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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Why does this distribution function depend on time and not temperature?

When reading Sterile neutrino hot, warm, and cold dark matter I came across the following momentum distribution function for a neutrino species $\alpha$: $$\tag{5.8} f(p,t) = \frac{1}{e^{E(p)/T + \...
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Convert propagators from Euclidean to Minkowski spacetime

I'm looking for a rule to "convert" the propagators of a quantum field theory formulated in Euclidean spacetime into those of the same theory but in Minkowski spacetime (with the $\operatorname{diag}(-...
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Why do people use real values for the Wick-rotated time $\tau$?

In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in ...
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Wick rotation on Ward identities

I'm having trouble performing a Wick rotation back to Minkowski spacetime ($\eta_{\mu\nu}=(-1,1,1,\dots)$), following page 19 in the lecture notes here by C.P. Herzog. I have this expression (equation ...
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Is there a link between the logistic differential equation and Fermi-Dirac statistics?

I was working out some statistical problems and I could not fail to notice that Fermi-Dirac distribution, $$f_{\rm Fermi-Dirac}(E)=\frac{N_{\rm sites}}{1+e^{\beta(E-\mu)}},$$ looks like the kind of ...
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Why is Euclidean Time Periodic?

I've been reading a bit about finite temperature quantum field theory, and I keep coming across the claim that when one Euclideanizes time $$it\to\tau,$$ the time dimension becomes periodic, with ...
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Imaginary time in quantum and thermodynamics

The following question is about chapter 2 of Sakurai's Modern Quantum Mechanics. In the section about propagators and Feynman path integrals (p. 113 in my edition) he gives the following example: $$ \...
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How to understand the path integral of $U(1)$ gauge field under Coulomb gauge?

I want to obtain Green's function of $U(1)$ gauge field under Coulomb gauge. For some reason, I want to finish it in Euclidean space, i.e. both time-space $x_\mu$ and field strength $A_\mu$, so that ...
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Why can you deform the contour in the integral expression for the Klein-Gordon propagator to get the Euclidean propagator?

I'm trying to understand the use of the Euclidean correlation functions in QFT. I chased down the problems I was having to how they manifest in the simplest example I could think of: the two-point ...
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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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In path-integral, when do we have to insert fact $i$ in front of the action $S$ in the exponent?

I have got stuck in these concepts for a fews days: Wick rotation, Euclidean spacetime and QED in gravity. Generally, in Minkowski space time, there is a factor $i$ in front of the action $S$, e.g., ...
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Polchinski Eq 3.2.4 and Eq 3.2.5: Deforming contours in path integral

Here is the section of the book I'm talking about. I'm confused about the following two points: (i) Why is the path integral oscillatory? (ii) What does it mean, "we can deform contours just as ...
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Question about if there can be a measure (possibly related to $d\mu$) in the Wick rotated path integral

In the Measure Theoretic subsection of https://en.wikipedia.org/wiki/Path_integral_formulation it is stated that sometimes the path integral must contain a measure that cannot be absorbed into the ...
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Question about how to (mathematically) interpret averages with the Feynman-Kac formula

I'm trying to understand the Feynman-Kac formula (https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) for the Wick-rotated Feynman path integral. Would it be correct to say that $$\langle \phi\...
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Does a partially traced density operator also become a Boltzmann density operator under Wick rotation to Euclidean space?

I know that, under the Wick rotation $(i\Delta t/\hbar,p_0)\to(-\beta,-ip_{0,E})$, Feynman's path integral supposedly transforms into the traced-over Boltzmann partition function, $trace(e^{-\beta H})=...
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Propagator in Wave Mechanics Laplace-Fourier transform

In my Modern quantum mechanics, J. J. Sakurai p.119-120, when considering the integral of the propagator $K$ in whole space, he gets: $$G(t)= \int d^3 x' K(\textbf{x'},t;\textbf{x'},0) = \sum_a \exp \...
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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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How well does the concept or model of imaginary time work?

In order to make the Minkowski metric, in special relativity, equivalent to the Euclidean metric, one idea is to allow time to take imaginary values. As far as I have learned about SR, it does make ...
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Wick rotation with magnetic fields

How does Wick rotation work with magnetic fields? Let us take up single-particle $d$+1 QM. Then the Euclidean time path integral is given (in $\natural$ units) by $$ \langle x| \exp(-t H) y\rangle = \...
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How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?

I came across the S. Coleman's seminal papers 'Fate of the false vacuum' (http://dx.doi.org/10.1103/PhysRevD.15.2929, http://dx.doi.org/10.1103/PhysRevD.16.1762) where he describes the tunneling ...
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How to understand “analytical continuation” in the context of instantons?

Since this is a subtle and interesting question to me. I will give a rather detailed description. I hope you can keep reading it and find it interesting too. For simplicity, in the following I will ...
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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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How do we perform 'time' translation in Euclidean QFT?

If we have an operator in a $1+1$ dimension QFT then we get the Hamiltonian, which comes from and generates translations in the $t$ direction and a momentum operator which comes from and generates ...
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What is the advantage of using imaginary units for time in the Minkowski Space rather than regular euclidian space as Lorentz used? [duplicate]

I do understand that Lorentz transformations became as a rotation of coordinates as of a hyperbolic rotation. But what is its advantage over real vector? What is the new thing that it introduces and ...
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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. If we replace $t$ by $it$ (wick rotation) we get again ...
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Why the imaginary unit in time axis? [duplicate]

Why time is not like other dimensions is a real amount? In relativity time axis is $i*c*t$, where $i$ is the imaginary unit and $c$ is light speed in free space. Did science or philosophy reached to ...
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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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Analytic Continuation: Replacement of $t \rightarrow - i \tau$ Mathematical Justification [duplicate]

It's commonly used in imaginary-time path integral that "analytic continuation" means replacing $t \rightarrow - i \tau$ or reparametrizing the theory in terms of imaginary time $\tau = i t$....
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Field strength and Levi-Civita tensor in Euclidean spacetime

I am trying to formulate gauge theory in Euclidean spacetime. I have Googled a lot of thing, but I cannot find any standard way. The following is what I am doing. Suppose in Minkowski spacetime, we ...
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Subtlety of analytic continuation - Euclidean / Minkowski path integral

I subconsciously feel not fully comfortable about Wick rotating or analytic continuation from Euclidean to Minkowski space. I simply wonder whether there is any subtlety here, and when we need to be ...
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Imaginary time & predictions

Is the imaginary time just a different convention to express the time evolution to make the calculations easier? Hawking also said that "It turns out that a mathematical model involving ...
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Does Wick rotation work for time-dependent Hamiltonian?

Consider a quantum system that is governed by a Hamiltonian with explicit time dependence $H(t)$. Is it always legitimate to perform a Wick rotation $t \rightarrow -i\tau$, and calculate the time-...
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What's wrong with using a vielbein to define Wick rotation?

Wick rotation is supposed to be a relationship between field theories with spacetime metrics of Lorentzian and Euclidean signature. I thought the definition of Wick rotation was settled, until I came ...
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Lorentz spinor in Lorentz $\rm Spin(3,1)$ signature and the real structure?

In this paper: J. Wang, X. Wen and E. Witten, "A new ${\rm SU}(2)$ anomaly", J. Math. Phys. 60 (2019) 052301, arXiv:1810.00844, it says the following in p.2, It says for $3+1$ dimensional ...
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Besides dim regularization, what are the advantages of Euclidean QFT?

Initially, I saw Wick rotation as a useful trick to apply dimensional regularization, but then I learned about instantons and how they only exist in Euclidean Yang-Mills. Also, I heard that path ...
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On the topic of imaginary-time

I apologize for my crude line of questioning, as I'm not well-versed in physics at all but it fascinates me. I was researching the concept of "imaginary-time" and the shuttlecock model of ...
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What is the correspondance between imaginary time and heat?

I apologize for my crude line of questioning, as I am not well versed in physics but there are concepts that interest me. I'm trying to understand the concept of imaginary-time, and I've read in ...
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Can a stress-energy tensor induce signature changes on the metric?

Suppose we use the signature of a Riemannian manifold $$ \eta^{\mu\nu}=\operatorname{diag}(+,+,+,+) $$ as the starting point to describe a 4d Euclidean version of general relativity. Alternatively one ...
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Is there any difference between signature $(1,1)$ and $(2,0)$ in 2D CFT?

Is there any difference between signature $(1,1)$ and $(2,0)$ in 2D CFT? The only thing I could thought of was that the previous one had Lorentz symmetry and the later one was Euclidean (rotation), ...
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What does the Temperature of a QFT physically mean?

In elementary statistical mechanics, one can think of temperature as arising from the average kinetic energy of particles in the ensemble. Is there a similar way to think about the temperature of a ...
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What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?

At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function: $$ Z=\int D\phi \exp (-\beta H[\phi]) \tag{1} $$ is a consequence of ...
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Why do we use the imaginary time evolution in simulations of some quantum system?

I realize that the imaginary evolution could help us to find the ground state for a system. However, I very puzzled why it works, and what the principle is back up there? I have done some searching on ...
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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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Wick Rotation & Scalar Field Value & Mapping

Wick Rotation helps to solve the problem of the convergence of the path integral, by changing the integral contour in the complex plane. But my question is: In the scalar field path integral, the ...
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Proving that a Wick rotation is valid for a quantum field theory

While trying to find out if there is a rigorous justification for Wick rotating a QFT, I came across this other question (link below [1]) that mentions the Osterwalder-Schrader Theorem that gives a ...
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What is QFT at finite temperature?

On the one hand, according to the Wick rotation that relates Statistical Field Theory and Quantum Field Theory, a finite temperature statistical system corresponds to a compact time quantum field ...
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Scalar field propagator in euclidean field theory

We have a scalar field propagator in minkowski space with signature $(+,-,-,-)$ as $$ G (k)={1\over k^2-m^2 }.$$ But in Euclidean space the scalar field propagator is $$G (k)={1\over k^2+m^2 }.$$ ...
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Under what assumptions does a state following the TDSE converge to its ground state?

Until $t=0$ a system is in an eigenstate $\psi_0(x)$ of the Hamiltonian $\hat{H}_0$. The time-evolution is the trivial phase factor. Now at $t=0$ the system changes to $\hat{H}$ (you can assume it is ...
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Choice of folliation in path integral

Assume we have a scalar field theory for a field $\phi$. Can we think of the Hilbert space as being spanned by states of the form $|\varphi\rangle$ for configurations $\varphi\in C^\infty(\mathbb{R}^3)...
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566 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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