Holger Neumann

Causality between preparation and registration processes in relativistic quantum theory

A causality postulate is considered which is based on the conception of systems that are prepared in some finite region of space-time and recorded in some other region. If these regions are spacelike separated, the recording apparatus should react as if no preparing apparatus were present, i.e., it should respond with at most some vacuum rate. The causality postulate is mathematically formulated within the framework of statistical theories. The connections with algebraic field theory are discussed and the relation between causality and spectral conditions is studied. General methods for constructing systems satisfying the causality postulate are given and applied in several examples.

The description of preparation and registration of physical systems and conventional probability theory

The connection of the structure of statistical selection procedures with measure theory is investigated. The methods of measure theory are applied in order to analyze a mathematical description of preparation and registration of physical systems that is used by G. Ludwig for a foundation of quantum mechanics.

A mathematical model for a set of microsystems

Starting with the usual Hubert space formulation of quantum mechanics we construct a mathematical model which proves that the set of axioms for the theoretical description of single microsystems developed by Ludwig is free of contradictions and admits of nontrivial solutions.

A new physical characterisation of classical systems in quantum mechanics

A physical characterisation of classical systems in quantum mechanics is given in terms of the set of ensembles in contrast to the well-known characterisations concerning the effects or observables: A quantum mechanical system is classical if and only if each two decompositions of every ensemble are compatible.

The space-time structure of quantum systems in external fields

An axiomatic foundation of a quantum theory for microsystems in the presence of external fields is developed. The space-time structure is introduced by considering the invariance of the theory under a kinematic invariance group. The formalism is illustrated by the example of charged particles in electromagnetic potentials. In the example, gauge invariance is discussed.

The Full Standard Model

We will now give a description of the classical field theory underlying the complete Standard model. We allow for curved space-time in the presence of gravity, considered as a non-dynamical background field (in Section 4 the gravitational field will be considered as a dynamical variable as well).

Standard Model Coupled with Gravity

In this section we combine the Yang-Mills type interactions (electromagnetic, weak and strong force) discussed in Section 9 with gravity, by treating the space-time metric g together with the 1-forms described in Proposition 15 as an independent dynamical variable. In this section we keep the notation introduced in Section 9 but for simplicity we put \( I_{q\ell } = 1 = \tilde I_{q\ell } \). According to the “Spectral Action Principle” (Section 6, [31]), the Lagrangian of the Standard Model coupled with gravity is expressed in terms of the spectral properties of the “perturbed” Dirac operator

Embedding of the classical into the quantum description of photons

\begin{gathered} \left( {\begin{array}{*{20}c} {D_{gq\ell } } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {J_{q\ell } D_{gq\ell } J_{q\ell } } \\ \end{array} } \right): = \left( {\begin{array}{*{20}c} {D_{q\ell } } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {J_{q\ell } D_{q\ell } J_{q\ell } } \\ \end{array} } \right) \hfill \\ + \left( {\begin{array}{*{20}c} {\omega _{q\ell } } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\tilde \omega _{q\ell } } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} 0 \\ { - \kappa ^2 J_{q\ell } } \\ \end{array} \begin{array}{*{20}c} {J_{q\ell } } \\ 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\omega _{q\ell } } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\tilde \omega _{q\ell } } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} 0 \\ { - \kappa ^2 J_{q\ell } } \\ \end{array} \begin{array}{*{20}c} {J_{q\ell } } \\ 0 \\ \end{array} } \right)^{ - 1} \hfill \\ \end{gathered}