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17
votes
5answers
11k views

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 ...
10
votes
6answers
2k views

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 $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
14
votes
1answer
697 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; ...
7
votes
3answers
847 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.
7
votes
2answers
4k 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 ...
15
votes
1answer
2k views

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 ...
7
votes
2answers
987 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 ...
6
votes
3answers
239 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 ...
1
vote
3answers
516 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 ...
10
votes
1answer
591 views

Subtlety of analytic continuation - Euclidean / Minkowski path integral

I subconsciously feel not fully comfortable about Wick rotating or analytic continuation from Euclidean to Minkowski space. I simply wonder whether there is any subtlety here, and when we need to be ...
7
votes
3answers
690 views

Applications of analytic continuation to physics

I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the ...
5
votes
0answers
402 views

Time Reversal, CPT, spin-statistics, mass gap and chirality of Euclidean fermion field theory

In Minkowski space even-dim (say $d+1$ D) spacetime dimension, we can write fermion-field theory as the Lagrangian: $$ \mathcal{L}=\bar{\psi} (i\not \partial-m)\psi+ \bar{\psi} \phi_1 \psi+\bar{\psi} ...
4
votes
1answer
214 views

About the gauge formalism in statistical quantum field theory

I would like to understand a bit more the aspects of the gauge theory in statistical field theory. In particular, I would like to understand how the replacement $\tau \rightarrow it/\hbar$ is ...
4
votes
1answer
125 views

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 ...
3
votes
0answers
211 views

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 ...
16
votes
4answers
591 views

What is a simple intuitive way to see the relation between imaginary time (periodic) and temperature relation?

I guess I never had a proper physical intuition on, for example, the "KMS condition". I have an undergraduate student who studies calculation of Hawking temperature using the Euclidean path integral ...
8
votes
2answers
295 views

The poles of Feynman propagator in position space

This question maybe related to Feynman Propagator in Position Space through Schwinger Parameter. The Feynman propagator is defined as: $$ G_F(x,y) = \lim_{\epsilon \to 0} \frac{1}{(2 \pi)^4} \int d^4p ...
2
votes
0answers
138 views

Feynman Propagator in Position Space through Schwinger Parameter

So I am aware of a thread at Propagator of a scalar in position space but it does not answer my question, which is more about poles in position space. Starting from $$D_F(x_1-x_2) = \int \frac{d^4 ...
5
votes
1answer
204 views

Problem understanding sign of volume integral in Minkowski space

My professor told me that a 4-dimensional Minkowski - Space Integral I was working on can be written as the product of a metric tensor and a scalar: $\int d^4 k \frac{k^\mu ...