Narrow Results

With the gas ionization experiments (Section 1.7) we know that atoms and molecules are made of nuclei and electrons that revolve around them. In this chapter we give representations of these microscopic objects and show how certain atoms or molecular fragments form bonds between them. From the description of the hydrogen atom, the lightest atom in Nature since it contains only one proton and one electron, we will develop a simple model for the bonds between atoms in molecules stable at 25°C. Without making complicated calculations we will be able to predict which bonds between atoms are possible a priori and which structures the molecules containing them have. Atoms are the parts of a Lego® set that can be assembled to make a large number of constructions, but not just any construction (three-dimensional objects with predefined geometries). In Chapter 7, we will develop a slightly more advanced model of the chemical bond. We will examine why some bonds are weaker than others, i.e. which bonds are more or less easily broken by heating…

Based on recent work, we discuss how the problem of infrared divergence in Nelson’s massless scalar field model can be tackled. We translate the problem originally formulated in terms of operators into a problem of stochastics by associating well specified processes to the particle and field operators. The ground state will be put in direct relationship with data given by Gibbs measures relative to these processes. When interpreted back, it turns out that in three dimensions the Nelson Hamiltonian has no ground state in Fock space and thus is not unitary equivalent with the Hamiltonian obtained from Euclidean quantization. In contrast, for dimensions higher than three the Nelson Hamiltonian does have a unique ground state in Fock space and the two Hamiltonians are unitary equivalent. We show how another representation for the 3D case can be constructed which eliminates infrared problems altogether.