The Amplitiuhedron - mathematical device or predicted real particle

[Math Amateur]

One of the things that pervades my mental life is whether theoretical physicists are implying that the ampituhedron is a (predicted) real particle or just a helpful mathematical device?

Can anyone shed light on this question ... ?

Peter

 

[Deveno]

As I understand it (poorly), it's not even a particle, but rather an "underlying mechanism" of particles.

Strictly speaking, at present it's a mathematical abstraction, but one that may serve as a MODEL of the underlying nature of time and space. That is to say, if it's an ACCURATE model, certain features of quantum physics are consequences of geometry.

The particular features affected are called locality and unitarity. Locality essentially says (as I understand it) particle interactions are "local", they happen in a certain small region of space-time (So shooting neutrinos at the moon, shouldn't cause interactions on the opposite side of the Earth (the side that can't see the moon)). Unitarity says (as I understand it, again, poorly) that the possible particle interaction outcomes sum to one (as with a probability distribution).

One of the main "problems" with quantum physics, is that at "very local" scales (less than a Planck length) there's no way for us to know "what is actually happening". Just trying to find out changes the interactions. So it has long been suspected that there is something "hidden to us" which makes this be so.

One possible interpretation of all this (and by no means the only one) is that what we call the "universe" may actually be a certain kind of geometric object, and the properties we observe are certain "facets" of this geometric object. I do not know "how many dimensional" such a proposed object would be (possibly infinite, though). You can imagine it like this:

Suppose our universe was a line (as we saw it). The actual underlying reality, might be a circle surrounded by all its tangent lines. What we see, the nature of our line-world, is actually because of which point on the circle we are tangent to.

The actual mathematics involved in this, is, as far as I know, fairly sophisticated. It involves investigating the symmetry structure of $SU(n)$, the group of unitary $n \times n$ complex matrices with determinant 1. In particular, $n$ is sought so that the Langragian (which can be thought of something that captures the dynamical behavior of a system) has symmetry group $SU(n)$, in other words, is invariant under these transformations.

The topological nature of $\Bbb R$ endows $SU(n)$ with a topological structure, so that it forms a Lie group (the older term was "continuous group" or "infinitesimal group"). There is a well-known correspondence of Lie groups with Lie algebras, so that purely algebraic investigations can be used to investigate the "more physical" Lie group.

To give an example: the symmetry group of the weak force is posited to be $SU(2)$. The corresponding Lie algebra is 2x2 complex matrices of the form:

$\begin{bmatrix}ia&-b+ci\\b+ci&-ia \end{bmatrix}: a,b,c \in \Bbb R$

Remarkably, this can also be thought of as the set of "pure quaternions", that is:

$q \in \Bbb H: \mathfrak{Re}(q) = 0$, given a suitable choice of 2x2 anti-hermetian matrices for $i,\ j$ and $k$ (traditionally, choosing $i:a = 1,b = c = 0,\ j:b = -1, a = c = 0$ and $k: c= 1, a = b = 0$, but many other choices are possible)

As I understand it, "supersymmetry" theories seek to couple individual symmetry (or gauge) theories in larger semi-simple Lie groups, this has been done with what is considered "complete success" in the electro-weak theory, and with "largely accepted success" in the Standard Model. One of the holy grails of physics is the formulation of a large enough symmetry group that would essentially, "dictate everything". It is not known if this is even possible, but many people believe it is.

Some of these theories would require more particles than we have currently observed. I want to point out, though, that the speculation of a new physical theory, is not necessarily the same thing is positing that new particles must exist (although often they are presented this way in the media). From what I can gather (and the mathematics is too heavy going, for me, at least) the amplituhedron isn't a gauge theory itself, but a construct that can give rise to gauge theories.