The
F-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the
F-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all
F-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated
F-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,'' or equivalently when the associated "twisted'' Frobenius-Schur (or "generalized'' Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose
F-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex
F-symbols.
A measurement is a physical operation in space and in time. The properties of a system are described in relation to measurements at a given time, and no value of the time is intrinsically distinguished from another by the results of measurements on an isolated physical system. The operators symbolizing analogous properties at different times must be related by a unitary transformation. The propagation in time of the disturbance produced by a measurement implies that physical quantities referring to different times are incompatible, in general. The development of the fundamental dynamical variables through an infinitesimal time interval is described by an infinitesimal unitary transformation, which implies first order differential equations of motion for these variables. The necessity of an even number of variables and the possibility of dividing them into two complementary sets also appears for variables of the second kind.
A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time-dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansions - one associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O( 0 ) approximation recovers the standard classical path approximation for line shapes. The higher-order O( 1 ) corrections arise from the noncommutative nature of the star product.