Did the Universe Begin? IX: More about Imaginary Time

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!

 

About Aron Wall

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at the Institute for Advanced Study in Princeton. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my first postdoc at UC Santa Barbara.
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4 Responses to Did the Universe Begin? IX: More about Imaginary Time

  1. TY says:

    Aron,
    I guess a book titled “Did the Universe Begin” is becoming long overdue.

    As I have followed your ideas and explanations in these 9 epic blogs (and with more on the way) on this very theme, I am starting to feel that there are good scientific reasons to believe that the universe had a beginning in time, which requires God as the explanation (not a gap filler).

    In a Brief History of Time, Hawking argues that the no boundary proposal does away with a Creator because a universe with no boundary or edge, a universe that is self-contained, has no beginning or end. But how robust is this conclusion drawn from so insecure (or unreal?) a notion as imaginary time (in the mathematical sense of complex numbers). We have also seen other sleight-of-hand attempts by cosmologist like Lawrence Krauss to remove God from creation in arguing spontaneous creation out of “quantum vacuum”. And you have refuted that rather succinctly in previous blogs.

    I am with those who theists who believe that God is required regardless of whether there is a beginning. Since God is omnipotent what prevent Him from creating a universe with an infinite past? I don't think this generalisation does away with the Kalam argument favoured by St Craig and similar arguments (thermodynamic beginning). Conversely, in Bayesian langauge, they increase the prior probability of the role of a Creator.

  2. abdul aziz says:

    Interesting it reminds me of a kind of imaginary time mentioned in my holy book . meaning that :

    when the Hour arrives the guilty ones will be deluded thinking they remained dead but for an HOUR but to those to whom knowledge and faith is given they will say : You remained dead until the Creator His decree but you used not to know .

    This means that when people will die there will be no passage of time perhaps but stil the soul wil perceive of some kind of time in the case of the wicked ones they will be deluded .

    Peace

  3. Aron Wall says:

    Dear Abdul,

    Thanks for your comment. I should say that my post concerns time which is "imaginary" in the technical mathematical sense of a square root of a negative number, which is unrelated to the meaning of "imaginary" in the sense of delusional or nonexistant.

    Nevertheless, you raise an interesting theological question about what is the state of people in between death and the General Resurrection of all human beings at the Final Judgement. From the scientific viewpoint we know that time (as we know it) is part of the physical universe, so if we talk about something outside of the physical universe it is not clear to what extent our temporal concepts still apply to the situation.

    Christians believe that the souls of the righteous dead are alive to God, that they are in Christ and therefore live with him. The challenge is to relate these promises to the doctrine that God will raise everyone from the dead physically at the end of this age of history.

    Conversely, there is one place in the New Testament which suggests the wicked may be punished immediately following death, but this is in the context of a parable---one of the many fictional story told by Jesus to illustrate a point---and there are very few details given.

    The bulk of the New Testament teaching about the afterlife (even in the context of most of the passages cited above) concerns the more important issue of what happens during and after the Resurrection from the dead. There is only a tiny amount about the time in between. Correct me if I am wrong, but I believe the same is true in the Qur'an.

    Peace to you too,
    Aron

  4. abdul aziz says:

    Thank you Aaron for your reply .

    The only necessity in which I believe is the Creator . Stephen Hawking said because there is a Law called gravity there is no need for God . I would argue if there was no physical world as we know it there would be no need for gravity lol.

    I am aware of the use of imaginary numbers however it seems that some people use this mathematical concept to negate any beginning of the physical word .

    Therefor I referred to the verse in quran in its literal sense .(being deluded)

    whenever one dies he enters the Barzakh according to Islam .

    Linguistically, “Barzakh” means a veil, barrier or partition between two things. So in context of resurrection it is the state between death and resurrection .

    They ask you, [O Muhammad], about the Hour: when is its arrival? Say, "Its knowledge is only with my Lord. None will reveal its time except Him. It lays heavily upon the heavens and the earth. It will not come upon you except unexpectedly." They ask you as if you are familiar with it. Say, "Its knowledge is only with Allah , but most of the people do not know."

    Allah also says in chapter 40 verse 57:

    The creation of the heavens and the earth is definitely a greater thing
    than creation of mankind, however, most of mankind knows not. Qur'an 40:57

    Qur'an Chapter 33 Verse 63
    Men ask you of the Hour, Say the knowledge of it is with God only,
    What can convey to you that may be the Hour is near.

    Qur'an Chapter 29 Verse 53
    They bid you to hasten on the doom.
    And if a term had not been appointed, the doom would have definitely come on them.
    And it will come upon them suddenly when they perceive not.

    Qur'an Chapter 53 Verse 57-59
    The threatened hour is near. None beside God can disclose it. Are you surprised then at this statement?

    So will it happen soon or over a long period of time? Only God knows , to us it is important to have prepared ourselves before entering that state , because as soon as one enters the Barzakh one's test of life test is over being at a believer or disbeliever .

    So whoever follows Jesus son of Mary(puh) should keep follow the commandments and never ascribe partners unto God or to associate Him with creation .

    from scientific perspective however it is difficult to prove resurrection directly . No man has been resurrected but resurrection from Islamic perspective is compared to clouds being lead to dead land and reviving the earth and there you have life .

    Another view is from Allah His words .

    Does man not consider that We created him from a [mere] sperm-drop - then at once he is a clear adversary?
    And he presents for Us an example and forgets his [own] creation. He says, "Who will give life to bones while they are disintegrated?"
    Say, "He will give them life who produced them the first time; and He is, of all creation, Knowing."
    [It is] He who made for you from the green tree, fire, and then from it you ignite.
    Is not He Who created the heavens and the earth able to create the like of them? Yea! and He is the Creator (of all), the Knower.

    I hope this answers your question . peace be with you Aaron .

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