(Mass-)Energy conservation holds for every time?

[Linker]

Hi,

 

in classical physics, in a closed system energy is conserved for every instant of time. But in quantum physics, quantum fluctuations (very short particle/energy creation or annihilation) on microscopic level can occur. But both theories have the same propertiy that the Noether theorem holds: If the theory is invariant under time translations the energy is conserved.

In every quantum field theory there is time translation invariance.

 

But are there theories that are time dependent (i.e. theories that break homogenity in time)? Why, the homogenity in time and homogenity in space are the most essential symmetries in physics (of course there are theories that violate symmetries, e.g. CPT-symmetry)?

 

[ajb]

In every quantum field theory there is time translation invariance.

I don't think this is true in full generality, only for those that have Poincare invariance and once we have split space-time into space and time.

 

But are there theories that are time dependent (i.e. theories that break homogenity in time)?

Right, at the classical level you would not expect energy conservation in such theories.

[Linker]

Which theories have no Poincare invariance?

[ajb]

Which theories have no Poincare invariance?

Quantum field theory on a curved background will not have Poincare invariance. If the space-time is "nice" then you can develop the calculations in more or less the same way as a flat space-time taking are of some subtleties. In full generality one has to turn to much more abstract ideas using operator algebras.

[Linker]

May be A, B initial conditions and C a phenomenon that occur if A and B are satisfied. Now, consider A' and B' as the same initial conditions, but translated arbritary in space and time. Then, these initial conditions (A' and B') will imply C' the same phenomenon C but translated arbritary in space and time.

 

Is there a reason why every Lagrangian is space/time translation invariant?

 

General relativity (and maybe Quantum gravity) are space-time-translation invariant theories, too, because the laws of gravity are space/time independent, right?

How does a quantum field theory in curved background look like?

[ajb]

Is there a reason why every Lagrangian is space/time translation invariant?

No and we can certainly write down Lagrangians that are not invariant under translations.

 

General relativity (and maybe Quantum gravity) are space-time-translation invariant theories, too, because the laws of gravity are space/time independent, right?

As general relativity is a theory about the local geometry of a space-time I am not sure what you mean by translational invariance here. You could be referring to the trivial fact that we always have theories that are invariant under passive diffeomrorphisms, that is we can always change coordinates. But this is not special to general relativity, it is the active diffeomorphisms that are more important in this context.

 

How does a quantum field theory in curved background look like?

For "nice" space-times almost the same as on flat space-time but there are some subtleties. In particular no single vacuum state is singled out.

[Linker]

Why the laws of gravity are space-time-translation invariant, too?

 

I don't know what is the fundamental reason fore space-time symmetry...

[ajb]

Are you asking about conservation of energy-momentum here?

[Linker]

Yes, I am asking why in general relativity case the covariant divergence of the energy-momentum tensor hold, too.

 

Assuming, a stone is falling down on earth on the time point T and given its trajectory x(T). Repeat the same initial conditions (stone has same mass, same shape, there are same environmental conditions, etc.) on time point T+DeltaT. Then x(T+dT)=x(T) even when there takes gravity place.

Or I say it better, if there is Newtonian gravity. Treating this phenomenon with General Relativity, the stone must increase its mass (because it accelerates and gains kinetic energy which is extra mass by special relativity) and there are some deviations from Newton's theory.

 

Can be really assumed x(T+dT)=x(T) in General Relativity case or is energy-momentum not really conserved by treating phenomena with General Relativity?

[ajb]

Yes, I am asking why in general relativity case the covariant divergence of the energy-momentum tensor hold, too.

 

....

 

Can be really assumed x(T+dT)=x(T) in General Relativity case or is energy-momentum not really conserved by treating phenomena with General Relativity?

This is a local condition and the global question is much more subtle. In fact in a general space-time it is not all clear what you would mean by energy and momentum globally.