Monday, October 08, 2007

Is an Omniscient God Incompatible with the Copenhagen Interpretation of Quantum Mechanics?

Yesterday was Neils Bohr's birthday. In addition to being one of the great minds in the history of physics, he also had a fascinating life, and bore an uncanny resemblance to the Chief in Get Smart.

He is one of the major figures in early quantum mechanics and a primary architect of the Copenhagen interpretation. In honor of the anniversary of his birth, I want to play with a question brought up by Elron over on View from the Edge last week. The question was whether quantum mechanics posed any issues for an omniscient God.

Quantum mechanics holds that physical systems are governed by a relation called the Schrodinger equation which traces the evolution of the variable which describes the state of the system. This variable, called the wave function, is a mathematical combination of all possible observable states of the system with a coefficient that ranges between zero and one attached to each possibility. At any given time the system will be in a given state which is the set of values for the coefficients and over time the state changes to new values for the coefficients in perfect accord with Schrodinger's equation. This is a completely deterministic development. No probability or uncertainty anywhere in the theory yet.

So, what does this mean the system looks like? This far, we just have mathematical equations, to give them meaning requires that we interpret the symbols. The most prominent interpretation of the formalism came out of the Bohr Institute in Copenhagen and took on the name the "Copenhagen Interpretation." In it, the wave function is held to be the complete description of the system in the world, in other words, the system is simultaneously in a state comprised of every possible value that observable properties could have. If we express the wave function in terms of position, the particle described is in a superposed state of every possible location at once.

The problem is that we never ever see the world in this superposed state. The instant we look at it, the Schrodinger equation breaks down, the wave function collapses, and the particle ends up in exactly one of its property states, in this case, at one place in space.

Here's the weird part -- we can't know which place it's going to end up. That is a completely random occurrence. That randomness, however, is constrained by the array of coefficients determined by the Schrodinger equation. You see, it turns out that the coefficient squared gives the probability that the particle will end up in that position. Over the long run, if we performed many, many, many runs of the experiment, we know what the spread of the outcomes will look like, but we can have no clue whatsoever on the Copenhagen interpretation of the outcome of any given flip of the quantum coin.

The problem is not that we just don't know enough. That's what Einstein thought. He considered the theory good as far as it goes, but that it didn't go far enough. He thought there were other hidden variables and that once we figured out where to look for them, we'd be able to figure out with certainty what slot in the carton the quantum egg would occupy. Turns out that it can't be so. In technical philosophy jargon, the underdetermination is ontological not epistemological; in ordinary language, its not that we just don't know enough to know which way the coin will fall, its that there is no fact of the matter about which value the system will assume, that is, where the particle will land, until the moment of observation, until the wave function collapses.

And this is where the theological question considered by Elron gets fun. The notion of omniscience has a classical sense of determinism built in. God knows everything there is to know and, like Laplace's demon, if you know every fact, that gives you a complete knowledge of the entire history of the universe in the past and future going direction. Every fact is connected to every other fact and the set of all facts comprises all possible knowledge about the universe. God knows all there is to know and all the is to know is all there is.

But in the Copenhagen interpretation, there is no fact to be known about the value of the observed variable for a system that will be observed until it is observed. God can know the complete set of all facts and this would not include how a given quantum system will react to be observed because there simply is no such fact. So, does the Copenhagen interpretation mean that God could know everything there is to be known and yet still not know some things? Is this modified omniscience really omniscience? If God is given the classical sense of omniscience, does that mean there has to be hidden variables of the sort Einstein wanted (in seeming violation of the Bell and Kochen-Specker theorems)?