Quantum Dynamics

  The purpose of this section is to show that the Quantum Chance formulation is mathematically equivalent to standard quantum mechanics in both the Schrödinger and Heisenberg pictures.

Now that it has been shown that the symmetries of \(Q(\mathcal{H})\) are also implemented by symmetries of \(\mathcal{H}\), time symmetry is used to introduce a dynamics for quantum systems. To define dynamical evolution, consider systems that are invariant under time translation. For such systems, there is no absolute time, only time differences. The change from time \(0\) to time \(t\) is given by the symmetry transform \(\alpha_t : Q \rightarrow Q\), since the structure of the system of property values is indistinguishable at two values of time. If the state evolves first for a time \(t\) and then the resulting state for a time \(t' \), then this yields the same result as the original state evolving for a time \(t + t'\). It is assumed that a small time-period results in slight changes in the probability of property values occurring.

  The passage of time is represented by a continuous additive group $\mathbb{R}$ of real numbers into the group ${\mathop{\rm Aut}\nolimits} (Q)$ of automorphisms of $Q$ under composition. That is, a map $\alpha :\mathbb{R} \to {\mathop{\rm Aut}\nolimits} (Q)$, such that

 \[ \alpha_{t+t' }= \alpha_t \circ \ \alpha_{t'} \]

and the state $p_{\alpha_t}(x) $ is a continuous function of $t$. 

 The image of $\alpha$ is then a continuous one-parameter group of automorphisms on $Q$.\footnote{The group ${\mathop{\rm Aut}\nolimits} (Q)$ may be taken to be a topological group by defining, for each $\epsilon>0$,  an $\epsilon$-neighbourhood of the identity to be $\{ \alpha\mid |p_\alpha(x)- p(x)| < \epsilon$  for all $x$ and $p \}$. It is possible to directly speak of the continuity of the map $\alpha$, in place of the condition that $p_{\alpha_t}(x)$ is continuous in $t$.}

It has been shown that an automorphism $\alpha$ corresponds to a unitary operator. Therefore, the time evolution of the state $p_{\alpha_t}$ corresponds to that of the density operator $\rho_t = u_t  \rho u_t^{-1}$ and by Stone's Theorem (see Hall [1], section, 10.15.)

 \[u_t = e^{-\frac{i}{\hbar}  Ht},\] 

 where $\hbar$ is a the reduced Plank constant, the value of which is determined by experiment and \(H\) is a self-adjoint operator with units of energy; so

 \[  \rho_t = e^{-\frac{i}{\hbar} Ht} \rho \ e^{\frac{i}{\hbar} Ht}.\]

 Differentiating both side by time, \(t\),          

 \[  \partial_t \rho_t = -\frac{i}{\hbar} [ H, \rho_t ].\]

 This is the Liouville-von Neumann Equation and by correspondence with the classical Liouville Equation, \(H\) is the Hamiltonian of the quantum object.

  Conversely, this equation yields a continuous representation of $\mathbb{R}$ into ${\mathop{\rm Aut}\nolimits} (Q(\mathcal{H}))$.

  For $\rho = \Pi_\psi $, a pure state, $\rho_t  = \Pi_{\psi(t)}$ and this equation reduces to the Schr\"{o}dinger Equation:

 \[ \partial_t \psi(t) =-\frac{i}{\hbar} H \psi(t).\]                                    

This equivalence with standard quantum mechanics shows that the theory of quantum chance has the same predictive power. What the formulation of in terms of quantum chance does is make clear is that although the time evolution of probabilities of potential values of quantum properties is well defined, the actualisation of the values is not covered by the theory.

[1] Hall, B. C., Quantum Theory for Mathematicians, Springer, 2013