Connes Lecture 7
Connes begins lecture 7 with a few remarks which are also stressed in the preface of the new book with M. Marcolli, namely that there are two very difficult but related problems behind his work and they are (i) the structure of spacetime and (ii) the prime numbers, or so called arithmetic site (recall that a site is a category equipped with a Grothendieck topology).
The lecture then begins with an analogy between the primes and elementary particles in QFT, which both form composites. Yesterday we saw that in higher topos theory this analogy might actually make some mathematical sense. But Connes discusses QFT, not simply quantum mechanics, so the composition of particles need not preserve particle number. Ordinary primes (in $\mathbb{N}$) are, as usual, cardinalities of sets. The Cartesian product of two sets results in the product of cardinalities, so any $n \in \mathbb{N}$ is represented by a Cartesian product of sets with a prime number of elements (we could always take, say, Saharan grains of sand, if we insist on constructing the sets in question). For basic quantum mechanical logic, Cartesian product is replaced by tensor product, but QFT requires a tensor product capable of shifting dimensions. This is the (higher categorical) Gray tensor product of Crans.
The lecture then begins with an analogy between the primes and elementary particles in QFT, which both form composites. Yesterday we saw that in higher topos theory this analogy might actually make some mathematical sense. But Connes discusses QFT, not simply quantum mechanics, so the composition of particles need not preserve particle number. Ordinary primes (in $\mathbb{N}$) are, as usual, cardinalities of sets. The Cartesian product of two sets results in the product of cardinalities, so any $n \in \mathbb{N}$ is represented by a Cartesian product of sets with a prime number of elements (we could always take, say, Saharan grains of sand, if we insist on constructing the sets in question). For basic quantum mechanical logic, Cartesian product is replaced by tensor product, but QFT requires a tensor product capable of shifting dimensions. This is the (higher categorical) Gray tensor product of Crans.
posted by Kea | 2:15 PM
5 Comments:
For infinite primes mappable to polynomial primes the analogy between elementary particles and primes is much more than mere analogy since infinite primes corresponds to Fock states of super-symmetric arithmetic QFT.
Besides free states also bound states which correspond to higher degree irreducible polynomails are present. And the many particle states and their bound states represent infinite primes at next level of hierarchy.
The assigment of finite prime to parton and collection of finite primes to particle (defining more complex infinite prime) is consistent with this interpretation.
Hi Matti, yes of course you are right. This was just one of those simple summary posts, for a wider audience.
Kea, you're becoming dangerously low brow. I almost understood parts of these last two posts! You're going to have to raise the bar another 10 or 20 IQ points; otherwise you're going to end up with the likes of me making intelligent comments.
I almost understood parts of these last two posts!
Wow, thanks Carl! Usually my poor communication skills (and innate stupidity) get in the way of comprehensibility. I'm looking forward to your next post on the derivation of the particle masses.
Kear,
sorry if I created kind of defensive impression! But I must be honest; I feel sometimes inpatience. I have been preaching so long time about these fascinating ideas.
I agree that your posts are becoming more and more comprehensible but I don't blame you for this: I think that this what unavoidably happens in successful communication;-).
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