Understanding Imaginary Time: A Simplified Explanation

[maximus]

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.

 

[Tail]

It's a pity there are no replies. I have exactly the same problem - I do not understand imaginary time...

 

[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.

 

[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).

 

[Ivan Seeking]

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.

hasn't time been imaginary ever since 1905?

Edit: perhaps that was a little vague. I mean, hasn't time been considered as having imaginary coordinates ever since special relativity was first published?

 

[heumpje]

Hi there,

I haven't been online for awhile but now that I am, obviously I will reply to such a profound question..;)

Imaginary time is just what it says: "imaginary"
The term comes from quantum mechanics/field theory and is often used when people are calculating transition probabilities, i.e. Path integrals.
In principle a number of definitions exist and I don't know the context you are working with, but I "use" imaginary time in the following case:

replace 1/kT --> it, in the expression exp (-H/kT)
So exp(-H/kt)--> exp(-iHt)

Using this, the partition function Z can be rewritten as:

Z=Tr(exp[-H/kT])=Tr(exp[-iHt])

Tr is the trace and is the sum of all diagonal elements:

Tr(A)=sum_i <x_i| A|x₀>

By introducing a imaginary time coordinate (here a "minute" of imaginary time corresponds with a chnage in inverse temperature) we can now write the partition function as a path integral. Imporatnt 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...

 

[einsteinian77]

It was only after Minkowski looked at special relativity did the imaginary time coordinate arose, i think.

 

[jeff]

Originally posted by heumpje ...replace 1/kT --> it, in the expression exp (-H/kT)
So exp(-H/kt)--> exp(-iHt)...

It's the other way round: Using a more explicit path integral notation, although we can think heuristically of the path integral Z = &int;D&phi;exp[i&int;d^dxL(&phi;,&part;&phi;)] 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 = &int;D&phi;exp[-&int;d^dx_EL_E(&phi;,&part;&phi;)] obtained from Z by analytic continuation via a wick rotation t&rarr;t_E &equiv; -it in which t_E is called imaginary time due to the factor of i, and d^dx_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.

 

[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?

 

[jeff]

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?

In going from real time t to imaginary time &tau; via t&rarr;&tau; &equiv; -it, the original temporal dimension becomes in effect just another spatial dimension, that is, imaginary time &tau; 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 x²+y²+z²-t². Taking t&rarr;&tau; &equiv; -it changes this to x²+y²+z²+&tau;² which is the distance from the origin of a point with coordinates (x,y,z,&tau;) in a space with four spatial dimensions.

It's main use is the one I alluded to in my response to heumpje's post.

 

[cityhunter]

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...

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?

 

[Tiberius]

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 = &int;D&phi;exp[i&int;d^dxL(&phi;,&part;&phi;)] 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 = &int;D&phi;exp[-&int;d^dx_EL_E(&phi;,&part;&phi;)] obtained from Z by analytic continuation via a wick rotation t&rarr;t_E &equiv; -it in which t_E is called imaginary time due to the factor of i, and d^dx_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.

Where can I get a bumper sticker with that on it?

 

[Tiberius]

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).

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.

 

[vedder]

But what if the "-2" were meant to represent a negative charge tiberius?

Pages 58-63 of Universe In A Nutshell has a real easy to understand explanation of this subject. It might give one a good base to then go on and contemplate jeff's excellent expose'.

 

[Tiberius]

Originally posted by vedder
But what if the "-2" were meant to represent a negative charge tiberius?

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".

So you're still not really representing any reality of something being "negative". For example, we could have just as easily called a charge of +1 a "Dolomar charge of 1" (just to make up a word). Then we could have called a charge of -1 a "Trinamar charge of 1". Then we'd have a law that states that Dolomar and Trinamar charges attract while Dolomar charges repel one another, as do Trinamar charges. It would all work just as well to describe what's happening, all without the concept of negative or positive. Here again, we use the concept of "negative" because its more convenient and less cumbersome, but that still doesn't mean negative numbers actually represent something directly.

This isn't important to mathematics and physics, but it is important when you're trying to philosophically conceptualize what is ACTUALLY being represented by the things we talk about in math and physics.

 

[vedder]

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".

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... let's 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.

 

[Imagine]

Bonjour,

Irrelevantly where you place positive real (up-north, down-south, down-north or up-south), negative is to positive as per squared imaginary is to squared real, on the same linear axis but at the opposite. BUT where is the imaginary relatively to real (east, west?). I remember to draw complex number onto orthogonal chart. Complex numbers have also conjugates which are mathematically usefull to find "real" length.

Mathematicians found answers to some mathematical interrogations as per "root of negative real number" or Lorentz transformation. Physicists applied the Lorentz transformation to respect relativity constraint (speed limit of event propagation). Engineers applied imaginary and complex numbers and functions to solve oscillatory systems (don't break it, that's oscillate!).

As physicist, I didn't find satisfactory explanation except that, in our world, physical observables are always squared things but, frequently, we are resolving in "mono" things, here interveins imaginary stuff, borrowed to maths to help us.

 

[Tiberius]

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... let's 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.

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.

Again, in the case of protons, neutrons, and electrons, you could say...

neutron: no charge
proton: 1 Dolomar charge
electron: 1 Trinamar charge

That might be more cumbersome but conceptually it's more representative of the reality of the situation. Electrons are real, as is their charge. But we CHOOSE to use the concepts of negative numbers, and we CHOOSE to apply them to electrons. All that is real in this situation is that there is a particular sort of force that can be stronger or weaker, and that interacts in certain ways with the forces of things like protons. We don't HAVE to call electron charges "negative". In fact, we COULD call electron charges +1 and Protons -1 if we wanted as well. It would be just as workable and "correct".

But my point is, because numbers can represent real things, and negative numbers actually don't represent OBJECTS, but higher concepts - THIS is why you get into problems when you're working out various equations involving negatives, and then you back out and think, wait a minute - "If I were to apply this to reality it would be nonsense!" So little contraptions have to be made up so that negative numbers will still function properly. I'm certain that all of mathematics could be reworked, and still function in every way, without the use of negative numbers. While conceptually and philosophically more closely symbolic of reality, it would be a LOT more cumbersome and difficult - and that's why you should keep us philosophers out of the math lab!

 

[vedder]

Naaaa, we need philosophers in the math lab to keep the mathematicians honest.

 

[Royce]

I agree, Tiberius, but pure math has no basis in reality. It doesn't matter if we count positive charges or negative charges, we still count with real positive numbers. We cannot count (-) anything.
If I count 100 electrons each with a - charge of 1 I end up this 100 negative charges or a minus charge of 100. I don't have in my hand
-100 electrons.

 

[sheldon]

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?

http://www.psyclops.com/hawking/resources/origin_univ.html

 

[Imagine]

Originally posted by sheldon
http://www.psyclops.com/hawking/resources/origin_univ.html

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 physics expressions? Somethings like negative energy, imaginary mass, imaginary particule/anti-particule, ...

 

[sheldon]

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 physics expressions? Somethings like negative energy, imaginary mass, imaginary particule/anti-particule, ...

here is some more food.

http://www.motherservice.org/Essays/Sri Aurobindo & Hyperspace.htm