4 Concluding Remarks

As argued in Ref. [3], the conceptual foundations of quantum mechanics are still the subject of debate, and thus ``it is important to introduce undergraduate physics majors to non-standard developments of quantum mechanics'' in the hope that exposure to such developments, early in the curriculum, might motivate some students to focus on the physical principles underlying the theory. In this paper we have taken up that challenging task by discussing a novel approach to the computation of the sum over histories in the path integral.
On the mathematical side, we have suggested a phase space formulation of the Feynman sum over paths in which the spatial trajectory ${\vec{\mbox{\boldmath {$y$}}}}(t)$ plays the role of a Lagrange multiplier enforcing the classical equation of motion (18) at the quantum level. Imposing such a constraint, with the aid of a Dirac delta-distribution, reduces the functional integral to an ordinary integral over the arbitrary values of the initial momentum. In the simple case of a free, non relativistic particle, the use of the phase space path integral, or the Lagrangian formulation of it, is largely a matter of choice, in the sense that one arrives at the same expression of the propagation kernel. In the case of relativistic extended systems, the use of the phase space path integral along the same lines discussed here, seems definitely advantageous, if not mandatory. On the physical side, however, our formulation of the path integral reflects a new mechanism of propagation which, in our view, sheds some new light on the connection between classical and quantum systems. The element of novelty can be summarized thus: it is well known that classical mechanics, which is a deterministic theory, demands that one definite value of the initial position and momentum be assigned a priori, and thereafter selects a single classical trajectory that satisfies the equation of motion. On the other hand, in quantum mechanics one cannot fully specify the value of the momentum of a precisely localized particle. The price to pay for specifying the particle position at ${\vec{\mbox{\boldmath {$x$}}}}_0$ when $t=0$, is that ${\vec{\mbox{\boldmath {$p$}}}}$ is completely arbitrary, and therefore one has to sum over all possible values of the three-momentum. This brings into the game the whole family of trajectories which satisfy the classical equations of motion. With all such channels of propagation open, a quantum particle turns to a ``wave-like'' propagation mode, exploring all of them by ``evolving simultaneously'' along all classical trajectories corresponding to all possible values and orientations of the momentum, up to the final point ${\vec{\mbox{\boldmath {$x$}}}}$ where the particle is detected. Thus, as stated in most textbooks of Quantum Mechanics, but never fully clarified, a quantum particle starts as a pointlike object precisely located at a point, propagates as a wave, only to reappear as a dot on a screen, at the point of detection. We have exploited this dual, wavelike, description of the family of classical trajectories by connecting the quantum evolution of a particle with the Jacobi formulation of Classical Mechanics, i.e., by deriving the diffusion equation from the propagation kernel, and vice versa. The main purpose of the whole exercise was to test the consistency of our approach, and its ability to reproduce some standard results obtained by different methods. Equally important, however, the objective was to clarify, with the familiar example of a free particle, the connection between the classical theory and the central idea upon which the path integral formulation is based, namely, that ``trajectories'' still play a role in quantum mechanics and that all paths, with their intrinsic randomness, contribute to the evolution of the wave function which is governed by the propagation kernel. Having said that, we may add that our mathematical analysis and its physical interpretation extend well beyond the case of a free, non relativistic particle. Indeed, the very possibility of formulating a quantum mechanics of closed strings and other extended objects[9], points to the effectiveness of our approach to the path integral for studying new fundamental physics. By an interesting feedback process, we have found that many new formal and physical properties of our approach to strings can be applied equally well to the case of a point particle which is now regarded as the object of lowest dimensionality in a hierarchy of geometric objects to which the same dynamical principles seem to apply. >From the present introduction, the interested reader may eventually progress to a more advanced discussion involving relativistic extended systems, such as strings, membranes and p-branes, with a change in notation rather than substance.