## God and Dice

We don’t spend much time thinking about what God does for recreation these days, but the great scientists from Newton to Einstein who have framed our Western cosmology were often deeply religious men. It is understandable that when they found three term equations that explained nearly everything, they believed these were insights to the mind of God.

1/r^2 Newton’s inverse square diminution of gravity.

F=ma Newton’s second law. Describes the relations between force, mass, and acceleration.

S=klogW Boltzmann’s formula for entropy.

E=hv Planck-Einstein energy of a photon, Planck’s Constant x  frequency.

E=mc^2 Einstein’s General Relativity. In units of the speed of light, energy=mass.

These are a few famous examples, and except Boltzmann, they are classical mechanics. In classical mechanics there is no need for probability, and there is no margin of error (except in measurement). The rules are absolute. Space is a field, or reference frame with fixed Cartesian coordinates, x,y, and z. Even today, you need nothing but classical mechanics to send a rocket to the moon. Except for a small deviation in the orbit of Mercury, classical mechanics describes the motions of the planets in our solar system.

The flamboyant Monsieur Laplace seems to have been the first to observe that using Newton’s laws, if you knew the position and momentum of every particle in the universe in the present moment, you could extend this information indefinitely into both the past and the future. You would know everything there was to know. (Momentum is a vector quantity that includes direction). Asked by Napoleon why his work contained no mention of God, he replied that he had no need.

In Special Relativity Einstein argued the concept of spacetime, where every xyz coordinate becomes a clock that is inseparable from the point. He showed that in a curved trajectory different observers would record different amounts of time passing. Space and time are relative.

In General Relativity Einstein argued that spacetime itself is warped by gravity, and that not only space and time, but energy and mass are relative. The only constant is the speed of light.

Newton’s and Einstein’s theories are classical, there is no probability, no statistics, and no dice. The title of this post refers to a quote from Albert Einstein made in a letter discussing statistical mechanics with Max Born. He said (in German),  that he believed God does not play dice with the universe. We will argue in favor of Einstein in this regard. We will argue that Boltzmann’s introduction of probability in his study of entropy set the stage for probabilistic notions in quantum mechanics. We will argue that both probabilistic approaches founder on an anthropocentric philosophical error.

It is ironic that the probabilistic approaches to entropy and quantum mechanics both begin with a wildly improbable assumption: that what humans can perceive somehow matters at a fundamental level in the universe. The basic problem is that the nanoscale world of molecules, atoms, and sub-atomic particles is largely invisible to us.

For Boltzmann, the solution was to count the ways the invisible components can be arranged and remain visibly the same. Entropy becomes the log of this number of ways the invisible components can be arranged multiplied by Boltzmann’s Constant (k); which is the ideal gas constant divided by Avogadro’s number. Boltzmann’s insight was that the behavior of gasses well understood at a practical level by engineers of the industrial revolution, was dictated by the interactions of the invisible molecules and atoms.

We can see above that it works very well for the density of the earth’s atmosphere. The molar concentrations of the invisible (to us) gasses in the atmosphere are well-known. If we suddenly developed the ability to see the different gasses, and they became distinguishable to us, nothing would change. Entropy here is the separation between the molecules in the air, density–regardless if we can see the molecules or not. Is it simply more probable that the spaces between molecules seek a higher entropy equilibrium as pressure is relieved, or is it certain they will do so?

Probability is a tool for the blind. If we can see it, it is classical; if not, we would have God play dice. We are blind at the scale of particles, atoms, and most molecules. Quantum mechanics is fundamentally the notion that the energy of electrons is not a smooth linear progression, but rather a stairway with discrete intervals or steps; quanta. Our efforts to locate electrons and other subatomic particles are hindered by the unfortunate circumstance that when we ping them we unavoidably alter them. Werner Heisenberg formulated this as his uncertainty principle. We can know either the positions or  momenta of invisible particles, but not both. This quandary evolved into the probabilistic notion of a wave function, where waves of probability are our only information until we collapse the wave to a particle by pinging it. It is even maintained that there is no such thing as where the particle actually is until it is observed.

Can you imagine anything more anthropocentric? If a tree falls in a forest and nobody hears it, did it really fall? Of course it did. It is not a matter of phase space between probability waves. It is easy to go and see many fallen trees that very likely nobody heard. Trees fall and  particles have positions and momenta regardless the perceptions of naked apes, or even – aliens capable of somehow garnering information on both position and momentum.

Take a macroscopic object, a watermelon. They differ somewhat in color, size, and skin pattern. These differences arise from somewhat different arrangements of components, but people will generally agree that a watermelon is a watermelon. A watermelon is composed of all manner of quarks held together in Protons and Neutrons by the strong nuclear force. These may be surrounded by electrons in quantized shells held to the protons by the electromagnetic force (and to both neutrons and protons weakly by gravity). These atoms may be bound to others by ionic or covalent bonds in complex molecules. Add these layers up and you sometimes get a watermelon.

When we observe a watermelon, we change the universe. Photons of visible light reflect from the surface of the watermelon and our retinas absorb these photons, which otherwise would have travelled to a different fate. Did the watermelon exist before we observed it? Did the photons exist before our retinas absorbed them, or were both the watermelon and the photons mere probability waves before we intervened? When we turn our back to the watermelon, our eyes no longer absorb the photons. Some may be reflected from our clothing, and others may be absorbed by our skin without sending information about the watermelon to our brain. We change the universe even when we don’t observe the watermelon, but the watermelon is still there.

Imagine that our vision deteriorates and we can no longer see individual watermelons, but can barely discern a large pile of watermelons. We will have lost a lot of information. Our degraded retinas absorb a different set of photons.We could then argue that the positions of the watermelons in the stack is a probability function, and there is no such thing as exactly where a watermelon is in the pile until we reach in blindly and grab one. This would be foolishness, but it is exactly what the probabilistic  Copenhagen Interpretation of quantum mechanics would have us do. If the pile of watermelons is subject to gravity and forms a cone, we could even develop formulas that determine the probability of grabbing a watermelon as being much higher when we reach blindly toward the bottom of the pile rather than the top.

By having God play dice, we elevate probability to the status of a force of nature, joining the strong, weak, electromagnetic, and gravitational forces. The field and waves are established in the form of the wave function. A probability particle can’t be far behind.