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I am reading Feynman's famous paper Space-Time Approach to Non-Relativistic Quantum Mechanics, and in section 7 he says:

Actually, Huygens' principle is not correct in optics. It is replaced by Kirchhoff's modification which requires that both amplitude and its derivative must be known on the adjacent surface. This is a consequence of the fact that the wave equation in optics is second order in time. The wave equation of quantum mechanics is first order in time; therefore, Huygens' principle is correct for matter waves, action replacing time.

What does Feynman actually mean by this? Why does Huygens need to be modified? What is Kirchhoff's modification? What is the difference in this sense between matter waves and light waves?

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As alluded to in the quotation, the time-evolution of non-relativistic matter waves is given by Schrodinger's equation, which is a linear, first-order differential equation in time. As such the future state of the system is fully specified if we specify a single boundary condition. On the other hand, photons, being massless, cannot be treated in a non-relativistic framework. As a result, the equations of motion for the photon field are second-order in time. This implies that two boundary conditions are required to specify the evolution of the system.

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