Some questions about QM

[Alan Grayson]

1) What necessitates the use of complex numbers (whereas in GR only real numbers are used)?

2) What necessitates the postulate that some, but presumably not all operators are non commuting?

3) With respect to 2), why is the non commuting difference i*h (or i*hbar)?

TY, AG

[John Clark]

We know that some observables (like momentum and position, and energy and time) are non commuting, therefore any theory that attempts to describe that behavior is going to need to have non commuting mathematical entities in it. And if you're going to explain the quantum interference effects that occur when two particles interact with each other then you're going to need to preserve both the wave phase and the amplitude of both particles, and that's easy to do in the complex plane. Actually for a long time physicists thought you could develop a quantum theory that used only real numbers, although it was less elegant and would make calculations far more difficult; and in some special situations that is true, but in 2021 it was proven that it's NOT generally true. 

Quantum theory based on real numbers can be experimentally falsified 

This is the abstract of the above paper: 

"While complex numbers are essential in mathematics, they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through theories. While most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. In fact, previous works showed that such "real quantum theory" can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Thus, are complex numbers really needed in the quantum formalism? Here, we show this to be the case by proving that real and complex quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment whose successful realization would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics."

 John K Clark    See what's on my new list at  Extropolis

6ab

 

[Brent Meeker]

Barandes formulation of QM avoids complex Hilbert space, but gives up describing paths thru configuration space.

Brent

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[Alan Grayson]

On Monday, March 17, 2025 at 3:36:31 PM UTC-6 Brent Meeker wrote:

Barandes formulation of QM avoids complex Hilbert space, but gives up describing paths thru configuration space.

Brent

What does "gives up" mean, that it can, or cannot describe paths through configuration space? AG