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The Quantum Vacuum

August 31, 2008

One fundamental feature of quantum field theory (QFT) is the notion that empty space is not really empty. Emptiness has been replaced by the concept of the vacuum ground state. It is this ground state which is responsible for a ubiquitous energy density that is ultimately believed to act as a contribution to the cosmological
constant appearing in Einstein’s field equations from 1917 onwards,

R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}+\Lambda g_{\mu\nu}

In 1916 Nernst, who was originally inspired by the new ideas of quantum theory and Plancks law for the radiation from a black body, put forward the proposition that the vacuum of spacetime is not empty but is, in fact, a medium filled with radiation containing a large amount of energy. One feature of this model was that the energy density of the vacuum was infinite, and even when a modest cutoff was proposed, the total energy content was still large. Nernst’ ideas about the vacuum were never used for any cosmological models as his interests were in chemistry, and in forming a model of the water molecule.

A more solid foundation for speculations of the energy density of the vacuum became available with the developments in QFT in which all the fields in nature are treated as a collection of quantized harmonic oscillators. The various amplitudes and frequencies of oscillation represent the different boson and fermion species we observe in nature.

The vacuum has all the quantum properties a particle may acquire, for example energy, spin and polarization. These quantities on average cancel each other out, with the exception of the vacuum expectation value of the energy \left<E_{vac} \right> . A consequence of the Heisenberg Uncertainty principle is that no field oscillator can ever be completely at rest. There will always be some residual `zero-point energy.’ Naively one can see this in

E=(n+\frac{1}{2})\hbar \omega .

For n=0 we are left with

E=\frac{1}{2}\hbar \omega .

One problem that arises frequently arises with regards to the ground state of the vacuum are the huge energies that are found. Pauli was concerned with the \textit{gravitational} effects of the zero-point energy.  Pauli’s calculation (mid-late 1920s) demonstrated that if the gravitational effect of the zero-point energies was taken into account  the radius of the universe would be smaller than the distance from the Earth to the Moon. Pauli’s calculation invloved applying a cut-off energy at the classical electron radius, which was considered to be a natural cut-off in his day. Pauli’s concern was readdressed by Straumann in 1999, who found that indeed, the radius of the universe would be \approx 31 \ km. Setting \hbar =c=1 the calculation reads

<\rho>_{vac}=\frac{8\pi}{(2\pi)^3}\int_{0}^{\omega_{max}}\frac{\omega}{2}\omega^2d\omega\frac{1}{8\pi^2}\omega_{max}^4

Inserting the appropriate cutoff,

\omega_{max}=\frac{2\pi}{\lambda_{max}}=\frac{2\pi m_e}{\alpha}

and plugging into

8\pi G \rho = \frac{1}{a^2}=\Lambda

where a is the radius of curvature obtained from solving Einsteins equation for a static dust filled universe. One obtains

a=\frac{\alpha^2}{(2\pi)^{2/3}}\frac{M_{pl}}{m_e^2}\approx 31 \ km.

One way to reconcile this inconsistency with the known size of the universe, as noted in Pauli’s \textit{Handbuch der Physik} is that it is more consistent to begin from the ansatz that the zero-point energy does not interact with the gravitational field. Indeed, the speculations of Dirac regarding the huge vacuum energy and also the final version of quantum electrodynamics (QED) constructed by Schwinger, Feynman and others never prompted any interest in the \textit{gravitational} consequences of these theories. This is not surprising when one considers that theoretical landscape of QED which was plagued with divergences in higher order calculations which preoccupied the community.

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