Saturday, February 27, 2021

Edwin's understanding of QM: Part I, physics vs mathematics

Today I have listened to a large number of interviews, especially on the brilliant Free Universum Radio hosted by the smart opera singer Ms Martina Kociánová. But I also saw a 10-day-old quantum mechanical video with Edwin Steiner.

It is 13 minutes long but if your brain is fast enough, you may speed up the video by a factor of 1.5 or 2.0.

In this case, I have carefully listened to everything and I may say: I approve this message. ;-) It's a very nice piece of work, Edwin.

In particular, he focuses on the definition of a physical theory – a set of rules how to produce quantitative statements about the real world that should be true (and this is verified experimentally). This is a seemingly trivial but important point that is often misunderstood, especially whenever a confusion about quantum mechanics emerges. In particular, Edwin also emphasizes that mathematical objects never have any automatic physical interpretation. Any physical interpretation of mathematical objects (or, inversely and almost equivalently, the representation of physical objects by the mathematical ones) must be postulated by physical axioms or derived from these physical axioms.

Then he also illuminates the key "positivist" point that our knowledge of the reality ultimately comes from observations, and if something comes from other channels (without observations), then we can't say it is a real justifiable knowledge. We should be skeptical about ideas that don't come from observations. And indeed, ideas about unobserved quantities have often turned out to be wrong (philosophically incorrect prejudices). Because of this central role played by observations, we must accept that words like "knowledge", "observation", "measurement result", and "uncertainty" are important and pre-exist: they must be considered independent of any particular theory. And again, when we add a physical theory, it is because we want rules to derive new true quantitative statements (often about the future) from others, initial ones (often observations that were done in the past).

In mathematics, we can prove the truthfulness of a theorem without "observations". In physics and natural sciences, our knowledge about reality does require (and depend on) the observations.

In more detail and with new clothes (if I watched carefully enough), he explained that our knowledge is encoded in numbers that are assigned to measured quantities. Our knowledge is made out of \(L=\lambda\) statements; the Greek letter is called lambda. This statement is translated to some initial conditions for something (like the wave function and density matrix in QM; but he doesn't make premature statements!) and this something may be evolved in time. The final form of this something is used to make predictions. Those may be compared to observations.

He correctly points out that the physical theory is not just the "differential equation for the evolution in time" but the whole \(2\times 2\) square, including the recipe what we observe, what the values may be, and how this knowledge is translated to/from the initial and final conditions of the mathematical object that is evolved in time. Again, the physical interpretation of (often time-dependent) mathematical objects cannot be considered "obvious". It doesn't pre-exist and the right connection may require quite some original and/or calm thinking – and it may change once we switch from one physical theory to another.

He shows what it means, using the example of the measurements and predictions of \(X,P\) in the 1D mechanics of a point-like particle. The uncertainty of continuous quantities was always nonzero because no apparatuses could ever be perfect. Steiner carefully figures out that the measurements of \(X,P\) and the error margins \(\Delta X, \Delta P\) that say that certain initial states cannot be distinguished, are theory-independent parts of the physical theory. Well, every theory that describes and predicts something dealing with measurements of \(X,P\) must involve these pieces. (So it's only theory-independent if we look at classical or quantum theories with \(X,P\), I would say.) The remaining 3 boxes in the \(2\times 2\) square are said to be different in classical and quantum physics.

This \(X,P\) stuff should be done in his Lecture 2. Here in the final 2 minutes, he reminds you once again to distinguish mathematical and physical objects. Quantum mechanics isn't stuff for philosophers' cocktail parties. It is a physical theory which means a complete set of rules for quantitative reasoning about the observable world.

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