Is imaginary time a scientifically serious proposal?

[kodama]

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 string theory and LQG, if there is imaginary time?

 

[snatchingthepi]

I have no answers to your direct questions, but I will mention as an aside that there's exists an "old" notation for the 4-dimensional space-time in SR that uses a complex coordinate for time.

http://en.wikipedia.org/wiki/Special_relativity_(alternative_formulations)#Minkowski_spacetime

 

[atyy]

I'm not sure about Hawking's proposal, but there is imaginary time in quantum field theory. It is just a mathematical trick at an intermediate step of the calculation. At the end of the calculation, one rotates from imaginary time back into real time. So imaginary time is a serious idea, but it does not alter the fundamentals of quantum mechanics in any way.

http://www.einstein-online.info/spotlights/path_integrals has information about inmaginary time in quantum mechanics, including some remarks on its use in quantum gravity.

 

[kodama]

what about imaginary space?

 

[atyy]

http://arxiv.org/abs/1301.5859
http://arxiv.org/abs/1303.4752
http://arxiv.org/abs/1305.2207

 

[kodama]

um thanks. how about imaginary energy and imaginary momentum and imaginary particles? the wave function

 

[atyy]

Well, if you read the papers above, you'll find complex momenta, eg. http://arxiv.org/abs/1301.5859 Eq 2.

 

[rootone]

Imaginary energy?
That's like saying imaginary mass, it's not a mathematical framework, it's something definitely real and measurable.

 

[kodama]

Well, if you read the papers above, you'll find complex momenta, eg. http://arxiv.org/abs/1301.5859 Eq 2.

i'm confused , i thought you don't regard LQG/SF as a plausible candidate of QG?

 

[kodama]

Imaginary energy?
That's like saying imaginary mass, it's not a mathematical framework, it's something definitely real and measurable.

since energy is measured in joules in SI unites in terms of time and imaginary time can be meaningful, imaginary energy might be meaningful in some theory.

 

[atyy]

i'm confused , i thought you don't regard LQG/SF as a plausible candidate of QG?

I'm a frequentist, so I'm allowed to be incoherent.

 

[Qiao]

I always saw imaginary time the same way as the imaginary number, you always use it in your derivations up until the end where you just rotate back to real time where you so somehow get the physical answer you needed. Just like that you alway take the absolute value of a function to remove the imaginary part.

 

[ohwilleke]

Not precisely on point, but worth mentioning that at least one of the four parameters of the CKM and PMNS matrixes include imaginary parts, which is necessary to express CP violation which is equivalent to time symmetry violation.

 

[kodama]

Not precisely on point, but worth mentioning that at least one of the four parameters of the CKM and PMNS matrixes include imaginary parts, which is necessary to express CP violation which is equivalent to time symmetry violation.

so complex general relativity could be viable theory, perhaps our universe also has imaginary space or time

 

[ohwilleke]

An alternative to Einstein's equations that involves imaginary components related to time in its equations could certainly be viable. Of course, in general, any set of equations described by 2N real variables can be described by N complex variables, and there are times when the coordinate switch makes sense, because the set of complex numbers is self-contained under many operations under which the set of real numbers is not self-contained.

I could imagine, for example, a set of GR-like equations with two observers, at least one of whom is moving at close to the speed of light or is in an intense gravity well, being described in a coordinate system in which (1) the experience of time from the perspective of the observer "at rest" in deep space from from any other matter-energy would be asymptotic to a real numbered time coordinate, and (2) the experience of time from the perspective of an observer hurling at almost the speed of light towards a black hole event horizon that the second observer had almost reached would be asymptotic to an imaginary numbered time coordinate.

The notion would be that while GR is background independent and the rate at which time progresses is particular to each observer, rather than universal and constant for all observers, that the rate at which time progresses is also not infinitely variable in GR. There is a floor and a ceiling which could be represented by a single quadrant in the real-imaginary time coordinate system, which would reduce the amount of theoretically possible variation in the rate at which time progresses between two observers (i.e. relative time rate parameter space) by 75%. The angle of the complex number relative to the origin of a complex valued time coordinate would indicate the "average" rate at which time was passing for a particular observer since leaving the point of origin, and the magnitude of the vector corresponding to the complex valued time coordinate would correspond to the amount of time elapsed for that observer since leaving the point of origin.

To determine the time elapsed by a person from the perspective of another observer with a common origin, you would project the observed person's vector on the observer's vector, which should be equivalent to taking the dot product of the two.

This constraint on the available parameter space for the rate at which time passes for observers relative to each other might provide a means by which one could address the perennial and vexing problem of ghosts in various GR-variants and quantum gravity theories. It could be that the physical reason that ghosts are such a problem in gravity theories that lack such constraints on relative time progression rate between observers, is that the ghosts are attributable to non-physical points in the real-imaginary time dimension plane. This would be similar to, but not identical to, the impact of the causality constraint in making Causal Dynamical Triangle gravity theories workable mathematically.

Another of the corollaries of a complex valued time variable like that is that it would imply an equivalence between every possible path from the origin to that point in the real-imaginary time plane, despite the infinite number of paths to that point. This equivalence, in turn, would allow people do to calculation in GR that were based on much more easily observable end points, and much more easy to infer or determine origin points, without having the know an impossible to observe path, in a theory that in its usual formulation is completely path dependent, short cutting a huge amount of calculations that would by hypothesis cancel out in this two-dimensional time version of the theory. Indeed, a transformation from real valued time coordinate to complex valued time coordinates might be just the thing to turn intractable equations in a non-commutative algebra into much more tractable commutative algebra.

Of course, it is also entirely possible that none of that would work, or that the way that an imaginary time dimension would be defined and the problems that it would solve could be completely different. But, this little thought experiment demonstrates that it is at least possible to imagine the existence of a theory in which time was a complex variable, rather than a real variable could make sense in some GR variant.

It is a bit misleading, loaded and even obscurantist to say that the universe has "imaginary time" or "imaginary space" dimensions. That is a description that confounds more than it clarifies. Instead, I think, it could be more useful and lucid to say that an equation that allows for complex values for a parameter related to what we intuitively understand to be time can be a useful tool to understanding how our universe in general, and GR phenomena, in particular, work.

 

[kodama]

so why is it difficult to extract real physics from complex-valued ashketar variables?

 

[member 11137]

Well I cannot directly give an answer to your last question. But at some fundamental level: I think it's important to permanently make a difference between the path we must follow (the mathematics including imaginary complex numbers) and the target of our walk (the real world where we obviously live in). Complex numbers are a part of the mathematical apparatus and play an important role in physics (electromagnetism, waves functions, a.s.a...). Mathematics are giving insigths on how the real world works but are not systematically to be confused or identified with it (in opposition with what I believed and wrote somewhere else). With different words: we sometimes introduce new mathematical technics with the purpose to reach a better description of the reality. Sometimes: it works. But sometimes we loose ourself inside the technicity we have introduced and we are no more able to perceive what is real and what was just a tool.

 

[Ilja]

Imaginary time is a nice mathematical method to identify the lowest energy states. Simply compute the standard Hamilton equation in imaginary time for some arbitrary or guessed initial values, and the limit will be the ground state, whatever you start with. The terms which survive the longest time are the lowest energy states. That means, the states which are the most important ones in the low energy considerations. With this in mind, it is also reasonable to add a small imaginary time to path integrals and so on, as a regularization. It suppresses exactly what one has to suppress in an effective field theory -- extremely large energies.

 

[Jimster41]

To the point of these last few posts. I find the "meaning" of $i$ really hard to manage.

A)"it is mere mathematical artifact, pure invention, just a wrench" - I've heard this often, and I've seen the formalism, used it minimally, passed the tests on it. I get it's just a 2d plane. But the "just"... I don't know what that intends, or means, when wondering about its significance. It seems unlikely given how ubiquitous manipulations via the complex plane, and especially complex exponents are, that the artifact, or invention is not somehow more important than "just" suggests. Plus it seems a little contrary to the purpose of math to declare which mathematical terms, manipulations, outcomes, etc, have identifiable implications or connections to experience, and which don't. Isn't math supposed to be the compass when we can't tell, pointing to perspectives we don't yet have?

B)"it is a mathematical invention, but one that has meaning, meaning maybe we aren't totally clear on, or it has whole categories of meaning which are a wild and wooly. IOW it's wrench, we have invented because it fits a bolt we found. The bolt is very much a thing in the world. Which is why our wrench works. So yeah it's just a wrench that works good on those bolts, but what the heck Are thos bolts?

Currently, in my hopeless quest to get a grasp on wtf Euler's formula, and all the other stuff going on with $i$ is, just well enough to not immediately drop any equation I come across them in (because I cannot read symbols that have no connotation for me, regardless how well I memorize their definitions) - I am fixated for better or worse on the idea that (among other things maybe) the complex plane and it's ilk are a way of carrying around system descriptions where the system has a "regular" dynamic character, but it also has important attendant properties of periodicity, from simple sine waves to complicated structures of hierarchical repetition, maybe also stuff like bifurcation, or replication of representation from stands for a "one" to stands for a set of perhaps indistinguishable, but still multiple "ones", like discrete scale invariance.

So far that is feeling like a screwdriver.

@ohwilleke I found a lot to think about in you post above, like the idea of spacetime having an elasticty processing function. One that bounds temporal curvature, and maybe does other stuff. Confusing, tantalizing.

 

[ohwilleke]

"I find the "meaning" of i really hard to manage."

Here's another way to think about it. The decision to call one direction negative and the other positive is often a completely arbitrary choice of coordinates. "i" is used to eliminate the mathematical difficulties which are purely a product of that choice being arbitrary.

For example, suppose that you have some accounting equations. Conventionally, credits are positive numbers and debits are negative numbers. But, it is necessarily true that what are credits and debits for you, are debits and credits, respectively, for someone lending money to you. If you owe someone $2,000, then you have a $2,000 liability and they have a $2,000 asset.

If taking the square root of a negative bank balance were something so strange that it really deserved the label "imaginary", but taking the square root of a positive bank balance were an ordinary and unexceptional thing, then that would be crazy if it made any difference, because the same quantity is a real number for someone on one side of the transaction, and an imaginary number for someone on the other side of the transaction.

Similarly, in electromagnetism, the convention that electrons have negative charge, and that protons have positive charge, is inherently arbitrary.

While it isn't always true that imaginary numbers come up when the "negative" value that one must take the square root of has a negative rather than a positive coordinate for purely arbitrary reasons, it is often the case that this is true when complex analysis is appropriate.

Very frequently, when imaginary numbers come up, it is possible to transform what you are working on with a coordinate transformation (for example, flipping positive and negative when the values are purely arbitrary anyway), then do the math, and then do a coordinate transformation back. Using "i" is an elegant, yet subtle, way to do the same thing without changing coordinate conventions.

Put another way, going back to the electromagnetic equations, you could have a dual real numbered coordinate system. Charges could have units of P (for proton) or E (for electron), and you could keep separate track of P units and E units, and then relate them with the equation that for P>E that total charge equal P-E in units of P, while for P<E that total charge equals E-P in units of E, and that when P=E that total charge is zero. Voila, no imaginary numbers required, and you'd just keep track of two sets of units for the same thing. Imaginary numbers allow us to condense P units and E units into a single unit of charge, with a sign convention that allows us to suppress the P and E units from our paperwork at the cost of having to use "i" as a unit a little bit of the time to retain the structure that we had when we were using P units and E units.

"i" isn't so much a number as it is a unit notation in situations when there are multiple, orthogonal kinds of units involved and we want the notation we look at to be as unit-free as possible.