Chapter I Dirac's Formalism According to Dirac and Its Relations with Linear Algebra

T. Banks

Jerome R. Busemeyer, Peter Bruza

What can you learn about quantum theory in one chapter without knowing any physics? A complete presentation of quantum theory requires an entire textbook, but our goal for this chapter is to provide only the essential elements of the theory that we feel are relevant for modelling behavioral phenomena. Just like classical probability theory, quantum theory is based on a small set of axioms used to assign probabilities to events. This chapter is limited to what is called the structural principles of quantum theory, whereas Chapter 8 will set out the dynamic principles. For simplicity, only finite state systems will be described, although quantum theory can also be applied to continuous state systems. Even though this book is limited to finite dimensions, the number of dimensions can be arbitrarily large. For finite state systems, the structural part of quantum theory is expressed in the formalism of linear algebra (for continuous state systems, it is expressed in the formalism of functional analysis). Consequently, a brief tutorial of linear algebra is presented along with our elementary introduction to quantum theory. Various notations are used to describe linear algebra, depending on the application field. Physicists like to use Dirac notation, invented by one of the founders of quantum theory, Paul Dirac (Dirac, 1958). Although the Dirac notation will be unfamiliar to many cognitive scientists, we will still use it as it is a succinct notation for expressing linear algebra. It helps the reader to see relations that are more difficult to identify using other formalisms.

Michaël Esfeld, Dirk-André Deckert, Dustin Lazarovici +2 more

This chapter shows that the Bohmian approach does the same service for quantum field theory (QFT) that it does for quantum mechanics. The mathematical problems left aside, when it comes to the ontology of QFT, one has to be concerned with what is going on in the world according to the theory. The chapter examines so-called Dirac sea model or hole theory, which was the one to predict the phenomenon of electron-positron pair-creation. In general, the Dirac sea model applies to any fermion sector and potentially any interaction between those fermions. One can emphasize the quasi-particles character of the excitations already in the mathematical formalism. The Bohmian approach vindicates the minimalist ontology of there being only matter points individuated by distance relations and the change of these relations also when it comes to our most advanced physical theory of matter—the standard model of QFT.

S.J.L. van Eijndhoven, J. de Graaf

Krishnan Thyagarajan, Ajoy Kumar Ghatak