What is Imaginary time: Definition and 20 Discussions
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.
Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the Imaginary unit i. Imaginary time is not imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers.
An idea I was thinking about for the last few days:
You want to calculate the ground state of some system from the Schrödinger eqn ##\hat{H}\left|\psi\right.\rangle = E\left|\psi\right.\rangle##. One way is to choose a trial state ##\left|\psi (t_0 )\right.\rangle## and use the TDSE to...
Suppose I want to find the ground states corresponding to several Hamiltonian operators ##\left\{ \hat{H}_i \right\}##, which are similar to each other. As an example, let's take the ##\hat{H}_i##:s to be anharmonic oscillator Hamiltonians, written in nondimensional form (##\hbar = m = 1##) as...
Why do the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form? – i.e. both exponentials of the Hamiltonian operator.
The Matsubara trick/method just takes this as a fact in thermal QFT; but...
Hi, I have been trying to use imaginary time propagation to get the ground state and excited states eigen function but the results I got is different from the analytical solution. I know that to get excited states, I should propagate 2 or more orthogonal functions depending on the number of...
I know what imaginary numbers are, but I'm struggling to understand why the Lorentz transformation makes a time-like dimension space-like. I suppose what I'm really asking is what is the difference between time-like and space-like. I've read that it has something to do with special relativity...
Homework Statement
I have the following massive spin-1 propagator-
$$ D^{\mu\nu}(k)=\frac{\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^2}}{k^2 - m^2} $$
I want to write down the propagator in the imaginary time formalism commonly used in thermal field theories.
Homework EquationsThe Attempt at a...
Regarding interacting green's function, I found two different description:
1. usually in QFT:
<\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>}
2. usually in quantum many body systems...
I've seen Stephen Hawking mention it. there's an article on it. Is imaginary time a scientifically serious proposal? what are the ramifications to physics, including general relativity and gravity, if we accept imaginary time? what about imaginary space? what about quantum gravity theories like...
I find this passage from A Brief History of Time a bit hard to believe. When he talks about using imaginary time for the purposes of calculation, is it the same like in the Schro eq which uses an imaginary number? How plausible is the following passage? Is using imaginary time a common practice...
In QM and QFT, imaginary time is used to make the oscillatory path integral converge, and also to handle terms that are not semibounded in Minkowski spacetime.
In CDT, imaginary time is also used after the path integral is restricted to "causal" configurations.
How is the oscillatory...
I recently read the book "A Brief History of Time"by Stephen Hawking, and in it he described the concept of imaginary time.
It had something to do with the squares of numbers being equal to negative numbers, which were called imaginary numbers. Also, he mentioned being able to travel in...
Hello everyone,
In Fermi Liquid Theory, I'm trying to understand what the relationship is between the Green's function and the average occupancy of levels. In my lecture they gave the relation
\left\langle n_k \right\rangle = G(k,\tau\rightarrow 0^+)
Anyone know where this comes from...
Imaginary numbers are a lot less mysterious than they sound. They are the result from trying to take the square root of a negative number. They are called “imaginary” because they don’t exist in the normal number system, normally you can’t take the square root of a negative number because the...
Steven Hawking writes in A Brief History of Time that time itself must sometimes have an imaginary component in order for Feynman's Sum-Over-Histories approach to work. Why, in a nutshell, is this so? Thanks in advance.
I have trouble understanding the concept of imaginary time. As I understand it (by reading other online sources), imaginary time was developed by Stephen Hawking and Hartle in their theory of quantum cosmology. Quantum cosmology is applying quantum mechanical principles to cosmology as a...
Hi, I am new here and have been lurking awhile. I have been reading Stephen Hawking Theory of Everything and it brought up imaginary time. ( I got this from the library and just saw online he did not endorse this book) It got me thinking. Time is based on Earth's observations to the solar...
This is from "Exploring Black Holes" by Taylor and Wheeler. It's a very good book but I struggle not with the math, but the explanations (sometimes)
On page B-13 is a frame called "Metric for the Rain Frame", which is a transformation of the Schwarzschild Metric from "bookkeeper coordinates"...
When I was reading the Landau's The Classical Theory of Fields, I found that when distance of four dimensional space is negative, the time's square should be a negative as well.Then time is an imaginary number. it's SPACELIKE. But what is imaginary time mean on earth?
They say that in that...
this is a concept which i don't believe has been properly described to me and i really don't understand well enough. can someone discribe it to me in relitivly simple terms.