Using imaginary time to unify QM and GR

[g.lemaitre]

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?

We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should have. One is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does not have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a couple of numbers, one representing the size of a wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. When one actually tries to perform these sums, however, one runs into severe technical problems. The only way around these is the following peculiar prescription: one must add up the waves for particle histories that are not in the “real” time that you and I experience but take place in what is called imaginary time. ... To avoid the technical difficulties with Feynman’s sum over histories, one must use imaginary time. That is to say, for the purposes of the calculation one must measure time using imaginary numbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be Euclidean, after the ancient Greek Euclid, who founded the study of the geometry of two-dimensional surfaces. What we now call Euclidean space-time is very similar except that it has four dimensions instead of two. In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by ordinary, real values of the time coordinate, it is easy to tell the difference the time direction at all points lies within the light cone, and space directions lie outside. In any case, as far as everyday quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate answers about real space-time.

Further on he writes:

in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like. But according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of which is the more useful description.

 

[Mark M]

Take a look at the wikipedia page on imaginary time:

http://en.m.wikipedia.org/wiki/Imaginary_time#section_2

Essentially, it's related to a concept pushed by Hawking and Jim Hartle. That is, the 'wavefunction of the universe'. Whereas wavefunctions for particles represent the probability of finding a particle in a particular location or having taken a certain path, the wavefunction of the universe would encode the probability of certain ways the universe could evolve.

Gravitationally singularities disappear in imaginary time, which I'd have to say is why Hawking says it is crucial.

 

[sheaf]

Is using imaginary time a common practice?

Yes, definitely. In addition to the quantum cosmology applications of Euclidean time mentioned by Mark M, there is also the widespread use of instantons as contributions to path integral calculations, for example in Yang Mills theory.

 

[sbrothy]

This is what is referred to as "Wick rotation" isn't it?

 

[Mark M]

This is what is referred to as "Wick rotation" isn't it?

Imaginary time can be obtained from real time in quantum mechanics by applying a Wick rotation by [itex] \frac {\pi} {2} [/itex] through the complex plane so that imaginary time, [itex] \tau [/itex], is given by [itex] \tau = it [/itex].

 

[cosmik debris]

This is what is referred to as "Wick rotation" isn't it?

Yes, it gives spacetime a positive definite signature (i.e. makes it euclidean). Einstein used it originally in relativity. It is more used by particle physicists.