LOG#230. Spacetime as Matrix.

Surprise! Double post today! Happy? Let me introduce you to some abstract uncommon representations for spacetime. You know we usually represent spacetime as “points” in certain manifold, and we usually associate points to vectors, or directed segments, as X=X^\mu e_\mu, in D=d+1 dimensional spaces IN GENERAL (I am not discussing multitemporal stuff today for simplicity).

Well, the fact is that when you go to 4d spacetime, and certain “dimensions”, you can represent spacetime as matrices or square tables with numbers. I will focus on three simple examples:

  • Case 1. 4d spacetime. Let me define \mathbb{R}^4\simeq \mathbb{R}^{3,1}=\mathbb{R}^{1,3}\simeq \mathcal{M}_{2x2}(\mathbb{C}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(1)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^3& x^1+ix^2\\ x^1-ix^2& x^0-x^3\end{pmatrix}=\begin{pmatrix} x^0+x^3& z\\ \overline{z}& x^0-x^3\end{pmatrix}}\end{equation*}

and where z\in\mathbb{C}=x^1+ix^2=x^1+x^2e_2=\displaystyle{\sum_{j=1}^2}x^je_j is a complex number (e_1=1).

  • Case 2. 6d spacetime. Let me define \mathbb{R}^6\simeq \mathbb{R}^{5,1}=\mathbb{R}^{1,5}\simeq \mathcal{M}_{2x2}(\mathbb{H}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(2)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^5& x^1+ix^2+jx^3+kx^4\\ x^1-ix^2-jx^3-kx^4& x^0-x^5\end{pmatrix}=\begin{pmatrix} x^0+x^5& q\\ \overline{q}& x^0-x^5\end{pmatrix}}\end{equation*}

and where q\in\mathbb{H} is a quaternion number q=x^1+ix^2+jx^3+kx^4=x^1+x^2e_2+x^3e_3+x^4e_4=\displaystyle{\sum_{j=1}^4}x^je_j, with e_1=1.

  • Case 3. 10d spacetime. Let me define \mathbb{R}^{10}\simeq \mathbb{R}^{9,1}=\mathbb{R}^{1,9}\simeq \mathcal{M}_{2x2}(\mathbb{O}) as isomorphic spaces, then you can represent spacetime X^\mu e_\mu=X as follows

(3)   \begin{equation*} \boxed{X=\begin{pmatrix} x^0+x^9& x^1+\sum_jx^je_j\\ x^1-\sum_jx^je_j& x^0-x^9\end{pmatrix}=\begin{pmatrix} x^0+x^9& h\\ \overline{h}& x^0-x^9\end{pmatrix}}\end{equation*}

and where h\in\mathbb{O} is

h=\displaystyle{\sum_{j=1}^8}x^je_j=x^1+x^2e_2+x^3e_3+x^4e_4+x^5e_5+x^6e_6+x^7e_7+x^8e_8 is an octonion number with e_1=1.

Challenge final questions for you:

  1. Is this construction available for different signatures?
  2. Can you generalize this matrix set-up for ANY spacetime dimension? If you do that, you will understand the algebraic nature of spacetime!

Hint: Geometric algebras or Clifford algebras are useful for this problem and the above challenge questions.

Remark: These matrices are useful in

  • Superstring theory.
  • Algebra, spacetime algebra, Clifford algebra, geometric algebra.
  • Supersymmetry.
  • Supergravity.
  • Twistor/supertwistor models of spacetime.
  • Super Yang-Mills theories.
  • Brane theories.
  • Dualities.
  • Understanding the Hurwitz theorem.
  • Black hole physics.
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