The Correspondence Principle

In 1920 Niels Hendrick David Bohr (1885 Oct 07 – 1962 Nov 18) explicitly laid out his correspondence principle, which he had used in his theorizing in one form or another since at least 1913. That principle necessitates that the subtheories that comprise the Map of Physics fit together as neatly as do the pieces of a jigsaw puzzle. In essence it does not concern the content of the Map so much as it concerns the coordinate frame on which the Map appears. It requires that the Map have no discontinuities because Nature admits no discontinuities in the laws of physics.

For physicists the phrase "correspondence principle" denotes the requirement that any new theory reduce to an older theory when it gets applied to the older theory’s domain of validity. Of course, if the older theory, such as Aristotelian physics, has no domain of validity, it has no correspondence with any theory that does have a domain of validity. For the purpose of correspondence in modern physics the oldest theories with nonzero domains of validity include Newtonian mechanics, Maxwellian electromagnetism, Newton-Young optics, and Carnot-Clausius thermodynamics. Those theories represent the low-hanging fruit of science, theories that fall into domains of validity that physicists can explore through human-scale experiments. Of the validity of those theories in the world of our experience we have no doubt, so they serve readily as touchstones against which we can test other theories.

As an example, consider dynamics.

Inspired by experiments with moving bodies, such as Galileo’s experiment of rolling balls down inclined planks, Isaac Newton defined a body’s quantity of motion (what physicists now call linear momentum) as the body’s mass and velocity taken together and then he defined force mathematically as anything that makes that quantity change. Equating the magnitude of force with the rate at which a forced body’s linear momentum changes enabled Newton to work out a mathematical description of how bodies move in response to the gravitational force that they exert upon each other. Later, physicists discovered that they could obtain a useful number, called energy, by integrating the description of the force acting on a body over a description of the path the body follows while the force acts. If the body can accelerate freely in response to the force, that number represents the body’s kinetic energy,

(Eq’n 1)

As we expect, experiments have confirmed the validity and usefulness of that equation.

In the theory of Relativity, in the absence of potentials, the total energy of a body consists of the body’s relativistic mass taken together with the square of the speed of light. The kinetic energy contained in the body consists of that total energy minus the body’s rest energy,

(Eq’n 2)

That equation bears little resemblance to Equation 1, but using Newton’s binomial series to represent the reciprocal square root has an astonishing result. In 1676 Isaac Newton discovered that

(Eq’n 3)

Making the substitution x=-v^{2}/c^{2} in that equation and
then plugging the result into Equation 2 yields

(Eq’n 4)

If the body’s velocity has a value extremely small relative to the speed of light, that equation becomes Equation 1, thereby showing that relativistic dynamics satisfies the correspondence principle.

Now we seem to have come to the physicists’ version of the
sorites paradox, the paradox of the heap (soros in Greek), which historians
attribute to Eubulides of Miletus (4^{th} Century BC). The classical
sorites paradox proceeds as follows: one grain of wheat is not a heap, two
grains of wheat are not a heap, and so on toward infinity. At no point do we say
that adding a grain of wheat has transformed the collection of grains into a
heap, but eventually we would have to admit that we have a heap of wheat. We
seem to have here a weird perversion of mathematical induction: a) a single
grain of wheat does not make a heap, b) if N grains of wheat do not make a heap,
then N+1 grains of wheat do not make a heap; thus, c) N grains of wheat do not
make a heap for all possible values of N.

In the physics version of the sorites paradox we say that a body moving at one meter per second is not relativistic (conforms to Equation 1), a body moving at two meters per second is not relativistic, and so on until we say that a body moving at 299,792,457 meters per second (one meter per second shy of the speed of light) is not relativistic. Clearly that reasoning comes to an absurdity. But consider the inverse reasoning. The statement that a body moving at 299,792,457 meters per second is relativistic leads, by subtraction, to the statement that a body moving at one meter per second is relativistic. Equation 4 confirms that Equation 2 remains true to Reality even at speeds very much smaller than the speed of light. Thus relativistic dynamics has its domain of validity over the entire range 0≤v<c, but classical dynamics does not. If we went to draw the Venn diagram for dynamics, the circle representing Newtonian mechanics would lie entirely inside the circle representing Relativity.

So we find ourselves back in Plato’s Cave. In this version classical dynamics corresponds to the shadows projected onto the wall and relativistic dynamics corresponds to the ideal objects passing in front of the fire. With respect to basic dynamics Relativity shows us the Platonic Form and Newtonian physics shows us the silhouette of that Form projected onto the world of our perceptions. In that metaphor the correspondence principle demands that our description of the Form have a shape that properly accounts for the shape of the classical silhouette.

In consideration of Quantum Mechanics the method seems to break down. Mathematically the quantum theory looks like a variation on optics, one that describes the wave-like propagation of particles of matter instead of the propagation of light. But the wave, which conforms to the requirements set by Schrödinger’s Equation, does not correspond to anything in the classical realm. The quantum theory only properly satisfied the correspondence principle when Max Born showed that the square of the wavefunction describes the probability density of finding the particle that it represents in a given state in accordance with Heisenberg’s indeterminacy principle and when Paul Ehrenfest then proved and verified that the expectation values of position and linear momentum calculated from that probability density conform to the requirements of Newtonian dynamics.

Again we catch a glimpse of a Platonic Form manifesting the fundamental laws of physics. Completely alien to the common sense that we derive from out direct experience of the world, the quantum theory describes the playing out of events over the entire range of possible actions, down to the minimum possible action, represented by Planck’s constant, that can occur in an event. As the amount of action in an event increases, coming into the realm of human experience, the quantum theory segues into Hamiltonian dynamics, another optics-like theory that we can derive from Newtonian dynamics. Again our theories of physics satisfy the correspondence principle.

In the realm of radiation thermodynamics the correspondence principle offers an example of how its demand that we reconcile differing theories leads us to discover new knowledge about the Forms of Reality. In the description of a beam of monochromatic light the electromagnetic theory gives us one value for the energy density of the radiation and thermodynamics, by way of the law of entropy and Adolfo Bartoli’s imaginary experiment demonstrating the existence of radiation pressure, gives us a different value for the energy density. The only process available for reconciling those differing calculations involves reconceiving light as having a nature analogous to that of cottage cheese; in that model light consists of curds of energy (photons) dispersed within a whey of electric and magnetic fields. The entropy calculation thus emerges as calculating the density of the energy in each photon and the electromagnetic calculation yields the average energy density throughout the beam. Through this model we reconceive the wave equation of light as analogous to Schrödinger’s Equation, yielding a wave function whose square describes the probability density of finding a photon at a given point and yields the expectation value (a kind of average) for the energy density in the beam.

The correspondence principle comes from the fact that Reality does not – indeed, cannot – contradict itself. That fact necessitates that theories of the operations of Reality possess the property of mutual consistency, which gives us the content of the correspondence principle. As we proceed from theories derived from our immediate percepts, the mural shadows in Plato’s Cave, to theories describing the Platonic Forms underlying Reality, the correspondence principle ensures that we will eventually come to a single theory of everything.

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