Stephen Hawking's Imaginary time

[TimBowe]

To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. One can think of ordinary real numbers such as 1, 2, -3.5, and so on as corresponding to positions on a line stretching from left to right: zero in the middle, positive real numbers on the right, and negative real numbers on the left.
Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don?t need a physical realization; one can?t have an imaginary number of oranges or an imaginary credit card bill.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?

Stephen Hawking
The Universe in a Nutshell


So is imaginary time the true physical quantity?

[TimBowe]

http://www.universe-review.ca/I15-54-NB.jpg

[Peeter]

This is really a reference to special relativistic distance, where the distance between a pair of "event"s (something with a time and place), can be expressed with imaginary time. Say for example, that you wake up a position X and time T (X = 0 say for your side of the bed), and T=7:30AM. Your wife, who is less lazy, gets up T' =6:00AM at X' = X+1.5 (where 1.5 meters is the width of your bed).

You can measure the time between these wake up events and you can measure the space between them, but in the mathematics of special relativity, the difference between the events is a composite difference between the two, that has a particular form.

The spatial distance, squaring (aka Pythagoras) to allow for three dimensions is:


(\text{spatial separation})^2 = X^2 - {X'}^2 = 0^2 - (1.5)^2 = -(1.5)^2


The time difference (squared) is


(\text{time separation})^2 = T^2 - {T'}^2 = 0^2 - (1.5 \text{hrs})^2 = -(1.5 \text{hrs})^2


The relativistic event separation, say s, (squared) is then


s^2 = (\text{spatial separation})^2 - c^2 (\text{time separation})^2 = (\text{how far distance that light travels in 1.5 hrs})^2 - (1.5 meters)^2


where c is a constant that adjusts the units so that the whole thing is measured as a spatial difference (it is also the speed of light).

You may ask why the time part is subtracted here and not added, but that's a question of a much bigger scope.

What the imaginary buys you is the mathematical nicety of working with all positive quantities, so the length of your wake up event can be written in two equivalent ways:


X^2 - c^2 T^2 = X^2 + (i c T)^2


With this the separation of two events becomes


s^2 = (X^2 - {X'}^2) + ((i c T)^2 - (i c T')^2)


Now time and space can be handled in a more uniform fashion.

[TimBowe]

Antimatter is a real physical entity that is conveyed by a formalism that uses imaginary numbers.

http://i889.photobucket.com/albums/ac96/TimBowe/image002.gif?t=1257635462

[Phrak]

Nice quotes of Hawking and all, but it's all for a popular audience. What else does he have to say about it?

The first questions one might want to ask are: Is this imaginary time a periodic dimension?, and if not, how it is compatible with the apparently observation of one dimension of time?

On top of that, the metric of spacetime would have an interesting form.

ds2 = dX2 - t2 + t'2,

where t' is imaginary time. How does this differ from a forth spatial dimension?

[TimBowe]

Nice quotes of Hawking and all, but it's all for a popular audience. What else does he have to say about it?

Well, what he has to say is that we don't know whether our physical theories should be expected to model reality or merely predict measurements.

We cannot ask if a model corresponds to reality, because we have no independent test of what reality is. All we can ask is whether the predictions of the model are confirmed by observation. Models of quantum theory use imaginary numbers, and imaginary time in a fundamental way. These models are confirmed by many observations.

Stephen Hawking

[Phrak]

This is more of the discourse for popular consumption. Nice potatoes, where's the meat?

[TimBowe]

Hawking thinks the universe may be finite and unbounded in space and time. Space-time forms a closed but unbounded surface. In the very early universe space was very compressed, in the Hartle-Hawking model it is a compact (closed) four-dimensional Euclidean space analogous to the surface of a sphere, whose boundary is a single three-dimensional space. Space-time forms a closed surface without an edge like the surface of the earth but in two more dimensions, space-time is finite but unbounded.

[TimBowe]

The Hartle and Hawking model states that space and time form the four-dimensional surface of a five-dimensional sphere.

In the Hartle-Hawking quantum proposal, the quantum state is the only boundary, probabilities are calculated from this state, only. Here, the three-dimensional compact boundary is like an absence of boundary, and the usual notion of time, and related notion of “becoming” disappears. In the four-dimensional internal manifold which fits on the boundary, time is an imaginary variable, and on equal footing with spatial dimensions. Their proposal is time-less, and the theory avoids initial singularity through a breaking of the notion of time. “Time ceases to be will be well defined in the very early universe just as the direction ‘north’ ceases to be well defined at the North Pole of the Earth….The quantity that we measure as time had a beginning, but that does not mean spacetime has an edge, just as the surface of Earth does not have an edge at the North Pole” The role of time orientation also disappears somehow, since the total wave function of the universe can be shown to be “time-symmetric” in this quantum cosmology.

[TimBowe]

The No Boundary proposal is confirmed by observations.

The "No Boundary" Proposal
As we trace the universe back in time to the singularity we not only find our laws of physics breaking down, but we are also left with the apparently unanswerable question of "what happened before the Big Bang?"
http://www.ipod.org.uk/reality/reality_hartle_hawking1.gif
In 1981, Stephen Hawking and James Hartle came up with an imaginative proposal which promised to avoid the singularity at the origin of the universe, and also gave a answer to the question of "what happened before the Big Bang?". But before we can consider the theory, we need to introduce a couple of concepts.
Firstly, we need to introduce the idea of a metric, which is a way of defining distance. In our (x, y, z) three-dimensional space, the formula for distance is provided by Pythagoras's theorem:
http://www.ipod.org.uk/reality/reality_pythagoras.gif
When we extend this notion to 4-dimensional spacetime, it might be imagined that the time axis is treated the same way (creating the Euclidean or Riemannian metric):http://www.ipod.org.uk/reality/reality_riemannian.gif
However, Einstein's theory of special relativity says that the clock which travels the furthest actually shows the smallest time measurement, not the largest. So we need to use the Lorentzian metric (the "time" element becomes negative):
http://www.ipod.org.uk/reality/reality_lorentzian.gif
Hartle and Hawking's proposal was to employ a mathematical transformation called a Wick rotation to modify the time axis to avoid the singularity. In the Wick rotation, the time axis is multiplied by the imaginary number i (the square root of minus one), in which case the Lorentzian metric is converted to a Euclidean metric:
http://www.ipod.org.uk/reality/reality_wick_rotation.gif
As a result of the Wick rotation, the time axis is converted to a complex number. The time axis is rotated 90° anticlockwise from the original time axis to become the imaginary time axis:
http://www.ipod.org.uk/reality/reality_hartle_hawking2.gif
The following diagram shows the resultant graph rotated by a further 90° (but this time clockwise) so that the imaginary time axis now points in the vertical direction - taking the place of the old time axis:
http://www.ipod.org.uk/reality/reality_hartle_hawking3.gif
It is as though when we travel back in time we find time itself curving round so that spacetime forms a smooth surface, instead of coming to a point singularity:
http://www.ipod.org.uk/reality/reality_hartle_hawking4.gif
So in the No Boundary proposal, there is no time before the Big Bang: time itself began with the Big Bang. Asking what came before the Big Bang is - in Hawking's words - like asking what lies south of the South Pole. The answer is nothing. And the question "what happened before the Big Bang?" is meaningless.

[TimBowe]

Is that a little more meat?

[atyy]

"Euclidean quantum gravity (with metric signature (++++)) gives different and (in d=4) wrong results
(=> don't do path integral Hawking's way!)"

On Loll's slide which she starts talking about at 26:55 of http://pirsa.org/09110045/

[marcus]

"Euclidean quantum gravity (with metric signature (++++)) gives different and (in d=4) wrong results
(=> don't do path integral Hawking's way!)"

On Loll's slide which she starts talking about at 26:55 of http://pirsa.org/09110045/

I agree. As far as I know, professionals have mostly avoided Hawking style quantum cosmology for around 10 years.

It was big in the 1980s. If anyone wants to dig back in the Spires database, I can get the keywords to use.

After 2000 or so, very few papers written, and those that were written did not get cited much. Hawking kind of dropped out, got ignored (except as beloved and admired public figure). The professional research community concerned with quantum cosmology moved on into other approaches, quite different from Hawkings 1980s ideas.

You are referring to Loll's slide number 7, I think, where she shows some of the things that go wrong with "Euclidean quantum gravity" which was Hawking's pet approach. There is more, but that is a good concise thing to point to. No need to beat an ex-horse.

We should give Tim Bowe some links to the current quantum cosmology literature so he can find out where the field has gone after 1990 or so.

[marcus]

Tim, you might want to take a look at Hawking's professional output in cosmology---to compare with the celebrity pop-sci.
http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+DK+COSMOLOGICAL+MODEL+AND+A+HAWKIN G&FORMAT=www&SEQUENCE=citecount%28d%29

This is whatever Spires database has with the keyword "cosmological model" and the author Hawking.
Almost all is from the 1980s. He has several titles that say Quantum Cosmology. and another (cited 446 times!) that says Quantum State of the Universe.

Here's a sample from the list of 26.

2) The Quantum State of the Universe. (446 cites)
S.W. Hawking, (Cambridge U.) . PRINT-84-0117 (CAMBRIDGE), Nov 1983. 28pp.
Published in Nucl.Phys.B239:257,1984.

16) Quantum Cosmology.
S.W. Hawking, (Cambridge U.) . Print-87-0166 (CAMBRIDGE), Dec 1986. 31pp.

18) Quantum Cosmology - Beyond Minisuperspace.
J. Halliwell, S. Hawking, (Cambridge U.) . 1985.
In *Rome 1985, Proceedings, General Relativity, Pt. A*, 65-83.

Many of the papers have several hundred citations (references to the work by other scholars.)

I see two out of the 26 that appeared after 2000. Each of the recent papers has on the order of 10 citations. Despite the author's celebrity, his work in the last ten years has not achieved much notice by the professional research community.
That is just a clue--there is more to say about this. And there are reasons that Hawking's cosmology ideas have tended to be neglected since the 1990s.

For example, the pathology that Renate Loll pointed out was discovered in the early 1990s by Jan Ambjorn and others, who were trying to develop Hawking's ideas about quantum gravity. They tried repeatedly to compute the universe Hawking style and were frustrated. Nothing they tried worked. In 1998 Ambjorn gave up on Euclidean quantum gravity and the Euclidean path integral, favored by Hawking. He teamed up with Loll and they started a new approach, which has been remarkably successful.

There is a lot to catch up on!

I have a link to an article by Renate Loll in my signature at the bottom of this post, to an article that appeared in the Scientific American. If you want a more technical introduction please ask!

[marcus]

What approaches to quantum cosmology have come to replace Hawking's, over the past 20 years?

Certainly that of Ambjorn and Loll is not the only one! I just mentioned it as an example following up on what Atyy said. But theirs is growing in recognition and an increasing number of researchers have joined in the effort, especially after their quantum universe model produced a classical one similar to ours in the largescale average---a result reported in 2008.

But there is also Steinhardt and Turok's approach---which avoids a need for inflation.

And there are various inflationary scenarios, notably by Alex Vilenkin and by Andrei Linde.

Aside from Ambjorn and Loll's all the approaches I can think of have something physical and describable happening before the big bang. All the approaches being actively investigated.
Particularly after 2005, experts in cosmology would typically not be inclined to say that "before the big bang" is a meaningless phrase analogous to "north of the north pole".
That is a dated aphorism. In one of his talks at Cambridge, Roger Penrose pinpointed when it went out of fashion. According to him the "north of the north pole" idea was dropped in 2005, sometime after Stephen Hawking used it on BBC television.

I would say much earlier than that, except in the pop-sci literature.

I didn't mention Loop cosmology yet. But it's bedtime here and that has to wait until tomorrow.

[TimBowe]

http://archive.sciencewatch.com/interviews/stephen_hawking2.gif

[TimBowe]

Stephen Hawking's quantum cosmology uses "imaginary time" as more fundamental than "real time".

The principle of causality only makes sense in real time where there are light cones at each spacetime point event. It is violated in imaginary time. The missing mass enigma may be a clue that real time is only the top of an iceberg. Most of the universe may still exist in imaginary time. Mathematically spacetime is described by a metric tensor which has a topological property called the "signature". The light cone only exists in real time where the signature is -+++. There is no light cone in imaginary time where the signature is ++++. The phase transition from imaginary to real time which starts the Big Bang is a topological transition. Is it controllable from the far future of real time in a self-consistent loop?

[Phrak]

Is that a little more meat?

Thanks, that's better. What causes a Wick rotation?

[TimBowe]

In order to get a better-behaved path integral, one has to do a Wick rotation to Euclidean space by introducing the imaginary time coordinate T = it. In other words you can rotate into the complex plane by introducing an imaginary time coordinate. But what is the physical meaning of this operation?

While the deep inner meaning of this trick is mysterious, it can be justified in a wide variety of contexts using the "Osterwalder-Schrader theorem".

The Osterwalder-Schrader reconstruction theorem says that it's possible to take all your calculations in imaginary time, do them there, and then bring the answers back to real time in a unique way.

[Phrak]

In order to get a better-behaved path integral, one has to do a Wick rotation to Euclidean space by introducing the imaginary time coordinate T = it. In other words you can rotate into the complex plane by introducing an imaginary time coordinate. But what is the physical meaning of this operation?

What you are descibing is a calculational tool. I think Hawking was talking about a physical change; somehow time, at some earlier locations on the spacetime manifold, was space-like, and now, it is timelike; the metric underwent a phase change in some manner. Is there any speculation on the cause of this, or did I misunderstand?

[TimBowe]

The dimension we call time becomes "fuzzy" and turns into a fourth spatial dimension.

The tiny compressed space-time with which the universe began (less than 10~33 cm radius) James Hartle and Stephen Hawking have produced, time-like world lines get bent into space-like directions, and even if each did have a first moment there would be no unique such.

[TimBowe]

Quantum physics introduces the new feature that the separate identities of space and time can be "smeared" or "blurred" on an ultramicroscopic scale.

http://www.dhushara.com/book/quantcos/alivtime/strt.jpg