Previously in series: Relative Configuration Space

Warning: The central idea in today's post is taken seriously by serious physicists; but it is not experimentally proven and is not taught as standard physics. Today's post draws heavily on the work of the physicist Julian Barbour, and contains diagrams stolen and/or modified from his book "The End of Time". However, some of the arguments here are of my own devising, and Barbour might(?) not agree with them.

I shall begin by asking a incredibly deep question:

What time is it?

If you have the excellent habit of giving obvious answers to obvious questions, you will answer, "It is now 7:30pm [or whatever]."

How do you know?

"I know because I looked at the clock on my computer monitor."

Well, suppose I hacked into your computer and changed the clock. Would it then be a different time?

"No," you reply.

How do you know?

"Because I once used the 'Set Date and Time' facility on my computer to try and make it be the 22nd century, but it didn't work."

Ah. And how do you know that it didn't work?

"Because," you say, "I looked outside, and the buildings were still made of brick and wood and steel, rather than having been replaced by the gleaming crystal of diamondoid nanotechnological constructions; and gasoline was still only $4/gallon."

You have... interesting... expectations for the 22nd century; but let's not go into that. Suppose I replaced the buildings outside your home with confections of crystal, and raised the price of gas; then would it be 100 years later?

"No," you say, "I could look up at the night sky, and see the planets in roughly the same position as yesterday's night; with a powerful telescope I could measure the positions of the stars as they very slowly drift, relative to the Sun, and observe the rotation of distant galaxies. In these ways I would know exactly how much time had passed, no matter what you did here on Earth."

Ah. And suppose I snapped my fingers and caused all the stars and galaxies to move into the appropriate positions for 2108?

"You'd be arrested for violating the laws of physics."

But suppose I did it anyway.

"Then, still, 100 years would not have passed."

How would you know they had not passed?

"Because I would remember that, one night before, it had still been 2008. Though, realistically speaking, I would think it more likely that it was my memory at fault, not the galaxies."

Now suppose I snapped my fingers, and caused all the atoms in the universe to move into positions that would be appropriate for (one probable quantum branch) of 2108. Even the atoms in your brain.

Think carefully before you say, "It would still really be 2008." For does this belief of yours, have any observable consequences left? Or is it an epiphenomenon of your model of physics? Where is stored the fact that it is 'still 2008'? Can I snap my fingers one last time, and alter this last variable, and cause it to really be 2108?

Is it possible that Cthulhu could snap Its tentacles, and cause time for the whole universe to be suspended for exactly 10 million years, and then resume? How would anyone ever detect what had just happened?

A global suspension of time may seem imaginable, in the same way that it seems imaginable that you could "move all the matter in the whole universe ten meters to the left". To visualize the universe moving ten meters to the left, you imagine a little swirling ball of galaxies, and then it jerks leftward. Similarly, to imagine time stopping, you visualize a swirling ball of galaxies, and then it stops swirling, and hangs motionless for a while, and then starts up again.

But the sensation of passing time, in your visualization, is provided by your own mind's eye outside the system. You go on thinking, your brain's neurons firing, while, in your imagination, the swirling ball of galaxies stays motionless.

When you imagine the universe moving ten meters to the left, you are imagining motion relative to your mind's eye outside the universe. In the same way, when you imagine time stopping, you are imagining a motionless universe, frozen relative to a still-moving clock hidden outside: your own mind, counting the seconds of the freeze.

But what would it mean for 10 million "years" to pass, if motion everywhere had been suspended?

Does it make sense to say that the global rate of motion could slow down, or speed up, over the whole universe at once—so that all the particles arrive at the same final configuration, in twice as much time, or half as much time? You couldn't measure it with any clock, because the ticking of the clock would slow down too.

Do not say, "I could not detect it; therefore, who knows, it might happen every day."

Say rather, "I could not detect it, nor could anyone detect it even in principle, nor would any physical relation be affected except this one thing called 'the global rate of motion'. Therefore, I wonder what the phrase 'global rate of motion' really means."

All of that was a line of argument of Julian Barbour's, more or less, Let us pause here, and consider a second line of argument, this one my own. That is, I don't think it was in Barbour's The End of Time. (If I recall correctly, I reasoned thus even before I read Barbour, while I was coming up with my unpublished general decision theory of Newcomblike problems. Of course that does not mean the argument is novel; I have no idea whether it is novel. But if my argument is wrong, I do not want it blamed on an innocent bystander.) So:

"The future changes as we stand here, else we are the game pieces of the gods, not their heirs, as we have been promised."

—Raistlin Majere

A fine sentiment; but what does it mean to change the future?

Suppose I have a lamp, with an old-style compact fluorescent bulb that takes a few seconds to warm up. At 7:00am, the lamp is off. At 7:01am, I flip the switch; the lamp flickers for a few moments, then begins to warm up. At 7:02am, the lamp is fully bright. Between 7:00am and 7:02am, the lamp changed from OFF to ON. This, certainly, is a change; but it is a change over time.

Change implies difference; difference implies comparison. Here, the two values being compared are (1) the state of "the lamp at 7:00am", which is OFF, and (2) the state of "the lamp at 7:02am", which is ON. So we say "the lamp" has changed from one time to another. At 7:00am, you wander by, and see the lamp is OFF; at 7:02am, you wander by, and see the lamp is ON.

But have you ever seen the future change from one time to another? Have you wandered by a lamp at exactly 7:02am, and seen that it is OFF; then, a bit later, looked in again on the "the lamp at exactly 7:02am", and discovered that it is now ON?

Naturally, we often feel like we are "changing the future". Logging on to your online bank account, you discover that your credit card bill comes due tomorrow, and, for some reason, has not been paid automatically. Imagining the future-by-default—extrapolating out the world as it would be without any further actions—you see that the bill not being paid, and interest charges accruing on your credit card. So you pay the bill online. And now, imagining tomorrow, it seems to you that the interest charges will not occur. So at 1:00pm, you imagined a future in which your credit card accrued interest charges, and at 1:02pm, you imagined a future in which it did not. And so your imagination of the future changed, from one time to another.

As I remarked previously: The way a belief feels from inside, is that you seem to be looking straight at reality. When it actually seems that you're looking at a belief, as such, you are really experiencing a belief about your beliefs.

When your extrapolation of the future changes, from one time to another, it feels like the future itself is changing. Yet you have never seen the future change. When you actually get to the future, you only ever see one outcome.

How could a single moment of time, change from one time to another?

I am not going to go into "free will" in today's blog post. Except to remark that if you have been reading Overcoming Bias all this time, and you are currently agonizing about whether or not you really have free will, instead of trying to understand where your own mind has become confused and generated an impossible question, you should probably go back and read it all again. For anyone who is just now joining us... perhaps I shall discuss the issue tomorrow.

Just remember Egan's Law: It all adds up to normality. Apples didn't stop falling when Einstein disproved Newton's theory of gravity, and anyone who jumped off a cliff would still go splat. Perhaps Time turns out to work differently than you thought; but tomorrow still lies ahead of you, and your choices, and their consequences. I wouldn't advise reworking your moral philosophy based on confusing arguments and strange-seeming physics, until the physics stops appearing strange and the arguments no longer seem confusing.

Now to physics we turn; and here I resume drawing my ideas from Julian Barbour.

For the benefit of anyone who hasn't followed the series on quantum mechanics, a very very quick summary:

In classical physics—the mistaken physics that was developed first historically, and matches human intuitions all too well—a particle is like a little billiard ball. A particle is in a single place in 3D space, and we can describe its position with three real numbers. In quantum physics, we need an amplitude distribution over all possible positions for the particle—a complex number for the particle being here, a complex number for the particle being there, and so on through all the positions in space; a continuous distribution. (Configurations and Amplitude.)

In classical physics, we can consider each particle independently. This particle is here, that particle is there. In quantum physics this is not possible; we can only assign amplitudes to configurations that describe the simultaneous positions of many particles. In fact, the only mathematical entities that actually have amplitudes are joint configurations of all the particles in the entire universe. (Joint Configurations.)

Above is a diagram that shows what a configuration space might look like for three particles, A, B, and C. ABC form a triangle in two-dimensional space. Every individual point in the configuration space corresponds to a simultaneous position of all the particles—above we see points that correspond to particular triangles i.e. joint positions of A, B, and C. (Classical Configuration Spaces; The Quantum Arena.)

The state of a quantum system is not a single point in this space; it is a distribution over this space. You could imagine it as a cloud, or a blob, or a colored mist within the space.

Here we see a relative configuration space, in which each axis is the distance between a pair of particles. This has some advantages I'm not going to recapitulate (it was covered in a previous post), so if you're dropping into the middle of the series, just pretend it's a regular configuration space.

We've just chopped up the pyramidal space you saw before, into a series of slices. In this configuration space, the slices near the bottom show all the particles close together (tiny triangles). As we rise up, the particles get further apart (larger triangles).

At the very bottom of the configuration space is a configuration where all the particles occupy the same position.

(But remember, it's nonsense to talk about an individual particle being anywhere in a configuration space—each point in the configuration space corresponds to a position of all the particles. Configuration space is not the 3D space you know. It's not that there are a bunch of particles resting in the same place at the bottom. The single bottom point corresponds to all the particles being in the same place in 3D space.)

Here we take a closer look at one of the slices of configuration space, and see a cloud of blue and red mist covering some of it. (Why am I only showing the cloud covering a sixth (exactly a sixth) of the triangle? This has to do with a symmetry in the space—exchanges of identical particles—which is not important to the present discussion.)

But there is your glimpse of some quantum mist—in two colors, because amplitudes are complex numbers with a real and imaginary part. An amplitude distribution or "wavefunction" assigns a complex number to every point in the continuous configuration space—a complex number to every possible configuration of all the particles.

Yesterday, I finished by asking how the state of a quantum system might evolve over time.

You might be tempted to visualize the mist churning and changing colors, as quantum amplitude flows within the configuration space.

And this is indeed the way that you would visualize standard physics.

Behold the standard Schrödinger Equation:

Here ψ(r, t) is the amplitude distribution over configuration space (r) and time (t). The left-hand side of the Schrödinger Equation is the change over time of the wavefunction ψ, and the right-hand-side shows how to calculate this change as the sum of two terms: The gradient of the wavefunction over configuration space (at that time), and the potential energy of each configuration.

Which is to say, the derivative in time of the wavefunction—the instantaneous rate of change—can be in terms of the wavefunction's derivative in space, plus a term for the potential energy.

If you tried to visualize Schrödinger's Equation—doesn't look too hard, right?—you'd see a blob of churning, complex mist in configuration space, with little blobs racing around and splitting into smaller blobs as waves propagated.

If you tried to calculate the quantum state of a single hydrogen atom over time, apart from the rest of the universe—which you can only really do if the hydrogen atom isn't entangled with anything—the atom's quantum state would evolve over time; the mist would churn.

But suppose you think about the whole universe at once, including yourself, of course. Because—even in the standard model of quantum physics!—that is exactly the arena in which quantum physics takes place: A wavefunction over all the particles, everywhere.

If you can sensibly talk about the quantum state of some particular hydrogen atom, it's only because the wavefunction happens to neatly factor into (hydrogen atom) * (rest of world).

Even if the hydrogen atom is behaving in a very regular way, the joint wavefunction for (hydrogen atom * rest of world) may not be so regular. Stars move into new positions, people are born and people die, digital watches tick, and the cosmos expands: The universe is non-recurrent.

Think of how the universal wavefunction ψ(r, t) might behave when r is the position of all the particles in the universe.

Let's call 9:00am the time t=0, measured in seconds.

At ψ(r, t=0), then, you are wondering what time it is: The particles making up the neurons in your brain, are in positions r you that correspond to neurons firing in the thought-pattern "What time is it?" And the Earth, and the Sun, and the rest of the universe, have their own particles in the appropriate r rest-of-universe . Where the complete r roughly factorizes as the product (r you * r rest-of-universe ).

Over the next second, the joint wavefunction of the entire universe evolves into ψ(r, t=1). All the stars in the sky have moved a little bit onward, in whatever direction they're heading; the Sun has burned up a little more of its hydrogen; on Earth, an average of 1.8 people have died; and you've just glanced down at your watch.

At ψ(r, t=2), the stars have moved a little onward, the galaxies have rotated, the cosmos has expanded a little more (and its expansion has accelerated a little more), your watch has evolved into the state of showing 9:00:02 AM on its screen, and your own mind has evolved into the state of thinking the thought, "Huh, I guess it's nine o' clock."

Ready for the next big simplification in physics?

Here it is:

We don't need the t.

It's redundant.

The r never repeats itself. The universe is expanding, and in every instant, it gets a little bigger. We don't need a separate t to keep things straight. When you're looking at the whole universe, a unique function ψ of (r, t) is pretty much a unique function of r.

And the only way we know in the first place "what time it is", is by looking at clocks. And whether the clock is a wristwatch, or the expansion of the universe, or your own memories, that clock is encoded in the position of particles—in the r. We have never seen a t variable apart from the r.

We can recast the quantum wave equations, specifying the time evolution of ψ(r, t), as specifying relations within a wavefunction ψ(r).

Occam's Razor: Our equations don't need a t in them, so we can banish the t and make our ontology that much simpler.

An unchanging quantum mist hangs over the configuration space, not churning, not flowing.

But the mist has internal structure, internal relations; and these contain time implicitly.

The dynamics of physics—falling apples and rotating galaxies—is now embodied within the unchanging mist in the unchanging configuration space.

This landscape is not frozen like a cryonics patient suspended in liquid nitrogen. It is not motionless as an isolated system while the rest of the universe goes on without it.

The landscape is timeless; time exists only within it. To talk about time, you have to talk about relations inside the configuration space.

Asking "What happened before the Big Bang?" is revealed as a wrong question. There is no "before"; a "before" would be outside the configuration space. There was never a pre-existing emptiness into which our universe exploded. There is just this timeless mathematical object, time existing within it; and the object has a natural boundary at the Big Bang. You cannot ask "When did this mathematical object come into existence?" because there is no t outside it.

So that is Julian Barbour's proposal for the next great simplification project in physics.

(And yes, you can not only fit General Relativity into this paradigm, it actually comes out looking even more elegant than before. For which point I refer you to Julian Barbour's papers.)

Tomorrow, I'll go into some of my own thoughts and reactions to this proposal.

But one point seems worth noting immediately: I have spoken before on the apparently perfect universality of physical laws, that apply everywhere and everywhen. We have just raised this perfection to an even higher pitch: everything that exists is either perfectly global or perfectly local. There are points in configuration space that affect only their immediate neighbors in space and time; governed by universal laws of physics. Perfectly local, perfectly global. If the meaning and sheer beauty of this statement is not immediately obvious, I'll go into it tomorrow.

And a final intuition-pump, in case you haven't yet gotten timelessness on a gut level...

Think of this as a diagram of the many worlds of quantum physics. The branch points could be, say, your observation of a particle that seems to go either "left" or "right".

Looking back from the vantage point of the gold head, you only remember having been the two green heads.

So you seem to remember Time proceeding along a single line. You remember that the particle first went left, and then went right. You ask, "Which way will the particle go this time?"

You only remember one of the two outcomes that occurred on each occasion. So you ask, "When I make my next observation, which of the two possible worlds will I end up in?"

Remembering only a single line as your past, you try to extend that line into the future -

But both branches, both future versions of you, just exist. There is no fact of the matter as to "which branch you go down". Different versions of you experience both branches.

So that is many-worlds.

And to incorporate Barbour, we simply say that all of these heads, all these Nows, just exist. They do not appear and then vanish; they just are. From a global perspective, there is no answer to the question, "What time is it?" There are just different experiences at different Nows.

From any given vantage point, you look back, and remember other times—so that the question, "Why is it this time right now, rather than some other time?" seems to make sense. But there is no answer.

When I came to this understanding, I forgot the meaning that Time had once held for me.

Time has dissolved for me, has been reduced to something simpler that is not itself timeful.

I can no longer conceive that there might really be a universal time, which is somehow "moving" from the past to the future. This now seems like nonsense.

Something like Barbour's timeless physics has to be true, or I'm in trouble: I have forgotten how to imagine a universe that has "real genuine time" in it.

Part of The Quantum Physics Sequence

Next post: "Timeless Beauty"

Previous post: "Relative Configuration Space"