# Minkowski vacuum: Poincare invariant, quasi-free state

• paweld
In summary, the discussion is about the Minkowski vacuum being Poincare invariant and a quasi-free state. The question is whether these two conditions fully define the Minkowski vacuum or if there are other states that fulfill these conditions. It is mentioned that in advanced QFT, there is an attempt to construct an "interacting representation" of the Poincare group and the vacuum is defined as the lowest-energy eigenstate of the full, interacting Hamiltonian. The question is whether a state that fulfills the two conditions mentioned has to be the Minkowski vacuum.
paweld
Minkowski vacuum is Poincare invariant and quasi-free state.
I wonder if these two conditions fully define it or there are more
states which fulfill these conditions (or maybe Poincare
invariance alone is sufficinet). Thanks for answers.

paweld said:
Minkowski vacuum is Poincare invariant and quasi-free state.
I wonder if these two conditions fully define it or there are more
states which fulfill these conditions (or maybe Poincare
invariance alone is sufficient).

I'm not entirely sure what you're asking. In advanced QFT, one attempts to
construct something called an "interacting representation" of the Poincare
group. (See Weinberg vol 1). The vacuum is still defined as the lowest-energy
eigenstate of the (full, interacting) Hamiltonian -- which often does not coincide
with the lowest-energy eigenstate of the free Hamiltonian.

My question is whether a state which fulfills two condition:
(1) its two point function is invariant under action of Poincare
group and
(2) all n-point function can be express in terms of two point function by
the sum over all parings (quasi-free state)
has to be Minkowski vacuum.

## 1. What is the Minkowski vacuum in quantum field theory?

The Minkowski vacuum is a state of minimum energy in quantum field theory, also known as the zero-point energy. It is a state of the quantum field where there are no particles present, and all fields are in their ground state. This state is Poincare invariant, meaning it is the same for all observers regardless of their relative motion.

## 2. What does it mean for the Minkowski vacuum to be Poincare invariant?

Poincare invariance means that the laws of physics remain the same for all observers who are in a state of constant, uniform motion. In the context of the Minkowski vacuum, this means that the vacuum state is unchanged for all observers regardless of their relative motion, making it a fundamental property of the vacuum state.

## 3. What is a quasi-free state in quantum field theory?

A quasi-free state is a state in quantum field theory that is close to but not exactly equal to the Minkowski vacuum. It is a state that contains a small number of particles or excitations, but the distribution of these particles follows the same statistical properties as the Minkowski vacuum. Quasi-free states are often used in calculations and approximations in quantum field theory.

## 4. How is the Minkowski vacuum related to the concept of vacuum energy?

The Minkowski vacuum is closely related to the concept of vacuum energy, also known as the cosmological constant. In quantum field theory, the vacuum state is not truly empty but contains fluctuations of the quantum fields. These fluctuations contribute to the vacuum energy, which has been observed to have a non-zero value and is responsible for the expansion of the universe.

## 5. What are the implications of the Minkowski vacuum being a quasi-free state?

The fact that the Minkowski vacuum can be approximated by a quasi-free state has important implications for calculations and predictions in quantum field theory. It allows for simplifications and approximations in complex calculations, making it a valuable tool for understanding the behavior of quantum fields. However, it is important to note that the true vacuum state is not exactly equal to a quasi-free state, and there may be subtle differences that can affect the accuracy of predictions.

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