AAWSAP DIRD, Traversable Wormholes, Stargates, and Negative Energy, April 2010

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It turns out that there are many different types of Casimir effects found in quantum
field theory (Reference 22-24, 28, 55). For example, if one introduces a single infinite
plane conductor into the Minkowski (flat spacetime) vacuum by bringing it adiabatically
from infinity so that whatever quantum fields are present suffer no excitation but
remain in their ground states, then the vacuum (electromagnetic) stresses induced by
the presence of the infinite plane conductor produces a Casimir effect. This result holds
equally well when two parallel plane conductors (with separation distance d) are
present, which gives rise to the familiar Casimir effect inside a cavity. Note that in both
cases, the spacetime manifold is made incomplete by the introduction of the plane
conductor boundary condition(s). The vacuum region put under stress by the presence
of the plane conductor(s) is called the Casimir vacuum. The generic expression for the
energy density of the Casimir effect is pcE = -A(ric)a4, where A= (,(D)/8rr.2 in spacetimes
of arbitrary dimension D (Reference 22-24). The appearance of the zeta-function (,(D)
is characteristic of expressions for vacuum stress-energy tensors, r::,;. In our familiar
four-dimensional spacetime (D = 4), A= rc2/720. To calculate r,:; for a given quantum
field is to calculate its associated Casimir effect.
Analogs of the Casimir effect also exist for fields other than the electromagnetic field.
When considering the vacuum state of other fields, one must consider boundary
conditions that are analogous to the perfect-conductor boundary conditions for the
electromagnetic field at the surfaces of the plates (Reference 22-24, 28). Other fields
are not electromagnetic in nature, that is to say they are non-Maxwellian, and so the
perfect-conductor boundary conditions do not apply to them. It turns out that complete
manifolds exhibit what is called the topological Casimir effect for any non-Maxwellian
fields. In order to define boundary conditions for other fields the conductor boundary
conditions are replaced and Minkowski spacetime by a manifold of the form 9t x r (i.e.,
a product space), where 9t is the real line defining the time dimension for this particular
product space and r is a flat three-dimensional manifold having any one of the
following topologies: 912 x S1, 9~ x T2, T3, 9t x K2, etc., 9t being the real line that defines
any linear space dimension (e.g., 9t = line, 9t2 = two-dimensional plane, etc.), T" being
the n-torus, K 2 the two-dimensional Klein bottle, S1 the circle, etc.
The case r = 9\2 x S1 has the closest resemblance to the electromagnetic Casimir effect,
the difference being that instead of imposing conductor boundary conditions, one
imposes periodic boundary conditions on some of the space coordinates in the three­
dimensional manifold. When imposing this topological constraint on the field theoretic
calculation of the topological Casimir effect (for linear massless fields), one finds that
the generic expression for the energy density is also pcE = - A(11c)a4, where
A= ±dr (rr.2/90), dr is the number of degrees of freedom (e.g., helicity states) per spatial
point, the plus sign holds for boson fields (giving a negative energy density) and the
negative sign for fermion fields (giving a positive energy density).
If one were to admit spin structure in the manifolds described above and the field is
spinorial, then there is another important subtlety that must be taken into account
when evaluating r::,; . However, this introduces an additional complexity involving the
relationship between the spin structure and the global structure (i.e., the configuration
space or fibre bundle) of the field in question whereby the topology not only of the base
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