Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

The use of recombinant activated factor VII for refractory bleeding post complex cardiothoracic surgery

We reviewed the outcome following use of recombinant activated factor VII (rVIIa) in patients with major bleeding post cardiothoracic surgery in our unit between January 2002 and July 2004. The unit consists of 16 cardiothoracic intensive care beds in a public metropolitan teaching hospital which serves as a referral centre for heart and lung transplant surgery Patients with refactory bleeding following cardiothoracic surgical procedures who were treated with rVIIa were identified. A total of 12 episodes of rVIIa use were recorded in ten patients, including three episodes with ventricular assist devices, and 5 heart and/or lung transplants. The median dose used was 85 mu g/kg. Chest tube drainage decreased in all patients following administration of rVIIa; median chest tube drainage decreased front 445 ml/h to 171 ml/h (P=0.03). Despite cessation of bleeding, mortality was high, when rVIIa was used after more than 24 hours. In six episodes, despite early rVIIa use (within six hours), continued bleeding necessitated return to theatre, where a surgical source of bleeding was found. In this small retrospective study, rVIIa significantly reduced bleeding that was refractory to standard blood product transfusion. In this series of patients., those that did not respond to rVIla early in the postoperative phase were found to have a surgical source of bleeding.