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Questions tagged [analyticity]

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39 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(...
115 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-...
29 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 ...
40 views

Analytic Elements Haag-Araki theory

Why the elements of a local von Neumann algebra cannot be analytic elements of the timelike translation?
43 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 ...
123 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 ...
46 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, ...
58 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 ...
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 ...
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 ...
137 views

100 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 ...
70 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 $\text{Im}\big[\Sigma(0)\big] = 0$,...
109 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 ...
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....
181 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\...
169 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 "$i\epsilon$...
189 views

629 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 ...
66 views

817 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 ...
228 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 ...
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 ...
612 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.
146 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 ...
148 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 ...
265 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 ...
197 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 ...
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 ...
219 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 ...
857 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 ...