QPL 09 II
Jamie Vicary bet me a bottle of wine that the fairy field would be found in the next few years. Actually, he was willing to bet on the next two years, but I let him off the hook on that one. (Unfortunately, my total stakes on this question include a little wine and around ten dollars, so even I can afford to lose.) At QPL, Vicary spoke about his characterisation of the complex numbers using natural structures in dagger monoidal categories with superposition (see this paper).
Superposition says that given two morphisms $f,g: A \rightarrow B$, there exists a morphism called $f+g$ and addition is commutative and associative. The crucial notion is that of a $\dagger$ limit for a diagram $D$, defined to be a limit $L$ such that the arrows $f_{S}: L \rightarrow D(S)$ satisfy
$\sum f_{s} \circ f_{s}^{\dagger} = 1_{L}$,
where the sum is over a set of source objects in $D$. This is a normalisation condition for superpositions. When all objects in a discrete diagram $D$ are sources, this reduces to the categorical biproduct $\oplus$. Given a category with a zero object and all finite biproducts (such as the category of Hilbert spaces) it turns out that there is a unique superposition rule.
One of the things that Vicary shows is that, for a category with tensor unit $I$ and all finite dagger limits, the semiring of scalars $I \rightarrow I$ has a natural embedding into a characteristic zero field. This relies on the decomposition of any non-zero ordinal $p: I \rightarrow I$ into a diagonal arrow
$I \oplus I \oplus \cdots \oplus I \rightarrow I$
and its adjoint codiagonal. So, if we want to work with finite fields of characteristic $p$, we can now identify exactly which pieces of complex number structure break down. For instance, there might be a zero map constructed from a finite diagonal and codiagonal on the unit object.
Superposition says that given two morphisms $f,g: A \rightarrow B$, there exists a morphism called $f+g$ and addition is commutative and associative. The crucial notion is that of a $\dagger$ limit for a diagram $D$, defined to be a limit $L$ such that the arrows $f_{S}: L \rightarrow D(S)$ satisfy
$\sum f_{s} \circ f_{s}^{\dagger} = 1_{L}$,
where the sum is over a set of source objects in $D$. This is a normalisation condition for superpositions. When all objects in a discrete diagram $D$ are sources, this reduces to the categorical biproduct $\oplus$. Given a category with a zero object and all finite biproducts (such as the category of Hilbert spaces) it turns out that there is a unique superposition rule.
One of the things that Vicary shows is that, for a category with tensor unit $I$ and all finite dagger limits, the semiring of scalars $I \rightarrow I$ has a natural embedding into a characteristic zero field. This relies on the decomposition of any non-zero ordinal $p: I \rightarrow I$ into a diagonal arrow
$I \oplus I \oplus \cdots \oplus I \rightarrow I$
and its adjoint codiagonal. So, if we want to work with finite fields of characteristic $p$, we can now identify exactly which pieces of complex number structure break down. For instance, there might be a zero map constructed from a finite diagonal and codiagonal on the unit object.
posted by Kea | 10:09 PM
9 Comments:
I haven't given up on my long term goal of understanding category theory by translating density operator formalism into that language. As far as this paper goes, the big difference is that density operator formalism avoids the Hilbert spaces. Instead, it looks only at the maps between them.
I believe I've seen a definition in category theory where one creates a category by looking at the arrows of another category.
The reason for the difference is that the usual quantum formalism represents a quantum state by a state vector which is an element of a Hilbert space. Separate from the state vector are the operators which act on state vectors and also on each other.
In density operator formalism, you associate the quantum states with operators (i.e. ket x bra products) so state vectors go away, all you have is operators.
In density operator formalism, the quantum states form a subset of all possible operators. Given an operator, it represents a quantum state if and only if it is a Hermitian primitive idempotent. One can then show that there is a corresponding state vector. In my papers I often loosen the Hermitian and primitive requirements. Right now I'm typing up a new paper which will explain some of this better.
For a density operator of the form rho(x,x'), the corresponding normalized wave function is given by
psi(x) = rho(x,x_0)/sqrt(rho(x_0,x_0))
where x_0 is a point in space where rho(x_0,x_0) is non zero. Such an x_0 exists because rho is primitive and so has trace = 1.
Of course the reverse definition is
rho(x,x') = psi(x) psi^*(x'), and this works for the above.
I still haven't had time to try to learn category theory (which will come once I've finished with the stack of QFT textbooks I have to read).
But if I can comment on "fairy fields", there must be some kind of field giving rise to mass as gravitationa charge for fermions and weak gauge bosons, so did your bet specify any particular Higgs theory? There are different Higgs theories, containing different predictions about how many Higgs bosons there are.
Mass has got to be acquired by massive particles somehow. A graviton has to interact with a particle which represents gravitational charge. Gravitational charge (mass) can't be the same thing as say electromagnetic charge, because masses are affected by motion in a different way to electromagnetic charges. If a discrete number of fixed-mass gravitational charges clump around each fermion core, "miring it" like some kind of treacle-like "Higgs field", you can predict all lepton and hadron inertial and gravitational masses.
The gravitational charges are have inertia because they are exchanging gravitons with all other masses around the universe, which physically holds them where they are (if they move, they encounter extra pressure from graviton exchange in the direction of their motion, which causes contraction, requiring energy; hence resistance to acceleration, which is just Newton's 1st law, inertia). I won't go into the gravity mechanism again, but the point is that the Higgs "fairy field" idea - while it is wrong in filling the entire vacuum (insiead of having a finite number of gravitational charges cluster around a fermion or weak boson), and it is very unhelpful for electroweak symmetry breaking in the standard model becaus it doesn't make hard falsifiable predictions - physically provides a vacuum picture that is helpful to some extent (although wrong in detail). Corrected, it helps predict masses.
An illustration of a miring particle mass model (with discrete number of 91 GeV mass particles surrounding the IR cutoff outer edge of the shell of the polarized vacuum around to give them mass) is linked here.
A PDF table comparing crude model mass estimates to observed masses of particles is linked here. There is evidence for the quantization of mass from the way the mathematics work for spin-1 quantum gravity. If you treat two masses being pushed together by spin-1 graviton exchanges with the isotropically distributed mass of the universe accelerating radially away from them (viewed in their reference frame), you get the expected correct a prediction of gravity as illustrated here. But if you do the same spin-1 quantum gravity analysis but only consider one mass and try to work out the acceleration field around it, as illustrated here, you get (if you use the black hole event horizon radius to calculate the graviton scatter cross-section) a prediction that gravitational force is proportional to M^2, which suggests all particles masses are built up from a single fixed size building block of mass.
If this comment comes across as too long, annoying, off topic or whatever, then feel free to delete it. I'm not implying that I have a complete understanding of mass yet (the identification of the number of mass particles to each fermion is crude, as are the derivations of some of the geometric factors like twice Pi from spin considerations in the accompanying blog posts), but the bits of evidence available do suggest that there is a "fairy field", albeit not any of the mainstream Higgs ones.
Carl, there has been some work done on the categorical density formalism. Read here:
Peter Selinger:
Dagger compact closed categories and completely positive maps
http://www.mscs.dal.ca/~selinger/papers.html#dagger
Me:
Axiomatic Description of Mixed States From Selinger's CPM-construction
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B75H1-4SY6BJB-2&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=24c2d2f2e35bbedf8e8a056fcb643934
(if you send me an emaiol I can send you the .pdf)
Peter Selinger:
Idempotents in dagger categories
http://www.mscs.dal.ca/~selinger/papers.html#idem
Me:
Complete Positivity without Positivity and Without Compactness
http://web.comlab.ox.ac.uk/publications/publication54-abstract.html
We also posted a paper at the arXiv to day on the arXiv entitled Classical and Quantum Structuralism which has some more on density operators and CPM.
I haven't given up on my long term goal of understanding category theory by translating density operator formalism into that language. As far as this paper goes, the big difference is that density operator formalism avoids the Hilbert spaces.
Indeed, in the context of noncommutative geometry, this would allow one to dispense with spectral triples (which include a Hilbert space), and focus purely on the C*-algebra, i.e., the operator algebra acting on itself rather than a separate Hilbert space.
Perhaps Bob has an operator-only categorical formalism, which is exactly what's needed in the description of extremal black holes with an entanglement interpretation.
Yes, as you know kneemo, I think the special sets of MUB/Clifford operators indicate arithmetic axioms quite unlike those for dagger monoidal categories.
And there are other reasons for trying to move away from the complex numbers. For example, the messy Chu space description of a Hilbert space over C suggests that C should only appear via a continuum heirarchy of structures beginning with small truth value sets. Since Spekken's toy model only requires four truth values, there is a good reason to think that the qubit stabiliser case should also be considered as expressing limited truth.
The entanglement measure is then obviously some kind of graphical invariant. You scoundrel, kneemo! You know that's what I'm working on.
The entanglement measure is then obviously some kind of graphical invariant. You scoundrel, kneemo! You know that's what I'm working on.
No, this is a different project. :)
Bob, thanks for the links.
Pure density operators (which are exactly equivalent to standard quantum mechanics) are a subset of the algebra of operators that happen to be Hermitian primitive idempotents. The usual generalization of pure density matrices is to mixed (statistical mixture) density matrices which are Hermitian and primitive (i.e. have unit trace) but not idempotent. I suppose these will be what is covered, mostly, but I see that one of the links addresses idempotency.
I'm interested in the generalization of pure density matrices to the cases of non Hermitian and non primitive and this may be a little different. The paper I'm writing up right now has to do with this. I more or less finished the section on types of density matrix generalizations but haven't spell checked it, etc.
Hi Carl. The idempotents in Selinger's paper don't refer to the fact that the density operators are idempotent but to the idempotent-splitting categorical construction which he uses to construct classical data types as a control structure. The nice thing about this construction is that it essentially works for any dagger symmetric monoidal category so it leaves plenty of degrees of axiomatic freedom. Which you can fill in with your favorite axioms and/or models and/or combinatorics, be it MUBs expressing complementarity, algebra enrichment if you want to stay closer to C*-algebras, Chu pairs of whatever kind, ... it doesn't take much effort to see that the construction is highly conceptual. Cheers, Bob.
Hmm. The Karoubi envelope stuff is interesting. Carl, check out the example on page 100 where there is a 3x3 circulant matrix.
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