The semantics of the canonical commutation relation

1.1 This paper is a part of a much broader project which aims to establish, for a typical ’co-ordinate algebra’ A in the sense of non-commutative geometry, or a scheme, a geometric counterpart VA. Here “geometric” should also be understood in some broad but well-defined sense. We believe this is possible in the setting of model theory, where a crucial new experience in generalisation of semantics of algebraic and analytic geometry, and applications of this, has been accumulated in the last 50 years. In fact, we aim to establish a rigorous mathematical duality