Posts Tagged ‘Riemann’


August 8, 2013

Abstract: simple considerations of a philosophical, non computational, nature, on Space, Time and the Quantum show that the former two are not basic (and that some apparently most baffling traits of the Quantum are intuitive!). Progress in knowledge of the interdependence of things should not be hampered by traditional prejudices. (Not an easy essay: readers are encouraged to jump around it like kangaroos!)


What is time? Today’s physics does not answer that question, it just computes with the notion as if it were obvious. To find out what time could be, a little bout of metaphysics different from the tentative one in today’s understanding of nature, is needed.

Einstein amplified the notion that the universe is about spacetime (x,t) in a reference frame F. He, and his friends Hilbert and Besso used the mathematical, and physical ideas, created by Riemann (and his Italian successors: Ricci, Levi-Civita, etc.)

"Solitary and Uncomprehended Genius"

Riemann: “Solitary and Uncomprehended Genius” (Poincaré said)

Lorentz discovered one had to assume that (x’,t’) in a moving frame F’ cruising by at a steady speed v is related to (x,t) in frame F according to the Lorentz transformations.

Lorentz got the Nobel Prize, for finding these (thanks to the recommendation of the towering Henri Poincaré); I am not pointing this out to compare the relative merits of celebrities, but to establish the hierarchy of the discoveries they made, and thus the logic therein. (Poincaré’s 1904“Principe de Relativite’” was firmly established before Einstein showed up on the scene, and the latter’s contributions, although enlightening, have been vastly overestimated.)

Not that the initial logic of a discovery always perdures, but sometimes it’s important. The Einstein cult has been obscuring reality; Einstein would have been the first one to decry it (Einstein basically ran away with the idea of Poincaré that the constancy of the speed of light, c, being always observed, was thus a fundamental law of physics, and made it the foundation of what Poincare’ called “Relativite'”).

Only by using the Lorentz transformations are the equations of electrodynamics preserved. In other words: only thus is the speed of light measured to be c in both F, using (x,t) and F’, using (x’,t’).

So what is time t?

According to the scheme in Relativity, it’s simple: given the sanctity of the speed of light, c, and space x, time can be measured by having a photon of light going between two perfect mirrors, and counting the impacts (that’s what is called a light clock; it’s very useful to derive most equations of Relativity).

Indeed space is measured by the time it takes light to go back and forth. This sounds like a circular logic: time is needed to measure space and space is needed, to measure time.

Does that mean one of the two, say, time, is derivative?

I used to think so (propped by the lack of time in Quantum Theory, see below). But, actually, no.

Indeed, time can be localized down to the proton scale.

One can measure time at that scale with how long it takes some elementary particle to decay. Or because to any particle is associated its De Broglie wave, hence a frequency (and that particle can be confined in as small a space as a proton).

Basically time can be measured at a point.

However, space, by definition is… non local (space is always an extent, all the more if time is used to measure it, thanks to c; technically my idea is that space depends upon the holonomy group, time does not; thus Minkowsky’s “spacetime” belongs to the dustbin!).

Thus the conceptual universe in which bask electromagnetism makes it look as if, somehow, time was more fundamental.

The situation is the exact opposite in Quantum Theory. Quantum Theory is full of entangled situations. Measure such a situation somewhere, and it changes all over. “Measure such a situation somewhere, and it changes all over” means that a Quantum Process is all over it. Whatever “it” is. Einstein called that “spooky interaction at a distance”. I call it the QUANTUM INTERACTION.

Einstein tried to escape the spookiness. Instead, I claim it should be embraced. After all, Quantum spookiness makes life possible.

We indeed know now that this spooky Quantum interaction is fundamental to life. It allows life to be more efficient than any understanding from classical mechanics could have it. Vision and the chlorophyll molecule use Quantum spookiness at a distance. This recent discovery did not surprise me at all. I fully expected it, just as I fully expect that consciousness will be revealed to be a Quantum effect (an easy prediction, at this point, in this Quantum universe!)

A computer using the Quantum Theory would be more efficient, for the same reason: the Quantum computer computes all over, in a non local way. (The computers we have now are just sleek electron-using versions of the classical computers the ancient Greeks had, with their little teethed wheels; the Quantum computer is founded on a completely different process.)

This “spooky” non locality has alarmed many a thinker. But notice this simple fact: space itself, even the classical space used in electromagnetism, is non local (as one uses light travel, plus time, to determine space).

So it’s only natural that space in Quantum Theory be non local too.

The “spookiness” is easily understood thus: spacetime physics a la Einstein and company singles out a particular interaction, electromagnetism, and the sanctity of c, to measure the universe with. Why this one, and not another of the fundamental interactions we know?

Quantum Theory (QT) gets out of this would-be choice by choosing none of the traditional forces to measure space with!

As QT has it, as it stands, QT does not need to measure the universe. (I believe it does, using the Quantum Interaction, and I can support that with impossible simultaneous measurements at great distances, but that’s another, more advanced set of considerations.)

Those who think thinking is reduced to computing will object that it is not the same type of non locality (the one I claim to see in classical space and the “spooky” one of Quantum space). Whatever: the non locality in quantum Theory does not depend upon light speed. That’s the important point.

There, the lesson cannot be emphasized enough: on the face of it, the basic set-up of Quantum Theory tells us that light, and, in particular light speed, is NOT fundamental.

This few observations above should they prove to be as deep and correct as I believe they are, show the power of the philosophical method, even in today’s physics. Some will scoff, but not consider carefully all the philosophy behind spacetime a la Einstein.

A warning for those who scoff about the importance of meta-physics: the founding paper of differential geometry in mathematics, and physics, was a lecture by Bernhard Riemann. It’s full of metaphysics and metamathematics, for the best.

The paper had just one equation (and it is a definition!)

That lecture was entitled “Über die Hypothesen welche der Geometrie zu Grunde liegen (“On The Hypotheses Which Underlie Geometry“). (Call these “hypotheses” meta-geometrical, metamathematical, or metaphysical.)

The lecture was published in 1868, two years after his author’s death (and 14 years after he gave it). Riemann’s main idea was to define manifolds and curvature. (Riemannian) manifolds were defined by a metric. Curvature ought to be a tensor, Riemann said, not just a simple number (scalar; as Gaussian curvature).

From top to bottom: positive, negative and no curvature.

From top to bottom: positive, negative and no curvature.

Riemann generalized the notion of curvature to any dimension, thanks to the Riemann Curvature Tensor (the simplified Ricci form of which appears in Einstein’s gravitational field equation).

Here is for some meta-physics; Riemann: “It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry… The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience.

Gauss, Riemann’s teacher, knew this so well that he had tried to measure the curvature of space, if any, using a triangle of tall peaks. Gauss found no curvature, but now we know that gravitation is best described as curved spacetime.

(This lack of Gaussian curvature shows that it’s not because situation is not found under some conditions that it is not there under other conditions; in biology the proof by Medawar that Lamarckism was false, using mice, for which he got the Nobel (being British, ;-)) comes to mind: no Lamarckism in Medawar experiments did not prove that there would be no Lamarckism in other experiments; now four Lamarckist mechanisms are known!)

Twentieth Century physics, in particular the theory of gravitation, exploits the following fact, understood by Riemann as he laid, dying from tuberculosis in Italy. Force is a tautology for geodesics coming closer (or not). Thus curvature is force.

Einstein remarkably said: “Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible.”

(I find this statement all the more remarkable and prophetic in that it is not in Einstein’s physics, and could not be, but rather in the one I would like to have, where fundamental dynamic processes literally create space…)

The fact that a tautology is at the heart of Einstein’s Theory of Relativity means that it explains nothing much! (Relativity fanatics are going to hate that statement!…although it describes very well what happens to objects evolving in spacetime, especially GPS, let it be said in passing.)

“Only to be deduced from experience”, said mathematician Riemann. What’s the ultimate experience we have? Quantum Theory. And what did we find QT said? You can’t measure with space, you can’t measure with time (although clearly the Quantum depends upon the differential topology of the situation, see the Bohm-Aharanov effect! where, by the way, the space metric is made fun of once again!)

Last splendid idea from Riemann (1854-1866):

“Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.”



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