- #1
SupremeFunky
- 3
- 0
Consider the following model.
Put a lattice of N electrical nodes on a sphere. The lattice doesn't have to be perfectly regular. Each node is connected to others by copper wires that run through the interior of the sphere. The wires do not interfere with each other.
In some initial state I, some of the nodes are turned on, and the rest of the nodes are turned off. Each node turns on the other nodes it is connected to, to various degrees, in the next time-step. (So at each successive time-step a node's value is the sum of the contributions from connected nodes that were on in the previous time-step). The set of wires that connects a given node to other nodes is given by a specification that is invariant with respect to where on the sphere the given node is, i. e. the specification is the same for all nodes.
An evolution is the set of states over time, given some initial state I. An evolution is local if the nodes connected to the given node are immediate neighbors (or a subset of them). (“Immediate neighbors” are algorithmically selected using the metric on the sphere.) An evolution is non-local otherwise.
The motivation for this is the following. For lawful evolutions that are non-local, and for many that appear local but are otherwise mysterious, one can give an initial state and a “non-local” specification of wires that gives rise to that evolution. So what? The ontological status of the wires is the same as the ontological status of the electrical nodes. If this sort of model applies to quantum mechanics, one could ontologically interpret some kinds of quantum phenomena using a classical ontology. Further, the physics inside the sphere contains only local correlations. If Bell's theorem does not apply, it would be because the locality takes place in a dimension other than where the non-locality lives.
What is the role played by probabilities? Granting that behavior can be interpreted geometrically, is there a cause for the fact that an electron was observed to have x-spin up, instead of x-spin down? Okay, here is a possible geometrical interpretation. Suppose you have 2 manifolds in space (R^3), one “over” the other as given by a gravitational gradient. You drop Buffon's needle from the top one. The (classical) probability it intersects parallel lines on the bottom manifold is 2l/pw. If the needle landing on a line represents an observation of x-spin up, and landing on a space between lines represents an observation of x-spin down, then the size of the spacing between the lines is a geometrical interpretation of the probabilities for observing x-spin up or else x-spin down. Note the electron does not take on a spin (eigen)value until it lands on the bottom manifold. If the two manifolds are really just a concave part of a larger one, the larger one can represent the laboratory space continuously.
It might be necessary for the specification to evolve in time or to vary continuously over the surface, but these still preserve the ontology.
It goes without saying that to get any of the details of quantum mechanics one would use complex, and complicated, manifolds.
Put a lattice of N electrical nodes on a sphere. The lattice doesn't have to be perfectly regular. Each node is connected to others by copper wires that run through the interior of the sphere. The wires do not interfere with each other.
In some initial state I, some of the nodes are turned on, and the rest of the nodes are turned off. Each node turns on the other nodes it is connected to, to various degrees, in the next time-step. (So at each successive time-step a node's value is the sum of the contributions from connected nodes that were on in the previous time-step). The set of wires that connects a given node to other nodes is given by a specification that is invariant with respect to where on the sphere the given node is, i. e. the specification is the same for all nodes.
An evolution is the set of states over time, given some initial state I. An evolution is local if the nodes connected to the given node are immediate neighbors (or a subset of them). (“Immediate neighbors” are algorithmically selected using the metric on the sphere.) An evolution is non-local otherwise.
The motivation for this is the following. For lawful evolutions that are non-local, and for many that appear local but are otherwise mysterious, one can give an initial state and a “non-local” specification of wires that gives rise to that evolution. So what? The ontological status of the wires is the same as the ontological status of the electrical nodes. If this sort of model applies to quantum mechanics, one could ontologically interpret some kinds of quantum phenomena using a classical ontology. Further, the physics inside the sphere contains only local correlations. If Bell's theorem does not apply, it would be because the locality takes place in a dimension other than where the non-locality lives.
What is the role played by probabilities? Granting that behavior can be interpreted geometrically, is there a cause for the fact that an electron was observed to have x-spin up, instead of x-spin down? Okay, here is a possible geometrical interpretation. Suppose you have 2 manifolds in space (R^3), one “over” the other as given by a gravitational gradient. You drop Buffon's needle from the top one. The (classical) probability it intersects parallel lines on the bottom manifold is 2l/pw. If the needle landing on a line represents an observation of x-spin up, and landing on a space between lines represents an observation of x-spin down, then the size of the spacing between the lines is a geometrical interpretation of the probabilities for observing x-spin up or else x-spin down. Note the electron does not take on a spin (eigen)value until it lands on the bottom manifold. If the two manifolds are really just a concave part of a larger one, the larger one can represent the laboratory space continuously.
It might be necessary for the specification to evolve in time or to vary continuously over the surface, but these still preserve the ontology.
It goes without saying that to get any of the details of quantum mechanics one would use complex, and complicated, manifolds.