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Complex measures are a thing in mathematics since… well, possibly since measure theory was. The thing is expressing this in a way which is understandable by a p

Complex measures are a thing in mathematics since… well, possibly since measure theory was. The thing is expressing this in a way which is understandable by a physicist (measure spaces are not exactly “simple”). https://web2.ph.utexas.edu/~gsudama/pub/1994_004.pdf https://web2.ph.utexas.edu/~gsudama/pub/1994_004.pdf

sigmoid10 4y ago
Measure theory itself only became a formal branch of mathematics around the turn of the 20th century and complex measures developed pretty much in the fairway of quantum mechanics (which appeared at the same time). This is in stark contrast to e.g. non-euclidean geometry, which had already been well formalised into its own branch decades before the invention of GR.
- mananaysiempre 4y ago
  Riemannian geometry in general (and not just geometry of specific homogeneous spaces like the hyperbolic plane) was still somewhat recent and hard at the time or at least had that reputation. I’ve heard that the notion of parallel transport on Riemannian manifolds, for one, was developed as a consequence of people making sense of GR, before that Christoffel symbols were apparently just ... a formal thing. (The general theory of connections on bundles certainly postdates GR although it does predate Yang–Mills.)
  - sigmoid10 4y ago
    To some extent, yes. But the point was that Einstein was probably only able to develop GR thanks to work by people like Riemann, who investigated these things not knowing where they'd lead half a century later. For QM it was almost the other way around and people really had to come up with and formalise a whole new branch of mathematics to make sense of the early experiments. Sometimes physics follows math and sometimes math follows physics. For QM it was closer to the latter.
- pfortuny 4y ago
  I know. I was trying to stress that it is not novel at all to think of “complex probability”.

sigmoid10 4y ago

Measure theory itself only became a formal branch of mathematics around the turn of the 20th century and complex measures developed pretty much in the fairway of quantum mechanics (which appeared at the same time). This is in stark contrast to e.g. non-euclidean geometry, which had already been well formalised into its own branch decades before the invention of GR.

mananaysiempre 4y ago
Riemannian geometry in general (and not just geometry of specific homogeneous spaces like the hyperbolic plane) was still somewhat recent and hard at the time or at least had that reputation. I’ve heard that the notion of parallel transport on Riemannian manifolds, for one, was developed as a consequence of people making sense of GR, before that Christoffel symbols were apparently just ... a formal thing. (The general theory of connections on bundles certainly postdates GR although it does predate Yang–Mills.)
- sigmoid10 4y ago
  To some extent, yes. But the point was that Einstein was probably only able to develop GR thanks to work by people like Riemann, who investigated these things not knowing where they'd lead half a century later. For QM it was almost the other way around and people really had to come up with and formalise a whole new branch of mathematics to make sense of the early experiments. Sometimes physics follows math and sometimes math follows physics. For QM it was closer to the latter.
pfortuny 4y ago
I know. I was trying to stress that it is not novel at all to think of “complex probability”.