A Reconstruction of Quantum Mechanics

Abstract

We show that exactly the same intuitively plausible definitions of state, observable, symmetry, dynamics, and compound systems of the classical Boolean structure of intrinsic properties of systems lead, when applied to the structure of extrinsic, relational quantum properties, to the standard quantum formalism, including the Schrödinger equation and the von Neumann–Lüders Projection Rule. This approach is then applied to resolving the paradoxes and difficulties of the orthodox interpretation.

Appendix: Summary Table of Concepts

	General mechanics	Classical mechanics	Quantum mechanics
Properties	$\sigma$-Complex	$\sigma$-Algebra	$\sigma$-Complex
	$Q = \cup B$, with $B$ a $\sigma$-algebra	$B(\Omega )$	$Q(\mathcal {H}\}$
States	$p: Q\rightarrow [0,1]$	$p: B(\Omega )\rightarrow [0,1]$	$w: \mathcal {H}\rightarrow \mathcal {H}$
	$p\mid B$, a probability measure	a probability measure	Density operator
			$p(x)=\mathrm{tr}(wx)$
Pure states	Extreme point		1 dim operator
	of convex set	$\omega \in \Omega$	i.e. unit $\phi \in \mathcal {H}$
			$p(x)=\left\langle x,x\phi \right\rangle$
Observables	$u: B(\mathbb {R})\rightarrow Q$	$f:\Omega \rightarrow \mathbb {R}$	$A: \mathcal {H}\rightarrow \mathcal {H}$
	homomorphism	Borel function	Hermitean operator
Symmetries	$\sigma :Q \rightarrow Q$	$h:\Omega \rightarrow \Omega$	$u: \mathcal {H}\rightarrow \mathcal {H}$
	automorphism	canonical	unitary or
		transformation	anti-unitary operator
			$\sigma (x)=uxu^{-1}$
Dynamics	$\sigma :\mathbb {R}\rightarrow \mathrm{Aut}(Q)$	Liouville equation	von Neumann
	representation	$\partial _t \rho =-[H, \rho ]$	–Liouville equation
			$\partial _t w_t=-\frac{i}{\hbar } [ H, w_t]$
Conditionalized	$p(x) \rightarrow p(x \mid y)$	$p(x) \rightarrow p(x \mid y)$	$w\rightarrow ywy / \mathrm{tr}(wy)$
states	for $x,y\in B$ in $Q$	$=p(x \wedge y)/p(y)$	von Neumann
	$p(x \mid y)=p(x \mid y)/p(y)$		–Lüders Rule
Combined	$Q_1 \oplus Q_2$	$\Omega _1 \times \Omega _2$	$\mathcal {H}_1 \otimes \mathcal {H}_2$
systems	direct sum of	direct product of	tensor product of
	$\sigma$-complexes	phase spaces	Hilbert spaces