# Does the law of large numbers explain why quantum mechanics leads to statistical regularities?

When the question of why chancy effects in quantum mechanics lead to statistical regularities is proposed, it is often answered using the law of large numbers.

When you have particles that can be located at with certain probabilities in the double slit experiment, of course, on aggregate you will have effects that seem deterministic. This is similar to how if you define an event that has two possible outcomes with 50% probability each, the law of large numbers dictates that with enough repetitions of this same event, about 50% of them will land on one of the binary outcomes.

But the law of large numbers would only work if the event just discussed is defined to have a probability distribution in the first place. As such, how can this be an explanation of why there is a probabilistic nature to the seemingly uncaused effects in quantum mechanics?

Note that this question is different from asking why certain probabilities exist instead of others, a question that I have asked before. One can presumably use the Schrödinger’s equation and the wave function to explain this.

Neither is this similar to asking why, for example, a radioactive atom decays at a certain time t. For this, atleast we have experiments that may explain why or atleast justify why one should believe this occurs for no (sufficient) cause (as per Bell’s theorem). But even if this does occur for no cause, it has no bearing on why multiple events occurring witj no cause should result in a probabilistic structure in the first place.

Why then is the world probabilistic fundamentally in the first place? Why not chaos? And why does this probabilistic structure evolve in a deterministic way? There is no rule stating that many things that happen with no cause must result in a probabilistic structure. So simply asserting this implication doesn’t work and neither is there any existing physical theorem that points to an answer here.

Are there any philosophers or even scientists that have pondered this or have tried to figure it out, or is this just another one of those things that is chalked up to happening “for no reason”.

(Explain philosophical significance of the question).

• a probabilistic law can indicate how measurements will "behave", it's not an explanation of a phenomenon. Commented Nov 12, 2023 at 19:05
• This question should be migrated to Physics SE. Commented Nov 13, 2023 at 17:30

## 5 Answers

The “law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of independent identical trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.”

The law of large numbers is a mathematical theorem. Independently, the result from physics is, that the average value of measurements of particles, which are prepared the same way, follows a probability distribution.

1. The law of large numbers does not explain why the laws of measurement in quantum mechanics do have a probabilistic character. We do not know the reason why!
2. Interestingly, the probability distribution, which follows from the Schroedinger wave function psi, develops in a deterministic way. Because the function psi obeys a simple linear, first-order differential equation. Also here, we do not know why?
3. Eventually, it is an open question, why psi develops deterministically, but single events may happen indeterministally.

All three questions are well-known to physicists, the issues are under discussion, but no proposed answer is commonly accepted. I do not know which major contributions to solving the issue have been made by philosophers; for a historical review see Max Jammer: The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective.

Added: A very condensed summary of the Copenhagen view has been given by Andrew Whitaker Einstein, Bohr and the Quantum Dilemma, p. 247:

The conventional [i.e. Copenhagen] view would be that one only has a distribution of positions after measurements of position have been made. Before any measurements any distribution is potential, and can only be made actual by measurement. At the time of the measurement [...] the situation changes from probabilistic to statistical.

# Reality comes first, models after

The Law Of Large Numbers does not dictate anything. As with any other model of reality, it simply fits observations of reality. The Law Of Large Numbers was chosen to describe these phenomena, because it fits.

So the question is: why does reality behave such that the Law Of Large Numbers fit as a model?

Answer: we simply do not know.

...yet.

• But the law of large numbers is a theorem, not a model?!
– Stef
Commented Nov 13, 2023 at 0:00
• I agree with the distinction that a theorem is not a model. However, the law of large numbers are about things that are models in practice. Things like random variables (i.e. measurable functions of an outcome space) and probability distributions are often what we use as models for observations and ensembles of observations that would make at some frequency. Commented Nov 13, 2023 at 19:07
• @Stef It is, and for that reason is exists without any context. All of mathematics is abstract. This in turn means it does not dictate how physical phenomena behaves. Commented Nov 15, 2023 at 12:47

For any model of quantum mechanics proposed to explain the behavior of exceedingly small physical systems to be judged potentially correct, it must smoothly cross over into the predictive realm of newton's laws as the size of the system is smoothly increased. This is called the correspondence principle and it comes about because as the number of particles in the system is increased, the average behavior of the system converges upon the "classical result".

Note that the predictions of quantum mechanics are statistical in the sense that they are expressed in probabilities. In this way, the law of large numbers is one of the things that drives the crossover phenomenon.

• Newtonian mechanics would be a derivative of quantum mechanics if the latter is true. But one cannot use an effect to explain the cause. Hence, this still seems to be the same kind of circular explanation I highlighted in the post. The question is why there is a probabilistic structure in the first place, not why probabilities lead to average behavior which is trivially explained by the law of large numbers.
– user62907
Commented Nov 12, 2023 at 18:32

Perhaps you might be reading too much into the meaning of the word probabilistic when you ask why does the world have to be probabilistic instead of chaotic. There is a sense in which you can't escape probabilities if you live in a world like ours which is not entirely random. If you have a large number of events of any type with no discernible cause, there is usually a way in which you can categorise them. Having decided on a set of categories to use, you can then count the numbers of events in each category, and lo and behold you have what I can consider to be a probability distribution. In that sense, probability is just categorising and counting- ie a human activity which can be performed in commotion with virtually anything.

Take the answers on this site, and suppose they arise from the more-or-less random urges of people to let off intellectual steam. I can still categorise them in lots of ways. I can categorise them by the number of paragraphs they contain, or their word length, of how many votes they have received, or how many comments, etc, or how many where written by people with silly joke names. And given that, I can talk about the probability, if you pick a question at random, of it having such a length, or so many paragraphs and so on.

I can't see how you can avoid probabilities in that sense, even if some degree of chaos is driving events.

I have pondered this, but I'm no scientist. This might be related to the question: "where does the universe end on either macro or micro scale?". But at any scale, any physical phenomenon has to manifest itself in occurrences, which are measured in numbers. The law of big numbers comes in when we try to describe, or in other words, summarize, the behavior, where we average out of these manifestations.

Also, this might help in your musings. Did you know that anything you look at changes state about 10^8 times per second due to the interference caused by atoms changing energy states at this frequency? If a human eye could resolve at this precision everything would be a blur. However, this interference chaos averages out to consistent wavelengths that can be perceived every second, never failing.