A mathematical statement of energy conservation can be given using the continuity equation in terms of the total energy(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{\partial}{\partial t}\iiint_V\epsilon dV+\iint_S \epsilon \mathbf{V\cdot dS}=0[/tex]

where t is time, V is a velocity vector, V is the volume of the system, dS is a point along the surface area, S, of the volume, and [tex]\epsilon[/tex] describes the total energy of a point in the volume which might be given as

[tex]\epsilon=\rho (e+0.5v^2 +\Phi)+h\nu[/tex]

where [tex]\rho[/tex] is the internal energy, v is the velocity, [tex]\Phi[/tex] is the potential energy of the body forces acting on the mass, h is plank's constant, and [tex]\nu[/tex] is light frequency.

So, the first equation tells us that the total energy of a volume, V, can only change if energy is transported into or out of it. I derived the first equation as an application to classical systems, but I am wondering if it will become problematic when talking about quantum mechanics or relativity.

My questions are:

1. Can this formalism be applied to quantum mechanics? I imagine that the uncertainty principle, and phenomena like quantum fluctuations and the production of virtual particles are problematic for this formulation that is based on the continuity equation.

Perhaps we must say, for instance, that you cannot consider a volume less than some length because of uncertainty, or something like that?

2. How do relativistic effects, such as time dilation, factor into the dependence on time in a formulation like this? It would seem problematic, for instance, to consider a large volume in which time proceeds at different rates due to their relative velocities.

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# Energy Conservation versus Quantum and Relativistic Effects

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