Lubos wrote a commentary about a new article of
Nima Arkani Hamed et al about the positivity of the amplitudes in in
the amplituhedron. Nima et al argue using explicit calculations that
positivity could be a quite general property: as if the amplitudes were
analogous to generalized volumes of higher-dimensional polyhedra.
The Grassmannian twistor approach considers also positive Grassmannians.
This is not the same thing but most be closely related to the positivity
of amplitudes. I recall that the core idea is however that one considers
higher-D analogs of polygons. Simple representative for positive space
polygon would be a triangle bounded by positive x- and y-axis and the
line y=-x. x and y coordinates are positive.
What is of course
certain is that the entire scattering amplitude cannot be positive since
it is complex number. Rather, it must decompose into a product of
"trivial" part determined by symmetries and non-trivial part which for
some reason must be positive. What does this mean? Lubos considers
this question in his posting: the idea is roughly that the real amplitude
in question is an exponential and this guarantees positivity. Bundle
theorist might speak about global everywhere non-vanishing section of
vector bundle having geometric and topological meaning in algebraic
geometry. Below I try to interpret positivity in terms of number
Does number theoretical universality
Number theoretical universality of physics
is one of the key principles of TGD and states that real physics must
allow algebraic continuation to p-adic physics and vice versa. I suggest
that number theoretical universality requires the positivity.
- The p-adicization of the amplitudes requires positivity. The
"non-trivial" factor of the amplitude is mapped to its p-adic counterpart
by a variant of canonical identification mapping p-adics to
non-negative reals and vice versa and is well-defined only for
non-negative real numbers. Hence positivity. This condition could also
explain why positive Grassmannians are needed: the preferred coordinates
must be positive in order to have well-defined p-adic counterparts and
The basic reason for positivity is that p-adic
numbers are not well-ordered. When one has two p-adic numbers with the
same p-adic norm, one cannot tell which of them is the larger one. This
makes it impossible to tell whether a given p-adic number is positive or
negative and one cannot talk about p-adic boundaries. p-Adic line segment
has no ends. As a consequence, definite integral is very difficult notion
p-adically although the notion of integral function can be defined. p-Adic
counterparts of differential forms are also far from trivial to define.
- What about the symmetry determined parts of amplitudes, which are
typically analogs of partial waves depending on angular coordinates and
are complex and cannot be positive? Also here one encounters a technical
problem related to another conceptual challenge of p-adicization
program: the notion of angle is not well-defined in p-adic context and
one can talk only about discrete phases. More precisely, one can
define exponential function exp(x) if x has p-adic norm smaller than one
but it does not have the properties of the ordinary exponential function
(it has p-adic norm 1 for all values of x). One can define also
trigonometric functions with the help of exp(ix) for p mod 4=3 formally
but the trigonometric functions are not periodic. Something is missing.
In the case of ordinary trigonometry the only way out is to
perform algebraic extension of p-adic numbers by adding roots of unity
representing phases associated with corresponding angles: phases replace
angles. For instance, Un=exp(i2π/n) exists in an algebraic
extension of p-adic numbers. Angle degrees of freedom are discretized.
In the case of hyperbolic geometry, one can add roots of e to
obtain p-adic counterparts of e(1/n) (note that ep
is ordinary p-adic number so that these extensions are finite-dimensional
algebraically and e is completely unique real transcendental in that it is
algebraic number p-adically!): this extension allows to get the
counterparts of ordinary exponential functions in extension.
One can also multiply the points of discretization by p-adic numbers of norm
smaller than 1 (or some negative power of p) to get something as near as
possible to continuum. Hence discretization in both hyperbolic and
trigonometric degrees of freedom by algebraic extension solves the
problems quite generally for amplitudes defined in highly symmetric
spaces. In Grassmann twistor approach one indeed considers projective
- Consider as an example the p-adic counterpart of Euclidian 2-space
with coordinates (ρ,φ). ρ is non-negative radial coordinate
and has p-adic counterpart obtained by canonical identification or its
variant. The values of φ are replaced with discrete phase factors
Un characterizing the values of φ coordinate. One has a
collection of n rays emanating from origin instead of entire plane. One
obtains infinite number of variants of E2 labelled by n
characterizing the angular resolution.
The p-adicization of
Cartesian representation of real Euclidian 2-space defined using (x,y)
coordinates would give only the first quadrant since negative x and y have
no p-adic counterparts. In both cases cognitive representations lose a
lot of information for purely number theoretical reasons. Cognitive analog
of Uncertainty Principle is suggestive.
- Obviously General Coordinate Invariance is broken at the level of
cognition which is actually not so surprising after all since the worlds
in which mathematician has chosen to used Cartesian resp. spherical
coordinates must differ in some delicate manner! Of course, the
resulting discretized spaces are very different.
How should one p-adicize?
The p-adicization of various spaces and amplitudes is needed. p-Adicization means that
one assigns to a real (or complex) number a p-adic variant by some rule.
In the case of trigonometric and hyperbolic angles one can use
discretization and algebraic extension but what about other kinds of
coordinates? There are two guesses.
The resolution of the problems is a compromize based on the use of two
cutoffs. In some length scale range above p-adic UV cutoff
LUV and scale L a direct correspondence between common
rationals is assumed. Between L IR cutoff LIR canonical
identification is used. Outside the range [LUV,
LIR] there is no correspondence and conditions like smoothness
dictate the details at both sides.
- Consider only rationals or their algebraic extension and maps
only them to their p-adic counterparts in the needed extension of p-adic
numbers. This correspondence is however extremely discontinuous since real
numbers which are arbitrary distant can be arbitrary near p-adically and
vice versa. What is nice that this map respects symmetries suggesting
that one has symmetries below some rational cutoff defining measurement
resolution. This conforms with the general philosophy about measurement
resolution realized in terms of inclusions of hyper-finite factors
realizing measurement resolution as analog of dynamical gauge symmetry.
- Use canonical identification mapping p-adic numbers to p-adic numbers
by canonical identification: x= ∑ xnpn →
∑ xnp-n is the first option. It indeed works
only for non-negative real numbers: hence positivity! Canonical
identification is continuous but does not respect differentiability nor
symmetries. Direct identification via common rationals in turn does not
p-Adic space-time surfaces
as cognitive maps of real ones and real space-time surfaces as correlates
for realized intentions
One wants to talk about real
topological invariants also in p-adic context: p-adic space-time surface
should be a kind of cognitive representation of real space-time surface.
- Space-time surfaces are extremals of Kähler action (real/p-adic)
but have a discrete set of common rational points. The notion of p-adic
manifold for which p-adic regions are mapped to real chart leafs (rather
than to p-adic ones!) formalizes this concept and allows to avoid the
problem that p-adic balls are either disjoint or nested so that the usual
construction of manifold does not make sense. The problem that there
exists an endless variety of coordinate choices and each would give
different notion of p-adic manifold.
- The cure comes from space-time as a surface property, and from
symmetries allowing to induce manifold structure from the level of
imbedding space the level of space-time surfaces. In TGD imbedding space
is M4 × CP2. CP2 allows very
natural p-adicizations by using complex coordinates transforming linearly
under maximal subgroup of isometries and one obtains discrete variants of
coset space with points labeled by phases in the algebraic extensions of
p-adic numbers. One can also have a generalization in which each
discrete point corresponds to a continuum of p-adic units.
case of M4 one has more options but if one requires that the
coordinate which is mapped by canonical identification to its real
counterpart, the natural choices is the cosmic coordinates assignable to
the future or past light-cone of causal diamond. Light-cone proper time
would correspond to non-negative coordinate and the remaining coordinates
would be ordinary angles hyperbolic angle and mappable to phase factors
and real exponential which exists if one introduces finite number of roots
of e and discretizes the hyperbolic angle. Note that p-adic causal
diamond (CD) is p-adically non-trivial manifold requiring two chart
leafs and this might deeply relate to the fact that state function
reductions occur to either boundary of CD.
- Surface property implies that the correspondence between real and
p-adic space-time surfaces is induced from that between the corresponding
imbedding spaces and thus dictated to high degree by symmetries.
Preferred imbedding space coordinates and their discretizations induce
coordinates and discretizations at space-time surfaces so that there is a
huge reduction in the number of different but cognitively non-equivalent
- This could make possible in practice to define the notion of p-adic
space-time surface as a cognitive map of real space-time surface and real
space-surface as a realization of intention represented by p-adic
space-time surface. The real extremal of Kähler action is mapped
to p-adic one only in a given resolution. A subset of discrete points of
the space-time surface is mapped to p-adic ones and vice versa and this
discrete skeleton is continued to a p-adic extremal of Kähler action.
The outcome is interpreted as cognitive representation or its inverse
(transformation of intention to action) and need not be unique. The
mapping taking p-adic skeleton to real one is up to some cutoff pinary
digit just identification along common rationals respecting symmetry and
above that canonical identification up to the highest allowed pinary
To sum up, number theoretic universality condition for scattering
amplitudes could help to understand the success of twistor Grassmann
approach. The existence of p-adic variants of the amplitudes in finite
algebraic extensions is a powerful constraint and I have argued that they
are satisfied for the polylogarithms used. Positive Grassmannians and
positivity of the amplitudes might be also seen as a manner to satisfy
For background see the chapter
Some fresh ideas about twistorialization of TGD or
Positivity of N=4 scattering amplitudes from number theoretical universality?.