A Reconstruction of Quantum Mechanics

Abstract

Almost a century after the mathematical formulation of quantum mechanics, there is still no consensus on the interpretation of the theory.

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
			$p(x)=\mathop {\mathrm{tr}}\nolimits (wx)$
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	$\sigma :\mathbb {R}\rightarrow \mathop {\mathrm{Aut}}\nolimits (Q)$	Liouville equation	von Neumann
	representation	$\partial _t \rho =-[H, \rho ]$	-Liouville equation
			$\partial _t w_t=-\frac{i}{\hbar } [ H, w_t]$
Conditionalized states	$p(x) \rightarrow p(x \mid y)$	$p(x) \rightarrow p(x \mid y)$	$w\rightarrow ywy / \mathop {\mathrm{tr}}\nolimits (wy)$
	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 systems	$Q_1 \oplus Q_2$	$\Omega _1 \times \Omega _2$	$\mathcal {H}_1 \otimes \mathcal {H}_2$
	direct sum of	direct product of	tensor product of
	$\sigma$-complexes	phase spaces	Hilbert spaces