vacuum state

In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. By definition, it contains no physical particles. The term "zero-point field" is sometimes used as a synonym for the vacuum state of an individual quantized field.

Non-vanishing vacuum state

If the quantum field theory can be accurately described through perturbation theory, then the properties of the vacuum are analogous to the properties of the ground state of a quantum mechanical harmonic oscillator (or more accurately, the ground state of a QM problem). In this case the vacuum expectation value (VEV) of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies (for example, Quantum chromodynamics or the BCS theory of superconductivity) field operators may have non-vanishing vacuum expectation values called condensates. In the Standard Model, the non-zero vacuum expectation value of the Higgs field, arising from spontaneous symmetry breaking, is the mechanism by which the other fields in the theory acquire mass.

The energy of the vacuum state

In many situations, the vacuum state can be defined to have zero energy, although the actual situation is considerably more subtle. The vacuum state is associated with a zero-point energy, and this zero point energy has measurable effects. In the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the vacuum state appears as the cosmological constant. An outstanding requirement imposed on a potential Theory of Everything is that the vacuum energy of the vacuum state must explain the physically observed cosmological constant.

The symmetry of the vacuum state

For a relativistic field theory, the vacuum is Poincaré invariant. Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEVs. The VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum has less symmetry than the theory allows, and one says that spontaneous symmetry breaking has occurred.

Notations

The vacuum state is written as $|0\rangle$ or $|\rangle$ . The VEV of a field φ, which should be written as $\langle0|\phi|0\rangle$ , is usually condensed to $\langle\phi\rangle$ .

Virtual particles

The uncertainty principle in the form $\Delta E\Delta t\ge\hbar$ implies that in the vacuum one or more particles with energy ΔE above the vacuum may be created for a short time Δt. These virtual particles are included in the definition of the vacuum.

References

M.E. Peskin and D.V. Schroeder, An introduction to Quantum Field Theory.

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