The differential-space theory of quantum systems

Norbert Wiener

Armand Siegel, Norbert Wiener

The "reduction of the wave packet" is an essential feature of the quantum description of measurement. Since it means an abrupt change in the wave function, it therefore also implies an abrupt change in the statistical description of the system. If the differential-space theory of quantum systems rests on a description of reality more detailed than quantum mechanics, it must contain an account of this change which is consistent with the strictly probabilistic point of view of the theory. It is here shown that the theory does satisfy this necessary condition. The need of a special postulate for the measurement process is eliminated in this formulation, and a new explicitness is introduced into the description.

W. Ochs

S. Wickramasekara

We construct two rigged Hilbert spaces that furnish differentiable representations of the causal Poincaré semigroup. These rigged Hilbert spaces provide the mathematical foundation for a theory of relativistic quasistable states that synthesizes the S-matrix description of resonance scattering with the Bakamjian–Thomas construction for interacting relativistic quantum systems.

W. Ochs

P. A. M. Dirac

Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation prin­ciple a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.