Qm Framework And Quantities

The arena. The non-relativistic quantum framework of this category uses as its spacetime the affine time space of Affine time space with Euclidean simultaneous slices and adapted charts: a quadruple $(V,\tau ,g,A)$ consisting of a four-dimensional real vector space $V$, a surjective linear map $\tau :V\to \mathbb{R}$ (the absolute-time covector), the three-dimensional kernel $K=\ker \tau$ (the spatial displacement space), a symmetric positive definite bilinear form $g:K\times K\to \mathbb{R}$ (the spatial metric), and an affine space $A$ modelled on $V$ whose elements are called events. When the framework speaks of "a particle at position $x$ at time $t$", of "the time interval between two events" or of "the distance between two simultaneous events", these are read in this structure.

Spatial orientation. In addition to the quadruple, fix the orientation of $K$ required by Affine time space with Euclidean simultaneous slices and adapted charts: an equivalence class of ordered bases, where two bases are equivalent exactly when their change-of-basis map has positive determinant. Every adapted chart uses a $g$-orthonormal basis in this fixed class. This chooses an orientation, not a distinguished basis.

Absolute time and slices. The absolute time of a displacement $w\in V$ is $\tau (w)$; two events $p,q\in A$ are simultaneous when $\tau (q-p)=0$, and simultaneity is an equivalence relation whose classes are the simultaneous slices. For a fixed adapted chart and its coordinates $(t,x)$ of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices, the slices are the level sets of $t$ and the number $t(p)$ is the time coordinate of the event $p$. The pair of events of one slice carries the slice distance $d_g(p,q)=\sqrt{g(p-q,p-q)}$; no distance is defined between events of different slices by this structure.

Smooth structure. $A$ is read with the affine smooth structure determined by adapted coordinates: for a nonempty open interval $I\subseteq \mathbb{R}$, a curve $\gamma :I\to A$ is of class $C^k$ when its coordinate representation $s\mapsto (t(\gamma (s)),x(\gamma (s)))$ in one adapted chart is of class $C^k$; the transition law between adapted charts of Transformation between adapted charts and its kinematic consequences makes this reading independent of the chart. Regularity of curves, of potential functions and of apparatus descriptions of the framework is stated with respect to this smooth structure.

What the arena does not contain. The data contain no inner product on $V$, no Lorentzian metric, no light cone, no causal classification, no universal speed and no curvature; they contain no distinguished event, no distinguished time vector and no distinguished spatial basis. In particular the arena alone does not select a frame: the inertial frames of the model and the transformations between them are fixed by the separate definition of the frames on this page and the accompanying postulate.

Status. These clauses define the spacetime objects of the model and their elementary readings. They are not a physical postulate that nature is such an arena and not a mathematical theorem; the corresponding adoption is made by the kinematics postulate of this page. No dynamics, force law, potential or matter model is fixed here, and no empirical measurement is recorded.

• The coordinate space of the Galilean transformations of The Galilean transformations of the coordinate space is a model of the coordinate form of this arena: an adapted chart identifies $A$ bijectively with $\mathbb{R}\times {\mathbb{R}}^3$, and the transition law between two such charts is one of those transformations, as recorded in Transformation between adapted charts and its kinematic consequences.
• Nothing in this definition fixes units. The readings of the time coordinate in seconds, of spatial coordinates in metres and of slice distances in metres are fixed by the unit and quantity conventions of Dimensions, SI unit conventions and quantity types of the framework.

Coordinate space. Let $C:=\mathbb{R}\times {\mathbb{R}}^3$, whose elements are written $(t,x)$ with $t\in \mathbb{R}$ and $x=(x_1,x_2,x_3)\in {\mathbb{R}}^3$. The Euclidean inner product and norm on the second factor are those of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$; addition and scalar multiplication on $C$ are componentwise, and $I_3$ is the identity matrix of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes.

The transformations. For a quadruple $\theta =(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) define the map $g_{\theta }:C\to C$ by

$g_{(b,a,v,R)}(t,x):=(t+b,Rx+vt+a),$

where $Rx$ and $vt$ are the usual matrix–vector products with $R=(R_{ij})$ and with the column vectors $x,v,a$. The four components of $\theta$ are called the time shift $b$, the spatial translation $a$, the boost velocity $v$ and the rotation $R$, and $\theta$ is called the parameter quadruple of $g_{\theta }$. Write

$G:=\{g_{(b,a,v,R)}:(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)\}$

for the set of all these maps and $\mathrm{\Theta }:=\mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$ for the parameter set. The elements of $G$ are the Galilean transformations of the coordinate space, and the assignment $\theta \mapsto g_{\theta }$ is the parameter map.

Elementary readings. For $R=I_3$ and $v=0$ the map $g_{(b,a,0,I_3)}$ is the pure time-and-space translation $(t,x)\mapsto (t+b,x+a)$; for $b=0$, $a=0$ and $R=I_3$ it is the pure boost $(t,x)\mapsto (t,x+vt)$; and for $b=0$, $a=0$, $v=0$ it is the rotation $(t,x)\mapsto (t,Rx)$ acting only on the spatial factor. The time coordinate of $g_{\theta }(t,x)$ depends only on $t$ and $b$, in accordance with the absolute-time convention of the model that consumes this definition.

What is and is not asserted. This definition introduces the maps and names their parameters. It asserts no property of the parameter map (such as injectivity), no closure of $G$ under composition or inversion, no continuity, smoothness or differentiability of the maps, and no physical interpretation: whether these maps describe changes between inertial frames of a physical model is a separate, explicitly adopted assumption of the physics items that consume this definition. The group structure of $G$ is proved in the group lemma of this page, and the units in which the parameters are read are fixed by the model's unit conventions, not here.

• The definition is purely mathematical. Its content is the set of maps displayed above, the naming of the parameters and the reading of the elementary cases; nothing is assumed about which coordinate systems a physical model uses.
• The composition and inverse computations of The Galilean transformations form a group of bijections of the coordinate space show that $G$ is a group of bijections of $C$ with respect to composition, with the parameters of a composite given by the displayed polynomial expressions. In the model, the composition of frame changes is used only through that lemma.
• Notation $Rx$ requires the identification of matrices over $\mathbb{R}$ with linear maps of ${\mathbb{R}}^3$; the products displayed are the usual ones of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes read componentwise, and the case $R\in SO(3)$ is the one of The orthogonal group $O(3)$ and the rotation group $SO(3)$.

In the notation of The Galilean transformations of the coordinate space, with $C=\mathbb{R}\times {\mathbb{R}}^3$, $G$ the set of Galilean transformations and $\theta =(b,a,v,R)$ the parameters of $g_{\theta }$:

1. The parameter map $\theta \mapsto g_{\theta }$ is injective: if $g_{(b,a,v,R)}=g_{(b^{\mathrm{\prime }},a^{\mathrm{\prime }},v^{\mathrm{\prime }},R^{\mathrm{\prime }})}$ then $b=b^{\mathrm{\prime }}$, $a=a^{\mathrm{\prime }}$, $v=v^{\mathrm{\prime }}$ and $R=R^{\mathrm{\prime }}$.
2. Each $g_{(b,a,v,R)}$ is a bijection of $C$, with inverse $g_{(b,a,v,R)}^{-1}=g_{(-b,-R^{\mathsf{T}}(a-vb),-R^{\mathsf{T}}v,R^{\mathsf{T}})}\mathrm{.}$
3. Composition of Galilean transformations is Galilean, with $g_{(b_2,a_2,v_2,R_2)}\circ g_{(b_1,a_1,v_1,R_1)}=g_{(b_1+b_2,R_2{a}_1+v_2{b}_1+a_2,R_2{v}_1+v_2,R_2{R}_1)},$ and $g_{(0,0,0,I_3)}$ is the identity map of $C$.
4. Consequently $G$ is a subgroup of the group of all bijections of $C$ under composition.

(a) Dimensions. Write $\mathrm{\ell }$, $\tau$, $\mu$, $\vartheta$ for the four independent dimension symbols of length, duration, mass and electric charge. A dimension is an element of ${\mathbb{Q}}^4$, written ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$ with $a,b,c,d\in \mathbb{Q}$; products and inverses of dimensions are componentwise addition and negation, and the dimensionless dimension ${\mathrm{\ell }}^0{\tau }^0{\mu }^0{\vartheta }^0$ is abbreviated $1$. The conventions assign to each quantity of the framework a dimension $[X]$, subject to $[XY]=[X][Y]$, $[X\mathrm{/}Y]=[X][Y{]}^{-1}$ and $[dX\mathrm{/}du]=[X][u{]}^{-1}$, with sums meaningful only between quantities of equal dimension, in which case the sum carries that dimension. For a rational power $q$, dimension powers mean $[X{]}^q={\mathrm{\ell }}^{qa}{\tau }^{qb}{\mu }^{qc}{\vartheta }^{qd}$ when $[X]={\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$. Rational exponents include the half-integer wave-function dimensions in clause (d); the integer-exponent dimensions form the subgroup ${\mathbb{Z}}^4$. These are consistency requirements on the conventions, not laws of nature.

For extensions using temperature, adjoin an independent dimension symbol $\mathrm{\Theta }$: the enlarged group is ${\mathbb{Q}}^4\times \mathbb{Q}$, written ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d{\mathrm{\Theta }}^e$, with the same componentwise rules. The preceding group embeds by $e=0$, so all previously assigned dimensions are preserved. Temperature has dimension $\mathrm{\Theta }$, the Boltzmann constant has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}{\mathrm{\Theta }}^{-1}$, and entropy in joules per kelvin has that same dimension; a dimensionless entropy remains dimensionless. This convention does not assign a temperature or an entropy to any system.

(b) Reference units. The metre (m), second (s), kilogram (kg) and coulomb (C) are the reference units of length, duration, mass and charge; a quantity of dimension ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$ is expressed in the unit ${\mathrm{m}}^a{\mathrm{s}}^b\mathrm{k}{g}^c{\mathrm{C}}^d$. Coordinates of an inertial frame are read with the time coordinate $t$ in seconds, $[t]=\tau$, and the spatial coordinates $x_i$ in metres, $[x_i]=\mathrm{\ell }$; slice distances of the arena are read in metres. In the temperature extension the reference unit of $\mathrm{\Theta }$ is the kelvin (K), so the general reference unit gains a factor ${\mathrm{K}}^e$.

(c) Defining constants of the current SI. The framework keeps $\mathrm{\hslash }$ explicit. In the current SI the constants $h$, $c$ and $e$ are exact defining constants, with the values $h=\mathrm{6,626}07015\times {10}^{-34}$ J s, $c=\mathrm{2,99792458}\times {10}^8$ m s${}^{-1}$ and $e=\mathrm{1,602}176634\times {10}^{-19}$ C, and $\mathrm{\hslash }:=h\mathrm{/}(2\pi )$. These are real scalar constants with dimensions and units $[h]=[\mathrm{\hslash }]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (J s), $[c]=\mathrm{\ell }{\tau }^{-1}$ (m s${}^{-1}$), and $[e]=\vartheta$ (C). The vacuum permeability ${\mu }_0$ is a positive real scalar that is experimentally determined in the current SI, with $[{\mu }_0]=\mu \mathrm{\ell }{\vartheta }^{-2}$ (N A${}^{-2}$ = H m${}^{-1}$); ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ is a positive real scalar with $[{\epsilon }_0]={\mu }^{-1}{\mathrm{\ell }}^{-3}{\tau }^2{\vartheta }^2$ (F m${}^{-1}$). These are unit-convention statements of the SI, not measurements reported by this framework; $c$ enters the framework only through the separate radiation assumptions adopted on their own pages, not in the nonrelativistic spacetime of The Newtonian spacetime arena of the framework.

(d) Dimensions and types of the framework's quantities. The dimension map is fixed on the following quantities.

1. State quantities: a position wave function of a system with $d$ spatial dimensions and $r$ internal components is an almost-everywhere equivalence class in $L^2({\mathbb{R}}^d;{\mathbb{C}}^r)$ with dimension ${\mathrm{\ell }}^{-d\mathrm{/}2}$, its norm $\mathrm{\parallel }\psi \mathrm{\parallel }$ is dimensionless, and an abstract normalized state and the density operators of the state postulate are dimensionless.
2. Probability: the outcome probability $\mu (B)$ of the observable postulate is a dimensionless number; a position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ has dimension ${\mathrm{\ell }}^{-d}$; a momentum wave function normalized against ${\mathrm{d}}^{d}p$ has dimension $(\mu \mathrm{\ell }{\tau }^{-1}{)}^{-d\mathrm{/}2}$; an amplitude for a finite outcome is a dimensionless complex number, while an amplitude read against a detector coordinate carries the square-root of the inverse of the detector measure.
3. Frame parameters (The Galilean transformations of the coordinate space): $[b]=\tau$, $[a_i]=\mathrm{\ell }$, $[v_i]=\mathrm{\ell }{\tau }^{-1}$ and $[R_{ij}]=1$.
4. Mechanical quantities: action $[\mathrm{\hslash }]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$; energy and the Hamiltonian eigenvalues $\mu {\mathrm{\ell }}^2{\tau }^{-2}$; momentum $\mu \mathrm{\ell }{\tau }^{-1}$; position components $\mathrm{\ell }$; angular momentum and spin $\mu {\mathrm{\ell }}^2{\tau }^{-1}$; mass $\mu$; velocity of a free spinless model $\mathrm{\ell }{\tau }^{-1}$; force $\mu \mathrm{\ell }{\tau }^{-2}$; a real potential energy $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.
5. Probability current: in the ordinary continuous-field regime, fix an integer $d\ge 1$, an open spatial region $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ and an open time interval $I\subseteq \mathbb{R}$. A probability current density is a declared continuous map $j:\mathrm{\Omega }\times I\to {\mathbb{R}}^d$, whose spatial components have dimension ${\mathrm{\ell }}^{1-d}{\tau }^{-1}$ and reference unit ${\mathrm{m}}^{1-d}{\mathrm{s}}^{-1}$. At fixed time it is a spatial vector field; its signed normal component is read as probability transported per unit spatial $(d-1)$-area and time. Explicitly, for a flat face $S=\{x:x_k=a,a_i\le x_i\le b_i(i\mathrm{\ne }k)\}\subset \mathrm{\Omega }$, with $a_i<b_i$ and chosen normal $n=ϵe_k$, $ϵ\in \{-1,1\}$, define its signed probability flux at $t\in I$ as ${\mathrm{\Phi }}_S(t):=ϵ{\int }_{{\prod }_{i\mathrm{\ne }k}[a_i,b_i]}{j}_k(x_1,\dots ,x_{k-1},a,x_{k+1},\dots ,x_d,t){\prod }_{i\mathrm{\ne }k}d{x}_i$. Here $e_k$ is the $k$th coordinate unit vector and the integral uses $(d-1)$-dimensional Lebesgue measure; for $d=1$ this means ${\mathrm{\Phi }}_{\{a\}}(t)=ϵj(a,t)$. The flux has dimension ${\tau }^{-1}$ and unit ${\mathrm{s}}^{-1}$; positive sign means transport in the chosen normal direction. Fluxes through curved surfaces, or currents with weaker regularity, require separately specified surface measures or traces. This fixes the quantity type and flux convention; it supplies neither a current associated to an arbitrary $L^2$ state nor a conservation or detector law. Each dynamical model supplies its current construction and balance hypotheses.
6. Electric quantities of the model parameters: charge has dimension $\vartheta$; a magnetic moment has dimension ${\mathrm{\ell }}^2\vartheta {\tau }^{-1}$, equivalently $\mathrm{A}{\mathrm{m}}^2=\mathrm{J}{\mathrm{T}}^{-1}$, and a gyromagnetic parameter $\gamma$ in the relation $\mathit{\mu }=\gamma \mathbf{S}$ has dimension ${\mu }^{-1}\vartheta$, equivalently ${\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$; these quantities appear only in the model assumptions that introduce them.
7. Statistical quantities: a mean has the dimension of its outcome coordinate, a variance the square of it and a standard deviation it; a mean, variance or higher moment exists only when the corresponding moment of the outcome probability is finite, so a symbolic average is not a number unless that integrability is established. Angles and phases are dimensionless under the radian convention.
8. Quantities of later extension assumptions - radiation frequencies and energies, electric dipole moments, cross sections, temperatures and entropies - are declared with their dimensions and units by the items that introduce those assumptions, under the rules fixed here; they are not part of the present list.

(e) Conventions only. No numerical value of any model parameter is fixed by this item: no mass, charge, spin label, potential or measured constant is asserted, and natural-unit shorthand (for example setting $\mathrm{\hslash }=1$ or $c=1$) is read as an abbreviation for the corresponding SI-dimensioned statement with the constant restored.

• The dimension rules fix bookkeeping only. They neither predict a value nor restrict which models are admissible; each later item that introduces an operation outside the product, quotient and derivative rules declares the dimensions of its outputs.
• The framework's exact constants of (c) are conventions of the current SI: they assert nothing about the validity of the model, and the measured constants of the model (masses, charges, magnetic parameters) are qualified empirical inputs wherever they enter.

Inertial frames. Fix the arena $(V,\tau ,g,A)$ of The Newtonian spacetime arena of the framework. An inertial frame of the framework is an adapted chart

$F=(o,u,(e_1,e_2,e_3))$

of that arena in the sense of Affine time space with Euclidean simultaneous slices and adapted charts: an event $o\in A$ (the spatial and time origin of the frame), a vector $u\in V$ with $\tau (u)=1$ (the frame's time vector), and an ordered $g$-orthonormal positively oriented basis $(e_1,e_2,e_3)$ of $K$ (the frame's spatial triad). The associated frame coordinates are the functions $(t,x):A\to \mathbb{R}\times {\mathbb{R}}^3$ of Existence and uniqueness of adapted coordinates, and the affine structure of simultaneous slices, defined by $p=o+t(p)u+{\sum }_i{x}_i(p)e_i$.

What kind of object a frame is. A frame is a coordinate chart of $A$: a bijection of $A$ with $\mathbb{R}\times {\mathbb{R}}^3$ carrying the affine structure, and in particular a global chart, not a local one. It is deliberately not an observer congruence (no family of curves indexed by spatial position, no four-velocity field) and not a time-varying orthonormal basis field: the triad $(e_1,e_2,e_3)$ is a fixed basis of $K$ and the time vector $u$ is fixed once the frame is chosen. A time-dependent family of bases or a congruence of observers is a different geometric object, and statements about them are not statements about frames unless a separate construction relates them.

Allowed changes of frame. Let $F$ and $F^{\mathrm{\prime }}$ be two inertial frames with coordinates $(t,x)$ and $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$. The change of frame from $F$ to $F^{\mathrm{\prime }}$ is the bijection of $\mathbb{R}\times {\mathbb{R}}^3$ that sends the $F$-coordinates of an event to its $F^{\mathrm{\prime }}$-coordinates. By clause 1 of Transformation between adapted charts and its kinematic consequences there are $b\in \mathbb{R}$, $a,v\in {\mathbb{R}}^3$ and $R\in SO(3)$ (The orthogonal group $O(3)$ and the rotation group $SO(3)$) with

$t^{\mathrm{\prime }}=t+b,x^{\mathrm{\prime }}=Rx+vt+a;$

that is, the change of frame is the Galilean transformation $g_{(b,a,v,R)}$ of The Galilean transformations of the coordinate space applied to the coordinates. The class of transformations between inertial frames is therefore the set $G$ of the Galilean transformations, which by The Galilean transformations form a group of bijections of the coordinate space is a group of bijections of the coordinate space: the composite of two frame changes is again a frame change, and the inverse of a frame change is a frame change. The parameters are the time shift $b$, the spatial translation $a$, the boost velocity $v$ and the rotation $R$ of the two frames.

Conventions. The spatial triad of every frame is positively oriented for the fixed orientation of $K$ used in the definition of adapted charts, so all frame changes have $R\in SO(3)$ rather than $R\in O(3)$ with negative determinant. The reading of the coordinates in seconds and metres, and the physical meaning of the parameters as measured quantities, are fixed by the unit conventions recorded on this page; no measured value of any parameter is asserted here.

Status. This definition fixes the frames and the class of frame changes of the model. It asserts no property of forces or of the motion of any body, selects no distinguished frame, and makes no empirical claim. The adoption of these frames as the frames of the physical model, together with the reading of apparatus motion as classical, is part of the kinematics postulate of this page.

• The distinction between chart and congruence matters for later claims: a change of frame changes the coordinate description of a state or of a trajectory but does not move the apparatus, and an accelerated congruence of observers is not a frame of the model.
• Two inertial frames related by a pure time shift, a pure spatial translation, a pure boost or a pure rotation are the elementary cases listed in The Galilean transformations of the coordinate space, and a general frame change is the composite of these.
• Every Galilean map is realized by a pair of frames. Fix $F=(o,u,(e_i))$ and any $(b,a,v,R)\in \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3\times SO(3)$. Define $e_j^{\mathrm{\prime }}:={\sum }_i{R}_{ji}{e}_i$, $u^{\mathrm{\prime }}:=u-{\sum }_i(R^{\mathsf{T}}v{)}_i{e}_i$, and $o^{\mathrm{\prime }}:=o-bu+{\sum }_i(R^{\mathsf{T}}(vb-a){)}_i{e}_i$. Then $\tau (u^{\mathrm{\prime }})=1$, and $(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }},e_3^{\mathrm{\prime }})$ is a positively oriented $g$-orthonormal basis of $K$ because its change-of-basis matrix is orthogonal with determinant $\det R=1$. Thus $F^{\mathrm{\prime }}:=(o^{\mathrm{\prime }},u^{\mathrm{\prime }},(e_j^{\mathrm{\prime }}))$ is an inertial frame. Substitution into $p=o+tu+{\sum }_i{x}_i{e}_i=o^{\mathrm{\prime }}+t^{\mathrm{\prime }}{u}^{\mathrm{\prime }}+{\sum }_j{x}_j^{\mathrm{\prime }}{e}_j^{\mathrm{\prime }}$ gives $t^{\mathrm{\prime }}=t+b$ and $x^{\mathrm{\prime }}=Rx+vt+a$. Hence every parameter quadruple in the definition of $G$ occurs as the coordinate change of two inertial frames; combined with the adapted-chart lemma's forward direction, the class of changes is exactly $G$.

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.

Graph-core convention used in the Statement. If $A:D(A)\subseteq H\to H$ is a densely defined linear operator, its graph norm on $D(A)$ is $\mathrm{\parallel }f{\mathrm{\parallel }}_A:=(\mathrm{\parallel }f{\mathrm{\parallel }}_H^2+\mathrm{\parallel }Af{\mathrm{\parallel }}_H^2{)}^{1\mathrm{/}2}$. A linear subspace $D_0\subseteq D(A)$ is a core for $A$ when it is dense in $D(A)$ in this graph norm; equivalently, the graph of $A{\mathrm{\mid }}_{D_0}$ is dense in the graph of $A$ in $H\times H$. A common core for a family of operators is one subspace that is a core for each operator in the family.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix $d\ge 1$ and $\mathrm{\hslash }>0$, let $Q_j,P_k$ be the position and momentum operators on $L^2({\mathbb{R}}^d;\mathbb{C})$ of Position and momentum operators on Euclidean L2, and let

$\mathcal{S}:=\mathcal{S}({\mathbb{R}}^d;\mathbb{C})$

be the Schwartz space of Schwartz space and its seminorms. Then:

In this operator statement, $\mathcal{S}$ denotes the image in $L^2({\mathbb{R}}^d;\mathbb{C})$ of the actual Schwartz functions under the quotient map by almost-everywhere equality (The space $L^p(\mu )$ as the quotient by null functions); the smooth representatives are used in the pointwise derivative calculations.

(a) $\mathcal{S}\subseteq D(Q_j)\cap D(P_k)$ for all $j,k$, and $Q_{j}f\in \mathcal{S}$ and $P_{k}f\in \mathcal{S}$ for every $f\in \mathcal{S}$. (b) for every $f\in \mathcal{S}$ the products $Q_j{P}_k$ and $P_k{Q}_j$ are defined at $f$ and $(Q_j{P}_k-P_k{Q}_j)f=i\mathrm{\hslash }{\delta }_{jk}f,{\delta }_{jk}=\{\begin{array}{ll}1,& j=k,\\ 0,& j\mathrm{\ne }k;\end{array}$ that is, the canonical commutation relation holds on the common core $\mathcal{S}$.

No statement is made about products of $Q_j$ and $P_k$ outside $\mathcal{S}$: the relation is asserted on the Schwartz core only, not as an identity of unbounded products on all of $L^2$.