## October 13, 2008

### The Nature of Time

#### Posted by John Baez

The Foundational Questions Institute is having an essay contest on The Nature of Time. The top prize is $10,000, the second prize is$5,000, and so on.

It’s a fascinating topic, but I can’t say I’m thrilled with most of the essays. In fact, that’s a polite way of expressing my feelings, in keeping with the civil atmosphere of this café. I’ll mention my favorite essay below, and keep quiet about the worst.

Maybe you could do better. In fact, maybe you should give it a try! Just make sure to submit your essay before December 1st, 2008.

I would write one myself, but I don’t have… time.

I haven’t read all the essays — so there could be some gems I haven’t seen. So far, my favorite is the one by my friend Carlo Rovelli:

• Carlo Rovelli, Forget time.

Abstract: Following a line of research that I have developed for several years, I argue that the best strategy for understanding quantum gravity is to build a picture of the physical world where the notion of time plays no role at all. I summarize here this point of view, explaining why I think that in a fundamental description of nature we must “forget time”, and how this can be done in the classical and in the quantum theory. The idea is to develop a formalism that treats dependent and independent variables on the same footing. In short, I propose to interpret mechanics as a theory of relations between variables, rather than the theory of the evolution of variables in time.

For those familiar with his work on quantum gravity and the ‘thermal time hypothesis’, there’s nothing drastically new about this essay. But, it makes a good case for the radical viewpoint that has motivated his work all along.

By the way: you can vote for your favorite essay! However, your vote will have no direct effect, unless you’re a member of the Foundational Questions Institute. It’s just like voting in the U.S. presidential election when you don’t live in a swing state.

Also by the way: I’m slightly suprised by the fact that right now, every essay except Rovelli’s has exactly 10 votes. How likely is that? Do you get 10 free votes just for playing?

Posted at October 13, 2008 10:16 PM UTC

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### Re: The Nature of Time

When I checked, most essays had less than ten public votes, one had ten and there were a few with more than ten.

Posted by: Bruce Westbury on October 14, 2008 7:32 AM | Permalink | Reply to this

### Re: The Nature of Time

Hmm. Now they all have 10 except for three that have more: Bundy’s, Miller’s, and Rovelli’s.

Posted by: John Baez on October 14, 2008 6:49 PM | Permalink | Reply to this

### Re: The Nature of Time

0 restricted votes • $\lt 10$ public votes

The symbol before 10 means “fewer than”.

Posted by: Simon Willerton on October 14, 2008 7:18 PM | Permalink | Reply to this

### Re: The Nature of Time

Simon inquired:

I think it’s my brain — you know, that inert lump of matter a few inches behind my glasses. I didn’t expect them to be saying “$\lt 10$” votes, so I didn’t see the ‘less than’ sign. I must have thought it was meaningless cruft just like the $\bullet$.

I feel happier now, though I wonder why you need to get 10 votes or more before you can see how many you got. Maybe it’s to avoid people’s feelings being hurt?

Posted by: John Baez on October 14, 2008 8:49 PM | Permalink | Reply to this

### Re: The Nature of Time

It would be desireable to have a really good answer to

Why Lorentzian signature?

And: what is Wick rotation really?

Consider 2-dimensional QFT. There are two kinds of precise axiomatic definitions to capture it:

- one is local nets of operators: this requires Lorentzian signature.

- the other is vertex operator algebras: this requires Euclidean signature – but comes in two copies: left and right chiral parts.

What’s the relation? Something like this:

consider the real 4-dimensional space $\mathbb{C} \times \mathbb{C}$ with canonical (complex) coordinates $z_1$ and $z_2$ and equipped with a symplectic form

$\omega : (\mathbb{C} \times \mathbb{C}) \times (\mathbb{C} \times \mathbb{C}) \to \mathbb{R}$ given by $\omega((z_1,z_2),(z'_1,z'_2)) = Re(z_1)Im(z'_2) + Re(z_2)Im(z'_1) - Re(z'_1)Im(z_2) - Re(z'_2)Im(z_1) \,.$

There are the following two interesting polarizations of $\omega$ (i.e. 2-dimensional sub-vectorspaces of $\mathbb{C}\times \mathbb{C}$ on which $\omega$ vanishes):

one is spanned by $\{ \partial_{Im(z_1)}\,, \partial_{Im(z_2)} \}$ call this $P_1$.

The other is spanned by $\{ \partial_{Re(z_1)} - \partial_{Re(z_2)} \;,\;\; \partial_{Im(z_1)} + \partial_{Im(z_2)} \} \,.$

The first polarization imposes the constraint: “both $z_1$ and $z_2$ are real”.

In the sense that: if you know a function which is annihilated by $P_1$ on the subspace given by $Im(z_1) = Im(z_2) = 0$ then you know it everywhere.

The second polarization imposes the constraint “$z_2$ is the complex conjugate of $z_1$”.

In the sense that: if you know a function which is annihilated by $P_2$ on the subspace given by $\bar z_2 = z_1$ then you know it everywhere.

The first case is the Lorentzian case: $Re(z_1)$ and $Re(z_2)$ play the role of lightcone coordinates on $\mathbb{R}^2$.

The second case is the Euclidean case: $z_1$ and $z_2 = \bar z_1$ play the role of the two “chiral” coordinates.

Here “play the role” means: in 2d QFT, when switching between the Lorentzian and the Wick-rotated Euclidean setup, this is how the two variables in each case are related.

Posted by: Urs Schreiber on October 14, 2008 10:00 AM | Permalink | Reply to this

### Re: The Nature of Time

One nice thing about these questions is that they don’t even involve general relativity. General relativity brings up a lot of the deep questions about time. But, you’re pointing out that even 2d special relativity holds its mysteries.

Posted by: John Baez on October 14, 2008 6:53 PM | Permalink | Reply to this

### Re: The Nature of Time

It would be desireable to have a really good answer to

Why Lorentzian signature?

I think posets have something interesting to say regarding this question (as you know). A Lorenztian signature arises naturally from a causal structure. A causal structure can arise naturally via many means. My favorite is via directed graphs.

Posted by: Eric on October 14, 2008 7:16 PM | Permalink | Reply to this

### Re: The Nature of Time

It would be desireable to have a really good answer to

Why Lorentzian signature?

I think posets have something interesting to say regarding this question (as you know).

Yes, indeed, good point. There are various approaches by various people to pin down what precisely it is that makes Lorentzian signature so special, or maybe exceptional.

One is the observation that, roughly, a Lorentzian metric can be encoded in a poset structure on spacetime plus a volume density. Since a poset is a rather fundamental concept, this might indicate a deeper hidden meaning.

Interestingly, at least when restricting to certain dimensions, there are also characterizations of rather different flavor. For instance 3+1-dimensional Minkwoski space is closely related to the quaternions.

A causal structure can arise naturally via many means. My favorite is via directed graphs.

For others, let me say that in detail:

for any natural number $n$, consider the directed graph whose vertices are $\mathbb{Z}^n$ and whose edges consist of those going from the origin to the standard basis elements and all those obtained from these by translation along them.

To avoid irrelevant technical issues, truncate this to some finite subgraph.

Now consider the “graph algebra” of this graph: the associative algebra generated from the vertices and the edges, where the product from the left of an edge with its source vertex is the edge itself, same for the product from the right with the target vertex, and where all other products between vertices and edges vanish. The product of two edges which are not composable vanishes, while the product of two edges which do touch is a new generator of the algebra. Then divide out the relation that all sums of generators of length more than one with the same source and target vanish.

This algebra then is a model for differential forms on $\mathbb{Z}^n$. $k$-forms are generators of length $k$. It is noncommutative, where the non-commutativity is “proportional to the lattice spacing” in an obvious sense.

Due to the non-commutativity, the differential is an inner graded derivation: it is the graded commutator with the “graph operator”: the algebra element which is the sum of all edges.

(Notice that, while I can’t claim to see a direct relation at the technical level, at the moral level this is not unlike the phenomenon that type III von Neumann factors have a “canonical time evolution”.)

So we have a noncommutative DGA.

By taking the generators to be an orthonormal basis, there is naturally the structure of a Hilbert space on this algebra, which corresponds to the Hodge inner product $(\alpha,\beta) = \int_X \alpha \wedge \star \beta$, for $\star$ the Hodge star of standard flat $\mathbb{R}^n$.

On the algebra side one changes this metric by deforming the inner product by a hermitean operator $\hat g$: $(\alpha,\beta)' := (\alpha, \hat g \beta)$.

There is one god-given such operator in the game: the graph operator plus its ajoint times the graph operator minus its adjoint (with respect to the original metric). One checks that the metric this operator induces is the standard Minkowski metric on $\mathbb{Z}^n$ which regards all the edges of the graph as timelike and future-directed (or all as past directed, depending on convention).

Then what we end up with is a lattice version of differential form on Minkowski space with $\hat g$ being the Krein operator which changes the Minkowski Hodge inner product to the Euclidean one.

By the way, I think it is not inconceivable that there is a nice generalization of the picture of Lie-$\infty$-theory obtained by generalizing DGCAs to possibly non-commutative DGAs as above.

Posted by: Urs Schreiber on October 15, 2008 9:45 AM | Permalink | Reply to this

### Re: The Nature of Time

Concerning the question of Wick rotation and passage from Euclidean to Lorentzian QFT:

Dirk Schlingemann, From Euclidean field theory to quantum field theory

which, after a quick review of the Osterwalder-Schrader theorem which relates Euclidean Schwinger correlation functions with Minkwoskian Wightman correlation functions, gives a definition of Euclidean nets of observables (section 2, page 12) and then reconstructs directly from that (without going through correlation functions) a Haag-Kastler local net on Minkowski space (section 3.2, p. 17).

(Currently I am somewhat perplexed by his Eulcidean locality condition on p. 12, though. But I may just need to read this in more detail and think about it…)

Posted by: Urs Schreiber on November 13, 2008 12:56 PM | Permalink | Reply to this

### Re: The Nature of Time

“your vote will have no direct effect, unless you’re a member of the Foundational Questions Institute”

The way I read it you also get the right to cast a vote that counts if you submit an essay and it gets approved, but you can’t vote for yourself.

“I can’t say I’m thrilled with most of the essays”

What in general don’t you like about them? Are they too speculative, too dull, not enough maths, not original, the crackpot index is too high/low, not crazy enough, not even wrong, something else?

I am surprised there are so few entries but perhaps there will be loads at the last minute.

Pity you are not going to give it a shot but I hope Greg Egan has time for one. The contest needs some good writers.

Posted by: PhilG on October 14, 2008 7:02 PM | Permalink | Reply to this

### Re: The Nature of Time

PhilG wrote:

What in general don’t you like about them? Are they too speculative, too dull, not enough maths, not original, the crackpot index is too high/low, not crazy enough, not even wrong, something else?

Let me just quote portions of a few of the abstracts. Can you — or anyone — guess what I think about each one?

Time is a functional entity, derived from the fundamental property of basic 3D matter particles to move at a constant linear speed. 3D matter has no ability to act or move.

We construct a relativistic theory in which time plays an active physical role in the cosmology and self-organized sustainability of the universe. ve physical role in the cosmology and self-organized sustainability of the universe. Our model, consistent with scale-invariant quantum field theory, compels an absolute unit of time. Results precisely account for current WMAP data.

By using the tools of global Lorentzian geometry I prove that, under physically reasonable conditions, the impossibility of finding a global time implies the singularity of spacetime.

The only observed relationship of time to space is a reciprocal relation, in the equation of motion. However, it seems absurd to think of space, defined as a set of points satisfying the postulates of geometry, as the inverse of time. Only when we view the observed increase of time, as the 0D inverse of the observed increase of 3D space, does it begin to make sense: The expanding universe is an expanding set of four-dimensional spacetime coordinates, just as Einstein conceived it, but this also may be its initial condition.

The following essay puts forward an academic conception of the here-and-now or “present moment.” It is then identified as coincident to both our subjective moments and Nature. We then use this conjecture to construct a scientific and objective world-view, based on our human biology, to answer the question “what is reality?”

We have recently introduced a consistent way of defining time relationally in general relativity. When quantum mechanics is formulated in terms of this new notion of time the resolution of the em measurement problem can be implemented via decoherence without the usual pitfalls. The resulting theory has the same experimental results of ordinary quantum mechanics, but every time an event is produced or a measurement happens two alternatives are possible: a) the state collapses; b) the system evolves without changing the state.

This essay takes the idea of time as a relation of distance and velocity. Then it shows how an observer traveling at a relative velocity measures the same velocity for objects that assume they are at rest, using geometry to find the affects of relativity. Then formulates a proof that v’=v so that the affects of time dilation and length contraction do not interfere with a body in motion so that Newton’s Second Law of Motion holds true without time being a force acting on a body in motion.

Posted by: John Baez on October 14, 2008 9:10 PM | Permalink | Reply to this

### Re: The Nature of Time

In general, to everybody who has serious stuff to offer:

in my humble opinion currently at FQXi they tend to have more resources to offer than there are worthwhile submissions. Serious researchers should take that as an opportunity…

Posted by: Urs Schreiber on October 14, 2008 10:48 PM | Permalink | Reply to this

### Re: The Nature of Time

Fortunately, even us non serious researchers have an opportunity to participate in this essay contest.

Posted by: Kea on October 15, 2008 1:37 AM | Permalink | Reply to this

### Re: The Nature of Time

I am again and again astonished how blind these people calling themselves serious researchers are to the deepest problems of recent day physics. The text written by these “serious” researchers about time or other profound problems is typically boring repetition of what was believed to be known already century ago.

I know that these people are not idiots as individuals: perhaps kind of institutional stupidity is in question.

Posted by: Matti Pitkänen on October 16, 2008 8:33 AM | Permalink | Reply to this

### Re: The Nature of Time

Final submissions are in and voting proper has begun.

If you didn’t like the look of the entries six weeks ago have another look now. There have been many more submissions including some “serious” stuff from the likes of Sean Carol, Julian Barbour, George Ellis, David Finklestein, Dean Rickles, David Hestenes, Paul Halpern etc.

Of course all of us authors will look forward to being judged on equal footing according to the scientific merit of our essays - or will the voting be like at the Eurovision song contest? We will see.

Coincidently for UK residents there is a Horizon program about Time on BBC2 tonight.

Posted by: PhilG on December 2, 2008 5:47 PM | Permalink | Reply to this

### Re: The Nature of Time

I found Carlo Rovelli essay lucid and brilliant as usual. However, I hit on this little text by Lee Smolin for Edge, http://www.edge.org/q2009/q09_9.html#smolin and I’m puzzled by the certainty of his sudden switch to a ‘new’ presentist and time realist viewpoint. These classical, newtonian ideas are coming back like a boomerang from some new theory? I would be grateful if anyone may offer an expert comment on this.

Posted by: Gianluca on January 5, 2009 1:46 AM | Permalink | Reply to this

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