cfgauss2718

Even if you could get around the payload and energy density (power autonomy) constraints, which are very stringent, this seems like a dead product. Sure, it’s a cool project for some controls engineers. But I can scarcely imagine the deafening noise this thing makes, and no doubt it would tend to kick up clouds of dust anywhere it went, which is hardly desirable from a health perspective. This kind of device would be a permanent irritant in almost any environment. Thankfully, buildings are already well equipped with things like elevators. I’ll keep my wheeled cart, thank you very much.

We know from the No Communication theorem that quantum entanglement does not transmit information between classical observers who have no role in preparing the initial state of the entangled elements. Why should one think that this No Communication theorem is applicable to the phenomenological description of brain behavior? It seems that the speed of nerve conduction is at odds with the time constants of brain synchronization. If we are to doubt that quantum entanglement plays a role in accounting for this discrepancy, then what are the assumptions that underlie this doubt, and are they reasonable?

This series by lecturer Frederic P. Schuller is astounding work of thorough mathematical exposition delivered with clarity and charisma. This body serves as an exemplary introduction and tutorial on mathematics spanning propositional logic, topology, multilinear algebra, and differential geometry, towards the fundamentals of Quantum Mechanics and Gauge theory.

In quantum field theory (standard model of particle physics), photons are the quanta of a field (naturally identified with the quantized electromagnetic field, or U(1) gauge field), which evolves in a nonlinear way that is coupled not only to itself, but also the electron field (as photons are the carrier of the electromagnetic interaction). It seems to me that the concept of “photon” (the particle) is one that is useful in some contexts (like modeling the possible interactions between two electrons in a Feynman diagram), but that the concept of photon is not a fundamental constituent of reality. Certainly, wave packets of the field can interact by way of superposition and thereby altering the space-time evolution of the field through its coupling constant.

I think are overstating the strength of the No Communication Theorem. It does not state that communication via quantum entanglement is impossible in all cases; the theorem is weaker. From the article you linked:

“Being only a sufficient condition there can be extra cases where communication is not allowed and there can be also cases where is still possible to communicate through the quantum channel encoding more than the classical information.”

Consider the quoted assumption (again from the article you shared)

“An important assumption going into the theorem is that neither Alice nor Bob is allowed, in any way, to affect the preparation of the initial state. If Alice were allowed to take part in the preparation of the initial state, it would be trivially easy for her to encode a message into it; thus neither Alice nor Bob participates in the preparation of the initial state.”

Now it’s not clear to me how to correctly map this assumption onto the papers context. Who are the observers? Are they a pair of neurons on the ends of the axon? Are they neighboring myelin sheaths? Is it fair to assert that these observers have no role in preparation of the initial quantum state? Seeing as the entangled photons in the paper are produced as the result of chemical reactions occurring within the confines of the myelin sheaths causing excitation of carbon atoms and subsequent radiation of infrared photons, do the particular assumptions of the No Communication Theorem (a thought experiment that translates those assumptions into definite mathematical propositions) make sense?

Where are we drawing the boundary between these classical observers and the quantum system? Are there even any classical observers at all?

Your yawning dismissal strikes me as an overconfident conclusion.

I love when grammatical mistakes become unintentional puns

Here’s maybe a useful example. Consider a scalar potential function F on R^3 that describes some nonlinear spring law. At a point p=(x,y,z), the differential dF can be thought of as a (1,0) tensor measuring the spring force. It acts on a particle at p moving with velocity v to give the instantaneous work of the particle on the spring dF(p)(v). Now, suppose that we want to know how this quantity changes when we vary the x coordinate. The x coordinate is also a function of p, we can represent its differential as dx, which is a co-vector(field). The quantity that captures this change can be thought of as a (1,1) tensor field, which is related to the stiffness of the spring potential in the x direction at each point p. In the usual undergraduate setting, this tensor field is given as the hessian of F, call this H. The action of this tensor looks like the product u^T H(p) v, where in our case, u^T = dx(p) = [1 0 0]. A good giveaway for when a “co-vector” appears in a tensor calculation is whenever there is a “row vector” in a matrix operation (most people identify “column” vectors with proper vectors). It’s helpful in this case that “row” rhymes with “co-“.

Can you provide some examples of important tensors in physics for which the underlying vector space is infinite dimensional? I’m most familiar with the setting of tensor fields on manifolds, in which case the vector bundle consists of finite dimensional vector spaces. Nevertheless, I suppose in the absence of a pseudo-Riemannian metric one lacks a natural isomorphism between vectors/dual vectors. Does this “bidual” distinction arise in that case as well?

Yes, the “multi linear map” definition is accessible to an undergraduate who has taken linear algebra. However, the more common meaning of tensor in physics, like the metric tensor of spacetime, requires some more sophisticated background to understand (differential geometry, Lie Groups come to mind).

Here here! Functions do not depend on your choice of coordinates, only the components of tensors do! I think this is why it’s important to keep covariance and contravariance in mind. While tensor(fields) do not depend on coordinates intrinsically, the way we represent them when doing calculations most certainly does, and this is usefully characterized by co/contravariance.

I think the point above is that in physics tensor is usually overloaded, and those practicing physicists when they speak of tensors are more often referring to tensor fields, and most often this is in a context with more geometric structure than is required by a tensor space in reference to a vector vector bundle. Typically they (physicists) are dealing with domains where the tensor space is in reference to the tangent bundle of a smooth manifold, with the prototypical example being the metric tensor(field) of space time in general relativity. Another prominent example may include tensor fields defined in reference to the tangent bundle of a group of gauge transformations, as in quantum electrodynamics, quantum chromodynamics, etc.

Obviously these things are not just useful to physics, but are indispensable, and so I think the assertion that only the definition of tensor that is useful to physics is the definition tensor=multilinear map is somewhat out of step. Perhaps it would be better to assert that the concept of multilinear map is essential to every useful definition of tensors in physics.

Very well, let’s just agree that in physics (r,s) tensors usually refer to sections of the tensor product of some fixed number of copies of the tangent bundle (r copies) and cotangent bundle (s copies) of a smooth manifold (almost always pseudo-Riemannian) and leave it there. Elementary!

Agreed, I can’t help but feel there is some overcompensation driving the style of writing. It was difficult to finish.

There are some interesting parallels to ideas in this article and IIT. The focus on parsimony in networks, and pruning connections that are redundant to reveal the minimum topology (and the underlying computation)is reminiscent of parts of IIT: I’m thinking of the computation of the maximally irreducible concept structure via searching for a network partition which minimizes the integrated cause-effect information in the system. Such redundant connections are necessarily severed by the partition.