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Kea
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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.
-----------------------------------------------------------------
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
U \vee \neg U = \textrm{true}
holds. Consider the following examples of its failure.
1. The collection of subspaces of a Hilbert space \mathcal{H} forms an algebra under intersection \wedge and U \vee V the smallest subspace containing both U and V. \neg is the orthogonal complement (hence the need for inner product). False is the zero subspace and true is the full
space \mathcal{H}.
2. The open sets of the two dimensional sphere under the
operations of intersection and union, with \neg U 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
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.
-----------------------------------------------------------------
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
U \vee \neg U = \textrm{true}
holds. Consider the following examples of its failure.
1. The collection of subspaces of a Hilbert space \mathcal{H} forms an algebra under intersection \wedge and U \vee V the smallest subspace containing both U and V. \neg is the orthogonal complement (hence the need for inner product). False is the zero subspace and true is the full
space \mathcal{H}.
2. The open sets of the two dimensional sphere under the
operations of intersection and union, with \neg U 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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