In this article I will show how the formalism of quantum mechanics can accommodate apparent randomness in fundamental particles without resorting to grand metaphysical claims about the existence of an "inherent randomness" in our universe. In brief: Temperature is a form of energy in a quantum system. The quantization of heat is sufficient to produce apparent randomness upon measurement. In-determinism remains in quantum systems for the following reason : when a particle gives its thermal energy back to the heat bath of the universe, there is no way to determine from that emission (and/or measurement) which exact absorption events initially caused the rise in temperature. I will begin by describing energy budgets as heat. Second, why and how measurement produces apparent randomness for an observer, and finally -- I will finish by describing why this is not a Hidden-variable theory.
Quantum Beggar's Hat
Imagine a scruffy homeless man who sits near a busy intersection and collects spare change in a hat throughout the day. At the end of each day, the beggar retrieves the money from the hat. But this is a magical quantum hat. Whenever he gets money out of it, it only ever comes in exact $1-dollar bills. He watches people put change into the hat all day long, but every time he retrieves it, he only sees 1's. Also, if people placed several dollars plus change, the extra change that was put in above the dollar mark cannot be retrieved. For example, if he gets $6.35 one day, he cannot get the 35 cents out of the hat. Nevertheless, he never loses money from this. Whenever he goes to retrieve the money in future days , the 35 cents 'rolls over' into the next day's "budget."
Quantum particles adsorb and emit energy in an analogous way with the beggar's hat. Energy is always conserved. Excess absorption of infrared photons will excite the particle to higher energies, until at which point it emits that heat energy back into the environment. Only an exact amount of energy can be released at this emission event, the same way the beggar's hat only gave back 1's. An "energy budget" is an energy metaphor for all the different ways a 1-dollar bill can be broken up into change.
- 4 quarters
- 2 quarters, 4 dimes, 2 nickles.
- 1 dime, 18 nickles.
- 3 quarters, 1 nickle, 20 pennies
- (... this list goes on and on)
The analogy in fundamental particles is that heating radiation can arrive at the particle and get absorbed practically from any direction, and from any location in the nearby universe. Emission of this energy at the "trip point" is analogous the beggar collecting his money at the end of the day. The surrounding universe is so large in relation to a particle that emission and absorption by the particle has no appreciable effect on the overall temperature of the nearby environment. In this sense, the surrounding environment is a heat bath : it has no budget, can give a near-infinite amount of thermal energy, receive an infinite amount back, and never seem to change temperature.
The quantum particle energy budget is fundamentally different from the beggar's hat in this aspect : The beggar remembers who paid him and when, and how much. A fundamental particle "forgets" all these events. While thermal energy always came from some direction at some specific emitter, and some specific energy, the particle has no knowledge of this after several absorption events. Further, and most importantly, a grad student who comes along and measures the particle's temperature cannot know these things either. Nor can the grad student "recover" that information.
Microscopically, if a particle in a surface has become hotter by some nearby large warm thing, in principle it is not possible to perform any experiment that can tell you which particles in the warm thing made particle X in the cold surface hotter. Due to conservation of energy, (and conservation of information) we know that some particular group of particles did the warming, but determining which ones is impossible. There are 292 ways to break up a 1-dollar bill into change. There are God-knows-how-many ways that a particle could absorb heat radiation.
In the very shortest time frames, the daily life of a fundamental particle may involve the absorption and reflection of radiation from the environment (such as what happens in mirrors). But most the time, the incoming radiation is thermal photons, which get absorbed into the particle's energy budget, and ratcheting the particle to a more excited state, in turn increasing its temperature. If the particle does not give back the energy fast enough, it continues to become more and more excited -- and the result is that we large macroscopic humans find that the particle is "hotter".
As described before , the hot particle re-emits only a precise energy back into the environment. This is the quantization of heat. Why that quantization can give rise to apparent randomness for an observer is described next.
Quantization and Measurement
All particles in our universe evolve and change according to a wave called the Schroedinger Wave. That wave is unitary and deterministic, and I will denote it . That is how all fundamental particles act in those strange spans of time in which they are not being measured. Whenever measured, the particle will suddenly snap into a special kind of state, called an eigenstate, which I will denote . (pronounced psi-ket ). The physics which underlies emission and absorption of energy-carrying particles also acts in accordance with the same essential scenario. Evolve according to , but during emission, snap into .
Absorption works similarly, but all manner of different absorbing events can constitute a "measurement", including raw impacts with other particles in which momentum is exchanged. In any everyday scenario at room temperature, fundamental particles never really "collide" per se but wip each other around via nuclear and magnetic forces. When a black shirt absorbs sunlight, the higher energy in one molecule will rebound against nearby molecules in the shirt, giving rise to the necessary mixing which raises the shirt's overall temperature. An iron hammer smashed into brick will raise the temperature of the stone molecules so high, that they will emit visible light photons --- we large humans call this a spark.
There are several different ways of describing in english what is going on during measurement. One can say
- "Energy is quantized."
- "The wave function collapses."
- "Only eigenstates of the system are admissible during measurement."
This quantization forces nature's hand. The particle cannot willy-nilly re-emit thermal energy at whatever amount it wants. It can only give it up to the environment at a particular . So what happens is the particle takes on energy, and holds onto it, becoming ever more excited until probability allows it to emit this energy back into the environment.
For ultra-cold particles, the underlying is a large, smooth and elegant wave form. But as the particle takes on more energy , becomes more garish and chaotic in shape. At very very high energies/temperatures , the particle can be boosted to certain 's that correspond to the emission of a red photon in the visible wavelength. Even higher, an ensemble of particles will glow "white hot" as they give off photons at all wavelengths. At that time, the wave is very chaotic in shape. Photons are being emitted at a rapid pace, and only in a stepwise fashion (visiting each lower .).
Emission is strictly bounded to the rules of discrete eigenstates, and emission is the principle way in which an observer will learn anything about a quantum system to begin with. However, absorption of outside energy can follow from a myriad of different physical pathways. When a physicist goes to measure a particle, he supposes that those readings are "random". Repeated experimental trials verify this "randomness". It is far too early , however, for the physicists to declare that fundamental particles operate by some kind of universal random oracle that is magically imbued into the fabric of the universe.
Not a Hidden-variable Theory.
The situation for a particle is that it is extremely tiny in size versus the size of the the entire universe. For this reason, incoming energy can come from anywhere, come from anything, and when it does come to be absorbed, this does not appreciably lower the energy of the "heat bath" that is the entire environment. The underlying wave is knocked to-and-fro by the incoming random bath. This knocking is evolving deterministically in time. A friendly grad student comes along and measures the particle, to find it in a discrete eigenstate. (He does not perceive the underlying chaos in the ).
Over several thousand measurements, he sees that the particle follows the Born Rule. The Born Rule dictates that the position at any given measurement will be random, but many measurements will start to form the particle in clusters around the square of . After integrating over the probability density, he gets an observable, and lo-and-behold that's how a real particle acts. This assumption of a uniformly-distributed random variable in his calculations is there, -- but why is it there? It is not because the universe is inherently random, but it is only there because of the energy budget, the quantization of heat, and the fact that incoming energy can come from anywhere else in the universe. Because all the energy gets blasted back out into the environment in manner, the individuated events which built up to that energy level are forever lost. They could not in principle be retrieved, any more than you could know what coins were in the beggar's hat that added up to the dollar bill.
The "variables" that are driving apparent randomness in quantum measurements are not hidden :: they are very real and they are the universe itself. They are the very real particles that initially gave up their thermal energy to the system we are about to measure. The time of arrival of photons is "random" (apparently) and the relative positions of the emitters is "random" (apparently) -- and even if they were highly ordered, the energy budget white-washes what they did precisely to the particle to raise its temperature.
At the end of the day, we realize that there is a very clean algorithm for how this "randomness" is being created . That algorithm does not require a magic oracle spitting out uncaused random values. Because is deterministic and continuous, it can be bumped up into very chaotic shapes correspding to higher temperature/energy states. That very real thing is the source of the randomness, not a hidden variable yet to be found by science.