Orthogonality and Symmetry

Quantum physics is a theory that takes some getting used to and its use may be accompanied by a certain discomfort until it has become routine. Over the years, countless approaches have been proposed to justify the peculiar formalism, but the discussion on the foundations is still ongoing. Our project work is intended to be a further contribution to this issue. We start at the lowest possible level. We are less concerned with physics itself than with the most important structures employed in quantum physics and we adopt a mathematical perspective. The basic model of quantum physics is the complex Hilbert space or, more generally, a C*-algebra. The definitions of both these structures are not well comprehensible without a firm mathematical background and the question arises as to whether the structures cannot be reduced to simpler ones. We take up the long-standing efforts that were once initiated by a work of Birkhoff and von Neumann, who proposed to describe the Hilbert space by a means of certain algebra that emerges from the set of its subspaces. Our work circles around the notion of orthogonality. The focus is thus on a concept that occurs in mathematics at numerous places. Although the term is linked to a clear geometric concept two vectors that form a right angle are orthogonal , its role in mathematics is not so easy to grasp. In quantum physics, the (pure) states of a physical system are described by vectors of a Hilbert space and orthogonality means that the transition probability from one state to the other as a result of a measurement is zero. Remarkably, this binary relation determines the structure; in a certain sense, the Hilbert space is reducible to the concept of orthogonality. Equipped with the orthogonality relation alone, a Hilbert space forms a so-called orthoset and everything else can be reconstructed from it. Part of the project is to describe the relevant type of orthosets in the simplest possible way. Similarly, a C*-algebra can also be assigned an orthoset, although in this case the orthoset alone is not sufficient for a description. A further part of the project is to find ways to extend the description without great complications. What kind of properties are eligible in order to specify a certain orthoset? This is where a further keyword comes into play: symmetry. Structure-preserving self-mappings are of central importance in physics and likewise in this project. The notion of an orthoset is as general as that of an undirected graph; the notion of symmetry refers to the vast theory of groups; both together, however, seem to offer an extraordinary potential when it comes to the characterisation of standard structures used in physics.