Time translation invariance and the vacuum state

[Jim Kata]

I kind of get the connection, but could someone elaborate the necessity for time translation invariance for the existence of a unique vacuum state.

 

[fermi]

Time translation invariance is neither necessary nor sufficient for the existence of a unique vacuum state. The vacuum state can be degenerate (in fact infinite-fold degenerate) without breaking the time-translation invariance. Conversely, if you violate time translations, you can still have a unique vacuum, depending on what the Lagrangian is.

The time translation invariance is directly related to energy conservation. Just like the spatial translation invariance is related to the momentum conservation. Without time translation invariance, the energy is not conserved. Therefore, even if you have a unique vacuum, its energy will not be conserved if the time translation invariance does not hold. In fact the vacuum at t=0 may not be the vacuum state any more at a later time.