hasn't time been imaginary ever since 1905?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.
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 heumpje ...replace 1/kT --> it, in the expression exp (-H/kT)
So exp(-H/kt)--> exp(-iHt)...
In going from real time t to imaginary time τ via t→τ ≡ -it, the original temporal dimension becomes in effect just another spatial dimension, that is, imaginary time τ has the properties of a spatial coordinate. This is easily understood in the context of flat spaces. The square of the distance from the origin of a point with coordinates (x,y,z,t) in the usual 4-dimensional minkowski space is x^{2}+y^{2}+z^{2}-t^{2}. Taking t→τ ≡ -it changes this to x^{2}+y^{2}+z^{2}+τ^{2} which is the distance from the origin of a point with coordinates (x,y,z,τ) in a space with four spatial dimensions.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?
I don't really understand imaginary time myself, but what do you mean by "there is no 'curved imaginary' space time). I read the following: "Imaginary time was introduced to avoid singularities, or points at which the spacetime curvature becomes infinite, that occur in ordinary time. Imaginary time too would be curved by matter in the universe and therefore would meet the three spatial dimensions to form a closed surface like that of Earth. This curved surface would not have a beginning or end, or indeed any boundaries or edges." from this website: http://library.thinkquest.org/27930/time.htm?tqskip1=1&tqtime=0708. It sounds like they are saying that imaginary space-time is curved, or am I missing something?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...
Where can I get a bumper sticker with that on it?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.
In a sense, really, all negative numbers would be "imaginary" (although not technically so), right? What I mean is, when you think of math and numbers as "representing something", then "2 oranges" actually represents something. "-2 oranges" doesn't really represent anything other than the position that you SHOULD have 2 oranges and you don't. You can't actually hold "-2 oranges" in your hand. In this sense, negative numbers themselves were really just something we made up as a convenience because it helps us solve certain equations easier - and that's why I say they're really "imaginary" in a sense, even if they don't technically count as Imaginary numbers. I think this is why we get into all sorts of problems when we try to find the square root of -4, and then have to make up even more stuff (2i) that will help us get where we need to be.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).
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
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".
No, +2 would not be imaginary. You have to understqand that the "+" is only there to differentiate it because we also use a "-". Numbers are imaginary in that they are symbols, but a normal positive number is a symbol that actually represents something more directly correlated to reality than does a negative number. Without any negatives, you'd simply have numbers, plain and simple. The oranges is a good example so I won't repeat it.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.