d'Alembert to Dyson
   Re: sources
d'Alembert to Dyson



d'Alembert

d'Alembert



search tips advanced search
site search by freefind



'D' wing of R&D


The laws of physics must be valid in all systems of coordinates. They must thus be expressible as tensor equations. Whenever they involve the derivative of a field quantity, it must be a covariant derivative. The field equations of physics must all be rewritten with the ordinary derivatives replaced by covariant derivatives. For example, the d’Alembert equation quablaV = 0 for a scalar V becomes, in covariant form

dAlembertian

Dirac

The D'Alembertian is the equivalent of the Laplacian in Minkowskian geometry. It is given by:

d'Alembertian

PlanetMath

blue sphere

wave equation
wave equation


Klein-Gordon equation
Klein-Gordon equation






Leonardo da Vinci
Leonardo da Vinci

Simplicity is the ultimate sophistication.





If the front of a building, or any piazza or field, which is illuminated by the sun, has a dwelling opposite to it, and if in front that does not face the sun you make a small round hole, all the illuminated objects will send their images through that little hole and will appear inside the dwelling on the opposite wall, which should be white. And there they will be, exactly and upside down ... If the bodies are of various colors and shapes, the rays forming the images will be of various colors
and shapes, and of various colors and shapes will be the representations on the wall.


da Vinci

projective geometry


During the Renaissance there was a fruitful collaboration between artists and mathematicians. Artists were developing a method for accurately projecting 3-dimensional objects onto a canvas. This led to a new type of geometric analysis, where the goal is to understand which geometric properties are invariant under change of perspective. In the world of art this method is known as the perspective drawing; in mathematics it is known as projective geometry.

The early students of this new theory were both artists and mathematicians. Among the most notable are Brunelleschi, Alberti, and Durer. The world of art has now moved on. Althugh perspective drawing is still taught in art schools, it is no longer a cutting-edge technology. On the other hand, projective geometry remains one of the cornerstones of modern mathematics. Some of the mathematicians who firrst recognized its wider utility and intrinsic beauty were Bezout, Pascal, and Fermat.

Hewitt



apple of my eye





De Broglie

Louis De Broglie


'D' wing of R&D










































Descartes
René Descartes






























PAM Dirac
Paul Dirac


Main menu













































Dirichlet


Dirichlet


'D' wing of R&D


The question which finally must be answered is knowing (Einstein has often emphasized this point) whether the present interpretation, which uses solely the psi-wave with its statistical character is a "complete" description of reality—in which case it would be necessary to assume indeterminism and the impossibility of representing reality on the atomic level in a precise way in the framework of space and time — or if, on the contrary, this interpretation is incomplete and hides behind itself, as the older statistical theories of classical physics do — a perfectly determinate reality, describable in the framework of space and time by variables which would be hidden from us, i.e., which would escape experimental determination by us.

De Broglie


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

While a proper understanding of M-theory still eludes us, much is now known about it. In particular the various geometric results that have emerged from string theory become related in interesting but mysterious ‘dualities’ whose real meaning has yet to be discovered.

Atiyah


prism


The aspects of things that are most important for us are hidden because of their simplicity and familiarity.
Wittgenstein


Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot- wave model, and the causal interpretation of quantum mechanics, is a version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952. It is the simplest example of what is often called a hidden variables interpretation of quantum mechanics. In Bohmian mechanics a system of particles is described in part by its wave function, evolving, as usual, according to Schrödinger's equation. However, the wave function provides only a partial description of the system.






If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things.

§

I am not merely present in my body as a sailor is present in his ship, but ... am very closely joined and, as it were, intermingled with it, so that I form with it a single entity.


Cartesian coordinates


The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another. And I had little difficulty in determining the objects with which it was necessary to commence, for I was already persuaded that it must be with the simplest and easiest to know, and, considering that of all those who have hitherto sought truth in the sciences, the mathematicians alone have been able to find any demonstrations, that is, any certain and evident reasons, I did not doubt but that such must have been the rule of their investigations.

Descartes



It seems clear that the present quantum mechanics is not in its final form [...] I think it very likely, or at any rate quite possible, that in the long run Einstein will turn out to be correct.


superposition


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

Harmonic Analysis


The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other. The idea is that a general sound wave is a superposition of harmonics, the way a symphony is a superposition of the harmonics corresponding to the notes played by various instruments. Mathematically, this means expressing a given function as a superposition of the functions describing harmonics, such as the familiar functions sine and cosine. Automorphic functions are more sophisticated versions of these familiar harmonics. There are powerful analytic methods for doing calaculations with these automorphic functions. And Langlands' surprising insight was that we can use these functions to learn about much more difficult questions in number theory.

Frenkel


Riemann discovered that the physics of music was the key to unlocking the  secrets of the primes. He discovered a mysterious harmonic structure that  would explain how Gauss's prime number dice actually landed when Nature  chose the primes.

§

What Riemann discovered was that Gauss's graph is like the fundamental note played by an instrument, but that there are special harmonic waves that, when added to this graph, gradually change it into the true graph or "sound" of the primes, just as the harmonics of the clarinet change the sine wave into the square wave.

du Sautoy


[The] Kaluza-Klein states of the compactified 11-dimensional  M-theory are equivalent to the nonperturbative 10-dimensional Dirichlet 0-brane  states.


Dirichlet gave the first rigorous proof of Fourier’s theorem. Riemann (1868) says that this paper was the first in which the fact that Fourier series are non-absolutely convergent was noticed and dealt with correctly.

Riemann states this in the historical introduction to his paper on Fourier analysis, which was his Habilitationsschrift (probationary essay) at Göttingen in 1854. It remained unpublished until 1868, when Dedekind discovered it after Riemann’s death. Riemann actually says more: He says that Dirichlet was the first person to discover the phenomenon of conditional convergence. This, however, cannot be true. Cauchy was evidently aware of the difference between absolutely and conditionally convergent series, and we already noted that Abel took careful account of this distinction in 1826. It is true, however, that Dirichlet begins his paper by pointing out that one of Cauchy’s attempted proofs of convergence for Fourier series fails on just this point. Riemann based his history in this paper on a long conversation with Dirichlet, so this probably represents Dirichlet’s memory of the essential difficulty in the proof. Dirichlet’s proof worked for functions that were “piecewise monotonic,” and guaranteed convergence at points of continuity of such functions. Dirichlet believed the proof could be extended to prove convergence for any continuous function.
Offner


Vibrating Drum

Vibrating strings and membranes produce
characteristic (
eigen) sound vectors.








Freeman Dyson
Freeman Dyson

Mind and intelligence are woven into the fabric of our universe in a way that altogether surpasses our comprehension.


This is the characteristic mathematical property of a classical field: it is an undefined something which exists throughout a volume of space and which is described by sets of numbers, each set denoting the field strength and direction at a single point in the space.

§

There is nothing else except these fields: the whole of the material universe is built of them.


field



The universe shows evidence of the operations of mind on three levels. The first level is elementary physical processes, as we see them when we study atoms in the laboratory. The second level is our direct human experience of our own consciousness. The third level is the universe as a whole. Atoms in the laboratory are weird stuff, behaving like active agents rather than inert substances. They make unpredictable choices between alternative possibilities according to the laws of quantum mechanics. It appears that mind, as manifested by the capacity to make choices, is to some extent inherent in every atom. The universe as a whole is also weird, with laws of nature that make it hospitable to the growth of mind.

§

It would not be surprising if it should turn out that the origin and destiny of the energy in the universe cannot be completely understood in isolation from the phenomena of life and consciousness.

§

Fourscore and seven years ago, Erwin Schrödinger invented wave-functions as a way to describe the behavior of atoms and other small objects. According to the rules of quantum mechanics, the motions of objects are unpredictable. The wave-function tells us only the probabilities of the possible motions. When an object is observed, the observer sees where it is, and the uncertainty of the motion disappears. Knowledge removes uncertainty. There is no mystery here.

Unfortunately, people writing about quantum mechanics often use the phrase "collapse of the wave-function" to describe what happens when an object is observed. This phrase gives a misleading idea that the wave-function itself is a physical object. A physical object can collapse when it bumps into an obstacle. But a wave-function cannot be a physical object. A wave-function is a description of a probability, and a probability is a statement of ignorance. Ignorance is not a physical object, and neither is a wave-function. When new knowledge displaces ignorance, the wave-function does not collapse; it merely becomes irrelevant.
Dyson