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Path Integrals and Holism

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Abstract

This paper argues that the path integral formulation of quantum mechanics suggests a form of holism for which the whole (total ensemble of paths) has properties that are not strongly reducible to the properties of the parts (the single trajectories). Feynman’s sum over histories calculates the probability amplitude of a particle moving within a boundary by summing over all the possible trajectories that the particle can undertake. These trajectories and their individual probability amplitudes are thus necessary in calculating the total amplitude. However, not all possible trajectories are differentiable, thus suggesting that they are not physical possibilities, but only mathematical entities. It follows that if the possible differentiable trajectories are taken to be part of the physical system, they are not sufficient to calculate the total probability amplitude. The conclusion is that the total ensemble is weakly non-supervenient upon the physically possible trajectories.

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Notes

  1. For a more comprehensive mathematical treatment see, among others: [3, 16, 45].

  2. To be more precise, in the classical limit \(\hbar \rightarrow 0\), the main constructive interfering paths are those in the infinitesimal neighborhood (in the order of the quantum of action) of the stationary paths.

  3. It can be shown by proving that it is the path that minimizes the action.

  4. For the present purposes, it is sufficient to take supervenience as a form of reductionism where the properties of set \(\alpha\) can be determined (reduced) by the properties of set \(\beta\).

  5. To be precise, [27] also distinguishes properties that are qualitative, i.e., properties that to be instantiated do not depend on the existence of a particular object or individual.

  6. We have been cutting around some corners and we have not covered the large and prolific debate about both atomism, supervenience and their relation with classical mechanics. See, among others: [23, 27, 32, 35, 37].

  7. The point was also raised in [22].

  8. The specification of the supervenience basis is necessary so as not to make supervenience trivial.

  9. To be precise, the system follows the path for which the action is stationary

  10. For the mathematical details see: (Wharton [49], p. 5).

  11. One can conceive of this last case as the difficulty of calculating the exact outcome of a dice-tossing because of the presence of variables such as the friction between the dice and the surface, the exact strength of the tossing, the imperfections of the shape of the dice and so on.

  12. The mathematical details do not concern us here, but can be found in: [7] and [10].

  13. For the details of the proof, see: Feynman et al. ([7], p. 176)

  14. For a better and more detailed exposition of such interpretation see: [12, 17, 18].

  15. Such an idea of probability fits more the objective probability expressed by the quote from Feynman we have introduced in the previous subsection. As such, it is also different from the epistemic probability advocated by Wharton.

  16. A realm is a set of decoherent coarse-grained alternative histories.

  17. I am leaving aside the mathematical details as not relevant for our purposes here.

  18. One might promptly note that if a reduction of the ensemble is possible, then the total amplitude does not depend on each single trajectory. One could consider this as a further weakening of the non-supervenience or, alternatively, simply take the ensemble to be the smallest set of trajectories necessary to obtain the appropriate total probability amplitude.

  19. We are assuming the condition that there are no obstacles within the boundary that could determine a bouncing of the particle.

  20. For a trajectory to be (non) physical corresponds to the concrete possibility for a particle to traverse it.

  21. A third possibility would be to discuss whether trajectories can be non-local. The argument would require a longer investigation on the non-local nature of quantum mechanics. Nonetheless, it is opinion of the author that this would lead to a case of holism by analogy with entanglement. We leave the issue to further investigations.

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  1. University of South Carolina, Columbia, USA

    Marco Forgione

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Forgione, M. Path Integrals and Holism. Found Phys 50, 799–827 (2020). https://doi.org/10.1007/s10701-020-00351-7

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