David Tong argues that quantum mechanics is ultimately continuous, not discrete.

In other words, integers are not inputs of the theory, as Bohr thought. They are outputs. The integers are an example of what physicists call an emergent quantity. In this view, the term “quantum mechanics” is a misnomer. Deep down, the theory is not quantum. In systems such as the hydrogen atom, the processes described by the theory mold discreteness from underlying continuity. … The building blocks of our theories are not particles but fields: continuous, fluid-like objects spread throughout space. … The objects we call fundamental particles are not fundamental. Instead they are ripples of continuous fields.

Source: The Unquantum Quantum, Scientific American, December 2012.

But what about the apparent discreteness of time and space at the Planck scale?

Well, yeah. The Schrödinger equation, for instance, is a differential equation whose solutions are continuous wave functions on the complex numbers. But those solutions do not transform continuously from one to another, and their energy eigenvalues are discrete. That discreteness emerges from the possible solutions to the Schrödinger equation; it’s not put there ab initio.

Fred: I’m no expert in such things, but I believe what David Tong is arguing is that discrete effects in quantum mechanics, including the ones you allude to, are consequences of a continuous theory, so the continuous theory is more fundamental.

This reminds me of the quip that all computers are analog computers because

digitalcomputers are made out ofanalogparts. But those analog parts depend ondiscretequantum phenomena. But those discrete quantum phenomena depend oncontinuousfields …One of the themes of modern math is that continuous and discrete as not such discrete categories as once thought. We’re continually finding connections between the two perspectives. (Puns intended.)

I found similar remarks on Lubos Motl’s blog:

“There is strong scientific evidence today that the world isn’t discrete (and it isn’t simulated).

We do encounter integers and discrete mathematical structures in physics but in all the cases, we may see that they’re derived or emergent. They’re just limited discrete aspects of a more general and more fundamental underlying continuous structure, or they’re a rewriting of a continuous structure into discrete variables (eigenstates in a discrete spectrum) which makes it impossible to understand the value of certain parameters.

Quite generally, if the Universe were fundamentally discontinuous, it couldn’t have continuous symmetries such as the rotational symmetry, the Lorentz symmetry, and even descriptions in terms of gauge symmetries (which aren’t real full-fledged symmetries but redundancies) would be impossible. In a fundamentally discrete world, many (or infinitely many) continuous parameters would have to be precisely fine-tuned for the product to “look” invariant under the continuous transformations.”

It gets really tricky because

“discrete quantum phenomena depend on continuous fields ”

where “continuous fields” isn’t something directly observable, but more of a mathematical tool.

Now it’s possible that there is some fundamental connection between the math tools and reality.

But I thought that, in general, dealing with continuum at a fundamental level is complicated because all sorts of infinities start to creep in.

Fred: You could say that whether or not nature is continuous, the quantum mechanical model of nature is continuous.

The math model is just a tool, their concepts are not the reality…

Think of classical mechanics. Everybody know Newton’s laws, based on the concept of force, but there are two, more modern, alternative formulations of classical mechanics (Lagrangian and Hamiltonian), they bypass the concept of “force”, instead use energy, and momentum… Their equations look very different too.

Now, if you don’t know the other formulations, you could argue that the nature of the universe is ultimatelly linked to the concept of “force”, because you see it all over, in the hard of your equations…

What I mean is, it’s perfectly possible that somebody device some day a formulation of quantum mechanics that begins with discrete fields… as long as it arrives to the same observable results, it’s correct!

Cheers,

Excellent. So it’s not just turtles all the way down. It’s like turtle and taffy and then turtle and taffy…

Thank you Cade! Yours is the only comment I understand! Actually, I didn’t understand the article either. I just have to be continuously discrete about my lack of knowledge.

But what if those continuous fields are made up of discrete units?

Quantum physics is quantum because the observable values are quantised, not because anything is discrete in the fields. That was the surprise, and why it got its name; classical physics is all continuous, and quantum physics just came up with quantisation in the middle, entirely unexpectedly.

The wave/particle duality is not so much about things being discrete on a fundamental level, but about fields and interactions being continuous yet only observable in discrete quantities. It is quantum as opposed to entirely continuous.

If we had only had discrete theories before that point, it would have been called Continuous Physics.