Tensors and second quantization

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

A complex-like calculus for spherical vectorfields

First, R^{1+d}, d in N, is turned into an algebra by mimicing the usual complex multiplication. Indeed the special case d = 1 reproduces C. For d > 1 the considered algebra is commutative, but non-associative and even non-alternative. Next, the Dijkhuis class of mappings (’vectorfields’) R^{1+d} ¿ R^{1+d}, suggested by C.G. Dijkhuis for d=3, d=7, is introduced. This special class is then fully characterized in terms of analytic functions of one complex variable. Finally, this characterization enables to show easily that the Dijkhuis-class is closed under pointwise R^{d+1}-multiplication: It is a commutative and associative algebra of vector fields. Previously it had not been observed that the Dijkhuis-class only contains vectorfields with a ’time-dependent’ spherical symmetry. Such disappointment was to be expected! The class of functions which are differentiable with respect to the algebraic structure, that we impose on R^{1+d}, contains only linear functions if d > 1. The Dijkhuis-class does not appear this way either! In our treatment neither quaternions nor octonions play a role

Matrix gauge fields and Noether's theorem

No abstract

A complex-like calculus for spherical vectorfields

First, R^{1+d}, d in N, is turned into an algebra by mimicing the usual complex multiplication. Indeed the special case d = 1 reproduces C. For d > 1 the considered algebra is commutative, but non-associative and even non-alternative. Next, the Dijkhuis class of mappings (’vectorfields’) R^{1+d} ¿ R^{1+d}, suggested by C.G. Dijkhuis for d=3, d=7, is introduced. This special class is then fully characterized in terms of analytic functions of one complex variable. Finally, this characterization enables to show easily that the Dijkhuis-class is closed under pointwise R^{d+1}-multiplication: It is a commutative and associative algebra of vector fields. Previously it had not been observed that the Dijkhuis-class only contains vectorfields with a ’time-dependent’ spherical symmetry. Such disappointment was to be expected! The class of functions which are differentiable with respect to the algebraic structure, that we impose on R^{1+d}, contains only linear functions if d > 1. The Dijkhuis-class does not appear this way either! In our treatment neither quaternions nor octonions play a role