1 Introduction

[page 626, §1]
[626.1.1] Applications of fractional time derivatives and engineering assume the existence of a physical time automorphism (time evolution) of observables, which for closed quantum many-body systems is usually given as a Hamiltonian-generated one-parameter group of unitary operators on a Hilbert space. [626.1.2] Dissipative processes, irreversible phenomena, decay of unstable particles, approach to thermodynamic equilibrium or quantum measurement processes are difficult to accommodate within this mathematical framework [1, 2, 3].

[page 627, §1] [627.1.1] Many theoretical approaches to these problems consider an “open” system (or subsystem) $S$ coupled to a “reservoir” $R$ , often viewed as a heat bath or as an apparatus for measurement[4, 3]. [627.1.2] A different physical interpretation with the same mathematical structure is to identify $S$ with a selection of macroscopic degrees of freedom of a large or infinite many body system $S\cup R$ , while $R$ corresponds to the large or infinite number of microscopic degrees of freedom. [627.1.3] It has remained difficult to find physical conditions which rigorously imply irreversibility for the time evolution of the subsystem [4, 5]. [627.1.4] One expects intuitively that separation of time scales will be important. [627.1.5] Relaxation processes in the reservoir $R$ are usually much faster than the characteristic time scale for the evolution of the system $S$ of interest. [627.1.6] Equally important for macroscopic dynamics and thermodynamic behaviour is scale separation in the size of $R$ and $S$ . [627.1.7] Memory effects are expected to arise from interaction between the system and the reservoir.

[627.2.1] Dynamical equations of motion for closed systems are frequently formulated as abstract Cauchy problems on some Banach space $\mathfrak{B}$ of states or observables $A\in\mathfrak{B}$

$\displaystyle\tau\epsilon\frac{\mathrm{d}}{\mathrm{d}t}A(t/\tau)$	$\displaystyle=\mathscr{L}A(t/\tau)$		(1a)
$\displaystyle A(t_{0}/\tau)$	$\displaystyle=A_{{0,\tau}}$		(1b)

where $A_{{0,\tau}}$ is the initial value, $t,t_{0}$ are time instants measured in units of $\tau$ seconds (such that $t/\tau\in\mathbb{R}$ ) and $\epsilon$ provides energy units (Joule) for the infinitesimal generator $\mathscr{L}$ (Liouvillian), which is a linear, often unbounded, operator with domain $D(\mathscr{L})\subset\mathfrak{B}$ .

[627.3.1] Existence of a physical time evolution is equivalent to the existence of global solutions of eq. (1) under various circumstances and assumptions such as physical constraints and boundary conditions. [627.3.2] It is well known that global solutions do not always exist, particularly when the system is infinite.

[627.4.1] Given a kinematical structure describing the states and observables of a physical system, the infinitesimal generator $\mathscr{L}$ in eq. (1) describes infinitesimal changes of these states and observables with time starting from an initial condition $A_{{0,\tau}}\in\mathfrak{B}$ . [627.4.2] Let me briefly recall the kinematical structures for classical mechanics, quantum mechanics and field theory [6, 7, 2]. [627.4.3] Observables and states in classical mechanics of point particles correspond to functions over and points in a differentiable manifold. [627.4.4] Rays in a Hilbert space and operators acting on them are the kinematical structure in quantum mechanics. [627.4.5] In field theory the observables form a C ${}^{*}$ -algebra of field operators and the states correspond to positive linear functionals on this algebra. [627.4.6] Automorphisms of the algebra of field operators in field theory, unitary operators on the Hilbert space in quantum mechanics and diffeomorphisms of the differentiable manifold in classical mechanics, represent the time evolution of the system as a flow on the kinematical structure. [627.4.7] Many theories of interacting particles are based on some Hamiltonian formalisms as in eq. (1) with a Hamiltonian $\mathscr{L}$ corresponding to a vector field in classical mechanics, a selfadjoint operator in quantum mechanics and some form of derivation on the algebra in field theories.

[627.5.1] Let $\mathfrak{B}=\mathfrak{A}$ be the C ${}^{*}$ -algebra of observables of a physical system. [627.5.2] Unless otherwise stated all C ${}^{*}$ -algebras will be assumed to have an identity. [627.5.3] Formally integrating eq. (1) gives

$\displaystyle\mathscr{T}^{{t/\tau}}\mathcal{K}_{{A_{{0,\tau}}}}\left(\frac{t_{0}}{\tau}\right)=T^{{t/\tau}}A_{{0,\tau}}$

(2)

[page 628, §0] where the maps $T^{{s}}\colon\mathfrak{A}\to\mathfrak{A}$ and $\mathscr{T}^{{s}}\colon\mathfrak{A}\to\mathfrak{A}$ are

$\displaystyle T^{{t/\tau}}A$	$\displaystyle=\exp\left(\frac{\mathscr{L}t}{\epsilon\tau}\right)A$		(3a)
$\displaystyle\mathscr{T}^{{t/\tau}}\mathcal{K}_{A}(s)$	$\displaystyle=\mathcal{K}_{A}\left(s+\frac{t}{\tau}\right)$		(3b)

and the orbit maps $\mathcal{K}_{A}\colon\mathbb{R}\to\mathfrak{A}$ are defined as

$\displaystyle\mathcal{K}_{A}(s)\colon s\mapsto T^{{s}}A$

(4)

for each fixed $A\in\mathfrak{A}$ , if $T^{{s}}$ with $s\in\mathbb{R}$ is a one-parameter familiy of *-automorphisms of $\mathfrak{A}$ . [628.0.1] Of course, the problem is to give meaning to the formal exponential in eq. (3a) such that the orbit maps $\mathcal{K}_{A}\colon\mathbb{R}\to\mathfrak{A}$ are continuous for every $A\in\mathfrak{A}$ .

[628.1.1] The one-parameter family $(T^{{s}})_{{s\in\mathbb{R}}}$ of *-automorphisms is expected to obey the time evolution law

$\displaystyle T^{{t/\tau}}T^{{s/\tau}}=T^{{(t+s)/\tau}}$

(5)

with $T^{0}=\mathbf{1}$ being the identity. [628.1.2] The continuity of the orbit maps may be rephrased as continuity of the maps $t\mapsto T^{t}$ from $\mathbb{R}$ into the space $\mathfrak{B}(\mathfrak{A})$ of all bounded operators on $\mathfrak{A}$ endowed with the strong operator topology [8, 9]. [628.1.3] The operator family $(T^{{s}})_{{s\in\mathbb{R}}}$ is then a strongly continuous one-parameter group ( $C_{0}$ -group) on $\mathfrak{A}$ .

[628.2.1] The time evolution of states is obtained from the time evolution of observables by passing to adjoints [10, 11]. [628.2.2] States are elements of the topological dual $\mathfrak{A}^{*}=\{\mathsf{z}\colon\mathfrak{A}\to\mathbb{C}:\mathsf{z}\text{~is linear and continuous}\}$ . [628.2.3] The notation $\langle\mathsf{z},A\rangle$ is used for the value $\mathsf{z}(A)\in\mathbb{R}$ of a selfadjoint $A\in\mathfrak{A}$ in the state $\mathsf{z}$ . [628.2.4] States are positive, $\langle\mathsf{z},A^{*}A\rangle\geq 0$ for all $A\in\mathfrak{A}$ , and normalized, $\|\mathsf{z}\|=\sup\{|\langle\mathsf{z},A\rangle|,\| A\|=1\}=1$ , linear functionals on the algebra $\mathfrak{A}$ of observables [6]. [628.2.5] The adjoint time evolution $T^{{*t}}\colon\mathfrak{A}^{*}\to\mathfrak{A}^{*}$ with $t\in\mathbb{R}$ consists of all adjoint operators $(T^{{t}})^{*}$ on the dual space $\mathfrak{A}^{*}$ [12, 10]. [628.2.6] Let $\mathsf{Z}\subset\mathfrak{A}^{*}$ denote the set of all states. [628.2.7] The orbit maps for states $\mathcal{K}_{\mathsf{z}}:\mathbb{R}\to\mathsf{Z}$ are defined as

$\displaystyle\mathcal{K}_{\mathsf{z}}(s)\colon s\mapsto(T^{{s}})^{*}\mathsf{z}=T^{{*s}}\mathsf{z}$

(6)

for states $\mathsf{z}\in\mathsf{Z}\subset\mathfrak{A}^{*}$ . [628.2.8] If $T^{{t}}$ is strongly continuous then

$|\left\langle(T^{{*t}}-\mathbf{1})\mathsf{z},A\right\rangle|=|\left\langle\mathsf{z},T^{t}A-A\right\rangle|\leq\|\mathsf{z}\|\;\| T^{t}A-A\|$

(7)

shows that the adjoint time evolution $T^{{*t}}$ is weak*-continuous in the sense that the maps $\left\langle A\right\rangle _{{\mathsf{z}}}\colon\mathbb{R}\to\mathbb{R}$

$\displaystyle t\mapsto\left\langle A\right\rangle _{{\mathsf{z}}}(t)=\left\langle\mathsf{z},T^{t}A\right\rangle=\left\langle T^{{*t}}\mathsf{z},A\right\rangle$

(8)

are continuous for all $A\in\mathfrak{A},\mathsf{z}\in\mathsf{Z}$ . [628.2.9] These maps are the time evolutions of all expectation values. [628.2.10] In other words for a $C_{0}$ -group $(T^{{s}})_{{s\in\mathbb{R}}}$ the orbit maps $\mathcal{K}_{\mathsf{z}}(s)$ are continuous from $\mathbb{R}$ into the space $\mathfrak{B}(\mathfrak{A}^{*})$ of all bounded operators on $\mathfrak{A}^{*}$ endowed with the weak* topology [8, 13] and the adjoint family $(T^{{*s}})_{{s\in\mathbb{R}}}$ is a $C_{0}^{*}$ -group. [628.2.11] Note, that the adjoint time evolution $T^{{*t}}$ need not be strongly continuous unless $\mathfrak{A}$ is reflexive. [628.2.12] The relation between the time evolution of states and observables is

$\displaystyle\left\langle\mathcal{K}_{\mathsf{z}}(t_{0}),T^{t}\mathcal{K}_{A}(t_{0})\right\rangle$	$\displaystyle=\left\langle\mathcal{K}_{\mathsf{z}}(t_{0}),\mathscr{T}^{t}\mathcal{K}_{A}(t_{0})\right\rangle$
	$\displaystyle=\left\langle\mathcal{K}_{\mathsf{z}}(t_{0}),\mathcal{K}_{A}(t_{0}+t)\right\rangle$
	$\displaystyle=\left\langle\mathcal{K}_{\mathsf{z}}(t_{1}-t),\mathcal{K}_{A}(t_{1})\right\rangle$	(9)
	$\displaystyle=\left\langle\mathscr{T}^{{-t}}\mathcal{K}_{\mathsf{z}}(t_{1}),\mathcal{K}_{A}(t_{1})\right\rangle$
	$\displaystyle=\left\langle T^{{*t}}\mathcal{K}_{\mathsf{z}}(t_{1}),\mathcal{K}_{A}(t_{1})\right\rangle$

[page 629, §0] where $t_{1}=t_{0}+t\in\mathbb{R}$ . [629.0.1] The adjoint time evolution of states is related to right translations along the orbits in state space in the same way as the time evolution of observables is related to left translations along orbits in the algebra.

[629.1.1] Equation (1a) combined with eq. (1) for the adjoint time evolution states formally the proportionality

$\displaystyle\pm\tau\frac{\mathrm{d}}{\mathrm{d}t}=\pm\frac{\mathscr{L}}{\epsilon}$

(10)

of the infinitesimal generator $\mathrm{d}/\mathrm{d}t$ of time translations and the infinitesimal generator $\mathscr{L}$ of changes of the physical system. [629.1.2] Independently of the manner in which one attaches a meaning to the formal exponential, equation (2) says that the time evolution of a physical system is a translation along orbits corresponding to the changes of the system, specifically

	$\displaystyle\text{(left shift along $\mathfrak{A}$-orbit)}=\text{(change of observable)}$		(11a)
	$\displaystyle\text{(right shift along $\mathsf{Z}$-orbit)}=\text{(change of state)}$		(11b)

where the first equation reflects the Heisenberg picture, while the second corresponds to the Schrödinger picture.

Author:	R. Hilfer
originally published in:	Mathematics, Vol. 3, pages 626-643 (2015)
originally submitted:	March 24th 2015