#### marcus

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We already had some brief discussion (2 posts) of this paper in the "Intuitive...Program" thread that is serving as a catch-basket for Loop-and-allied Quantum Gravity links. It probably could use its own thread, to allow space for more discussion, if desired. After I posted this mention of the new paper, selfAdjoint responded drawing some connections to interesting mathematics----I quote his full post in a moment. First, here's the paper:

Daniele Colosi, Carlo Rovelli

http://arxiv.org/abs/gr-qc/0409054

---exerpt from conclusions section---

...the distinction between global and local states can therefore be safely neglected in concrete utilizations of QFT. However, the distinction is conceptually important because it bears on three related issues: (i) whether particles are local or global objects in conventional QFT; (ii) the extent to which the quantum field theoretical notion of particle can be extended to general contexts where gravity cannot be neglected; and furthermore, more in general, (iii) whether particles can be viewed as the fundamental reality (the “ontology”) described by QFT. Let us discuss these three issues separately. ...

...Can we base the ontology of QFT on local particles? Yes, but local particle states are very different from global particle states. Global particle states such as the Fock particle states are defined once and for all in the theory, while each finite size detector defines its own bunch of local particle states. Since in general the energy operators of different detectors do not commute ([H

The world is far more subtle than a bunch of particles that interact.

---end exerpt from Colosi-Rovelli---

https://www.physicsforums.com/showthread.php?p=313419#post313419

Daniele Colosi, Carlo Rovelli

**Global particles, local particles**http://arxiv.org/abs/gr-qc/0409054

---exerpt from conclusions section---

...the distinction between global and local states can therefore be safely neglected in concrete utilizations of QFT. However, the distinction is conceptually important because it bears on three related issues: (i) whether particles are local or global objects in conventional QFT; (ii) the extent to which the quantum field theoretical notion of particle can be extended to general contexts where gravity cannot be neglected; and furthermore, more in general, (iii) whether particles can be viewed as the fundamental reality (the “ontology”) described by QFT. Let us discuss these three issues separately. ...

...Can we base the ontology of QFT on local particles? Yes, but local particle states are very different from global particle states. Global particle states such as the Fock particle states are defined once and for all in the theory, while each finite size detector defines its own bunch of local particle states. Since in general the energy operators of different detectors do not commute ([H

_{R}, H_{R'}] nonzero) there is no unique “local particle basis” in the state space of the theory, as there is a unique Fock basis. Therefore, we cannot interpret QFT by giving a single list of objects represented by a unique list of states. In other words, we are in a genuine quantum mechanical situation in which distinct particle numbers are complementary observables. Different bases that diagonalize different H_{R}operators have equal footing. Whether a particle exists or not depends on what I decide to measure. In such a context, there is no reason to select an observable as “more real” than the others.The world is far more subtle than a bunch of particles that interact.

---end exerpt from Colosi-Rovelli---

https://www.physicsforums.com/showthread.php?p=313419#post313419

selfAdjoint said:Apropos of this Colosi-Rovelli attempt to generalize particles, see today's post on Peter Woit's Not Even Wrong about Grothendiek and his toposes. He generalized the Nullstellenstatz view, which you have explicated so clearly, Marcus, in which the points of a continuum are represented as the prime ideals of the algebra of continuous functions on it. G. represents points of a space X as sheafs over X; a sheaf is a kind of category, and this leads to G.'s definition of topos, which we have had some discussion about in connection with Chris Isham's papers. G. was looking to define the "group of a point", and he actually reached a good definition.

Much of the perceived beauty of string theory is actually the beauty of G. and his generation's work in topology and algebraic geometry, which people like Witten have scarfed up and instantiated in physical models, orbifolds, for example. See the survey of this work by Jacques Cartier which Woit gives a link to.

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