From amplituhedron to associahedron

Lubos has a nice article (see this) explaining the proposal represented in the newest article by Nima Arkani-Hamed, Yuntao Bai, Song He, Gongwang Yan (see this). Amplituhedron is generalized to a purely combinatorial notion of associahedron and shown to make sense also in string theory context (particular bracketing). The hope is that the generalization of amplituhedron to associahedron allows to compute also the contributions of non-planar diagrams to the scattering amplitudes - at least in N=4 SYM. Also the proposal is made that color corresponds to something less trivial than Chan-Paton factors.

The remaining problem is that 4-D conformal invariance requires massless particles and TGD allows to overcome this problem by using a generalization of the notion of twistor: masslessness is realized in 8-D sense and particles massless in 8-D sense can be massive in 4-D sense.

In TGD non-associativity at the level of arguments of scattering amplitude corresponds to that for octonions: one can assign to space-time surfaces octonionic polynomials and induce arithmetic operations for space-time surface from those for polymials (or even rational or analytic functions). I have already earlier demonstrated that associahedron and construction of scattering amplitudes by summing over different permutations and associations of external particles (space-time surfaces). Therefore the notion of associahedron makes sense also in TGD framework and summation reduces to "integration" over the faces of associahedron. TGD thus provides a concrete interpretation for the associations and permutations at the level of space-time geometry.

In TGD framework the description of color and four-momentum is unified at the level and the notion of twistor generalizes: one has twistors in 8-D space-time instead of twistors in 4-D space-time so Chan-Paton factors are replaced with something non-trivial.

1. Associahedrons and scattering amplitudes

The following describes briefly the basic idea between associahedrons.

1.1 Permutations and associations

One starts from a non-commutative and non-associative algebra with product (in TGD framework this algebra is formed by octonionic polynomials with real coefficients defining space-time surfaces as the zero loci of their real or imaginary parts in quaternionic sense. One can indeed multiply space-time surface by multiplying corresponding polynomials! Also sum is possible. If one allows rational functions also division becomes possible.

All permutations of the product of n elements are in principle different. This is due to non-commutativity. All associations for a given ordering obtained by scattering bracket pairs in the product are also different in general. In the simplest case one has either a(bc) or (ab)c and these 2 give different outcomes. These primitive associations are building bricks of general associations: for instance, abc does not have well-defined meaning in non-associative case.

If the product contains n factors, one can proceed recursively to build all associations allowed by it. Decompose the n factors to groups of m and n-m factors. Continue by decomposing these two groups to two groups and repeat until you have have groups consisting of 1 or two elements. You get a large number of associations and you can write a computer code computing recursively the number N(n) of associations for n letters.

Two examples help to understand. For n=3 letters one obviously has N(3)= 2. For n=4 one has N(4)=5: decompose abcd to (abc)d and a(bcd) and (ab)(cd) and then the 3 letter groups to two groups: this gives 2+2+1 =5 associations and associahedron in 3-D space has therefore 5 faces.

1.2 Geometric representation of association as face of associahedron

Associations of n letters can be represented geometrically as so called Stasheff polytope (see this). The idea is that each association of n letters corresponds to a face of polytope in n-2-dimensional space with faces represented by the associations.

Associahedron is constructed by using the condition that adjacent faces (now 2-D polygons) intersecting along common face (now 1-D edges). The number of edges of the face codes for the structure particular association. Neighboring faces are obtained by doing minimal change which means replacement of some (ab)c with a(bc) appearing in the association as a building bricks or vice versa. This means that the changes are carried out at the root level.

1.3 How does this relate to particle physics?

In scattering amplitude letters correspond to external particles. Scattering amplitude must be invariant under permutations and associations of the external particles. In particular, this means that one sums over all associations by assigning an amplitude to each association. Geometrically this means that one "integrates" over the boundary of associahedron by assigning to each face an amplitude. This leads to the notion of associahedron generalizing that of amplituhedron.

Personally I find it difficult to believe that the mere combinatorial structure leading to associahedron would fix the theory completely. It is however clear that it poses very strong conditions on the structure of scattering amplitudes. Especially so if the scattering amplitudes are defined in terms of "volumes" of the polyhedrons involved so that the scattering amplitude has singularities at the faces of associahedron.

An important constraint on the scattering amplitudes is the realization of the Yangian generalization of conformal symmetries of Minkowski space. The representation of the scattering amplitudes utilizing moduli spaces (projective spaces of various dimensions) and associahedron indeed allows Yangian symmetries as diffeomorphisms of associahedron respecting the positivity constraint. The hope is that the generalization of amplituhedron to associahedron allows to generalize the construction of scattering amplitudes to include also the contribution of non-planar diagrams of at N=4 SYM in QFT framework. 2. Associations and permutations in TGD framework

Also in the number theoretical vision about quantum TGD one encounters associativity constraints leading to the notion of associahedron. This is closely related to the generalization of twistor approach to TGD forcing to introduce 8-D analogs of twistors (see this).

1. By M⁸-H duality (H=M⁴× CP₂) the scattering are assignable to complexified 4-surfaces in complexified M⁸. Complexified M⁸ is obtained by adding imaginary unit i commutating with octonionic units I_k, k=1,,..,7. Real space-time surfaces are obtained as restrictions to a Minkowskian subspace complexified M⁸ in which the complexified metric reduces to real valued 8-D Minkowski metric. This allows to define notions like Kähler structure in Minkowskian signature and the notion of Wick rotations ceases to be ad hoc concept. Without complexification one does not obtain algebraic geometry allowing to reduces the dynamics defined by partial differential equations for preferred extremals in H to purely algebraic conditions in M⁸. This means huge simplifications but the simplicity is lost at the QFT-GRT limit when many-sheeted space-time is replaced with slightly curved piece of M⁴.
2. The real 4-surface is determined by a vanishing condition for the real or imaginary part of octonionic polynomial with RE(P) and and IM(P) defined by the composition of octonion to two quaternions: o= RE(o)+ I₄IM(o), where I₄ is octonionic unit orthogonal to a quaternionic sub-space and RE(o) and IM(o) are quaternions. The coefficients of the polynomials are assumed to be real. The products of octonionic polynomials are also octonionic polynomials (this holds for also for general power series with real coefficients (no dependence on I_k). The product is not however neither commutative nor associative without additional conditions. Permutations and their associations define different space-time surfaces. The exchange of particles changes space-time surface. Even associations do it. Both non-commutativity and non-associativity have a geometric meaning at the level of space-time geometry!
3. For space-time surfaces representing external particles associativity is assumed to hold true: this in fact guarantees M⁸-H correspondence for them! For interaction regions associativity does not hold true but the field equations and preferred extremal property allow to construct the counterpart of space-time surface in H from the boundary data at the boundaries of CD fixing the ends of space-time surface.

   Associativity poses quantization conditions on the coefficients of the polynomial determining it. The conditions are interpreted in terms of quantum criticality. In the interaction region identified naturally as causal diamond (CD), associativity does not hold true. For instance, if external particles as space-time surfaces correspond to vanishing of RE(P_i) for polynomials representing particles labelled by i, the interaction region (CD) could correspond to the vanishing of IM(P_i) and associativity would fail. At the level of H associativity and criticality corresponds to minimal surface property so that quantum criticality corresponds to universal free particle dynamics having no dependence on coupling constants.

4. Scattering amplitudes must be commutative and associative with respect to their arguments which are now external particles represented by polynomials P_i This requires that scattering amplitude is sum over amplitudes assignable to 4-surfaces obtained by allowing all permutations and all associations of a given permutation. Associations can be described combinatorially by the associahedron!

   Remark:. In quantum theory associative statistics allowing associations to be represented by phase factors can be considered (this would be associative analog of Fermi statistics). Even a generalization of braid statistics can be considered.

Nima et al talk also about color structure of the scattering amplitudes usually regarded as trivial. It is claimed that this is actually not the case and that there is non-trivial dynamics involved. This is indeed the case in TGD framework. Also color quantum numbers are twistorialized in terms of the twistor space of CP₂, and one performs a twistorialization at the level of M⁸ and M⁴× CP₂. At the level of M⁸ momenta and color quantum numbers correspond to associative 8-momenta. Massless particles are now massless in 8-D sense but can be massive in 4-D sense. This solves one of the basic difficulty of the ordinary twistor approach. A further bonus is that the choice of the imbedding space H becomes unique: only the twistor spaces of S⁴ (and generalized twistor space of M⁴ and CP₂ have Kähler structure playing a crucial role in the twistorialization of TGD. To sum up, all roads lead to Rome. Everyone is well-come to Rome!

See the articles Does M 8 − H duality reduce classical TGD to octonionic algebraic geometry? and From amplituhedron to associahedron.

Addition: Marni Lee Sheppard wrote a thesis in which the notion of associahedron appeared. I remember discussions in some net group. Her motivations came from category theory. Marni had bad luck. Big boys rarely remember who proposed the idea first if she/he is not a name.

See the chapter Does M⁸-H duality reduce classical TGD to octonionic algebraic geometry? or the article From amplituhedron to associahedron.