I like this! Something I would add at some point before unitarity is that there is another type of universe that we almost inhabit, where your vectors of states have real positive coefficients that sum to 1, and your evolution matrices are Markovian (i.e., have positive coefficients and preserve the sum of coordinates). In a certain sense in such a universe it’s weird to say “the universe is .3 of this particle being in state 1 and .7 of it being in state 2”, but if we interpret this as a probability, we have lived experience of this.
Something that I like to point out that clicked for me at some point and serves as a good intuition pump, is that for many systems that have a real and quantum analogue, there is actually an interpolated collection of linear dynamics problems like you described that exactly interpolates between quantum and statistical. There’s a little bit of weirdness here, BTW, since there’s this weird nonlinearity (“squaring the norm”) that you need to go from quantum to classical systems. The reason for this actually has to do with density matrices.
There’s a whole post to be written on this, but the basic point is that “we’ve been lied to”: when you’re introduced to QM and see a wavefunction ψ, this actually doesn’t correspond to any linear projection/disentanglement/etc. of the “multiverse state”. What instead is being linearly extracted from the “multiverse state” is the external product matrixψψ†, which is the n×n complex-valued matrix that projects to the 1-dimensional space spanned by the wave function. Now the correction of the “lie” is that the multiverse state itself should be thought of as a matrix. When you do this, the new dynamics now acts on the space of matrices. And you see that the quantum probabilities are now real-valued linear invariants of this state (to see this: the operation of taking the outer product with itself is quadratic, so the “squared norm” operators are now just linear projections that happen to have real values). In this picture, finding the probability of a measurement has exactly the same type signature as measuring the “probability of an event” in the statistical picture: namely, it is a linear function of the “multiverse vector” (just a probability distribution on states in the “statistical universe picture”). Now the evolution of the projection matrix still comes from a linear evolution on your “corrected” vector space of matrix states (in terms of your evolution matrix U, it takes the matrix M to UMU†, and of course each coefficient of the new matrix is linear in the old matrix). So this new dynamics is exactly analogous to probability dynamics, with the exception that your matrices are non-Markovian (indeed, on the level of matrices they are also unitary or at least orthogonal) and you make an assumption on your initial “vector” that, when viewed as a matrix, it is rank-1 complex projection matrix, i.e. has the form ψψ†. (In fact if you drop this assumption of being rank-1 and look instead at the linear subspace of matrices these generate—namely, Hermitian matrices—then you also get reasonable quantum mechanics, and many problems in QM in fact force you to make this generalization.)
I like this! Something I would add at some point before unitarity is that there is another type of universe that we almost inhabit, where your vectors of states have real positive coefficients that sum to 1, and your evolution matrices are Markovian (i.e., have positive coefficients and preserve the sum of coordinates). In a certain sense in such a universe it’s weird to say “the universe is .3 of this particle being in state 1 and .7 of it being in state 2”, but if we interpret this as a probability, we have lived experience of this.
Something that I like to point out that clicked for me at some point and serves as a good intuition pump, is that for many systems that have a real and quantum analogue, there is actually an interpolated collection of linear dynamics problems like you described that exactly interpolates between quantum and statistical. There’s a little bit of weirdness here, BTW, since there’s this weird nonlinearity (“squaring the norm”) that you need to go from quantum to classical systems. The reason for this actually has to do with density matrices.
There’s a whole post to be written on this, but the basic point is that “we’ve been lied to”: when you’re introduced to QM and see a wavefunction ψ, this actually doesn’t correspond to any linear projection/disentanglement/etc. of the “multiverse state”. What instead is being linearly extracted from the “multiverse state” is the external product matrix ψψ†, which is the n×n complex-valued matrix that projects to the 1-dimensional space spanned by the wave function. Now the correction of the “lie” is that the multiverse state itself should be thought of as a matrix. When you do this, the new dynamics now acts on the space of matrices. And you see that the quantum probabilities are now real-valued linear invariants of this state (to see this: the operation of taking the outer product with itself is quadratic, so the “squared norm” operators are now just linear projections that happen to have real values). In this picture, finding the probability of a measurement has exactly the same type signature as measuring the “probability of an event” in the statistical picture: namely, it is a linear function of the “multiverse vector” (just a probability distribution on states in the “statistical universe picture”). Now the evolution of the projection matrix still comes from a linear evolution on your “corrected” vector space of matrix states (in terms of your evolution matrix U, it takes the matrix M to UMU†, and of course each coefficient of the new matrix is linear in the old matrix). So this new dynamics is exactly analogous to probability dynamics, with the exception that your matrices are non-Markovian (indeed, on the level of matrices they are also unitary or at least orthogonal) and you make an assumption on your initial “vector” that, when viewed as a matrix, it is rank-1 complex projection matrix, i.e. has the form ψψ†. (In fact if you drop this assumption of being rank-1 and look instead at the linear subspace of matrices these generate—namely, Hermitian matrices—then you also get reasonable quantum mechanics, and many problems in QM in fact force you to make this generalization.)
That is… a very interesting and attractive way of looking at it. I’ll chew on your longer post and respond there!