The Fundamental Quantum

In this essay I will deduce the old quantum theory from the finite-value theorem, the fundamental mathematical theorem underlying the Map of Physics, and the principle of least action. We will find that Leukippos and Demokritos had the right idea when they conceived their doctrine of the atomos, the uncuttable. But we will obtain results far stranger than anything that Leukippos and his pupil could have imagined.

The Finite-Value Theorem Again

In my original essay on this topic I applied it to limit the size of space, making space finite and, therefore, relativistic. Now I want to argue the other end of the scale of size.

Consider how we measure things. We proceed by defining some perfectly arbitrary standard and then counting the number of copies of that standard that we would have to bring together to equal the measured quantity, implicitly assuming that we could divide the object of our measurement into pieces equal to one copy of the standard. When I say that a certain block of ice ponders one hundred kilograms, I imply that someone could cut that block into one hundred equal pieces, each of which ponders one kilogram. If I then say that the same block ponders one hundred thousand grams, I imply that someone could cut it into pieces that each ponder one gram. Like Leukippos, I contemplate cutting the block into ever smaller pieces and find that I must come to an end of that process, that I must come to some uncuttable piece of that block: I find that I cannot cut the block into pieces of infinitesimal size.

Suppose that I could cut the block into pieces of infinitesimal size. I would then have an infinite number of pieces (because the infinitesimal is the reciprocal of an infinity). I could then remove an infinite number of pieces from the block and still have an infinite number of pieces left over: I could actually double the size of the block and double its mass, as measured by other standards, just by rearranging the pieces. Such a thing, as I have shown, cannot be true of any object or property subject to a conservation law.

We express the conservation law pertaining to some conserved property X as X=0; that is, we say that we have no difference in X as measured at two different times. To maintain the truth of that statement we must also state that we can never have an infinite amount of X (the infinity in question being Georg Cantor's Aleph-Null, the smallest of the infinities that Cantor studied). But we must also state that we cannot divide X into infinitesimal bits, because then we would have an infinite number of them and in accordance with that count, we would have an infinite amount of X and thus have the potential to violate the conservation law. Thus we find that the finite-value theorem also gives a finite-divisibility theorem.

We also express certain laws in terms of achieving an extremum, the principle of least action being the best known example. For that we have the mathematical statement S=0; that is, we say that we have no difference between the action played out on the true path of a body and the action played out on a nearby variant path. Here we see an example of physics in the subjunctive mood, the physics of "as if". In this case we must treat the path that the body follows as if the body followed it and also followed a number of nearby alternate paths.

That aspect of our subjunctive physics may seem bizarre. The subtraction of an actual quantity from a "nearby" potential quantity, the actual action of the body as distinct from a potential action, seems illegitimate. We seem to be combining the real and the unreal in the proverbial apples and oranges manner. Logic tells us that we cannot do such a thing unless the potential quantity is, in some sense, at least partially realized. As we shall see the aleatoric field that accompanies every particle provides just such a partial realization.

The Principle of Least Action

Now let's apply the new finite-value theorem, in the form of the finite-divisibility theorem to the principle of least action. We start by eliminating the potential energy from the action integral via the relation U = E-T. We thus have

(Eq'n 1)

That's the quantity that we subject to the finite-divisibility theorem, which we express by writing

S = nh,

(Eq'n 2)

in which n represents a finite whole number and h represents a small, but nonzero and non-infinitesimal number (Planck's constant), the smallest possible change in the action of a body or system. In any physical process a change in the system manifesting that process must proceed in integer multiples of that little number that we represent with aitch. Now let's think about the simplest systems.

Consider an observer tracking a single particle moving through space. That observer uses measurements to assign to that particle both a position (along the x-axis in this example) and a linear momentum (in the x-direction). Thus as the particle traverses the distance x between two points, Point-A and Point-B, (that we can position as precisely as we like, because space has the character of a continuum of an infinite number of points), with linear momentum p it enacts an action that the observer calculates as

S = px.

(Eq'n 3)

Equation 2 tells us that the action that the observer has so determined must equal some whole number multiple of Planck's constant.

Now imagine that a second observer moves parallel to the x-axis at some speed V, which speed we make very much less than the speed of light so that we have a non-relativistic situation. Using the Galilean transformation, we can calculate the linear momentum of the particle and the distance that the particle traverses between points A and B as measured by that second observer. We have

p' = p ± mV = p ± Δp

(Eq'n 4)

and

x' = x ± Vt = x ± Δx.

(Eq'n 5)

With those equations (in which m represents the mass of the particle and t represents the time the particle needs to go from A to B) the second observer calculates the action

S' = p'x' = px ± xΔp ± pΔx + ΔpΔx.

(Eq'n 6)

Now we let V approach zero. As p and x become progressively smaller S' comes closer in value to S. But the finite-divisability theorem tells us that S' cannot differ from S by anything less than Planck's constant, so we must assert as true to Reality the statement that the smallest increment of action in Equation 6 must equal Planck's constant at the very least. That leads us to state that

ΔpΔx = h.

(Eq'n 7)

That equation gives us the mathematical statement of Werner Heisenberg's indeterminacy principle. It tells us that two observers who believe themselves to occupy the same inertial frame cannot pin down a particle's momentum and location with arbitrary precision.

If Existence did not make that statement true to Reality, then we could give one particle an action S and then nudge the apparatus that projected it to an infinitesimally different velocity and give a second particle an action S+δS with δS less than Planck's constant. Indeterminacy means that we can't do that and we can't know enough about the particles' momenta and locations to know how we can't do it. However, we will be able to deduce how Existence structures Reality to make that statement true; more precisely, we will deduce how Existence structures particles to make that statement come true to Reality, but we will do that in another essay in this series.

First Application: The Old Quantum Theory

Given that the most fundamental particles must possess some property to distinguish them from nothings, we must find a basis for that fundamental property on which all observers can agree. We need some kind of an absolute and rotary motion fits the requirement. Rotary motion differs from linear motion primarily in comprising an absolute state of motion. Because motion in a circle necessitates acceleration (toward the center of the circle), which is an absolute state of motion, we can refer rotation and revolution to an absolute state of rest, the state in which the centripetal acceleration equals zero. We can thus refer angular displacement of a body moving in a circle to that absolute frame of rest. But since we have revolution and rotation as absolute states of motion, we infer that spin will suffice to distinguish the most fundamental bodies (particles) from nothings. Actually, we know of two features that distinguish particles from nothings for all observers (because they are based on absolute references): travel at the speed of light (photons) or spin (electrons), though some particles can have both features (neutrinos).

As I implied above we want to consider the quantization of actions involving angular momentum, the measure of rotary motion. If we have a body moving in a circular path, then we have

S = ∫Ldθ,

(Eq'n 8)

in which L represents the body's angular momentum relative to the center of the circle. We want to apply that formula to one basic unit of angular momentum, represented by Planck's constant.

How far apart can we put the endpoints of the traverse on which a body enacts that unit of action? Though we may speak of greater angles, just as we speak of sunrise or the flight realm beyond the speed of light, such angles do not exist in Reality. We have, in this Universe at least, a maximum angle beyond which nothing can turn: bodies going round and round merely repeat their traverses of the full circle (2π radians). But that maximum angle then necessitates a minimum angular momentum, less than which we cannot have in Reality. We have S = L2π. By the quantum rule, then, we must have

L = nh/2π = nħ

(Eq'n 9)

and, at minimum, when n = 1,

ΔL = ħ,

(Eq'n 10)

with the aitch-bar representing the reduced Planck's constant that we use in discussing the rotary aspects of the quantum theory or those parts involving trigonometric functions.

That analysis means that no angular momentum can change by less than one unit of aitch-bar. That proposition means, in turn, that all angular momenta can only exist as integer multiples of aitch-bar. If Existence did not make that proposition true to Reality, then we could have two rotary systems, turning in opposite senses, with angular momenta that differ by less than aitch-bar. We might put those systems together in a compound system that possesses a net angular momentum less than aitch-bar and then contrive to transfer that net angular momentum to another system, in violation of the rule expressed in Equation 10.

Note that Equation 8 looks like it should lead us to a rotary analogue of Heisenberg's indeterminacy principle. But it doesn't. We have no indeterminacy in the angular momenta available to rotary systems, but rather sharply defined angular momenta that truly deserve the name quanta. However, if we could somehow contrive to limit the extent of space as Reality limits the angular length of a circle (as in a deep, square potential well), we would have similarly well-defined linear momenta.

Now consider a special kind of rotary system, a body spinning on an axis. What angular momenta can it have? Let's assume that the angular momentum in question inheres in the particle; that is, assume that we cannot in any way transfer that angular momentum to another particle. We can, nonetheless, change the particle's angular momentum by the simple expedient of turning the particle's axis through some angle. If the particle carries the minimum possible amount of spin, then a flip of radians must represent a change of one aitch-bar of angular momentum. In that case the particle carries one-half aitch-bar of angular momentum as its inherent spin. From that analysis we infer that fundamental particles carry spins that equal integer multiples of one-half aitch-bar.

These new rules now let us work out Niels Bohr's model of the atom from a simple description of the atom as comprising a set of light particles (electrons) revolving about a heavy nucleus. This gives us the essence of the Old Quantum Theory, the theory of the electronic structure of atoms and of the transitions of spectroscopy.

Clearly each electron must have an angular momentum of n about the nucleus. That fact constrains the orbit that we calculate from the attractive force of Coulomb's law. As in the classical case we must equate the attractive force exerted between an electron and the atom's nucleus and the centripetal force necessitated by the electron's angular momentum. For the simple case of the hydrogen atom we have the centripetal force as

(Eq'n 11)

and the electric force as

(Eq'n 12)

The forces that those equations describe balance each other when

(Eq'n 13)

If we let n=1, then that equation describes the
Bohr radius of the hydrogen atom, about r=0.55x10^{-10} meter. Orbits in
which the electron has greater angular momentum have radii that are square
multiples of that basic radius.

In its orbit an electron has an electric potential energy of

(Eq'n 14)

in which equation

(Eq'n 15)

If, somehow, an electron can change its angular momentum and leap to another orbit, then its energy must change by the amount

(Eq'n 16)

If we use the Planck relation (E=hν, which we feign not to know in this essay), then we recognize that equation as expressing the frequencies of light emitted by hydrogen, which pattern Johann Jacob Balmer found by induction in the 1880's.

That fact means that the electrons revolving in an atom can only absorb or emit energy in certain specific values. We see one way, then, in which the quantum theory constrains the structure of matter to a small set of forms. However, we have not explained why electrons all follow orbits with different angular momenta (Pauli's exclusion principle does that) or why the electrons in the lowest orbit don't jump into the nucleus, thereby collapsing the atom (the weak force answers that question). But we have much more to deduce before we can answer those questions via the axiomatic-deductive method.

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