What are the calculations for Vacuum Energy? In wiki the Vacuum Energy in a cubic meter of free space ranges from $10^{-9}$ from the cosmological constant to $10^{113}$ due to calculations in Quantum Electrodynamics (QED) and Stochastic Electrodynamics (SED).
I've looked at Baez and references given on the wiki page but none of them give a clear working for how these values are derived.
Can someone point me in the right direction, as to how values like $10^{-9}$ are derived from the cosmological constant; OR $10^{113}$ due to calculations in Quantum Electrodynamics?
 A: You can understand the origin of these numbers from a simple consideration of dimensional analysis, and the cosmological data available. This keeps the answer intuitive, and any more complicated derivation will not change the answer substantially.
The first of your numbers, $10^{-9}$ Joules per cubic meter, is simply an empirical measurement in the framework of the Lamda-CDM model. Measurements of the CMB (WMAP), combined with type Ia supernovae, tell us that this is about the energy density of the universe, and that most of the energy density is in the form of dark energy. We assume that the dark energy comes from a cosmological constant $\Lambda$.
In natural units where $\hbar$ and $c$ are set equal to 1, a length is essentially an inverse energy. So in these units $\Lambda$ is about $10^{-46}$ GeV$^{4}$.
Here comes the essential point: If we consider the Planck mass to be the natural energy scale for the vacuum energy, then the ratio of the observed energy density in the cosmological constant is too small by 122 orders of magnitude (and this is the origin of the second number - it just comes from taking the planck mass to the fourth power in natural units). 
So, the fundamental puzzle is, why is $\Lambda$ so small compared to the 'natural' scale we would expect? One way out is to argue that a different energy scale other than the planck mass is what we should be comparing $\Lambda$ to.
A: The vacuum energy for a free field is the ground state energy of each field oscillator, ${1\over 2} \omega$, summed over all the modes. For a cubic periodic box of side-length L, you get
$$\sum_k {1\over 2} \sqrt{k^2+m^2}$$
Where the sum is all k's in an infinite size 3d cubic lattice where each k component is an integer multiple of $2\pi\over L$. When you make L big, this makes the k-lattice continuous, and the sum turns into an integral:
$$({L\over 2\pi})^3 \int \sqrt{k^2+m^2} d^3k $$
If you put a cutoff $\Lambda$, the result diverges as
$$ E\propto V \Lambda^4 $$
so that the energy density is proportional to the fourth power of the momentum cutoff. This integral reproduces the dimensional expectation when $\Lambda$ is the Planck length.
For interacting field theories, the vacuum energy is the sum of all vacuum loop Feynman diagrams. In the free case, the loop is just a single propagator joined to itself (this is a very degenerate Feynman diagram). The sign of Fermion and Boson loops are opposite, and Fermionic oscillators with the most natural definition of energy give an opposite sign vacuum energy in each oscillator. In a supersymmetric theory, when the Hamiltonian is presented in the form that preserves supersymmetry, the vacuum energy is zero. This is the only principle that we know today that can control the cosmological constant.
the problem is that SUSY is broken in our world at a cutoff scale of about a Tev, so the cancellations in SUSY are not exact. This means that the residual non-SUSY vacuum energy has to cancel from the scale of the Higgs (at least) to the scale of the observed cosmological constant, which is many orders of magnitude smaller.
You can't just get rid of vacuum energy by a natural statement that the vacuum has zero energy, because the vacuum in QCD (and in the Higgs mechanism) is full of crud. There is a pion condensate, a gluon condensate, and a Higgs condesate at the very least, and if you make everything cancel for our exact values of the masses of the quarks and leptons, if you change the mass of the quark, the condensate energy density changes in extremely complicated ways, so that the subtraction constant must be tuned to a magical value with no dynamical explanation.
Weinberg suggested that this is an anthropic accident--- that we need to have a low cosmological constant to evolve intelligent life. This predicts that the cosmological constant should be of the exact same order as the density of matter today, after life has evolved, but no lower, since it doesn't need to be any lower. This is what is observed, so Weinberg might be right, and there might be no explanation for the cosmological constant.
If Weinberg is right, the string vacuum that describes our universe will be very special--- it will be a non-supersymmetric vacuum with an accidentally small cosmological constant. If this is an accident with no rhyme or reason, then it will be very useful in picking out the right vacuum. We'll know we have it when it produces the right cosmological constant.
A: The first number is not vacuum energy actually. It is the energy of the cosmological repulsive field. This field is one of the forces of matter, possibly a repulsive component of gravitation or a separate force. 
