I Saw a Cat Who Wasn't There...

One of the many mysteries of quantum mechanics is why things that are very small act so weirdly, compared to everyday, normal-sized things. After all, everyday, normal-sized things are made up of the very small things, so we might expect them to behave like they are groups of the very small things. But they don't, and physicists have long scratched their collective heads as to why that is.

Strike a Superpose

One of those weird things very small objects do is called "quantum superposition." Physical systems, like the incredibly tiny subatomic particles we call electrons, can exist in a bunch of different states. They have different properties, like their electrical charge or energy level, which is called "spin." Each of these properties can show up in different ways: Electrons have a negative electrical charge, but they can have different energy levels. The combination of a certain energy level and the negative charge is a state.

Lots of different experiments, both physical and "thought-experiments," have shown that electrons and many other very tiny subatomic particles show up in one state or another depending on which experiment is being done. An experiment designed to show electrons exist in state A, for example, whatever that state is, will find electrons are in state A. But an experiment on the same electrons that's supposed to show they're in state B will show they are indeed in state B. The electrons seem to hang out in some kind of in-between condition that collapses into one state or another when they are measured.

A physicist named Edwin Schrödinger devised a thought-experiment that showed just how wacky a situation superposition was (it's a thought-experiment because doing it for real would be mean and get you arrested). Suppose, he said, there was a cat inside a closed box. Without opening the lid, you can't see the cat. Now suppose there is a mechanism that would release poison gas into the box and ice poor Whiskers, depending on whether a specific nuclear particle decayed or not. That decay is completely random; it may happen but it isn't guaranteed to happen.

This means, Schrödinger said, that Whiskers is either alive or he's dead, depending on whether the nuclear decay has happened. But the observer doesn't know which until the box opens, meaning that for all practical purposes, to all outside observers Whiskers is both dead and alive at the same time and won't "collapse" into one state or another until someone opens the box lid and looks. One purpose of Schrödingers' experiment was to try to illustrate how superposition works, of course. But another was to establish how different the quantum subatomic world was from the world we experience through our senses and measurements. Measurement and experimentation force the electron into one state or another as the wave function of its superposition collapses. The state really does depend on the measurement. But Whiskers is really alive or really toast, no matter whether we look in the box or not. We don't know which one he is, but that doesn't change that he is one or the other. But on the quantum scale, the electron really isn't one or the other until we look at it.

So one might expect these macroscopic things to behave like an aggregation of quantum-scale things, because that's what they are. Only they don't: superposition doesn't occur on the macroscopic scale. We do not need the wave function of a tuna fish sandwich to collapse in order to detect or experience it. Scientists have never really been able to explain very well why this is.

Super-solving?

Schrödinger did more than imagine experiments to tick off PETA. He also developed Schrödinger's Equation, which can be solved to describe quantum behavior for limited systems, such as a single particle. Particle physicist and retired Anglican priest John Polkinghorne has described it as a matter of language -- what can't be expressed with words can be expressed with equations. So while everyday language (and everyday heads) can't get clear on something being both a particle and a wave at the same time, mathematical language can. This is what Schrödinger's Equation does.

In principle, the equation should be able to describe not simply one particle but larger groups of particles, such as cats or people. Even though we never see larger objects demonstrate superposition or quantum indeterminancy, the equation should be able to describe them in that way. But even the top supercomputers balk at trying to solve Schrödinger's Equation for anything larger than a few thousand subatomic particles. You can see why. The equation for one electron suggests one of two solutions; one state or another, particle or wave. The equation for two electrons has four solutions: A is a particle and B is a wave, A is a particle and B is a particle, A is a wave and B is a particle or A is a wave and B is a wave. It only gets worse, from the point of view of those solving the equations. Three electrons means eight possible solutions, four electrons would mean 24 possible solutions, and so on. When we realize that everyday objects like cats, people and Twinkies consist of billions of atoms, each with its own number of electrons being quantumly uncertain, it becomes pretty clear why it would be difficult for is to figure out superposition for those everyday objects. The math is too complex and like a juggler working one too many things into the pattern, even a computer loses track of what it's doing.

It might seem like the answer is just to wait for better, faster computers. Schrödinger's Equation is not an inherently unsolvable problem like computing the last digit of pi. It has a solution, but like many solvable equations it gets pretty tough to work when the terms get really big. A + B = C, for example, is a solvable equation: Given any two of the terms, the third can be computed. If I know A=1 and C=3, then I can figure out that B=2. Now, as the terms get larger and larger, the solving takes more and more time. But today's computers can solve equations and make calculations that would have locked up a TRS-80 in a heartbeat, so more powerful computers could be the answer. That would work, except for one reality: Even the simplest math computation takes time and energy.

Because we can almost intuitively solve something like "1+1=2" faster than we can say it, we may overlook this, but a finite amount of time elapsed between when I started to say the equation and when I solved it. We could obviously see that a complex equation would take more time, but even a simple equation takes more time as the terms get larger. Simple addition of two single digit numbers takes next to no time at all, but make those even two-digit numbers and it takes an extra beat. More digits will involve still more time. Computers work much much faster than we do, but they still require measurable time to figure answers. It's possible to imagine a fairly simple equation that uses terms so large it could take a computer a very long time to solve it. In fact, there are some equations that would take more time to solve than the universe may have left. The so-called "heat death" of the universe is immensely far into the future, but if current cosmology has it right, there will come a time when the universe has used up all its energy and will be for all intents and purposes "dead." The equation may have a solution, but the universe has said we are out of time and have to put our pencils down even though we are not finished.

If I use a pencil and paper to work a problem, my body uses energy to physically move my hand around the sheet of paper. Even if I solve it "in my head," energy is expended, since the brain cells that do the calculations work by electrical impulses (albeit somewhat slowly, in some cases). Calculators solve math problems more quickly, but they use more energy to do it. Computers work even more quickly than calculators, but only at the cost of a still higher energy use. That energy produces heat, which is why computer towers have fans in them to cool off the processors. A supercomputer can do millions of operations in an eyeblink, but requires huge air-conditioning units to keep it cooled because its operations require -- you guessed it -- even more energy.

The universe has a finite amount of energy. The first law of thermodynamics says that energy can't be created or destroyed, only transformed from one form to another. The second law says that at each transformation, the energy involved is less and less useful for work, meaning more and more energy on hand that can't be used to do things -- like mathematical calculations. This non-usable energy is called "entropy," and in every closed system, entropy increases over time. This is how the universe winds down in the manner mentioned above. And so our solvable equation with the too-large terms in it goes unsolved because the universe runs out of time to solve it and energy to perform the calculations -- in fact, the energy used in the attempt to solve it speeds up the process of dying! Had I known about entropy while a student I might have tried to point out that math homework hastened the death of the universe but I suspect I would not have been heeded.

So, is Schrödinger's Equation potentially solvable for a macroscopic object? It would seem, given infinite energy and time, the answer would be yes. But the universe contains finite energy and will exist for only a finite time, meaning that Schrödinger's Equation won't be solved for a macroscopic object even if it could be.

And in this case, the lack of a solution means that macroscopic objects don't exhibit quantum behavior and they're particles or they're waves, period. Schrödinger's Equation is not just a blackboard decorator -- it describes things that actually happen. If it can't be solved that means what it's supposed to describe can't happen.

Invoking a Deity

But suppose there was infinite energy -- call it "omnipotence." And suppose there was infinite time -- call it "eternity." That's not precisely what eternity is, but in this instance we can use that word. And suppose there was a being that possessed omnipotence and existed in eternity...but you see where I'm going. Macroscopic objects, like, say, people, and systems that contain them like, say, universes, don't show quantum behavior in situations with finite energy and time, giving them a sense of determinancy. Smack one billiard ball with another and it will carom off in a predictable direction rather than disappear and materialize on Neptune. The chance that it would do so exists, but it's so small that the universe is almost certainly too limited to contain that particular outcome. But if an omnipotent being in eternity sees that same action, then such a being could indeed see the interplanetary outcome, or ones even weirder than that.

A lot of times our talk of God suggests that God knows the future before it happens. Such a view is implied if we claim God is all-knowing or omniscient. This leaves us with questions, though, such as why God would have created human beings with free will knowing that we would abuse it and cause others to suffer. But the omniscient God would also know of possible futures in which an individual chose to use his or her free will to not harm someone else. And, in fact, of all possible futures, deriving from all possible choices and all possible combinations of matter and every possible solution to Schrödinger's Equation. The omniscient God could know this because God's perspective is not limited by entropy or time, and thus equations unsolvable in our universe would be solvable for God.

So we don't say that God knows the future; God actually knows all possible futures.

The thought-experiment Schrödinger created is not technically an example of a macroscopic object showing quantum behavior. Remember, the poison gas was released or contained based on the actions of a single atom. The cat's survival is affected by quantum behavior, but the cat itself remains non-quantum, whether it has passed on from poison or not. Schrödinger said that because of the weirdness of quantum indeterminancy, an observer could not know if the cat was alive or dead before the box opened. Macroscopic quantum behavior leads to an even weirder question than Schrödinger's. He only asked if the cat was alive or dead. The macroscopic question is, "Is the cat even inside the box?"

And it would seem that the answer, in the fullest sense, is "God only knows."


(This piece came after some reflection on reading this paper at The Physics arXiv Blog at medium.com. Although it deals more with one of the mathematical features of macroscopic quantum indeterminancy, the P≠NP problem, this wound up headed a different way. Thanks for reading, if you have!)