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.
180
questions
2
votes
0answers
60 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 ...
1
vote
0answers
37 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 ...
3
votes
1answer
63 views
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. $(11.2.5)$
$$
\begin{align}
\Pi^{\rho\sigma} (q) = \frac{-ie^2}{(2\pi)^4} \int_0^...
3
votes
1answer
79 views
Equilibrium points of bounce/instanton solution after Wick's rotation
In Coleman's paper Fate of the false vacuum: Semiclassical theory while working out the exponential coefficient for tunneling probability through a potential barrier, he studies the problem with Wick'...
5
votes
1answer
268 views
Is a Wick rotation a change of coordinates?
My understanding is that a Wick rotation is a change of coordinates from $(t,x) \rightarrow (\tau , x)$ where $\tau = i t$. In the $(t,x)$ coordinate system, the Minkowski metric has components $ \...
0
votes
0answers
68 views
The imaginary time [duplicate]
Some people work with the interpretation that the time basis vector has magnitude sqrt(-1) to justify the negative sign in a -+++ Minkowski metric signature.
I came across a Youtube comment that ...
2
votes
2answers
119 views
Wick rotation: still trouble in getting how it works
I'm preparing my second exam in QFT and I still have trouble in getting the Wick rotation and its analytic continuation. I know that this topic have been discussed a lot in previous threads, but I ...
0
votes
1answer
98 views
Lorentz transformations: new actual notation for a $4$-vector [duplicate]
For the Lorentz trasformations I use this notation
\begin{equation*}
\left\{\begin{aligned}
x&=\gamma (x'+\beta ct')\\
y&=y'\\
z&=z'\\
ct&=\gamma (ct'+\beta x')\\
\end{aligned}\right.
...
0
votes
0answers
36 views
Wick rotation graphically
We have to evaluate
$$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.$$
If we now multiply the ...
2
votes
1answer
64 views
Wick rotation convergence. Functions in the integrand
Performing a Wick rotation over an integral is not equivalent to just a change of variable $t \to \mathrm{i}t = \tau$, after that we rotate the complex plane so that
$$\mathrm{i} \int_{-\infty}^{\...
2
votes
1answer
228 views
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 ...
1
vote
1answer
46 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, ...
0
votes
1answer
57 views
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 ...
1
vote
2answers
42 views
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 ...
3
votes
1answer
168 views
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 ...
1
vote
1answer
66 views
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 ...
1
vote
0answers
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 $E^i_M = F^{0i}_M = i F^{0i}_E = i E^i_E$, ...
3
votes
0answers
56 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 ...
1
vote
1answer
102 views
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
...
0
votes
1answer
67 views
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 ...
5
votes
1answer
194 views
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 ...
0
votes
0answers
126 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. ...
4
votes
1answer
224 views
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 ...
0
votes
0answers
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 ...
1
vote
2answers
144 views
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?
2
votes
1answer
159 views
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 ...
3
votes
1answer
128 views
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 ...
2
votes
2answers
161 views
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 ...
2
votes
0answers
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 ...
3
votes
2answers
195 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 ...
3
votes
0answers
111 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 ...
3
votes
0answers
99 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 ...
2
votes
0answers
229 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 ...
1
vote
0answers
156 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 ...
2
votes
1answer
60 views
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 ...
1
vote
0answers
116 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
$$Z = ...
3
votes
0answers
189 views
Extra $i$ in grand canonical partition function: why the Wick rotation?
Going through my notes I stumbled upon something I can't wrap my head around.
I'd like to write the grand canonical partition function for a system of identical charged particles (charge $e$) ...
3
votes
0answers
201 views
Is Wick rotation of loop integrals legitimate?
In Feynman diagram calculations, we seem to invariably Euclideanise loop integrals in order to exploit the resulting spherical symmetry. This Wick rotation is simply a deformation of the contour; ...
5
votes
1answer
296 views
Wick-rotating the Fourier transform of $\mu+1$ propagators
In Equation (8) of this paper by Groote et. al., we are given the following Euclidean identity:
$$
\int \frac{d^{4}\mathbf{p}_{\mathrm{E}}}{(2\pi)^{4}} \frac{e^{ i \mathbf{p}_{\mathrm{E}} \cdot \...
4
votes
0answers
92 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 ...
7
votes
1answer
303 views
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
$$Z=\int\mathcal{D}\Phi\,e^{-S_E[\...
1
vote
0answers
92 views
How to numerically implement a Wick rotation?
I'm solving a Schroedinger-type differential equation using numerical methods (RK4 for precision, explicit Euler to get a rough idea). I have an initial condition to start. I understand that replacing ...
3
votes
1answer
141 views
Does imaginary time actually not “generalize to curved space”, or is it merely messy to generalize?
I've run into the statement that imaginary time does not generalize to curved space, and other contradictory statements that imaginary time can be used with curved space but only awkwardly. Which is ...
3
votes
0answers
145 views
Equivalence of $d$ dimensional quantum system to $d+1$ dimension stats system
" There are close analogies between quantum field
theories in d dimensions and classical statistical mechanics in d + 1."
What does this statement imply and from where does this extra dimension ...
3
votes
2answers
418 views
Wick rotations, convergence and Feynman propagators?
It is said (in e.g. Hawking, 1979, Euclidean quantum gravity) that the integral:
$$ \int \mathcal{D}\phi \exp(iS[\phi])\tag{1} $$
for real fields in Minkowski space does not converge, but the Wick ...
asked Dec 17 '17 at 11:38
Quantum spaghettification
1,3267 gold badges31 silver badges96 bronze badges
3
votes
2answers
214 views
Why is the path integral action not real in most condensed matter QFT?
The partition function in statistical physics $Z=Tr\exp(-\beta H)$, with the second quantized Hamiltonian like
$$
H=\sum \epsilon_k c_k^\dagger c_k
+...$$
can be represented by using the path-...
2
votes
0answers
228 views
Causality and wick rotation
What is the connection between causality and wick rotation?
I came across implication of this connection multiple times but can't find a rigorous explanation.
For example in the answer to Wick ...
16
votes
2answers
4k views
Gaussian integral with imaginary coefficients and Wick rotation
Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
2
votes
0answers
404 views
Temperature of Kerr black hole and conical singularity
For spherical static black holes, for example, the metric may take the form
\begin{equation}
ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega^2_d
\end{equation}
One can use conical singularity method to ...
-2
votes
3answers
421 views
Why did Einstein put a negative sign in the Pythagorean theorem? [duplicate]
In 4-dimensional spacetime, when we study the spacetime interval, why did Einstein put a negative sign in it?
$$x_1=x$$
$$x_2=y$$
$$x_3=z$$
$$x_4=ct$$
$$ds^2=dx^2+dy^2+dz^2-(cdt)^2$$
