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In some places it is stated that one needs to include all paths in the path integral approach to quantum mechanics. But in the implementations I have seen one has been content with paths that goes in small steps along an operator, and not included paths that for instance goes to another galaxy and draws Mona Lisa and then goes somewhere else et cetera et cetera and then goes to the end point. So I assume there is some guiding principle or perhaps some bounds that show what kind of paths and how many paths are sufficient to bring the error down to an acceptable level?

It seems reasonable to me that the particle moves slower than the speed of light, for example. And that it doesn't teleport or branch off into fewer/more trajectories (unless that is needed for chemistry).

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  • $\begingroup$ The small steps are in time $\delta t$. The position basis completeness relation, however, $\int dx |x \rangle \langle x |$, spans the complete space of states. $\endgroup$
    – Avantgarde
    Oct 6, 2018 at 7:38
  • $\begingroup$ Bruce: path weighing? Is it possible to get a probability distribution of where the particle can go conditioned on where the particle is at that moment? I thought the whole integral was the probability. $\endgroup$
    – Emil
    Oct 6, 2018 at 10:18
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    $\begingroup$ For a mathematically technical discussion of you question see physics.stackexchange.com/questions/185445/… $\endgroup$
    – isometry
    Oct 6, 2018 at 12:31

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It depends on Lagrangian. In cases of physical systems, Lagrangian has kinetic energy part and another following from some potential energy. In such systems this terms has certain property: kinetic energy is quadratic in velocity. When potential energy is absent - as in a case of free particle for example - action integral is Gaussian. In such case only small amounts of trajectories, related to maximum of Gaussian density, gives value of the integral itself. When potential energy terms are present, usually such potential has an maximum etc. Most of the time such maximum may be approximated by parabola ( quadratic terms again).

In both cases the stationary phase techniques are used, and only trajectories close to classical trajectory gives account to the sum.

In purely mathematical cases it may be required to include all the paths as you mentioned. And of course for some systems, even physical ones, there may be more sophisticated examples, where for example potential energy part is flat, or system has not only potential energy terms but some topological bounds included. In such cases various subsets of all possible trajectories set has to be included in calculations.

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