Monday, February 5, 2024

What's the difference, said Heisenberg

From a math site:
In the 1960s Friedrichs met Heisenberg and used the occasion to express to him the deep gratitude of mathematicians for having created quantum mechanics, which gave birth to the beautiful theory of operators on Hilbert space. Heisenberg allowed that this was so; Friedrichs then added that the mathematicians have, in some measure, returned the favor. Heisenberg looked noncommittal, so Friedrichs pointed out that it was a mathematician, von Neumann, who clarified the difference between a self-adjoint operator and one that is merely symmetric. "What's the difference," said Heisenberg.

- story from Peter Lax, Functional Analysis (slightly edited for length)

There is the difference between a physicist, and a mathematical physicist.

John von Neumann wrote a 1932 book on quantum mechanics, and turned it into a real theory.

To a physicist, an observable is a symmetric operator, because those are the ones that give real values, and only real values are observed. To von Neumann, an observable is a self-adjoint operator on a Hilbert space, where some additional technical requirements are needed in order to prove the spectral theorem.

I am not trying to say that Heisenberg was stupid. But it is striking that a world-famous physicist could get a Nobel Prize for using operators as observables, and still be oblivious to the formal mathematical definition found in textbooks. We cannot expect physicists to understand mathematical subtleties.


  1. Roger,
    Many mathematicians don't know what mathematical subtleties are either, namely the biggest subtlety of all, that math isn't reality, things don't elide between abstract and actual. The field of math is ironically accurate, as it IS now an endless field filled grown fallow with fast growing weeds that push out useful ideas and rabbit holes that go unto infinity.

    OH OH! you say, BUT I can describe it with MATH! Sure you can, you can also describe anything you imagine with it (Think 'The landscape' problem and probabilities), which IS the problem...potentially 10^272000 or more problems.

    Reality isn't anything you can imagine, it's a very tiny subset. Think of a potential lottery winner even in reality... one in seven million... ok, you COULD win, but betting on winning it in order to claim success? Mathematical Sleight of hand is great for cheating at cards and snowing illiterate government grant writers, not so much explaining how things work.

  2. What Hilbert and what Banach, Roger?

    Space is blue and birds fly through it.