“But you have correctly grasped the drawback that the continuum brings. If the molecular view of matter is the correct (appropriate) one, i.e., if a part of the universe is to be represented by a finite number of moving points, then the continuum of the present theory contains too great a manifold of possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with the quantum theory. The problem seems to me how one can formulate statements about a discontinuum without calling upon a continuum (space-time) as an aid; the latter should be banned from the theory as a supplementary construction not justified by the essence of the problem, which corresponds to nothing “real”. But we still lack the mathematical structure unfortunately. How much have I already plagued myself in this way!”

It's from a 1916 letter to Hans Walter Dällenbach, a former student of Einstein. (Unfortunately the letter is not available online.) I hadn't been aware Einstein thought (at least then) that a continuous space-time is not “real.” It's an interesting piece of history.

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I am really impressed that Einstein, back in 1916, had the modern insight that quantum mechanics implies granular space-time (or at least, the difficulty of a continuous model.) He may even have been thinking about how gravity could work into this. (Note the existence of early thought experiments like "Einstein's box" involving gravity and QM.)

I wonder whether this might have been connected to the hidden-variable quantum theory he was toying with early on. (I'm not sure whether it this time or not -- and haven't had quite enough coffee to try to figure it out . . . )

Physicalist,

That's an interesting comment, for I was thinking about hidden-variable theories when I came across this quotation...

Bee,

Taking a quick look, it seems that the paper I had in mind was from 1927. But still, I would not be at all surprised to learn that he had similar worries/thoughts in mind a decade earlier.

What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics?

See for example "The Foundation of the General Theory of Relativity” On page 185 Einstein says "

the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy". It’s a concentration of energy that causes gravity. Matter only causes gravity because of the E=mc² energy content.In Einstein’s 1920 Leyden address he talks about the stress-energy of space itself, and says its inhomogeneous:

"

This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that “empty space” in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions has, I think, finally disposed of the view that space is physically empty.

"But therewith the conception of the ether has again acquired an intelligible content, although this content differs widely from that of the ether of the mechanical undulatory theory of light. The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities, but helps to determine mechanical and electromagnetic events.

Note that gravitational energy is problematical. Penrose talks about this in his big opus

The Road to Reality: A Complete Guide to the Laws of the Universe. Don't have quote, but he notes (and has been discussed here) that space-time ripples don't carry energy in the conventional sense of EM waves, and even to define total energy in GR context is ambiguous etc.In dense aether theory the space-time must always remain inhomogeneous at least a bit - or it couldn't serve for energy spreading via transverse waves.

Try to imagine Universe like the dense gas. If this Universe would be quite homogeneous, no energy could propagate through at distance along density fluctuations. But if such density fluctuations would exist, then the space-time formed by them would always contain some insintric inhomogeneity, at least infinitesimal one.

It's similar to observation of laser beam in foggy atmosphere: if this atmosphere would be completely smooth, the beam would remain invisible for us. But if such beam is visible, we can be sure, some portion of energy will be always dispersed. Then the visibility of things is conditioned by their inhomogeneity - and the same applies to space, regarding its ability to propagate energy at distance.

And why we are perceiving space-time with transverse waves only? These waves are always spreading with the lowest speed possible through inhomogeneous environment, thus making Universe as large, as possible.

I think one should keep in mind that when Einstein wrote this he did not (and could not) know that discrete states of matter (a state with 17 electrons, for eaxmple) are excitations of a continuous quantum field. So maybe space time is discrete, but certainly not for the reason Einstein mentions.

THE SURFACE of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a “ neighbouring” one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing jumps.” I am sure the reader will appreciate with sufficient clearness, what I mean here by “neighbouring” and by “jumps” (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.Albert Einstein (1879–1955).Relativity: The Special and General Theory.1920.XXIV. Euclidean and Non-Euclidean Continuum

I wonder if someone can explain Einsteins description above.

Somehow this reminds me of Joao's description of Einstein's Bovine dream. How perception when looking at an aquarium from different angles. Can't quite remember it in full....hmmmm.

Why invent topology?

Best,

oops...I mean how did we discover topology?:)

Objectively if one is considered with phenomenology how is such an expression given that one could see this in a way that one could by analogy accepted differing version to encapsulate this movement of expression?

2.3.1

The Word "continuum"

Many people use the phrase "the continuum" to mean the real number line but be aware that there are many types of continua, including:

* One dimensional linear continua - line, line segment, curve (path of a point)

* Euclidean space - one, two, three, or n dimensional

* Einstein's four-dimensional spacetime

* String theory's ??-dimensional spacetime

* Any manifold (definition of `manifold' is at UC Davis and Eric's Treasure Trove)

Of course I tangle with the definitions and it's applications in the real world. A single photon?

Best,

The Single Photon Experiment at Rowan University is a Success!

A FAREWELL TO COPENHAGEN?

Calculus assumes sufficient magnification smooths a function into analysis. Fractal and self-similar functions do not smooth. A discontinuum Velcros with a coherent discontinuous probe. Atomic mass distributions self-similarly sum to centimeter radii as single crystals.

Inverse chiral atomic mass distributions experience inverse torques translating through discontinuous space. Chiral emergence is four atoms, a ~0.3 nm diameter sphere for light elements. Silicon dioxide obtains as an amorphous glass. It also crystallizes in enantiomorphic space groups P3(1)21 (right-handed screw axis) and P3(2)21 (left-handed screw axis). Stereogram.

HK Moffat, "Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory," pp. 175-6 in R Balian & J-L Peube (eds),

Fluid Dynamics(Gordon and Breach, 1977)http://www.igf.fuw.edu.pl/KB/HKM/PDF/HKM_027_s.pdf

pdf pp. 25-27, calculation of the chiral case.

Otherwise identical solid single crystal quartz balls, space groups P3(1)21 and P3(2)21, plus an amorphous fused silica ball are coated with superconductor and Meissner effect-levitated in hard vacuum at 45 degrees latitude. Levitated superconducting dual sphere gravimeters are unremarkable. Crystal balls counter-rotating with diurnal phasing, fused silica as zero rotation control, will detect a discontinuum.

Somebody should look. The worst it can do is succeed.

Wow, I get a completely different feeling. Did I get it it wrong Bee, as follows:

What I get is he's busting on Copenhagen again, the Interpre... , sorry, I meant the Copenhagen MIS-interpretation of Quantum Mechanics (but bad as it is, its the best we got. Damned positivists.).

However, Copenhagen wasn't postulated yet in 1917, so that can't be it.

I can only conclude the unknown, again and unfortunately. People put quotes around words for different reasons. It's probably not proper Englanderish.

Note that for a discrete fractal model of nature's physical structure and its 4d S-T representation, the physical "manifold" is neither continuous nor discrete.

The distinctions between "continuous" and "discrete" are resolution-dependent. For example, it depends on the usual S-T resolution and also on what portion(s) of the multi-leveled hierarchy of systems you choose to model.

So is the model continuous or discrete? Well, it is neither. And it is approximately both, but in the final analysis it is fractal, i.e., discretely self-similar.

This new way of seeing things may take a while to be processed, but it offers remarkable new scientific possibilities.

Plato, I think the Afshar experiment is important and likely proves his point (in some manner) but I don't know of any new findings of importance since 2004-5. I think now it's more a matter of interpretative argument, or am I missing a new wrinkle in that?

Captcha = "kidin", is Nature trying to tell me something ...

“reality cannot at all be represented by a continuous field."

A. Einstein in The Meaning of Relativity (Princeton Science Library, 1988). p. 160.

Hi Bee,

An interesting quote and one of which I wasn’t aware of and yet an opinion that can be garnered by reading other things that Einstein wrote; with the one Christine offered being just one example. This in effect is why I believe Einstein couldn’t bring himself to take Bohm’s treatment of QM seriously, as the wave component would necessarily have been comprised materially of something that represents being a continuum. This being true, not withstanding that he had also given up on believing any theory being able to contain only observable quantities and still remain consistent, as I had pointed out to Kris in an earlier comment.

So in the end it appears that as most physicists today he subscribed only to nature being represented by theories of a singular ontology. For instance, even with the differences between approaches like LQG and String theory, with the latter being background dependant and the former not, each still in essence are singular ontologically.

Best,

Phil

It must be difficult to know exactly when one has subtly projected the map upon the terrain. It would seem that, topologically, what appears continuous from one vantage might appear discrete from another.

Quite correct, Don.

Consider the Sun observed with 3 different sets of observational parameters: lower spatial resolution and upper spatial scale limit, similar upper and lower temporal constraints, and finally upper and lower mass scale constraints. We set and change these 6 parameters without thinking much about the significance of what we are doing.

From a huge distance the Sun can appear to be a point source.

For a different choice of resolution and scale constraints, the Sun can be modeled as huge dense sphere of fluid emitting a thin high-speed fluid (solar wind).

With a much finer scale of resolution, the Sun is a vast and unbelievably complex collection of discrete particles.

This alternation between "discrete" and "continuous" descriptions probably repeats many, many times over throughout nature's hierarchy, if not infinitely.

Thus in the ancient battle between Discrete and Continuous models, there is a new contender - a "great mathematical hope":

Fractal!

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