Vacuum energy and normal ordering

Place a real scalar field in a periodic box and keep finitely many modes. Its vacuum energy is then a finite sum of half the positive frequency of each retained oscillator. Enlarging the box increases the number of modes proportionally to volume. Dividing by volume removes that extensive factor, but increasing the momentum cutoff still adds more short-distance modes per unit volume. These are logically distinct limits and should be controlled separately.

For a massless field with a spherical sharp spatial cutoff, the density grows as the fourth power of cutoff in three space dimensions. This simple estimate diagnoses ultraviolet sensitivity. A spatial cutoff selects a frame, so its raw stress tensor cannot be assumed to have Lorentz-invariant vacuum equation of state. A regulator is a definition of an intermediate calculation, and symmetry-compatible renormalization requires more than reading a divergent energy integral as a measured cosmological constant.

For bosonic free modes, normal ordering puts creation operators to the left of annihilation operators without adding commutators during the definition of the ordered expression. Comparing it with the original operator then introduces the commutator terms. For a single oscillator coordinate, the ordered square differs from the original square by its vacuum variance. The resulting operator has zero vacuum expectation, even though the original coordinate continues to fluctuate in the vacuum.

For a field, the analogous contraction at coincidence is ultraviolet singular. One can first smear separated fields or retain a cutoff, subtract the chosen vacuum two-point function, and then define a Wick polynomial through an appropriate limit. The subtraction depends on the selected state and its singularity structure. A different background or vacuum need not use the same contraction. Normal ordering therefore records a reference construction rather than an assertion that local products of distributions were already ordinary operators.

Higher powers have several possible pair contractions, so subtracting only their vacuum mean is generally insufficient to obtain the Wick-ordered operator. A regulated fourth power contains a quadratic subtraction and a constant term with combinatorial coefficients. These terms follow from reorganizing creation and annihilation operators. They matter when matrix elements between excited states are computed, not merely when the vacuum expectation is checked.

In flat-space free dynamics a constant energy subtraction leaves evolution of observables unchanged. Comparing boundary geometries, however, requires a common physical subtraction and regulator; independently erasing each geometry's vacuum energy would discard the difference of interest. Interacting theories need composite-operator renormalization and parameter counterterms beyond free normal ordering. Coupling to gravity adds a renormalized cosmological term and stress-tensor conditions. None of those tasks is completed merely by setting a free Fock-vacuum expectation to zero.