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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8
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2answers
654 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
137 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
393 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
153 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 ...
2
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2answers
254 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 ...
3
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1answer
711 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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1answer
95 views

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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0answers
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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1answer
600 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
299 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 ...
5
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1answer
312 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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0answers
163 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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0answers
91 views

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 ...
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1answer
1k views

Why doesn't Wick rotation work for this integral?

I thought that for momentum integrals in Minkowski space, the Wick rotation to Euclidean space $k_0 \to ik_0$ allows one to write (let's say $f$ comes with an $i\epsilon$ prescription): $$\int_{\...
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1answer
405 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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0answers
145 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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1answer
166 views

How is a pseudo-Euclidean metric superior to Minkowski's complex metric? [duplicate]

This is my second attempt to get a meaningful response from you guys on this issue. The SR invariance formula makes space-like intervals imaginary (e.g., the distance $x$ in a given frame has ...
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1answer
356 views

Gaussian integrals in Feynman and Hibbs

I was going through the calculation of the free-particle kernel in Feynman and Hibbs (pp 43). The book describes $$ \left(\frac{m}{2\pi i\hbar\epsilon}\right)\int_{-\infty}^{\infty}\exp\left(\frac{im}...
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2answers
740 views

Minkowski's complex Euclidean space vs. the real pseudo-Euclidean version

The SR invariance formula makes space-like intervals imaginary (e.g., the distance $x$ in a given frame has interval $ix$). Yet modern physicists consider it bad form to define the distance itself as $...
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2answers
1k views

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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1answer
1k views

Path integral and imaginary time in Quantum and Statistical Mechanics

I have come across the path integral formulation of quantum mechanics, and have found plenty of websites, papers and book chapters explaining the relation to statistical mechanics. The general ...
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1answer
440 views

Schroeder's Minkowski Space Integral - Concerns about Wick Rotations

In the Appendix of Peskin & Schroeder's "An Introduction to Quantum Field Theory" there is a list of integrals in Minkowski space. Of particular interest to me is the integral (A.44): $$ I(\Delta) ...
3
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1answer
243 views

Sign of the coupling constant for a $\phi^4$ interaction

The vacuum-vacuum transition for a simple bosonic $\phi^4$ theory is typically written as $$ \langle0|0\rangle = \int[D\phi]\ \exp\left[-i\int (L_0+L_\mathrm{int}) d^4x \right], \tag{1} $$ Where $L_0$ ...
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1answer
376 views

Does the imaginary time path integral for the partition function imply that temperature set a characteristic time scale for quantum systems?

In the ordinary path integral, the action is an integration over the time your interested in. In quantum statistical mechanics the integration is over an imaginary time with the limit $\frac{\beta}{\...
4
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1answer
139 views

Are statistical fields commutative?

In both statistical field theory and quantum field theory one computes average values / time ordered expectation values of functionals of fields with the path integral. I have two related questions: ...
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1answer
96 views

Sign ambiguity in the transition from Minkowski space to Euclidean space

In the metric convention $(+,-,-,-)$, the spacetime interval is given by $$x^2=x_\mu x^\mu=(x^0)^2-|\textbf{x}|^2=t^2-|\textbf{x}|^2$$ in the units $c=1$. To make the theory Euclidean one considers ...
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1answer
2k views

How can I find the Euclidean action?

How can I show that $$S_E[x]=\int_{t_i}^{t_f} dt \left(\frac{m}{2}\dot{x}^2+V(x)\right),$$ starting from the definition of transition amplitude $$A=\langle x_f\,|\,e^{-\frac{i}{\hbar}(t_f-t_i)\hat{H}}...
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1answer
1k views

How does Euclidean Quantum Field Theory describe tunneling?

We know that Euclidean QFT originates from path integral formalism of $$\langle\phi_f|e^{-\beta\hat{H}}|\phi_x\rangle.\tag{1}$$ We can understand that for $\beta\rightarrow\infty$, we can obtain the ...
3
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1answer
163 views

Spinor vacuum energy

I'm reading the calculation in the book Quantum field theory in a nutshell of A. Zee of chaoter II.5 In this chapter the vacuum energy is calculated through the path integral approach. At some point ...
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0answers
132 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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1answer
726 views

The computation of the propagator in two dimensions

I did the computation of the propagator in two dimensions at (19.26) in Peskin & Shroeder as follows. First I performed a Wick rotation. \begin{alignat}{2} \int\frac{d^2 k}{(2\pi)^2}e^{-ik\cdot (...
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3answers
448 views

Evaluation of functional determinants

Consider the evaluation of the following functional determinant: $$\text{log}\ \text{det}\ (\partial^{2}+m^{2})$$ $$=\text{Tr}\ \text{log}\ (\partial^{2}+m^{2})$$ $$= \sum\limits_{k} \text{log}\ (-...
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1answer
308 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 \...
14
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1answer
4k views

Connection between Schrödinger equation and heat equation [duplicate]

If we do the wick rotation such that τ = it, then Schrödinger equation, say of a free particle, does have the same form of heat equation. However, it is clear that it admits the wave solution so it is ...
9
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2answers
471 views

The idea of analytical continuation method to solve the Klein-Gordon equation, how and why?

For simplicity, let's consider a two dimensional version of Klein-Gorden equation: $$ (\partial_t^2-\partial_x^2-\partial_y^2+m^2) G(\vec{x},t) = -\delta(\vec{x})\delta(t) $$ From the previous posts: ...
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0answers
170 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=\...
4
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2answers
265 views

Rewriting bosonic action in Altland and Simon Chapter 4

In page 179 of Altland and Simon, Condensed Matter Field Theory, the author obtained the action \begin{equation} S[\theta]=\frac{1}{2\pi}\int dx\,d\tau\,\left[(\partial_x\theta)^2+(\partial_\tau\...
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2answers
3k views

Problems while Wick rotating the path integral

I am trying to begin from the path integral of QM and write the Euclidean version of it performing the Wick rotation but it seems that I am missing a few things. For simplicity I work on 1 dimension ...
5
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1answer
539 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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0answers
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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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1answer
192 views

A problem in an integration related to Wick rotation

In quantum field theory, we often calculate some integrations using Wick rotation. In the following, I will carefully deal with an integration involving Wick rotation. In the end, I have found that I ...
8
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1answer
497 views

Lack of Maslov index in the path integral formalism

Introduction Consider Feynman's famous path integral formula \begin{equation} K(x_a,x_b) = \int \mathcal{D}[x(t)] \exp \left[ \frac{i}{\hbar} \int_{t_a}^{t_b} dt \, \mathcal{L}(x(t),\dot{x}(t),t) \...
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1answer
590 views

What is Hawking's, “No Boundary Conditions”? [closed]

In his "No Boundary Conditions", is Hawking stating that time is eternal? And what is the difference between Real Time and Imaginary Time? Is he saying there are two different arrows of time, and ...
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1answer
2k views

Quantum field theory: zero vs. finite temperature

I have recently been made aware of the concept of thermal field theory, in which the introductory statement for its motivation is that "ordinary" quantum field theory (QFT) is formulated at zero ...
2
votes
1answer
378 views

Sign of Wick rotation [closed]

Suppose you have the integral $$i \int^\infty_{-\infty} L_M(t) dt$$ and that $L_M$ contains two poles: when $t>0$ the pole lies above the t-axis and when $t<0$ the poles lies below the t-axis. ...
11
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3answers
1k views

Performing Wick Rotation to get Euclidean action of a scalar field $\Psi$

I'm working with the signature $(+,-,-,-)$ and with a Minkowski space-time Lagrangian $$ \mathcal{L}_M ~=~ \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi. $$ The Minkowski action is $$...
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1answer
401 views

What is the time imaginary method? [closed]

I have to submit homework about the scheme which solves the time-independent Schrödinger equation and finds the ground state by the imaginary time method. I know the substitution $-\mathrm{i}\tau\...
5
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1answer
1k views

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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2answers
245 views

Is this “modified wave equation” (with opposite signed derivatives) used?

This is the wave equation: $$(\partial_t^2 - \nabla^2) \psi = 0$$ What is the following equation? $$(\partial_t^2 + \nabla^2) \psi = 0$$ What's it like? What can you do with it? Does it show up ...
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0answers
321 views

Complex Space Time - Mathematical Foundations [duplicate]

I am really curious as to what the current research is in complex space time. Because in "The theory of Everything". Stephen Hawking does talk about imaginary time. Is there any mathematical ...