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Chapter 11 Vector Valued Generalized Functions and Free Fields

DOI: 10.1016/s0304-0208(08)70993-9
  • Mathematics


Abstract The free fields operators are vector valued distributions so we need to extend the concepts of generalized functions to the vector valued case. For a given test function ϕ ɛD(Rn) their values are unbounded operators on the Fock space. However they are elements of some natural algebras of unbounded operators on the Fock space which are not normed algebras. Actually such algebras are bornological algebras and so we introduce the concept of generalized functions valued in a bornological algebra which is a rather straightforward extension of the scalar case considered before. We extend the concepts of pointvalues, integration and their applications. In these extensions the multiplication of the fren fields operators - which is a starting point for the computations of Q.F.T. — makes sense and we study the Hamiltonian formalism of the free fields. The Hamiltonian and Lagrangian densities — which are polynomials involving the free fields and their derivatives-make sense as generalized functions while they were meaningless within Distribution Theory. The heuristic computation of the Hamiltonian operator — which is the integral over R3 of the Hamiltonian density — leads formally to an “infinite quantity” called the “0-point energy” which is classically suppressed by the Physicists The Hamiltonian density should be integrable over R3 and for this we need to use a special concept of generalized functions obtained with the choice of particular sets Aq (i.e. the elements of these Aq must satisfy some additional assumptions). After this choice we obtain the result heuristically obtained if we suppress the above infinity in the formal computations. This example is basic because using our theory of generalized functions more complicated “infinite quantities” of Renormalization Theory will give rise to the same phenomenon.

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