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Questions tagged [analyticity]

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44
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7answers
8k 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
90 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
107 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
23 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 ...
2
votes
1answer
78 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
43 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 ...
3
votes
1answer
162 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
65 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 ...
5
votes
0answers
55 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 $\text{Im}\big[\Sigma(0)\big] = 0$,...
1
vote
0answers
84 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
48 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....
1
vote
1answer
150 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
146 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
143 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
152 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
87 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
128 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
106 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
104 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 ...
2
votes
0answers
193 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 ...
4
votes
2answers
143 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) - \...
18
votes
5answers
592 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
65 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)=\...
8
votes
2answers
748 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 ...
10
votes
2answers
739 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 ...
2
votes
1answer
348 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
264 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 ...
3
votes
2answers
356 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 ...
0
votes
1answer
638 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
202 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 ...
14
votes
2answers
3k 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 ...
1
vote
3answers
498 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.
1
vote
1answer
136 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 ...
1
vote
1answer
137 views

Contour Integration in Deriving Coulomb's Law from Classical Field Theory

This is much more on the mathematical side on the derivation, so I may be barking up the wrong Stack Exchange for this question; however, I have a curiosity about a contour integration performed in ...
0
votes
1answer
173 views

Why does a shuttlecock flip head down even when dropped head up? [duplicate]

If a shuttlecock is dropped(not giving it any angular momentum wrt it's COM) head up(i.e. the heavier part up) without tilting it, it always falls head down(i.e. the heavier part down). This fact it's ...
3
votes
0answers
168 views

Sommerfeld expansion for temperature dependent spectral function

Consider the integral \begin{align} I(\beta)=\int_{-\infty}^{\infty}\frac{\text{d}\epsilon}{\pi}\frac{\rho(\epsilon,\beta)}{1+e^{\beta \epsilon}} \end{align} where $\rho(\epsilon,\beta)$ is a generic ...
0
votes
3answers
1k views

Can scalar quantities have a direction?

Current is a scalar yet it has direction. But ideally, only vectors should possess a sense of direction. Does this mean that even scalars can possess a sense of direction? If yes, can someone give me ...
2
votes
0answers
191 views

Where is the method of Contour integration used in physics? [closed]

Complex numbers have a wide variety of application in physics and so must be contour integration but where do we exactly apply the principles of contour integration, residues and poles in the field of ...
5
votes
3answers
756 views

Free energy functions are analytic or non-analytic in phase transitions?

I already saw this Phys.SE post and it seems perfectly reasonable that the free energy describing a system must be a non-analytic function in order to display a phase transition. An analytic ...
0
votes
1answer
49 views

X-ray Photoelectron Spectroscopy

Is XPS already an old method? not a lot of new research is produced recently about it. Besides, you can carry out the same analysis by several cheaper methods. Do you think the chemical analysis by ...
2
votes
1answer
81 views

How do we prove analyticity of Schwinger functions?

Starting from Wightman axioms, we can define the Schwinger functions as the Wick-rotated Wightman functions (as for instance is explained in the book by R. Haag, Local Quantum Physics). The Schwinger ...
2
votes
1answer
52 views

Non-analyticity of the modulus function

I'm studying the Ginzburg-Landau theory of superconductors, and I have read that the free energy can not depend on odd powers of the modulus of the order parameter $\psi$ because $| \psi| = \sqrt{\...
4
votes
1answer
860 views

An integral in Peskin's Quantum Field Theory P. 27

In Peskin's QFT P. 27, there is an integral $$-\frac{i}{2 (2 \pi )^2 r}\int_{-\infty}^{\infty} \mathrm{d}p\frac{p\ e^{ipr}}{\sqrt{p^2+m^2}}.\tag{2.51a}$$ He said that in order to push the contour up ...
6
votes
1answer
835 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} \...
2
votes
0answers
116 views

Is the free energy density analytic in temperature at the Kosterlitz-Thouless phase transition?

I know that the KT transition is infinite-order so the free energy density is a smooth (i.e. infinitely differentiable) function of temperature, but is the function actually analytic at the critical ...
0
votes
1answer
154 views

A problem in an integration related to Wick rotation

In quantum field theory, we often calculate some integrations using Wick rotation. In the following, I will carefully deal with an integration involving Wick rotation. In the end, I have found that I ...
28
votes
2answers
2k views

How can dimensional regularization “analytically continue” from a discrete set?

The procedure of dimensional regularization for UV-divergent integrals is generally described as first evaluating the integral in dimensions low enough for it to converge, then "analytically ...
3
votes
0answers
145 views

Non-equivalence between $\omega \to \omega \pm i\varepsilon$ and Cauchy principle value

I am looking to gain a more rigorous and deeper understanding as to how an $i\varepsilon$ prescription actually changes the end result of a divergent integral, specifically in regards to Green's ...
9
votes
0answers
696 views

Why does analytic continuation as a regularization work at all?

The question is about why analytical continuation as a regularization scheme works at all, and whether there are some physical justifications. However, as this is a relatively general question, I ...
4
votes
1answer
315 views

Boundary conditions in holomorphic path integral

Consider the holomorphic representation of the path integral (for a single degree of freedom): $$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^...