AAWSAP DIRD, Concepts for Extracting Energy from the Quantum Vacuum, April 2010

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Appendix: The QCD Bag Model
When considering the effective masses of quarks bound within hadrons, it is common to
think of the constituent masses of the quark and antiquark pair as their zero-point
energy (ZPE) when they are bound by the confining potential 25 (acting between a quark
and an antiquark) with an energy spectrum that corresponds to the masses of the
observed mesons. For charm, and heavier quarks, it appears that the total ZPE is not
much different from the masses of the lowest-lying meson states. This picture also
holds true for baryons. The quark-gluon model for hadrons is called "the bag model."
Even in an "empty" bag-that is, one containing no quarks-there will be nonzero fields
present because of quantum zero-point fluctuations (ZPF) . This gives rise to a zero­
point (ZP) or Casimir energy inside the empty bag. The estimated total ZP/Casimir
energy of the confined gluon field inside a spherical bag is EzP;:;: +0. 7/a (or +0.7nc/a in
MKS units), where a is the radius of the bag in Gev-1 units. EzP is numerically the entire
story because the ZPE contribution of the confined fermion (quark) field is far smaller
than EzP (the leading approximation per degree of freedom is down by two orders of
magnitude) . A typical empty bag has an estimated radius a ;:;: 2.6 Gev-1 (0.5 fm), so EzP
~ 10-1 GeV (or 10-11 J), and this result remains approximately true if one uses a= 5.07
Gev-1 (1 fm) for a nucleon-sized bag. If the bag contains quarks, then this result does
not change because the quarks don't affect the ongoing quantum ZPF inside the bag
due to asymptotic freedom.
The inventor of the empty bag model (Ken Johnson) proposed that space is filled with
closely packed empty bags, and that the energy of space filled with contiguous bags is
simply the sum of the field energies contained within each bag. But this is not widely
accepted since the phenomenologically preferred model for the exterior "ordinary"
vacuum is given as follows.
QCD color confinement in hadrons is approximated by the phenomenologically
successful "bag model." In this model, the "ordinary" vacuum external to hadrons is a
perfect color magnetic (or chromomagnetic) conductor; that is, the chromomagnetic
permeability ~L is infinite, while the chromomagnetic vacuum in the interior of the bag is
characterized by µ = 1. This implies that the color electric (Eqco) and magnetic (Sqco)
fields are confined to the interior of the bag, and that they satisfy the following
boundary conditions on its surface 5: n . Eqco Is = 0, n x Sqco Is = 0, where n is a unit
normal vector to 5. In other words, this model defines QCD vacua that coexist in two
phases: 1) an ordinary vacuum exterior to the bag, impenetrable to color; and 2) a
vacuum interior of the bag, in which the Yang-Mills fields that carry color (gluons)
propagate freely. Both phases are separated by the surface boundary of the bag upon
which the Yang-Mills and fermion (quark) field satisfy the aforementioned boundary
25 Quarks carry "color charge" as well as electric charge. Color charge is considered to be the "true" charge of
strong interactions. Gluons are the "photons" of strong interactions, and color is exchanged by eight bicolored
gluons, which are massless and have spin 1. Color interactions are assumed to be a copy of electromagnetic
interactions. Theoretical and phenomenological studies found that the confining potential V(r) lies between a
Coulomb and a harmonic oscillator potential: V (r) = - (4a s I 3r ) + kr, where r is the rad ial distance between
confined quarks, k is a constant parameter, and { is the color factor associated with as for the case of quark­
antiquark pair confinement in mesons. For the case of baryons (qqq), the color factor in V(r) is replaced by f .
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