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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How to simplify this integral using Wick rotation?

The integral is $$\sum_{n=1}^{\infty}\int \frac{d^4k}{(2\pi)^4}\frac{1}{2n}Tr\left[\left(\frac{ia\!\not\! k\,\mathcal{C}}{k^2+i\epsilon}\right)^{2n}\right]$$ I have to simplify it using Wick rotation, ...
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How to Wick rotate differential forms?

The bosonic sector of the Cremmer-Julia-Sherk (CJS) 11D supergravity action is $$ eR-\frac12 F\wedge *F-\frac16 A\wedge F\wedge F,$$ where $F=dA$ is a 4-form field strength. How would one perform a ...
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Minkowski and Euclidean 4 vectors

I am reading these lecture notes. On page 114 they define Minkowskian space time coordinates: $$\mathcal{X}=(\mathcal{X}^0,\mathcal{X}^1,\mathcal{X}^2,\mathcal{X}^3)=(t,x^i),$$ where $x^1=x,x^2=y,x^3=...
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Poincaré Symmetry becoming Mobius Symmetry for Euclidean Theory on Riemann Sphere

I've just started reading some introductory notes by Goddard and Gaberdiel on CFTs. The authors start by considering a Euclidean signature meromorphic field theory on the Riemann sphere. They state ...
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Schwinger-Keldysh contour and $i\epsilon$ prescription

In Tom Hartmann's notes on path integrals, he describes the Schwinger-Keldysh (or "in-in") formalism for calculating vacuum correlators in QFT. He explains that Lorentzian time-ordered ...
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Harmonic oscillator propagator in Euclidean time

I'm following Nastase's book on Quantum Field Theory but this question is just about quantum mechanics in the path integral formalism. In chapter 8 he considers the propagator equation for a harmonic ...
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Fourier transform of Wick rotated functions

I am learning the imaginary time formalism of thermal field theory / reviewing the Euclidean formalism of quantum field theory. One thing that appears to be left implicit in many treatments is a ...
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Double Wick Rotation of de Sitter Metric with Closed Slicing

The de Sitter spacetime (with closed spherically symmetric slicing) has the metric $$\text{d}s^2 = -\text{d}t^2 + \alpha^2\cosh(t/\alpha)^2\text{d}\Omega_{d-1}^2$$ where $\text{d}\Omega_{d-1}^2$ is ...
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Fermions in Euclidean vs Lorentzian Signature

We know that in Lorentzian signature, fermions are representations of \begin{equation} Spin(3,1)\cong SL(2,\mathbb{C})\cong SU(2)\times SU(2)^* \end{equation} where crucially left/right handed ...
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Feynman diagrams, can't Wick-rotate due to poles in first and third $p_0$ quadrants?

I have a confusion about relating general diagrams (involving multiple propagators) in Minkowski vs Euclidean signature, which presumably should be identical (up to terms which are explicitly involved ...
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Does a $d$-dimensional stat-mech theory necessarily has a $(d-1)$-dimensional quantum theory equivalence?

A $d$-dimensional stat-mech theory on a lattice usually can be represented by a $d$-dimensional tensor network. Taking a row/slice of tensors ($M$ tensors or sites) as the transfer matrix (MPO in 2$d$ ...
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Do the Ward identities contain contact terms in Euclidean QFT?

In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to ...
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Doubts about the "periodicity trick" to compute temperature

The "periodicity trick" is a mysterious way to compute some sort of temperature associated to a Rindler-like spacetime. Suppose there exist coords $R\in(0,\infty), \eta\in(-\infty,\infty)$ ...
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What is the relation between the partition function from Stat. Mech. And the Path Integral? [duplicate]

Beside the fact that they look identical when you take imaginary time in the path integral formulation. I understand we doing statistics and we are just integrating over all states with a relative ...
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Special Orthogonal Group in Euclidian and Minkowski spacetimes

I apologize if my question seems a little half-baked. I was wondering if while working with a QFT, one can make transitions from imaginary time to real time and thereby changing the underlying ...
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Quantum to classical mapping

I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works. Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
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How does the $+i\varepsilon$ prescription in the propagator comes from analytic continuation of the Euclidean 2-point function?

Let $S_0[\phi]$ be the action for a real Klein-Gordon field $$S_0[\phi]=\dfrac{1}{2}\int d^Dx \phi(x)(\Box-m^2)\phi(x)\tag{1}.$$ If we try to construct the generating functional $Z_0[j]$ we find that ...
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Euclidean space to Minkowski spacetime

Can you continuously deform (i.e., shrink, twist, stretch, etc. in any way without tearing) four-dimensional Euclidean space to make it four-dimensional Minkowski spacetime?
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Sign Wrong in Effective Action of Bosonic String?

In David Tong's Lectures in String Theory Chapter 7 he sketches a derivation of the low-energy effective action of the bosonic string $(7.16)$: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\...
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Was time statistical?

In QFT the change to the Euclidean picture the $$\exp[iS]$$ looked like the partition function in statistical mechanics, and thereby the approaches in the statistical mechanics might be uaeful to that ...
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In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?

When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
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Why isn't there an $i$ in the Riemannian path integral but there is one in a Pseudo-Riemannian path integral?

In David Skinner's lecture note on AQFT he gives the definition of the path integral as $$ \int_{\mathcal{C}} [\mathcal{D}\phi] \exp\left(-\frac{1}{\hbar}S[\phi]\right) $$ and later states that the ...
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Why is imaginary time evolution non-unitary?

If I have a Hamiltonian $H$, the corresponding time-evolution operator is $e^{-iHt}$. If one defines the evolution operator in imaginary time, one uses $e^{-H\tau}$, where $\tau = it$. It is commonly ...
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How to analytically continue Schwinger functions?

To get Wightman functions $W(t_1, \dots, t_{k-1})$ from Schwinger functions $S(\tau_1 = i t_1, \dots)$, we use analytical continuation. But I don't think this is simply an issue of plugging $it_a$ for ...
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How to obtain imaginary time Green's function from real time Green's function?

Take the following real time time-order Green's function as an example: \begin{equation} i G(x, t)=\left\{\begin{array}{l}\frac{1}{\alpha \sqrt{t}} \exp \left[-\beta \frac{x^{2}}{t}\right], t>0 \\ \...
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Why can we do Wick rotations even if $\Delta<0$?

The above page is from Schwartz QFT. Why can we do Wick rotations even if $\Delta<0$?
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Kinematics of Scattering Amplitudes in $\left(2, 2\right)$ Signature within the Amplituhedron

I am just working my way through the concepts of Amplituhedron and often stumble across the phrase [...] in $\left(2,2\right)$ signature $\lambda$, $\tilde{\lambda}$ are real and independent [...] ...
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Peskin and Schroeder Page no. 95 Feynman Diagrams

From Peskin and Schroeder Page no. 95, ... First, what happened to the large time $T$ that was taken to $\infty(1- i\epsilon)$? We glossed overit completely in this section, starting with Eq. (4.43). ...
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(Non-)Hermiticity of Dirac operator

I have a Dirac operator given by \begin{equation} D\!\!\!/[A, A^{5}]=\gamma^\mu D_\mu=\gamma^\mu (\partial_{\mu} - {\rm i} A_{\mu} - {\rm i} \gamma_{5} A_{\mu}^{5}), \end{equation} where $A_{\mu}$ ...
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The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions

Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions) My question is that what does the conjugacy mean ...
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What is the physical difference between the Euclidean and the Lorentzian path integral?

This is a specific example of the broader question of why should physics change with metric signature basically. Based on a talk by Daniel Harlow, I am generally wondering what exactly makes the ...
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Complex time theories with spacetime $\mathbb{R}^3\times\mathbb{C}$

Are there any well-developed (string?..) theories assuming that, what we perceive as a (3+1) Minkowskian manifold, is a projection/compactification of a 5-dim spacetime, locally obtained via ...
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In the Wick rotated path integral, are the paths functions of an imaginary time variable?

Consider the following action: \begin{equation} S=\int_{-\infty}^{\infty}[\frac{1}{2} \dot x^2(t)-V(x(t))]dt. \end{equation} I promoted $t$ to a complex parameter, and calculate the action over the ...
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Error of $-i$ factor in light cone indices in conformal field theory in Becker's book

In Becker's book of String theory Ch-$3$ I'm getting an error of factor $-i$ in the definition of lightcone indicies after Wick rotation. The convention of the book is following $\sigma_{\pm}=\tau\pm\...
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Different versions of Schwinger parameterization

One common used trick when calculating loop integral is Schwinger parameterization. And I have seen two versions among wiki, arxiv and lecture notes. $$\frac{1}{A}=\int_0^{\infty} \mathrm{d}t \ e^{-tA}...
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Can you perform a Wick rotation if the poles are on the imaginary axis?

I know you can perform a Wick rotation whenever the poles are outside the contour but what happens if the poles are on the imaginary axis? Can you do it anyway?
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Wick rotation for calculation of the heat kernel for massive scalar field in curved spacetime

Let $(\mathcal{M},g)$ be a pseudo-Riemannian manifold. I am interested in the calculation of $\det(\square_g+m^2)$, more precisely in the evaluation of the partition function: \begin{equation*} Z[g]=\...
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What is the equivalent of causality in Euclidean field theory?

In Wick rotated quantum field theory where $t$ becomes $it$ and it has Euclidean metric signature. What would be the equivalent statement that events outside each others light-cones are disconnected ...
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Identifying Lorentzian conformal group after radial quantization

Consider radial quantization in Euclidean signature, in $d$ dimensions. We obtain a Hilbert space with a (non-Hermitian) representation of the Euclidean conformal algebra $so(1,d+1)$, with these ...
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8 votes
2 answers
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Lorentz vs. Euclidean invariance for hard momentum cutoff in QFT

Several accounts of QFT allege that using a hard momentum cutoff $p^2<\Lambda^2$ breaks Lorentz invariance. For instance, see Schwartz's book, p833, or Weinberg p14, or answers here. But I don't ...
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How to Wick rotate the Yang-Mills instanton winding number?

How to Wick rotate the instanton number of Yang-Mills theory? (Related to the earlier question Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength?) My question is particularly about ...
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How do we Wick rotate the Maxwell $U(1)$ gauge theory's field strength $F$? [duplicate]

How do we Wick rotate the Maxwell $U(1)$ gauge theory's field strength, say in 3 space and 1 time dimensions? Suppose we start with a Lorentz signature with coordinates $(x_0, x_1, x_2, x_3)$, then we ...
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Wick rotation from Minkowski Dirac theory to Euclidean Dirac theory: $\gamma^{0} = -i\gamma^{4}$

I am reading Path Integrals and Quantum Anomalies by Kazuo Fujikawa and Hiroshi Suzuki. In chapter 4.2 they calculate the self-energy of photon for QED and say that the actual calculation is performed ...
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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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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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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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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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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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How well does the concept or model of imaginary time work? [duplicate]

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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