Algebra of physical space

In physics, the name "algebra of physical space" (APS) originally stems from the use of the Clifford or geometric algebra Cl_3,0(R), also written ${\displaystyle \mathbb {G} _{3}}$ or ${\displaystyle \mathbb {R} _{3}}$, of three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). Although, recent research has adopted the name "APS" as a synonym for Cl_3,0(R) in general contexts.

The Clifford algebra Cl_3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C²; further, Cl_3,0(R) is isomorphic to the even subalgebra Cl^[0]
_3,1(R) (also ${\displaystyle \mathbb {G} _{3,1}^{+}}$) of the Clifford algebra Cl_3,1(R) (also ${\displaystyle \mathbb {G} _{3,1}}$), and to the even subalgebra Cl^[0]
_1,3(R) (also ${\displaystyle \mathbb {G} _{1,3}^{+}}$) of the spacetime algebra Cl_1,3(R) (also ${\displaystyle \mathbb {G} _{1,3}}$).

The APS can be used to construct a compact, unified, and geometrical formalism for both classical and quantum mechanics. This blurs the line between what is traditionally considered classical or quantum.

The APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl_1,3(R) of the four-dimensional Minkowski spacetime.