By S. T Hu

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**Example text**

In many structures, the position dimension s = 3 is not distinguished and can be generalized to any natural number Ss ∼ = R, = Rs . In contrast to time T ∼ the transfer of the natural R-structures to positions with s > 1 is not obvious and, in general, not unique. Position is related to the real numbers by a scalar product (metric), a symmetric bilinear form associating a real number to two position vectors with the condition that nontriviality of a position translation is equivalent to a strictly positive length (norm): σ( , ) : S × S −→ R, σ(x, y) = σ(y, x) = xa yb σ ab , x 2σ = σ(x, x) > 0 ⇐⇒ x = 0.

2. , in position space. The related isomorphism uses the scalar-product-induced position-momentum isoδ morphism S ↔ ST (for simplicity σ ∼ = δ = 13 ), concatenated with a volumeelement-induced isomorphism (axial vector isomorphism): δ S ⊗ ST ←→ S⊗S ←→ S, a abc b abc b c ↔ −ϕa pa . ϕa O = −ϕa p ⊗ xc ↔ −ϕa p ⊗ p The “inﬁnitesimal” rotation of a translation x is expressible by the vector product ϕa Oa (x) = −ϕ × x. In a box-diagonal form, the rotation axis can be chosen to deﬁne the third axis exp ϕO3 = 0 0 ϕ ∼ = ϕO3 = cos ϕ − sin ϕ sin ϕ cos ϕ 0 0 0 −ϕ 0 0 ϕ 0 0 0 0 0 0 1 ∈ SO(2) ⊂ SO(3), ∈ log SO(2) ⊂ log SO(3).

The area ab of any parallelogram b with one corner on the hyperbola, one the intersection point of the asymptotes and sides parallel to the asymptotes is unchanged for all transformations h(a, b) ∈ SL(R2 ). Here ab characterizes the angle between the two asymptotes. Hyperbolas with equal sides a = b are the analogue to the circle. A real 4-dimensional vector space has many Lorentz metrics g. All bases and all Lorentz metrics arise from a ﬁxed basis {pj }3j=0 and a ﬁxed metric g: GL(R4 ) • {pj }3j=0 GL(R4 ) • g ∼ = GL(R4 ), ∼ = GL(R4 )/O(1, 3).