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Footnotes Section 1 1) a Vivid Account of the of the Idea, Influenced Footnotes Section 1 1) A vivid account of the of the idea, influenced particularly by remarks of Dirac [2J, has been given by FeYnman himself in [3J. 2) For the physical foundation see the original work of FeYnman and the book by FeYnman and Hibbs (Ref. [1J). Also e s g , [7J. 3) Actually, Holder continuous of index 1ess than 1/2 , see e s g , [8J. 4) For the definition by a "sequential limit", in more general situations, see e.g. [11J. 5) See also e.g. the references given in [41J and [42J. 6) The results are also briefly announced in [42J. 7) A theory of related pseudomeasures has inbetween been developed by P. Kree. See e.g. and references therein. 8) Besides the topics touched in this brief historical sketch of the mathematical study of FeYnman path integrals of non relativistic quantum mechanics there are others we did not mention, either because they concern problems other than those tackled later in this work or because no clear cut mathematical re­ sults are available. Let us mention however three more areas in which Feynman path integrals have been discussed and used, at least heuristically. a) Questions of the relation between FeYnman's quantization and the usual one: see e.g. [10J,6), [31J, [35J. b) FeYnman's path integrals on functions defined on manifolds other than Euclidean space, in particular for spin part­ icles. Attempts Using the sequential limit and analytic continuation approaches have been discussed to some extent, see e.g. [6J,7), [36J and references given therein. For - 116 - the analytic continuation approach there is available the well developed theory of Wiener integrals on Riemannian manifolds, see e.g. [37]. c) As mentioned before, an important application of FeYnman path integrals is in the discussion of the classical limit, where 11" o. In [41] we tackle this problem and we refer to this paper and [42J also for references (besides e.g. [1], [2], [5J, [7J, [38J). 9) The Wightman axioms for a local relativistic quantum field theory (see e.g. [40J) have been proved, in particular. 10) The space cut-off has now been removed [44J. 11) See also [45]. 12) We did not mention here other topics which have some rela- tions to FeYnman's approach to the quantization of fields, for much the same reason as in the preceding footnote 8). For discussion of problems in defining FeYnman path inte- grals for spinor fields see e.g. [36J and references given therein. For the problem of the formulation of FeYnman path integral in general relativity see e.g. [39J, [4J,2), and references given there. 13) Similar results hold for a system of N non relativistic quantum mechanical particles, moving each in d -dimensional space, interacting through a superposition of \I -body poten- tials (\I = 1 ,2, ••• ) allowed in particular to be translation invariant. 14) The same results hold for a system of N anharmonic oscil- lators, with anharmonicities given by superpositions of \I - body potentials, as in the preceding footnote. Section 3 1) By the translation invariance of the normalized integral we have - 117 - JLlyl2 JL\y+y 12 2tl Je f(y)dy = Je2rJ. 0 f(Y+Yo)dy, de J(g where Yo is any element of Hence im t . t ,... 2ti. So -tJ V(y(T))dT S e e 0 cp(Y(O))dY, y(t)",x although defined by replacing in its argument YeT) by y( T) + x, and interpreting the result as i\Y12 J f( y )dy, is actually independent of the chosen Je particular translation, which could be replaced by x + Yo ' wi th y arbitrary, and only depends on its value o e for T '" t, which is required to be x , 2) The above Theorem 3.1 has been stated for the Schrodinger equation of a particle moving in Rn under the action of the potential V. From the proof it is evident that the same results hold for a system of N non relativistic quantum mechanical particles moving each in d dimensional space, thus with n =Nd, under the influence of a poten- n) tial V which is only restricted to belong to (R but is otherwise arbitrary, hence can be e.g. a sum of \I-body translation invariant potentials (\I '" 1 , 2, ••• ). In this case 2 ii2 n 1 ,,2 11 is replaced by -"'7 E- -- and by 1., =1 m.1. " xi2 1 dY1 dY2 (Y1'Y2) =ii st aT·MdT"dT, where is the matrix o (M)ij '" mi 6i j, i, j = 1, ••• ,n. 3) The elementary definitions of mathematical scattering theory are e.g. in [47]. For recent work see e.g. [32]. 4) This assumption is actually enough for proving the complete- . ness of the wave operators, in the sense that Range W = + Range W_, see [48]. A slightly weaker condition, sufficient - 118 - for the existence of the wave operators, is ([49J): 2 JIV(x) 1 (1 +Ixl )2-n+E: dx < co, > O. See also [11J, 3). 5) With obvious modifications the results hold also for the N particle system of footnote 2) of this section. Section 4 1) The square root \(1/2rri)B\t is given by the formula .1 / t - i;f:signB 1(1/2TTi)B\z = (1/2TT)n 2 1 IBII e , where (2n)n/2 is the positive root, I IBI It is the posi- tive root of the absolute value of the determinant of B and sign B is the signature of the form B. This is according to the formula 1 S I(1 /2TTi)BIZ e dx 1. Rn Section 5 1) This condition is however not necessary for the mathematics involved, so that all results actually hold also for com- plex V. 2) In Section 5, 6, 7 all results are stated for the case of an anharmonic oscillator. From the proofs it is evident that they hold also for N anharmonic oscillators, with anharmo- nicity V which is a superposition of v-body potentials (v = 1,2, ••• ), which can also be translation invariant. Section 8 1) See e.g. [53J, [25J, 2) • 2) See e.g. [54J. - 119 - Section 9 1) In this case 60 is the Feynman i.e. the causal propagator. For a discussion of its role in relativistic local quantum field theory see [55]. 2) As mentioned in the introduction, models with these inter- actions and their limit when the space cut-off is removed have been studied before, see [30], [45] and re- ferences given therein. References. Section 1. (1) R.P. Feynman, Space-time approach to quantum'mechanics, Rev. Mod. Phys. £2, 367-387 (1948). (Based on Feynman's Princeton Thesis, 1942, unpublished). The first applications to quantum field theory are in R.P. Feynman, The theory of positrons, Phys. Rev. 76, 749-759 (1949); Space-time approach to quantum electro- dynamics, Phys. Rev. 76, 769-789 (1949); of the quantum theory of electro- magnetic interaction, Phys. Rev. §Q, 440-457 (1950). (All quoted papers are also reprinted in: J. Schwinger, Selected Papers on Quantum Electrodynamics, Dover, New York (1958). See also: R.P. Feynman - A.R. Hibbs, Quantum mechanics and path integrals. MacGraw Hill, New York (1965). R.P. Feynman, The concept of probability in quantum mechanica, Proc. Second Berkeley Symp.Math. Stat. Prob., Univ. Calif. Press (1951), pp. 533-541. [2) P.A.M. Dirac, The Lagrangian in quantum mechanics, Phys. Zeitschr. d. Sowjetunion, 2, No 1, 64-72 (1933). P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, IV. Ed. (1958), Ch. V, § 32, p. 125. P.A.M. Dirac, On the analogy between classical and quantum mechanics, Rev. Mod. Phys. 11, 195-199 (1945) [3] R.P. Feynman, The development of the space-time view of quantum electrodynamics, in Nobel lectures in Physics, 1965, ElseVier Publ. (1972). - 121 - (4] 1) F. Dyson, Missed opportunities, Bull. Amer. Math. Soc. 78, 635-652 (1972). 2) Also e.g. L.D. Faddeev, The Feynman integral for singular Lagrangians, Teor. i Matem. Fizika, 3-18 (1969) (transl. Theor. Mathem. Phys. 1, 1-13 (1969)). (5] I.M. Gelfand - A.M. Yaglom, in functional spaces, J. Math. Phys. 48-69 (1960) (transl. from Usp. Mat. Nauk. 11, Pt. 1 ; 77-114 (1956)). (6] Exclusively concerned with reviewing work on specifically mathematical problems: 1) E.J. McShane, Integral devised for special purposes, Bull. Am. Math. Soc. §2, 597-627 (1963). 2) L. Streit, An introduction to theories of integration over function spaces, Acta Phys. Austr. Supple g, 2-20 (1966) • 3) Yu L. Daletskii, Integration in function spaces, in Progress in Mathematics, Vol. 4, Ed. R.V. Gamkrelidze (transl.), Plenum, New York (1969), 87-132. The following reviews are somewhat less concentrated on specific mathematical problems, and include also some discussions of heuristic work: 4) E.W. Montroll, Markoff chains, Wiener integrals and quantum theory, Commun. Pure Appl. Math. 2, 415-453(1952). 5) r,n. Gelfand-A.M. Yaglom, see (5]. 6) S.G. Brush, Functional integrals and statistical physics, Rev. Mod. Phys. 33, 79-92 (1961). 7) C. Marette DeWitt, fonctionnelle de Feynman. Une introduction, Ann. Inst. Henri Poincare, 11, 153-206 (1969) • 8) J. Tarski, Definitions and selected applications of Feynman-type integrals, Presented at the Conference on functional integration at Cumberland Lodge near London, 2-4 April, 1974, Hamburg Universitat, Preprint, May 1974. (7) 1) C. Marette, On the definition and approximation of Feynman's path integrals, Phys. Rev. 81, 848-852 (1951). 2) W. Pauli, Ausgewahlte Kapitel aus der Feldquantisierung, ausg.
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