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# 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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### Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ds^{2} = -(c^{2}dx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}$$ ...
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### The meaning of imaginary time

What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics,...
2k views

### Wick rotation and spinors

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
992 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 ...
2k views

### Special relativity and imaginary coefficient of the time coordinate

I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary ...
7k views

### Wick rotation in field theory - rigorous justification?

What is the rigorous justification of Wick rotation in QFT? I'm aware that it is very useful when calculating loop integrals and one can very easily justify it there. However, I haven't seen a ...
3k views

### Minkowski spacetime: Is there a signature (+,+,+,+)?

In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if ...
5k views

### Schrödinger equation derivation and Diffusion equation

I am aware of the debate on whether Schrödinger equation was derived or motivated. However, I have not seen this one that I describe below. Wonder if it could be relevant. If not historically but for ...
5k views

### What is imaginary time? [duplicate]

I am not professional physicist; but I am curious about Stephen Hawking's "imaginary time". It would be better to elaborate exactly what it is. I am not confused because of the word "imaginary" but I ...
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### Euclidean derivation of the black hole temperature; conical singularities

I am studying the derivation of the black hole temperature by means of the Euclidean approach, i.e. by Wick rotating, compactifying the Euclidean time and identifying the period with the inverse ...
2k 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 ...
4k views

### Gaussian integral with imaginary coefficients and Wick rotation

Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
3k views

### How to Perform Wick Rotation in the Lagrangian of a Gauge Theory (like QCD)?

I'm studying Lattice QCD and got stuck in understanding the process of going from a Minkowski space-time to an Euclidean space-time. My procedure is the following: I considered the Wick rotation in ...
4k views

### How exact is the analogy between statistical mechanics and quantum field theory?

Famously, the path integral of quantum field theory is related to the partition function of statistical mechanics via a Wick rotation and there is therefore a formal analogy between the two. I have a ...
1k views

### Does Wick rotation work for quantum gravity?

Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.
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### 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 $\tau$ ...
534 views

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### Can we obtain non-Lorentzian metric from Lorentzian metric, through renormalization methods?

Since low-energy, non-relativistic thermal field theories are defined in Euclidean spacetime, while high-energy relativistic theories are define in Minkowski spacetime, I was wondering if there are ...
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### Moments of a Distribution via Laplace Transforms and Wick Rotations [closed]

On a mathematical level, the statistical mechanical partition function is just a Laplace transform of the microcanonical probability distribution, i.e. it's moment generating function. Understanding ...
724 views

### Wick rotation and the arrow of time

It is well known that we can switch from a statistical system to a quantum mechanical system by a Wick rotation. Has this rotation some implication on the way the time flow? namely, this is an ...
332 views

### Does Schrödinger equation have dual-property with Heat equation?

I have experimental data that Schödinger equation maintains high frequencies, while heat equation low. Does Schrödinger equation have some duality property with heat equation?
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### Two math methods apply the same loop integral lead different results! Why?

I tried to adopt the cut-off regulator to calculate a simple one-loop Feynman diagram in $\phi^4$-theory with two different math tricks. But in the end, I got two different results and was wondering ...
231 views

### Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

Consider a momentum integral of the form $$I=\int d^3k\int dk^0\frac{1}{k^4}$$ where $k^2=(k^0)^2-(\vec{k})^2$ and the integral over $k^0$ runs from $-\infty$ to $+\infty$. This integral is common in ...
Performing a Wick rotation over an integral is not equivalent to just a change of variable $t \to \mathrm{i}t = \tau$, after that we rotate the complex plane so that \mathrm{i} \int_{-\infty}^{\...
### Wick rotation vs. Feynman $i\varepsilon$-prescription
The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...