Questions tagged [wick-rotation]

Wick rotation substitutes an imaginary-number variable for a real-number time variable to map an expression or a problem in Minkowski space to one in Euclidean space which are easier to evaluate or solve. Use for all types of rigid analytic continuation maps.

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154 views

Dirac equation without $i$

In Witten's review paper "Fermion path integrals and topological phases", the Dirac equation (Eq(2.2)) is $$(\gamma^{\mu}D_{\mu}-m)\psi=0$$ which appears very strange to me. Initially I thought this ...
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Convergence of the path integral

In P&S 9.3 the path integral $$ Z[J]=\int {\cal D}\phi \exp[i\int d^4x ({\cal L} + J\phi)]$$ of the (Minkowski) $\phi^4$-theory when subjected to a Wick-rotation (change of the integration path ...
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Can there exist a contrived case for the Wick rotation in position space to be meaningful?

Question So this is a complementary question to this. From the answers it seems can't one use a Wick rotation of position to create something physically meaningful. Does this hold even in contrived ...
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In light of Wick rotations is position and time on the same footing in QFT?

Taken from here Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/ℏ$ But I was under the impression position ...
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Confusion about gamma matrices in Euclidean spacetime

I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime. I'd like some clarification as to how to go from Minkowski to ...
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“Imaginary-time” argument in high energy physics

In many-body physics, there are many "imaginary-time" techniques, such as Matsubara Green's function, imaginary-time path integral and others. It seems that these concepts are frequently used in ...
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Wick Rotation and sign of the integrand in Weinberg's book

I'm studying from Weinberg's QFT volume 1, chapter 11. I have a problem with equation $(11.2.7)$. Starting from eq. $(11.2.5)$ $$ \begin{align} \Pi^{\rho\sigma} (q) = \frac{-ie^2}{(2\pi)^4} \int_0^...
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Equilibrium points of bounce/instanton solution after Wick's rotation

In Coleman's paper Fate of the false vacuum: Semiclassical theory while working out the exponential coefficient for tunneling probability through a potential barrier, he studies the problem with Wick'...
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Is a Wick rotation a change of coordinates?

My understanding is that a Wick rotation is a change of coordinates from $(t,x) \rightarrow (\tau , x)$ where $\tau = i t$. In the $(t,x)$ coordinate system, the Minkowski metric has components $ \...
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The imaginary time [duplicate]

Some people work with the interpretation that the time basis vector has magnitude sqrt(-1) to justify the negative sign in a -+++ Minkowski metric signature. I came across a Youtube comment that ...
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Wick rotation: still trouble in getting how it works

I'm preparing my second exam in QFT and I still have trouble in getting the Wick rotation and its analytic continuation. I know that this topic have been discussed a lot in previous threads, but I ...
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Lorentz transformations: new actual notation for a $4$-vector [duplicate]

For the Lorentz trasformations I use this notation \begin{equation*} \left\{\begin{aligned} x&=\gamma (x'+\beta ct')\\ y&=y'\\ z&=z'\\ ct&=\gamma (ct'+\beta x')\\ \end{aligned}\right. ...
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Wick rotation graphically

We have to evaluate $$i\int_{-\infty}^{\infty} f(t) dt.$$ We can make a change of variable $t\mapsto -i\tau$, which results in $$\int_{-i\infty}^{i\infty} f(-i\tau) d\tau.$$ If we now multiply the ...
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Wick rotation convergence. Functions in the integrand

Performing a Wick rotation over an integral is not equivalent to just a change of variable $t \to \mathrm{i}t = \tau$, after that we rotate the complex plane so that $$\mathrm{i} \int_{-\infty}^{\...
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Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

Consider a momentum integral of the form $$I=\int d^3k\int dk^0\frac{1}{k^4}$$ where $k^2=(k^0)^2-(\vec{k})^2$ and the integral over $k^0$ runs from $-\infty$ to $+\infty$. This integral is common in ...
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Can conformal transformations in $\mathbb{R}^{1,1}$ be analytically continued to $\mathbb{R}^{2,0}$?

In 1+1 dimensions, 2D Minkowski space, a conformal transformation is given by two real functions (of one variable). After Wick rotating the time dimension, giving us 2 dimensional Euclidean space, ...
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What's the relation between Euclidean and Minkowski entities in lattice field theory?

To my understanding, lattice QFT basically continues the time $t$ (and fields depend on it) in Minkowski space action to imaginary time $\tau\equiv it$. But normally when we do calculations in lattice ...
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Convergence Property of Path-Integral

Let the action be $$S= \int \bigg\{ \frac{1}{2} \big(\frac{dX}{dt}\big)^2 - V(X) \bigg\} d\tau$$ and the corresponding Path-Integral $$Z= \int DX(t) e^{iS}.$$ Since the convergence is not clear we ...
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What is the physical interpretation of the Wick rotation?

What is the physical interpretation of the Wick rotation? How is it that we can just propose there's a new time coordinate tau? Are physicists saying time is modeled by an imaginary number? Isn't ...
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Imaginary time concept in S.Hawking's No Boundary proposal, extra-time dimensions and the Big Bang

In this post I will be refering to S.Hawking's lecture: http://www.hawking.org.uk/the-beginning-of-time.html I have a couple of questions regarding the Imaginary time and the Big Bang. In the ...
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Time Reversal of electric field in Euclidean signature (Wick Rotation)

This is a follow up to this question: How to Perform Wick Rotation in the Lagrangian of a Gauge Theory (like QCD)? I am wondering if their (6), using that $E^i_M = F^{0i}_M = i F^{0i}_E = i E^i_E$, ...
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Difference between different approximations in QM and other definition of the integral

I am currently studying the path integral formalism by myself and I am a bit lost within all the different way to solve the integrals we have. I have one big question: It sounds maybe a bit strange ...
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Integral and Wick rotation (Srednicki ch75)

I was reading chapter 75 of Srednicki's QFT book and I ran into this statement. To determine the value of its integral, we make a Wick rotation to euclidean space, which yields a factor of i as ...
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Question about differentiating wrt. momentum in Srednicki chapter 14

I am having a bit of trouble following a simple integral from the book on QFT by Mark Srednicki - free draft can be accessed at http://web.physics.ucsb.edu/~mark/qft.html - and I was hoping you could ...
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
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A Naive Question about Delta Function and Wick Rotation

A delta function can be written as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp\,e^{ipx}.$$ I have a very poor understanding of the Wick rotation technique used in quantum field theory. ...
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Feynman $i\varepsilon$-prescription in path integral by adding an imaginary part to time

It is known that the well-definiteness of the path integral leads to the Feynman's $i\varepsilon$-prescription for the field propagator. I've found many ways of showing this in the literature, but it ...
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Would the time dependent wave equation of a free particle be the same in 4D Euclidean spacetime as it is in 4D Minkowski spacetime?

In 4d minkowski spacetime the time dependent wave equation of a free particle is $$\Psi(x,t) = \left({a \over a + i\hbar t/m}\right)^{3/2} e^{- {x^2\over 2(a + i\hbar t/m)} }.$$ I notice that there ...
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Should we start from Euclidean QFT if we are to be rigorous? [closed]

Path integral is only rigorous in Euclidean QFT. This suggests that one should start from Eucliden QFT and transport back the results back into Minkowski time. Is this how I should think of QFT?
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What is the link between statistical and QFT correlation functions?

I'm studying statistical mechanics in particular correlation function: https://en.wikipedia.org/wiki/Correlation_function_(statistical_mechanics) and I have understood it. Now searching on internet ...
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Wick rotation of the propagator in quantum mechanics

I am told that making the substitution $t\to-i\tau$, or a 'Wick rotation', can be used to study the propagator in imaginary time, making some problems easier. For example, this source proposes that we ...
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Functional derivative for the same function expressed before and after Wick rotation

This question arises when I'm reading section "3.3.1 Minkowski Space" of page 16-17 of the following document: http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf On page 17, they ...
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Deriving the generating function in Minkowski space

I will be referring to the following document (page 16-17): http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf I would like to understand the expression of the generating ...
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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 ...
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Can we do a Wick rotation by an angle not being $\pi/2$?

If a state obeys an evolution equation, we can replace t by -t. we get another equation and it is interesting to study its solutions. it we replace t by it (wick rotation) we get again another ...
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Analytic cotinuation between Minkowskian and Euclidean space, and causality

We can flip between Minkowkian and Euclidean signature by Wick rotation, and it is a well defined operation, provided there are no non - trivial singularities. Now, Unitarity in Minkowskian space ...
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The way momentum space integrals tend to infinity

At the beginning of chapter 15 of Schwartz, he states that $$\int d^4k \frac{k^2}{k^4}=\int \frac{d^4k}{k^2}\sim \int k\ dk. $$ I don't see how he got this at all. Isn't this just the integral \...
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Two possibilities for Wick rotation

$\newcommand{\ld}{\mathcal{L}}\newcommand{\adj}[1]{#1^\dagger}\newcommand{\dc}[1]{\overline{#1}}\newcommand{\Psi}{\varPsi}\newcommand{\dd}{\mathrm{d}}$Take a typical Lagrangian density defined over ...
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D'Alembertian Green's Function and Wick Rotation

Consider the wave equation: $$ \square A(t,x^i) = S(t,x^i) , $$ where $\square = -\partial_\mu \partial^\mu = \partial_t ^2 - \nabla^2 $, $S(t,x^i)$ is the source term and $A(t,x^i)$ is the field of ...
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Reality properties of auxiliary fields after Wick rotation

I was reading the treatment of the large $N$ limit of the Non-Linear Sigma Model (NLSM) in Peskin & Schroeder, Sec. 13.3, and I noticed something strange in the evaluation of the path-integral by ...
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Euclidean QFT definition

I have a question on Euclidean field theories and their relationship with QFT defined on a Minkowski spacetime. In order to compute the generating function $Z$, one has to compute the integral $$Z = ...
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Extra $i$ in grand canonical partition function: why the Wick rotation?

Going through my notes I stumbled upon something I can't wrap my head around. I'd like to write the grand canonical partition function for a system of identical charged particles (charge $e$) ...
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Is Wick rotation of loop integrals legitimate?

In Feynman diagram calculations, we seem to invariably Euclideanise loop integrals in order to exploit the resulting spherical symmetry. This Wick rotation is simply a deformation of the contour; ...
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Wick-rotating the Fourier transform of $\mu+1$ propagators

In Equation (8) of this paper by Groote et. al., we are given the following Euclidean identity: $$ \int \frac{d^{4}\mathbf{p}_{\mathrm{E}}}{(2\pi)^{4}} \frac{e^{ i \mathbf{p}_{\mathrm{E}} \cdot \...
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Connection between contra-/covariant vectors in SR and complex numbers?

If we take a spacetime with one spatial dimension, we can write a vector as $A^\mu=(t, x)$. This is a contravariant vector, and we can calculate the covariant vector by multiplying it with the ...
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Is there any physical meaning for such a correlation function?

Consider a thermal scalar field theory, we have the partition functional $$Z={\rm tr}(e^{-\beta H}).$$ We can build this theory as an Euclidean quantum field theory $$Z=\int\mathcal{D}\Phi\,e^{-S_E[\...
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How to numerically implement a Wick rotation?

I'm solving a Schroedinger-type differential equation using numerical methods (RK4 for precision, explicit Euler to get a rough idea). I have an initial condition to start. I understand that replacing ...
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Does imaginary time actually not “generalize to curved space”, or is it merely messy to generalize?

I've run into the statement that imaginary time does not generalize to curved space, and other contradictory statements that imaginary time can be used with curved space but only awkwardly. Which is ...
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Equivalence of $d$ dimensional quantum system to $d+1$ dimension stats system

" There are close analogies between quantum field theories in d dimensions and classical statistical mechanics in d + 1." What does this statement imply and from where does this extra dimension ...
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Wick rotations, convergence and Feynman propagators?

It is said (in e.g. Hawking, 1979, Euclidean quantum gravity) that the integral: $$ \int \mathcal{D}\phi \exp(iS[\phi])\tag{1} $$ for real fields in Minkowski space does not converge, but the Wick ...