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Loop Effect of $\phi$ Propagator in $t$-channel of scalar $\phi^3$ theory [closed]

In Schwartz's QFT chapter 16, he calculates the loop effect (vaccum polarization) of $\phi$ propagator in $\phi^3$ theory, with the choice of Pauli-Villars regulator, the scattering amplitude would be ...
Ting-Kai Hsu's user avatar
4 votes
2 answers
210 views

Splitting Scalar into Holomorphic and Anti-Holomorphic Parts

I am reading Tong’s string theory lecture notes. On page 78, he splits the 2d free scalar into left- and right-moving parts, seemingly using the classical equation of motion as justification. Why is ...
Sam's user avatar
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2 votes
1 answer
38 views

Complex BCFW-shift of Parke-Taylor amplitude

(This question stems from problem 3.3 of Elvang's and Huang's "Scattering Amplitudes in Gauge Theory and Gravity" book). Consider the Parke-Taylor amplitude given as \begin{equation} A_n[1^- ...
MathZilla's user avatar
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0 votes
0 answers
52 views

Analytic continuation Matsubara/imaginary-time to retarded function in complex time domain

In linear response theory, one may either use the real-time retarded correlation function, or analytically continue to imaginary time/frequency to use the Matsubara Green's function instead. While ...
evening silver fox's user avatar
1 vote
1 answer
51 views

On complex impedance representation and Riemann surfaces

We know that a complex number, $z=re^{i\phi}$, can be represented with infinitely many phases, $\phi+2\pi n$, for integer $n$, as can be easily seen from the equivalent picture of a vector on the ...
user135626's user avatar
11 votes
1 answer
293 views

Analytical continuation as regularization in Quantum Field Theory, the remaining questions

There is an old question posted (Regularization) which did not get an answer, about the validation of analytic continuation as regularization. It did get some discussion in the comments, referring to ...
Jos Bergervoet's user avatar
1 vote
1 answer
89 views

Kramers-Kronig relations for a Gaussian function

Consider a function of a complex variable $\omega$ which is given by $f(\omega) = e^{-\omega^2/2}$. This function is symmetric, holomorphic everywhere, and vanishes as $|\omega| \rightarrow \infty$. ...
user19642323's user avatar
1 vote
0 answers
148 views

How is Wick rotation an analytic continuation?

Wick rotation is formally described by the transformation $$t \mapsto it.$$ In many place it is stated more rigorously as an analytic continuation into imaginary time. I understand why we do it but ...
CBBAM's user avatar
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1 vote
0 answers
102 views

Discontinuity of the scattering amplitude and optical theorem

The generalized optical theorem is given by: \begin{equation}\label{eq:optical_theorem} M(i\to f) - M^*(f\to i) = i \sum_X \int d\Pi_X (2\pi)^4 \delta^4(p_i-p_X)M(i\to X)M^*(f\to X).\tag{Box 24.1} ...
Andrea's user avatar
  • 53
4 votes
1 answer
271 views

Branch cut of a one-loop bubble diagram after cutting a single propagator

I am trying to understand Cutkosky cutting rules and generalized unitarity. Consider the article https://arxiv.org/abs/0808.1446 by Arkani-Hamed, Cachazo & Kaplan. In chapter 5.1 equation 133, the ...
Andrea's user avatar
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1 vote
0 answers
61 views

Does an initially analytic wavefunction remain analytic under time evolution?

My question has to do with when "mathematically nice" properties of a wavefunction (e.g. analyticity) are preserved under time evolution. Consider the Schrodinger equation $i\frac{d}{dt}|\...
C.M.O.B.'s user avatar
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1 vote
0 answers
88 views

What is the position-space form of the photon propagator in axial gauge?

I'm interested in the form of the photon propagator in position space, when expressed in an axial gauge $ n \cdot A =0$, where in the case I am interested in, $n^\mu = \{1,0,0 \dots, 0\}$ (for a $D$-...
NoName's user avatar
  • 63
-2 votes
1 answer
109 views

About Second-Order Poles of Matsubara Sum

I would like to ask about the calculation regarding Matsubara sum of the form \begin{equation} \frac{1}{\beta}\sum_{i\omega_n} \frac{1}{(i\omega_n-\xi)^2} \end{equation} which is a second order pole ...
HereXD's user avatar
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3 votes
1 answer
327 views

General interpretation of the poles of the propagator

I am somewhat familiar with the fact that the poles of the Feynman propagator in QFT give the momentum of particle states. I'm also familiar with the KL spectral representation in that context (See ...
P. C. Spaniel's user avatar
1 vote
1 answer
165 views

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 ...
Adrien Martina's user avatar
1 vote
0 answers
24 views

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 ...
gaugedude's user avatar
1 vote
0 answers
108 views

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 ...
John Doe's user avatar
1 vote
2 answers
308 views

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 ...
freude's user avatar
  • 1,725
2 votes
0 answers
111 views

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 ...
alpha's user avatar
  • 83
0 votes
0 answers
130 views

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 ...
Bastam Tajik's user avatar
  • 1,268
2 votes
0 answers
186 views

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 $$G_+^R = {}_M \langle 0 | \phi(x) \phi^{\dagger}(x') | 0 \rangle_M$$ $...
Brain Stroke Patient's user avatar
6 votes
2 answers
229 views

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, ...
Jakob KS's user avatar
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0 votes
1 answer
86 views

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 ...
0 votes
1 answer
149 views

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). ...
3 votes
1 answer
123 views

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\...
Nakshatra Gangopadhay's user avatar
2 votes
0 answers
92 views

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 =...
anon1802's user avatar
  • 1,330
3 votes
1 answer
265 views

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 ...
user119706's user avatar
1 vote
0 answers
181 views

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) \, ,\...
Entang1ed's user avatar
6 votes
1 answer
176 views

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(...
YONGAO's user avatar
  • 145
7 votes
2 answers
586 views

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{...
hwan's user avatar
  • 169
1 vote
0 answers
28 views

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 $...
Tonymowgli's user avatar
2 votes
1 answer
172 views

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^...
arow257's user avatar
  • 1,055
2 votes
2 answers
125 views

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 ...
Panopticon's user avatar
0 votes
0 answers
109 views

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 ...
JeffK's user avatar
  • 697
2 votes
1 answer
181 views

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)^...
schris38's user avatar
  • 3,992
1 vote
0 answers
109 views

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 \...
김승현's user avatar
3 votes
0 answers
135 views

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 ...
critical_Exponent's user avatar
1 vote
0 answers
45 views

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 ...
Ventura's user avatar
  • 11
5 votes
1 answer
711 views

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 ...
Miero Patteucci's user avatar
0 votes
0 answers
59 views

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 ...
Hans Schmuber's user avatar
4 votes
1 answer
317 views

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 ...
Evangeline A. K. McDowell's user avatar
0 votes
0 answers
37 views

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(\...
user avatar
2 votes
1 answer
61 views

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 ...
user avatar
0 votes
1 answer
203 views

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,...
agaminon's user avatar
  • 1,775
2 votes
1 answer
178 views

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 ...
Ruben Campos Delgado's user avatar
6 votes
0 answers
320 views

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 ...
nodumbquestions's user avatar
1 vote
0 answers
191 views

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 ...
3KidsInATrench's user avatar
5 votes
3 answers
276 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 ...
ummg's user avatar
  • 1,215
15 votes
2 answers
1k views

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 ...
Arturo don Juan's user avatar
3 votes
2 answers
172 views

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<...
Anzu Ariake's user avatar