A History of Feynman’s Sum over Histories in Quantum Mechanics

Exact calculations of Feynman’s path integrals (defined on a time lattice) are mainly based on recurrence integral formulas in which the convolution of two functions having a common feature retains the same feature. Therefore, exactly soluble path integrals in quantum mechanics may be classified by their recurrence integral formula used in the calculation. According to this classification, there are three types of path integrals: (i) Gaussian path integrals, (ii) Legendrean path integrals, and (iii) Besselian path integrals. The Gaussian path integrals are calculated by the well-known convolution of two Gaussian functions which produces recurrently another Gaussian function. Path integrals of this type have been widely used in semiclassical approaches, coherentstate path integrals, applications in statistical physics, field theory and many others areas. The Legendrean path integrals are based on the convolution integral for zonal spherical functions (generalized Legendre functions), which are particular matrix elements of unitary irreducible representations of Lie groups. An elementary type of the Legendrean path integrals appears in the angular path integral in 3-dimensional polar coordinates. More generally, path integrals of this type are useful for systems with certain symmetries of kinematical or dynamical origin. For details see the review [523]. The Besselian path integrals are based on Weber’s integral formula for Bessel functions [771] or the group composition law (convolution) for particular unitary representations of an element of SU(1, 1) in a continuous base. Radial path integrals are of the Besselian type and may be associated with a certain dynamical group or spectrum generating algebra [523]. The books of Feynman and Hibbs [340] and of Schulman [828] discuss mainly path integrals of the Gaussian type which are undoubtedly most im-