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Mathematical Problems on Generalized Functions and the Can– Mathematical problems on generalized functions and the canonical Hamiltonian formalism J.F.Colombeau [email protected] Abstract. This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a “formal way”- as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on C ∞ functions and on bounded operators, and can be considered at a purely mathematical level, ignoring physics in a first step. The mathematical tools to be used are nonlinear generalized functions, unbounded operators on a Hilbert space and computer calculations. This text is the improved written version of a talk at the congress on linear and nonlinear generalized functions “Gf 07” held in Bedlewo-Poznan, Poland, 2-8 September 2007. 1-Introduction . The Heisenberg-Pauli calculations (1929) [We,p20] are a set of 3 or 4 pages calculations (in a simple yet fully representative model) that are formally quite easy and mimick calculations on C ∞ functions. They are explained in detail at the beginning of this text and in the appendices. The H-P calculations [We, p20,21, p293-336] are a basic formulation in Quantum Field Theory: “canonical Hamiltonian formalism”, see [We, p292] for their relevance. The canonical Hamiltonian formalism is considered as mainly equivalent to the more recent “(Feynman) path integral formalism”: see [We, p376,377] for the connections between the 2 formalisms that complement each other. The H-P calculations start with irregular distributions that are “formally” multiplied, exponentiated,… Since 1955 it has been taught to physicists that mathematicians have “proved” (i.e. with a rigorous mathematical proof!) that there will never exist a nonlinear theory of generalized functions, in any mathematical context possibly different from distribution theory [S2,p10]. Thus this made hopeless to give the H-P calculations a rigorous mathematical sense. The above “proof” is discussed in this text. The H-P calculations make sense mathematically in the nonlinear theory of generalized functions: [C-G-P]. But only a too coarse work has been done. The H-P calculations request a deeper study: they concern generalized functions whose values are unbounded operators on a Hilbert space, which makes them more intricate than scalar valued calculations [C1,C2,C4], as done in continuum mechanics [C3,C8], and general relativity [S- V]. Numerous presumably easy problems of pure mathematics and other ones presumably difficult are stated. Further, a reader interested in computer calculations can attempt the calculation of the new numerical predictions suggested by the mathematical understanding of the H-P formalism. Absolutely no knowledge of physics is needed. It suffices to know the definitions of generalized functions in the “simplified” or “special” case [G-K-O-S, p10] and to be interested in unbounded operators on a Hilbert space. [C-G-P] should be reconsidered 1 in view of problems concerning domains of operators and numerical calculations (from imaginary exponentials of unbounded self-adjoint operators). These problems should be solved before a presentation to theoretical physicists (i.e. mathematicians in pure mathematics and other ones in computer calculations should clarify simple mathematical models before one could propose theoretical physicists to reproduce their work on physically significant theories not accessible to mathematicians). Even if a future String Theory replaces Quantum Field Theory a mathematical understanding of QFT would be needed [Wi]. So the proposed works and problems are presumably important for the future. They are suited for mathematicians unaware of physics. 2-The Heisenberg-Pauli calculations: overall view. One has a specific Hilbert space called “the Fock space” denoted by IF (IF is a Hilbertian direct sum of spaces of square integrable functions, so a very familiar object for mathematicians, see Appendix 1 and [C-G-P]), and a Φ ∈ 3 ∈ family of explicitely defined objects 0 (x, t) (x IR , t IR) imagined to be linear operators Φ on the Hilbert space IF for each (x,t) (see Appendix 1 and [C-G-P]). This family { 0 (x, Φ t)} ( x∈IR 3 ,t∈IR ) - equivalently the map 0 - is called “the free field operator”. After 1955 it was Φ recognized that the free field operator 0 is a distribution in the variable x (and a usual function in the t variable) whose “values” are densely defined unbounded linear operators on IF : more precisely there is a dense vector space D in IF such that, for any test function ϕ , Φ ϕ 0 ( ,t) maps D into D. The H-P calculations, or “canonical Hamiltonian formalism”, are described as follows on the simplest model close to physics. Let τ , m, g ∈IR. One sets: ∂ ∂ τ Φ ξ τ 2 Φ ξ τ 2 2 Φ ξ τ 2 H 0 ( ) := ∫ {½. ( 0 ( , ) ) +½. ∑ ( 0 ( , ) ) + ½. m .( 0 ( , ) ) + ∂ ≤µ≤ ∂ ξ∈IR 3 t 1 3 xµ Φ ξ τ N +1 ξ g/(N+1).( 0 ( , ) ) }d where the powers mean composition of operators. For a mathematician the main point is that this formula involves products of irregular distributions (these products do not make sense Φ within distribution theory): 0 (x,t) may be considered intuitively as a distribution as irregular as δ (x) the Dirac delta distribution (further, whose values are operators on a Hilbert space). This formula also involves an unjustified integration. Then one sets: Φ τ τ τ Φ τ τ τ (x, t, ) := exp(i(t– )H 0 ( )) o 0 (x, ) o exp(–i(t– )H 0 ( )). Of course the exponentials are à fortiori not defined within the distributions: they are τ imagined as unitary operators on IF (because H 0 ( ) looks symmetric). There is also a Φ τ problem of composition of operators because of 0 (x, ). Then calculations mimicking calculations on C ∞ functions give that Φ (x,t, τ ) is solution of the Cauchy problem called “interacting field equation” (the “formal proof” is in section 3 below): ∂ 2 ∂ 2 “Theorem ”1. Φ (x, t, τ ) = Φ (x ,t, τ ) – m 2 Φ (x,t, τ ) – g.( Φ (x,t, τ )) N , ∂ 2 ∑ ∂ 2 t 1≤µ≤3 xµ Φ τ τ Φ τ (x, , ) = 0 (x, ), 2 ∂ ∂ Φ (x, τ ,τ ) = Φ (x, τ ). ∂t ∂t 0 It is a “wave” equation with nonlinear second member. Since the initial condition is a pair of irregular distributions the solution is not expected more regular than a distribution for which the nonlinear term does not make sense in distribution theory (with further a “big” problem due to the fact one is confronted with unbounded operators). The exponential exp(it P 0 ) of the energy operator P 0 is immediately defined from an explicit formula that causes no mathematical problem: appendix 2 and [C-G-P ]. It is a unitary operator on IF and it maps D into D. We set: τ τ τ S τ (t):= exp(i(t– )P 0 ) o exp(–i(t– )H 0 ( )). Recall that the second exponential is not defined mathematically within the distributions. S τ (t) is imagined as a unitary operator on IF . Then one has (from calculations mimicking calculations on functions: the “formal proofs” of th. 2 and 3 are in appendix 2): Φ τ −1 Φ “Theorem ” 2 . (x, t, ) = (S τ (t) ) o 0 (x, t) o S τ (t). This formula suggests that the numerical results of the theory are the limits when τ → −∞ ,t → +∞ of the scalar products | F ,S (t)F |, F , F ∈D (probability that an initial 1 τ 2 IF 1 2 state F2 become F1 after interaction). One obtains (from formal calculations) that S τ (t) is solution of the differential equation: d g + “Theorem ” 3. S (t) = – i ( Φ (ξ ,t)) N 1 dξ S (t) τ + ∫ 0 o τ dt N 1 ξ∈IR 3 S τ (τ ) =Identity operator. Φ ξ N +1 This ordinary differential equation (in which ( 0 ( ,t)) does not make sense in distribution theory) is the starting point of an attempt (called “perturbation theory”) of calculation of S τ (t) by developing it in powers of the “coupling constant” g , when g is small. Already before 1950 the results were formidable: for instance in the well known case of the anomalous magnetic moment of the electron the numerical value issued from the above calculations [with the coupled Maxwell-Dirac equations as interacting field equation – the H- P calculations are exactly similar to those in this paper–, and after other unjustified “manipulations” in a power series development (in powers of g)] is 1.00115965246 ± 20 in the last digits (i.e. ± 20.10 −11 ; this imprecision is due to an imprecision on the physical constants like m, g above), while the experimental value (about 1950) was 1.00115965221 ± 4 in the last digits. This appears as the evidence that, although mathematically meaningless within distribution theory, the H-P calculations contain a “deep truth” in physics and, for those who believe mathematics and physics are connected, should make sense mathematically in some way. 3 3-The Heisenberg-Pauli calculations: details. In order to understand the H-P calculations we give the detailed calculations in proof of theorem 1: the passage from a formula to the following one is immediate. Concerning the free field it is convenient to set ∂ Π (x, t) := Φ (x, t). 0 ∂t 0 Φ From the explicit formula of 0 one has the canonical commutation relations ([A,B] := A o B - B o A), see Appendix 1: Φ Φ [ 0 (x, t), 0 (x’, t)]=0, (0) [ Π 0 (x, t), Π 0 (x’, t)] =0, Φ δ [ 0 (x, t), Π 0 (x’, t)]=i (x–x’)Id, with δ = the Dirac delta distribution.
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