The basic laws of classical physics relied upon the principle ”Natura non facit saltus” (nature does not make jumps), transmitted from ancient philosophy. The underlying assumption was the existence of a space-time continuum and all changes in nature should occur continuously within this space-time continuum. Starting towards the end of the 17’th century, the classical laws governing these changes were expressed in form of differential equations or variational principles, where infinitesimally small changes of various physical variables are related to each other. Typically these differential equations of classical physics possessed exact solutions for given initial and boundary conditions, at least in principle. This led to the general conclusion that nature is deterministic; the state of nature at any given time was believed to be related in a unique way to its state at any past or future time. Even if the development of statistical thermodynamics related probabilities to thermodynamic variables, these probabilities were meant to describe insufficient knowledge of details due to the large numbers of microscopic particles involved, but deterministic behavior of all individual processes was not questioned. ...No, this is a very strange statement of the QM basic principles.

Born, Werner Heisenberg and Pascual Jordan in 1925, are contained in:

The basic principles of Quantum physics:

- On the microscopic level all elementary changes in nature are discontinuous,

consisting of quantized steps: ”quantum transitions”.

- The occurrence of these quantum transitions is not deterministic, but governed by probability laws.

(He means atomic level, not microscopic level.)

Position, momentum, time, frequency, energy, etc. are not necessarily discrete. The wave function for an electron is typically a continuous function of space and time, and progresses smoothly according to a differential equation. So it is just wrong to say everything is discontinuous.

The emphasis on probabilities is also misleading, as he concedes that other physics theories also hinge on probabilities. He distinguishes QM by whether "insufficient knowledge" is involved, but that is just a philosophical issue.

The path towards the future Quantum Theory is defined asSome problems have a discrete spectrum, but all this talk about a discrete manifold is misleading.

”the systematic transformation of classical mechanics into a discontinuous atomic mechanics..... the new mechanics replaces the continuous manifold of (classical) states by discrete manifold, which is described by ”quantum numbers”....

quantum transitions between different states are determined by probabilities...

the theoretical determination of these probabilities is one of the profound tasks of Quantum Theory....”.

The essence of QM is that the observables are represented by non-commuting linear operators.

The core of the theory is continuous, with continuous spacetime, continous wave functions, etc. For all we know, light is continuous, but only shows properties at discrete frequencies sometimes because some continuous problem has a discrete spectrum. Even then, that spectrum can often be perturbed by applying an electric or magnetic field.

1. No, light does show discreteness in the low-flux experiments, whether you regard the photon as spatially delimited or not. Radiation and absorption processes involve not just jumps between different energy eigenstates (as captured by the creation/annihilation operators) and therefore specific frequencies, but also a certain temporal discreteness to them.

ReplyDeleteThe temporal discreteness of the photon is an experimentally observed fact. Experimentally, there is a small but finite time duration over which the process of detecting any single photon occurs, and there also are finite temporal gaps (I suppose themselves obeying the Poisson statistics) between two successive detection events. (No process occurs in zero time, and further, events become discrete because of the time-gaps between them.)

Now, sure, when it comes to photons, the case for a supposed spatial discreteness is not quite on an equal footing.

In theory, you may consider the fact that the nuclei of atoms are localized and isolated (i.e. spatially distinguishable), and use this fact to deny spatial discreteness to the photon. According to this argument, the photon does appear to be spatially discrete, but that is only because, in this argument, it goes from one atom to another atom and the two atoms are spatially isolated (i.e. they are discrete). However, continuing the same argument, the actual transmission of the EM energy occurs only as a continuous flow-like process between the two. In other words, you may offer an analogy to pouring of a drink from a bottle to a glass, and contrast it from putting a basketball into the ring. The ball has discreteness, but the flow of the drink doesn't (at least not for the ``middle'' duration of the process). You may argue that way.

But no matter how you look at it, temporally, the photon always remains discrete---whether there is perturbation or not, and whether the photon propagation dynamics is ballistic or diffusive.

2. As to ``Natura non facit saltus'', things aren't always as simple as they might initially appear to the layman.

Take a piece of paper and fold it to produce a sharp fold. The paper remains continuous (it has not been torn apart, i.e. the C0 continuity exists), but the two parts are clearly distinguishable from each other because of the fold separating the two. Now, assume the fold to be infinitely sharp, i.e., a lack of differentiability. If you now take a pencil and trace a continuous curve on the folded paper, even if you don't ever have to lift the pencil off the paper, its crossing over at the fold can still regarded as a jump---a discrete jump from one part to another. Conclusion: Smoothness (or a lack of it) can be used in order to define a jump, even if a basic continuity is assured.

3. Yes, the core of the QM theory _is_ continuous. Doesn't TDSE govern the quantum jumps?

4. It's only today that I realized that ``natura non facit saltus'' is 17th century (Leibnitz) in origin, not ancient Greek (say, Aristotle). (At least that's what the Wiki leads one to believe.)

--Ajit

[E&OE]

I agree with most of what you say. And yes, a jump in the derivative is a jump, even if there is no jump in the function.

ReplyDeleteBut all those experimental proofs of the temporal discreteness of the photon depend on some material absorbing the photon and turning it into an electrical signal. That means that the photon had to strike some atom, and knock an electron into a higher energy level. Quantum mechanics teaches that those energy levels are discrete. So it is possible that the photon is not discrete, but it looks discrete because we only have discrete means for observing it.

If I only ever listened to music on my (digital) computer, I might conclude that all music is discrete at 44kHz.

Whatever a photon is, it is not a particle in the classical Newtonian sense. If you say it is a particle, you have to give it some sort of blurry existence by adopting some probability model for where it is and what it is doing.

Roger, math is not nature and it makes no sense to claim something is continuous if the mathematical definition defines it has infinite discreteness. Again, you ignore this but apparently your UCLA education didn't give you any idea that continuity and discreteness are not opposites. I'm used to debating fifth-rate mathematicians, so this is just getting to be a copy-and-paste for me. You claim positivism but continuity is never provable. Physics is computational and MIT's Wen even says chiral fermions are simulatable with lattice gauge theories. Grow up.

DeleteDear Roger,

Delete1. Earlier (i.e. up to c. 2015-end), I used to take the photon as spatially discrete (like a ball). (I was thinking under the influence of Feynman.) Now, especially after going through and thinking about W. E. Lamb, Jr.'s ``Anti-Photon'' paper, I now do appreciate your argument (which I have presented in my comment above, in the passage from ``Now, sure, ... '' to ``...You may argue that way.'')

This argument is excellent. But does it fully settle the matter? I was not immediately sure, and so, though I am more or less fully in favor of it, I presented it only as an argument.

If you ask why I hesitate: How about a global function that carries a hump? The hump makes it distinguishable (due to discontinuity in some n-th order derivative(s)), but the function remains continuous everywhere and hence, global (with a potentially infinitely large support). With such a hump, can't one still describe the photon as discrete, even spatially?

Keeping that point aside, our present differences, I think, are only in wording. You talk in terms of a material absorbing a photon (and turning it into an electrical signal), whereas I talk only of the initial part of the process, and so put it in terms of _an_ atom radiating (I meant emitting) or absorbing a photon, but without as yet touching on knocking an electron loose and that event leading to a catastrophic cascade involving a large number of atoms and electrons on the absorber side. (And what about on the emission side?)

2. Actually, I am in the process of writing my next blog post in my series about the physical view underlying QM. I will come to some of these matters in detail when I post it. I have written down some points and notes, already. (I have decided to cease blogging until the time that I land a Mech. Engg. Professor's job in Pune. By now I am _very_ likely to, but unless it actually happens...)

But here is a rapid preview, simply because with you, I don't have to spend time on building the context.

The wavefunction for all the electrons in the universe as one; so, all the theoretical issues eventually to be explained in reference to that one wavefunction. The issues of: the meaning of photons of radiation and their propagation (both now being taken as abstract views of certain local adjustments in the universal electron cloud over time, because the photon number is not conserved); the discrete-or-continuous character of the cavity radiation graph; the relation between n electrons and their singleton (or shared) n-dimensional wavefunction (singleton, as in C++ programming); the relation between the n-dimensional wavefunction and the 3D ambient space; ionization and de-ionization; effective (or net) electron travel (as in a CRT tube); photon propagation and the constant effective speed of light; catastrophic cascades (near the atom of absorption); scattering as a grosser-scale description of many simpler processes; etc. (Spinors, polarization, entanglement, anti-bunching and all will simply have to wait! Not to mention the relativistic (Dirac's) QM!!)

All in all, I think we are in a pretty good agreement, though there still remain a lot of issues and details to think through (esp. for me).

Bye for now.

Best,

--Ajit

[E&OE]

Matthew, you are right that a positivist has to concede the possibility of discrete electrons because continuity cannot be proven. I said, "For all we know, light is continuous". I meant that light can be interpreted as continuous, but discreteness is also possible.

ReplyDeleteThey don't have to "concede" anything because no one has ever given a clear idea of what it truly means to be continuous in nature. I reject the dichotomy. There are parts and wholes. Continuous analysis is a degenerate case of discrete analysis. Limit arguments are just finite reasoning in disguise and so is any of analysis that makes sense. Math is mostly about numbers and numbers (that anyone can actually write down) are necessarily about chopping up the world into intervals and pieces. Otherwise, we are talking about whole objects that are not divisible. Show me a real number and then we'll talk...

Delete