Quantum Numbers and Rules

Philip Goyal, Kevin H. Knuth, John Skilling

Complex numbers are an intrinsic part of the mathematical formalism of quantum theory and are perhaps its most characteristic feature. In this article, we show that the complex nature of the quantum formalism can be derived directly from the assumption that a pair of real numbers is associated with each sequence of measurement outcomes, with the probability of this sequence being a real-valued function of this number pair. By making use of elementary symmetry conditions, and without assuming that these real number pairs have any other algebraic structure, we show that these pairs must be manipulated according to the rules of complex arithmetic. We demonstrate that these complex numbers combine according to Feynman's sum and product rules, with the modulus-squared yielding the probability of a sequence of outcomes.

Klaus Hentschel

Galina A. Nemova

Klaus Bärwinkel, Peter Hage, Heinz–Jürgen Schmidt +1 more

We suggest a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings. This rule generalizes the well-known results of Marshall, Peierls, Lieb, Schultz, and Mattis for even rings. Our rule is confirmed by numerical investigations and rigorous proofs for special cases, including systems with a Haldane gap for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Implications for the total spin quantum number S of relative ground states are discussed as well as generalizations to the XXZ model.

Lev I. Plimak

It is shown that provided two devices are distinguishable, their electromagnetic interactions are in essence classical. This results in the existence of simple correspondence rules connecting q- and c-number statistical descriptions of macroscopic optical arrangements. These rules allow formulas found by c-number techniques to be used as precise quantum relations. For the vast majority of real devices, including all linear devices and real light detectors regardless of their internal system, their c-number macroscopic description can be directly employed in QED. The problem of locality of QED is discussed and a rigorous theory of electrical-feedback arrangements in optics is presented.