# Why geometrically does the orientation of a scalar particle plane wave relate to its velocity

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In this question I am trying to think about quantum field theory geometrically. I abbreviate "in the zero speed direction" to "vertically" and "in the equal-time plane" to "horizontally".

I base my question of the following understandings. A correction of any of these would (helpfully) pre-empt my question:

• A quantum field of a specific particle variety in a single particle state may be in a plane wave state for that particle (idealised situation where the whole field contains one particle and it is in an exactly known abstract (reference frame independent) momentum state).
• The field mode of such particle may be geometrically transformed via Lorentz boost, yielding another plane wave, to get its appearance in another inertial frame.
• The rest mass/energy corresponds to the frequency of the field in the frame in which the frequency is least. Equivalently the vertical spacing (period) is most. In this frame the wavefronts are oriented horizontally. The higher the frequency / smaller the vertical spacing, the greater the energy. If the particle is massless there is no such frame.
• The total energy in a frame corresponds to the frequency / closeness of vertical spacing in that frame.
• The momentum in a frame corresponds to the closeness of horizontal spacing in that frame ('horizontal version of the energy').

That in itself seems fine to me. However I can't reconcile these definitions of energy and momentum with velocity. In a simple sense the velocity of something is how much spatial translation per unit time translation is required to keep it invariant. So I would expect a vertically (in a spacetime sense) aligned object to be stationary. A sloped configuration to have a faster speed the more the slopes deviate from vertical. But this is the opposite situation from the one implied above. Instead of vertical features a zero momentum has horizontal features (wave fronts). I understand the relativistic energy-velocity-momentum relationship is v proportional to p / E. Or equivalently v proportional to period / wavelength. Again this is the opposite of what I would expect, which would be for increasing wavelength that stretches the phenomena horizontally to increase speed and the opposite for increasing period.

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