The Born-Infeld action is a sort of "square root" of the standard action term. This has a number of features which places gauge field actions on a same polynomial or linear footing with Dirac fields. This is then used to study Hawking-Moss tunneling physics.
Tunneling usually involves how a quantum particle may traverse a tunneling region with a potential energy larger than the kinetic energy of the particle. The Schrodinger equation (SE)
iħ∂ψ/∂t = (ħ^2/2m)∂^ψ/∂x^2 V(x)ψ
in one dimension is the classical example in Mertzbacher and other texts. For a stationary phase ψ(x,t) = ψ(x)exp(-iEt/ħ) and we have a basic position dependence with ψ(x) ~ exp(ikx), the SE is easily seen as
Eψ = (ħk)^2/2mψ V(x)ψ,
And the solution for the wave vector k or momentum p = ħk is
p = sqrt{2m}sqrt(E - V(x)).
The tunneling probability may be explicitly computed by knowing the form of V(x) and working out boundary conditions, which is not conceptually difficult but a bit tedious to work through. The form of the momentum p here though is imaginary if V(x) is larger than E, and the tunneling probability is greater than zero. Now the form of the wave function with this imaginary p is of the form ψ ~ exp(-|p|x/ħ), which for the magnitude |p| very large (equivalently large V(x)) is a rapidly dropping to zero exponential. So we don't expect a significant tunneling process.
If the potential is very large at its peak V_{max} ~ 2mc^2 for m the mass of an electron there is a probability that the e-e^ pair created here will annihilate the e^- at one side of the potential with the e^ and the pair generated e^- escapes to the other side. Since this is a quantum process then what ever information is carried by the initial e^- is the same as on the e^- which has tunneled through. This is a sort of resonance phenomenon.
So what does this have to do with cosmologies and the landscape? Hawking, Halliwell and others proposed a version of the "no-boundary cosmology" where a spacetime cosmology with a particular arrow of time and a CPT violation (say left handed) is mirrored by another cosmology with an oppositely directed arrow of time and a CPT violation which is opposite (say right handed). So we might think of these as a cosmology and "anti-cosmology" with opposite quantum numbers or topological indices and ..., all which make things cancel out to zero. So we then have a huge landscape with a large potential barrier. The basic Friedmann-Lemaitre-Robertson-Walker metric has a tunneling barrier, which is a local aspect of the more general landscape potential barriers. So the symmetrical portion of a cosmology can be then this curious anti-cosmology.
Now consider a region near a black hole singularity. There the tidal forces of gravity are enormous and a wave function is then squeezed. The phase space volume it occupies is "squashed," and this means its uncertainty in certain directions becomes very large while the conjugage momenta uncertainties becomes very small. As such a patch or region near the singularity is sufficiently quantum uncertain that it may become lost in this quantum noise. In effect this patch and the quantum vacuum energy it contains has quantum tunneled out of the universe which contains this black hole, or it "ventures" into this potential hill. For a large potential hill there are virtual quantum pairs of these cosmologies (universe plus anti-universe) or a virtual biverse. Will this little patch or bubble of vacuum energy near the singularity has some probability of annihilating with the anti-universe which then lets the virtual universe escape beyond the potential well. This patch or virtual bubble of spacetime then becomes the "seed" for a nascent cosmology.
The Born-Infeld action is the machinery one can use to examine this type of tunneling process. The square root in the computation of the momentum above is generalized into a type of action which has a square root of ordinary action or Hamiltonian (energy) terms.
Cheers LC
