# 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 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. I have this expression (equation 53 from the ...
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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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ) that mentions the Osterwalder-Schrader Theorem that gives a ...
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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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### 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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### 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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