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have been writing some Idea-section at causal perturbation theory
(currently this has much overlap with locally covariant perturbative quantum field theory, eventially the latter will contain more stuff)
Here’s a question: is there a version of the Bohr topos approach for causal perturbation theory,
but with values in formal power series algebras (as befits a perturbation theory) instead of C*-algebras
? (to quote from causal perturbation theory)
Good question, I have been wondering about this myself.
Of course the definition of the Bohr topos itself goes through for any kind of algebra. One would have to see that one gets something that properly reflects the idea of an “infinitesimal neighbourhood in phase space”.
In fact there are several open questions on the structural nature of perturbative QFT. Bohr toposes might help, or other ideas might help.
For instance despite the remarkable result of Collini, that the traditional construction of causal perturbation theory is just Fedosov deformation quantization, it is so only after restricting to a sub-algebra of the smooth functions on phase space. In fact even the Poisson bracket (the Peierls bracket) is not defined on all smooth funtions on phase space. Deformation quantization has really no explanation for this, by default the prescription asks to deform the full algebra of functions. Here one just restricts by hand in an ad hoc way, since it does not work on the full algebra.
But geometric quantization has an explanation for phenomena like this this. First of all geometric quantization knows that the Poisson bracket exists only on the Hamiltonian functions, those smooth functions $f$ such that there exists a smooth vector field $v$ such that $d f = \omega(v,-)$, for $\omega$ the pre-symplectic 2-form. In finite dimensions this is no constraint on $f$, but here it is. Moreover, geometric quantization knows that not all such Hamiltonian functions are to be quantized, but just a subspace which respects a given polarization.
This should be checked: What are possible polarizations on the Peierls-bracket phase space of field theory, and which subspace of the Hamiltonian functions respect these. Maybe that turns out to be the microcausal functionals, that would provide a nice conceptual explanation for the ad hoc restriction to this algebra.
In view of this it seems suggestive that the phase space of a field theory on a globally hyperbolic spacetime is always almost Kähler, due to the existence of Hadamard states, whose 2-point functions serve as the compatible almost Kähler metrics. Now in finite dimension, it is (compact) almost Kähler phase spaces on which geometric quantization becomes equivalent to push-forward in K-theory. Maybe this should be telling us regarding quantization of the field theory.
Now in finite dimension, it is (compact) almost Kähler phase spaces on which geometric quantization becomes equivalent to push-forward in K-theory.
Which reference for this you suggest ?
Back to #2, #3, here is a thought:
If we really go for the Bohr topos, then maybe there is a direct route from geometric pre-quatization.
Recall (say from the discussion at order-theoretic structure in quantum mechanics) that one point of the Bohr topos perspective is to make Gleason’s theorem manifest. This in turn realizes the old observation that led Jordan to consider Jordan algebras, namely the informal observation that with self-adjoint operators regarded as observables, then there is actually no corresponding physical meaning to the operation of taking the product of two such operators, which is formalized by the curious fact that a state in the sense of operator algebra is a linear function on the algebra of operators/observables, which does in fact satisfy no condition concerning the algebra structure except positivity. Gleason’s theorem captures this by saying that quasi-states are already equivalent to states, where a quasi-state is a function on the algebra which is required to be linear and positive only on every commutative subalgebra. All this says that the full algebra structure on the usual algebra of observables is to some extent spurious, and of course the Bohr topos makes this manifest.
Anyway, here is finally the thought: Given this, and given that geometric pre-quantization gives us the quantomorphism group in a very robust way (in that we know to construct the quantomorphism groups in vast generality: non-perturbatively, for field theories which are only locally variational, for higher gauge symmetry included, etc. pp.) and given that the abelian subgroups of the quantomorphism group correspond to the commutative subalgebras of the algebra of observables, maybe the Bohr topos really wants to be built not from the poset of commutative subalgebras of obervables (since we just saw that the algebra structure on observables is spurious as far as quantum physics goes), but directly from the system of abelian sub-groups of the quantomorphism group.
I am already half-way on vacation, but now I realize that I need to check out but forgot to download an institute copy of Il’in-Slavnov 78, doi. Could somebody with access be so kind to send me a pdf copy? Thanks!
;-)
given that the abelian subgroups of the quantomorphism group correspond to the commutative subalgebras of the algebra of observables
If you have time, is there a pointer as to how this works?
Thanks, David!
is there a pointer as to how this works?
Ah, no, that was just a rough idea I had. And maybe it’s wrong.
Next day I had another idea, which looks better: Given that the symplectic form on the covariant phase space is the transgression of the “Lepage gerbe” of the field theory to a Cauchy surface (by section 1.4 of arXiv:1601.05956), the result of arXiv:1603.09626 suggests to find an analogous de-transgression of the Hadamard states (regarded as almost Kähler metric on the covariant phase space) to non-skew tensors on the jet bundle. That should then be the right “higher polarization”. But I still need to understand what these de-transgressions would be, if they exist.
Yes, I was wondering (for different reasons) how one might get a construction for the quantomorphism group analogous to how $\widehat{\Omega G}$ is used to build $String_G$.
So what I am referring to here is one step beyond the quantomorphism group, to the actual quantum information.
Regaring $String_G$, what i can offer is that this turns out to be the Heisenberg 2-group inside the quantomorphism 2-group of the WZW-gerbe-with-connection, regarded as a prequantum 2-bundle (p. 42 of arXiv:1304.0236)
I was more thinking of the general case, whereby there is a natural principal bundle with structure group the Hamiltonian symplectomorphism group, for instance, such that its lifting gerbe uses the quantomorphism group in the same way the basic gerbe on G can be constructed using the principal $\Omega G$-bundle $PG\to G$ and the central extension of $\Omega G$. Not a deep application, just a source for examples for gerbes with “natural” geometric origins.
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