Questions tagged [analyticity]

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Are all Local Observables Measured on Gibbs States Analytic as a Function of Temperature Away from Phase Transitions?

Let $\rho(\beta)=e^{-\beta H}/Z$ be the Gibbs state of a quantum Hamiltonian, and $H$ is some local Hamiltonian on $N$ particles, and $Z(\beta)$ is its partition function. Suppose I measure some local ...
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1 answer
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Smooth vs analytic spacetimes

Recently in more technical settings (I was learning algebraic QFT), I encountered the term "real analytic" manifolds (Lorentzian manifolds, to be precise). This is in contrast to smooth ...
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How to Understand the First Term in the Calabrese-Lefevre Distribution?

I am currently reading the following paper and I am trying to understand the first term in equation (6) (reproduced below): $$ P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
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2 votes
1 answer
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How to Prove A Claim made to Construct the Calabrese-Lefevre Distribution?

My question is a mathematical one based on this physics paper. Suppose that $\lambda_i $ is an eigenvalue of a reduced density matrix. Up to a normalization factor, the distribution of eigenvalues is ...
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1 answer
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Two Liouville's theorem

Within the context of Hamiltonian mechanics and phase spaces I have learnt that the phase space distribution function is constant, in other words, that the "volume" of any region is constant,...
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1 answer
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Path integral with double integration involving the free particle case

Suppose we have the path integral: \begin{equation} Z=\int \mathcal{D}x\mathcal{D}y\,\exp\left[-\frac{a}{2}\int_0^1 dt\,\left(\,\dot{x}(t)^2-\,\dot{y}(t)^2\right)\right]. \end{equation} The ...
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2 votes
0 answers
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Schwinger-Keldysh contour and $i\epsilon$ prescription

In Tom Hartmann's notes on path integrals, he describes the Schwinger-Keldysh (or "in-in") formalism for calculating vacuum correlators in QFT. He explains that Lorentzian time-ordered ...
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1 vote
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Fourier transform of Wick rotated functions

I am learning the imaginary time formalism of thermal field theory / reviewing the Euclidean formalism of quantum field theory. One thing that appears to be left implicit in many treatments is a ...
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5 votes
3 answers
238 views

Hilbert transform in soliton paper

I asked this question over at the Mathematics SE, see here, but have not gotten any responses, so I figured I might as well try here as well. While the question is mathematical, it does appear in a ...
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9 votes
2 answers
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Feynman diagrams, can't Wick-rotate due to poles in first and third $p_0$ quadrants?

I have a confusion about relating general diagrams (involving multiple propagators) in Minkowski vs Euclidean signature, which presumably should be identical (up to terms which are explicitly involved ...
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3 votes
2 answers
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Why can an analytic continued Hamiltonian have squared integrable eigenfunctions?

In 1D quantum mechanics, there are no bound states and there are resonant states for the following potentials: $$ W(q)=\frac{1}{2}q^2-gq^3,\tag{1.3.2} $$ $$ W(q)=\frac{1}{2}q^2+\frac{g}{4}q^4,\; g<...
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Analytic continuation of Feynman amplitudes seems ill-defined

I was reading Peskin & Schroeder's book on Quantum Field Theory and on chapter 7, "The optical theorem for Feynman diagrams" (page 232) they extend analytically the Feynman amplitude $i \...
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Null tetrad for a metric with most of the metric components non zero

I am working on a metric which is basically $g = g_0 + h_{\mu \nu}$ where $g_0$ is the Kerr metric up to order a (taking $a^2 = 0$) and $h_{\mu \nu}$ denotes perturbed metric. Hence the complete ...
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1 vote
1 answer
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Holomorphicity of Functions of unitary matrices

I am studying lattice QCD and there I encounter functions of unitary matrices. For ex. The action, $S = \sum$Tr( plaquettes), where each plaquette, $P$ is written as, $$ P = U_{\mu}(x)U_{\nu}(x+\mu){U}...
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Why do we say $\exp(-iHt)$ is holomorphic for $t$ in the lower half plane when $H>0$?

I've seen this statement in many papers. However the function $e^{-iHt}$ satisfies the Cauchy-Riemann condition for all t. So why do we say $e^{-iHt}$ is holomorphic in the lower half plane ...
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How does the $+i\varepsilon$ prescription in the propagator comes from analytic continuation of the Euclidean 2-point function?

Let $S_0[\phi]$ be the action for a real Klein-Gordon field $$S_0[\phi]=\dfrac{1}{2}\int d^Dx \phi(x)(\Box-m^2)\phi(x)\tag{1}.$$ If we try to construct the generating functional $Z_0[j]$ we find that ...
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Does the Harmonic conjugate of the Hamiltonian have to do with the Lagrangian?

Say, we have a Hamiltonian $H(x,p)$. We find a function $G(x,p)$ such that the function $H(x,p)+iG(x,p)$ has a complex derivative. $G$ is then the harmonic conjugate of $H$. Since the change in $H(x,p)...
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7 votes
1 answer
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Does Feynman's path integral include complex trajectories?

The WKB approximation provides the correct exponential decay of eigenstates inside classically forbidden regions if one allows classical momenta to be imaginary. The typical example is a double well ...
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3 votes
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Do finite sized 1D Hamiltonians have free energies which are analytic everywhere in the complex plane?

It's well known that 1D classical and quantum short-ranged Hamiltonians have free energies which are analytic/holomorphic everywhere as a function of inverse temperature $\beta=1/k_BT$ (see Araki, &...
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How to analytically continue Schwinger functions?

To get Wightman functions $W(t_1, \dots, t_{k-1})$ from Schwinger functions $S(\tau_1 = i t_1, \dots)$, we use analytical continuation. But I don't think this is simply an issue of plugging $it_a$ for ...
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1 answer
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Residue of the Fermi Distribution Function

In the "Lecture notes on many-body theory" by Michele Fabrizio, it is stated: How we do show that the Fermi distribution function $f(z)$ has residue $-T$? In the examples on Wikipedia, the ...
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1 answer
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Complex time theories with spacetime $\mathbb{R}^3\times\mathbb{C}$

Are there any well-developed (string?..) theories assuming that, what we perceive as a (3+1) Minkowskian manifold, is a projection/compactification of a 5-dim spacetime, locally obtained via ...
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Expression for the causal retarded potential for $t<0$ must give $0$ but my calculation produces a nonzero result. What's the mistake?

This question was previously asked here in the Mathematics StackExchange but using a slightly different notation. But I did not find the answer I was looking for or rather got two very different ...
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2 votes
1 answer
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Can you perform a Wick rotation if the poles are on the imaginary axis?

I know you can perform a Wick rotation whenever the poles are outside the contour but what happens if the poles are on the imaginary axis? Can you do it anyway?
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Kramers-Kronig relations for geometric series

Suppose $\phi(z)$ is an analytic function in the upper complex plane, so it satisfies the Kramers-Kronig relations, i.e. \begin{equation} \Re\phi(w) = \frac{1}{\pi}\int \frac{\Im\phi(x)}{x-w} dx \end{...
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1 vote
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How "smooth" is the evolution of the universe? [duplicate]

Mathematicians have developed different definitions for how "smooth" a function might be. A function (e.g. from or to $\mathbb{R}^n$ or $\mathbb{C}^n$) might be continuous, or once-...
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Improper Integrals and Contour Integration

I am reading a physics paper which employs contour integration to evaluate some integrals and I am a little confused about something. According to the author if we want to integrate some function from ...
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4 votes
0 answers
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Has it been proven that causality means the S-matrix is analytic?

In books like 'the analytic S-matrix' they give justification that causality implies analyticity however they also said it hasn't been explicitly proven. This book was written a while ago, has it been ...
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Polchinski Eq 3.2.4 and Eq 3.2.5: Deforming contours in path integral

Here is the section of the book I'm talking about. I'm confused about the following two points: (i) Why is the path integral oscillatory? (ii) What does it mean, "we can deform contours just as ...
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5 votes
3 answers
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Why can you deform the contour in the integral expression for the Klein-Gordon propagator to get the Euclidean propagator?

I'm trying to understand the use of the Euclidean correlation functions in QFT. I chased down the problems I was having to how they manifest in the simplest example I could think of: the two-point ...
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Use of Cauchy's integral formula in the derivation of the Feynman propagator

In deriving the Feynman propagator in Timo Weigand's 2014 QFT2 notes, at the top of page 37, (equation 1.170), we use Cauchy's integral formula: $$g(z_0)=\frac{1}{2\pi i}\oint_{C_1}\frac{g(z)}{z-z_0}\...
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1 vote
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Gaussian Integral with Complex Parameters -- Divergence and Convergence

Although this is more of a mathematical question, I will in what follows refer to an answer of @Qmechanic that has been posted in this forum (I am sorry for creating a new post for this, but I ...
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3 votes
1 answer
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Any good references on the analytic structure of scattering amplitudes? [duplicate]

In papers they often say things about the analytic structure of S matrices - things like resonances are poles on the unphysical sheet, particle channels cause a square root branch cut etc. I've seen ...
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Analytic Continuation: Replacement of $t \rightarrow - i \tau$ Mathematical Justification [duplicate]

It's commonly used in imaginary-time path integral that "analytic continuation" means replacing $t \rightarrow - i \tau$ or reparametrizing the theory in terms of imaginary time $\tau = i t$....
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2 votes
2 answers
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Can't the heat equation be inverted?

I have heard people say that the heat equation isn't invertible because it smooths out irregularities that can not be recovered by backwards time evolution. But is this so? I will now argue that it ...
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2 votes
1 answer
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Independence of $z$ and $z^*$ in coherent states

In the book of Lowell Brown on QFT its mentioned that $$\int_{\mathbb{R}^2} \frac{dq'dp'}{2\pi} e^{(-z^{*}z + z^{*}_1z + z^{*}z_2)} = e^{z^{*}_1z_2}\tag{1.8.12}$$ where $$z=\frac{q'+ip'}{\sqrt{2}} \...
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5 votes
1 answer
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Divergent Energies and Analytical Continuation - Two questions on the inverted harmonic oscillator and the inverted double well

I have two questions on the general topic of energy potentials that diverge at infinity. First of all, the inverted harmonic oscillator. I found this post on Physics SE, Inverted Harmonic oscillator. ...
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0 votes
1 answer
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Example of analytic time evolution with a Pauli Hamiltonian

I'm looking for any (non-trivial*) time-independent Hamiltonian expressed in the Pauli basis (with analytically known real coefficients), which unitary time evolves some analytic initial state to some ...
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1 vote
1 answer
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Non-analytic functions and non-local Lagrangians

Infinite sums of increasingly higher-order derivatives, when present in Lagrangians, are typically taken as a sign of nonlocality. This is supposed to rule out fractional, negative and exotic (for ...
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5 votes
2 answers
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How is the free energy of Kosterlitz-Thouless transition smooth yet non-analytic?

Here is an answer by @tparker which makes the following remark "... a Kosterlitz-Thouless transition, at which the free energy density is smooth but non-analytic..." The expression for the ...
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1 vote
0 answers
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Analytic Continuation of the $S$-Matrix and Analytic Continuation of Spinors

Broadly speaking, what is actually being done when one analytically continues the external particles' momenta in scattering amplitudes in QFT? Some associated more detailed questions: When thinking of ...
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2 answers
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Poles of a transmission coefficient

I stumbled on the question I can't quite grasp: What is the meaning of poles for transmission probability $T(E)$? $$ T(E) = \left( 1+\frac{1}{4}\frac{V_0^2}{E (E+V_0)} \sin^2 \left(\frac{2 a}{\hbar }\...
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1 vote
1 answer
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Why is (anti)-holomorphicity considered in physics?

As a mathematician, holomorphicity is an extremely good property that provides rigidity, finite dimensionality, algebraicity. etc to whatever theory that's considered. I'm curious about why (anti-)...
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1 answer
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Electric dipole moment (EDM) and dilogarithm with complex argument

In the paper https://arxiv.org/abs/1310.1385 it's calculated the electron electric dipole moment (eEDM) in terms of a function $f_1(x)$ defined as $$ f_1(x) = \frac{2x}{\sqrt{1 - 4x}}\left[Li_2\left(...
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4 votes
2 answers
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Wick Rotation & Scalar Field Value & Mapping

Wick Rotation helps to solve the problem of the convergence of the path integral, by changing the integral contour in the complex plane. But my question is: In the scalar field path integral, the ...
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0 votes
1 answer
186 views

Confusion about dimensional regularization

I am recently trying to understand dimensional regularization in the context of quantum field theory. So to solve an integral $$ \int_{\mathbb R^d} \frac{\text d ^d p}{(2 \pi)^d} \frac{1}{(p^2 + m^2)^...
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2 votes
1 answer
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What paths are allowed in the Fourier form of the Dirac Delta distribution?

In this form of the Dirac Delta distribution $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$ can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
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2 votes
2 answers
487 views

Klein-Gordon equation multiple Green's functions

I am trying to understand Green's functions for a Klein-Gordon equation: $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +m^2) \phi(\vec{x},t) = 0$ and $ (\frac{\partial^2}{\partial t^2} - \nabla^2 +...
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1 vote
1 answer
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Evaluating integral for Friedel oscillation using branch cuts

I am finding some difficulties understanding the following problem. I have the following logarithm for which I have to identify branch cuts: $$\lim_{\epsilon\rightarrow0}\ln{\frac{(p+2p_F)^2+\epsilon^...
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1 answer
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Proof for getting delta function on $t \to t_0 $ from the equation of the propagator for the free particle in 1 dimension

From Sakurai's quantum mechanics equation 2.5.16 give propagator for a free particle in 1 dimension. Equation 2.5.16 is $$K (x^",t;x',t_0)=\sqrt {m\over {2\pi i\hbar (t-t_0)}} \exp \Biggl [{im (x^"...
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