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.

 

[sir-pinski]

Time translation invariance is a fundamental concept in physics that states that the laws of nature remain unchanged regardless of when they are applied. In other words, the physical properties of a system do not change over time. This concept is closely related to the existence of a unique vacuum state.

The vacuum state is the lowest energy state of a system, and it is often referred to as the “ground state”. It is the state in which the system has no excitations or particles present. In quantum field theory, the vacuum state is also known as the state of minimum energy, and it is invariant under time translation.

The necessity for time translation invariance for the existence of a unique vacuum state can be understood in the following way. In quantum field theory, particles and excitations are described as fluctuations in the vacuum state. These fluctuations can be thought of as disturbances in the underlying fabric of space-time. However, if the laws of nature were not time translation invariant, these fluctuations would change over time, and the vacuum state would not remain constant. This would mean that the vacuum state would not be unique, as it would depend on the time at which it is measured.

Furthermore, time translation invariance is closely related to the principle of energy conservation. Since the vacuum state is the state of lowest energy, any fluctuations or excitations that occur in the system must conserve energy. This is only possible if the laws of nature are time translation invariant. If they were not, energy conservation would not hold, and the vacuum state would not be unique.

In summary, time translation invariance is necessary for the existence of a unique vacuum state because it ensures that the laws of nature remain unchanged over time, and that energy is conserved. Without this fundamental principle, the concept of a vacuum state would not hold, and the underlying fabric of space-time would not be stable.