Vacuum energy refers to the underlying energy present in empty space, or "vacuum." In quantum field theory, even in a perfect vacuum devoid of matter, there are still fluctuations due to the Heisenberg uncertainty principle. These oscillations happen because pairs of virtual particles can spontaneously form and annihilate within very short time periods.
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Vacuum energy is the energy assigned to a theory's vacuum state. In a free bosonic field description it includes the sum of oscillator zero-point energies, formally . The continuum limit of that expression is generally divergent and requires a prescription, rather than direct identification with a finite measured energy density.
Adding a constant to a nongravitational Hamiltonian changes an overall evolution phase but not transition probabilities. Energy differences between boundary configurations can nevertheless have physical effects, as in the Casimir effect. When gravity is included, a vacuum energy density contributes to the gravitational source and is related to the cosmological constant. Normal ordering removes a reference-theory constant by convention; it does not solve every interacting or gravitational vacuum-energy problem.