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At computational trilogy I made explicit the quantum version – here.
As with the classical trilogy, it’s evident once you think about it.
Can we think of anything as neat as:
A proof of a proposition. (In logic.)
A program with output some type. (In type theory and computer science.)
A generalized element of an object. (In category theory.)
Something to do with your propositions as projections idea?
Yes, it’s exactly the same dictionary. Just that the types/objects/spaces in the fibers are now linear.
(And we notice that for “generalized element” we may equivalently say “cocycle” in all cases: classical, parameterized, quantum.)
Perhaps a little speculative for this time in the morning, but if we follow Susskind in
The universe is filled with subsystems, any one of which can play the role of observer. There is no place in the laws of quantum mechanics for wave function collapse; the only thing that happens is that the overall wave function evolves unitarily and becomes more and more entangled. The universe is an immensely complicated network of entangled subsystems, and only in some approximation can we single out a particular subsystem as THE OBSERVER.
is there are quantum trilogy which is not classically controlled? Or is it that just ’quantum plain’?
Let me lay out the picture that I have at the moment, also so that you see where you fit in:
Hypothesis H, full quantum version:
For
a $G$-orbi $\mathbb{R}^{10,1\vert \mathbf{32}}$-fold $\mathcal{X}$ (spacetime)
with geometric torsion-free $\mathrm{Spin}(10,1)$-structure (i.e. field of super-gravity)
further lifted to tangential $\mathrm{Sp}(2)$-structure $\tau$ (anomaly cancellation),
consider the cocycle space $\mathbf{\pi}_G^\tau(\mathcal{X})$ of $\tau$-twisted equivariant differential 4-Cohomotopy (i.e. the C-field configuration G-space)
and then the $G$-equivariant $E$-cohomology cocycle space of its loop space $\mathbf{E}^\bullet_G\big( \Omega \mathbf{\pi}_G^\tau(\mathcal{X}) \big)$ (quantum states)
regarded as a $G$-equivariant spectrum (e.g. with $E = H\mathbb{C}$ and $\mathcal{X}$ a $D6 \perp D8$ spacetime as in SS19 this has homotopy groups the weight systems on horizontal chord diagrams)
hence as a spectral Mackey functor
hence as a parameterized spectrum (over $G$-sets) equipped with consistent orientation for pull/push transformations
hence as a cohomological QFT-via-secondary-integral-transforms in the sense of Sec. 7 of Quantization via Linear homotopy types.
Unwinding the definitions, this QFT encodes
quantum operations between quantum states of C-field configurations
associated with $H$-fixed loci of the original spacetime orbifold $\mathcal{X}$.
But since these loci are conical black brane loci (as in HSS 18) in the spacetime,
we see that we have a “classically controlled quantum system” where the classical control parameter says where in a holographic system we are:
in the bulk ($1$-fixed locus)
or on a brane ($H$-fixed locus)
and if so on which of possibly several intersecting ones (choice of subgroup $H \subset G$).
Now take this mathematical picture and compare to the current folklore in string theory. Within the large hand-waving error bars of the latter, this could match well, I think.
Quite a vision!
we see that we have a “classically controlled quantum system” where the classical control parameter says where in a holographic system we are
I think this is the part I’ve heard least about before.
I remember you giving an abstract general nPOV on holography, way back. Yes, here at holographic principle of higher category theory. Untouched for nearly a decade.
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