Choice of Geodesic Interaction

We have seen last time that describing many particles moving on a manifold is best done by keeping on each cell a tag telling how many particles there are there. In other words, we do not evolve a sequence of maps $X_k: \{1,\dots,n\} to M$ but a divisor $X: M \to \mathbb{Z}$ . This is more effective if we deal with lots of particles.

The free geodesic flow is a permutation on the frame bundle $\pi: P \to F$ , where $F$ is the set of facets. Now, if we have a divisor $X: P \to \mathbb{Z}$ which evolves in time. If we want to have a model with interaction, we need to decide what to do on each $x \in F$ . Take the sum of all divisor values $X(x) = \sum_{p \in P, \pi(p)=x} X(p)$ at a position x, then use this number to decide how the particles on this cell $x \in F$ evolve. This needs to be simple and natural. The most obvious thing is to take a completely multiplicative function $\phi$ and have $\phi(X(x))$ decide how the dynamics on this cell works. Why multiplicative functions? Because primes in division algebras have affinities to particles. I will talk more about the quaternion case next week. It is nice for example that every Baryon or Meson has a Lipschitz prime quark there, so that it still makes sense to talk about divisors (quaternion primes are either Lipschitz primes a+ib+jc+kd or then Hurwitz primes a+ib+jc+kd +(1,1,1,1)/2 and in the later case we would not have a particle association (what is a half particle?) In any case, we can also in the quaternion division algebra see particles evolve; but now, the Baryons and Mesons can decay and combine. Also, Leptons and Hadrons can interact. It is this number theoretical “allegory” which made us choose a multiplicative function for interaction.

The most natural completely multiplicative number theoretical function that also works on the Gaussian integers and Quaternion integers is the Liouville function $\lambda(x) = (-1)^{\Omega(x)}$ , where $\Omega(x)$ counts the number of primes in x. The quaternion integers do not form a unique factorization domain but almost do by the Conway-Smith theorem. The number of primes in x is unambiguous. We can now use $\lambda(x)$ on how to turn the geodesic evolution on the cell x. The simplest way is to have every particle which has position $x \in F$ to evolve with the same rule depending on how many particles there are at x.

Here is the code which generated the animation above. There is a flag “free” if set True will have the particles not interact. The first few seconds of the clip shows the non-interacting case, where each particle moves on a geodesic. In the second case, the particles interact with the Liouville interaction rule.

free=True;d=0.3;M=100; Q=Table[0,{M},{M}];
P:=Table[If[Random[]<d && (k-M/2)^2+(l-M/2)^2<10,Random[Integer,1000],0],{k,M},{l,M}];
A123=P;A132=P;A213=P;A231=P;A312=P;A321=P; B123=P;B132=P;B213=P;B231=P;B312=P;B321=P;
AAAA=A123+A132+A213+A231+A312+A321;BBBB=B123+B132+B213+B231+B312+B321;
TX[A_]:=Transpose[RotateLeft[ Transpose[A]]];TY[A_]:=RotateLeft[ A];
SX[A_]:=Transpose[RotateRight[Transpose[A]]];SY[A_]:=RotateRight[A];
f[A_]:=Table[a=A[[i,j]];If[a==0,0,(LiouvilleLambda[a]+1)/2],{i,M},{j,M}];

DoIt:=Block[{},MA=f[AAAA];NA=(1-Q)-MA;MB=f[BBBB];NB=(1-Q)-MB;If[free,MA=1-Q;NA=Q;MB=1-Q;NB=Q];
 C123=TY[MB*B213];C231=SX[MB*B321];C312=MB*B132;C213=SX[MB*B231];C321=MB*B312;C132=TY[MB*B123];
 XB123=TX[MA*A132];XB231=MA*A213;XB312=SY[MA*A321];XB213=MA*A123;XB321=SY[MA*A231];XB132=TX[MA*A312];
 XA123=C123;XA132=C132;XA213=C213;XA231=C231;XA312=C312;XA321=C321;
 C123=TY[NB*B132];C231=SX[NB*B213];C312=NB*B321;C213=SX[NB*B312];C321=NB*B123;C132=TY[NB*B231];
 YB123=TX[NA*A321];YB231=NA*A132;YB312=SY[NA*A213];YB213=NA*A231;YB321=SY[NA*A312];YB132=TX[NA*A123];
 YA123=C123;YA132=C132;YA213=C213;YA231=C231;YA312=C312;YA321=C321;
 A123=XA123+YA123;A132=XA132+YA132;A213=XA213+YA213;A231=XA231+YA231;A312=XA312+YA312;A321=XA321+YA321;
 B123=XB123+YB123;B132=XB132+YB132;B213=XB213+YB213;B231=XB231+YB231;B312=XB312+YB312;B321=XB321+YB321;
 AAAA=A123+A132+A213+A231+A312+A321;BBBB=B123+B132+B213+B231+B312+B321;]

DynamicModule[{},Dynamic[DoIt;MatrixPlot[AAAA+BBBB,ImageSize->1000]]]

Author: oliverknill