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While there is this standard derivation of Schrödinger equation from Feynman path ensembles, can we also derive/imagine Klein-Gordon, Dirac equations through path ensembles?

The main difficulty seems that Schrödinger derivation uses these diffusion-like paths of infinite velocity, which are non-relativistic. It could be repaired by going to phase-space like in Langevin equation (e.g. https://doi.org/10.1103/PhysRevA.96.052116, simple simulators), but such phase-space Schrödinger equation is more complicated. Maybe there are different ways to overcome it?

Regarding Dirac equation, it works on bispinor encoding spatial direction (of spin), so the object in path ensembles would need to recognize both position and spatial direction.

Qmechanic
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Jarek Duda
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1 Answers1

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  1. OP is starting out by recalling the equivalence between the Schrödinger equation and the Feynman path integral for the 1st quantized non-relativistic point particle, cf. e.g. this & this Phys.SE posts.

    OP is apparently pondering if the same can be done for the Klein-Gordon & Dirac equation in the 1st quantized formalism? This is an interesting question, although one should be aware of the shortcomings of the 1st quantized formalism, such as, e.g., the inability to describe particle creation & annihilation.

    (NB: There are well-known action principles and 2nd quantized path integrals for the Schrödinger, Klein-Gordon & Dirac fields, cf. e.g. this Phys.SE posts, but this is presumably not what OP is asking about. For a relation between 1st and 2nd quantization, see e.g. this Phys.SE post.)

  2. To get the (massive) complex Klein-Gordon equation from a 1st quantized path integral, one should not surprisingly use the action for a relativistic (massive) point particle, cf. e.g. this Phys.SE post.

  3. To get the Dirac equation from a 1st quantized path integral is more tricky. One approach is via a so-called spinning particle with ${\cal N}=1$ SUSY, cf. e.g. this Phys.SE post and links therein.

Qmechanic
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