Nothing brings out the silliness in smart people like Quantum Mechanics; a subject I always associate with … R. A. Fisher. I confess to liking Fisher more than Bayesians should. Unlike the forgettable p-value conjurers I’ve known in person, Fisher’s writing portrays a thinking, creative scientist and feels oddly familiar. So what does Fisher have to do with Quantum Mechanics?
If you take the Lagrangian for Schrödinger’s Equation and replace the wave function with where is the probability distribution, you get:
While the quantity being averaged over is similar to the classical Hamilton-Jacobi equation:
Which makes the whole thing reminiscent of the constrained maxent procedure from Statistical Mechanics. It’s almost as though Fisher’s information was being extremized subject to a classical constraint. Note Fisher’s Information is a quadratic approximation to entropy and similar “information” terms appear in the lagrangians for the Pauli, Klein-Gordon, and Dirac equations as well.
Since Fisher’s Information is related to the Cramer-Rao bound, its not surprising it’s mathematically related to the Uncertainty Principle. In fact, Fishers’ Information can be used to derive an improved uncertainty relation,
All of this is incredibly suggestive, but unfortunately it’s gone nowhere. Both Fisher’s Information and Schrodinger’s Equation originated around the same time (1920′s) and many must have seen the connection, but no one’s been able to make anything out of it.
There is this work by B. Roy Frieden, which claims Fisher’s Information explains everything, but unfortunately this seems to be the work of a crank. I’m not sure about Frieden’s background, but he seems to have been a well respected Physicist who became familiar with Fisher’s information from applied work, possibly in image reconstruction, and noticed its presence in Quantum Mechanics. I don’t think he was a fully fledged Bayesian, but he seems to have run in Bayesian circles.
His attempt is obviously well meant, but I doubt this would have been published if he hadn’t already been an established scientist.
Anyway, Fisher’s Information may have nothing to do with the foundations of Quantum Mechanics, but it’s an idea worth keeping in mind.