A friend called me the other day and asked, “What is ‘Larsonian physics,’ as distinguished from ‘Newtonian physics?’” To compare the two systems (see here), it is helpful to understand Hestenes’ description of Newtonian physics. He writes: 1
Newtonian mechanics is, therefore, more than a particular scientific theory; it is a well defined program of research into the structure of the physical world…. [The foundation of the program to this day is that] a particle is understood to be an object with a definite orbit in space and time. The orbit is represented by a function x = x(t), which specifies the particle’s position x at each time t. To express the continuous existence of the particle in some interval of time, the function x(t) must be a continuous function of the variable t in that interval. When specified for all times in an interval, the function x(t) describes a motion of the particle.
This continuous function, representing the motion of an object from one location to another over time, which “expresses the continuous existence of a particle” and forms the foundation of Newtonian physics, is replaced in Larsonian physics, as developed here at the LRC, by a progression of space and time, independent of any object. When either the space or time aspect of the space|time progression oscillates, at a given point in the progression, a point in space (or time) is established, the position x of which is represented by the continuous function x = x(t) = 0 (or x = x(s) = 0); that is, the point’s spatial (or temporal) position, in the progression, no longer changes with the progression, due to the oscillation. In other words, it becomes stationary (non-progressing) in space (or time); Its fixed position, relative to the progression, is actually generated and maintained dynamically by its oscillation (as a 1D analogy, we can think of a stationary fish swimming against the current of a flowing stream.)
Thus, in the LRC’s Larsonian physics, the continuous function expresses a spatial or temporal location that had no prior existence, while in Newtonian physics, the continuous function expresses a change between pre-existing locations. Subsequently, the change that Einstein introduced into the Newtonian program replaced Newton’s ideas of absolute time and absolute space, so that spatio-temporal locations are no longer pre-existent in the Newtonian system, but are dynamically generated from a gravitational field in general relativity.
Of course, the fixed background of locations in Newton’s concept is still necessary to define motion in the quantum field theory of particle physics, where only Einstein’s special relativity is employed and his general relativity is ignored, causing the immense trouble with the attempts to unify modern physics, which is the subject of this blog: The theory of gravity generates the space and time continuum, while the theory of matter pre-supposes it, even though they are both field-theoretic constructs.
However, the important thing to understand about the LRC’s Larsonian physics is that the dynamic creation of points out of the space|time progression provides the basis for a physics of the discretium, without the need to resort to the continuous field concept, or, as Einstein would have expressed it, it provides a basis for an algebraic physics, as opposed to a physics of the continuum, or a geometric physics.
Though it’s not widely known, Einstein actually would have preferred a discretium based, or algebraic physics, but was unable to find a way to get to such a system. He was convinced, as today’s physicists are too, now, after pursuing unification as diligently as he did (even though he was mocked for doing so at the time) that the space continuum is “doomed,” as Witten puts it.2 In fact, according to Arkani-Hamed, “The idea that [the space continuum] might not be fundamental is pretty well accepted…”3 in the legacy physics community.
But, in his day, Einstein suffered alone, “plagued” with his thoughts that the assumption of a space and time continuum was probably the wrong approach, given that physical phenomena are quantized. Nevertheless, all the while, he is celebrated as the champion who revolutionized continuum physics. John Stachel, of the University of Boston’s Center for Einstein Studies, who first discovered this other side of Einstein,” explains: 4
If one looks at Einstein’s work carefully, including his published writings, but particularly his correspondence and reminiscences of conversations with him, one finds strong evidence for the existence of another Einstien, one who questioned the fundamental significance of the continuum. His skepticism seems to have deepened over the years, resulting late in his life in a profound pessimism about the field-theoretical program, even as he continued to pursue it.
What Einstein would have discovered, had he lived to study the algebraic physics that we are developing at the LRC, is that, while the continuum is something that can be conceived by the human mind, it isn’t necessary to conceive of it as an a priori construction needed to develop a discrete set of points, which was the great obstacle that baffled him. As Stachel points out, Einstein wrote to Walter Dallenbach, confirming that his former student had also correctly grasped the “drawback” of the continuum, which drawback is essentially that it seems that one needs to have a continuum in order to have a discontinuum:
The problem seems to me [to be] 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, [a construction] which corresponds to nothing “real.” But we still lack the mathematical structure unfortunately. How much have I already plagued myself in this way.
Of course, the mathematics of the time were still going the opposite way. Mathematicians were happily following Dedekind and Cantor in constructing a continuum (infinite sets and smooth functions) from a discontinuum (discrete numbers). In fact, Stachel, speculates that Einstein’s doubts about the reality of the continuum stem, in part, from his reading of Dedekind, from whom he borrows his oft used phrase “free inventions of the human mind,” that did anything but endear him to Larson. Dedekind argues against the continuum by insisting that discontinuity in the concept of space does not affect Euclidean geometry in the least:
For a great part of the science of space, the continuity of its configuration is not even a necessary condition…If anyone should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and call attention to the fact that a far advanced, refined, scientific, training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations also irrational, and besides algebraic, transcendental quantitative relations, are conceivable.
Of course, it was highly unlikely that Einstein was aware that Dedekind’s intellectual journey into irrational numbers and infinite sets began some fifty years previously with his exposure to Hamilton’s work, who had defined irrational numbers, but in the context of numbers derived, not from the abstract notion of a set, but from the intuition of the progression of time. And while Hamilton’s work on irrational numbers in his “Algebra of Pure Time,” is little regarded today, who could have known that it would have been eventually synthesized by Clifford with the Grassmann numbers as an “operationally interpreted” number, leading to Hestenes’ pioneering work in the recognition of geometric algebra as the unification of geometry and algebra through the geometric product.
While the bottom line can only be sketched at this point, all indications are that the mathematical structure, which Einstein pined for, that would enable him to be able to define a discontinuum, without the aid of a continuum, appears to be at hand. To be sure, he outlined major conceptual obstacles with both concepts in his letter to Dallenbach:
Yet, I see difficulties of principle here too. The electrons (as points) would be the ultimate entities in such a system (building blocks). Are there indeed such building blocks? Why are they all of equal magnitude? Is it satisfactory to say: God in his wisdom made them all equally big, each like every other because he wanted it that way? If it had pleased him, he could also have created them different. With the continuum viewpoint one is better off in this respect, because one doesn’t have to prescribe elementary building blocks from the beginnning. Further, the old question of the vacuum! But these considerations must pale beside the overwhelming fact: The continuum is more ample than the things to be descibed.
Thus, Einstein was left to plague himself with these thoughts, but now with a knowledge of Hamilton’s “algebraic numbers,” Larson’s “speed displacements,” and Clifford’s “operational interpretation” of numbers, and his multi-dimensional algebras, there is much more to work with than there was in Einstein’s day. Perhaps, it’s time to now stand with the legendary icon of physics and say:
I hope my friend would understand.
[O]ne does not have the right today to maintain that the foundation [of physics] must consist of a field theory in the sense of Maxwell. The other possibility leads, in my opinion, to a renunciation of the space-time continuum and to a purely algebraic physics.
Logically, this is quite possible: The system is described by a number of integers; “Time” is only a possible viewpoint, from which the other “observables” can be considered - an observable logically coordinated to all the others. Such a theory doesn’t have to be based upon the probability concept…
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