Two space and two time dimensions - simplified

 Whenever we start working on some project, at first everything we do, seems to be, and often is, very complicated. It is only with time, and with experience, we learn how to simplify our task, sometimes in an extraordinary way. We find shortcuts, we invent tricks, we learn from other experienced people whom we meet due to our unceasing efforts.

I asked AI to provide an example illustrating my preamble above, and here is the result:

"Historical Example: Richard Feynman and the Manhattan Project

Feynman simplified one group’s approach

When Richard Feynman joined the Manhattan Project at Los Alamos during World War II, he was of the brightest minds of the century. At first, the work was incredibly complicated: the project involved calculations of neutron diffusion, nuclear reactions, and bomb design—concepts that were at the cutting edge of physics, and far from straightforward.

Feynman found himself and others bogged down with repetitive, error-prone hand calculations. Every problem seemed daunting. But over time, through relentless experimentation and observation, Feynman began finding clever shortcuts.

One breakthrough came when he noticed that some of the mathematical tables used for calculations were riddled with mistakes. Instead of doing everything manually, he started looking for patterns in the errors and designed faster ways to check and simplify the calculations. He even taught himself how to use mechanical computing devices better than the engineers assigned to operate them, often streamlining the whole process.

In a particularly telling anecdote, Feynman simplified one group’s approach to neutron diffusion by realizing they were unnecessarily solving a full partial differential equation when an approximation and symmetry argument would give an answer almost instantly. This insight not only saved time but brought clarity to the underlying physics.

What had started out as complex and overwhelming became manageable—even elegant—through experience, effort, and creative shortcuts."

In a series of previous posts we were discussing the space R^2,2, two dimensional space and two-dimensional time, to study the conformal compactification of R^1,1, a toy spacetime with only one space dimension. We played with the Clifford algebra Cl(2,2), and we have a somewhat strange doubling. Matrices representing R22 vectors were block-off diagonal, matrices representing Spin(2,2) were block diagonal. They contained a lot of zeros! Why do we need all these zeros? Can't we get the desired result without all these zeros? The Eureka came onto me only yesterday. I checked if my discovery has no errors, and it seems that all works as desired. At the same time we are getting a new insight into the internal machinery of the whole structure. Which makes me happy. Doing all this I recalled one of the songs that I like. It has these words.

And happiness is close, happiness is far.
It is difficult and easy to find

It sounds much better in the original Russian (Роксана Бабаян):

А счастье близко, счастье далеко.
Его найти трудно и легко.

You can find the song online.

There is a movie "Maestro". We have another annoying doubling there: "American composer Leonard Bernstein (Bradley Cooper) lives a double life." The same with our Cl(2,2). We do need all this suspicious doubling. We do not need all these unnecessary zeros. So let us simplify everything from scratch. What we need is R^2,2 and Spin(2,2) isomorphic to SL(2,R) x SL(2,R).

The solution.

R^2,2 is already a Clifford algebra! Namely it is the Clifford algebra Cl(2,0) aka Cl(2). Excellent notes  "Clifford algebra, geometric algebra, and applications" by Douglas Lundholm and Lars Svensson provide the hint in Exercise 2.5, p. 13:

Exercise 2.5. Find an R-algebra isomorphism
G (R²)) → R^{2 × 2} = { 2 × 2 matrices with entries in R } .
Hint: Find (preferably very simple) 2 × 2 matrices γ¹, γ² which anti-commute and satisfy γ₁² = γ₂² = 1_{2 × 2} .

Note. Sec. 2.2 of this paper describes "Combinatorial Clifford algebra" - the concept I was not aware of before.

We will do it in details in the next post.

P.S. 18-07-25 18:19 The next notebook will use the calculation in this Mathematica notebook:

(* Cl(2,0) *)

e1={{1,0},{0,-1}};

e2={{0,1},{1,0}};

e12=\[Omega]=e1.e2;

Id=IdentityMatrix[2];

e3=-Id;

e4=e12;

x=x1 e1+x2 e2+x3 e3+x4 e4;

y=y1 e1+y2 e2+y3 e3+y4 e4;

(* Grade involution *)

\[Alpha][z_]:=\[Omega].z.Inverse[\[Omega]];

(* Reversion *)

t[z_]:=Transpose[z];

(* Clifford conjugation *)

\[Nu][z_]:=Transpose[\[Alpha][z]];

(* Quadratic norm *)

\[CapitalDelta][z_]:=Simplify[\[Nu][z].z];

Simplify[\[CapitalDelta][x.y]-\[CapitalDelta][x].\[CapitalDelta][y]]

TrueQ[Simplify[\[CapitalDelta][x.y]]==Simplify[\[CapitalDelta][x].\[CapitalDelta][y]]]

Simplify[\[CapitalDelta][x]]

P.S. 20-07-25 16:06 Here is the code for the wonderful free computer algebra system Reduce, doing the same, except the last three lines. Verifying the multiplicative property of Delta will be a good Exercise for my Readers in using Reduce. Learn by examples. Here is one. First there are definitions. The last four lines are examples of the results of calculations based on these definitions.

e1 := mat((1, 0), (0, -1));

e2 := mat((0, 1), (1, 0));

e12 := e1 * e2;

omega := e12;

id := mat((1,0),(0,1));

e3 := id;

e4 := e12;

invomega := -omega;

xmat := x1*e1 + x2*e2 + x3*e3 + x4*e4;

ymat := y1*e1 + y2*e2 + y3*e3 + y4*e4;

procedure gradeinvolution(z); 

begin

return omega*z*invomega;

end;

procedure reversion(z); 

begin

return tp(z);

end;

procedure cliffconj(z); 

begin

return reversion(gradeinvolution(z));

end;

procedure delta(z); 

begin

zc:= cliffconj(z);

return zc * z;

end;

gradeinvolution(xmat);

reversion(xmat);

cliffconj(xmat);

delta(xmat);

end;

On Windows PC save the code in a file, for instance  as C:\cl2.red. After installing the software, open CSL Reduce program. Editor window will open. Type there

in "c:\cl2.red";

Selecting your file from the File menu will give the same result it will write the "in" line for you. Press Enter. The file will open and will be executed. Results of calculations will be displayed. New input line will appear. Write a new command, for instance

x*y;

Press Enter to display the result. Here x and y are matrices, so the result will be the matrix product of x and y. Have fun. I will be using lot of Reduce in the future. It is an old, but absolutely fantastic free software! You can find on the web  guides with examples, for using Reduce, in English and in Russian.

Note: While Mathematica requires "Simplify" command to simplify the raw results, Reduce does it automatically. Mathematica can display 2x +x as s result. Reduce will display 3x. Reduce has even special Clifford algebra package, but we will not need it. We will simply play with matrices.

d easy to find