Graph Algebras for Quantum Theory

We consider algebraic structure of Quantum Theory and provide its combinatorial representation. It is shown that by lifting to the richer algebra of graphs operator calculus gains simple interpretation as the shadow of natural operations on graphs. This provides insights into the algebraic structure of the theory and sheds light on the combinatorial nature and philosophy hidden behind its formalism. Introduction. – Quantum Theory seen in action is an interplay of mathematical ideas and physical concepts. From the present-day perspective its formalism and structure is founded on theory of Hilbert spaces [1]. According to a few basic postulates physical notions of a system and apparatus, as well as transformations and measurements, are described in terms of operators. In this way algebra of operators constitutes the proper mathematical framework within which quantum theories are built. Structure of this algebra is determined by two operations, addition and multipliaction of operators, which lie at the root of all fundamental aspects of Quantum Theory [2]. Physical content of Quantum Theory comes from beyond the abstract mathematical formalism. It is provided by the correspondence rules assigning operators with physical quantities. This is always an ad hoc procedure invoking concrete representations of the operator algebra chosen to best reflect physical concepts related with the phenomena under investigation. Physics of the last century developed several such schemes e.g. in terms of matrices (Heisenberg), differential equations (Schrödinger), path integrals (Feynman), phase space (Wigner), etc. Various realizations of the operator algebra invoke different physical ideas, and hence are capable of describing diverse physical situations the key factor in successful application of abstract mathematical notions in description of the physical world. Interest in combinatorial representations of mathematical entities stems from a wealth of concrete models they provide. Their convenience comes from simplicity, which being based on elementary notion of enumeration directly appeals to intuition often rendering invaluable interpretations illustrating abstract mathematical constructions. This makes combinatorial perspective particularly attractive to quantum physics in its active pursuit of proper understanding and new viewpoints on fundamental phenomena. In this Letter we are interested in combinatorial representation of the operator algebra of Quantum Theory which will be recast in the language of graphs. In some respects this draws on the Feynman idea to represent physical processes as diagrams used as a bookkeeping tool in the perturbation expansions in field theory [3]. Combinatorial approach, however, has much more to offer if applied to the overall structure of Quantum Theory seen from the algebraic point of view. We shall show that by lifting to more structured algebra of graphs abstract operator calculus gains straightforward interpretation as the shadow of natural operations on graphs. This not only provides interesting insights into the algebraic structure of a theory but also sheds light on the intrinsic nature and philosophy hidden behind the formalism of Quantum Theory. Quantum Theory as Algebra of Operators. – General setting for Quantum Theory consists of specifying Hilbert space H of the system and identifying operators with physically relevant quantities. Operators acting in H naturally make an algebra with addition and multiplication which we denote by O. Most interesting structures in O are, of course, those generated by operators having physical interpretation. They usually originate from considering some observable of interest along with operations causing changes in the state of a system.