Textbooks like Andrei Moroianu’s Lectures on Kahler Geometry and Mikio Nakahara’s Geometry, Topology and Physics are helpful in joining these pieces together, but definitely there is at present no single textbook (or article either) that grinds through all the details. It would have to be a long one.
For young researchers especially, the present literature gap is perhaps a good thing!
I don’t think you actually answered any of my questions; was that deliberate? Anyway, it seems that (1) the general description in terms of Kähler manifolds is a somewhat nonstandard way of formulating “ordinary” quantum mechanics; (2) J does indeed play the role of i, kinda, since one way you can think about Kähler manifolds is that you start with a symplectic manifold and then give it a local complex structure; (3) yes, M is basically a phase space; (4) you see some great significance in the idea that some Lie derivative of J might be nonzero, but haven’t so far explained (a) whether that is a possibility within standard QM or a generalization beyond standard QM, or (b) along what vector field V you’re taking the Lie derivative (it looks—though I don’t understand this stuff well at all—as if it’s more natural to take the derivative of something else along J, rather than the derivative of J along something else), or (c) why you regard this as importance.
And I still don’t see that there’s any connection between this and Scott’s stuff about free will. (That paragraph you added—is it somehow suggesting that “dynamic-J methods” for simulation can somehow pull out information that according to Scott is in principle inaccessible? Or what?)
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!
Gjm asks “Along 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 manifold for which Bob’s simulation cannot be verifiably distinguished from Alice’s simulation? In particular, do polynomially-many 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 practical engineering consequences? 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 algebraic state-spaces induces Knightian undertainty in quantum unravelings that is computationally indistinguishable from the dynamical effects that are associated to primordial freebits.”
Are these questions interesting? Here is it neither feasible, nor necessary, nor desirable that everyone think alike!
JLM, the mathematically natural answer to your questions is:
• the quantum dynamical framework of (say) Abhay Ashtekar and Troy Schilling’s Geometrical Formulation of Quantum Mechanics arXiv:gr-qc/9706069v1, and
• the quantum measurement framework of (say) Carlton Caves’ on-line notes Completely positive maps, positive maps, and the Lindblad form, both pullback naturally onto
• the varietal frameworks of (say) Joseph Landsberg’s Tensors: Geometry and Applications
Textbooks like Andrei Moroianu’s Lectures on Kahler Geometry and Mikio Nakahara’s Geometry, Topology and Physics are helpful in joining these pieces together, but definitely there is at present no single textbook (or article either) that grinds through all the details. It would have to be a long one.
For young researchers especially, the present literature gap is perhaps a good thing!
(Who’s JLM?)
I don’t think you actually answered any of my questions; was that deliberate? Anyway, it seems that (1) the general description in terms of Kähler manifolds is a somewhat nonstandard way of formulating “ordinary” quantum mechanics; (2) J does indeed play the role of i, kinda, since one way you can think about Kähler manifolds is that you start with a symplectic manifold and then give it a local complex structure; (3) yes, M is basically a phase space; (4) you see some great significance in the idea that some Lie derivative of J might be nonzero, but haven’t so far explained (a) whether that is a possibility within standard QM or a generalization beyond standard QM, or (b) along what vector field V you’re taking the Lie derivative (it looks—though I don’t understand this stuff well at all—as if it’s more natural to take the derivative of something else along J, rather than the derivative of J along something else), or (c) why you regard this as importance.
And I still don’t see that there’s any connection between this and Scott’s stuff about free will. (That paragraph you added—is it somehow suggesting that “dynamic-J methods” for simulation can somehow pull out information that according to Scott is in principle inaccessible? Or what?)
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!
(reposted with proper nesting above)
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 practical engineering consequences? 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 algebraic state-spaces induces Knightian undertainty in quantum unravelings that is computationally indistinguishable from the dynamical effects that are associated to primordial freebits.”
Are these questions interesting? Here is it neither feasible, nor necessary, nor desirable that everyone think alike!