The standard references on quaternionic QM are:

* D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, Journal of Mathematical Physics 3, 207 (1962)

* D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Some physical consequences of general Q-covariance, Helvetica Physica Acta 35, 328-329 (1962)

* D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Principle of general Q-covariance, Journal of Mathematical Physics 4, 788-796 (1963)

A standard structural result in the order-theoretic vein which separates Reals, Complex Numbers and Quaternions from ”non-classical fields” is:

* M. P. Soler (1995) Characterization of Hilbert spaces with orthomodular spaces, Comm. Algebra 23, pp. 219–243.

It does this relative to the order-theoretic characterization of Hilbert spaces:

* C. Piron (1964, French) Axiomatique Quantique, Helv. Phys. Acta 37, pp. 439-468.

* I. Amemiya and H. Araki (1966) A Remark on Piron’s Paper, Publ. Res. Inst. Math. Sci. Ser. A 2, pp. 423–427.

* C. Piron (1976) Foundations of Quantum Physics, W. A. Benjamin, Inc., Reading.

A nicely written recent survey on this stuff is:

* Isar Stubbe and B. van Steirteghem (2007) Propositional systems, Hilbert lattices and generalized Hilbert spaces’, chapter in: Handbook Quantum Logic (edited by D. Gabbay, D. Lehmann and K. Engesser), Elsevier, to appear. Download-able from http://www.win.ua.ac.be/~istubbe/

It is not clear to me how exactly this order-theoretic stuff relates to the *thick* categorical axiomatics for QM John mentioned above. One key difference is that in the order-theoretic axiomatics one failed to find an abstract counterpart to the Hilbert space tensor product. (ie without having to say that we are working in the lattice of closed subspaces of a Hilbert space) On the other hand, the categorical approach starting from symmetric monoidal categories takes that description of compound systems as an a priori. Singling out the complex numbers is done in terms of two involutions on morphisms, one covariant and one contravariant, where the covariant one capture complex conjugation ie the unique non-trivial automorphism characteristic of complex numbers. The contravariant one captures transposition and together they make up the adjoint.

## Re: This Week’s Finds in Mathematical Physics (Week 251)

Scott Aaronson pointed out this nice webpage:

As he says:

I’ll add this link to the body of week251, and also this paper:

The quantum de Finetti theorem is a generalization of the classical de Finetti theorem. Both classical and quantum de Finetti theorems are about $n$ copies of a system sitting side by side in an ‘exchangeable’ state: a state that’s not only invariant under permutations of the copies, but lacking correlations between the different copies!

Here’s the quantum de Finetti theorem. Suppose you have an

exchangeabledensity operator $\rho_n$ on $H^{\otimes n}$ — that is, one such that for each $N \ge n$, there’s a density operator $\rho_N$ on $H^{\otimes N}$ which 1) is invariant under permutations in $S_N$ and 2) has $\rho_n$ as its marginal, meaning that$Tr (\rho_N) = \rho_n$

where $Tr$ is the partial trace map sending operators on $H^{\otimes N}$ to operators on $H^{\otimes n}$. Then, $\rho_n$ is a mixture of density matrices of the form $\rho \otimes \cdots \otimes \rho$: a tensor product of $n$ copies of a density matrix on $H$.

This is completely plausible if you know what all this jargon means.

And now for the punch line:

This theorem wouldfailif we did quantum mechanics using the real numbers!Of course, this is related to the fact I mentioned in week250, namely that for real quantum mechanics, ‘the whole is more than the product of its parts’ in a more severe way than for complex quantum mechanics.