In these lectures I will discuss the two kinds of evolution of
The first type
concerns evolution of closed,
isolated1 quantum systems that evolve
under the action of prescribed external forces, but are not disturbed by
observations, and are not coupled thermodynamically or in some other
irreversible way to the environment. This evolution is governed by the
Schrödinger equation and is also known as a unitary or, more
generally, as an automorphic evolution. In contrast to this idealized
case (only approximately valid, when irreversible effects can be neglected),
quantum theory is also concerned with a different kind of change of state.
It was first formulated by J. von Neumann (cf. [30, Ch. V. 1] and is
known as von Neumann - Lüders projection postulate. It tells us
roughly this: if some quantum mechanical observable is being measured,
then - as
a consequence of this measurement - the actual state of the quantum system
jumps into one of the eigenstates of the measured
observable. 2 This jump was thought to be abrupt and take no time at all
, it is also known as
reduction of the wave packet. Some physicists feel quite uneasy about this
von Neumann's postulate, to the extent that they reject it either as too
primitive (not described by dynamical equations) or as unnecessary. We will
come to this point later, in Chapter 3, when we will discuss piecewise
stochastic processes that unite both kinds of evolution.
- CONTINUOUS evolution of closed systems
- STOCHASTIC evolution of open systems.
It is rather easy to explain to the mathematician the Dirac equation -
it became already a part of the mathematical folklore. But Dirac's equation
belongs to field theory rather than to
quantum theory. 3 Physicists are being
taught in the course of their education rather early that every attempt at a
sharp localization of a relativistic particle results in creation and
annihilation processes. Sometimes it is phrased as: "there is no relativistic
quantum mechanics - one needs to go directly to Relativistic Quantum Field
Theory". Unfortunately we know of none non-trivial, finite,
field theory in four dimensional space-time continuum. Thus we are left
with the mathematics of perturbation theory. Some physicists believe
that the physical ideas of relativistic quantum field theory are sound, that
it is the best theory we ever had, that it is "the exact description of
the difficulties we have with it are only temporary, and that they will be
overcomed one day -
the day when bright mathematicians will provide us with new, better, more
powerful tools. Some other say: perturbation theory is more than sufficient
for all practical purposes, no new tools are needed, that is how physics
is - so mathematicians better accept it, digest it, and help the physicists
to make it more rigorous and to understand what it is really about. Still some
minority, also believe that it is only a temporary situation, which
one day will be resolved. But the resolution will come
owing to essentially
new physical ideas,
and it will result in a new quantum paradigm, more appealing than the
present one. It should perhaps be not a surprise if, in an appropriate sense,
these points of view will turn out to be right. In these lectures we will be
concerned with the well established Schrödinger equation, which is at the
very basis of the current quantum scheme, and with its dissipative
generalization - the Liouville equation.
In these equations we assume that we
know what the time is. Such a knowledge is negated in special
relativity, 5 and this results
in turn in all kinds of troubles that we are facing since the birth of
Einstein's relativity till this day.
The Schrödinger equation is more
difficult than the Dirac one, and this for two reasons:
first, it lives on the
background of Galilean relativity - which have to deal with much more
intricate geometric structures than Einstein relativity. 7 Second,
Schrödinger's equation is about Quantum Mechanics and we have to take
care about probabilistic interpretation, observables, states
etc. - which is possible for Schrödinger equation but faces
problems in the first-quantized Dirac theory.
Let me first make a general comment about Quantum Theory. There are physicists
who would say: quantum theory is about calculating of Green's Functions
- all numbers of interest can be obtained from these functions and all the other
mathematical constructs usually connected with quantum theory are superfluous
and unnecessary! It is not my intention to depreciate the achievements of Green's
Function Calculators. But for me quantum theory - as any other physical theory
- should be about explaining things that happen outside of us - as long
as such explanations are possible. The situation in Quantum Theory today,
more than 60 years after its birth, is that Quantum Theory explains much less
than we would like to have been explained. To reduce quantum theory to Green's
function calculations is to reduce its explanatory power almost to zero. It may
of course be that in the coming Twenty First Century humanity will unanimously
recognize the fact that `understanding'was a luxury, a luxury of the "primitive
age" that is gone for ever. But I think it is worth while to take a chance and
to try to understand as much as can be understood in a given situation. Today
we are trying to understand Nature in terms of geometrical pictures and
random processes8 More specifically, in quantum theory, we are
trying to understand in terms of observables, states, complex
Feynman amplitudes etc. In the next chapter, we will show the way that leads
to the Schrödinger Equation using geometrical language as much as
possible. However, we will not use the machinery of geometrical quantization
because it treats time simply as a parameter, and space
as absolute and given once for all. 9On the other hand, geometrical quantization
introduces many advanced tools that are unnecessary for our purposes, while at
the same time it lacks the concepts which are important and necessary.
Before entering the subject let me tell you the distinguishing feature of the
approach that I am advocating, and that will be sketched below in Ch. 2: one obtains
a fibration of Hilbert spaces over time. There is a distinguished family of local trivializations, a family
For each , the Hilbert space is a Hilbert space of sections of a complex line bundle over
. A space-time observer (that is, a reference frame) allows us to
identify the spaces for different -s, while a gauge allows us to identify the fibers. Schrödinger's dynamics
of a particle in external gravitational and electromagnetic fields is given by
a Hermitian connection in Solutions of Schrödinger equation are parallel sections
of Thus Schrödinger equation can be written as
- space-time observer
or, in a local trivialization, as
where will be a self-adjoint operator in . 10Gravitational and electromagnetic forces are
coded into this Schrödinger's connection. Let us discuss the resulting structure.
First of all there is not a single Hilbert space but a family of Hilbert
spaces. These Hilbert spaces can be identified either using an observer
and gauge or,better, by using a background dynamical connection. It is
only after so doing that one arrives at single-Hilbert-space picture of the textbook
Quantum Mechanics - a picture that is the constant source of lot of confusion.
In Quantum Mechanics we have a dual scheme - we use the concepts of
observables and states. We often use the word measurement
in a mutilated sense of simply pairing an observable with a state to get the expected result - a number
One comment is in place here: to compare the results of actual measurements with
predictions of the theory - one needs only real numbers. However experience
proved that quantum theory with only-real-numbers is inadequate. So, even if the
fundamental role of in Quantum Theory is far from being fully understood - we
use in Quantum Theory only complex Hilbert spaces, complex algebras etc.12However, usually, only real numbers are at the
Now, it is not always clear what is understood by states and observables.
There are several possibilities:
Figure: There are several possibilities
of understanding of state and observables. They can be instant , and thus
time-dependent, or they can be time-sections - thus time- independent
As it was already said, it is usual in standard presentations of the quantum
theory to identify the Hilbert spaces . There are several options there. Either we identify them
according to an observer (+ gauge) or according to the dynamics. If we identify
according to the actual dynamics, then states do not change in time - it is always
the same state-vector, but observables (like position coordinates) do change in
time - we have what is called the Heisenberg picture. If we identify them according
to some background "free" dynamics - we have so called interaction picture.
Or, we can identify Hilbert spaces according to an observer - then observables
do not change in time, but state vector is changing - we get the Schrödinger
However, there is no reason at all to identify the -s. Then dynamics is given by parallel transport operators:
The Schrödinger equation describes time evolution of pure states of a
quantum system, for instance evolution of pure states of a quantum particle, or
of a many body system. Even if these states contain only statistical information
about most of the physical quantities, the Schrödinger evolution of pure
states is continuous and deterministic . Under this evolution Hilbert
space vector representing the actual state of the system13 changes continuously with time, and with it
there is a continuous evolution of probabilities or potentialities,
but nothing happens - the formalism leaves no place for events.
Schrödinger equation helps us very little, or nothing at all, to understand
how potential becomes real. So, if we aim at understanding of this
process of becoming, if we want to describe it by mathematical equations and
to simulate it with computers - we must go beyond Schrödinger's dynamics.
As it happens, we do not have to go very far - it is sufficient to relax only
one (but important) property of Schrödinger's dynamics and to admit
that pure states can evolve into mixtures. Instead of Schrödinger equation
we have then a so called Liouville equation that describes time evolution of mixed
states. It contains Schrödinger equation as a special case.14It was shown in  that using the Liouville type
of dynamics it is possible to describe coupling between quantum systems and classical
degrees of freedom of measurement devices. One can derive also a piecewise deterministic
random process that takes place on the manifold of pure states. In this way one
obtains a minimal description of "quantum jumps" (or "reduction of wave packets")
and accompanying, directly observable jumps of the coupled classical devices.
In Ch. 3 simple models of such couplings will be discussed. The interested reader
will find more examples in Refs. -.15In particular in  the most advanced model of this
kind, the SQUID-tank model is discussed in details.
Galilean General Relativity is a theory of space-time structure, gravitation and
electromagnetism based on the assumption of existence of an absolute time function.
Many of the cosmological models based on Einstein`s relativity admit also a distinguished
time function. Therefore Galilean physics is not evidently wrong. Its predictions
must be tested by experiments. Galilean relativity is not that elegant as the
one of Einstein. This can be already seen from the group structures: the homogeneous
Lorentz group is simple, while the homogeneous Galilei group is a semidirect
product of the rotation group and of three commuting boosts. Einstein`s theory
of gravitation is based on one metric tensor, while Galilean gravity needs both:
space metric and space-time connection. Similarly for quantum mechanics: it is
rather straightforward to construct generally covariant wave equations for Einstein`s
relativity, while general covariance and geometrical meaning of the Schrödinger
equation was causing problems, and it was not discussed in textbooks. In the following
sections we will present a brief overview of some of these problems.
Let us discuss briefly geometrical data that are needed for building up generally
covariant Schrödinger's equation. More details can be found in Ref. .16
Our space-time will be a refined version of that of Galilei and of Newton, i.e.
space-time with absolute simultaneouity. Four dimensional space-time is fibrated over one-dimensional time The fibers of are three-dimensional Riemannian manifolds, while the basis is an affine space over By a coordinate system on we will always mean a coordinate system adapted to the fibration. That means: any two
events with the same coordinate are simultaneous, i.e. in the same fibre of
Coordinate transformations between any two adapted coordinate systems are of the
We will denote by the time form Thus in adapted coordinates .
is equipped with a contravariant degenerate metric tensor
which, in adapted coordinates, takes the form
where is of signature We denote by the inverse matrix. It defines Riemannian metric in the fibers of
We assume a torsion-free connection in that preserves the two geometrical objects and . 17The condition is equivalent to the conditions on the connection coefficients. Let us introduce
the notation Then is equivalent to the equations:
Then, because of the assumed zero torsion, the space part of the connection can
be expressed in terms of the space metric in the Levi-Civita form:
>From the remaining equations:
we find that the -symmetric part of is equal to otherwise the connection is undetermined.
We can write it, introducing a new geometrical object , as
where is antisymmetric. Notice that is not a tensor, except for pure space transformations or time
A basis in is called a Galilei frame if , and if are unit space-like vectors. If and are two Galilei frames at the same space-time point, then
they are related by a transformation of the homogeneous Galilei group :
where and is an orthogonal matrix. The bundle of Galilei frames is a principal bundle.
The homogeneous Galilei group acts on in two natural ways: by linear and by affine transformations.
The first action is not effective one - it involves only the rotations:
The bundle associated to this action can be identified with the vertical subbundle
of - i.e. with the bundle of vectors tangent to the fibers of .
acts also on by affine isometries :
To this action there corresponds an associated bundle, which is an affine bundle
over the vector bundle . It can be identified with the subbundle of consisting of vectors tangent to , and such that or, equivalently, as the bundle of first jets of
sections of . We will call it .
We will denote by the coordinates in corresponding to coordinates of .
The connection can be also considered as a principal connection in the bundle
of Galilei frames. It induces an affine connection in the affine bundle . As a result, it defines a natural
-valued one-form on . It can be described as follows: given a vector tangent to at it projects onto . Then is defined as the difference of and the horizontal lift of . It is a vertical tangent vector to , and can be identified with an element of . In coordinates:
There is another -valued one-form on , namely the canonical form . Given at , we can decompose into space- and time-component along . Then is defined as its space component. In coordinates:
Then, because the fibers of are endowed with metric , we can build out the following important two-form on :
The following theorem, proven in , gives a necessary and sufficient
condition for to be closed.
Let be a principal bundle over and let be its pullback to . We denote by and the associated Hermitian line bundles corresponding to
the natural action of on . There is a special class of principal connections on
, namely those whose connection forms vanish on vectors
tangent to the fibers of . As has been discussed in  specifying such a connection on
is equivalent to specifying a system of connections
on parameterized by the points in the fibers of . Following the terminology of  we call such a connection
universal.19The fundamental assumption that leads to the
Schrödinger equations reads as follows:
Quantization Assumption: There exists a universal connection whose curvature is .20
We call such an a quantum connection. >From the explicit form of one can easily deduce that is necessarily of the form
where parameterizes the fibres of ,
and is a local potential
As it is shown in , there exists a natural -invariant metric on of signature . Explicitly
Using this metric we can build out a natural Lagrangian for equivariant functions
or, equivalently, for sections of the line bundle . The Euler-Cartan equation for this Lagrangian will prove to be nothing
but the Schrödinger equation. Notice that the action of group on defines an Killing vector field for which is isotropic. Therefore the above construction
can explain why the approach of  works.
More precisely, the construction leading to the generally covariant Schrödinger-Pauli
equation for a charged spin particle in external gravitational and electromagnetic field can
be described as follows.
The contravariant metric , ,
can be obtained from the following Clifford algebra of complex matrices:
One takes then 5-dimensional charged Dirac operator and considers spinors that are equivariant with respect to
the fifth coordinate
This first-order, four-component spinor (called Lévy-Leblond equation in
Ref. ) equation reduces then easily
to the second-order, two-component Schrödinger Pauli equation with the correct
We finish this section with pointing to the Ref. , where Schrödinger's quantization
is discussed in details and where a probabilistic interpretation of generally
covariant Schrödinger equation is given using the bundle of Hilbert spaces. The parallel transport induced by the
quantum connection is shown in  to be directly related to Feynman
Replacing Schrödinger's evolution, which governs the dynamics of pure states,
by an equation of the Liouville type, that describes time evolution of mixed states,
is a necessary step - but it does not suffice for modeling of real world events.
One must take, to this end, two further steps. First of all we should admit that
in our reasoning, our communication, our description of facts - we are
using classical logic. Thus somewhere in the final step of transmission
of information from quantum systems to macroscopic recording devices and further,
to our senses and minds, a translation between quantum and classical should
take place. That such a translation is necessary is evident also when we consider
the opposite direction: to test a physical theory we perform controlled
experiments. But some of the controls are always of classical nature
- they are external parameters with concrete numerical values. So, we need to
consider systems with both quantum and classical degrees of freedom, and we need
evolution equations that enable communication in both directions, i.e. :
- flow of information from quantum to classical
- control of quantum states and processes by classical parameters .
We begin with a brief recall of relevant mathematical concepts. Let be a - algebra. We shall always assume that has unit . An element is positive, , iff it is of the form for some . Every element of a -algebra is a linear combination of positive elements. A linear
functional is positive iff implies . Every positive functional on a -algebra is continuous and Positive functionals of norm one are called
states. The space of states is a convex set. Its extremal points are called
pure states. The canonical GNS construction allows one to associate
with each state a representation of on a Hilbert space , and a cyclic vector such that Irreducibility
of is then equivalent to purity of
Quantum theory gives us a powerful formal language and statistical algorithms
for describing general physical systems. Physical quantities are coded
there by Hermitian elements of a -algebra of observables, while information about their values
(quantum algorithms deal, in general, only with statistical information) is coded
in states of . Pure states correspond to a maximal possible information.
For each state and for each the (real) number is interpreted as expectation value of observable in state while
is the quadratic dispersion of in the state It is assumed that repeated measurements of 22made on systems prepared in a state
will give a sequence of values so that approximately and If is Abelian, then it is isomorphic to an algebra of functions
Then pure states of are dispersion free - they are parameterized by points and we have This corresponds to a classical theory:
all observables mutually commute and maximal possible information is without any
statistical dispersion. In the extreme opposition to that is pure quantum
theory - here defined as that one when is a factor, that is has a trivial centre. The centre of a -algebra is defined as In general If Z (A) = A - we have pure classical theory. If - we have pure quantum theory. In between we have
a theory with superselection rules. Many physicists believe that the ''good
theory'' should be a ''pure quantum'' theory. But I know of no one good reason
why this should be the case. In fact, we will see that cases with a nontrivial
are interesting ones. Of course, one can always argue that
whenever we have an algebra with a nontrivial centre - it is a subalgebra of an
algebra with a trivial one, for instance of - the algebra of all bounded operators on some Hilbert
space. This is, however, not a good argument - one could argue as well that we
do not need to consider different groups as most of them are subgroups of - the unitary group of an infinite dimensional Hilbert
space - so why to bother with others?
Let be -algebras. A linear map is Hermitian if It is positive iff implies Because Hermitian elements of a -algebra are differences of two positive ones - each positive
map is automatically Hermitian. Let denote the by matrix algebra, and let be the algebra of matrices with entries from Then carries a natural structure of a -algebra. With respect to this structure a matrix from is positive iff it is a sum of matrices of the form If is an algebra of operators on a Hilbert space , then can be considered as acting on Positivity of is then equivalent to or equivalently, to for all
A positive map is said to be completely positive or, briefly, CP
by , is positive for all When written explicitly, complete positivity is equivalent
for every and In particular every homomorphism of
algebras is completely positive. One can also show that if
either or is Abelian, then positivity implies complete positivity. Another
important example: if is a algebra of operators on a Hilbert space , and if then is a CP map The celebrated Stinespring theorem gives us
a general form of a CP map. Stinespring's construction can be described as follows.
Let be a CP map. Let us restrict to a case when Let be realized as a norm closed algebra of bounded operators on
a Hilbert space . One takes then the algebraic tensor product and defines on this space a sesquilinear form by
This scalar product is then positive semi-definite because of complete positivity
of Indeed, we have
Let denote the kernel of . Then is a pre-Hilbert space. One defines
a representation of on by One shows then that is invariant under , so that goes to the quotient space. Similarly, the map defines
an isometry We get then on the completion of
(Stinespring's Theorem) Let
-algebra with unit and let
be a CP map. Then there exists a Hilbert space
and a bounded linear map
is an isometry iff
is unital i.e. iff
maps the unit of
into the identity operator of
are separable, then
can be taken separable.
The space of CP maps from to is a convex set. Arveson  proved that is an extremal element of this set iff the representation above is irreducible.
A dynamical semigroup on a -algebra of operators is a strongly continuous semigroup of CP maps of A into itself.
A semigroup is norm continuous iff its infinitesimal generator is bounded as a linear map We then have
The right hand side is, in this case, a norm convergent series for all
real values of however for negative the maps , although Hermitian, need not be positive.
Evolution of observables gives rise, by duality, to evolution of positive functionals.
One defines Then preserves the unit of iff preserves normalization of states. A general form of a generator
of a dynamical semigroup in finite dimensional Hilbert space has been derived
by Gorini, Kossakowski and Sudarshan , and Lindblad  gave a general form of a bounded
generator of a dynamical semigroup acting on the algebra of all bounded operators
It is worthwhile to cite, after Lindblad, his original motivation:
"The dynamics of a finite closed quantum system is conventionally represented
by a one-parameter group of unitary transformations in Hilbert space. This formalism
makes it difficult to describe irreversible processes like the decay of unstable
particles, approach to thermodynamic equilibrium and measurement processes . It seems that the only possibility of introducing an irreversible
behaviour in a finite system is to avoid the unitary time development altogether
by considering non-Hamiltonian systems. "
In a recent series of papers [7,8,9,10] Ph. Blanchard and the present
author were forced to introduce dynamical semigroups because of another difficulty,
namely because of impossibility of obtaining a nontrivial Hamiltonian coupling
of classical and quantum degrees of freedom in a system described by an algebra
with a non-trivial centre. We felt that lack of a dynamical understanding of quantum
mechanical probabilistic postulates is more than annoying. We also believed that
the word ''measurement'' instead of being banned, as suggested by J. Bell [5,6], can be perhaps given a precise
and acceptable meaning. We suggested that a measurement process is a coupling
of a quantum and of a classical system, where information about quantum state
is transmitted to the classical recording device by a dynamical semigroup of the
total system . It is instructive to see that such a transfer of information
can not indeed be accomplished by a Hamiltonian or, more generally, by any automorphic
evolution23. To this end consider an algebra with centre Then describes classical degrees freedom. Let be a state of then denotes its restriction to Let be an automorphic evolution of and denote Each is an automorphism of the algebra and so it leaves its centre invariant: The crucial observation is
that, because of that fact, the restriction depends only on as the evolution of states of is dual to the evolution of the observables in This shows that information transfer from the total algebra
to its centre is impossible - unless we use more general, non-automorphic
>From the above reasoning it may be seen that the Schrödinger picture,
when time evolution is applied to states, is better adapted to a discussion of
information transfer between different systems. The main properties that a dynamical
semigroup describing time evolution of states should have are: should preserve convex combinations, positivity and normalization.
One can demand even more - it is reasonable to demand a special kind of stability:
that it should be always possible to extend the system and its evolution in a
trivial way, by adding extra degrees of freedom that do not couple to our system.24That is exactly what is assured by complete
positivity of the maps One could also think that we should require even more, namely
that transforms pure states into pure states. But to assume that
would be already too much, as one can prove that then must be dual to an automorphic evolution. It appears that
information gain in one respect (i.e. learning about the actual state of the quantum
system) must be accompanied by information loss in another one - as going from
pure states to mixtures implies entropy growth.
We will apply the theory of dynamical semigroup to algebras with a non-trivial
centre. In all our examples we will deal with tensor products of and an Abelian algebra of functions. The following
theorem by Christensen and Evans  generalizes the results of Gorini,
Kossakowski and Sudarshan and of Lindblad to the case of arbitrary -algebra.
We will apply this theorem to the cases of being a von Neumann algebra, and the maps being normal. Then can be also taken normal. We also have so that We will always assume that or, equivalently, that Moreover, it is convenient to introduce then from we get and so Therefore we have
(Christensen - Evans) Let
be a norm-continuous semigroup of CP maps of a
- algebra of operators
Then there exists a CP map
into the ultraweak closure
and an operator
such that the generator
is of the form:
where denotes anticommutator. Of particular interest to us will
be generators for which is extremal 25. By the already mentioned result of Arveson
 this is the case when is of the form
where is an irreducible representation of on a Hilbert space and is a bounded operator (it must be, however,
such that ).
We consider a model describing a coupling between a quantum and a classical system.
To concentrate on main ideas rather than on technical details let us assume that
the quantum system is described in an -dimensional Hilbert space and that it has as its algebra of observables Similarly, let us assume
that the classical system has only a finite number of pure states Its algebra of observables is then isomorphic to For the algebra of the total system we take which is isomorphic
to the diagonal subalgebra of Observables of the total system are block
where are operators in 26Both and can be considered as subalgebras of consisting respectively of matrices of the form and States of the quantum system are represented by positive,
trace one, operators on States of the classical system are -tuples of non-negative numbers with States of the total system are represented
by block diagonal matrices with and For the expectation value we have Given a state of the total system, we can trace over the quantum system to get
an effective state of the classical system or we can trace over the classical system
to get the effective state of the quantum system 27
Let us consider dynamics. Since the classical system has a discrete set of pure
states, there is no non-trivial and continuous time evolution for the classical
system that would map pure states into pure states. As for the quantum system,
we can have a Hamiltonian dynamics, with the Hamiltonian possibly dependent on
time and on the state of the classical system As we already know a non-trivial coupling
between both systems is impossible without a dissipative term, and the simplest
dissipative coupling is of the form where is an irreducible representation of the algebra in a Hilbert space and is a linear map. It is easy to
see that such an is necessarily of the form:
where is an block matrix with only one non-zero entry. A more general
CP map of is of the same form, but with having at most one non-zero element in each of its rows.
Let us now discuss desired couplings in somewhat vague, but more intuitive, physical
terms. We would like to write down a coupling that enables transfer of information
from quantum to classical system. There may be many ways of achieving this aim
- the subject is new and there is no ready theory that fits. We will see however
that a naive description of a coupling works quite well in many cases. The idea
is that the simplest coupling associates to a property of the
quantum system a transformation of the actual state of the classical system.
Properties are, in quantum theory, represented by projection operators.
Sometimes one considers also more general, unsharp or fuzzy properties.
They are represented by positive elements of the algebra which are bounded by
the unit. A measurement should discriminate between mutually exclusive
and exhaustive properties. Thus one usually considers a family of mutually orthogonal
projections of sum one. With an unsharp measurement one associates a
family of positive elements of sum one.
As there is no yet a complete, general, theory of dissipative couplings of classical
and quantum systems, the best we can do is to show some characteristic examples.
It will be done in the following section. For every example a piecewise deterministic
random process will be described that takes place on the space of pure states
of the total system 28 and which reproduces the Liouville evolution
of the total system by averaging over the process. A theory of piecewise deterministic
(PD) processes is described in a recent book by M. H. Davis . Processes of that type, but without
a non-trivial evolution of the classical system, were discussed also in physical
literature - cf. Refs [13,17,18,20]. 29We will consider Liouville equations of the
where in general and the can explicitly depend on time. The will be chosen as tensor products , where act as transformations 30 on classical (pure) states.
First, we consider only one orthogonal projector on the two-dimensional Hilbert space To define the dynamics we choose the coupling operator
in the following way:
The Liouville equation (
) for the density matrix of the total system reads now
For this particularly simple coupling the effective quantum state evolves independently of the
state of the classical system. One can say that here we have only transport of
information from the quantum system to the classical one. We have:
The Liouville equation (
) describes time evolution of statistical states of the total system.
Let us describe now a the PD process associated to this equation. Let be a one-parameter semigroup of (non-linear) transformations of
rays in given by
Suppose we start with the quantum system in a pure state , and the classical system in a state (resp. ). Then starts to evolve according to the deterministic (but non-linear
Schrödinger) evolution until a jump occurs at time . The time of the jump is governed by an inhomogeneous Poisson process with
the rate function . Classical system switches
from to (resp. from to ), while jumps to , and the process starts
again. With the initial state being an eigenstate of , , the rate function is approximately constant and equal to . Thus can be interpreted as the expected time interval between
the successive jumps.
More details about this model illustrating the quantum Zeno effect can be found
in Ref. .
Using somewhat pictorial language we can say that in the previous example
each actualization of the property was causing a flip in the classical system. In the present example,
which is a non-commutative and fuzzy generalization of the model discussed in
, we consider in general fuzzy, properties The Hilbert space can be completely arbitrary, for instance -dimensional. We will denote The -s need not be projections, and the different -s need not to commute. The classical system is assumed to have states with thought of as an initial, neutral state. To each actualization of
the property there will be associated a flip between and Otherwise the state of the classical system will be unchanged.
To this end we take
The Liouville equation takes now the following form:
We will derive the PD process for this example in some more details, so that a
general method can be seen. First of all we transpose the Liouville equation so
as to get time evolution of observables; we use the formula
In the particular case at hand the evolution equation for observables looks almost
exactly the same as that for states:
Each observable of the total system defines now a function on the space of pure states of the total system
We have to rewrite the evolution equation for observables in terms of the functions
To this end we compute the expressions Let us first introduce the Hamiltonian
vector field on the manifold of pure states of the total system:
Then the terms can be written as We also introduce vector field corresponding to non-linear evolution:
Then evolution equation for observables can be written in a Davis form:
where is a matrix of measures, whose non-zero entries are:
The symbol denotes here the Dirac measure
We describe now PD process associated to the above semigroup. There are one-parameter (non-linear) semigroups acting on the space of pure states of the quantum system
If initially the classical system is in a pure state , and quantum system in a pure state , then quantum system evolves deterministically according to the
semigroup : . The classical system then jumps at
the time instant , determined by the inhomogeneous Poisson process with rate function
. If the classical system was in
one of the states , then it jumps to with probability one, the quantum state jumps at the same time to
the state . If, on the other hand, it
was in the state , then it jumps to one of the states with probability . The quantum
state jumps at the same time to Let
Then is the distribution of - the first jump time. More precisely, is the survival function for the state :
Thus the probability distribution of the jump is and the expected jump time is The probability that the jump will occur between and , provided it did not occur yet/, is equal to . Notice that this depends on the actual state . However, as numerical computation show, the dependence
is negligible and approximately jumps occur always after time . 31
In the previous example the coupling between classical and quantum systems
involved a finite set of non-commuting observables. In the present one we will
go to the extreme - we will use all one-dimensional projections in the
coupling. One can naturally discover such a model when looking for a precise answer
to the question:
how to determine state of an individual quantum system
For some time I was sharing the predominant opinion that a positive answer to
this question can not be given, as there is no observable to be measured that
answers the question: what state our system is in ?. Recently Aharonov
and Vaidman  discussed this problem in some
details. 32The difficulty here is in the fact that we have
to discriminate between non-orthogonal projections (because different
states are not necessarily orthogonal), and this implies necessity of
simultaneous measuring of non-commuting observables. There have been many papers
discussing such measurements, different authors taking often different positions.
However they all seem to agree on the fact that predictions from such
measurements are necessarily fuzzy. This fuzziness being directly related
to the Heisenberg uncertainty relation for non-commuting observables. Using methods
and ideas presented in the previous sections of this chapter it is possible to
build models corresponding to the intuitive idea of a simultaneous measurement
of several non-commuting observables, like, for instance, different spin components,
positions and momenta etc. A simple example of such a model was given in
the previous section. After playing for a while with similar models it is natural
to think of a coupling between a quantum system and a classical device that will
result in a determination of the quantum state by the classical device. Ideally,
after the interaction, the classical "pointer" should point at some vector in
a model Hilbert space. This vector should represent (perhaps, with some uncertainty)
the actual state of the quantum system. The model that came out of this simple
idea, and which we will now discuss, does not achieve this goal. But it is instructive,
as it shows that models of this kind are possible. I believe that one day somebody
will invent a better model, a model that can be proven to be optimal, giving the
best determination with the least disturbance. Then we will learn something important
about the nature of quantum states.
Our model will be formulated for a -state quantum system. It is rather straightforward to rewrite it for
an arbitrary -state system, but for we can be helped by our visual imagination. Thus we take for the Hilbert space of our quantum system. We can
think of it as pure spin . Pure states of the system form up the manifold which is isomorphic to the -sphere . Let , denote the Pauli -matrices. Then for each the operator has eigenvalues . We denote by the projection onto the -eigenspace.
For the space of pure states of the classical system we take also - a copy of . Notice that is a homogeneous space for . Let be the invariant measure on normalized to . In spherical coordinates we have . We denote the Hilbert space
of the total system, and by its von Neumann algebra of observables. Normal
states of are of the form
We proceed now to define the coupling of the two systems. There will be two constants:
The idea is that if the quantum system is at some pure state , and if the classical system is in some pure states , then will cause the Hamiltonian rotation of around with frequency , while will cause, after a random waiting time proportional to , a jump, along geodesics, to the "other side" of . The classical transformation involved is nothing but a geodesic
symmetry on the symmetric space . It has the advantage that
it is a measure preserving transformation. It has a disadvantage because overjumps .
- - regulating the time rate of jumps
- - entering the quantum Hamiltonian
We will use the notation to denote the rotation of around . Explicitly:
For each we define by
Using -s we can define Lindblad-type coupling between the quantum
system and the classical one. To give our model more flavor, we will introduce
also a quantum Hamiltonian that depends on the actual state of the classical system;
thus we define
Our coupling is now given by
Notice that and . Now, being -invariant, it must be proportional to the identity. Taking its
trace we find that
Explicitly, using the definition of , we have
Notice that for each operator we have the following formula: 33
If , that is if we neglect the Hamiltonian part, then using this
formula we can integrate over to get the effective Liouville operator for the quantum state
with the solution
It follows that, as the result of the coupling, the effective quantum state undergoes
a rather uninteresting time-evolution: it dissipates exponentially towards the
totally mixed state , and this does not depend on the initial state of the classical
Returning back to the case of non-zero we discuss now the piecewise deterministic random process of
the two pure states and . To compute it we proceed as in the previous example, with
the only change that now pure states of the quantum and of the classical
system are parameterized by the same set - in our case. To keep track of the origin of each parameter we will
use subscripts as in and . As in the previous example each observable of the total system determines a function by
The Liouville operator , acting on observables, can be then rewritten it terms
of the functions :
where is the Hamiltonian vector field
is known as the transition probability between the two quantum states.
The PD process on can now be described as follows. Let
and be the initial states of the quantum and of the classical
system. Then the quantum system evolves unitarily according to the quantum Hamiltonian
until at a time instant a jump occurs. The time rate of jumps is governed by the homogeneous
Poisson process with rate . The quantum state jumps to a new state with probability distribution while jumps to and the process starts again (see Fig.
Figure: The quantum state jumps to a new state with probability distribution while jumps to .
Acknowledgements: This paper is partially based on a series of publications
that were done in a collaboration with Ph. Blanchard and M. Modugno. Thanks are
due to the Humboldt Foundation and Italian CNR that made this collaboration possible.
The financial support for the present work was provided by the Polish KBN grant
No PB 1236 on one hand, and European Community program "PECO" handled by the Centre
de Physique Theorique (Marseille Luminy) on the other. Parts of this work have
been written while I was visiting the CPT CNRS Marseille. I would like to thank
Pierre Chiapetta and all the members of the Lab for their kind hospitality. Thanks
are due to the Organizers of the School for invitation and for providing travel
support. I would like to express my especially warm thanks to Robert Coquereaux
for his constant interest, criticism and many fruitful discussions, and for critical
reading of much of this text. Thanks are also due to Philippe Blanchard for reading
the paper, many discussions and for continuous, vital and encouraging interaction.
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- ... isolated1
- Emphasized style will be used in these notes for concepts that are
important, but will not be explained. Sometimes explanation would
need too much space, but sometimes because these are either primitive
or meta-language notions.
- ... observable.2
- Lüders  noticed that this formulation
is ambiguous in case of degenerate eigenvalues, and generalized it to cover also
- ... theory.3
- In these lectures, "quantum theory" usually means "quantum mechanics", although
much of the concepts that we discuss are applicable also to systems with infinite
number of degrees of freedom and in particular to quantum field theory.
- ... other4
- Including the author of these notes.
- ... relativity,5
- There exists however so called "relativistic Fock-Schwinger proper time formalism"
[23, Ch. 2-5-4] where one writes Schrödinger
equation with Hamiltonian replaced by "super-Hamiltonian, and time replaced by
- ... day.6
- One could try to "explain" time by saying that there is a preferred time direction
selected by the choice of a thermal state of the universe. But that is of no help
at all, until we are told how it happens that a particular thermal state is being
- ... relativity.7
- In group theoretical terms: the proper Lorentz group is simple, while the
proper Galilei group is not.
- ... processes8
- The paradigm may however change in no so distant future - we may soon try
to understand the Universe as a computing machine, with geometry replaced
by geometry of connections, and randomness replaced by a variant of algorithmic
- ... all.9
- Cf. e. g. Ref. [33, Ch. 9].
- .... 10
- A similar idea was mentioned in . For a detailed description of
all the constructions - see the forthcoming book 
- ... confusion.11
- Notable exceptions can be found in publications from the Genevé school
of Jauch and Piron.
- ... etc.12
- Quaternionic structures, on the other hand, can be always understood as complex
ones with an extra structure - they are unnecessary.
- ... system13
- Some physicists deny "objectivity" of quantum states - they would say that
Hilbert space vectors describe not states of the system, but states of knowledge
or information about the system. In a recent series of papers (see  and references therein) Aharonov
and Vaidman  attempt to justify objectivity
of quantum states. Unfortunately their arguments contain a loophole.
- ... case.14
- It should be noted, however, that Schrödinger equation describes evolution
of state vectors. and thus contains direct information about phases.
This information is absent in the Liouville equation, and its restoration (e.
g. as it is with the Berry phase) may sometimes create a non-trivial task.
- Cf. also the recent (June 1994) paper "Particle Tracks, Events and Quantum
Theory", by the author.
- The reader may also consult , where a different approach, using
dimensional reduction along a null Killing vector, is discussed.
- .... 17
- Some of these assumptions are superfluous as they would follow anyhow from
the assumption in the next paragraph.
- .... 18
- Notice that because is not a tensor, the last condition need not be, a priori, generally
- ... universal.19
- i.e. universal for the system of connections.
- ... .20
- We choose the physical units in such a way that the Planck constant and mass of the quantum particle are equal to 1.
- ... factor.21
- For an alternative detailed derivation see 
- ... 22
- Our point is that "measurement" is an undefined concept in standard quantum
theory, and that the probabilistic interpretation must be, because of that, brought
from outside. What we propose is to define measurement as a CP semigroup
coupling between a classical and a quantum system and to derive the probabilistic
interpretation of the quantum theory from that of the classical one.
- ... evolution23
- For a discussion of this fact in a broader context of algebraic theory of
superselection sectors - cf. Landsman [25, Sec. 4. 4]. Cf. also the no-go
result by Ozawa 
- ... system.24
- That requirement is also necessary to guarantee physical consistency of the
whole framework, as we always neglect some degrees of freedom as either irrelevant
or yet unknown to us.
- ... extremal25
- It should be noticed, however, that splitting of into and is, in general, not unique - cf. e. g. Refs  and [32, Ch. III. 29-30].
- ... 26
- It is useful to have the algebra represented in such a form, as it enables us to apply
the theorem of Christensen-Evans.
- ... 27
- One can easily imagine a more general situation when tracing over the classical
system will not be meaningful. This can happen if we deal with several phases
of the quantum system, parameterized by the classical parameter . It may then happen that the total algebra is not the tensor
product algebra. For instance, instead of one Hilbert space we may have, for each value of a Hilbert space of dimension .
- ... system28
- One may wonder what does that mean mathematically, as the space of pure states
of a algebra is, from measure-theoretical point of view, a rather
unpleasant object. The answer is that the only measures on the space of pure states
of the quantum algebra will be the Dirac measures.
- ...car,dal,dum,gar. 29
- Thanks are due to N. Gisin for pointing out these references.
- ... transformations30
- Or, more precisely, as Frobenius-Perron operators. Cf. Ref. for definition and examples of
Frobenius-Perron and dual to them Koopman operators.
- .... 31
- This sequence of transformation on the space of pure states of the quantum
system can be thought of as a nonlinear version of Barnsley's Iterated Function
System (cf. e. g. 
- ... details.32
- I do not think that they found the answer, as their arguments are circular,
and they seem to be well aware of this circularity.
- ... formula:33
- The formula is easily established for of the form , and then extended to arbitrary operators by linearity.