This simple and beautiful PI talk by Freddy Cachazo is a must see! He begins with some history on the S-matrix and proceeds to whiz through twistors and MHV diagrams.

I saw this talk and thought about blogging it. It concentrates on analytic functions. From a pure density matrix point of view, it would be nicer to think of density matrices.

One gets the analytic functions by multiplying by the initial and final states in the form of spinors. The stuff between them is in the form of operators or matrices.

The idempotency equations I'm always working on correspond to taking the pole part of the S-matrix and ignoring the incoming and outgoing states.

Maybe that needs better explanation. If there is a resonance state R, that fits between incoming state I and outgoing state O, then we would write the Feynman diagram as ORRRRRRRRRRI That is, the resonance survives for some time.

The above will interfere with other similar, that is, we will end up summing over things like ORRRRRRRRRRI + ORRRRRRRRRRRI + ORRRRRRRRRRRRI + etc.

So if we solve the equation RR = R we automatically get resonances which we interpret as particles.

Cool, Carl. I guess the talk is more interesting to those who already know how amazing MHV diagrams are....

The conversation with Moffat was very funny. That is, C: There were no Feynman diagrams ... we're in the 1950s here. M: Er. Well .... Feynman diagrams go back to the 1940s. C: (shrugs) Oh, well, remember that we are now very against QFT ....

Indeed, Cachazo made it seem as if the use of complex momenta was inevitable and natural. From a purely mathematical standpoint, it very much is.

Carl mentioned the connection to pure density matrices. One goes about doing this by representing a momentum vector as a matrix bi-spinor (hep-th/0504194, pg. 5). Then our momentum is lightlike if the corresponding matrix bi-spinor has determinant zero, implying the rank is less than or equal to one. Hence, our momentum vector is lightlike when the matrix bi-spinor it can be written as the outer product of a spinor and conjugate spinor. Such lightlike bi-spinors are just pure density matrices, which are idempotent for normalized momenta.

As the MHV amplitudes (pg. 14) are written in terms of lightlike bi-spinors, they are actually amplitudes over pure density matrices in projective space. After transforming our amplitudes to twistor space, our MHV vertices in Minkowski space are mapped to lines (CP^1's) in projective twistor space (CP^3). In string theory, the lines (degree one, genus zero curves) are interpreted as instantons (hep-th/0312171, pg. 37) Twistor fields are tubes connecting two spheres (degree one instantons), yielding configurations which are topologically just deformed two-spheres themselves. The corresponding diagrams have more in common with worldsheet diagrams than the typical Feynman diagrams.

## 4 Comments:

I saw this talk and thought about blogging it. It concentrates on analytic functions. From a pure density matrix point of view, it would be nicer to think of density matrices.

One gets the analytic functions by multiplying by the initial and final states in the form of spinors. The stuff between them is in the form of operators or matrices.

The idempotency equations I'm always working on correspond to taking the pole part of the S-matrix and ignoring the incoming and outgoing states.

Maybe that needs better explanation. If there is a resonance state R, that fits between incoming state I and outgoing state O, then we would write the Feynman diagram as

ORRRRRRRRRRI

That is, the resonance survives for some time.

The above will interfere with other similar, that is, we will end up summing over things like

ORRRRRRRRRRI +

ORRRRRRRRRRRI +

ORRRRRRRRRRRRI +

etc.

So if we solve the equation

RR = R

we automatically get resonances which we interpret as particles.

Cool, Carl. I guess the talk is more interesting to those who already know how amazing MHV diagrams are....

The conversation with Moffat was very funny. That is,

C: There were no Feynman diagrams ... we're in the 1950s here.

M: Er. Well .... Feynman diagrams go back to the 1940s.

C: (

shrugs) Oh, well, remember that we are now very against QFT ....Remote KeaElvis has left the building!

Congratulations!

Indeed, Cachazo made it seem as if the use of complex momenta was inevitable and natural. From a purely mathematical standpoint, it very much is.

Carl mentioned the connection to pure density matrices. One goes about doing this by representing a momentum vector as a matrix bi-spinor (hep-th/0504194, pg. 5). Then our momentum is lightlike if the corresponding matrix bi-spinor has determinant zero, implying the rank is less than or equal to one. Hence, our momentum vector is lightlike when the matrix bi-spinor it can be written as the outer product of a spinor and conjugate spinor. Such lightlike bi-spinors are just pure density matrices, which are idempotent for normalized momenta.

As the MHV amplitudes (pg. 14) are written in terms of lightlike bi-spinors, they are actually amplitudes over pure density matrices in projective space. After transforming our amplitudes to twistor space, our MHV vertices in Minkowski space are mapped to lines (CP^1's) in projective twistor space (CP^3). In string theory, the lines (degree one, genus zero curves) are interpreted as instantons (hep-th/0312171, pg. 37) Twistor fields are tubes connecting two spheres (degree one instantons), yielding configurations which are topologically just deformed two-spheres themselves. The corresponding diagrams have more in common with worldsheet diagrams than the typical Feynman diagrams.

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