Vacuum in QFT: Fock space or effective potential?

In summary, there are two ways to define a vacuum in QFT. The first definition is a state $|0\rangle$ in Fock space that satisfies $a_k|0\rangle = 0$ for all annihilation operators $a_k$ and creation operators $a_k^{\dagger}$. The second definition is a state $|0\rangle$ in Fock space that satisfies $V_{eff}'(\phi_c) = 0$ for the effective potential $V_{eff}$ and $\langle 0|\hat{\phi}|0\rangle = \phi_c$. These two definitions may not seem compatible at first, but it is possible for the second definition to follow from the first if Fock
  • #1
Krudak Krudak
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There seem to be two ways of defning what a vacuum is in QFT:

1. It is state $|0\rangle$ such that $$a_k|0\rangle = 0$$ for all anihilation operators $$a_k$$, with creation operators $$a_k^{\dagger}$$. Thus, it is defined in Fock space.

2. It is state $$|0\rangle$$ such that derivative $$V_{eff}'(\phi_c) = 0$$ for effective potential $$V_{eff}$$ with $$\langle 0|\hat{\phi}|0\rangle = \phi_c$$.

Two definitions do not seem to be completely compatible at first glance, but it seems that at least second definition follows from first definition when Fock space can be defined. What exactly is happening?

Or I can ask my question differently as follows:

It is known that even for Minkowski spacetime, there exist vacua other than usual Poincare-invariant vacuum often just called as vacuum. As far as my understanding goes, the usual unique Poincare-invariant vacuum locally minimizes effective potential. Do other vacua also locally minimize effective potential?
 
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  • #2
Other vacuums typically do not minimize effective potential.
 
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FAQ: Vacuum in QFT: Fock space or effective potential?

What is a vacuum in quantum field theory (QFT)?

A vacuum in QFT refers to the lowest energy state of a quantum field. It is the state with no particles present and is often denoted as |0⟩. However, this does not mean that the vacuum is completely empty, as it may contain virtual particles and fluctuations in the field.

What is Fock space in QFT?

Fock space is a mathematical framework used to describe the quantum states of a system with an infinite number of particles. In QFT, it is used to describe the different possible particle states that can exist in a given field, including the vacuum state.

How is the vacuum defined in terms of Fock space?

In Fock space, the vacuum state is defined as the state with no particles present. This is represented by the vacuum vector |0⟩, which is the first basis vector in Fock space. All other states with particles present are constructed by acting on the vacuum state with creation operators.

What is the effective potential in QFT?

The effective potential in QFT is a term used to describe the potential energy of a field in the presence of interactions with other fields. It takes into account the effects of virtual particles and fluctuations in the field and is often used to calculate the behavior of a system at different energy scales.

How is the vacuum affected by the effective potential?

The vacuum can be affected by the effective potential through the presence of virtual particles and fluctuations in the field, which can change the energy of the vacuum state. In some cases, the effective potential can also lead to spontaneous symmetry breaking, where the vacuum state is no longer invariant under certain transformations.

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