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I am wondering if there are any open questions about the structure of quantum mechanics. If so, how do you know that this is an open question? Topics that come to mind are electron spin, probability amplitudes, and decoherence. I am interested in the foundations of the theory, and have studied quantum mechanics at the graduate level, yet I am somewhat curious about and unsure of the viewpoints of other scientists at this point in time (January 2019).

Are there any open questions in theoretical quantum physics?

There is at least this million dollar open question:

The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap": the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.

There would be large numbers of open questions in quantum mechanics, both technical and conceptual.

An example of an unanswered conceptual question would be, what is quantum chaos, i.e. the quantum counterpart of chaos theory? People have thought about this for at least thirty years, they have all sorts of ideas, but it seems (to me at least, and not just to me) that something is still lacking.

An example of an unanswered technical question would be, anything that we don't know how to calculate, or any strictly mathematical question that is relevant to physics and which we don't know the answer to.

However, you express an interest in foundations and axioms. So perhaps you are really interested in things like, what is quantum mechanics actually saying about reality, and, can it possibly be the entire truth about physics? My answers to those questions are, it gives you conditional probabilities for physical possibilities, and, no, it is not the last word.

But my point really is that in the current state of knowledge, you need to arrive at some opinion of your own (even if it's just a tentative opinion) regarding those questions, if you are going to think independently about quantum mechanics; and you cannot rely on learning what the consensus is, in order to determine what opinions you should have, because there is no consensus.

By the time you have that sorted out - it is partly a philosophical task, and partly a practical one - you will be ready to investigate any of the thousand-and-one unsolved technical and conceptual problems of quantum physics. But the odds are that in getting to that level, you will already have spotted open questions that you find personally compelling.