Quantum physics tells us that weird things happen on the very small scale. Particles popping into brief existence out of nowhere, or interfering with other particles or even themselves like a wave, or tunneling across a barrier; this sort of thing.
Quantum physics also tells us that the same laws apply, in principle, at the macroscopic scale. It’s “just” that at that scale, the probabilities of something like this happening become infinitesimally small, so we don’t get to see such events in our daily lives. But then again, the world is a big place, and humans have been around observing things for quite a while. So could it be that somewhere, at some moment in human history, a low-but-not-quite-zero-probability macroscopic quantum event of the sort mentioned above occurred? Are there any weird recorded incidents, possibly something that would have been written off as “a miracle” in the past, where historians, or quantum physicists (or both, or either) seriously entertain the hypothesis that a quantum effect at the scale visible to the naked eye was occurring?
Pretty much the answer is no. I was in an early college class one time with a sharp kid, who insisted that a theoretical box of air will at some time, MUST have all of the air molecules randomly on one side of the box, leaving a vacuum on the other side. He neglected to understand the instantaneous nature of such an occurrence and that if it did happen, would be undetectable over any reasonable time scale.
Same here. “IF” anything like you suggest actually did happen, it would be so fleeting as to be unnoticed.
There’s a temptation among humans to think that all big numbers are the same. The probability of a macro-scale quantum event is 1 over a big number, and the number of people, or of seconds, or whatever, in human history is also a big number, therefore their product should be about 1. But not all big numbers are the same, and in fact, there’s more variability between big numbers than there is among small numbers. The probability of a macro-scale quantum effect happening spontaneously is so low that it’s almost certainly never happened in the entire Universe.
I remember the joke in class about this being “the odds of X happening just once in the lifetime of the universe is infinitesimal.” Usually in regards to something like a monkey typing at random producing a work of Shakespeare.
1 year = 3.154E7, universe15,000,000,000 years, so about 4.5E17 seconds.
At least conceptually I could imagine a quantum event occurring at quantum scale at just the right time & place to trigger a macroscopic event that was about to occur anyhow.
As a specific but made up example, consider lightning. Slowly but surely separated charges are developing then eventually enough voltage gathers to overcome the air resistance and ZAPPP!!!
Might one or another virtual electron popping into existence been the straw which broke the jam and triggered the discharge?
Conceptually yes. Would / Could we ever know? No.
But other than this sort of thing, a quantum scale trigger to a macroscale event powered by ordinary macroscale forces …
As everyone has already said, there’s just no statistical way to have enough quantum things occur the right way in the right place at the right time to add up all by themselves to a detectable macroscale event.
A quantum effect happening at a small scale but triggering something at a large scale is something that happens all the time. In fact, you could say that everything is an example of that.
That’s not the same thing as a single quantum event itself being large.
As an undergraduate, I took Statistical Physics (or maybe it was Thermal Physics, I don’t remember the exact course name). The first day, the professor took us through the calculation of the probability of all the gas in a container ending up on one side of the container for a fixed amount of gas in a fixed size container. The probability ended up being negligible over many lifetimes of the universe. I think he did that to short circuit the “sharp kid” remarks (we were all sharp kids at that school).
On a side note, that professor was awarded the Nobel Prize this year. Now you know…the rest of the story.
Anyway, the sidenote has a point. This year’s Nobel was awarded for work demonstrating Macroscopic Quantum Tunneling (MQT) in superconducting devices, which was a first step towards Quantum Computing’s qubits.
Superconducting materials allow creation of macroscopic quantum objects and direct manipulation and observation of their properties. Which means that the use of superconducting magnetometers to explore and detect vast mineral resources worth billions of dollars was, in fact, a macroscopic quantum effect in human history.
Oh, right, I knew there was a qualifier I was forgetting to include. As I said, the probability of it happening spontaneously is negligible. But it can be engineered to happen.
In the notation I am thinking of ℵ1 = 2^ℵ0, ℵ2 = 2^ℵ1, etc. so we are saying the same thing - or is my assumption that your leftmost equals sign was supposed to be a greater than sign wrong?
Well, you could use a random number generator that relies on quantum randomness to make a decision that impacts the real world. Get an odd number, you eat a burger; an even number, a pizza.
Like I said, small-scale quantum events triggering macro-scale events are ubiquitous.
The first sign should definitely be a greater-than sign. As to whether ℵ1 = 2^ℵ0 etc., that’s an indeterminate question in mathematics: The rules are equally consistent whether it’s true or not, so you’re free to add an axiom either way. The axiom that ℵ1 = 2^ℵ0 is known as the continuum hypothesis. Formally, though, aleph-0 is defined as the smallest cardinal number that’s larger than all integers, aleph-1 is defined as the smallest cardinal number that’s larger than aleph-0, etc. Sometimes one will instead see beth numbers, where beth-0 is equal to aleph-0, but the others are defined by exponentiation as you described: Then, the continuum hypothesis is the axiom that aleph-1 = beth-1.
It has to be an equals sign, not greater, because the number of closed curves is the number of continuous functions S^1\to\mathbb{R}^2, which is (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0}.
As for 2^{\aleph_0}=\aleph_1,and 2^{\aleph_1}=\aleph_2, this Generalized Continuum Hypothesis sort of thing is independent of the default axioms of set theory, so you need to state explicitly if you wish to assume anything like that.
