It seems that 20th C philosophers of science have traditionally demanded that predictions be worded as real-valued probabilities of outcomes. I tend to think that ultimately QM will be understood differently — I believe, with only intuition and casual hearsay to support me — that a deeper physical understanding will involve discarding the reals in favor of much more intricate algebraic structures. (Some of this move is happening, and I don’t know enough about Quantum Geometry to comment on what the underlying structures actually are.) It’s certainly the case, though, that human experience is well (although not perfectly) described and approximated by the reals, so demanding that probabilities be real seems reasonable.

So what? Understanding sigfpe’s points at face value, it seems that fundamental particles don’t experience the world in real probabilities, but in complex ones. Somehow, of course, these complexities, so to speak, disentangle as the number of particles approaches infinity, so that we get classical events.

A big part of this is that thinking of complex probabilities models one too many dimensions: for any given event, there’s an overall non-physical phase factors. So if I have a bunch of particles, except in a carefully constructed situation on average all the individual particles, etc., will be come with randomly distributed phases, which will ultimately cancel out.

Why do we take norms at the end? I say this is exactly _because_ of this overall phase factors: at the end of the day, the argument of your “probability” is meaningless. And the only good way, with current mathematical technology, of throwing away that one piece of information is by taking absolute values.

The situation is similar, in my mind at least, to gauge (and other symmetry) transformations. To wit: in EM, for instance, physicists use the (lorentz)vector field A_\mu to describe the state of the universe, even though A_\mu + \d_\mu\Phi will describe the state exactly as well, for any (lorentz) scalar field \Phi. During a calculation, which \Phi (which “gauge”) you pick is important, and there are “interference patterns” that you would not predict if you through away the gauge information for each constituent part (roughly equivalent to calculating as if you have a bunch of “classical” particles instead of “quantum” particles).

To make another physics analogy, GR notwithstanding the total potential energy is non-physical (in fact, passing from Lagrangians to Actions a la Feynman translates any constant shift of the potential into a phase shift): only the relationships between different potential energies matter. For potential energies, there is _no_ natural way to through away that bit, except by taking derivatives, and so its no wonder that Newton talked about the patently physical Force rather than Potential Energy. And yet the mathematics is so nice if potential energies had full real meaning that all of physics these days is worded in terms of lagrangians and hamiltonians rather than forces, that it’s almost as if potentials really exist.

Just like QM makes it seem that complex probabilities *almost* exist in their own right.

P.S. I was going to mention, too, that complex amplitudes can’t be completely physical because there’s no way of picking *i* or –*i*. Which is sortof true, but not a good proof. For one: places like Spec(\R[x]) (i.e. the orbifold of \C under complex conjugation) do a good job of collapsing such concerns, and still preserving enough underlying structure that one might construct a natural and coherent system of arithmetic in which to do calculations (it hasn’t been done yet, to my knowledge). But also: I would read any (unpatched) CPT violation as essentially the universe distinguishing between *i* and –*i*, and I don’t think this is an out-of-the-question possibility.

But I’ll certainly check out the Hacking at some point! (And I’ll have to check out your paper too.)

]]>The ‘calculus’ that you use to derive these amplitudes is formally very similar to probability theory. In particular these amplitudes act formally like probabilities up until the very last step where you translate them into ‘normal’ probabilities by using |.|^2. For example, speaking crudely, the amplitude of “A or B”, where A and B are disjoint, is the sum of the separate amplitudes, and the amplitude of “A and B”, where A and B are ‘independent’, is the product of the amplitudes. So we’re not simply doing ordinary probability theory but working with some kind of complex square root of the ordinary probability. An amplitude really is a different kind of thing. For example P(A or B) can be 0 even though P(A)>0 and P(B)>0, something impossible for ordinary probabilities.

If you have any interest in alternative probability theories I strongly urge you to investigate QM – or maybe quantum computing, because that isolates the part of QM that has consequences for the philosophy of probability without you having to get bogged down in the subjects of traditional physics like masses, momenta and energies.

Actually I started writing a document a while back giving a basic introduction to quantum mechanics from the point of view of QM being formally almost identical to a probability theory. The intention was to give people with no physics background, but at least some intuition about probability, an easy “way in” to understand QM. I must finish it some time. I think what I was writing was completely uncontroversial, I was just highlighting the formal mathematical similarity without making any philosophical claims. The simlarity is obscured by most physics textbooks.

]]>I suppose though that there is no core sense of what probability really is – it’s just a family of related notions, like credence, partial entailment, partial belief, propensity, chance, and perhaps other notions that happen to satisfy (something very much like) the Kolmogorov axioms. These others with strange sets of numbers are just slightly more exotic members of the family.

]]>I’ve also heard of some proposals that would …expand to complex numbers…I think most of these suggestions occur for purely mathematical reasons

Expanding to complex numbers is the essence of quantum mechanics and they are used because they actually work in the real world. One of the most bizarre things about quantum mechanics is that the “possible worlds” become a little more actual than possible and participate in interactions in an observable way. Hence phenomena like the double slit experiment or even “counterfactual computation”.

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