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The path integral formulation of Quantum Mechanics is related to Huygens principle, as stated by Feynman in his seminal article [1] and widely commented since then. However Huygens principle does not hold in even dimensions, see this web page for example, or this Physics SE. Can we conclude that path integral only work in odd dimensions then?

[1] Feynman, R. P. (Apr. 1948). “Space-Time Approach to Non-Relativistic Quantum Mechanics”. In: Rev. Mod. Phys. 20 (2), pp. 367–387

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    $\begingroup$ There are other justifications for the path integral formulation of QM aside from Huygen's Principle, such as a least-action approach. The fact that Huygen's Principle doesn't apply doesn't mean that a path-integral formulation can't be done in 2-d, but it does impact what that path-integral formulation looks like. $\endgroup$ – Paul Jul 6 '18 at 13:35
  • $\begingroup$ AFAIK, the path integral formulation is the lagrangian formulation of QM (instead of the Schrödinger's hamiltonian formulation) and has nothing to do with Huygens principle (even if this was an initial or historical justification). The path formulation should be valid for any dimensions. $\endgroup$ – Cham Jul 6 '18 at 15:44
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I fear you are misreading Feynman. He tells you explicitly that he is discussing propagation of the Schroedinger equation, first order in the time, and not the 2nd order d'Alembertian whose Green's functions exhibit the peculiarities he himself warns you about.

He follows Dirac's epochal (1933) Physikalische Zeitschrift der Sowjetunion 3, 64–72 to actually demonstrate that if the amplitude of the wave is given on any "surface", its value at a short time after this is the sum of all contributions from all points of the surface at the original time, each contribution delayed in phase by an amount proportional to the action S of that segment, classically (extremally). This is the essence of QM amplitudes multiplied, concatenated, and summed over, $$ \langle q_t| q_T\rangle =\int \langle q_t| q_m\rangle dq_m\langle q_m| q_{m-1}\rangle dq_{m-1} ... \langle q_2| q_1\rangle dq_1 \langle q_1| q_T\rangle,$$ eqn (11) of the above; and of Huygens' principle; today it is just about the second thing we all learn about QM, cf Dirac's QM book, §32. This is, in fact, up to normalization, the very definition of the path integral, and it works in any and all dimensions. This is what he calls the strong Huygens principle--definitely not what your links dictate.

He then "excuses" himself for repeating Dirac's "very beautiful" destructive interference of the nonextremal paths and the dominance of the classical limit, the "reason" classical mechanics is extremal.

Now, do you see why it is straightforward to multiplex the 1-d free propagator found this way to an arbitrary number of dimensions, even and odd?

  • In fact, Gutzwiller 1988 derives the Huygens principle out of the path integral, including the dimensional weakening you mention.
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