I don’t know much about Clifford algebras. But do you really need them here? I thought the standard formulation of abstract quantum mechanics was that every system is described by a Hilbert space, the state of a system is described by a unit vector, and evolution of the system is given by unitary transformations. The Born probabilities are concerned with the question: if the state of the universe is the sum of civi where vi are orthogonal unit vectors representing macroscopically distinct outcome states, then what is the subjective probability of making observations compatible with the state vi? The only reasonable answer to this is |ci|2, because it is the only function of i that’s guaranteed to sum to 1 based on the setup. (I don’t mean this as an absolute statement; you can construct counterexamples but they are not natural.) By the way, for those who don’t know already, the reason that |ci|2 is guaranteed to sum to 1 is that since the state vector ∑civi is a unit vector,
Of course, most of the time when people worry about the Born probabilities they are worried about philosophical issues rather than justifying the naturalness of the squared modulus measure.
I don’t know much about Clifford algebras. But do you really need them here? I thought the standard formulation of abstract quantum mechanics was that every system is described by a Hilbert space, the state of a system is described by a unit vector, and evolution of the system is given by unitary transformations. The Born probabilities are concerned with the question: if the state of the universe is the sum of civi where vi are orthogonal unit vectors representing macroscopically distinct outcome states, then what is the subjective probability of making observations compatible with the state vi? The only reasonable answer to this is |ci|2, because it is the only function of i that’s guaranteed to sum to 1 based on the setup. (I don’t mean this as an absolute statement; you can construct counterexamples but they are not natural.) By the way, for those who don’t know already, the reason that |ci|2 is guaranteed to sum to 1 is that since the state vector ∑civi is a unit vector,
1=∥∑civi∥2=∑⟨civi,cjvj⟩=∑ci¯¯¯¯cj⟨vi,vj⟩=∑ci¯¯¯¯cjδi,j=∑ci¯¯¯¯ci=∑|ci|2.
Of course, most of the time when people worry about the Born probabilities they are worried about philosophical issues rather than justifying the naturalness of the squared modulus measure.