# Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, z$ using an arbitrary frame of reference $(O, e_x, e_y, e_z)$ and then associated to 3 different (mutually commutating) operators $Q_x, Q_y, Q_z$ (of the Hilbert Space $\mathcal{H}$). Now, the arbitrariness of direction is not such a big deal because one can define an object $Q = e_x \otimes Q_x + e_y \otimes Q_y + e_z \otimes Q_z$ that is independent of it and that respects (I believe) the invariants of standard (= newtonian-space-timed) QM. However I feel unconfortable with the fact it still arbitrarily depends on the origin $O$ whereas space (at a given $t$) has a structure of a 3d affine space rather than a 3d vector space.

In this context, the following question arose to me. Suppose that you have a n-dimensional $\mathbb{K}$-affine space $\mathcal{A}$, can you use it inside a tensor product? I believe that you can actually define a product with a unital associative $\mathbb{K}$-algebra $\mathcal{Z}$ (for example an operator algebra in the case of QM) as below.

If we agree to note $\Delta\mathcal{A}$ the vector space under $\mathcal{A}$, let's consider $\mathcal{A} \times (\Delta\mathcal{A} \otimes \mathcal{Z})$ and define the equivalence relation $\sim$ by: $$(P,U) \sim (Q,V) \Leftrightarrow U - V = (Q-P) \otimes 1_\mathcal{Z}$$ Then let us define $\mathcal{A} \boxtimes \mathcal{Z} = (\mathcal{A} \times (\Delta\mathcal{A} \otimes \mathcal{Z})) / \sim$.

This object has a canonical structure of an affine space. I think it satisfies the following properties:

• $\dim(\mathcal{A} \boxtimes \mathcal{Z}) = \dim(\mathcal{A}) \cdot \dim(\mathcal{Z})$,
• $\Delta (\mathcal{A} \boxtimes \mathcal{Z}) = \Delta \mathcal{A} \otimes \mathcal{Z}$,
• $(\mathcal{A} \oplus \mathcal{B}) \boxtimes \mathcal{Z} = \mathcal{A}\boxtimes \mathcal{Z} \oplus \mathcal{B} \boxtimes \mathcal{Z}$ for $\mathcal{A}$ and $\mathcal{B}$ affine spaces,
• $\mathcal{A} \boxtimes (\mathcal{Z} \oplus \mathcal{Y}) = \mathcal{A} \boxtimes \mathcal{Z} \oplus \mathcal{A} \boxtimes \mathcal{Y}$ for $\mathcal{Y}$ and $\mathcal{Z}$ unital associative algebras,
• $(\mathcal{A} \boxtimes \mathcal{Z}) \boxtimes \mathcal{Y} = \mathcal{A} \boxtimes (\mathcal{Z} \otimes \mathcal{Y})$.

If for a given $P \in \mathcal{A}$ we note $P \boxtimes 1_\mathcal{Z}$ the equivalent class of $(P,0)$ every element of $\mathcal{A} \boxtimes \mathcal{Z}$ can be written $P \boxtimes 1_\mathcal{Z} + U$. There is also a sort of conjugation from the group of invertible $\mathcal{Z}^*$ : $$z \in \mathcal{Z}^*, T \in \mathcal{A} \boxtimes \mathcal{Z} \mapsto T^z = z \cdot T \cdot z^{-1}$$ as well as a commutator: $$z \in \mathcal{Z}, T \in \mathcal{A} \boxtimes \mathcal{Z} \mapsto [z, T] = z \cdot T - T \cdot z$$ whereas neither $z \cdot T$, nor $T \cdot z$ are well defined operations.

In the end I feel this structure may be pretty interesting but I cannot find any mention on internet involving both "affine space" and "tensor product". Hence my question:

• Is this operation valid according to you?
• Is it already known? Does it have another name?
• Is it just a special case of a more general operation (with homogenous spaces or affine manifolds for example)?
• Any known references about something similar?

Edit Mar 28 2015

Although quantum mechanics was the context that triggered my reflections, it is not the only raison d'être of my question. I'm interested in understanding this kind of operations as pure algebraic objects, independently of the fact it might be used in QM or not.