Questions tagged [analyticity]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
3answers
729 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
153 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
169 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
361 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
213 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
2k 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
249 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
918 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
55 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
90 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
53 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
1k 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 ...
7
votes
1answer
1k 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
132 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
172 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 ...
30
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
158 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 ...
11
votes
0answers
803 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
356 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^...
7
votes
1answer
793 views

Strange use of complex analysis in Weinberg QFT 1?

In the beginning of chapter 3 on scattering theory in Weinberg's QFT book there is a use of the Cauchy residual theorem that I just cannot get. First some notation, we are looking at states that are ...
1
vote
1answer
1k views

Concrete example of the application of complex analysis in electrostatics [closed]

I've heard complex analysis can be useful in solving electrostatics problems, but despite doing some research I was unable to find any concrete examples. Would anyone be able to provide a simple ...
3
votes
0answers
336 views

How exactly analyticity of S-matrix comes from causality principle?

Recently I've read that analyticity of S-matrix ($S(k)$, where $k$ corresponds to momentum, may be analytically extended into complex values of momentum) comes from causality principle. How to prove ...
5
votes
2answers
297 views

Elementary question about endpoint singularities

In George Sterman's book "An Introduction to Quantum Field Theory", on pages 413-414, there is a description of the endpoint singularity. One begins with the function $$ I(w) ~=~ \int_{\zeta_a}^{\...
1
vote
0answers
77 views

How to parametrize off-shellness?

The energy of a massive on-shell particle of mass $m$ and three-momentum $\vec{p}$ satisfies $$E_\vec{p} = \sqrt{\vec{p}^2+m^2}. $$ What would be the analogous expression for an off-shell particle? ...
4
votes
0answers
99 views

Are there relativistic theories with spacetime modelled on $\mathbb C^4$ rather than real Minkowski space $\mathbb R^4$?

Does anybody know of references to theories where relativity & spacetime is modelled on a (complex/Kähler) manifold which is locally diffeomorphic to $\mathbb C^4$ rather than $\mathbb R^4$, hence ...
3
votes
1answer
631 views

Must the wavefunction be (real) analytic?

In order to show the preservation of normalization of the wave function (in one dimension for now), one shows that the time differential is zero, which entails the following step: $$ \frac{d}{dt}\...
1
vote
1answer
147 views

0+0 self interacting QFT - $ e^{-\sin^2 x}$ type integral — Bessel function expansion around infinity

In a physics paper (here) I found this variant of the Bessel function of the first kind. $$ \tag{1} Z(g) ~=~ \frac{1}{\sqrt{g}} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-\frac{1}{2g} \sin^2 x} \, dx ...
10
votes
2answers
710 views

Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT

This maybe a very naive question. I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic ...
4
votes
2answers
849 views

Lippmann-Schwinger Equation with Outgoing Solutions

I'm reading about Green's functions and how the Lippmann-Schwinger equation eventually leads to the textbook expression for the form of wavefunctions in the far radiation zone after scattering by a ...
7
votes
1answer
715 views

From Minkowski to Euclidean Time in Path Integrals

I'm trying to prove the following equality: $$ <x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge x\left(t_{i}\right)=x_{i}\...
1
vote
0answers
614 views

Formula for residence time/turnover rate with unsteady state

I'm not sure this is the correct place to ask my question... but maybe someone could still help me. I'm looking for a way to calculate the residence time/turnover rate. I have the production and ...
0
votes
1answer
306 views

Casimir effect when the zeta function has a pole

let us suppose the Casimir force/ Vaccumm energy of a certain system $$ E_{casimir}= \sum _{n} \omega _{n} $$ which is formally equal to $ E_{casimir}= \zeta _{spec}(-1) $ with $ \zeta (s)_{...
3
votes
0answers
849 views

$\mathrm{i}\epsilon$ prescription makes a function analytical?

I've seen this everywhere where they say "Analytic continuation is obtained by the usual $\mathrm{i}\epsilon$ prescription..." but how is that? How do you analytically continue (say) $\ln x$ with ...
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 ...
17
votes
5answers
556 views

Does the mass point move?

There is a question regarding basic physical understanding. Assume you have a mass point (or just a ball if you like) that is constrained on a line. You know that at $t=0$ its position is $0$, i.e., $...
5
votes
2answers
662 views

Is there a physical motivation to study finite fields?

Clearly finite groups are of immense value in physics and these are also substructures of fields. However I never came across any computations involving finite fields at university and so I never ...
3
votes
1answer
2k views

Deriving the Sommerfeld expansion by contour integration (Le Bellac p. 277)

In Le Bellac's statistical physics book he derives the Sommerfeld expansion by a contour integral. The idea is to expand integrals of the type $I(\beta)\equiv \int_{0}^{\infty}d\epsilon\, \frac{\...
0
votes
1answer
375 views

Two similar questions related to analytic continuation of a complex variable and its conjugate

See the scan attached below. Brown, in his QFT book, argues a certain way to do an integral. I understand that 1.8.13 or equivalently 1.8.14 can be performed once analytic continuation is done. I also ...
10
votes
2answers
1k views

Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
9
votes
1answer
972 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
8
votes
1answer
283 views

Are Born-Oppenheimer energies analytic functions of nuclear positions?

I am looking for references to bibliography that explores the smoothness and analyticity of eigenvalues and eigenfunctions (and matrix elements in general) of a hamiltonian that depends on some ...
10
votes
2answers
5k views

Is there an analytical solution for fluid flow in a square duct?

I couldn't find one but assumed it must exist. Tried to find it on the back of an envelope, but got to an ugly differential equation I can't solve. I'm assuming a square duct of infinite length, ...
25
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 ...
6
votes
1answer
2k views

Analytic continuation of imaginary time Greens function in the time domain

Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature $$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$ ...
9
votes
1answer
735 views

What is the significance of the branch cut in renormalization group logarithms?

What is the physical significance of the branch cut in renormalization group logarithms? (Is this just an avatar of the optical theorem, or is there something to be understood about these logarithms ...
1
vote
2answers
3k views

Inverted Harmonic oscillator

what are the energies of the inverted Harmonic oscillator? $$ H=p^{2}-\omega^{2}x^{2} $$ since the eigenfunctions of this operator do not belong to any $ L^{2}(R)$ space I believe that the spectrum ...
0
votes
2answers
491 views

vector cross products

Lets say you have a free particle in a rotating frame of reference with constant angular velocity $\mathbf{\omega}$. By free, I mean there are no real forces on it. Lets call the moving system "primed"...
7
votes
3answers
758 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 ...
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 ...
33
votes
4answers
2k views

Is the world $C^\infty$?

While it is quite common to use piecewise constant functions to describe reality, e.g. the optical properties of a layered system, or the Fermi–Dirac statistics at (the impossible to reach exactly) $T=...

1 2