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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3answers
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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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Question about how to (mathematically) interpret averages with the Feynman-Kac formula
I'm trying to understand the Feynman-Kac formula (https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula) for the Wick-rotated Feynman path integral. Would it be correct to say that $$\langle \phi\...
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Does a partially traced density operator also become a Boltzmann density operator under Wick rotation to Euclidean space?
I know that, under the Wick rotation $(i\Delta t/\hbar,p_0)\to(-\beta,-ip_{0,E})$, Feynman's path integral supposedly transforms into the traced-over Boltzmann partition function, $trace(e^{-\beta H})=...
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How well does the concept or model of imaginary time work?
In order to make the Minkowski metric, in special relativity, equivalent to the Euclidean metric, one idea is to allow time to take imaginary values. As far as I have learned about SR, it does make ...
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In path-integral, when do we have to insert fact $i$ in front of the action $S$ in the exponent?
I have got stuck in these concepts for a fews days: Wick rotation, Euclidean spacetime and QED in gravity.
Generally, in Minkowski space time, there is a factor $i$ in front of the action $S$, e.g., ...
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Wick rotation with magnetic fields
How does Wick rotation work with magnetic fields?
Let us take up single-particle $d$+1 QM. Then the Euclidean time path integral is given (in $\natural$ units) by $$ \langle x| \exp(-t H) y\rangle = \...
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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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56 views
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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1answer
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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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66 views
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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2answers
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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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83 views
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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1answer
87 views
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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1answer
92 views
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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142 views
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 [1]) that mentions the Osterwalder-Schrader Theorem that gives a ...
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202 views
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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90 views
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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Choice of folliation in path integral
Assume we have a scalar field theory for a field $\phi$. Can we think of the Hilbert space as being spanned by states of the form $|\varphi\rangle$ for configurations $\varphi\in C^\infty(\mathbb{R}^3)...
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234 views
Is there a link between the logistic differential equation and Fermi-Dirac statistics?
I was working out some statistical problems and I could not fail to notice that Fermi-Dirac distribution,
$$f_{\rm Fermi-Dirac}(E)=\frac{N_{\rm sites}}{1+e^{\beta(E-\mu)}},$$
looks like the kind of ...
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1answer
362 views
Wick-rotated quantum computers e.g. to be realized with Ising-like systems?
Quantum mechanics is equivalent with Feynman path ensemble, which after Wick rotation becomes Boltzmann path ensemble - and e.g. Ising model is a basic condensed matter model, which is assumed to use ...
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Why does the quantum expectation value depend only on the diagonals of the Hamiltonian in the long time limit?
I'm trying to understand the eigenstate thermalization hypothesis more and I keep coming across a limit I don't understand.
If the initial state of a system is in it's energy eigenstate basis as $|\...
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Wick's rotation for the harmonic oscillator. Explicit computation
I'm stuck in this computation; it shouldn' be difficult but it's always better to check with a lot of details these things. Consider the propagator for the harmonic oscillator Ashok p.55 bottom of the ...
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1answer
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Convert propagators from Euclidean to Minkowski spacetime
I'm looking for a rule to "convert" the propagators of a quantum field theory formulated in Euclidean spacetime into those of the same theory but in Minkowski spacetime (with the $\operatorname{diag}(-...
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1answer
90 views
How could a relation of the form $t=-i\tau$ hold with both $t$ and $\tau$ being real?
The physical time is a real quantity. But in quantum field theory, whenever we find oscillatory exponentials in time and we cannot literally take the limit $t\to \infty$, we make a change of variable, ...
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Notation on four-vectors using imaginary spacelike components [duplicate]
Can one just change the notation of four vectors so as instead of having $$ X^{\mu} =(X^0, \vec{X})$$we define $$ X^{\mu}=(X^0,i\vec{X})?$$ This way we could use the Euclidean metric instead of $$g^{\...
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1answer
245 views
On the Euclidean action for QCD
The Euclidean action for QCD reads, (see e.g., Eq. (45) in "ABC of instantons" by Novikov, Shifman, Vainshtein, and Zakharov)
$$S_E=\int d^4 x\left[\frac{1}{4}G^a_{\mu\nu}G^a_{\mu\nu}+\psi^\dagger(-i\...
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1answer
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What is QFT at finite temperature?
On the one hand, according to the Wick rotation that relates Statistical Field Theory and Quantum Field Theory, a finite temperature statistical system corresponds to a compact time quantum field ...
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Explain Imaginary Time and Temperature [duplicate]
I was amazed to learn that we can use Imaginary unit iota into physical quantities like time and Temperature but how exactly?
The explanation was not something I would say stellar so I am hoping can ...
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379 views
Dirac equation without $i$
In Witten's review paper "Fermion path integrals and topological phases", the Dirac equation (Eq(2.2)) is
$$(\gamma^{\mu}D_{\mu}-m)\psi=0$$
which appears very strange to me. Initially I thought this ...
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1answer
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Convergence of the path integral
In P&S 9.3 the path integral
$$ Z[J]=\int {\cal D}\phi \exp[i\int d^4x ({\cal L} + J\phi)]$$
of the (Minkowski) $\phi^4$-theory when subjected to a Wick-rotation (change of the integration path ...
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2answers
278 views
In light of Wick rotations is position and time on the same footing in QFT?
Taken from here
Wick rotation connects statistical mechanics to quantum mechanics by
replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/ℏ$
But I was under the impression position ...
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Confusion about gamma matrices in Euclidean spacetime
I have encountered a number of sources with differing definitions of the transition from Minkowski spacetime to Euclidean spacetime.
I'd like some clarification as to how to go from Minkowski to ...
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“Imaginary-time” argument in high energy physics
In many-body physics, there are many "imaginary-time" techniques, such as Matsubara Green's function, imaginary-time path integral and others. It seems that these concepts are frequently used in ...
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Wick Rotation and sign of the integrand in Weinberg's book
I'm studying from Weinberg's QFT volume 1, chapter 11. I have a problem with equation $(11.2.7)$.
Starting from eq. $(11.2.5)$
$$
\begin{align}
\Pi^{\rho\sigma} (q) = \frac{-ie^2}{(2\pi)^4} \int_0^...
