applications of complex numbers. However, I have not yet seen any application of complex analysis. The full name for complex analysis is "...
5
votes
3answers
2k views

Renormalization condition: why must be the residue of the propagator be 1

In on-shell (OS) scheme, one of the renormalization conditions is that the propagator, say, a scalar theory $$\frac{1}{p^2+m^2-\Sigma(p^2)-i\epsilon}$$ must have a unit residue at the pole of ...
1
vote
0answers
41 views

Proving that $$ [\phi(\vec{x}, 0), \phi(\vec{x}, t)] \sim e^{-i m t}-e^{+i m t} $$ in QFT

So far, I get the following (for the left term in the integral, $d$=3): \begin{equation} \begin{aligned} \Delta_{+}(x) &= \int \frac{\mathrm{d} \vec{p}^{d}}{(2 \pi)^{d} 2 e(\vec{p})} \exp (-i t e(...
3
votes
1answer
116 views

Why are worldsheets of strings _holomorphic_?

Disclaimer. I am a mathematician (algebraic geometer) who knows nothing about physics. Even worse, I might have major misconceptions about the objects I'll ask about. The level of the question is pop-...
0
votes
0answers
30 views

Double Sommerfeld Expansion

Consider the Fermi-Dirac expansion for an arbitrary function $f(\epsilon)$: $I(f)=\int_0^\infty d\epsilon\frac{f(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}$ The large $\beta$ expansion of this quantity ...
1
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0answers
40 views

Analytic Elements Haag-Araki theory

Why the elements of a local von Neumann algebra cannot be analytic elements of the timelike translation?
0
votes
0answers
45 views

Schwinger paper about charge in QED

I would like to find the paper by Schwinger where he discusses analytic properties of physical values in the limit $e\rightarrow 0$ and physical meaning of $e^2$ sign. I know only that this paper was ...
2
votes
2answers
147 views

Wick rotation: still trouble in getting how it works

I'm preparing my second exam in QFT and I still have trouble in getting the Wick rotation and its analytic continuation. I know that this topic have been discussed a lot in previous threads, but I ...
1
vote
1answer
47 views

Can conformal transformations in $\mathbb{R}^{1,1}$ be analytically continued to $\mathbb{R}^{2,0}$?

In 1+1 dimensions, 2D Minkowski space, a conformal transformation is given by two real functions (of one variable). After Wick rotating the time dimension, giving us 2 dimensional Euclidean space, ...
2
votes
1answer
61 views

2-sheeted Riemann surface with 2 branch cuts and Torus

A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that ...
46
votes
7answers
9k views

Why does Taylor’s series “work”?

I am an undergraduate Physics student completing my first year shortly. The following question is based on the physical systems I’ve encountered so far. (We mostly did Newtonian mechanics.) In all of ...
2
votes
2answers
93 views

Infinite sum: Renormalisation

Trying to do the calculation made in a physics article Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling (page 10 to go from equation 56 to 57), I ...
3
votes
1answer
149 views

Complex integration in Peskin and Schroeder

In Peskin and Schroeder, I have a problem with a claim in equation (2.54), which I will rewrite more concisely here. He claims that we have the following equality : $$ \frac{1}{2E_p}e^{-iE_p(x_0)}-\...
1
vote
0answers
39 views

WKB near turning point by means of complex integration (Landau & Lifshitz, Quantum Mechanics) [duplicate]

The question is basically about section 47. in Landau's Quantum Mechanics (non-relativistic theory) everything is fine until the sentence (at the beginning of the second page) where he says: it is ...
6
votes
1answer
984 views

Landau & Lifshitz's Approach (contour method) on the WKB connection formulas

Background of the question (see pp. 161, section 47 in Landau & Lifshitz's quantum mechanics textbook Vol3, 2nd Ed. Pergamon Press). We a following potential well $$U(x)\leq E \quad\text{for} \...
3
votes
1answer
207 views

Applying Kramers-Kronig relation to a simple damped oscillator

I just discovered the Kramers-Kronig relation and am trying to apply it to a simple damped oscillator of the form subjected to an impulse at $t=0$, which is a causal system: $$m\ddot x + c\dot x + k ...
2
votes
1answer
97 views

Fourier transform property in Feynman 1986 Dirac Memorial Lecture

In his famous 1986 Dirac Memorial Lecture, Feynman refers to a Fourier transforms theorem holding in case F(w) satisfies "certain properties", while being restricted to positive frequencies only: ...
1
vote
0answers
51 views

Four point function with complex momenta?

It is well known that the four-point function $$\int_{\mathbb{R}^{3,1}}\frac {d^4 q}{((q+p_1)^2-i\epsilon)((q+p_2)^2-i\epsilon)((q+p_3)^2-i\epsilon)((q+p_4)^2-i\epsilon)}$$ can be computed using the ...
2
votes
1answer
479 views

$i\epsilon$ in the expression of Feynman Propagator

In Peskin, the Feynman's propagator for a real scalar field is first presented in a form without $i\epsilon$ \begin{equation} D_F(x-y)=\int\frac{dp^3}{(2\pi)^3}\int\frac{dp^0}{2\pi}\frac{ie^{-i(x-y)}}...
2
votes
1answer
102 views

Contour Integration in Schwartz

In Matthew Schwartz's QFT text, on page 39, he has the following contour integral: $$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$ This can be split into two terms, one ...
2
votes
1answer
343 views

$i\varepsilon$ in momentum space propagator; is it actually needed?

In (say) phi-4 theory the momentum space propagator is given by: $$\frac{i}{p^2-m^2+i\varepsilon}$$ where I am using the signature $(+---)$. Now momentum space we can do momentum space integrals using ...
6
votes
0answers
72 views

Self-energy that does not obey sum rule

Analytically, I calculated a self-energy $\Sigma(\omega)$, for which I verified that 1) $\text{Im}\big[\Sigma(\omega)\big] \leq 0$ for all $\omega$ and specifically $\text{Im}\big[\Sigma(0)\big] = 0$,...
4
votes
2answers
171 views

Radial ordered commutation relation

In the book Conformal Field Theory of Francesco, Mathieu and Sénéchal, in Sec. 6.1.2, the authors state that the integral $$ \oint_w \mathrm{d}z~ a(z)b(w) ~=~ \oint_{C_1} \mathrm{d}z~ a(z)b(w) - \...
1
vote
0answers
123 views

What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
0
votes
1answer
54 views

Importance of analytic solutions to Hamiltonians

Why is it important to attempt to find an analytic solution for any theoretical model? It usually happens that many of the hamiltonians written to model the system may not usually have exact solutions....
10
votes
2answers
995 views

Where is the Feynman Green's function in quantum mechanics?

In quantum field theory, the Feynman/time ordered Green's function takes the form $$D_F(p) \sim \frac{1}{p^2 - m^2 + i \epsilon}$$ and the $i \epsilon$ reflects the fact that the Green's function is ...
1
vote
1answer
183 views

Diagonalization of a matrix with operators as elements

How to diagonalize a hamiltonian matrix that has differential operators as elements? My matrix looks something like: \begin{bmatrix} A \frac{d^{2}}{{d\theta}^{2}}+ B_{1} & a\cos{(b\theta +c)}\\ a\...
2
votes
0answers
175 views

Feynman propagator for Dirac fields and $i\epsilon$ prescription for analytic continuation

The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$...
4
votes
1answer
196 views

Where are the poles of the one-particle Green's function located in the complex plane?

This post is a followup question to: How to get an imaginary self energy? In the cited post, the two following representations for the one-particle Green's function are shown: $$G(k,\omega) = \...
1
vote
2answers
233 views

Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis. There was an important equation called Cauchy-Riemann equation. When I saw it at first, I recalled a book's sentence stating ...
3
votes
0answers
102 views

Use of Cutkosky rule, the Optical Theorem and Regge trajectories in pp scattering total cross-section calculation

Cutkosky rule states that: $$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$ putting $a=b=p$ in Cutkosky rule we deduce the ...
2
votes
3answers
131 views

Faster-than-light communication using Taylor's theorem? [closed]

I was thinking about Taylor's theorem and how if a function $f(x)$ is analytic at a point $a$ and one can measure all the derivatives at $a$, $f^{(n)}(a)$ then one knows the complete behaviour of the ...
3
votes
0answers
112 views

Can we do a Wick rotation by an angle not being $\pi/2$?

If a state obeys an evolution equation, we can replace t by -t. we get another equation and it is interesting to study its solutions. it we replace t by it (wick rotation) we get again another ...
0
votes
1answer
121 views

Analyticity of the generalized susceptibility in the linear response theory

In linear response theory, the generalized susceptibility $\chi(\omega)$ is defined as $$\chi(\omega)=\int\limits_{0}^{\infty}\phi(t) e^{i\omega t} dt, ~~t\geq 0\tag{1}$$ where $\phi(t)$ is the ...
1
vote
1answer
147 views

Analytical continuation of 2,3,4-point integrals

I was reading a paper that gives a nice collection of all scalar integrals that crop up in QCD loop calculations. Such integrals are computed in some kinematic region and then the authors say the ...
0
votes
1answer
870 views

Matsubara Summation and Contour Integration

I've been reading parts of the book 'Ultracold Quantum Fields' by Henk Stoof and in Chapter 7 I came across something which I don't understand. This chapter is about the functional-integral ...
2
votes
0answers
235 views

Two possibilities for Wick rotation

$\newcommand{\ld}{\mathcal{L}}\newcommand{\adj}[1]{#1^\dagger}\newcommand{\dc}[1]{\overline{#1}}\newcommand{\Psi}{\varPsi}\newcommand{\dd}{\mathrm{d}}$Take a typical Lagrangian density defined over ...
18
votes
5answers
637 views

Why is analyticity a good mathematical assumption in general relativity?

In general relativity, real-variable analytic continuation is commonly used to understand spacetimes. For example, we use it to extend the Schwarzschild spacetime to the Kruskal spacetime, and also ...
0
votes
0answers
67 views

How to properly understand the residue in the LSZ theorem?

The LSZ theorem for a scalar field reads $$ \mathcal M=\lim_{p^2\to m^2}\left[\prod_{i=1}^n(p^2-m^2)\right]\tilde G(p_1,\dots,p_n) $$ where $G$ is the $n$-point function, to wit, $$ G(x_1,\dots,x_n)=\...
16
votes
2answers
4k views

Gaussian integral with imaginary coefficients and Wick rotation

Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
10
votes
2answers
1k views

What is the intuition behind Kramers-Kronig relations?

I have heard that Kramers-Kronig relations constrains the real and imaginary parts of complex permittivity $\varepsilon= \varepsilon^{'} + j\varepsilon^{''}$. What is the intuition behind this ...
14
votes
7answers
1k views

Binary Black Hole Solution of General Relativity?

This is rather a technical question for experts in General Relativity. An accessible link would be an accepable answer, although any additional discussion is welcome. GR has well known solutions ...
3
votes
2answers
432 views

Wick rotations, convergence and Feynman propagators?

It is said (in e.g. Hawking, 1979, Euclidean quantum gravity) that the integral: $$ \int \mathcal{D}\phi \exp(iS[\phi])\tag{1} $$ for real fields in Minkowski space does not converge, but the Wick ...
2
votes
0answers
230 views

Causality and wick rotation

What is the connection between causality and wick rotation? I came across implication of this connection multiple times but can't find a rigorous explanation. For example in the answer to Wick ...
1
vote
3answers
650 views

Complex Variable Book Suggestion

What book should I choose to learn complex analysis as a physics Undergrad. I only want to use one book which will contain everything I need.
24
votes
3answers
3k views

What do the poles of a Green function mean, physically?

Is there a physical interpretation of the existence of poles for a Green function? In particular how can we interpret the fact that a pole is purely real or purely imaginary? It's a general question ...