Tensors and second quantization

TL;DR: In this article, a six-tuple formulation of the Canonical (Anti-) Commutation Relations (C-Commutation Relations) has been proposed, where the creation operator C and its adjoint (formula), the annihilation operator, are considered.

Abstract: Starting from a pair of vector spaces (formula) an inner product space and (formula), the space of linear mappings (formula), we construct a six-tuple (formula). Here (formula) is again an inner product space and (formula) the space of its linear mappings. It is required that (formula), as linear subspaces. (formula) Further, (formula) and (formula) denotes a lifting map (formula) such that, whenever (formula) solves an evolution equation (formula) then any product of operator valued functions (formula) solves the associated commutator equation in (formula), (formula) Furthermore, (formula). We also note that (formula) represents the state of k identical systems ’living apart together’. Cf. the free field ’formalism’ in physics. Such constructions can be realized in many different ways (section 2). However in Quantum Field Theory one requires additional relations between the creation operator C and its adjoint (formula), the annihilation operator. These are the so called Canonical (Anti-)Commutation Relations, (section 3). Here, unlike in books on theoretical physics, the combinatorial aspects of those 1This note is meant to be Appendix K in the lecture notes ’Tensorrekening en Differentiaalmeetkunde’. restrictions are dealt with in full detail. Annihilation/Creation operators don’t grow on trees! However, apart from the way of presentation, nothing new is claimed here. This note is completely algebraic. For topological extensions of the maps C; A to distribution spaces we refer to Part III in [EG], where a mathematical interpretation of Dirac’s formalism has been presented.