The n-Category Café

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

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.

“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.

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

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.

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.

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.