1. Introduction
Applications of fractional time derivatives in physics 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. Dissipative processes, irreversible phenomena, the decay of unstable particles, the approach to thermodynamic equilibrium or quantum measurement processes are difficult to accommodate within this traditional mathematical framework [
1,
2,
3].
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 [
3,
4]. 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
, while
R corresponds to the large or infinite number of microscopic degrees of freedom. It has remained difficult to find physical conditions that rigorously imply irreversibility for the time evolution of the subsystem [
4,
5]. One expects intuitively that the separation of time scales will be important. Relaxation processes in the reservoir
R are usually much faster than the characteristic time scale for the evolution of the system
S of interest. Equally important for macroscopic dynamics and thermodynamic behavior is scale separation in the size of
R and
S. Memory effects are expected to arise from interaction between the system and the reservoir.
Dynamical equations of motion for closed systems are frequently formulated as abstract Cauchy problems on some Banach space
of states or observables
where
is the initial value,
are time instants measured in units of
τ seconds (such that
) and
ϵ provides energy units (Joule) for the infinitesimal generator
(Liouvillian), which is a linear, often unbounded, operator with domain
. The existence of a physical time evolution is equivalent to the existence of global solutions of Equation (1) under various circumstances and assumptions, such as physical constraints and boundary conditions. It is well known that global solutions do not always exist, particularly when the system is infinite.
Given a kinematical structure describing the states and observables of a physical system, the infinitesimal generator
in Equation (1) describes infinitesimal changes of these states and observables with time starting from an initial condition
. Let me briefly recall the kinematical structures for classical mechanics, quantum mechanics and field theory [
2,
6,
7]. Observables and states in classical mechanics of point particles correspond to functions over and points in a differentiable manifold. Rays in a Hilbert space and operators acting on them are the kinematical structure in quantum mechanics. In field theory, the observables form a C
-algebra of field operators, and the states correspond to positive linear functionals on this algebra. 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. Many theories of interacting particles are based on some Hamiltonian formalisms as in Equation (1) with a Hamiltonian
corresponding to a vector field in classical mechanics, a self-adjoint operator in quantum mechanics and some form of derivation on the algebra in field theories.
Let
be the C
-algebra of observables of a physical system. Unless otherwise stated, all C
-algebras will be assumed to have an identity. Formally, integrating Equation (1) gives
where the maps
and
are
and the orbit maps
are defined as
for each fixed
, if
with
is a one-parameter family of *-automorphisms of
. Of course, the problem is to give meaning to the formal exponential in Equation (3a), such that the orbit maps
are continuous for every
.
The one-parameter family
of *-automorphisms is expected to obey the time evolution law
with
being the identity. The continuity of the orbit maps may be rephrased as continuity of the maps
from ℝ into the space
of all bounded operators on
endowed with the strong operator topology [
8,
9]. The operator family
is then a strongly continuous one-parameter group (
-group) on
.
The time evolution of states is obtained from the time evolution of observables by passing to adjoints [
10,
11]. States are elements of the topological dual
. The notation
is used for the value
of a self-adjoint
in the state z. States are positive,
for all
, and normalized,
, linear functionals on the algebra
of observables [
6]. The adjoint time evolution
with
consists of all adjoint operators
on the dual space
[
10,
12]. Let
denote the set of all states. The orbit maps for states
are defined as
for states
. If
is strongly continuous, then
shows that the adjoint time evolution
is weak*-continuous in the sense that the maps
are continuous for all
. These maps are the time evolutions of all expectation values. In other words, for a
-group
, the orbit maps
are continuous from ℝ into the space
of all bounded operators on
endowed with the weak* topology [
8,
13], and the adjoint family
is a
-group. Note, that the adjoint time evolution
need not be strongly continuous unless
is reflexive. The relation between the time evolution of states and observables is
where
. 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.
Equation (1a) combined with Equation (9) for the adjoint time evolution states formally the proportionality
of the infinitesimal generator
of time translations and the infinitesimal generator
of changes of the physical system. 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
where the first equation reflects the Heisenberg picture, while the second corresponds to the Schrödinger picture.
2. Problems and Objective
There are several unsolved problems with the mathematical framework described in the Introduction, particularly when the system is infinite. The following examples of open problems stem from three different areas of theoretical physics.
- (A)
Open quantum systems: For open quantum systems with infinite reservoirs,
is the C
-algebra of bounded operators on the Hilbert space
of the system
in a suitably-chosen GNS-representation. The states
can be identified with trace class operators on
if Tr denotes the trace operation. The unitary time-automorphisms
of
are generated by a Hamiltonian
H of
written formally in terms of Hamiltonians
of
S,
of
R and their interaction
. It is well known [
2] that the automorphisms
on
do not induce automorphisms on the algebra
of the open subsystem, because the time evolution will mix the Hilbert spaces
and
of the system and the reservoir. Even if the initial state is prepared as a product state
, the subsystem evolutions defined by
do not form groups:
because of memory effects accumulating from the mixing of the system and the reservoir whenever there is a nonvanishing interaction [
2]. Here,
denote the density matrices of the system and the reservoir. The trace
integrates out the reservoir degrees of freedom.
- (B)
Classical dynamical systems: In classical systems, it is well known [
14] that the orbits in the abelian algebra
of functions on phase space cannot always be defined for all
and for all initial conditions
in the thermodynamics limit. The integration of Equation (1) does not generally give a dynamical flow of time for all initial conditions, and the problem is to find sufficiently large subsets of
, such that catastrophic behavior is absent and a unique orbit exists for all
.
- (C)
Quantum field theory: For quantum field theories or infinite systems, the Stone–von Neumann uniqueness breaks down. Haag’s theorem shows that the determination of a suitable representation of the canonical commutation relations becomes a dynamical problem, if the vacuum states for different couplings are different. Non-normal states arise that yield representations assigning different values to global observables, like densities. Due to the problem of inequivalent representations, it is not possible to represent the time evolution as a group of unitary transformations within a single representation, because the representation algebra may change into an inequivalent representation as time evolves.
One objective of this paper is to suggest that these three open problems are, in fact, related to each other, even though they seem to be unrelated at first sight.
The present article suggests that the common denominator of Problems (A)–(C) associated with the mathematical framework described in the Introduction is the concept of time flow as a translation, implicitly assumed on the left-hand side of Equation (1). The common origin of Problems (A)–(C) emerges from studying the following two general questions associated with Equation (1).
Problem 1. Are there global solutions of equation (1), i.e., solutions for all ?
If global solutions and, hence, a group of *-automorphisms on exist, then this implies a continuous time evolution for all states . This means a time evolution independent of the state, which is not to be expected for general infinite systems without rescaling time. Rescaling of time is also expected to be necessary for establishing hydrodynamic limits governing invariant states.
Problem 2. If global solutions of Equation (1) exist, how can invariant solutions still change with time?
Local stationarity (invariance) in time arises from the underlying dynamics. Local stationarity in time is necessary, if thermodynamic observables, such as temperature, pressure or densities, are to provide an approximate representation of the physical system that changes slowly on long time scales. Hence, one has to study the set of stationary states that are invariant under the time evolution. If the thermodynamic observables change, then there must exist many invariant states and many possible time averages, i.e., the time averages are not unique.
The objective of this paper is to introduce a framework in which questions concerning the abundance of time-invariant states and their embedding in the set of all states can be posed mathematically in a proper way.
3. Almost Invariant States
Strictly stationary or invariant states [
15] are an idealization. In experiments, stationarity is never ideal, but only approximate. Expectation values are uncertain within the accuracy of the experiment. Experimental accuracy depends on the response and integration times of the experimental apparatus.
These experimental restrictions suggest to focus on a class of states that are stationary (invariant) only up to a given experimental accuracy
ε. To do so, recall the definition of invariant (stationary) states. A state
is called invariant if
holds for all
and
,
i.e., if the expectation values
of all observables
are constant. The set of invariant states
over
is convex and compact in the weak*-topology [
6]. The same holds for the set of all states
. Invariant states are fixed points of the adjoint time evolution
, as seen from Equation (8). Because invariant states are fixed points of
, they are of limited benefit for a proper mathematical formulation of the problems discussed above. Once an orbit in state space reaches an invariant state, it remains forever in that state and cannot leave it.
Almost invariant states are based on states whose expectation values are of bounded mean oscillation (BMO). A state
is called a BMO-state if all maps
have bounded mean oscillation for all
. The Banach space
of functions with bounded mean oscillation on ℝ is defined as the linear space
where
is the space of locally-integrable functions
. The BMO-norm is defined as
where
denotes intervals of length
and
denotes the average of
f over the interval
I. The set of all BMO-states
is convex by linearity. As a subset
of a weak* compact set, it is itself weak* compact. Hence, a decomposition theory into extremal BMO-states exists by virtue of the Krein–Milman theorem. The set of invariant states is identified through
as a subset
.
A BMO-state will be called
ε-almost invariant or almost invariant with accuracy
ε if the expectation of all observables are stationary to within experimental accuracy
ε. More precisely, the set
of all
ε-almost invariant states is defined as
as a family of subsets of
. For small
, these states are almost invariant. The accuracy
ε measures temporal fluctuations away from the time average.
The following inclusions of classes of states used in the following are summarized for orientation and convenience
where
and the set of KMS-states
at inverse temperature
are defined as states
, such that the KMS-condition [
16]
holds for all
and
. The KMS-states are invariant states for all
, but KMS-states for infinite volume systems at different
β are often disjoint [
16]. For
, the KMS-states are trace states,
i.e.,
holds for all
. Because KMS-states are Gibbs states, they are usually interpreted as equilibrium states with extremal states corresponding to pure thermodynamic phases [
16].
4. Indistinguishability of States
Experimental uncertainties limit also the ability to distinguish different states. Two states are experimentally indistinguishable (or metrologically equivalent) if they cannot be distinguished by measurements. Let
denote the maximal number of experiments that can be performed to distinguish the states of the system. Let
with
denote the observables in these experiments, and let
(
) be the experimental resolutions or accuracy that can be attained for
. Two states
with
for all
are called metrologically equivalent or experimentally indistinguishable with respect to the observables
. The sets of indistinguishable states
are
η-neighborhoods of z in the weak* topology [
17]. The algebra
generated by the elements
will be called the macroscopic algebra.
In the following, and will be assumed. The η-neighborhoods of ε-almost invariant states, i.e., the sets with for small , will be the candidates for local (in time) stationary states.
5. Invariant Measures on BMO-states
The set of BMO-states is weak*-compact. Its open subsets are the elements of the weak*-topology restricted to . They generate the σ-algebra of Borel sets on . Let denote an invariant state, so that Equation (15) holds for all . An invariant probability measure on corresponding to the invariant can be constructed with the help of a resolution of the identity on .
Let
denote the cyclic representation canonically associated with an invariant state
and the time evolution
on
. It is uniquely determined by the two requirements
for
,
and
for
. Let
denote the scalar product in
.
A resolution of the identity ([
13] p. 301) on the Borel
σ-algebra
is a mapping
with the properties
,
Each is a self-adjoint projector.
If , then
For every
and
, the set function
defined by
is a complex regular Borel measure on
.
Because the projectors are self-adjoint, the set function
is a positive measure for every
. For
, the resulting measure
is an invariant probability measure on the measurable space
associated with the invariant BMO-state
. The triple
is a probability space. The probability measure
is invariant under the adjoint time evolution
on
.
6. Almost Invariance and Recurrence
To discuss the question of how invariant states can evolve in time (Problem 2), consider two invariant states
and the straight line segment
connecting
and
. Of course,
. In practical applications,
might be a more or less general subset of
, e.g., a KMS-state in
. Straight line segments of invariant states are expected to be physically important for phase transformations at thermodynamic coexistence. Define a weak*-neighborhood
of
ε-almost invariant
η-indistinguishable states near
. Depending on the invariant states
and the macroscopic algebra
of interest, a similar weak*-neighborhood
can be defined for other subsets of
.
The time translations
with time scale
τ translate any initial state
along its orbit according to
where
denotes the initial instant,
and
the time scale. Discretizing time as
with
, such that
produces discretized orbits
,
for all
as iterates of
. For every initial state
, define
as the first return time of
into the set
. For all invariant
, one has
. For states
that never return to
, one sets
. For all
, let
denote the subset of states with recurrence time
with
interpreted as
The states
generate a one-parameter family of resolutions of the identity resulting in a one-parameter family of measures on
denoted as
with
. Their mixture
is again an invariant measure on
. The numbers
define a discrete probability density on
. It may be interpreted as a properly-weighted probability of recurrence into the neighborhood
of the straight line segment
.
7. Results
The time evolution of almost invariant states can be defined by the addition of random recurrence times. Let
be the probability density of the sum
of
independent and identically-distributed random recurrence times
. Let
from Equation (39) be the common probability density of all
. Then, with
and
,
is an
N-fold convolution of the discrete recurrence time density in Equation (39). The family of distributions
obeys
for all
, and the discrete analogue of Equation (5)
holds for all
. Because the individual states in
are indistinguishable within the given accuracy
η, but may evolve very differently in time, it is natural to define the duration of time needed for the first recurrence (a single time step) as an average
over recurrence times. If a macroscopic time evolution with a rescaled time exists, then one has to rescale the sums
and the iterations
in the limit
with suitable norming constants
.
Theorem 3 Let be the probability density of specified above in (41).
If the distributions of converge to a limit as for suitable norming constants , then there exist constants and , such thatwhereif diverges, whileif converges. For ,
the function is .
For ,
the function for andfor .
Proof. The existence of a limiting distribution for
is known to be equivalent to the stability of the limit [
18]. If the limit distribution is nondegenerate, this implies that the rescaling constants
have the form
where
is a slowly varying function [
19], defined by the requirement that
holds for all
. That the number
α obeys Equation (47) is proven in [
18] (p. 179). It is bounded as
, because the rescaled random variables
are positive.
To prove Equation (46), note that the characteristic function of
is the
N-th power
because the characteristic functions
of
are identical for all
. Inverse Fourier transformation gives
where
and
ξ was substituted with
. Let
denote the characteristic function of
, so that
holds.
Following [
20], the difference
in (46) can be decomposed and bounded from above as
with constants
to be specified below. The terms involving
from the second and third integral have been absorbed in the fourth integral. The four integrals are now discussed further individually.
The first integral converges uniformly to zero for , because belongs to the domain of attraction of a stable law with index α, as already noted above.
To estimate the second integral, note that the characteristic function
belongs to the domain of attraction for index
α if and only if it behaves for
as [
20]
where
and
is a slowly varying function at infinity obeying
By the representation theorem for slowly varying functions ([
21] p. 12), there exist functions
and
, such that the function
can be represented as
for some
where
is measurable and
, as well as
hold for
. As a consequence
so that with
and
is obtained for
. Therefore, there exists for any
a positive number
independent of
N, such that
for sufficiently large
N. If
N is sufficiently large, it is then possible to choose an
(and find
), such that
and this converges to zero for
.
The third integral is estimated by noting that
for
. Hence, there is a positive constant
, such that
for
. Consequently, with Equation (50),
converges to zero as
Finally, the fourth integral converges to zero, because the characteristic function is integrable on ℝ. In summary, all four terms in Equation (56) vanish for , and Equation (46) holds. □
Equation (46) implies
for sufficiently large
N and all
τ. Inserting this into Equation (45) gives
For
, the average return time
is proportional to the discretization
τ. In the case
, the average time
for return into the set
in a single step diverges. This suggests an infinite rescaling of time as
for
. This rescaling of time combined with
was called the ultra-long-time limit in [
22]. In the ultra-long-time limit
with
one finds from Equation (67) the result
for sufficiently large
N and
τ. The limit gives rise to a family of one-parameter semigroups
(with family index
α and parameter
h) of ultra-long-time evolution operators
which are convolutions instead of translations. Note that
because
and
. The rescaled age evolutions
are called fractional time evolutions, because their infinitesimal generators are fractional time derivatives [
22,
23].
The result shows that a proper mathematical formulation of local stationarity requires a generalization of the left-hand side in Equation (1), because Equation (1) assumes implicitly a translation along the orbit. In general, the integration of infinitesimal system changes leads to convolutions instead of just translations along the orbit [
22,
23]. Of course, translations are a special case of convolutions, to which they reduce in the case when the parameter
α approaches unity. For
, one finds
and therefore
is a right translation. Here,
is an age or duration. This shows that also the special case of induced right translations does not give a group, but only a semigroup.
8. Discussion
The introduction of the sets of ε-almost invariant BMO-states with has provided a mathematical framework in which questions concerning the abundance of time-invariant states and their embedding in the set of all states can be posed mathematically in a proper way. The class of BMO-states reflects in its definition the experimental reality that observations are always performed by integration of experimental data over time intervals. BMO-states allow for singular expectation values, thereby establishing a general framework to discuss Problems 1 and 2 above.
There exists a direct relation between Theorem 3 and the BMO-states. It is given by Equation (32), which directly determines the values of α and D, as well as the function Λ in Theorem 3 and Equation (70).
The result in Equation (70) shows that the left-hand side in a coarse-grained or rescaled version of Equation (1) may not always be a time translation along the orbits of the original unscaled dynamics. Instead, the left-hand side is in general the infinitesimal generator of a convolution along time rescaled orbits of ε-almost invariant states. The orbits of ε-almost invariant states can approach the manifold of invariant states of the physical system or subsystem of interest at every point for any length of time without being trapped.
As discussed above, the result in Equation (70) implies a general concept of time flow and, hence, provides a new perspective on the issue of irreversibility [
22,
24,
25]. It suggest a reformulation [
25,
26] of the much discussed irreversibility problem. The normal problem can be stated as:
Problem 4 (The normal irreversibility problem). Assume that time is reversible. Explain how and why time irreversible equations arise in physics.
The assumption that time is reversible,
i.e.,
, is made in all fundamental theories of modern physics. The explanation of macroscopically irreversible behavior for macroscopic nonequilibrium states of subsystems is due to Boltzmann. It is based on the applicability of statistical mechanics and thermodynamics, the large separation of scales, the importance of low entropy initial conditions and probabilistic reasoning [
27].
The problem with with assuming
is that an experiment (
i.e., the preparation of an initial state within an infinity of
η-indistinguishable initial states for a dynamical system) cannot be repeated yesterday, but only tomorrow [
25]. While it is possible to translate the spatial position of a physical system forward and backward in space, it is not possible to translate the temporal position of a physical system backwards in time. Translating an experiment backward in time is not the same as reversing the momenta of all particles in a physical system, as emphasized in [
25,
26]. These observations combined with Equations (71) and (72) suggest to reformulate the normal irreversibility problem above as:
Problem 5 (The reversed irreversibility problem). Assume that time evolution is always irreversible. Explain why time reversible equations are more frequent in physics.
The reversed irreversibility problem was introduced in [
25]. Its solution is given by Theorem 3 combined with two additional facts. Firstly, ultra-long-time evolutions with
are always irreversible, while those with
may be irreversible or reversible, depending on the operator on the right-hand side of Equation (1). Secondly, the set of recurrence time distributions
in the domain of attraction for the case
comprises all distributions whose first moment
exists, independent of their tail behavior. Contrary to this, the domain of attraction for the case
is restricted to those
with the correct tail behavior. Thus, the domain of attraction is much larger for
than for
. This explains why equations of motion with time reversal symmetry arise more frequently.
Because anomalous time evolutions from Equation (70) with
must be expected on theoretical grounds, they are attracting increasing experimental interest [
15,
28]. For the example of broadband dielectric spectroscopy in glasses, generalized relaxation functions and susceptibilities based on Equation (70) have already been successfully compared to experiments [
23,
29,
30,
31,
32]. Theoretical, mathematical and experimental studies are encouraged to further explore the consequences of the generalized concept.