The MHV ideas are concerned, typically, with scattering amplitudes of gluons in Yang Mills theories. Most of the foundational work has been done with $\mathcal{N}=4$ supersymmetric Yang Mills theory, though I believe there have been extensions beyond this.
The problem addressed is that you have n gluons meeting at a vertex, some incoming, some outgoing and you want to calculate the scattering amplitude. The assumption is that we wish to look at very high energy processes, so the gluons can be effectively treated as massless. Therefore, to specify an incoming or outgoing gluon we need only give its (null) momentum and its helicity (we should also need color, but this can be effectively put to one side by means of color ordering).
Computing the amplitude the traditional way using Feynman diagrams rapidly becomes intractible as the number of gluons considered increases (see slides 4 and 5 for the five gluon case in this talk by Zvi Bern). The answer was conjectured (I think) by Parke and Taylor and was an extremely simple formula:
$$A(1+, 2+, ....j-,...k-,...n+) = \frac{\langle i,j \rangle^4}{\prod_{k=1}^{n}\langle k, k+1 \rangle}\delta^4(\sum_{k=1}^{n}\lambda^A_k{\tilde{\lambda}}^{A'}_k)$$
Here the lambdas are Weyl (two) spinors. A null vector $p^a$ can be written as a product of an unprimed and complex conjugate primed spinor. $p^a = \lambda^A \bar{\lambda}^{A'}$. Here the notation $\langle i,j \rangle := \epsilon_{AB}\lambda_i^A \lambda_j^B$ is used, where $\epsilon_{AB}$ is the antisymmetric two - spinor. The formula gives the amplitude for n gluons, two of which (j and k in this example) have the opposite helicity from the others. (The cases where none or one of the helicities is different have vanishing amplitudes).
The Parke Taylor formula was verified, for smallish numbers of gluons by direct calculation.
If you go away from the MHV criterion (but stay at tree level), and have, say, 3+ and 3- helicity gluons scattering, it came as a great surprise, that there was STILL a way to write the scattering amplitude in a neat and concise form. This came about through the use of the BCFW recursion relations. Here, the MHV amplitudes are treated as building blocks and connected together in various ways.
The interesting thing about these scattering amplitude formulas is that they exhibit lots of symmetries. They have a conformal symmetry - the conformal group acts on them in a natural way. They also have "dual conformal symmetry" - if you take the gluon momenta, they add up to zero (obviously). If you represent this by placing the momentum vectors head-to-tail in momentum space and label the points where the vectors meet, then, there is a conformal group action on THIS space too. So the big question is - where are all these symmetries coming from?
It would be instructive if the problems could be reformulated in a way that makes these symmetries explicit from the outset. One clue about this was given when Witten reformulated the MHV amplitude problem in twistor space. When this was done, for the tree level amplitudes, the twistors corresponding to the gluons are found to lie on a line ($\mathbb{C}P^1$) in twistor space. As is well known in twistor lore, lines in twistor space are parametrized by points in spacetime. Here the point in spacetime is the scattering vertex. Other scattering amplitudes, including ones with loops, corresponded to the cases where the gluons corresponded to twistors lying on curves of other genus in twistor space. In this context, a key thing about twistor space is that it has a natural conformal group action. So performing computations on twistor space is going to make conformal symmetry manifest at each step, and you're naturally going to end up with conformally symmetric amplitudes.
For a reasonably up to date picture of these "twistor-inspired" reformulations of the scattering problems see here (I only understand about 10% of it).
Getting to your question - about why these methods work - well, according to Nima Arkani-Hamed, it's because the conventional route via Feynman diagrams forces manifest spacetime locality and unitarity at each step in the calculation. This comes at the expense of making the conformal invariance manifest. A more natural way for these gluon scattering calculations to be performed is to make manifest the (dual super-) conformal invariance during the calculation steps, and just check that unitarity is respected by the result. Indeed his view is that this is indicative that, ultimately, when we find the right way to formulate physics, the spacetime picture will be emergent rather than fundamental, hence his slogan "spacetime is doomed".
I believe these techniques (MHV vertices/BCFW recursion relations) have even been used to compute backgrounds for the LHC experiments, so they're of much more than just academic interest. The feeling is that if we understood better WHY they work, we'd have a big arrow pointing towards a useful reformulation of physics, and this arrow is pointing away from spacetime as the framework of choice.
This post imported from StackExchange Physics at 2014-03-22 17:21 (UCT), posted by SE-user twistor59
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