Classical Mechanics: Continuous or Discrete universe

[qttv]

Good morning.

The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics.

The use of mathematical functions (continuous) to describe the evolution over time of quantities such as position, velocity, acceleration, energy, has been introduced since Newton's time. However, when using a calculator, the mathematical functions of physical use are subjected to a necessary discretization which involves a certain error. My question is: is the reality in which the mentioned physical quantities are discreet? Could we conceive the environment in which a body moves, with a certain trajectory, like a three-dimensional screen composed of Pixels? In this case, the use of the integral calculation would result in a mathematical error, in exactly the opposite way to the discretization process that is conducted in a computer.

My physics professor said that reality is continuous, but I do not think that this concept can be assimilated by the human mind. I do not want to come to the treatment of space-time, but I believe that the paradoxes of Zeno are sufficient to agree that the physical greatness with which we deal every day is of a discrete nature.

Quantum mechanics confirms that entities such as energy and speed should be understood as discrete (just think of the "quantum" of energy), therefore it is possible that my question can be answered already in this. However, since the school years the use of continuous mathematics is taught but not justified. Is it possible that the universe is discrete, but composed of such a high number of stencils that any error is insignificant for classical mechanics, which deals with the macroscopic world?

Thanks a lot!

 

[muscaria]

I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics.

However, when using a calculator, the mathematical functions of physical use are subjected to a necessary discretization which involves a certain error. My question is: is the reality in which the mentioned physical quantities are discreet? Could we conceive the environment in which a body moves, with a certain trajectory, like a three-dimensional screen composed of Pixels?

From the point of view of classical mechanics, any calculations and "necessary discretisation" are imposed by you, independent of how things operate. In other words, you are simply choosing a scale to calculate things - where you effectively have infinite resolution and can zoom in as much as you want. So more akin to scalable vector graphics (SVG) than pixels.

 

[qttv]

From the point of view of classical mechanics, any calculations and "necessary discretisation" are imposed by you, independent of how things operate. In other words, you are simply choosing a scale to calculate things - where you effectively have infinite resolution and can zoom in as much as you want. So more akin to scalable vector graphics (SVG) than pixels.

The electric field is also schematised as a continuum, but instead is composed of photons. Yet calculations performed by the integral method are considered accurate. Perhaps because the mistake committed is definitely negligible?
Well, is then possible that the continuous maths help us to work with an "infinite resolution", but the universe has a finite one?
In other words, if you work with a gravity discrete field, the trajectory of a satellite will be different from the one calculated with a continuous model. Maybe that, anyway, the discrete resolution of the universe is so high that we can't perceive the difference between a continuous result and a discrete result?

 

[microsansfil]

but I do not think that this concept can be assimilated by the human mind. !

From a mathematical point of view yes

Best regards
Patrick

 

[mattt]

"Discrete" or "Continuous" are mathematical concepts of the mathematical structures that the human mind creates. The most you can ask is, if this or that mathematical creation, is more or less useful for us to "understand some things" or if they are used to make more or less accurate predictions about measurements in future observations/experiments.

The two types can be equally useful sometimes.

By the way, the spectrum of a given Self Adjoint Operator on a Hilbert Space, in general, has a Pure Point part (that you could interpret as "discrete"), an Absolute Continuous part and a Singular Continuous part, and the most useful (by far) mathematical structure that the human mind has created (to model the "Universe"), that give rise to Relativistic Quantum Field Theory, is "pouring" with mathematical things that you, intuitively, instinctively, would call "continuous".

On the other hand, there are serious (and brilliant) people that are working on models of our Universe in terms of "a Giant Quantum Computation (with a finite number of qubits anyway)" or even "a Giant Classical Computation (Cellular Automata)".

At the end of the day, the most we can do is decide honestly what model is more useful or satisfactory for us.

 

[PeterDonis]

is the reality in which the mentioned physical quantities are discreet?

We don't know for sure since we can't probe arbitrarily small distance scales. The smallest distance scale we can currently probe is about the size of an atomic nucleus. That's about 20 orders of magnitude larger than the Planck length, which is the usual estimate for the distance scale where, if there is discreteness in spacetime, it ought to start showing up in experiments. The best we can say right now is that treating spacetime as continuous works well enough for all conceivable purposes at this point, and is a lot simpler mathematically.

Is it possible that the universe is discrete, but composed of such a high number of stencils that any error is insignificant for classical mechanics, which deals with the macroscopic world?

It's not just the macroscopic world; as above, treating spacetime as continuous works well at the smallest scales we can currently probe, where you need quantum mechanics to properly model things.

 

[bhobba]

The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world.

It is? I must have missed the memo

Seriously this is an advanced level thread and you need an advanced level answer. The following videos will illuminate the question of why in QM things are sometimes discreet (well mostly they are) and occasionally not, despite the continuous nature of Schrodinger equation:

My physics professor said that reality is continuous, but I do not think that this concept can be assimilated by the human mind.

As far as we can tell today he is correct.

What continuous is was sorted out by guys like Weierstrass in the 19th century when modern analysis was founded. It resolved Zeno's pardox etc and puts continuity on a completely rigorous and exact footing. You need to study a book on analysis, such as the following which Dr Trench has kindly made available free on his website:
http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

The short answer is the least upper bond axiom (LUB) of real numbers. Real numbers and rationals have exactly the same axioms except for this one thing. But what a 'thing' it is, it allows all sorts of fascinating definitions and proofs, such as defining convergence, continuity, exactly what an integral is - all sorts of stuff. Only by analysis can your question be answered. It resolves Zeno's paradox trivially - the race has an upper bound so by the LUB axiom a least upper bound. That is the point, time, whatever way you want to look at it, the race finishes, as it must because its modeled by real numbers.

How does one get classical mechanics from QM. Well you use Feynman's path integral approach to get, for classical objects, the principle of least action (classically, except for paths with stationary action there is always a very close path that cancels that path. For stationary action the very close paths are the same so reinforces it). Now from that alone, and symmetry considerations (think Noether) you get classical mechanics. Its not usually done that way, but the great Lev Landau wrote a book doing just that:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

It will blow your mind - as it did mine - enough said - its up to you now. He doesn't explicitly use Noether, but the effect is the same - you are wide eyed at the true power of symmetry. A science adviser here, when he teaches this stuff, his class just sits there in stunned silence as natures true secret is revealed. You will never be the same again.

Thanks
Bill

 

[A. Neumaier]

is the reality in which the mentioned physical quantities are discreet?

Reality is not a physical notion. In your question you refer to time and indirectly (via pixels) to space. The following, taken from one of my contributions to SE Physics, discusses the problem for time, but everything said also holds for space.

• As we cannot resolve arbitrarily small time intervals, what is ''really'' the case cannot be decided.
• But in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous.
• Physics would become very awkward if expressed in terms of a discrete time: The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.
• If time appears discrete (or continuous) at some level, it could still be continuous (or discrete) at higher resolution. This is due to general reasons that have nothing to do with time per se. I explain it by analogy: For example, line spectra look discrete, but upon higher resolution one sees that they have a line width with a physical meaning.
• Thus one cannot definitely resolve the question with finitely many observations of finite accuracy, no matter how contrived the experiment.

Thus, in a sense, Nature is extremely discreet (sic!), in that it does not allow us to decide this question.

In any case, the most successful physical models of reality are all continuous.

 

[Boing3000]

But in classical and quantum mechanics (i.e., in most of physics), time is treated as continuous.

Excuse me to ask a question that will certainly sound silly but, can't the "renormalization" or the cut-offs needed in the continuous theory related to the impossibility for those n'th order possibility to exist (no more "pixel" to fit).
Otherwise asked, do all the "off-shelves" particles "last" for duration above the Planck scale ?

 

[OCR]

...discreet (sic!)...

Thank you... .

 

[A. Neumaier]

Excuse me to ask a question that will certainly sound silly but, can't the "renormalization" or the cut-offs needed in the continuous theory related to the impossibility for those n'th order possibility to exist (no more "pixel" to fit).
Otherwise asked, do all the "off-shelves" particles "last" for duration above the Planck scale ?

Even a QFT with cutoff is a continuum theory, with a continuous spacetime.

And virtual particles don't last at all; see my insight articles.

 

[Zafa Pi]

The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics.

You might want to check out the thread: https://www.physicsforums.com/threads/fundamental-theorem-of-quantum-measurements.937882/page-3
It deals with the same issues.

 

[qttv]

Thanks very much to everyone!