Helmholtz observed long ago that “similar light produces, under like conditions, a like sensation of color.” color is, of course, one of the “secondary qualities” of Locke, which he thought had only a mental existence. Why, then, discuss color in a scientific forum? Well, because color is simply the wavelength of light, right? Well, no, contrary to what “everyone knows,” it is not. We can both broaden and tighten his observation from with a little help from Heisenberg, and say that the same state vector [psi], acted upon by the same operators A, B, C, produces the same spectrum of secondary qualities. That is, the new state vector [phi] has an entirely predictable spectrum of these properties. Looking ahead a bit, recall that spectral analysis is all to do with matrices.
What has been gained from casting the observation from Helmholtz in the language of Heisenberg? Perhaps a great deal, for as the mathematician Steen reminds us, early on in the history of quantum theory, the “mathematical machinery of quantum mechanics became that of spectral analysis.”
A slightly subtler but fundamentally significant consideration attends the symmetries embodied by the vectors and (matrix) operators — symmetries manifest in the secondary qualities — for here the subject opens out into geometry, the action principle, gauge theory, Noether's theorem, and the Lagrangian and Hamiltonian formulations of the equations of motion — and so all of physics and much of mathematics.
A foundational question arises: Do the “secondary” symmetries contribute to the action, thus making them plausible candidates for supplying the values of “hidden” variables? Hearkening back to Maxwell on the absolute simplicity of these sensations, it seems clear enough that, in some sense, the secondaries are "elements" of reality.
In adopting Heisenberg’s matrix mechanics, we need only enlarge the number of dimensions needed to incorporate these readily observed “elements of reality,” but these days we are quite accustomed to dealing with additional dimensions. At a glance, it seems as though our use of matrices might find a simple, intuitive interpretation in M(atrix) theory.
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Helmholtz observed long ago that “similar light produces, under like conditions, a like sensation of color.” color is, of course, one of the “secondary qualities” of Locke, which he thought had only a mental existence. Why, then, discuss color in a scientific forum? Well, because color is simply the wavelength of light, right? Well, no, contrary to what “everyone knows,” it is not. We can both broaden and tighten his observation from with a little help from Heisenberg, and say that the same state vector [psi], acted upon by the same operators A, B, C, produces the same spectrum of secondary qualities. That is, the new state vector [phi] has an entirely predictable spectrum of these properties. Looking ahead a bit, recall that spectral analysis is all to do with matrices. What has been gained from casting the observation from Helmholtz in the language of Heisenberg? Perhaps a great deal, for as the mathematician Steen reminds us, early on in the history of quantum theory, the “mathematical machinery of quantum mechanics became that of spectral analysis.” A slightly subtler but fundamentally significant consideration attends the symmetries embodied by the vectors and (matrix) operators — symmetries manifest in the secondary qualities — for here the subject opens out into geometry, the action principle, gauge theory, Noether's theorem, and the Lagrangian and Hamiltonian formulations of the equations of motion — and so all of physics and much of mathematics. A foundational question arises: Do the “secondary” symmetries contribute to the action, thus making them plausible candidates for supplying the values of “hidden” variables? Hearkening back to Maxwell on the absolute simplicity of these sensations, it seems clear enough that, in some sense, the secondaries are "elements" of reality. In adopting Heisenberg’s matrix mechanics, we need only enlarge the number of dimensions needed to incorporate these readily observed “elements of reality,” but these days we are quite accustomed to dealing with additional dimensions. At a glance, it seems as though our use of matrices might find a simple, intuitive interpretation in M(atrix) theory.