Special Lecture Weak points of the current mathematical apparatus of Quantum Physics, which should be improved with the help of mathematicians

2006-10-18 16:30 - 2006-10-18 17:30
Faculty of Science Building #3 Room 508
Nguyen Vinh Quang (I.of Physics and Electronics, Vietnam.)
It is pointed out that there are many weak points of the currentmathematical apparatus of Quantum Physics, which should be improvedwith the help of mathematicians.1. The closure relation, the Green function theory and the basicpostulate of Quantum field theory.I analyze in detail three different formulations for the completeness ofthe orthonormal basis in Quantum Physics, which are using equality ofthe functions (1st formulation), equality of the operators (2ndformulation), and equality of the functionals (3rd formulation which iscurrently widely used in quantum physics). The theorem: “the 3rdformulation is not equivalent to the 1st and 2nd ones” is proved byexplicitly showing that the reasoning of Landau and of others is notrigorous. Consequently, several important relations obtained by usingthe closure relation in the Green function theory and Quantum fieldtheory should be corrected.2. The relations concerning with the delta function Dirac.It is explicitly shown that many relations concerning the deltafunction Diracare, in fact, not accurately and not sufficiently provedyet. For example, while it is necessary to verify both properties of theDirac delta function (the point distribution property and the functionalproperty), only one property has been checked.3. The uncertainty relations and the uncertainty principle.It is explicitly pointed out that the reasoning of Neumann, of Weyl, ofFadeev and of others in proving the Heisenberg uncertainty relations isnot rigorous. It is explicitly shown that there exist physical states(normalized to 1) in which the Robertson- Schrodinger and Heisenberguncertainty relations are invalid, namely, the mean values of thephysical operators are infinite. Consequently, these relations cannot beconsidered as mathematical representation of a general physicalprinciple. An explanation by the theory of functional analysis isgiven: for these states even the definition of the uncertainty notionthrough the dispersion notion in the probability theory is irrelevant.4. The hermiticity of the physical operators.I present some counter-examples, which explicitly show that one mustverify the hermiticity of the physical operators, which may be changeddue to the unbounded property of the Hamiltonian, position and momentumoperators, and their functions.5. The Dirac transition theory and the Fermi golden rule.It is explicitly shown that there are several essential weak points inthe Dirac perturbation theory. Consequently, the Fermi “golden rule” ismathematically incorrect.