Is there a fundamental connection between energy, momentum, and mass in physics?

[StatusX]

Mass-energy equivalence is fundamental in relativity, but it seems like energy and momentum are also different aspects of the same thing. They've each got very important conservation laws. In SR, the space coordinates of the four-momentum give the momentum while the time coordinate gives energy. In QM, -ih d/dx is the momentum operator, and ih d/dt is the energy operator. P=h/wavelength and E=h/period. It seems that what momentum is to space, (mass-) energy is to time. What is behind this symmetry? Is there a mass-energy-momentum equivalence principal?

Also, I notice that p=mv classically, where v is in units of distance/time. It almost looks like v is a conversion from time units to space units, where mass-energy is the time unit and momentum is the space unit. Is this a valid way of looking at it?

 

[pervect]

Mass-energy equivalence is fundamental in relativity, but it seems like energy and momentum are also different aspects of the same thing. They've each got very important conservation laws. In SR, the space coordinates of the four-momentum give the momentum while the time coordinate gives energy. In QM, -ih d/dx is the momentum operator, and ih d/dt is the energy operator. P=h/wavelength and E=h/period. It seems that what momentum is to space, (mass-) energy is to time. What is behind this symmetry? Is there a mass-energy-momentum equivalence principal?

Also, I notice that p=mv classically, where v is in units of distance/time. It almost looks like v is a conversion from time units to space units, where mass-energy is the time unit and momentum is the space unit. Is this a valid way of looking at it?

You are on the right general track. You might want to look at Wikipedia's article on theorem[/URL]

The derivation of Noether's theorem requires that one use the Lagrangian or Hamiltonian formulation of physics - one way of describing this is that physics is described by the principle of "least action". There is a much more formal definition on the wikipedia web page - it's formal to the point of unintelligibility to the non-Phd, unfortunately.

Anyway, given this basic formulation, one can say that energy conservation is associated with time translation symmetry, and that momentum conservation is associated with space translation symmetry.

Because the Lorentz transform "mixes" time and space together, it also "mixes" momentum and energy together.