

Smart Remarks 1: Einstein, Bohr, Bohm, Feynman, Dirac, Atiyah & Weyl 




We are accustomed to regarding as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perception. Einstein
Bohr Every
great and deep difficulty bears
[It] was found possible to account for the atomic stability, as well as for the empirical laws governing 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 socalled stationary quantum states and that, in particular, the spectra were emitted by a steplike 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 It is just like the mathematics of the addition of vectors, where (a, b, c) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians. In fact, Schrödinger wrote a wonder ful paper on color vision in which he developed this theory of vector analysis as applied to the mixing of colors. Feynman
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. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory. 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. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory. Dirac
Gauge theory first appeared in physics in the early attempt by H. Weyl to unify general relativity and electromagnetism. Weyl had noticed the conformal invariance of Maxwell's equations and sought to exploit this fact by interpreting the Maxwell field as the distortion of relativistic length produced by moving around a closed path. Weyl's interpretation was disputed by Einstein and never generally accepted. However after the advent of quantum mechanics with its allimportant complex wavefunctions it became clear that phase rather than scale was the correct concept for Maxwell's equations, or in modern language that the gauge group was the circle rather than the multiplicative numbers. Atiyah
Our discussion will focus on compactified string theory, which as we shall see, requires the compact portion of spacetime to meet certrain stringent constraints. Although there are more general solutions, we shall study the case in which the extra “curledup” dimensions fill out an ndimensional manifold that has the following properties: • it is compact, where d = n/2. For much of these lectures, n will be 6 and hence d = 3. Manifolds which meet these conditions are known as CalabiYau manifolds, for reasons which will become clear shortly. Greene
Simple CaabiYau space 

If
we knew what it was we were doing, I believe that the first step in the setting of a "real external world" is the formation of the concept of bodily objects and of bodily objects of various kinds. Out of the multitude of our sense experiences we take, mentally and arbitrarily, certain repeatedly occurring complexes of sense impression (partly in conjunction with sense impressions which are interpreted as signs for sense experiences of others), and we attribute to them a meaning—the meaning of the bodily object. Considered logically this concept is not identical with the totality of sense impressions referred to; but it is an arbitrary creation of the human (or animal) mind. On the other hand, the concept owes its meaning and its justification exclusively to the totality of the sense impressions which we associate with it. Einstein
Bohr suggests that thought involves such small amounts of energy that quantumtheoretical limitations play an essential role in determining its character. Bohm
Many of the foundations of wave mechanics are based on the analyses and equations that Rayleigh derived for the theory of acoustics in his book The Theory of Sound.
Erwin Schrödinger, a pioneer in quantum mechanics, studied this book
and was familiar with the perturbation methods it describes. Masters
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
To monochromatic light corresponds in the acoustic domain the simple tone. Out of different kinds of monochromatic light composite light may be mixed, just as tones combine to a composite sound. This takes place by superposing simple oscillations of different frequency with definite intensities. Weyl
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. Weyl
§
If is a derived quantity instead of a fundamental one, our whole set of ideas about uncertainty will be altered: is the fundamental quantity that occurs in the Heisenberg uncertainty relation connecting the amount of uncertainty in a position and in a momentum. This uncertainty relation cannot play a fundamental role in a theory in which itself is not a fundamental quantity. I think one can make a safe guess that uncertainty relations in their present form will not survive in the physics of the future. Dirac
Dirac
The characteristic of an ndimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colors, tones) may be specified by the giving of n quantities, the "coordinates," which are continuous functions within the manifold. Weyl Weyl
Epistemologically it is not without interest that in addition to ordinary space there exists quite another domain of intuitively given entities, namely the colors, which forms a continuum capable of geometric treatment. §
Weyl




