# Can Nature take exact random samples?

1. Jun 15, 2013

### Stephen Tashi

Human being's can't take exact random samples from continuous distributions like the uniform distribution on [0,1].

If we attempt to make measurements of physical pheonomena, we are limited to finite precision. Hence it isn't possible do empirical tests of properties involving exact sample values (such as "Is the value of the sample an irrational number?").

I'm curious if there is any physical evidence that Nature takes exact random samples from a continuous distirbution. Is there some kind of indirect argument that she does? (I"m hoping to hear that there is some example in QM where the failure to sample an exact value would show up in some macroscopic result or at least some result that could be detected without making an exact measurement. )

2. Jun 15, 2013

### Staff: Mentor

Would Bell's theorem apply here?

3. Jun 16, 2013

### Jano L.

I think the continuity of the probability distribution as used daily is a property of the theory, rather than the property of Nature. Whether the Nature really realizes all events from the continuous probability distribution or just some dense subset of it is hard to check experimentally - we stumble upon the problem that all measurement results are rational numbers.

In practice, if we do not have the reason to think otherwise, we usually adopt the attitude that if it is allowed mathematically in the model, it can happen in reality as well. Nature excluding some points from happening, for example, irrational points would be very strange a would go against physical laws. For example, if particles could not adopt irrational coordinates, this would be frame-dependent rule, as what is irrational in one coordinate frame, may be rational in another one.

4. Jun 16, 2013

### Stephen Tashi

I won't be inflamed if that is true, but I think that it would be hard to prove. It seems less controversial to say that any measurements we make are recorded in a finite number of symbols. The symbols often do denote rational numbers, but there are finite symbols for irrational numbers also. For example, if I have a 2D grid and I am using a ruler to measure points along the diagonal line x = y , I can take put the 0 mark of the ruler at (1,1) and measure a rational distance D to another point on the line and record the distance from the origin as sqrt(2) + D.

(Of course, I can't put the end of the ruler at exactly (1,1). However, that only shows that the actual distance I recorded as sqrt(2) + D might be rational number. It doesn't show that it must be a rational number.)

It would not necessarily be a frame dependent rule to say that in a given coordinate system, Nature never selects a value that is a rational number. In a different coordinate sysetm we must say that there is a set of numbers that Nature never selects, but that set doesn't have to be the set of numbers that are rational numbers when expressed in those coordinates.

5. Jun 16, 2013

### Stephen Tashi

My question isn't whether there is a set of exact values that Natures selects and another set of exact values that she doesn't. My question is whether she ever selects any exact values at all.

For example, in the popularized description of the double slit experiment when a particle passes through an apparatus in a state that has neither a definite position or momentum. Then it hits a screen and (in the popularized version) attains an exact position at an exact time.

We can't actually measure the exact position and exact time. We can imagine that Nature did, at some exact time, select an exact position for the particle. Can we also imagine that Nature only selected some set of possible positions at possible times?

One pesudo-practical way to express the belief that Nature selects exact positions is to assert that if we imagine a progression of human technolgy in measuring things, then as our precison increases, the measurements we get will more and more approach those predicted by a theory that assumes Nature selects an exact position. However, given the role that the experimenter plays in QM, I don't know if this is merely a belief that applying more technoglogical intervention forces Nature into a smaller and smaller range of possibilities.

If we can imagine some "physical record" of the past capable of recording an exact value and imagine more and more technology being applied to measure what's in that record (without disturbing the information in it) then we can express the belief that more precise measurments will converge toward some exact value selected by Nature in the past. However, I don't know if assuming the existence such a physical record is free of logical contradictions.

Last edited: Jun 16, 2013
6. Jun 16, 2013

### 256bits

Suppose, just for an example and sake of argument, Nature does really exclude the square root of 2 as an exact value. So any particle that moves along the line and does want to come to position P ( coordinates (1,1) on the x-y axis ) will not be able to know when it has arrived, or has surpassed the position. On the other hand, another particle moving along the x-axis 1 unit and then along the y-axis 1 unit can reach P exactly - the second particle never encounters the value of square root of 2,

Unless of course one argues that √2/3, √2/4, ... are points on the segment 0 to 1 that the second particle does encounter and have to be excluded as valid points that Nature can accept. Since the real numbers are uncountable, it follows that Nature would be excluded from an uncountable numner of values, in other words an infinite set.

I am not sure about this, but would not then the set of excluded values be as large as the set of accepted values.

7. Jun 16, 2013

### Stephen Tashi

Again, my question isn't whether certain exact values are included or excluded. My question is whether any exact value is ever selected in the first place.

In regards to your question, there is mathematics that deals with infinities of the same and different sizes. In that mathematical system there is nothing paradoxical about subtracting an infinity of points from an infinite set and still having an infinite set of points left over - if that's where you think a paradox lies.

8. Jun 16, 2013

### 256bits

No. the paradox, whether it realy is a paradox, would be that by just excluding only one possible number, there would be as many acceptable Nature values as not, but that is not your question.

I suppose what you are asking is say X being dicrete, is the range of values X*0.9999...< X < X*1.0000...1 that which Nature can accept as being good enough for the value X.

An interesting concept.

9. Jun 16, 2013

### glappkaeft

The first two (x*0.9999... and x) are identical and the third (x*1.0000....1) is not a valid way to write any type of real number.

10. Jun 16, 2013

### kith

If your experimental resolution is good enough, you have to consider quantum effects. So I try to answer from the point of view of QM.

As you probably know, the notion of measurements is a bit vague in QM. If we don't use the axiomatic cookbook-like approach to measurements, but try to describe them dynamically, the timescale of a measurement is given by the so-called decoherence time. This time depends on the interaction between the measurement apparatus and the system of interest and is many magnitudes shorter than typical time resolutions in most experiments.

If the time resolution Δt of your experiment is better than the decoherence time, you cannot describe the process as a complete measurement in the textbook sense, because you cannot describe your system as being in a single state after time t0+Δt. You still have interference effects. These effects typically decay exponentially with the decoherence time as the decay constant. So I would say the idea that an exact state is selected at some exact time is not true in QM.

Last edited: Jun 16, 2013
11. Jun 16, 2013

### Crazymechanic

as to the OP , I understand your thought but just as a side point , your question implies and you say "nature selects"
Now is nature self aware or even conscious to make a certain value or decision.Well I guess we can only speculate about that as we will probably never know , all we know that there are some constants and phenomenon and ways things tend to happen we call them rules.

Also speaking about numbers and the way "nature" sees them to my understanding is kinda funny because mathematics no matter how good is only a human tool for understanding the surroundings better.I doubt that even if nature would be self aware or a giant organism or call it whatever you like she would operate numbers or formulas the way we understand them.

The sun is shining today just as good as it was thousands of years ago.Before someone ever had the chance to calculate the distance or the radiation intensity etc.
Now I think that nature isn't doing her phenomenon to match certain elegant numbers of our theories rather the rules turn out the way that our math tends to be elegant when done the right way like in a gravitational equation or some other thing.Why is that so , who knows.

Can nature choose between any number or make measurements between our smallest possible measurements , well I think she can , can the electron tell you his exact position when passing through a doubleslit ?Well I think he could if he would be self aware.
Imagine if the walls could talk ....

12. Jun 17, 2013

### sophiecentaur

This all brings to mind Zeno's Paradox, which implies that motion is impossible. It's not Nature we are discussing here. It's our Take on Nature. Two different things entirely.

13. Jun 17, 2013

### Stephen Tashi

Intuitively, I like that answer, but I don't understand it completely. Is there any way to distinguish between "an exact state is not selected at an exact time" and "an exact state is not selected (ever)"?

The QM-for-everyman description of QM says that the state of particle is defined by a wave function and if we measure its position, then it's momentum becomes uncertain. I suppose people debate whether this means that the particle had a "definite, but unknown" momentum or whether it simply has no definite momentum. I think the more common view is that it does not have a definite momentum - i.e. that Nature does not select an exact momentum at the time when we force her to select an exact position. After the time of the measurement, a revised wave function arises and both position and momentum again become "uncertain", which I'll take to mean that Nature has not selected any exact value for them.

But in that QM-for-everyman description, it is assumed that there is an exact measurement of the particle position at some exact time. Since such a measurment cannot be made by humans, can we tell in some macroscopic way whether Nature was forced to make such a measurement? For example, if a particle in a double slit experiment hits a screen and leaves a dot on a photographic plate then someone might say "See that dot? That proves that at some time in the past, Nature was forced to select an exact position for the particle." Can that claim be verified in some way? Or does the dot on the photographic plate merely show that the wave function of the particle was highly constrained by the measurment? I imagine it is simple to do mathematical computations if we assume that hitting the screen "collapsed" the wave function and gave the particle a definite position. But is a complete collapse macroscopically distinguishable from a "near collapse"?

One point of view is that no experiment can every measure an exact position at an exact time. Therefore no wave function is every forced to actually collapse. Another point of view is that there is sufficient technology to cause the wave function to collapse but the technology is not sufficient to reveal the exact information about the "measurement" that caused the collapse. So there is a distinction between such a "measurement" and "the information we have about the measurement".

14. Jun 17, 2013

### Jano L.

I am afraid that no, we cannot verify that Nature obeys mathematics exactly. We always have some uncertainties in experiments.

I think it shows that particle interacted, was somewhere in the dot, which is very small volume but still not a mathematical point.

Incidentally, it seems that one cannot have collapse of $\psi$ to dimensionless point in theory based on the Born interpretation of $|\psi|^2$, since there is no function whose square is a delta distribution.

15. Jun 17, 2013

### D H

Staff Emeritus
No! The uncertainty principle puts a limit on the product of the uncertainties in a simultaneous measurement of position and momentum, or any two conjugate variables. An exact measurement (which we can't do) of one variable would smear the conjugate variable over all space.

A better example in your QM-for-everyman description is radioactive decay. The decay time of a radioactive nucleus is exponentially distributed. Does nature truly draw the decay time from an exponential distribution? That's what the model says, but it's just a model. Physics is a suite of models that describe what we observe experimentally. We don't know what nature truly does.

16. Jun 17, 2013

### Staff: Mentor

You are trying to apply the classical concept of "measurement" to qm systems. Measurement implies observing something I such a way that the disturbance to the observed system is negligably small.

In qm, the least disturbing thing we can do to a system is to hit it with a photon. Unfortunately, that leaves the system in a significantly different state after the hit.

Your questions seem to relate to an overly literal interpretation of "exact measurement". The best answer is to expunge that phrase from your vocabulary.

17. Jun 17, 2013

### Stephen Tashi

The fact that a measurement in QM disturbs the system being measured is a different property than whether the measurement causes Nature to select an exact value for a state variable when it causes this disturbance. Isn't the textbook presentation of a measurement that it produces a definite value of one of the state variables? - not an "approximate value" or a concentrated probability distribution of values? I'm only using the phrases like "exact measurement" to emphasize that the information that human beings have about measurements isn't precise. We can still imagine that our conducting a measurement produced one particular value of a state variable even if we do not know exactly what that value was. Alternatively, perhaps we can imagine that our conducting a measurement never attains the ideal of a textbook "measurement". If it never does then (anthropomorphically) Nature never takes a random sample from a probablity distribution to establish the (exact) value of one of the state variables.

Last edited: Jun 17, 2013
18. Jun 17, 2013

### Staff: Mentor

Might you mean definite rather the exact?

We can make an experiment to definitely measure the spin of an electron that was prepared in a known state. But we must align the axis of the measurement apparatus suitably.

If that is not it,the I confess that I don't understand what you're asking.

19. Jun 17, 2013

### Stephen Tashi

Isn't that why we must have a "collapse" of the wave function? Does the question of whether Nature every really draws a random sample for (say) position amount to asking whether the wave function every really collapses? (If so then I suppose I'll never get a definite answer - that's a complicated debate!)

There can be one debate on whether nature truly draws the decay time from an exponential distribution and a more fundamental debate on whether nature truly draws (an exact) decay time from any distribution. If we assume that a value is drawn from a distribution then I agree we can't prove the distribution is exactly an exponential distribution. However, I'm hoping that the less specific question of whether an exact decay time exists can be investigated in some macroscopic or axiomatic way.

20. Jun 17, 2013

### Stephen Tashi

I don't see how a numerical value can be "definite" without being "exact" or vice-versa. So I'll be happy to accept the term "definite".

Is that an example of taking a sample from a discrete random variable? Without getting into any debate on probability vs determinism, I don't see anything controversial about models that portray Nature as taking samples from discrete random variables, just as I don't see anything controversial about humans doing it by throwing dice or flipping coins. The title of the thread didn't include the proviso "from continuous random variables", but I tried to make it clear in the first post that this is the type of sampling I'm asking about.

21. Jun 17, 2013

### Staff: Mentor

Definite refers to the probability of observing a particular result. Exact refers to the accuracy or precision of the measurement.

Flipping a coin gives an indefinite result, maybe h maybe t. But the observation of h or t after each flip is exact.

A photon definitely has frequency f, but you can never measure exactly what f is.

22. Jun 17, 2013

### Stephen Tashi

I'll accept that as your own definition of the terms. However, I myself, would never use the phrase "definite result" to refer to a "probability of a result". And I wouldn't use "exact" to mean anything about "precision". To me, the statement that there is a certain "precision" to measurement is an admission that it is not "exact".

23. Jun 17, 2013

### D H

Staff Emeritus
No measurement is exact. You are asking a metaphysical question, Stephen.

Physicists model things as living in some continuous space. They tacitly assume that time and space derivatives are a valid representation, as are the continuous probability distributions. That time and space are discrete is a fringe stance in physics.

24. Jun 17, 2013

### Stephen Tashi

That is true, but my question is about the phenomena that are being measured, not about the information produced by a measurement.

That may be the case, I agree. I was just hoping the question had already been worked out or at least debated along specific lines. One type of argument would be that there is no distinction between the information produced by a measurement and the underlying phenomena that is measured - that it is meaningless to talk about an underlying phenomena and the only reality is the information in the measurement. (From that point of view, a particle doesn't have a unique "real" position when we measure position.)

I'm sure you'll hear from the fringe about that remark. As for my question, I'm not saying that an inability to take an (exact) sample from a continuous distribution implies that the correct model is a discrete distribution. Perhaps no (exact) sample is ever taken from either kind of distribution. The information that humans get from a measurement of a continuous quantity is some range of possible values or some probability distribution of values. Perhaps the underlying pheonomena being measured is correctly modeled by a range of values or a post-measurement probability distribution of values rather than by thinking of it as a unique but unknown value.

25. Jun 18, 2013

### kith

I think this thread belongs to the Quantum Physics subforum because all answers refer to QM. There are probably quite a few possible contributors who are not frequenting the General Physics forum. Maybe a mentor can move it back?

Theoretically, I would say no. Practically, I don't think there's a way to distinguish "small but finite" from "actually zero". So once the relevant quantities have decayed "enough", we can model the situation with exact states.

But your wording is a bit odd, because a system in superposition is already in an exact state. It just isn't in an eigenstate of the observable you are going to measure. The notion whether a single state is selected from a set of possible states during a measurement (be it exact or only approximate) is already controversial. What QM does say is that you usually cannot assign a unique state to a system after it has started to interact with another system - like a measurement apparatus. There are different ways how to interpret this.

The Copenhagen interpretation says because we perceive a single outcome, there has to be at least an effective and approximate selection. It is meaningless to talk about what "really happens" to the system during a measurement. The interpretation of de Broglie-Bohm adds hidden variables, so there is an exact state at all times but we can't know it. The Many Worlds interpretation says that the universe has an exact state at all times. If you want to describe a part of the universe, there are many equally real alternative states. A measurement doesn't select a state but adds new worlds. Because we perceive only one of these worlds, we get the appearance of collapse.

If the system is in an exact state after the measurement this state will evolve to an exact state for any subsequent time, as long as no interactions with other systems take place. The question now is, are position and momentum fundamental properties of the system or is only the quantum state fundamental? If we take the latter viewpoint, momentum and position are emergent properties of systems which have interacted in a special way. A measurement apparatus then does not probe the momentum of a system but defines the oberservable momentum through it's interactions with the system. I think this is along the line of reasoning of Bohr and it is similar to the argument in your last post.

Last edited: Jun 18, 2013