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.

 

[charng]

I can confirm that there is indeed a fundamental connection between energy, momentum, and mass in physics. This connection is described by the famous equation E=mc^2, which is known as the mass-energy equivalence principle. This principle states that mass and energy are two forms of the same physical quantity and can be converted into one another.

In classical mechanics, momentum is defined as the product of an object's mass and velocity. This is represented by the equation p=mv, where p is momentum, m is mass, and v is velocity. In this context, mass and momentum are two separate quantities, but they are closely related through the object's velocity.

However, in relativity and quantum mechanics, we see that momentum and energy are intimately connected. In special relativity, the four-momentum vector includes both spatial momentum and energy, with the time coordinate representing energy. This symmetry between momentum and energy is also seen in quantum mechanics, where the momentum operator and energy operator are related by the same constant, Planck's constant (h).

The idea of v being a conversion between time and space units is an interesting way of looking at it, but it may not be entirely accurate. Velocity is a fundamental quantity that describes an object's motion in space and time, and it is not necessarily a conversion factor between time and space units. However, the concept of mass-energy and momentum being two different aspects of the same thing is valid and reflects the underlying symmetry in nature.

In conclusion, the fundamental connection between energy, momentum, and mass is a key concept in physics and is described by the mass-energy equivalence principle. This principle highlights the deep connections between seemingly separate physical quantities and helps us understand the underlying symmetries in the universe.