Group Theory and Uncertainty

Giovanni Stegel

Michael Martin Nieto

From the motivation of Schroedinger, that of finding states which follow the motion which a classical particle would have in a given potential, we discuss generalizations of the coherent states of the harmonic oscillator. We focus on a method which is the analytic complement to the group theory point of view. It uses a minimum uncertainty formalism as its basis. We discuss the properties and time evolution of these states, always keeping in mind the desire to find quantum states which follow the classical motion.

Mark S. Hillery, Leonard D. Mlodinow

As was originally pointed out by Caves, squeezed light can be used to increase the accuracy of interferometric phase measurements. This analysis was carried further by Yurke, McCall, and Klauder, who used group theory to examine the ultimate sensitivity of conventional interferometers and also those constructed from four-wave mixers. Here we consider interferometers composed of mixtures of beam splitters, four-wave mixers, and degenerate parametric amplifiers. We show that each interferometer can be characterized by an uncertainty relation whose minimum uncertainty states can be used to explore its ultimate accuracy. The variables in the uncertainty relation are quadratic in the mode operators, which allows us to connect the study of improved interferometric measurements to recent work on higher-order squeezing.

Zen Revista, 10 PHYSICS

Quantum mechanics' formal core is a handful of objects---the wavefunction that encodes a state, the operators that act on it, the commutators that measure their incompatibility, and the uncertainty principle that the commutators imply---and the theory's history is the story of how those objects were given. This article presents a narrative review of the primary literature that gave them, from Heisenberg's 1925 umdeutung---the reinterpretation that replaced orbits with arrays of transition amplitudes---and Schrödinger's 1926 quantization as an eigenvalue problem, which wrote the wave equation and proved the equivalence with matrix mechanics, through Born's 1926 probability interpretation of the wavefunction, Heisenberg's 1927 uncertainty paper, Robertson's 1929 general commutator inequality, Pauli's 1926 solution of the hydrogen spectrum by the new mechanics, Bohr's 1928 quantum postulate and complementarity, Weyl's 1928 group theory and quantum mechanics, Dirac's 1930 Principles, which organized the formalism's transformations, and von Neumann's 1932 mathematical foundations, which gave the theory its Hilbert space, to the Einstein-Podolsky-Rosen 1935 argument that tested what the formalism claims about reality. The synthesis is organized around three themes: the state's representations, in which matrix and wave mechanics were shown to be one theory; the uncertainty structure, in which commutators became the measure of incompatibility and the uncertainty principle its theorem; and the formalism's completion, in which Hilbert space, transformations, and the measurement problem gave the objects their rigorous home. It is concluded that the formal core's construction is physics' most consequential algebra---the theory's content located not in its pictures but in its operators' relations.