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maximus
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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.
Originally posted by sheldon
Imaginary numbers can be used to help explain tunnelling, a quantum mechanical process in which, for instance, a particle can spontaneously pass through a barrier. In trying to unify general relativity with quantum mechanics, physicists used a related idea in which they would measure time with imaginary numbers instead of real numbers. By using this so-called imaginary time, physicists Stephen Hawking and Jim Hartle showed that the universe could have been born without a singularity.
Originally posted by heumpje ...replace 1/kT --> it, in the expression exp (-H/kT)
So exp(-H/kt)--> exp(-iHt)...
Originally posted by Imagine heumpje and jeff, I didn't fully understand your explanations but it seams, for me, that your usage of "imaginary time" aims only to simplify problem resolution, am I wrong?
Originally posted by heumpje
Hi there,
mporatnt to note is that this unrelated to relativity, in the sense that imaginary time is completely independent of normal space-time coordinates. (there is no "curved imaginary" space time)
Hope this helps...
Originally posted by jeff
It's the other way round: Using a more explicit path integral notation, although we can think heuristically of the path integral Z = ∫Dφexp[i∫d^{d}xL(φ,∂φ)] as converging due to the cancellation among the oscillatory phase factors for different paths, for greater mathematical rigour, we consider instead the euclideanized version Z_{E} = ∫Dφexp[-∫d^{d}x_{E}L_{E}(φ,∂φ)] obtained from Z by analytic continuation via a wick rotation t→t_{E} ≡ -it in which t_{E} is called imaginary time due to the factor of i, and d^{d}x_{E} is the measure on ordinary d-dimensional euclidean space (as opposed to the original minkowskian space). This transformation is well-defined since Z is analytic and analytic continuations of analytic functions are unique.
Originally posted by sheldon
just incase you don't know what imaginary numbers are.If you start with any “real” number and multiply it by itself, you get a positive number. For instance, 2 times 2 equals 4 but so does -2 times -2. That means the square root of 4 equals both 2 and -2. But what would the square root of -4 be? Mathematicians invented imaginary numbers to answer this question, defining the number i to equal the square root of -1 (making the square root of -4 equal to 2i).
Originally posted by vedder
But what if the "-2" were meant to represent a negative charge tiberius?
In that case wouldn't +2 be an imaginary number as well? Since we could logically say 'that a "positive" charge, is really just an opposite charge of a given strength.'Again, I think if you think conceptually about the fact that numbers are just symbols and ask youself what you're really representing with them, it seems to me that a "negative" charge, is really just an opposite charge of a given strength. We could have just as easily chosen to symbolize - as + and + as -. Just as we could have just as easily have decided that we were going to think of south as being "up" and always drawn our maps with australia at the top and north america at the bottom. It would be just as "correct".
Originally posted by vedder
In that case wouldn't +2 be an imaginary number as well? Since we could logically say 'that a "positive" charge, is really just an opposite charge of a given strength.'
I mean... if we were to call all negative numbers imaginary it would seem a bit arbitrary to me considering where i quoted you.
The oranges analogy is logically self sufficient. But i feel it to be inappropriate when considering things such as... lets say proton(1), neutron(0), and electron(-1), especially if we consider these things to be real. Is the electron imaginary? I think if one were to call all negative numbers imaginary, one would be forced to call all numbers imaginary. And "in a sense" i suppose they are. But, i'll leave that discussion to the philosophy forum i think.
Originally posted by Imagine
Bonjour,
Maximus, I like your well-defined "Location"
Ivan, your edit is fully justified, time was imaginary timinfinitely long before 1905.
Sheldon, I would appreciate to know more about "... showed that the universe could ..." Do you have Hawking or Hartle writing which shows the pertinence of imaginary time benefit in physics?
heumpje and jeff, I didn't fully understand your explanations but it seams, for me, that your usage of "imaginary time" aims only to simplify problem resolution, am I wrong?
Is someone, somewhere, sometime, already considered integrating the imaginary part (i) into the time metric instead of considering them apart?
Nice reading, merci!Originally posted by sheldon
http://www.psyclops.com/hawking/resources/origin_univ.html
Originally posted by Imagine
Nice reading, merci!
I liked the history resume. I am not a fan of maths but a fan of physics. Is it possible to find some writings about this Hawking "exercise" of integrating imaginary time into physic expressions? Somethings like negative energy, imaginary mass, imaginary particule/anti-particule, ...