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3
votes
1answer
165 views

Amplitude $\langle0|e^{-iHT}|0\rangle$ in A. Zee's QFT In A Nutshell

In his Quantum Field Theory In a Nutshell, in page 12, (Second Ed), A Zee says that conventionally, the amplitude $\langle0|e^{-iHT}|0\rangle$ is denoted by $Z$. In the next paragraph, he considers ...
1
vote
0answers
34 views

Anomaly and Weyl spinors

I try to better understand anomalies in QFT and I've got a question concerning derivation of axial anomaly in Terning's lectures (page 12) Consider a theory of Weyl fermions coupled to a gauge field ...
5
votes
1answer
731 views

Analytic continuation of imaginary time Greens function in the time domain

Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature $$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$ ...
13
votes
1answer
655 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; ...
-1
votes
1answer
50 views

Euclidean classical action

This is the Euclidean classical action $S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))$. It would be nice if somebody could explain the structure of the potential. I don't ...
3
votes
0answers
117 views

Why is imaginary time “outdated”? [closed]

I was looking at reviews for Sakurai's Quantum Mechanics textbook, and some mentioned it being outdated, specifically mentioning his use of imaginary time. Is this idea deliberately avoided in modern ...
3
votes
0answers
62 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 ...
2
votes
2answers
174 views

Complex numbers in quantum mechanics and in special relativity

Is there a physical relation between the use of complex numbers for the wavefunction in (non-relativistic) quantum mechanics and in special relativity (as formulated in the setting of Minkowski ...
17
votes
5answers
10k 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 ...
1
vote
0answers
19 views

Geodesic approximation and Euclidean continuation

I recently read many articles in the context of the AdS/CFT correspondance in which the geodesic approximation is used (see for example section 3.5 here). The correlator between two boundary operators ...
0
votes
1answer
203 views

Is imaginary time a fifth dimension? [duplicate]

I've read that by introducing the concept of imaginary time, the dimension of time can be treated like a spatial dimension mathematically. Assuming, without imaginary time, one considers the universe ...
4
votes
3answers
494 views

Can a Wick rotation be performed on the Pauli algebra to get from $+++$ to $+--$ signature?

Wick rotation makes sense for the Dirac algebra: it has $+---$ and $++++$ and $----$ signatures. Just wondering if one can Wick rotate the Pauli algebra from the standard $+++$ Pauli matrices to ...
5
votes
1answer
191 views

Black hole temperature in an asymptotically de Sitter spacetime

I am trying to calculate the Hawking temperature of a Schwarzschild black hole in a spacetime which is asymptotically dS. Ignoring the 2-sphere, the metric is given by ...
3
votes
1answer
154 views

Why use a particular regularization for $\int_0^\infty \mathrm{d}x\,e^{i p x}$?

There are many badly defined integrals in physics. I want to discuss one of them which I see very often. $$\int_0^\infty \mathrm{d}x\,e^{i p x}$$ I have seen this integral in many physical problems. ...
2
votes
3answers
164 views

Root of $i$, which one to take?

The propagator of a free particle in 1d is $$ K(x_b, t_b; x_a, t_a ) = \sqrt{\frac{m}{2\pi i \hbar (t_b-t_a)}} \exp \left [ \frac{i m (x_b-x_a)^2}{2 \hbar (t_b-t_a)} \quad \right ] .$$ It looks ...
3
votes
1answer
70 views

Why is the value of the action integral in general relativity the same on all regions that are homologous?

In their famous paper Action integrals and partition functions in quantum gravity, Gibbons and Hawking argue that in order to avoid the singularity of a Schwarzschild black hole you can complexify ...
1
vote
3answers
351 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 ...
4
votes
1answer
128 views

Imaginary time is to inverse temperature what imaginary entropy is to …?

The Wick-Rotation rotates imaginary time into inverse temperature (as can be seen from its "rotating" the Schrödinger equation into the heat equation). Now since entropy is temperature's conjugate, I ...
6
votes
1answer
147 views

Srednicki's book chapter 8

Reading first page in chapter 8 of Srednicki's it reads: To employ the $\epsilon$ trick, we multiply $H_0$ with $1-i\epsilon$. The results are equivalent to replacing $m^2$ with $m^2-i\epsilon$. ...
10
votes
0answers
129 views

LSZ reduction vs adiabatic hypothesis in perburbative calculation of interacting fields

As far as I know, there are two ways of constructing the computational rules in perturbative field theory. The first one (in Mandl and Shaw's QFT book) is to pretend in and out states as free ...
1
vote
0answers
105 views

Feynman's $i\epsilon$ prescription in path integrals (Mark Srednicki)

On page 63 in M.S. book , why m^(-1) goes to (1-iε)m^(-1) or m -> (1+iε)m and how can i verify eq.(7.3)? On page 63 writes : Looking at $H(P,Q)= \frac{1}{2m} P^2 +\frac{1}{2}mω^2Q^2$ we see that ...
4
votes
1answer
108 views

Evolution of harmonic oscillator in path integral formulation

The unnormalized ground state of the harmonic oscillator (choosing units such that $m = \hbar = \omega = 1)$ is $$\tag{1}\psi(q,t) = \exp(-q^2/2-it/2).$$ The transition function is ...
3
votes
2answers
209 views

Transition amplitudes by functional methods in QFT

I am following section 9.2 in Peskin and Schroeder in which the Feynman rules are derived for scalar fields. They define (in eqn (9.14), page 282) the transition amplitude from $\vert\phi_a\rangle$ ...
2
votes
1answer
114 views

Is it okay to Wick rotate to give the negative of the Euclidean metric? Also, could we make the space-like coordinates imaginary instead?

There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + ...
4
votes
1answer
220 views

Wick Rotation in Curved space

So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical ...
7
votes
3answers
775 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.
1
vote
0answers
86 views

Wick rotation and special relativity

CMIIW, but as I understand it, Wick rotation replaces the Minkowski basis (t,x,y,z) with the Euclidean basis (it,x,y,z). Suppose that $t_2=t_1 \cosh \beta+x_1 \sinh \beta$ and $x_2=t_1 \sinh \beta+x_1 ...
0
votes
0answers
98 views

Minkowski to Euclidean

When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now ...
2
votes
0answers
86 views

Does anybody know of a source that explains Wick rotation for fermions in 3-dimensional spacetime?

I've been looking for a long time and I've not had a lot of luck. I've found sources that use fermions in 3d Euclidean space but I can't find any that explain the Wick rotation from Minkowski space. ...
36
votes
4answers
1k 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 ...
7
votes
2answers
3k 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 ...
6
votes
3answers
200 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 ...
6
votes
2answers
140 views

Yang-Mills existence and mass gap

In the Clay institute problem description of the Yang-Mills existence and mass gap problem it states that the quantum Yang Mills needs to be formulated in $\mathbb{R}^4$ space. I was wondering whether ...
3
votes
0answers
166 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 ...
7
votes
0answers
112 views

Conditions permitting rotation to imaginary time

I often see that action is written with a Euclidean metric instead of the original Minkowski metric. My question is basically this : Under what conditions is okay to make a wick rotation? I am ...
2
votes
0answers
88 views

Relationship between the Black-Scholes model and path integrals

This question was inspired by some interesting comments by Rod Vance on this answer: Minkowski spacetime: Is there a signature (+,+,+,+)? Could you (Rod), or someone else, expand on these comments ...
5
votes
1answer
188 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 ...
0
votes
0answers
56 views

A question about sign in Euclidean path integral

I have a question about the sign in the Euclidean path integral in Polchinski's string theory vol I, p 337. In page 335, Polchinski introduced path integral in Euclidean space $$ \langle q_f, U| ...
10
votes
5answers
1k 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 $ ...
2
votes
0answers
52 views

What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann ...
4
votes
1answer
254 views

From Minkowski to Euclidean Time in Path Integrals

I'm trying to prove the following equality: $$ <x_{f},\, it_{f}|x_{i},\, it_{i}>=\mathcal{N}\int_{\left\{ x\in\mathbb{R}^{\mathbb{R}}:\, x\left(t_{f}\right)=x_{f}\wedge ...
6
votes
2answers
737 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 ...
1
vote
1answer
97 views

Why can we simply absorb the positive coefficient of $i\epsilon$ in a propagator?

As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification. In Peskin page 286, he did this: ...
6
votes
1answer
334 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 Minkwoski space-time to an Euclidean space-time. My procedure is the following: I considered the Wick rotation in ...
7
votes
3answers
534 views

Solving Quantum Tunnelling Without Wick Rotation

Edit It seems that I haven't written my question clearly enough, so I will try to develop more using the example of quantum tunnelling. As a disclaimer, I want to state that my question is not about ...
4
votes
1answer
193 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 ...
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votes
2answers
181 views

Gaussian Integral by Substitution [duplicate]

I am trying the derive a path integral representation. I understand this involves Gaussian integrals of the form: $$\int_{-\infty}^\infty e^{-x^2}\text dx=\sqrt\pi$$ However, I am trying to evaluate ...
7
votes
3answers
578 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 ...
4
votes
1answer
190 views

Hermitian conjugation in Radial Quantization

I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger ...
10
votes
1answer
525 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 ...