Abstract
We show that via the Grassmann-Pl\"ucker relations, the various apparent unrelated concepts, such as duality, matroids, qubits, twistors and surreal numbers are, in fact, deeply connected. Moreover, we conjecture the possibility that these concepts may be considered as underlying mathematical structures in quantum gravity.
Highlights
Frontiers in PhysicsThe pair (A, +) determines a dual structure through the map ∗ : A → A if satisfies the following axioms: Duality, Matroids, Qubits, Twistors, Surreal Numbers (I) ∗∗A = A (∀A ∈ A)
We show that via the Grassmann-Plücker relations, the various apparent unrelated concepts, such as duality, matroids, qubits, twistors, and surreal numbers are, deeply connected
It is a fact that the duality concept is everywhere in both mathematics and physics
Summary
The pair (A, +) determines a dual structure through the map ∗ : A → A if satisfies the following axioms: Duality, Matroids, Qubits, Twistors, Surreal Numbers (I) ∗∗A = A (∀A ∈ A). The two axioms (III) and (IV) are similar to the definition of a field in number theory For these reasons one it is straightforward to verify that the integer Z and the real numbers R are dual structures. It turns out that (16) is true for a general completely antisymmetric object F (d-form) when its dual is defined in terms of the ε-symbol. Let us explain how the Grassmann-Plücker relation (7) is connected with qubit theory [see [23] and references therein] For this purpose consider the general complex state | ψ >∈ C2N.
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