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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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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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Questions with Instanton Approach to Euclidean Path Integral

I'm reading through a section from Altland's and Simons' book on Condensed Matter Field Theory, and ran into a section I'm having trouble with (section 3.3 p. 115-117 if you have it, otherwise I try ...
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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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Dimensional regularization $\phi_4$ theory $i$ factor

When I apply dimensional regularization to one loop correction of $\phi_4$ theory, it seems that I missed some factor of $i$: We have $-i\lambda \int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2-m^2}$ from the ...
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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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203 views

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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Splitting kinetic and potential terms in imaginary time action

I have the following expression: $$S=T+V$$ $$T=\frac{m}{\tau_0+it}((x_1-x_0)^2+(x_2-x_1)^2)+\frac{m}{2(\tau_1-it)}(x_2-x_0)^2$$ $$V= \frac{(\tau_0+it)}{2}(\frac{k_0 x_0}{2}+\frac{k_0 x_2}{2}+k_0 ...
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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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50 views

Are all Euclidean $n$-dimensional Fourier transform results true for Minkowski space as well?

Consider the ordinary (Euclidean) Fourier transform in terms of variables $\mathbf{y} = (y_{1}, \ldots, y_{N})$ and $\mathbf{q} = (q_{1}, \ldots, q_{N})$ in $\mathbb{R}^{N}$ and the function $f:\...
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126 views

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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2answers
301 views

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 ...
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Why is the path integral action not real in most condensed matter QFT?

The partition function in statistical physics $Z=Tr\exp(-\beta H)$, with the second quantized Hamiltonian like $$ H=\sum \epsilon_k c_k^\dagger c_k +...$$ can be represented by using the path-...
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157 views

Causality and wick rotation

What is the connection between causality and wick rotation? I came across implication of this connection multiple times but can't find a rigorous explanation. For example in the answer to Wick ...
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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 ...
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289 views

Temperature of Kerr black hole and conical singularity

For spherical static black holes, for example, the metric may take the form \begin{equation} ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2_d \end{equation} One can use conical singularity method to ...
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Why did Einstein put a negative sign in the Pythagorean theorem? [duplicate]

In 4-dimensional spacetime, when we study the spacetime interval, why did Einstein put a negative sign in it? $$x_1=x$$ $$x_2=y$$ $$x_3=z$$ $$x_4=ct$$ $$ds^2=dx^2+dy^2+dz^2-(cdt)^2$$
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233 views

Least action principle in imaginary time

In quantum mechanics, the amplitude of wave function propagation can be found using the Feynman's path integral $$ \langle z'|e^{-itH/\hbar}|z\rangle=\int\limits_{x(0)=z\\x(t)=z'} Dx(t')\: \exp\left\{\...
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311 views

Euclidean QFT commutator vanishes for all spacetime separations?

In Minkowski spacetime, the commutator of the Klein-Gordon field operator with itself at different spacetime points evaluates to the advanced minus retarded Green's function of the classical theory, ...
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1answer
89 views

How to expand the Dirac equation?

I've been reading a little bit of the Dirac Equation and I'm a little confussed about how it shall be expaned. The dirac equation has the form of $$i\hbar \gamma^{\mu} \partial_{\mu} \Psi -mc \Psi = ...
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2answers
233 views

Finding classical action in tunneling problem

In QM: I am trying to show that the minimum action for a classical path going between two potential wells (centered at $\pm L$) in a dbl-well potential is $$S_{classical} = \int_{-L}^{L} dx' \sqrt{...
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1answer
95 views

Topological problems with Lorentzian metric on worldsheet

In string theory we study maps $X: \Sigma \to M$, where $\Sigma$ is the two dimensional worldsheet of the string and $M$ is the target manifold. When studying non-linear sigma models, for instance ...
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2answers
137 views

Metric tensor and imaginary time

I just started a re-reading of the Conformal Field Theory yellow book by Di Francesco et al. In chapter two, after defining imaginary time $\tau$ as $t=-i\tau$, the authors state that the metric ...
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1answer
209 views

Wick rotation - time and what else changes?

For aid of example consider two quantities the four-momentum $\tilde P$ and a time-independent four potential $\tilde A$. Now if a wick's rotation was carried out by simply replacing $it$ with $\tau$ ...
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Finite temperature $\mathcal{N} = 4$ SYM on ${\bf S}^3$

Consider the following paragraph taken from page 3 of Edward Witten's paper on Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories. To study the theory at finite ...
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Euclidean fermion propagator

I want to write the fermion propagator $$ i\dfrac{p^\mu\gamma_\mu+m}{p^2-m^2} $$ in Euclidean space. In Minkowski, the conventions are $g^{\mu\nu}=\{+,-,-,-\}$; $\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^...
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300 views

Boundary conditions on the Euclidean Schwarzschild black hole

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes. The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{...
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1answer
152 views

Boundary conditions for the gravitational path integral

This question is based on page 68 of Thomas Hartman's notes on Quantum Gravity and Black Holes. To evaluate a path integral in ordinary quantum field theory, we integrate over fields defined on a ...
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1answer
195 views

Book recommendation relating QFT in statistical physics and particle physics

I know that QFT is heavily used in statistical physics but, as a former particle physicists, I search a book that would nicely bridge the two different perspectives, especially when it comes to the ...
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Lorentzian path integral for string theory and causality

Is the Lorentzian path integral in string theory well defined, as opposed to the usual Euclidian path integral that is commonly used for simplicity? The path integral is roughly $$\sum_{\mathbf{\...
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What happens when there are different classical trajectories before and after Wick rotation? [duplicate]

Recently I read the path integral of double well tunnelling. I am puzzled about the Wick rotation calculation. For example, I choose potential like $V(x)=(x^2-1)^2$ and Lagragian $L= \frac{1}{2} \dot ...