Abstract
This paper fails to derive quantum mechanics from a few simple postulates. However, it gets very close, and does so without much exertion. More precisely, I obtain a representation of finite-dimensional probabilistic systems in terms of Euclidean Jordan algebras, in a strikingly easy way, from simple assumptions. This provides a framework within which real, complex and quaternionic QM can play happily together and allows some (but not too much) room for more exotic alternatives. (This is a leisurely summary, based on recent lectures, of material from the papers arXiv:1206:2897 and arXiv:1507.06278, the latter joint work with Howard Barnum and Matthew Graydon. Some further ideas are also explored, developing the connection between conjugate systems and the possibility of forming stable measurement records and making connections between this approach and the categorical approach to quantum theory.)
Highlights
Introduction and OverviewWhatever else it may be, Quantum mechanics (QM) is a machine for making probabilistic predictions about the results of measurements
That if we can motivate a representation of physical systems in terms of HSD order-unit spaces, we will have “reconstructed” what with a little license we might call finite-dimensional Jordan-quantum mechanics
At present, I do not know of any non-dagger-compact monoidal probabilistic theory satisfying the other assumptions. Another question is whether there exist examples other than real QM of non-locally-tomographic, but still dagger-compact, monoidal probabilistic theories satisfying the assumptions of Theorem 2
Summary
Whatever else it may be, Quantum mechanics (QM) is a machine for making probabilistic predictions about the results of measurements. As I will discuss below, one can construct mathematically-reasonable theories that embrace finite-dimensional quantum systems of all three types Another shortcoming, not related to the exclusion of real and quaternionic QM, is the technical assumption (explicit in [10] for bits) that all positive affine functionals on the state space taking values between zero and one correspond to physically-accessible “effects”, i.e., possible measurement results. U is the Jordan unit, and E+ is the cone of squares It seems, that if we can motivate a representation of physical systems in terms of HSD order-unit spaces, we will have “reconstructed” what with a little license we might call finite-dimensional Jordan-quantum mechanics.
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