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Aharonov to Churchland


Yakir Aharonov

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In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning.

In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the SchrÖdinger equation for a wave. This SchrÖdinger equation is obtained from a canonical formalism, which cannot be expressed in terms of the fields alone, but which also requires the potentials. Indeed, the potentials play a role, in the SchrÖdinger equation, analogous to that of the index of refraction in optics.

Aharonov & Bohm

EM Potential

We now proceed to the derivation of the laws of geometrical optics from the laws of wave optics. A rigorous and general derivation, however, requires a considerable amount of advanced mathematics so that we shall not go into details here. We shall be satisfied with observing the essential points by taking the simple example given above.

The problem is again that of the refraction of monochromatic light [...]


holds, as is apparent from the figure. In other words, the refractive index n is inversely proportional to the wavelength of the light wave in the respective media; i.e.,

n = k''/l

The proportionality constant k'', however, may depend on the frequency or, in other words, on the color of the light.


Furthermore, and now this is the point, this is the punch
line, the symmetries determine the action. ~Weinberg

Michael Atiyah
Michael Atiyah

In ordinary Minkowski space R3,1, the electromagnetic force is described by a 2-form (skew-symmetric 2-tensor) omega. In this notation, Maxwell’s equations in vacuo are

domega = 0, domega = 0,

where now ∗ is defined using the Lorentz metric in R3,1. Formally they are the same as Hodge equations for forms of degree 2 on a 4-dimensional Riemannian manifold, but here the space is ordinary Minkowski space, not the 4-dimensional Euclidean space. From this it appears that Maxwell’s equations, which unified electricity and magnetism, also encode a duality between electricity and magnetism in the sense that the ∗ operator interchanges both aspects. Physically, this is a very fundamental fact of the universe.


Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities.


The principle of duality in projective geometry states that we can interchange point and line in a theorem about figures lying in one plane and obtain a meaningful statement. Moreover, the new or dual statement will itself be a theorem—that is, it can be proven. On the basis of what has been presented here we cannot see why this must always be the case for the dual statement. However, it is possible to show by one proof that every rephrasing of a theorem of projective geometry in accordance with the principle of duality must be a theorem. This principle is a remarkable characteristic of projective geometry. It reveals the symmetry in the roles that point and line play in the structure of that geometry.


JS Bell
JS Bell

[...] conventional formulations of quantum theory, and of quantum field theory in particular, are unprofessionally vague and ambiguous. Professional theoretical physicists ought to be able to do better. Bohm has shown us a way.

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Theoretical physicists live in a classical world, looking out into a quantum-mechanical world. The latter we describe only subjectively, in terms of procedures and reults in our classical domain. This subjective description is effected by means of quantum-mechanical state functions state functions, which characterize the classical conditioning of  quantum-mechanical systems and permit predictions about subsequent events at the classical level. [...]

Now nobody knows just where the boundary between the classical and quantum domains is situated. Most feel that experimental switch settings and pointer readings are on this side. But some would think the boundary nearer, others would think it farther, and many would prefer not to think about it.

JS Bell

Quantum theory is the most successful scientific theory of all time. Many of the great names of physics are associated with quantum theory. Heisenberg and Schrödinger established the mathematical form of the theory, while Einstein and Bohr analysed many of its important features. However, it was John Bell who investigated quantum theory in the greatest depth and established what the theory can tell us about the fundamental nature of the physical world.

Moreover, by stimulating experimental tests of the deepest and most<br> profound aspects of quantum theory, Bell's work led to the possibility of exploring seemingly philosophical questions, such as the nature of reality, directly through experiments.


Non-locality in quantum theory is discussed in terms of the global form of the wave function, and as a subtle set of necessary and sufficient conditions on the 2-matrix or reduced density matrix. In addition to manifesting itself through the well known Bell's inequalities non-locality also appears as a macroscopic coherence length in condensed and coherent systems. By examining the structure of the 2-matrix, a connection between these two forms of non-locality is made. It is suggested that subtle enfolded orders and non-local forms may have a wider implication and be relevant for a variety of living systems.

F. David Peat

It is shown that the matrix models which give non-perturbative definitions of string and M-theory may be interpreted as non-local hidden variables theories in which the quantum observables are the eigenvalues of the matrices while their entries are the non-local hidden variables. 

Lee Smolin

'D' wing of R&D

Niels Bohr

In our description of nature the purpose is not to disclose the real essence of phenomena but only to track down as far as possible relations between the multifold aspects of our experience.

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The mathematical machinery of quantum
became that of spectral analysis... ~Steen

[It] was found possible to account for the atomic stability, as well as for the empirical laws govern- ing the spectra of the elements, by assuming that any reaction of the atom resulting in a change of its energy involved a complete transition between two so-called stationary quantum states and that, in particular, the spectra were emitted by a step-like process in which each transition is accompanied by the emission of a monochromatic light quantum of an energy just equal to that of an Einstein photon.


Bohr suggests that thought involves such small amounts of energy that quantum-theoretical limitations play an essential role in determining its character.


Niels Bohr brainwashed a whole generation of physicists into believing that the problem (of the interpretation of quantum theory) had been solved fifty years ago.


David Bohm
David Bohm

One may then ask what is the relationship between the physical and the mental processes? The answer that we propose here is that there are not two processes. Rather, it is being suggested that both are essentially the same.

Bohm & Hiley

But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the ‘observer,’ could be eliminated. ...

But why then had Born not told me of this ‘pilot wave’? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing ‘‘impossibility’’ proofs, after 1952, and as recently as 1978? ... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

JS Bell

Max Born
Max Born

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Anyone dissatisfied with these ideas may feel free to assume that there are additional
parameters not yet introduced into the theory which determine the individual event.

I am now convinced that theoretical physics is actually philosophy. [...] I believe there is no philosophical high-road in science, with epistemological signposts. No, we are in a jungle and find our way by trial and error, building our road behind us as we proceed.


Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory yields a lot, but it hardly brings us any closer to the secret of the Old One. In any case I am convinced that He doesn't play dice.

Einstein, Letter to Born

Eugenio Calabi

red sphere green sphere blue sphere

Lie group methods have proven to play a vital role in modern research in computer vision and engineering. Indeed, certain visually-based symmetry groups and their associated differential invariants have, in recent years, assumed great significance in practical image processing and object recognition.


Our approach to differential invariants in computer vision is governed by the following philosophy. We begin with a finite-dimensional transformation group G acting on a space E, representing the image space, whose subsets are the objects of interest. In visual applications, the group G is typically either the Euclidean, affine, similarity, or projective group. We are particularly interested in how the geometry, in the sense of Klein, induced by the transformation group G applies to (smooth) submanifolds contained in the space E. A differential invariant I of G is a real-valued function, depending on the submanifold and its derivatives at a point, which is unaffected by the action of G. In general, a transformation group admits a finite number of fundamental differential invariants, I1 ... IN, and a system of invariant differential operators D1 ... Dn, equal in number to the dimension of the submanifold, and such that every other differential invariant is a function of the fundamental differential invariants and their successive derivatives with respect to the invariant differential operators. This result dates back to the originalwork of Lie (1884) [...] For example, in the Euclidean geometry of curves in the plane, the group action is provided by the Euclidean group consisting of translations and rotations, and every differential invariant is a function of the Euclidean curvature and its derivatives with respect to Euclidean arc length.

Calabi et al.


Elie Cartan

Elie Cartan is one of the most influential of 20th-century geometers. At one point he had an intense correspondence with Einstein on general relativity.  His "Cartan geometry" refraction idea is an approach to the concept of parallel transport that predates the widely used Ehresmann approach (connections on principal bundles).  It simultaneously generalizes Riemannian geometry and Klein's Erlangen program, in which geometries are described by their symmetry groups:


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David Chalmers

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The abstract notion of information, as put forward by Claude E. Shannon of MIT, is that of a set of separate states with a basic structure of similarities and differences between them. We can think of a 10-bit binary code as an information state, for example. Such information can be embodied in the physical world. This happens whenever they correspond to physical states (voltages, say); the differences between them can be transmitted along some pathway, such as a telephone line.

color vectors

We can also find information embodied in conscious experience. The pattern of color patches in a visual field, for example, can be seen as analogous to that of pixels covering a display screen. Intriguingly, it turns out that we find the same information states embodied in conscious experience and in underlying physical processes in the brain. The three-dimensional encoding of color spaces, for example, suggests that the information state in a color experience correspond directly to an information state in the brain. We might even regard the two states as distinct aspects of a single information state, which is simultaneously embodied in both physical processing and conscious experience.


Patricia Churchland
Patricia Smith Churchland

A complex of coefficients of this type is comparable with a matrix such as occurs in linear algebra. ~Heisenberg


A trained-up network is one in which, for appropriate input vectors, the network gives the correct response, expressed in terms of an output vector. Training up a network involves adjusting the many weights so that this end is achieved. This might be done in a number of different ways. One might hand-set the weights, or the weights might be set by a back- propagation of error or by an unsupervised algorithm. Weight configurations too are characterizable in terms of vectors, and at any given time the complete set of synaptic values defines a weight state space, with points on each axis specifying the size of a particular weight. [...]

It is conceptually efficient to see the final resting region in weight space as embodying the total knowledge stored in the network. Notice that all incoming vectors go through the matrix of synaptic connections specified by that weight-space point. [...]

A matrix is an array of values, and the elements of an incoming vector can be operated on by some function to produce an output vector.


Paul Churchland
Paul Churchland

When a state is formed by the superposition of two other states, it will have properties that are in some vague way intermediate between those of the original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process.

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What we are looking at, then, is a multistage device for successively transforming an initial sensory activation vector into a sequence of subsequent activation vectors embodied in a sequence of downstream neuronal populations. Evidently, the basic mode of singular, ephemeral, here-and-now perceptual representation is not the propositional attitude at all; it is the vectorialinference drawn from one propositional atttitude to another; it is the synapse-induced transformation of one vectorial attitude into another, and into a third, a forth, and so on, as the initial sensory information ascends the waiting information-processing hierarchy. attitude. And the basic mode of information processing is not the

That highly trained processing hierarchy embodies the network's general background knowledge of the important categories into which Nature divides itself and many of the major relations between them. That is to say, the brain's basic mode of representing the world's enduring structure is not the general or universally quantified propositional attitude at al; it is the hard-earned configuration of weighted synaptic connections, those that transform the activation vectors at one neuronal population into the activation vectors of the next. It is these myriad connections that the learning process was originally aimed at configuring, and it is these connections that subsequently do the important computational work of the matured network.


[The] tensor calculus emerges as the natural framework with which to address such matters ...

unit sphere


color sphere