(adsbygoogle = window.adsbygoogle || []).push({}); Happy New Year - reply to Lubos

HAPPY 2005 EVERYBODY

Let us welcome the International Year of Physics! There is a

poster on the wall in the corridor here, saying "Help make 2005

another miraculous year".

Well, I don't know that I can do much - but one can try.

I've posted here, because it takes days for my posts to appear on Strings.

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Shuzan held out his short staff and said: "If you call this a

short staff, you oppose its reality. If you do not call it a

short staff, you ignore the fact. Now what do you wish to call

this?"

Zen koan

In classical (Boolean) logic the law of the excluded middle

[tex]U \vee \neg U = \textrm{true}[/tex]

holds. Consider the following examples of its failure.

1. The collection of subspaces of a Hilbert space [itex]\mathcal{H}[/itex] forms an algebra under intersection [itex]\wedge[/itex] and [itex]U \vee V[/itex] the smallest subspace containing both U and V. [itex]\neg[/itex] is the orthogonal complement (hence the need for inner product). False is the zero subspace and true is the full

space [itex]\mathcal{H}[/itex].

2. The open sets of the two dimensional sphere under the

operations of intersection and union, with [itex]\neg U[/itex] the

interior of the complement (it can't be the complement because

that's not an open set). False is the empty set, and true the

whole sphere.

This second example is interesting because it describes the two

sphere as a category (with an arrow between open sets when there

is an inclusion) which underlies the sheaf of germs of analytic

functions...anyway, think of it as the celestial sphere of an

observer in GR and it turns up in twistor theory.

One might also consider the axiom of choice. The 'local' version

of this also forces Booleanness on one's logic.

So we're going to have to be rather careful with which mathematics

we use if we want a description of the quantum world that is

sophisticated enough to describe decoherence, for instance.

As far as I understand it (correct me if I'm wrong, Lubos) String

theorists are quite keen on twistors at the moment, but they do

not address these questions.

Kea

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# Happy New Year

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