[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)
coupled to a “reservoir”
,
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
with a selection of
macroscopic degrees of freedom of a large or infinite
many body system
,
while
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
are usually much faster than the characteristic
time scale for the evolution of the system
of interest.
[627.1.6] Equally important for macroscopic dynamics
and thermodynamic behaviour
is scale separation in the size of
and
.
[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 of
states or observables
![]() |
![]() |
(1a) | |
![]() |
![]() |
(1b) |
where is the initial value,
are time instants
measured in units of
seconds (such that
)
and
provides energy units (Joule)
for the infinitesimal generator
(Liouvillian),
which is a linear, often unbounded, operator with
domain
.
[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
in eq. (1)
describes infinitesimal changes of these states and
observables with time starting
from an initial condition
.
[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
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 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
![]() |
(2) |
[page 628, §0]
where the maps
and
are
![]() |
![]() |
(3a) | |
![]() |
![]() |
(3b) |
and the orbit maps are defined as
![]() |
(4) |
for each fixed , if
with
is a one-parameter familiy of *-automorphisms of
.
[628.0.1] Of course, the problem is to give meaning to the formal exponential
in eq. (3a) such that the orbit maps
are continuous for every
.
[628.1.1] The one-parameter family of
*-automorphisms is expected
to obey the time evolution law
![]() |
(5) |
with being the identity.
[628.1.2] The continuity of the orbit maps may be rephrased
as continuity of the maps
from
into the space
of all bounded operators
on
endowed with the strong operator topology [8, 9].
[628.1.3] The operator family
is then
a strongly continuous one-parameter group (
-group) on
.
[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
.
[628.2.3] The notation
is used for
the value
of a selfadjoint
in the state
.
[628.2.4] States are positive,
for all
, and normalized,
,
linear functionals on the algebra
of observables [6].
[628.2.5] The adjoint time evolution
with
consists of all adjoint operators
on the dual space
[12, 10].
[628.2.6] Let
denote the set of all states.
[628.2.7] The orbit maps for states
are defined as
![]() |
(6) |
for states .
[628.2.8] If
is strongly continuous then
![]() |
(7) |
shows that the adjoint time evolution is weak*-continuous in
the sense that the maps
![]() |
(8) |
are continuous for all .
[628.2.9] These maps are the time evolutions of all expectation values.
[628.2.10] In other words for a
-group
the orbit maps
are
continuous from
into the space
of all bounded operators on
endowed with the
weak* topology [8, 13] and the adjoint
family
is a
-group.
[628.2.11] Note, that the adjoint time evolution
need not be strongly continuous
unless
is reflexive.
[628.2.12] The relation between the time evolution of states
and observables is
![]() |
![]() |
||
![]() |
|||
![]() |
(9) | ||
![]() |
|||
![]() |
[page 629, §0]
where .
[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
![]() |
(10) |
of the infinitesimal generator of
time translations and
the infinitesimal generator
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
![]() |
(11a) | ||
![]() |
(11b) |
where the first equation reflects the Heisenberg picture, while the second corresponds to the Schrödinger picture.