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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Wick rotation of CFT three-point function
Let $\langle O_1\cdots O_n\rangle$ be a Euclidean CFT$_d$ correlation function. I know that we can analytically continue to Lorentzian signature as follows. Let $x_i = (\tau_i,\mathbf{x}_i)\in\mathbb{...
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Wick Rotation vs Sokhotski-Plemeli Method to compute internal loop of Feynman correlators
When computing loop integrals in QFT, one often encounters integrals of the form
$$\int_{-\infty}^\infty\frac{dp^4}{(2\pi)^4}\frac{-i}{p^2+m^2-i\epsilon},$$ where we are in Minkowski space with metric ...
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Confusion about choosing an Euclidean world sheet metric in String Theory path integral
When it comes to construct a well-defined path integral for the Polyakov action in Bosonic String Theory, most authors assume that the world sheet metric $g$ is Riemannian (i.e. it has Euclidean ...
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Visualizing CTC - is it related to a "periodic wick rotation"?
As far as I understand Wick rotation, it means the mathematical transformation $$ ct → jct $$
Where $j$ is imaginary unit. While reading on CTC (closed timelike curves) in the Gödel metric I came ...
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How can we use saddle point approximation for a bounce solution which is not even a strict local minimum of the Euclidean action?
In calculating the false vacuum decay, the main contribution to the imaginary energy part of the Euclidean path integral comes from the bounce solution. And we somehow apply saddle point approximation ...
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Justification for Wick rotation for topological insulator
In Appendix B of the paper (1), the authors compute the second Chern number $C_2$ of a band structure by manipulating the ground- and excited-state projection operators $P_{\text{G}}(\mathbf{k})$ and $...
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Strange Wick rotation in the computation of string partition function
In order to compute the one-loop vacuum-to-vacuum amplitude for the bosonic string, one runs into
\begin{equation}
Z(\tau) = V_D (q \bar{q})^{-D/24} \int \frac{d^Dk}{(2 \pi)^D} \exp({- \pi \alpha^\...
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Embedding diagram in west coast signature
Assume you have a metric in (+,-,-,-) signature,
$$d s^2=e^{2 \Phi(r)} d t^2-\frac{d r^2}{1-\frac{b(r)}{r}}-r^2 d \Omega^2.$$
To embed it, we take $t=$Constant, $\theta=\pi/2$ slice,
$$d s^2=-\frac{d ...
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Validity condition for Wick rotation?
I'm reading page 193 of section 6.3 of the QFT textbook by Peskin and Schroeder. There are two integrals that we need to evaluate for the calculation in this section. (here, $\Delta>0$)
$$\int\frac{...
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Fourier Transform of temperature Green Function
I am doing a calculation involving a temperature Green function for some operator $\hat{A}$:
$$G_{\hat{A}}(\tau)=-\Big\langle{T_\tau\big(\hat{A}(\tau)\hat{A}^\dagger \big)} \Big\rangle=-\theta(\tau)&...
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Regularization of loop integrals with Feynman slash
As the title suggests, I am trying to compute a loop integral with a Feynman slash in the numerator, like
$$\int\frac{d^Dq}{(2\pi)^D}\cdot\frac{q_\mu\gamma^\mu}{\left(q^2+\Delta+i\epsilon\right)^3}$$
...
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Is there ever a situation where statistical electrodynamics is needed?
I.e. compute the Euclidean path integrals of QED/the statistical field theory of electrodynamics? I have never seen anyone discuss this anywhere and I am wondering why? What if there is just an ...
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Euclidean Black hole diagram
I am trying to understand how the Euclidean "cigar" is built. I understand how and why the time is periodic, as for the radius of the cigar I am confused, it should be constant far from the ...
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Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?
In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
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Taking imaginary time to show damping relation in path integral formalism
I'm having trouble with a step in the reasoning in pg.9 of Bailin & Love - Introduction to gauge field theory:
To find the ground-state to ground-state amplitude we have a term:(given a basis of ...
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Radial quantization and time order
In CFT, one ususally begins quantization by defining radial ordering on the complex plane, with the notion of radial ordering being the equivalence of time ordering. This is often "motivated"...
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Dimensional regularisation and Wick theorem [duplicate]
Consider an integral:
$$I^{ij}=\int\frac{d^d\textbf{p}}{(2\pi)^d}p^i p^j f(\textbf{p}^2).$$
How can we show that this is equal to:
$$I^{ij}=\frac{\delta^{ij}}{d}\int\frac{d^d\textbf{p}}{(2\pi)^d}\...
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Temperature of quantum fields and periodicity
I have read this PSE post Finite Temperature Quantum Field Theory, saying that
In a QFT at finite temperature, we consider the Euclidean time to be periodic, i.e. we consider a theory on the manifold ...
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Why does Wick rotation appear like an ordinary substitution in this example?
I've seen across several posts, that Wick rotation is not an ordinary substitution. Instead we're rotating the contour of integral and analytically continuing time $t$ to include imaginary time $-i\...
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What do you get when you Wick rotate classical electrodynamics?
Say you have some arbitrary distribution of 4-current density and Faraday tensor in Minkowski space, which satisfies Maxwell's equations and the Lorentz force law. Has it ever been found worthwhile ...
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How can there be total destructive interference after a Wick rotation?
My question is exactly this one from Physics Forums, but I don't see any duplicates on SE and it doesn't seem to have gotten a clear answer there.
If the Wick rotation switches out the complex $e^{iS/\...
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Calculation of one-loop diagram in $\phi^4$ theory
In Folland's book Quantum Field Theory, page 207, he gives the value of the amputated one-loop $\phi^4$ diagram as
$$I(p) = \frac{(-i\lambda)^2}{2} \int \frac{-i}{-q^2 + m^2 - i\epsilon} \cdot \frac{-...
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Planck constant imaginary instead of imaginary PDE coefficients in the Schrödinger equation
Trying to get a first understanding of QM. The Schrödinger equation in standard form for $\Psi$
$$ i \hbar\frac{\partial }{\partial t} \Psi(x,t)
=\left[-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial t^...
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Time-Ordered Propagator in Euclidean Space
I saw a paper stating that in Euclidean signature, the Feynman propagator $G_E$ is related to the Wightman functions $W_{\pm}$ via
$$
G_E (x) = \Theta(\tau) \, W_+ (x) + \Theta(-\tau) \, W_- (x) \, ,\...
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How are Schwinger and Wightman functions used in practice?
In Reed & Simon's Methods of Mathematical Physics Volume II, they define a (Hermitian scalar) quantum field theory to be the quadruple $\langle \mathcal{H}, U, \varphi, D\rangle$ that satisfies ...
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Why do we call it "Euclidean Quantum Gravity" instead of "Riemannian Quantum Gravity"?
Euclidean quantum gravity is an approach to quantum gravity based on working with Riemannian-signature manifolds and eventually relating the results to our Lorentzian spacetime by means of analytic ...
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Path Integral for Unruh Effect
In derivation of Unruh effect, according to arxiv 2108.09188, we have
$$
\langle\phi_L|\exp(-\pi H)|\phi_R\rangle=\int_{\phi=\phi_R}^{\phi=\phi_L} D\phi e^{-S_E}\propto \int_{lower\space half\space ...
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Why we can Wick rotate momentum axis for correlation function?
In QFT writtern by Peskin and Schroeder, in page 293, PS wick rotate both time axis and momentum axis of correlation function of Klein-Gordon field, ie
$$D_F=<0|T\phi(x_1)\phi(x_2)|0>=\int\frac{...
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Do we wick rotate momentum axis on correlation function?
In QFT written by Peskin and Schroeder, it is discussed how correlation function is evaluated in Euclidean space, on page 292 to 293,
In (9.48)
$$<\phi (x_{E1})\phi(x_{E2})>=\int \frac{d^4k_E}{(...
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Why do we Wick rotate before regularizing Feynman diagrams?
In Folland's Quantum Field Theory he mentions that we can apply Feynman's formula (Feynman parameterization) to either the Wick rotated integrals or the non-Wick rotated integrals corresponding to ...
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Correlators on the Euclidean section of a black hole
In the standard construction of the Euclidean section of a Schwarzschild black hole, we start with the exterior metric in Schwarzschild coordinates:
$$\tag{1} ds^2 = -(1-r_s/r)dt^2 + (1-r_s/r)^{-1}dr^...
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Euclidean Time Feynman Path Integral as Stochastic Differential Equation
For a quantum system with Lagrangian $L(x, \frac{dx}{dt})$ we can represent the action of a path $\mathbf{x}$ as $$S(\mathbf{x}) = \int_0^{t} L(\mathbf{x}(s), \mathbf{\frac{dx}{dt}}(s)) ds.$$ Then, ...
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Performing Wick rotation under conjugation
See the formulas (95) and (96) of this notes https://arxiv.org/abs/1602.07982. When one try to perform the Wick rotation $t=-i\tau$ to the field in Minkowski/Lorentzian spacetime
$$\mathcal{O}_L(t, \...
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Hawking temperature for an asymptotically $\text{AdS}_5$ black hole with a constant magnetic field
I'm trying to calculate the Hawking temperature for an asymptotically $\text{AdS}_5$ black hole with a constant magnetic field. Suppose that the metric and magnetic field of this background are given ...
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What would be the problems with supposing spacetime to be fundamentally Riemannian, not Lorentzian?
I'm thinking about the Wick rotation. My question may be similar to this one but I don't think it's a duplicate, though you can judge that.
Suppose we take the Wick rotation as an indication that ...
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Wick rotation in Peskin and Schroeder's QFT
I know there are many similar analysis about this topic, like here, here, many of them are answered by Qmechanic, excellent answer!
I have checked most of these posts, but I still don't clearly ...
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Green Function in Euclidean space time
My question based on Ashok Das's "Finite Temperature Field Theory", page 12-13.
The book assume that in bosonic Klein-Gordon theory, zero temperature Green function satisfies (metric in ...
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Can the QFT path integral be re-expressed using a real, positive-definite function of the action? [duplicate]
This question is based on my rather shaky grasp of QFT, so if I'm missing a key concept then just let me know!
If you're deriving the Schrodinger equation from the path integral as Feynman did, then ...
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Continuum limit of lattice field theory
Just a simple question for lattice QCD experts, is continuum limit of lattice field theory a relativistic quantum field theory?
Because i heard that lattice QCD is done in imaginary time, producing a ...
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Imaginary velocity components in geodesic [duplicate]
When we try to find the geodesic of a partical at rest, in the second term of the geodesic equation we use dt/dtau = 1. Shouldn’t it be i (for imaginary number), since the time component of the 4-...
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Hamiltonian and Lagrangian for a particle on a ring [duplicate]
In the book Condensed Matter Field Theory (A. Altland & B. Simons)(page 498, 2nd edition) they suggest the following Hamiltonian and Lagrangian for a particle on a ring in the presence of a ...
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Why does the Lagrangian have $O(4)$ symmetry after Wick rotating (previously Lorentz symmetry)?
Pertaining to the answer within link.
Why is it the case, that for Lorentz invariant Lagrangian $\mathcal{L}$, after Wick rotation, the $O(4)$ invariance is established, thus manifesting itself as ...
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Wick rotation of contravariant vector field holomorphic?
Suppose one has a contra variant vector field in Minkowski spacetime $A^\mu : \mathbb{R}^{1+3} \to \mathbb{R}$ for each component $\mu$ inside some path integral. I assume, that one can analytically ...
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Physics in Euclidean spacetime [duplicate]
I just have a very small and naive Question.
In my PhD I work on different Toy models which are implemented on the lattice.
In order to do so one performs a Wick rotation from minkowski to euclidean ...
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On symmetry of Lorentz matrix
For Lorentz transformations, if we put $x^1=x$ and $x^2=ct$ and restrict ourselves to $2D$ we get
$$x'=\gamma(x-\beta ct) \tag{1} $$
$$ct'=\gamma(ct-\beta x) \tag{2} $$
The matrix associated with this ...
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Chiral symmetry of the Euclidean action for fermions
In the literature, such as QFT Volume-II by Weinberg, p.368, the chiral anomaly is derived using Euclidean path integral. To formulate the question, let's start with the Minkowski space with signature ...
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Is the Schrödinger equation the heat equation with imaginary constants?
Playing around with the Schrödinger equation, I separated the time partial derivative this way:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}-\frac{i}{\hbar}V\...
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Is the value of the action important?
I know that the action, in Classical Mechanics, is a functional of the path of a physical system, such that
"the path actually followed by a physical system is that for which the action is ...
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Why is the Vacuum state got by a limit to imaginary time?
Given a (non-relativistic) propagator $K_t(A,B)$ giving the 'conditional amplitude' to go from state $B$ to state $A$ in time $t$, it is known that one can find the vacuum wavfunction by (independent ...
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Wick's rotation, unitary transformation and symmetry
The Wick's rotation $W$ facilitates dealing with integrals in the Minkowski space by rotating time into the Euclidean space. As this rotation in time is performed within integrals, one can view that ...
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