Vacuum energy in the bag model

Abstract

The vacuum energy of the Yang-Mills field is examined for the conditions of the bag model. The dominance of high-frequency effects results in a vacuum energy that decomposes naturally into a volume energy, a surface energy, and higher shape energies. These quantities are identified with the parameters of the bag model. The imposition of confining boundary conditions for all frequencies is shown to be inconsistent since this would result in the bag constant and certain of the shape tensions being infinite. The manner in which the boundary conditions should be relaxed at high frequency is discussed. The most naive procedure for relaxing the boundary conditions, which is to apply confining conditions only on modes of frequency less than some cutoff frequency, results in a negative bag constant and surface tension and would render the vacuum unstable against the spontaneous breaking of Poincaré invariance. Consideration of the manner by which the interacting electromagnetic field avoids a similar instability suggests that a more realistic way to relax the boundary conditions on the bag surface is to endow the vacuum exterior to the bag with a frequency-dependent dielectric constant and magnetic permeability. In this picture the stability of the vacuum is restored, the surface tension is finite and positive, and the bag constant is zero at least to lowest order in the coupling. It is pointed out that the fermion contributions to the bag constant and the surface tension may relate to the spontaneous breaking of chiral invariance. The aim throughout is to examine the bag model, as it relates to vacuum energy, strictly in its own terms with an emphasis on questions of principle. All too often is heard the alibi that since the theory itself is only approximate, the mathematics need be no better. In truth the opposite follows. Granted that the model represents but a part of nature, we are to find what such an ideal picture implies, a result strictly derived serves to test the model; a false result proves nothing but the failure of the theorist. To call an error by a sweeter name does not correct it. The oversimplification or extension afforded by the model is not error: the model, if well made, shows at least how the universe might behave, but logical errors bring us no closer to the reality of any universe. (Truesdell and Toupin 1960).