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 , in 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 as isomorphic spaces, then you can represent spacetime as follows
and where is a complex number ().
- Case 2. 6d spacetime. Let me define as isomorphic spaces, then you can represent spacetime as follows
and where is a quaternion number , with .
- Case 3. 10d spacetime. Let me define as isomorphic spaces, then you can represent spacetime as follows
and where is
is an octonion number with .
Challenge final questions for you:
- Is this construction available for different signatures?
- 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.
- Twistor/supertwistor models of spacetime.
- Super Yang-Mills theories.
- Brane theories.
- Understanding the Hurwitz theorem.
- Black hole physics.