The vacuum state of a quantum field is defined as the lowest-energy state of the field, devoid of any particles or excitations. It represents the "ground state" or the state of minimum energy for the field. In quantum field theory, fields are described as operators that act on the vacuum state to create particles or excitations.
To understand the vacuum state, let's consider a simple example of a scalar field. A scalar field is a field that associates a scalar value (a number) with each point in spacetime. The field can be thought of as oscillating at different points in space and time.
In quantum field theory, the scalar field is quantized, meaning that it is treated as a collection of harmonic oscillators, one for each mode of the field. Each oscillator can have different energies corresponding to different numbers of particles. The vacuum state is defined as the state in which all the oscillators are in their lowest-energy, or ground state.
Mathematically, the vacuum state is denoted by |0⟩ or |vac⟩. It satisfies the property that when a field operator acts on it, it produces zero particles:
ϕ(x)|vac⟩ = 0,
where ϕ(x) is the field operator at spacetime point x.
However, it's important to note that the vacuum state is not an empty state in the sense that it has no properties. It still has quantum fluctuations associated with it due to the uncertainty principle. These fluctuations can be thought of as virtual particles popping in and out of existence.
The vacuum state is a fundamental concept in quantum field theory and plays a crucial role in understanding particle physics and the quantization of fields.