Classical apparatus and preparation trajectories of the framework

Classical trajectories. Fix the arena of The Newtonian spacetime arena of the framework and an inertial frame $F$ of Inertial frames of the framework and the class of frame changes with coordinates $(t,x)$. A classical trajectory (also called a worldline of the model) is a pair $(I,\gamma )$ consisting of a nondegenerate open interval $I\subseteq \mathbb{R}$ and a map $\gamma :I\to A$ such that:

1. Parameter condition. The parameter of the curve is the frame time: $t(\gamma (s))=s$ for every $s\in I$. In the coordinates of the frame this reads $\gamma (s)=(s,x(s))$ with $x:I\to {\mathbb{R}}^3$.
2. Regularity. The coordinate map $x:I\to {\mathbb{R}}^3$ is of class $C^2$, so that the coordinate velocity $w(s):=x^{\mathrm{\prime }}(s)$ and the coordinate acceleration $a(s):=x^{\mathrm{\prime }\mathrm{\prime }}(s)$ exist as continuous curves; derivatives are those of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral and the chain rule used to differentiate compositions is The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$.

Apparatus and preparation constituents. The macroscopic parts of a laboratory arrangement - sources, slits and collimators, shutters, field-generating coils and magnets, screens and detector housings - are described in the model by classical trajectories: to each such constituent and each inertial frame there is assigned a trajectory in the above sense, with the same event of the arena carrying its position at the frame time. A preparation device is such a constituent together with the classical trajectory assigned to it.

Quantum particles. A trajectory in the above sense is not assigned to a quantum particle of the model: the state of a particle between preparations and detections is given by a state in the sense of the state postulate of this page, and the model asserts no definite trajectory of a particle between measurements. A detected particle event is a localized detection record; its distribution is computed with the model's measurement assumptions, not by reading off a worldline.

Status. This definition fixes the mathematical object used for apparatus descriptions and its regularity, and records the two restrictions above: apparatus motion is described classically, and no worldline is assigned to an unmeasured quantum particle. It adopts no force law determining the apparatus trajectories, no interaction between apparatus and particle other than those specified on the pages that state a measurement model, and no empirical claim. The identification of a particular laboratory device with a trajectory of the model is a physical assumption with uncertainty, in the sense of the preparation and apparatus postulate of this page.

• The parameter condition is a genuine restriction: under an affine reparametrization $s\mapsto \alpha s+\beta$ of a trajectory, the new parameter again equals the frame time only when $\alpha =1$ and $\beta =0$; free reparametrizations are not part of the model.
• No causal or velocity restriction is imposed on classical trajectories by this definition; the framework is non-relativistic and sets no speed bound. Constraints imposed by a particular apparatus model are stated where that model is introduced.
• Dimensions of the derivatives. The parameter $s=t(\gamma (s))$ has dimension $\tau$ and each coordinate $x_i$ has dimension $\mathrm{\ell }$ under Dimensions, SI unit conventions and quantity types of the framework. Therefore each coordinate velocity component $w_i=d{x}_i\mathrm{/}ds$ has dimension $\mathrm{\ell }{\tau }^{-1}$, and each coordinate acceleration component $a_i=d^2{x}_i\mathrm{/}d{s}^2$ has dimension $\mathrm{\ell }{\tau }^{-2}$. These are the dimensions of the frame-coordinate derivatives; they add no force law or speed bound.