Scattering Amplitudes For All Masses together with Spinsa few days ago. They claim to arrive at something that may move considered remarkable: to generalize the spinor-indices-based uprising inwards the scattering aAmplitude manufacture of the previous xv years to the representative of particles of whatsoever volume together with spin, together with to deduce approximately properties of all possible particle theories out of their novel formalism.
Is it possible? Does it work? What tin plough over the sack they learn?
First, they stay restricted to the representative of on-shell, i.e. scattering amplitudes, non full general off-shell, i.e. Green's functions. They direct keep a cute self-motivating semi-heuristic declaration why they don't lose whatsoever generality past times this constraint: the actual off-shell amplitudes are beingness experimentally measured past times the analysis of approximately on-shell scattering that involves the particles equally good equally approximately novel really heavy particles, namely the detectors together with other apparatuses.
Nice. I guess that the numbers showed on the apparatuses' displays must move considered equally labeling dissimilar particle species, non only polarizations of spin. If your Geiger-Müller counter shows "5" at the get-go together with measures something together with shows "6" at the end, it was a scattering inwards which the "Geiger-Müller-counter-type-5 particle species" collided alongside approximately modest particles, got annihilated, together with produced a similar large "*-6 counter" particle. Cute. ;-)
Second, the massless spinor-indices-based minirevolution depended on the possibility to write a massless momentum vector as\[
p^\mu = \sigma^{\mu}_{\alpha \dot\alpha} \lambda^\alpha \tilde \lambda^{\dot \alpha}.
\] Because the \(2\times 2\) matrix is written equally a tensor production of 2 vectors, its rank is only ane together with the determinant is thence zero. But the determinant is \(p^\mu p_\mu\) past times approximately simple calculus together with then the particle has to move massless.
How arrive at you lot write a to a greater extent than full general momentum which is timelike? Well, ane \(\lambda \tilde \lambda\) isn't plenty but the inwardness of 2 such products is enough: a time-like vector may move written equally a inwardness of 2 light-like ones. This generalization has been considered past times numerous people I direct keep met – but they never got equally good far. Arkani-Hamed, Huang, together with Huang are mature together with hard-working researchers inwards this area, however, together with then they didn't only appear all the massive generalizations of the rules to move straightforward together with autumn into their laps. They were intensely thinking together with deriving together with they did derive approximately things.
So they write the amplitudes inwards the spinor variables. Influenza A virus subtype H5N1 particle alongside spin \(S\) adds \(2S\) spinor indices to the scattering amplitude, a symmetrization may move assumed inwards the irreducible case. There is approximately ambiguity inwards the alternative of the spinors \(\lambda^I_\alpha\) together with \(\lambda^I_{\dot \alpha}\) for a particle, \(I=1,2\) represents the index needed because nosotros inwardness 2 products of spinors. The time-like momentum (a non-singular, rank-two matrix alongside 2 2-valued spinor indices) may move used to convert dotted together with undotted indices for a unmarried particle to each other.
OK, together with then their starting dot is to imagine that all the scattering amplitudes inwards a theory alongside particles of full general spins together with masses are beingness rewritten inwards damage of diverse rational functions of products of the spinor-index-based variables. The Lorentz invariance constrains what is allowed. Also, the amplitudes may direct keep diverse singularities together with Nima together with his collaborators direct keep gained an incredible amount of sense inwards figuring out which singularities may move acquaint inwards the amplitudes, which can't, together with and then on.
In this massive case, they were only going to rederive similar conclusions nearly approximately to a greater extent than full general Ansätze which include a larger release of spinors and/or indices. At the end, they seem confident that the switch to the massive representative doesn't cripple the substitution methods that worked inwards the massless case. Massive particles don't request the estimate invariance for consistency because the petty grouping is \(SU(2)\) together with entirely relates "positive-norm" polarizations.
But approximately of the constraints move to a greater extent than intense if you lot switch from the massless realm to the massive one.
Using their novel formalism, they review approximately of the full general theorems together with wisdoms nearly the allowed spins together with masses of particles. Influenza A virus subtype H5N1 coupling of iii spin-1 particle is impossible. Yang's theorem: a massive spin-1 particle cannot decay to 2 photons: derived using a novel formalism. With gravity (spin-two massless fields), massless higher-spin particles are impossible. The Weinberg-Witten theorem, rederived.
More cutely together with beyond quantum plain theory, they inquire whether a massive particle of spin exceeding 2 may move elementary. The describing word "elementary" agency that it tin plough over the sack be inwards isolation – together with then its volume may move parameterically separated from the masses of all other particle species. The answer turns out to move No. They acquire it past times studying the \(E\to \infty\) or, equivalently, \(m\to 0\) bound of the scattering aAmplitude involving such a hypothetical simple particle. Some singularity analogous to the massless particles' singularities materializes inwards the scattering aAmplitude together with that singularity may move used to struggle that an additional particle species whose volume is comparable to the same \(m\) has to exist. The master copy ane couldn't be inwards isolation.
Weakly-coupled string theory is compatible alongside this conclusion, of course, because it gives you lot a whole infinite tower of massive particles. You can't cherry-pick them ane past times one. You either direct keep to bring the whole bundle that string theory gives you, or you lot direct keep to kicking the bucket (or at to the lowest degree close up). One may run across that Arkani-Hamed together with Huang squared gave a novel derivation of the qualitative holding of string theory. With approximately optimism, ane could struggle that past times adding a few to a greater extent than derivations similar that, ane could derive that "all string theory is mathematically forced upon us" past times pure mathematical idea – an analysis of the scattering amplitudes written inwards their spinor-based formalism.
The massless twistor minirevolution already had lots of odd expressions, lots of indices, geometric shapes of large dimensions inwards spaces whose dimensions were fifty-fifty higher – equal to products of approximately numbers. It was already complicated. To play the game started past times this paper, you lot request to swallow an additional collection of indices – \(I=1,2\) for each particle that betoken the 2 products of spinors you lot request to depict the momentum, together with the corresponding increase of possible forms of the amplitudes. They acquire into appear less terrifying than it is past times using bold fonts for massive variables together with suppressing approximately petty grouping indices.
You shouldn't appear this manufacture to technically simplify, at to the lowest degree non inwards a foreseeable future. But its applicability is almost for sure beingness expanded together with equally they (and I) mentioned, in that place are signs that they tin plough over the sack derive approximately statements that move beyond the listing of known theorems inwards quantum plain theory, that actually move beyond the green quantum plain theory thinking itself. Similar methods are actually applicable to the S-matrix of plain theories alongside "infinitely many species" together with string theory is a representative. The stringy S-matrix is complicated, involves the ratios of gamma-function-like entities, together with it must display lots of exceptional features relatively to "random functions of a similar type" that follow from the consistency of string theory.
Their formalism may move a tool to brand approximately or all of these features "comprehensible" together with accessible to a straightforward, albeit inwards no trend technically easy, analysis.
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