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I just finished reading Richard Feynman's lectures on Quantum Electrodynamics (QED: The Strange Theory of Light and Matter) and it fascinated me. However, there's an unanswered question I have from reading it.

If, as Feynman argues, "light doesn't really travel only in a straight line; it 'smells' the neighboring paths around it and uses a small core of nearby space" (though it is overwhelmingly likely to appear to be traveling in a straight line over long distances), how does this not defy the law of conservation of momentum? My understanding is that this law applies to light as well, with a momentum defined by p = E/c. If so, light curving would clearly defy it, would it not?

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    $\begingroup$ A path integral is a mathematical object that just happens to describe the physics of an electromagnetic field properly. It does not give you a self-consistent interpretation of quantum mechanics in terms of classical mechanics of point particles that travel on well defined paths. Feynman himself warns against taking the picture of path integrals beyond its range of applicability. $\endgroup$ – CuriousOne Sep 8 '15 at 13:11
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Given you've read only QED, this is a highly astute question.

Conservation laws in the quantum world work a little differently from classical conservation grounded on Noether's theorem (there is a kind of quantum analogue in the Ward-Takahashi identity).

If a quantum entity has state $|\psi\rangle$, then conserved quantities are measurement means defined by $\langle\psi|\hat{p}|\psi\rangle$, where $\hat{p}$ is any non-time-varying observable that commutes with the quantum Hamiltonian. The momentum observable is one such observable. This is nicely summarized by Ehrenfest's theorem $\mathrm{d}_t\,\langle\psi|\hat{p}|\psi\rangle = \langle\psi|[\hat{H},\,\hat{p}]|\psi\rangle$.

Accordingly, when the photon "sniffs neighboring paths", which is Feynman's lay explanation for when the photon is in a superposition of different momentum eigenstates (going many directions at once), it is only the mean of the momentum measurement that is conserved. No violation of conservation happens simply by dint of there being "many different paths"; it is only the mean measurement we need to look at.

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