Karl-Erik Eriksson

RELATIVISTIC KINEMATICS FOR N-PARTICLE SCATTERING STATES. II. QUANTUM MECHANICAL TREATMENT.

The generalization of grand angular momentum to N-particle states with relativistic energies achieved classically in a previous paper is now transcribed into quantum mechanical language. The grand angular momentum are the generators Lij of a O(3N - 3)-invariance group of the invariant mass operator for N free particles. A radial momentum is defined as well as O(3N - 3)-invariant operators conjugate to radial momentum or total mass. A construction is made of wave functions spanning irreducible representation spaces of O(3N - 3) which are labelled by the eigenvalues of togetherness Λ2 = ½ Σij Lij2. Finally the sum of all those spaces is shown to be irreducible under a group O(3N - 3, 1)/t (where t is time reversal) which is an invariance group of the mass operator and its canonical conjugate.

Quantum statistics of quantum measurement

An investigation is made of the interaction of a microsystem μ, subject to measurement and a macroscopic apparatus M responsible for preparation of μ and measurement on μ of an observable T with a finite number of non-degenerate eigenvalues. It is shown that the statistics of stochastic properties in M leads to a bifurcation process which forces μ into an eigenstate |jμ of T combined with registration in M of the measurement result Tj with branching ratios in agreement with the usual postulate. The process may be fast, so that only a relatively small number of degrees of freedom in M are active in the bifurcation.

Random Walk in a Simplex – The Way "God Plays Dice"?

For a microscopic quantum-mechanical system we consider the measurement of an observable A which is assumed to characterize the system uniquely. The measurement process takes place through interaction between the microscopic system and the measurement apparatus, which in turn is influenced by external fields. The detailed influence from the initial state of the measuring apparatus and from the external fields on the density matrix of the microscopic system is stochastic and may be taken into account as a random walk in the simplex characterizing its diagonal elements (in the representation where A is diagonal). The diffusion equation corresponding to this random walk is derived. It is shown that the corners are attractors of this process, i.e. each single measurement ends up in an eigenstate of A. The relative frequencies for the different results are given by the squared moduli of the corresponding amplitudes. The reduction of the wave packet is thus the result of a statistical process.

Parametrization of Scattering Amplitudes: Definition of Crossing Symmetric Measure and Inertial Tensor

The parametrization of N-point functions and scattering amplitudes in a relativistic particle theory is briefly discussed. A democratic parametrization is suggested. In connection with this a 4-dimensional tensor of inertia is defined in close connection to the Gram matrix. A measure is introduced in the (non-physical) Euclidean part of momentum space.