Questions tagged [analyticity]
The analyticity tag has no usage guidance.
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Polchinski's doubling trick for extending open string theory to the whole complex plane
Open string theory can be described on the upper-half complex plane. To simplify the description of open string theory, Polchinski asserts (eq. 2.6.28 in his Vol. I String Theory book) that it is ...
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Discontinuities in the $u$ channel
if we consider a 2-to-2 scattering, we have normally an $s$ channel a $t$ channel and $u$ channel. In CMS frame $s$ is positive and $t$ and $u$ negative, by crossing symmetry there are kinematics ...
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Is it possible to determine a final orientation from an initial angular velocity and constant angular acceleration analytically?
I am looking to model the rotation of a ball over time. I have the following information:
an initial orientation, as a quaternion
an initial angular velocity, as X/Y/Z components, fixed to the global ...
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Analyticity in the upper half plane and causality
Can you, please, help me to understand the following
How is the analyticity of a complex-valued function in the upper half plane related to causality and Kramers-Kronig relations? Namely, why is it ...
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Proof of commutation relation in Lattice Vertex Operator Algebra
In DGM [1] on page 548 below Equation 5.4, it is claimed that the operators $\frac{dX^j(z)}{dz}$ and $\frac{dX^k(\zeta)}{d\zeta}$ commute, where
\begin{equation}
X^j(z)=q^j-i p^j \log z+i \sum_{n \neq ...
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Is the $S$-Matrix analytic in Planck constant?
Taking the scattering amplitude as a function of $\hbar$, is such function necessarily analytic in this variable.
Suppose I'm concerned with Relativistic Quantum Field Theory.
In QED, the tree level ...
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Properties of analytic continuation of two point/ Wightman function
In this paper, the author considers Wightman functions calculated on an accelerating detector for a massless scalar field, namely
$$G_+^R = {}_M \langle 0 | \phi(x) \phi^{\dagger}(x') | 0 \rangle_M$$
$...
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Are band structures non-analytic only at degenerate points?
The electronic properties of (crystalline) solids is typically described in terms of the electronic band structure, which reveals many properties of the electronic structure such as the band gap, ...
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Reference to understand this branch cut question
I am currently reading a physics paper in which the authors have complexified an ordinary differential equation (ODE). They mention the following statement in the paper:
"These branch points ...
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Complex Analysis books for Physics
I am now in my 6th semester of my physics bachelor and now I'm searching for a complex analysis book.
It shouldn't be too long and deep and not too "mathematical" (I don't need every proof). ...
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Why does Wick rotation appear like an ordinary substitution in this example?
I've seen across several posts, that Wick rotation is not an ordinary substitution. Instead we're rotating the contour of integral and analytically continuing time $t$ to include imaginary time $-i\...
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Analytic continuation of the many-body spectral density
For an observable $A$, define the real-time autocorrelation function
$$
C(t) = \langle A A(t) \rangle_{\beta} = \dfrac{1}{Z} \mathrm{Tr}\left[ e^{-\beta H} A e^{i H t} A e^{-i H t}\right],
$$
with $Z =...
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Connection between the Beta Function and Residue Theorem?
When we define the bare coupling in Minimal Subtraction we write it as a Laurent series where the analytic part is identified with the finite, renormalized coupling and the nonanalytic part is ...
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Time-Ordered Propagator in Euclidean Space
I saw a paper stating that in Euclidean signature, the Feynman propagator $G_E$ is related to the Wightman functions $W_{\pm}$ via
$$
G_E (x) = \Theta(\tau) \, W_+ (x) + \Theta(-\tau) \, W_- (x) \, ,\...
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Condition to the holomorphy of a complex function
In Witten's note https://arxiv.org/abs/1803.04993, during the proof of Reeh-Schlieder theorem, he made an arguement that considering a function
$$g(u)=\langle\chi|\phi(x_1)\dots e^{\mathrm{i}Hu}\phi(...
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How to integrate "recursive" pressure/temperature relations?
I hope the term recursive is correct in this context.
The Clausius-Clapeyron relation says that:
$\frac{dP}{dT} = \frac{L}{T\Delta v}$
Where P is the pressure, L is the latent heat of vaporization, T ...
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Integration along real axis with singularities
I'm trying to calculate Green function of wave equation
$\begin{align}
\bigg(\nabla^2 - \frac{\partial ^2}{\partial t^2}\bigg)G(\textbf{x},t;\textbf{x'},t')=\delta^3(\textbf{x-x'})\delta(t-t')
\end{...
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Express cross section in terms of tranverse momentum [closed]
im new here.
I have an expression for a differential cross section of a process two in two at fixed center of mass energy $\sqrt{s}$, $\frac{d\sigma}{dt}$, in terms of the three Mandelstam variables $...
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Double poles in propagators
I'm curious as to how to interpret double poles in the propagator. In general, the poles of a propagator tell us the mass.
For example, for a free, massive scalar
$$\mathcal{L}=\frac{1}{2}\phi(\Box-m^...
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How to interpret multivalued fields in 2D CFT?
In the notes I've seen on 2D Conformal Field Theory, we derive the Witt Algebra by considering infinitesimal transformations of the form
\begin{align}
z' &= z + \epsilon z^{n+1}
\end{align}
which ...
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Is it legitimate to use analytic continuation to equate a diverging series with a finite number in a physical theory of nature?
Analytic continuation can be used in mathematics to assign a finite value to an infinite series that diverges to infinity. Is it correct and legitimate to equate this value to a diverging infinite ...
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Schwinger and Hadamard functions derivation in Birrell's and Davies' book
In Birrell's and Davies' book on "Quantum Fields in Curved Space", and in particularly in Chapter 2.7, the authors claim that from the expression
$$\mathcal{G}(x,x')=
\int\frac{d^nk}{(2\pi)^...
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Peculiar calculation of the Klein-Gordon Propagator
I am reading Peskin & Schroeder's QFT textbook (page 29~30).
Here, to calculate Klein-Gordon Propagator, author computes following integral.
$$\left< 0 | [\phi(x), \phi(y)]|0\right> = \int \...
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Zeros of multiplicative wave function renormalization
It is probably needless to recall here that the Reimann zeta function $$\zeta(s)=\sum_{n=1}^\infty n^{-s}$$ and its generalizations are among the central objects of study in mathematics.
The main open ...
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Physics in Euclidean spacetime [duplicate]
I just have a very small and naive Question.
In my PhD I work on different Toy models which are implemented on the lattice.
In order to do so one performs a Wick rotation from minkowski to euclidean ...
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Wightman Green function derivation
In the book N. D. Birrell and P. C. W. Davies, "Quantum Fields in Curved Space" at pp. 52-53 the four dimensional (positive frequency) Wightman Green function is expressed in position space ...
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Are all Local Observables Measured on Gibbs States Analytic as a Function of Temperature Away from Phase Transitions?
Let $\rho(\beta)=e^{-\beta H}/Z$ be the Gibbs state of a quantum Hamiltonian, and $H$ is some local Hamiltonian on $N$ particles, and $Z(\beta)$ is its partition function. Suppose I measure some local ...
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Smooth vs analytic spacetimes
Recently in more technical settings (I was learning algebraic QFT), I encountered the term "real analytic" manifolds (Lorentzian manifolds, to be precise). This is in contrast to smooth ...
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How to Understand the First Term in the Calabrese-Lefevre Distribution?
I am currently reading the following paper and I am trying to understand the first term in equation (6) (reproduced below):
$$
P(\lambda) = \delta(\lambda_\text{max} - \lambda) + \frac{b \Theta(\...
user261609
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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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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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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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Schwinger-Keldysh contour and $i\epsilon$ prescription
In Tom Hartman'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 vacuum ...
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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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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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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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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:
$$
W(q)=\frac{1}{2}q^2-gq^3,\tag{1.3.2}
$$
$$
W(q)=\frac{1}{2}q^2+\frac{g}{4}q^4,\; g<...
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Analytic continuation of Feynman amplitudes seems ill-defined
I was reading Peskin & Schroeder's book on Quantum Field Theory and on chapter 7, "The optical theorem for Feynman diagrams" (page 232) they extend analytically the Feynman amplitude $i \...
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Null tetrad for a metric with most of the metric components non zero
I am working on a metric which is basically $g = g_0 + h_{\mu \nu}$ where $g_0$ is the Kerr metric up to order a (taking $a^2 = 0$) and $h_{\mu \nu}$ denotes perturbed metric. Hence the complete ...
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Holomorphicity of Functions of unitary matrices
I am studying lattice QCD and there I encounter functions of unitary matrices. For ex. The action, $S = \sum$Tr( plaquettes), where each plaquette, $P$ is written as,
$$ 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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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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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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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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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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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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