Undivided Looking

In this post, I've put some more technical details about what the concept of imaginary time means, to help clarify the previous post about the Hartle-Hawking No Boundary Proposal.  If you don't want to have to understand equations, skip this.

First of all, a bit of remedial math.  There are a lot of functions which (even if they teach them to you in school as being functions of real numbers) actually make sense when extended to complex numbers of the form $z = x + iy$.  I already had to say something about complex numbers earlier in this series.  If you know how to add, subtract, multiply, and divide complex numbers, you can pretty easily make sense out of polynomial fractions like $f(z) = z^3 + z / (z^2 - 1)$, but you can also make sense out of things like sines and cosines and exponentials.  For example, if we take an exponential of an imaginary number we get

$$e^{iy} = \cos(y) + i \sin(y).$$

This formula allows you to turn all sines and cosines into exponentials, enormously simplifying trigonometry by making it so you don't have to memorize a bunch of weird trig identities.  So even though they call them complex numbers, they actually make your life simpler!

So when you see something in a scientific equation like $e^{ix}$, that looks like an exponential, but the power is imaginary, that's really something that's spinning around in the complex plane as you change $x$, without growing or shrinking in its absolute size.  It is a general rule that things which oscillate in the real direction correspond to things which exponentially grow and/or shrink in the imaginary direction, and vice versa.

This process of extending functions to the complex plane is called analytic continuation, and functions which can be so continued are called (wait for it!) analytic.  (Not all functions are analytic: those which suffer from abrupt changes, like the absolute value function $|x|$, are not.  $|x$ changes unpredictably at $x = 0$; if someone told you what it looks like for $x < 0$, and you tried to extrapolate it to $x > 0$ you'd guess wrong.

Now it turns out that there is a close mathematical connection between quantum mechanics and thermodynamics (a.k.a. statistical mechanics).  Quantum mechanics is all about how the phase of a wavefunction oscillates around as time passes.  The rate at which the phase spins around is proportional to the energy $H$ of the state, as told to us by Schrödinger's equation:

$$H \Psi = i \hbar (d/dt) \Psi.$$

If you solve this equation, you find that a state with definite energy $H = E$ spins around as time passes like $\Psi(t) = \Psi(0) e^{iEt/\hbar}$, where $\hbar$ is Planck's constant.

On the other hand, statistical mechanics is all about thermal equilibrium states, and the rule of thermal equilibrium is that the probability to be in a given state falls off exponentially with the energy.  The probability is proportional to $p = e^{-E/T}/Z$, where $T$ is the temperature, and $Z$ is an extra random thing called the "partition function'' you throw in to normalize the probabilities so they add up to 1. It turns out that states like these maximize the entropy given how much entropy they have.  If you squint these two exponentials they start looking quite similar to each other, if only you can accept the mystical truth that inverse temperature is like imaginary time:

$$1/2T = it,$$

where the factor of 2 comes from the fact that the probability is the absolute value squared of the wave function.

If you start with an initial condition where all states have equal probability, and "evolve'" for a finite quantity of "imaginary'" time, you end up with a thermal state ( after normalizing the total probabilities to be 1 at the end).  Better still, if you start with (almost any) state and evolve for an infinite amount of imaginary time, you end up with the "vacuum" state of lowest energy, all other states being exponentially damped by comparison to that one.

Well, this may seem like a bit of mumbo-jumbo, but with the help of that complex number math I mentioned above, you can actually put it on a fairly rigorous footing, for ordinary QM systems, and even for quantum field theories.  So of course, Hartle and Hawking had to be more bold than that, and try to apply this idea in the context of quantum gravity.

In quantum gravity (to the extent that we understand it), the dynamics are not governed by an ordinary Hamiltonian.  Instead they are governed by a Hamiltonian constraint:

$$H \Psi = 0,$$

also known as the Wheeler-DeWitt equation.  This equation seems to say that nothing changes with time, but it really means that the choice of time slice is arbitrary and has no coordinate-invariant meaning.

Now the Hartle-Hawking prescription is really just a clever way to calculate one particular state which (at the level of formally manipulating equations that we can't really make sense of) solves the Wheeler-DeWitt equation.

It tells us the wavefunction of the universe, expressing the "quantum amplitude" for any possible metric of space at one time to exist.  (The quantum amplitude is just a term for the complex number saying what the wavefunction is for a particular possibility to occur.  Take the absolute value squared and you get the probability.) Since there are many ways to slice spacetime into moments of time, all of them have to exist side-by-side in this wavefunction, late moments in time no less than early ones.  That's what it means to solve the Wheeler-DeWitt equation!

It's not the only solution to the Wheeler-DeWitt equation, but it's an especially nice one.  In some ways it is like a "vacuum" state of the theory, one especially nice state to which others may be compared.  (In other ways, it's more like a thermal state, due to the fact that there is only a finite amount of imaginary time evolution, before one reaches the end of imaginary time).

In order to calculate the Hartle-Hawking amplitude that a given geometry for 3 dimensional space (call it $\Sigma$) will appear ex nihilo (as it were), all you have to do is this:

1. Consider the space of all 4 dimensional curved spatial geometries whose only boundary is $\Sigma$,
2. For each geometry, integrate the total value of the Ricci scalar $R$ over the 4 dimensional geometry, call that the action $S$, and assign to that geometry the value $e^{-S}$.
3. Figure out how to integrate $e^{-S}$ over the infinite dimensional space of all possible 4 dimensional geometries.  This requires choosing a measure on this space of possibilities, which is quite tricky for infinite dimensional spaces,
4. Cleverly dispose of several different kinds of infinities which pop up, and
5. Consider all possible choices of $\Sigma$ and figure out how to normalize it so that the total probability adds to 1 (nobody knows how to do this properly either).

Good luck!

 

The last bit of evidence from physics which I'll discuss is the "no-boundary" proposal of Jim Hartle and Stephen Hawking (and some related ideas).  The Hartle-Hawking proposal was described in Hawking's well known pop book, A Brief History of Time.  This is an excellent pop description of Science, which also doubles as a somewhat dubious resource for the history of religious cosmology, as for example in this off-handed comment:

[The Ptolemaic Model of Astronomy] was adopted by the Christian church as the picture of the universe that was in accordance with Scripture, for it had the great advantage that it left lots of room outside the sphere of fixed stars for heaven and hell.$^{[citation\,needed!]}$

Carroll, after making some metaphysical comments about how outdated Aristotelian metaphysics is, and how the only things you really need in a physical model are mathematical consistency and fitting the data—this is Carroll's main point, well worthy of discussion, but not the subject of this post—goes on to comment on the Hartle-Hawking state in this way:

Can I build a model where the universe had a beginning but did not have a cause? The answer is yes. It’s been done. Thirty years ago, very famously, Stephen Hawking and Jim Hartle presented the no-boundary quantum cosmology model. The point about this model is not that it’s the right model, I don’t think that we’re anywhere near the right model yet. The point is that it’s completely self-contained. It is an entire history of the universe that does not rely on anything outside. It just is like that.

Temporarily setting aside Carroll's comment that he doesn't actually think this specific model is true—we'll see some possible reasons for this later—the first thing to clear up about this is that the Hartle-Hawking model doesn't actually have a beginning!  At least, it probably doesn't have a beginning, not in the traditional sense of the word.  To the extent that we can reliably extract predictions from it at all, one typically obtains an eternal universe, something like a de Sitter spacetime.  This is an eternal spacetime which contracts down to a minimum size and then expands: as we've already discussed in the context of the Aguirre-Gratton model.

This is because the Hartle-Hawking idea involves performing a "trick", which is often done in mathematical physics, although in this case the physical meaning is not entirely clear.  The trick is called Wick rotation, and involves going to imaginary values of the time parameter $t$.  The supposed "beginning of time" actually occurs at values of the time parameter that are imaginary!  If you only think about values of $t$ which are real, most calculations seem to indicate that with high probability you get a universe which is eternal in both directions.

Now why is the Hartle-Hawking model so revolutionary?  In order to make predictions in physics you need to specify two different things: (1) the "initial conditions" for how a particular system (or the universe) starts out at some moment of time, and (2) the "dynamics", i.e. the rule for how the universe changes as time passes.

Most of the time, we try to find beautiful theories concerning (2), but for (1) we often just have to look at the real world.  In cosmology, the effective initial conditions we see are fairly simple but have various features which haven't yet been explained.  What's interesting about the Hartle-Hawking proposal is that is a rather elegant proposal for (1), the actual initial state of a closed universe.

One reason that the Hartle-Hawking proposal is so elegant is that the rule for the initial condition is, in a certain sense, almost the exact same rule as the rule for the dynamics, except that it uses imaginary values of the time $t$ instead of real values.  Thus, in some sense the proposal, if true, unifies the description of (1) and (2).  However, the proposal is far from inevitable, since there is no particularly good reason (*) to think that this special state is the only allowed state of a closed universe in a theory of quantum gravity.  There are lots of others, and if God wanted to create the universe in one of those other states, so far as I can see nothing in that choice would be inconsistent with the dynamical Laws of Nature in (2).

(Hawking has a paragraph in his book asserting that the proposal leaves no room for a Creator, but I'll put my comments on that into a later post!)

In the context of a gravitational theory, imaginary time means that instead of thinking about metrics whose signature is $(-, +, +, +)$, as normal for special or general relativity, we think about "Euclidean" (or "Riemannian") signature metrics whose signature is $(+, +, +, +)$.  So we have a 4 dimensional curved space (no longer spacetime).

The assumption is that time has an imaginary "beginning", in the sense that it is finite when extended into the imaginary time direction.  However, because there is no notion of "past" or "future" when the signature of spacetime, it's arbitrary which point you call the "beginning".  What's more, unlike the case of the Big Bang singularity in real time, there's nothing which blows up to infinity or becomes unsmooth at any of the points.

All possible such metrics are considered, but they are weighted with a probability factor which is calculated using the imaginary time dynamics.  However, there are some rather hand-waving arguments that the most probable Euclidean spacetime looks like a uniform spherical geometry. The spherical geometry is approximately classical, but there are also quantum fluctuations around it.  When you convert it back to real time, a sphere looks like de Sitter space: hence the Hartle-Hawking state predicts that the universe should look have an initial condition that looks roughly like de Sitter space, plus some quantum fluctuations.

I say handwaving, because first of all nobody really knows how to do quantum gravity.  The Hartle-Hawking approach involves writing down what's called a functional integral over the space of all possible metrics for the imaginary-time goemetry.  There are an infinite-dimensional space of these metrics, and in this case nobody knows how to make sense of it.  Even if we did know how to make sense of it, nobody has actually proven that there isn't a classical geometry that isn't even more probable than the sphere.  Worst of all,  it appears that for some of the directions in this infinite dimensional space, the classical geometries are a minimum of the probability density rather than a maximum!  This gives rise to instabilities, which if interpreted naively give you a "probability" distribution which is unnormalizable, meaning that there's no way to get the probabilities to add up to 1.

So Hartle and Hawking do what's called formal calculations, which is when you take a bunch of equations that don't really make sense, manipulate them algebraically as if they did make sense, cross your fingers and hope for the best.  In theoretical physics, sometimes this works surprisingly well, and sometimes you fall flat on your face.

Unfortunately, it appears that the predictions of the Hartle-Hawking state, interpreted in this way, are also wrong when you use the laws of physics in the real universe!  The trouble is that there are two periods of time when the universe looks approximately like a tiny de Sitter space, (a) in the very early universe during inflation, and (b) at very late times, when the acceleration of the universe makes it look like a very big de Sitter space.  Unfortunately, the Hartle-Hawking state seems to predict that the odds the universe should begin in a big de Sitter space is about $10^{120}$ times greater than the odds that it begins in the little one.  That's a shame because if it began in the little one, you would plausibly get a history of the universe which looks roughly like our own.  Whereas the big one is rather boring: since it has maximum generalized entropy, nothing interesting happens (except for thermal fluctuations).  St. Don Page has a nice article explaining this problem, and suggesting some possible solutions which even he believes are implausible.

Alex Vilenkin has suggested a different "tunnelling" proposal, in which the universe quantum fluctuates out of "nothing" in real time rather than imaginary time.  This proposal doesn't actually explain how to get rid of the initial singularity, and requires at least as much handwaving as the Hartle-Hawking proposal, but it has the advantage that it favors a small de Sitter space over a big one.  From the perspective of agreeing with observation, this proposal seems better.  And it has an actual beginning in real time, something which (despite all the press to the contrary) isn't true for Hartle-Hawking.

(*) There is however at least one bad reason to think this, based on a naive interpretation of the putative "Holographic Principle" of quantum gravity, in which the information in the universe is stored on the boundary.  A closed universe has no boundary, and therefore one might think it has no information, meaning that it has only one allowed state!  (The argument here is similar to the one saying the energy is zero.)  At one time I took this idea seriously, but I now believe that such a strong version of the Holographic Principle has to be wrong.   There are lots of other contexts where this "naive" version of the Holographic Principle gets the wrong answer for the information content of regions, and actual calculations of the information content of de Sitter-like spacetimes give a nonzero answer.  So I'm pretty sure this isn't actually true.