This is a Quora answer that I wrote three years ago, to a question about wave-particle duality, the effects of observation on the system being observed, and Einstein's views.

I prefer not to think of quantum physics too much in terms of this wave-particle duality business, because it hides something far more fundamental. Namely that at the quantum level, physics is not characterized by numbers, but rather, by non-commuting quantities, which Dirac called q-numbers.

This has numerous consequences, not the least of which is that because of the rules of arithmetic that apply to these quantities, not all of them can be simultaneously number-valued. In short, when you observe, say, the momentum of an electron (that is to say, you set up an experiment in which the electron’s momentum interacts with a classical apparatus, forcing the momentum to be in a so-called eigenstate, i.e., be number-valued) its position cannot be number-valued: this electron at this time has no classical position. This is important to emphasize: it is not the limitations of our instrumentation or our inability to measure something. You cannot measure what does not exist.

This is why it is fundamentally wrong to say that measuring the electron’s momentum, say, affects its position and introduces an uncertainty. This was, of course, Heisenberg’s view but we have come a long way since Heisenberg. Quantum reality is something much more profound: in an experimental configuration in which the electron’s momentum interacts with a classical apparatus, the electron has no classical position at all.

So what does it have instead? This electron would have a position characterized by a q-number. The q-number does not tell you what the position is; but it can tell you what the probabilities are of measuring specific values of position, that is, the probability of finding the electron in different places.

And when you look at the governing equation of these probabilities, it will typically be a wave equation. This wave equation tells you the probability that, given prior measurements (e..g, that prior measurement of the electron’s momentum) how the probability of finding the electron evolves from place to place and from time to time. But if there is, in fact, a measuring apparatus that interacts with the electron’s position at some point in space and time, that will constrain the position of the electron to be number-valued then and there. So it will be observed, as always, as a point-like particle with a definite position even though at no other time did it have a classically defined position.

There is, by the way, a fairly revealing classical analogy to all this. Never mind position and momentum. Think instead about an ordinary sound and two of its properties: the time when it is heard and its frequency. When a sound is a perfect sine wave, its duration is infinite, so the time when it is heard is ill-defined. Conversely, think of a momentary sound like a gunshot. Its timing is very well defined but in the frequency domain, that loud pop is a combination of a myriad frequencies; there is no well-defined pitch. The two quantities are related to each other by a so-called Fourier transform, just like the position and momentum of an elementary particle are related to each other by a Fourier transform. So when one of the two has a well-defined classical value (i.e., it is represented by a so-called Dirac delta function) the other would be a sine wave of sorts.

Einstein’s views… Einstein mostly objected to the idea that quantum physics is about probabilities, as he believed firmly that physical reality exists independent of our ability to observe it. I think that if he lived longer and had been given a chance to become familiar with quantum field theory, especially its modern formulations, he would have welcomed it. In particular, he would have liked the idea that in a typical quantum field theory calculation, there are no probabilities. Everything is exact, including conservation laws that are exactly satisfied, or the calculation of various cross-sections that an interaction can have. This would have made it a lot clearer that probabilities enter the scene when we introduce the fiction of a classical measuring apparatus, manifested in the form of “external legs” of a Feynman diagram, for instance. Alas, Einstein died 62 years ago, when QFT was still in its infancy, and its spectacular successes (in particular, the foundational role it plays in the Standard Model of particle physics) were still many years, even decades away.