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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818 views

Time Reversal, CPT, spin-statistics, mass gap and chirality of Euclidean fermion field theory

In Minkowski space even-dim (say $d+1$ D) spacetime dimension, we can write fermion-field theory as the Lagrangian: $$ \mathcal{L}=\bar{\psi} (i\not \partial-m)\psi+ \bar{\psi} \phi_1 \psi+\bar{\psi} ...
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1answer
225 views

Weak gravity limit of (Einstein-Hilbert + matter) action

The problem Consider the following euclidean action $$ S_E = - \int_{\mathcal{M}} d^4x \sqrt{g} \left [\frac{R}{2 \kappa} +\mathcal{L}_M \right ] + S_{GHY},$$ where $S_{GHY} = -\int_{\partial \...
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614 views

Time Reversal in Euclidean Spacetime - unitary or antiunitary?

(pre-request) We know that time reversal operator $T$ is an anti-unitary operator in Minkowsi Spacetime. i.e. $$ T z=z^*T $$ where the complex number $z$ becomes its complex conjugate. See, for ...
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98 views

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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1answer
344 views

What is the proper definition of Schwinger functions?

In the book Mathematical Aspects of Quantum Field Theory (2010) on page 159-160 in chapter 6.6 Wick rotations and axioms for Euclidean QFT the following is stated: We have seen earlier in this ...
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57 views

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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2answers
214 views

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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112 views

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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101 views

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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192 views

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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208 views

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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147 views

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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1k views

Feynman Propagator in Position Space through Schwinger Parameter

So I am aware of a thread at Propagator of a scalar in position space but it does not answer my question, which is more about poles in position space. Starting from $$D_F(x_1-x_2) = \int \frac{d^4 ...
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207 views

Geodesic approximation and Euclidean continuation

I recently read many articles in the context of the AdS/CFT correspondance in which the geodesic approximation is used (see for example section 3.5 here). The correlator between two boundary operators ...
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832 views

$\mathrm{i}\epsilon$ prescription makes a function analytical?

I've seen this everywhere where they say "Analytic continuation is obtained by the usual $\mathrm{i}\epsilon$ prescription..." but how is that? How do you analytically continue (say) $\ln x$ with ...
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174 views

Intuition behind the notion of reflection positivity

I came across Yuji's question. I'm finding it difficult to parse the meaning behind what's said on Wikipedia. Could someone give an explanation of the concept involved? I would also appreciate ...
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65 views

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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58 views

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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234 views

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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230 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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408 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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155 views

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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135 views

Why density matrices in QFT are calculated only by going to Euclidean metric?

I have been reading a few papers on entanglement entropy. I noticed that whenever people calculate either the density matrix or the reduced density matrix of a specific region, it is usually done by ...
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152 views

Continuation to Euclidean BTZ Black hole

BTZ black hole in Lorentzian signature is given by $$ ds^2= -fdt^2+f^{-1}dr^2+r^2(d\phi + N^{\phi} dt)^2 $$ $$ f=-M+r^2+\frac{J^2}{4r^2},~~~~~ N^{\phi}=-\frac{J}{2r^2} $$ $f$ can be wriitten as $$ f=\...
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859 views

The Feynman propagator and the $i\epsilon$ prescription

The Feynman propagator is usually represented in the i-epsilon form and texts solve the integral in this form (as opposed to doing the Feynman (time-ordered) contour on the real axis). Restricting ...
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134 views

Anomaly and Weyl spinors

I try to better understand anomalies in QFT and I've got a question concerning derivation of axial anomaly in Terning's lectures (page 12) Consider a theory of Weyl fermions coupled to a gauge field $...
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321 views

Does anybody know of a source that explains Wick rotation for fermions in 3-dimensional spacetime?

I've been looking for a long time and I've not had a lot of luck. I've found sources that use fermions in 3d Euclidean space but I can't find any that explain the Wick rotation from Minkowski space. ...
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166 views

What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann ...
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106 views

Is Wick rotation invariant under proper conformal transformations?

Is Wick rotation invariant under proper conformal transformations? Why or why not? Does Wick rotation apply to conformal field theories? $(1-i\epsilon )T$ is not invariant under proper conformal ...
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96 views

Wick rotation for FRW in quantum gravity

There is no timelike Killing vector for FRW cosmologies. In the path integral formalism, is it possible to Wick rotate for quantum cosmology in quantum gravity? If yes, how? If no, how does one work ...
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1answer
206 views

Charge Conjugation of massive Dirac spinor in 3 dimensions with Euclidean signature

In 2+1 dimensional massive Dirac equation (Minkowski signature), we can define the charge conjugation operator so that the equation can be symmetric under it. However, the charge conjugation does not ...
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38 views

“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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1answer
47 views

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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30 views

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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163 views

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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117 views

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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92 views

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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423 views

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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110 views

Having trouble evaluating a spacetime integral, using Feynman parameterization and wick rotation

I am trying to evaluate the following integral, $$\int \frac{d^4k}{(2\pi)^2k^2} \left(\frac{\mu}{4M} \right)^2 \frac{k^4 + \frac{1}{3}\vec{k}(k^2 - 2Mk_0)}{(k^4 - 4m^2k_0^2)(k^2 - 2Mk_0)} .$$ My ...
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355 views

Feynman's $i\epsilon$ prescription in path integrals (Mark Srednicki)

On page 63 in M.S. book , why m^(-1) goes to (1-iε)m^(-1) or m -> (1+iε)m and how can i verify eq.(7.3)? On page 63 writes : Looking at $H(P,Q)= \frac{1}{2m} P^2 +\frac{1}{2}mω^2Q^2$ we see that ...
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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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37 views

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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131 views

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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34 views

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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326 views

Minkowski to Euclidean

When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now $(+,+,+...