Vacuum energy ($E_0$)

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 $\frac{1}{2}{\sum }_{\mathbf{k}}\hslash {\omega }_{\mathbf{k}}$. 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.

The Casimir effect is a force associated with the change in electromagnetic field fluctuations produced by material boundaries. For ideal perfectly conducting parallel plates separated by $a$ at zero temperature, the attractive pressure isReal measurements require finite-conductivity, temperature and geometric corrections. A classic experiment used a torsion-pendulum electromechanical system to measure the force between nearby conducting surfaces. The experimental comparison concerns the force difference produced by a specified geometry; it does not make an arbitrary absolute vacuum-energy offset observable.^[ref]

The Casimir energy is the configuration-dependent part of a quantum field's vacuum energy after a specified subtraction or renormalization. Only an energy difference or its derivative is physically meaningful in the idealized calculation; the separate zero-point sums are divergent.

For the electromagnetic field between two infinite perfectly conducting parallel plates at zero temperature and separation $a$, the renormalized energy per area is $E/A=-{\pi }^2\hslash c/(720a^3)$. Differentiation gives the attractive pressure $-\partial (E/A)/\partial a=-{\pi }^2\hslash c/(240a^4)$. Real materials, finite temperature and geometry alter this result. The Casimir effect therefore tests a response to changed boundaries, not a direct weighing of an unregularized infinite energy.

For finitely many free physical degrees of freedom with nonnegative masses, assign positive weights to bosons and negative weights to fermions. With a common spherical ultraviolet cutoff, the unsubtracted vacuum energy density is finite precisely when the three displayed supertraces vanish. Expanding $\sqrt{p^2+m^2}=p+m^2/(2p)-m^4/(8p^3)+O(p^{-5})$ proves that they cancel the quartic, quadratic and logarithmic ultraviolet divergences. The remaining radial integral converges as ${\int }^{\infty }{p}^{-3}dp$. Equal-mass boson–fermion pairing is sufficient but stronger than the three conditions; a renormalized vacuum energy with a free counterterm need not satisfy them.

The vacuum energy density is the bulk limit ${\rho }_{\mathrm{vac}}={\lim }_{\mathcal{V}\to \infty }{E}_0(\mathcal{V})/\mathcal{V}$, with boundary effects removed and a fixed convention for the Hamiltonian. In an effective field theory at a homogeneous vacuum it is the value of the renormalized scalar potential. A global supersymmetry algebra expresses the regulated vacuum energy as a sum of supercharge norms, giving nonnegative density; the scalar potential of supergravity instead includes a negative gravitational term.

On a one-dimensional interval with Dirichlet boundary conditions, $k_n=n\pi /L$ and ${\omega }_n^2=k_n^2+m^2$. The massive-field vacuum energy density can depend on position even though its integral is the oscillator sum ${\sum }_n{\omega }_n/2$. All terms require a common ultraviolet regulator when vacuum energies in different geometries are compared.

If a free real scalar field's three-momentum ultraviolet cutoff is identified with the absolute vacuum expectation value of another scalar field, the displayed scalar potential is even and has its unique global minimum at zero. It grows as ${\phi }^4/(16{\pi }^2)$ for $|\phi |\gg m$, but as $m|\phi {|}^3/(12{\pi }^2)$ for $|\phi |\ll m$ with $m>0$. The positive derivative with respect to $|\phi |$ proves that this contribution alone does not produce spontaneous symmetry breaking. This cutoff identification is a model assumption, distinct from deriving a complete effective potential for a dynamical field.

Counting two transverse polarizations and zero-point energy $\hslash ck/2$ per mode gives $u=\hslash c{k}_{\max }^4/(8{\pi }^2)=2{\pi }^2\hslash c/{\lambda }_{\min }^4$. This is a bare cutoff-dependent mode sum, not a classical incident radiation bath or a renormalization-independent absolute observable.

A fermionic mode with canonical anticommutation relations $\{b,b^{†}\}=1$ has the symmetrically ordered Hamiltonian operator $H=\frac{E}{2}(b^{†}b-b{b}^{†})=E(b^{†}b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A quantum harmonic oscillator with bosonic canonical commutation relations contributes $+E/2$ instead. For a Dirac field, sum over both particle and antiparticle modes and over spin: there are four independent oscillators per spatial momentum in four spacetime dimensions. Comparing bosonic and fermionic vacuum energies requires a common regularization in quantum field theory, matching degrees of freedom and a stated vacuum-energy subtraction convention.