Questions tagged [analyticity]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
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=...
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} \...
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 ...
17
votes
2answers
5k 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 ...
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., $...
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, ...
10
votes
2answers
713 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 ...
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 ...
4
votes
2answers
271 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 ...
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 ...
18
votes
5answers
664 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 ...
10
votes
2answers
1k 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 ...
7
votes
1answer
716 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}\...
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
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 ...
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 ...
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 ...
5
votes
2answers
196 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) - \...
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 ...
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}\...
2
votes
0answers
144 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 ...
2
votes
1answer
57 views

Is this a holographic principle?

The idea of the holographic principle is that all the data about what's inside a volume can be discribed by fields on it's boundary. But... isn't this just obvious calculus? e.g. take a field in ...
2
votes
1answer
58 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 $ ...
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 ...
1
vote
2answers
276 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 ...