Gjm asks “Along what vector field V are you taking the Lie derivative?
The natural answer is, along a Hamiltonian vector field. Now you have all the pieces needed to ask (and even answer!) a broad class of questions like the following:
Alice possesses a computer of exponentially large memory and clock speed, upon which she unravels the Hilbert-space trajectories that are associated to the overall structure
), where M is a Hilbert-space (considered as a manifold), g is its metric, omega is its symplectic form, J is the complex structure induced by
), and
) are the (stochastic,smooth) Lindblad and Hamiltonian potentials that are associated to a physical system that Alice is simulating. Alice thereby computes a (stochastic) classical data-record as the output of her unraveling.
Bob pulls-back
) onto his lower-dimension varietal manifold (per Joseph Landsberg’s recipes), upon which he unravels the pulled-back trajectories, thus obtaining (like Alice) a classical data-record as the output of his unraveling (but using far-fewer computational resources).
Then It is natural to consider questions like the following:
Question For hot noisy quantum dynamical systems (like brains), what is the lowest-dimension varietal state-space for which Bob’s simulation data-record cannot be verifiably distinguished from Alice’s simulation data-record? In particular, do polynomially-many varietal dimensions suffice for Bob’s record to be indistinguishable from Alice’s?
Is this a mathematically well-posed question? Definitely! Is it a scientifically open question? Yes! Does it have engineering consequences (and even medical consequences) that are practical and immediate? Absolutely!
What philosophical implications would a “yes” answer have for Scott’s freebit thesis? Philosophical questions are of course tougher to answer than mathematical, scientific, or engineering questions, but one reasonable answer might be “The geometric foaminess of varietal state-spaces induces Knightian undertainty in quantum trajectory unravelings that is computationally indistinguishable from the Knightian uncertainty that, in Hilbert-space dynamical systems, can be ascribed to primordial freebits.”
Are these questions interesting? Here it is neither feasible, nor necessary, nor desirable that everyone think alike!
The natural answer is, along a Hamiltonian vector field. Now you have all the pieces needed to ask (and even answer!) a broad class of questions like the following:
Alice possesses a computer of exponentially large memory and clock speed, upon which she unravels the Hilbert-space trajectories that are associated to the overall structure
Bob pulls-back
Then It is natural to consider questions like the following:
Is this a mathematically well-posed question? Definitely! Is it a scientifically open question? Yes! Does it have engineering consequences (and even medical consequences) that are practical and immediate? Absolutely!
What philosophical implications would a “yes” answer have for Scott’s freebit thesis? Philosophical questions are of course tougher to answer than mathematical, scientific, or engineering questions, but one reasonable answer might be “The geometric foaminess of varietal state-spaces induces Knightian undertainty in quantum trajectory unravelings that is computationally indistinguishable from the Knightian uncertainty that, in Hilbert-space dynamical systems, can be ascribed to primordial freebits.”
Are these questions interesting? Here it is neither feasible, nor necessary, nor desirable that everyone think alike!