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\centerline{{\bf ON THE PROBLEM OF UNIVERSALITY OF QUANTUM THEORY}%
\footnote{$^{\dag})$}{Supported from Grant No. 201/93/2245 of the Czech Grant
Agency.}}
\vskip.7true cm
\centerline{\srm PAVEL B\'ONA}
\vskip3true pt
\centerline{\sit Department of Theoretical Physics, Comenius University}
\centerline{\sit 842 15 Bratislava, Slovakia}
\centerline{\srm and}
\centerline{\sit Department of Mathematics, Silesian University}
\centerline{\sit 746 01 Opava, Czech Republic}
\vskip.5true cm
\centerline{\srm ABSTRACT}
\vskip3true pt
{\leftskip=3true pc\rightskip=3true pc \noindent\baselineskip=12true pt
\srm
Two aspects of the problem of ``objectification'' in
quantum mechanics (QM) are considered:\nl
(1) The question of description of the usually requested nonexistence of
interference of macroscopically different states, and\
(2) the possibility of distinction of Genuine mixture (or ``Gemenge'')
from the Elementary mixture (or ``improper mixture'').
Some possibilities for a necessary modification of QM are considered.
Especially modifications of the set of ``observables'' allowing, e.g.,
description of quantal measurement process in framework of
systems with finite number of degrees of freedom are discussed.
An idealised solvable
model of measurement process in quantum theory is presented.
A nonlinear version of quantum mechanics (NLQM -- a consistent
generalisation of the Weinberg's one) describing different
evolutions for Genuine and Elementary mixtures is presented.
``Relative isolation'' of quantal systems is defined in
this connection. NLQM contains QM, classical mechanics,
and some ``quasiclassical'' theories as its exact subtheories.\par}
\baselineskip=14true pt
\vskip15true pt
\leftline{\bf 1. Introduction }
\vskip2.2true pt
Quantum theory is formulated in terms of statistical distributions of
values of observed quantities. The probability distributions of QT cannot
be endowed with a ``simultaneous ignorance interpretation'':
It cannot be reformulated in
terms of Kolmogorovian model of probability theory describing statistical
distributions on a {\sl firmly given} set of elementary events (representing in
applications a ``background reality'' -- some sort of ``Universe of all
possible events''); Kolmogorovian model works well, however, in classical
theory (CT) of physical phenomena, where an ``elementary event'' determines
sharp values of all possible quantities. On the other hand, classical
description of a ``background'' (resp. a ``pretheory'', cf. [11]) for QT
seems to be a necessary prerequisite for introduction into QT such
basic concepts
as reference frames, basic observables, preparations, and measurements;
hence, mere definition of ``a microsystem'' depends crucially on a use of
classical concepts. No clear boundary is known in physics between
phenomena described
by CT, resp. QT. Moreover, interests of physicists are attracted presently
by ``mesoscopic phenomena'' for description of which both CT and QT are
useful. Hence, the requirement of {\sl universality} of an adequate
physical theory containing both quantum and classical concepts is quite
actual. We expect that QT can be a basis for such a universal theory.
CT cannot be included into the traditional quantum mechanics (QM)
of systems with finite number of degrees of freedom, cf. [2]: Even
classical ``observables'' are not present in QM. Hence, QM needs some
modifications (resp. extensions) to become a universal QT. We intend to
consider here some questionable features of QM which could be, perhaps,
modified without changing any essential and successful feature of standard
QT, with the aim to reach universality of (such a minimal modification of)
QT, cf. also [3].
One of those essential features of QT which should be kept in its universal
modification (let us call UQT the intended universal theory) is,
according to present author's opinion, the irreducibly probabilistic
character of QT. Hence, also description of classical phenomena in
framework of UQT should be probabilistic. This might seem to contradict to
everyday experience with ``individual objects'' which can be observed
repeatedly as ``the same object'' with unchanged (sharp) values of
all their relevant quantities. The mentioned ``relevant quantities'' seem
to be stable also \wrt measurements of any other quantities of the observed
empirical object. (Warning: It should be distinguished between theoretically
defined ``physical system'' specified by a given set of ``observables'',
``states'', etc., and an ``empirical object'' specified by some empirical
procedures, some features of which -- e.g. some specific observable quantities --
serve as an empirical realization of the theoretical system; that
``empirical object'' can be used for an empirical realization of also other
theoretically defined ``systems'' which, e.g., are described by larger sets
of ``observables''.) It seems to be no serious problem to reformulate CT in
terms of very narrow (Kolmogorovian) statistical distributions, to match
just described intuition. More problems with intuition arise in trials
of description of ``macroscopically unstable'' situations, e.g., of the
process of quantal measurement. In such situations, QT leads (e.g., via a
version of the ``reduction postulate'') to statistical distributions of
values of macroscopic (classical) quantities with large dispersions, and,
in the same time, QT doesn't contain any (although {\sl intuitively desirable })
possibility of dealing with individual events (e.g., with the process of
acquirement of a specific pointer position). We
shall accept here such a state of affairs as granted: We believe that there
are physical situations (as in the case of measurement of some
quantities of a microscopic system) in which there is impossible to forecast
deterministically a motion of otherwise classical objects (e.g., the
pointer). The instability of situations allows us to foresee only a time
evolution of statistical ensembles (in the sense of Genuine mixtures)
of states of macroscopic (classical) systems, hence we cannot
theoretically describe observed individual results of a measurement. This
could be considered as a source of effective irreversibility (and of a loss
of information contained in theoretical description of actual states)
in formally time-reversible theory. (Let us mention here, however, that
also in pure CT there are unstable situations, e.g. systems with unstable
flows -- see Anosov diffeomorphisms, etc., where the formally deterministic
description is practically as stochastic, as the above mentioned quantum
processes; this ``practical stochasticity'' is due to our restricted
ability of distinction between points in a phase space corresponding to
different initial conditions of mutually quickly diverging trajectories.)
Let us mention that a sort of the desired UQT can be obtained in framework
of \Ca ic theory of infinite quantal systems (cf. [12], [13]): CT can be
included there together with QT, but the unstable processes of the type of
quantal measurement are lasting there for infinite time interval, hence
no evolution of irreversible situations is described by those theories.
Another generalization of QM is the form of nonlinear quantum mechanics
(NLQM) described in the last section of this paper: It also includes QM
together with CM ($\equiv$ classical mechanics), but the problem of
description of measurement is not solved yet in that framework.\par
\vskip8true pt
\leftline{\bf 2. Controversial ``Observables''}
\vskip2.2true pt
In standard formulations of QM, any selfadjoint operator on separable
Hilbert state-space \H\ of the considered system represents an
``observable''. No differences of observability of different observables
are formulated on this general level.
Hence, the set of all bounded ``observables'' consists of all symmetric
bounded operators $\mLH_s$. The set \LHs is the same for all
finite-particle systems (with some modifications for systems of identical
paricles, or systems with absolutely conserved charges): A structurless
point particle is described by the same set of abstract ``observables'' as a
collection of $10^{25}$ atoms. In practice, however, some ``basic
observables'' of a given system are picked out, e.g. generators of $2f+1$ --
dimensional Weyl-Heisenberg group and a few other operators (e.g.
generators of Galilean group representation) for a system with $f$ degrees
of freedom. This shows that in practice some ``hierarchy of observability''
of formally given operators is respected in QM.
A possible restriction can be anticipated for measurability of
``observables'' of multiparticle systems (let us keep in mind -- for
simplicity -- mutually distinguishable particles only, e.g. particles
placed in sites of a crystalline solid):
To measure interference of two distinct permutationally invariant
multiparticle states it would be necessary to measure very--high--order
correlation functions, those of the order of number of
particles present in the system, cf. also [3]. To illustrate this idea,
let us take a large system to be described by tensor-product Hilbert space
$\mH^N:= \otimes ^N_{p=1}\mH_p\ni\Phi, \Psi;\ \Phi := \otimes ^N_{p=1}
\varphi_p,\ \Psi := \otimes ^N_{p=1}\psi_p$, with $\varphi_p$ and
$\psi_p \in \mH_p$. Assume that $\Vert \mph p~ \Vert = \Vert \psi_p\Vert=1$,
and $\vert\lb\mph p~ \vert\psi_p\rb\vert\le\lambda<1$ for all $p$. Let us
consider the
observables of the form
$$A:=\sum_{p_1,p_2,\dots p_k} X^1_{p_1}\otimes X^2_{p_2}\otimes \dots\otimes
X^k_{p_k}\otimes I_{\{N\backslash \{p_1,p_2,\dots p_k\}\}}\eqno(1)$$
with $X^j_p$ being selfadjoint in ${\cal L(H}_p)$ of the norm $\leq 1$. Let
$k(\ll N)$ be fixed. Then, for $N\to\infty$, one has
$$\vert\lb\Psi\vert A\Phi\rb\vert0$. If there would not be any other observables
of the considered system, then (2) would present a kind of ``effective
disjoitness'' of the
states described by $\Psi$ and $\Phi$ \wrt the chosen set of observables
of the form (1). Conventional QM admits, however, observables with
$k$ in (1) of the order of $N$: $k \approx N$.
Such ``observables'' (i.e. products of
large number of one particle operators) seem to be practically
unobservable, [4]. This might be a reason of ``practical classicality''
of some subsystems (determined by their observables, etc.) of large quantal
systems.
The following note (although related to previous ones) could further
support the requirement for extension of QT by some criteria for
``observability of observables'': Projection operators are used to be
considered
to represent ``questions'' in QT with ``yes''--``no'' answers (cf. [5],
[6], [8]). Their interpretation is, however, determined by interpretation
of vectors in corresponding subspaces of states in \H. But interpretation
of vectors in \H\ is equally problematic as the interpretation of operators:
vectors correspond to ``preparation procedures'', whereas operators are
ascribed to ``registration procedures'', cf. [1], [10], [11]. Hence, all
systems with any finite number of degrees of freedom are described formall
by ``the same number of questions'', what seems to be absurd from the point
of wiev of physical intuition. (The situation is here
analogous as that one occuring at ``counting points'' on the real line,
resp. on its intervals of nonzero lenght:
mathematically there is no difference, but endowing the reals by a metric a
natural difference is created.)
Our last argument for necessity of distinction between ``degrees of
observability'' of different formally defined ``observables'' is, perhaps,
the most ``physical'' one but, simultaneously, it seems to be most
difficult to be
taken seriously in building of a general theory. We propose here a natural
{\sl principle of measurability} according to which the set of (really
observable) ``observables'' is determined as the set of those selfadjoint
operators which can be measured by a ``realistic'' apparatus. I.e., to any
``observable'', an apparatus can be (in principle, at least theoretically)
constructed by a use of observed interactions only (electroweak,
gravitational, etc.). One could pose the problem alternatively:
Let us
look for observables which can be measured by a given set of apparatuses
(trying to construct the ``realistic'' ones). An (unrealistic) example
of such a construction is given in the next section.
It might be superfluous to mention here, that a consistent restriction of
the set of observables in QT could be a difficult task, since there are
strong and naturally-looking requirements on that set, e.g. its
Jordan--algebraic, or quantum logic structures, cf. [5], [16]. The NLQM of
the last section represents an extension of the sets of
observables, states, etc. which introduces, however, a richer structures
into these sets, hence offers some possibilities to attempt to solutions to
our present problem.
%%
\hhead 3. An Observable Determined by Quantum Measurements~
We shall present here a model of an infinite size apparatus (formulated
rigorously in \Ca ic terms, [7]) detecting a scalar particle moving
on a line. This has to illustrate how a measured
quantity can be defined from the interaction of the apparatus
with the measured ``microsystem'' (this
scalar nonrelativistic particle interacts with {\it first two}
spins in the chain).
The state
space of a particle (modelled also as a quantum of scalar Fermi field)
is simply $\mH\equiv\mLq$. A suitably chosen element $\sigma\in\mLq$
describes also interaction with the apparatus ``placed'' on the line within the
set $supp\/(\sigma)\subset {\bf R}$. The apparatus is modelled by a semiinfinite
spin chain described by the spin creation - annihilation operators
$a_n^{(*)}\ (n\in{\bf Z}_+)$, and by the Hamiltonian (cf.[17])
$$H_{\infty}:=\sum_{n\in{\bf Z}_+}a_n^*a_n(a_{n+1}+
a^*_{n+1})a_{n+2}a^*_{n+2}.\eqno(3)$$
Let us introduce now the model which represents a nonideal measurement.
The microsystem (particle) is described by the Fermi
field described by annihilation operators $b(\varphi)$, with $\varphi\in\mLq$.
The total Hamiltonian is now
$$H:=H_{\infty}+d\Gamma(\hat p^2)+W,\eqno(4)$$
where the interaction W is defined by
$$W:=\gamma(a^*_1+a_1)a_2a^*_2\otimes b^*(\sigma)b(\sigma),\eqno(5)$$
with $\gamma \in {\bf R}$ and $\sigma \in \mLq,\
\Vert\sigma\Vert=1$. The symbol $d\Gamma (\hat p^2)$ denotes the second
quantized, [14], one particle
Hamiltonian $\hat p^2$, with $\hat p:=-i\partial_x$.
Let the chain be at $t=0$ in its ``vacuum'', i.e. all spins are pointig
down. Let the initial state of the particle be described by an arbitrary
normalized wave function $\mph0~\in\mLq$. (We do not use here the ``incoming''
asymptotics for $t\to-\infty$
of the scattering theory.) Denoting by \om t~ the state (i.e. the
expectation functional)
of the total system in time t, and by $A$ an arbitrary observable of
that composed system, one can obtain that
$$\bar\omega (A) := \lim_{t\to\infty}\mom t~(A)\eqno(6)$$
exists, and one arrives for suitable parameters of W at:
$$\bar\omega
=(w\mom1{chain}~+(1-w)\mom0{chain}~)\otimes\mom0{field}~,\
w\not\equiv 0,\eqno(7)$$
where \om j{chain}~ $(j=0,1)$ denotes the state of the chain with all spins
pointing down, resp. pointing up.
The field state \om0{field}~
is the Fermi vacuum. We obtained here a mixture of two disjoint states.
The probability $w$ of switching the apparatus can be expressed explicitly in
terms of dynamics of the chain, as well as of the particle-chain interaction
parameters, cf. [7]. It can be written in the form
$$w=\lb\mph0~\vert\ \Lambda\mph0~\rb.\eqno(8)$$
Here $\Lambda$ is a positive operator of the norm $\Vert \Lambda\Vert \leq1$
acting on the Hilbert space \Lq. This operator represents the measured
(unsharp) observable of the particle.
Its explicit expression can be written in the form
$$\Lambda = \gamma^2 (F^+_{\sigma})^*\hat f F^+_{\sigma}.\eqno(9)$$
Here $F^+_{\sigma}\in{\cal L(H,}L^2({\bf R},dt))$, with
$$F_{\msg}^+(\msg'):=\vartheta(t)\lb\beta_0\otimes\Omega(\msg)\vert\ U_t
\beta_0\otimes\Omega(\msg')\rb,\eqno(10)$$
where $\vartheta(t)=1$ for $t>0,\ \vartheta(t)=0$ for $t<0$;
$U_t := exp(-itH)$ is the total system's evolution operator,
$\beta_0$ is the spin--chain ``vacuum'', and $\Omega(\sigma')$ is the
one--particle state with the particle wave--function $\msg'$;\
$\hat f \in {\cal L}(L^2({\bf R},dt))$ is the convolution operator with the
function $f: t \mapsto f(t)$ determined as a diagonal matrix element of
the free evolution $U^0_t:= exp(-itH_0)$ of the total system with
$H_0:=H-W$:
$$f(t):=\lb\beta_1\otimes\Omega(\msg)\vert
\ U^0_t\beta_1\otimes\Omega(\msg)\rb,$$
with $\beta_1:=a_1^*\beta_0,\ \ a_m \beta_0:=0$ for all $m\in {\bf Z_+},\
\Omega(\msg):=b^*(\msg)\Omega_0$, and $b(\msg')\Omega_0:=0$ for all
$\msg'\in \mH$.
The operator $F_\msg^+$ can be found explicitly in terms of the free evolution
and of the interaction parameters, [7], hence $\Lambda$ is a known
``effect''
defining an ``unsharp observable''. Physical interpretation of this
observable is ``in principle'' given by the measurement process just described.
%\vfil\break
%%
\hhead 4. On Different Interpretations of Mixed States~
Let us consider a microsystem in (conventional formulation of) QM: It is
characterized by irreducibility of its set \LH of ``observables''. Hence,
any vector $\varphi \in \mH$ represents a pure state. Let \rh~ be any
density matrix on \H\ different from one--dimensional projections. Then \rh~
can be nontrivially decomposed in many different ways
into a convex combination
$$\mrh~:=\sum_j\lambda_j\mrh j~\eqno(11)$$
of density matrices \rh j~. Different decompositions (11) can correspond
to decompositions \wrt different, mutually incompatible (i.e. mutually
noncommuting) commutative subsets of observables. Such ``incompatible
decompositions'' cannot be both interpreted as Genuine mixtures although
they are in the framework of QM {\sl of the considered microsystem}
mutually equivalent. This indicates that for a separately taken isolated
microsystems the {\sl concept of Genuine mixture (i.e. ``Gemenge'') is
meaningless} in the framework of conventional QM.
On the other side, for classical systems one has instead of \rh~ a
probability measure on its phase space, and this is uniquelly decomposable
into its extremal (i.e. pure) components -- into an integral of Dirac
measures concentrated in points of the phase space.
In the case of quantal systems with (some kind of) superselection rules,
e.g. in the \Ca ic description of infinite systems, there are distinguished
unique decompositions of states (e.g. the central decomposition, cf. [14]).
In the both last mentioned situations the components of the decompositions
are mutually ``noninterfering'', i.e. the considered systems (also in any
of their empirical realizations) cannot be coupled to any other system in
such a way where a quantummechanical interference would be observed between
states of the composed system corresponding to different components of the
decomposition.
To be more specific, let us try to express some ideas in terms of QM.
Let us consider one--dimensional projections
$P_{\mph j~}$ on \H\ corresponding to
an orthonormal set $\{\mph j~, j=1,2,\dots\}$
of vectors in \H. In the case if a ``classical'' subsystem is present in
the considered system, the vectors \ph j~
for different values of $j$ can correspond to different superselection sectors
(e.g. to different values of some classical quantity) and, in that case,
there is no ``observable'' $B\in\mLH$ such that
$$\lb \mph j~\vert\ B\mph k~\rb \neq 0,\ \ {\rm for}\ some\ \ j\neq k.\eqno(12)
$$
For a microscopic system there are always $B\in\mLH$ such that (12) is
valid. Let now $\mrh j~ := P_{\mph j~}$ in (11). Then there is always
possible to find (in mathematical theory of QM) a system with its Hilbert
space ${\cal K}$ and an orthonormal basis $\{ \psi_j, j=1,2,\dots\}
\subset {\cal K}$ such that \rh~ is obtained as the partial trace
(cf. [8])\ \wrt ${\cal K}$ from the vector--state $\Psi$,
$$\Psi := \sum_j\sqrt{\lambda_j}\ \mph j~\otimes \psi_j
\ \in\ \mH\otimes \mK.\eqno(13)$$
If, for any choice of ``the system \K '', and for any ``observable $R$'' of the
``combined system $\mH\otimes\mK$'', one has
$$\lb \Psi\vert\ R\Psi\rb = \sum_j \lambda_j\ \lb\mph j~\otimes\psi_j
\vert\ R\mph j~\otimes\psi_j\rb,\eqno(14)$$
we say that the states $P_{\mph j~}$ of ``the system \H'' are mutually
noninterfering. This is the case of the mentioned belonging of different
\ph j~ into different superselection sectors in otherwise standard QM, [18].
Hence, for macroscopic (classical) systems the concept of Genuine mixture
(cf. also its rough description in [3]) makes sense: The distinguished
unique decomposition of the state (let us denote it \rh~ also in general
situation) into mutually noninterfering components \rh j~ can be
interpreted in terms of a statistical ensemble of states \rh j~ describing
a collection of systems (also in their empirical meaning, if interpreted)
each of which ``really occurs'' in one of the states \rh j~. Let us note
here that concepts like ``existence'', ``real occurence'', or ``being'' of
some specific ``objects'' seem to be of
profound macroscopic (and, conceptually: of classical) character:
Their empirical definition rests on repeatability of empirical measurement
procedures and on reproducibility of their (individual -- sharp) results
\wrt all their (macroscopic) observables. If these ``classical features''
of quantummechanically desribed macrosystems would be only ``effective'' in
the sense of considerations of preceding sections, then also such concepts
as ``existence'' could be only effective.
On the other hand, we would like to have a concept of Genuine mixture also
for microsystems, because there are situations which intuitively fit in
those described above for macrosystems with classical behaviour. E.g., we
can prepare a collection of microsystems by choosing during
the preparation a procedure preparing the state
$P_{\mph j~}$ with probability $\lambda_j$, for all $j$'s. Let us see, however,
that now
we have not an ensemble of separated microsystems, but, at the best,
an ensemble of couples {\sl microsystem \& preparation device}, and the
``Genuine interpretation'' is possible due to the presence of macroscopic
devices in macroscopically distinct states correlated with the states of
the microsystem in the sense of eq. (13). We can ask, however, whether
there is some possibility of distinction of the last mentioned situation
(in which the density matrix of the micro -- ensemble was \rh~ from (11))
from that one described by (13) with both subsystems being microscopic, and
the partial density matrix being again the same \rh~. Since microsystems
can be coupled both with microsystems, as well as with macrosystems, this
question addressed to the state \rh~ of a separately considered system
in an instant of time has (at least in
the framework of QM) negative answer. Another possibility arises if we
assume that there is present an interaction of the considered microsystem
with the correlated macrosystem where interaction Hamiltonian depends on
values of the macroscopic quantity distinguishing different \rh{j}~ in
(11).
Then we could ask for future time
evolution of the state \rh~, and we could look
for some characteristic differences
between evolutions in the cases when the considered microsystem is
correlated with macrosystems (with different
values of the macroscopic quantity) and in the cases in
which such a correlation is missing (or the microsystem's states \rh j~ are
all correlated with a unique value of the
macroscopic quantity determining the interaction). Since general
evolution of the state (13)
of the coupled system might lead to an arbitrary change of its partial trace
\rh t~, we have to restrict our interest to some class of evolutions where
the posed question could obtain a reasonable meaning.
Let us choose a class of such (sub--)\/systems
in QM (or also in UQT) which satisfy the following two conditions.\nl
(a)\ \ Time evolution of the chosen systems (in considered global
situations including also description of other relevant systems) can be
described by unitary operators, i.e.
$$\mrh t~ = U(t,t_o;\mrh{t_o}~)\mrh{t_o}~U(t,t_o;\mrh{t_o}~)^{-1},\ \ t\in{\bf
R},\ \mrh{t_o}~\in {\cal T}_{+1}{\cal (H)}\equiv \mSs.\eqno(15a)$$
The possible dependence of $U$ on \rh{t_o}~ leads, however, to a problem:
The ambiguity of the expression of state \rh~ in various convex
combinations could lead to ambiguity in time evolutions, if calculated as
the evolution of the components of the convex decompositions.
The possibility of the \rh{t_o}~ -- dependence of $U$ will be interpreted
here as a possible presence of the above described correlations of microscopic
states with (at least weakly interacting) macrosystem's states, cf. [21].
We shall assume further: \nl
(b)\ \ The function $\mrh~\mapsto U(t_1,t_2;\mrh~)$ is defined on \Ss\ for
all $t_j\in {\bf R}\ (j=1,2)$ and satisfies a `cocycle property' (up to
possible phase factors):
$$ U(t_2,t_1;\mrh{t_1}~)U(t_1,t_0;\mrh{t_0}~)\equiv
U(t_2,t_0;\mrh{t_0}~),\ U(t,t;\mrh~)\equiv I.\eqno(15b)$$
Now it is natural to define two kinds of (mixed) states of
the considered (sub--)systems:\nl
(i)\ \ Density matrices describe indecomposable quantal states
evolving according to (15), and they should be dealt within the theory as
a whole (their components in formal decompositions (11) have no
interpretation, hence for this states eq. (11) is physically
irrelevant).\nl
(ii)\ States corresponding to Genuine mixtures
are described by {\sl specific decompositions} of the form (11) (here
different components \rh j~ correspond to different sharp values of classical
quantities of a correlated macrosystem); in these states each component of the
(specific!) decomposition (11) evolves separately according to (15).
The systems satisfying the conditions (a) and (b) could be imagined
intuitively as microsystems moving in an external (classical) field which
can depend moreover on the microscopic state of the system. Some
correspondence between actual states of microsystems and their macroscopic
environment can be understood due to necessity of preparation of
specified microstates by macrodevices. Since there is no evolution of
quantum correlations changing spectral characteristics of density matrices
of states of systems fulfilling (a) and (b), we shall call such systems to
be {\sl relatively isolated}. Genuine mixtures correspond to
convex combinations (11) of microstates correlated with macroscopically
different states of the ``relevant macrosystems''.
Systems satisfying nontrivially the above conditions are, e.g., individual
quantal systems of an infinite collection of equal quantal
systems interacting mutually
by specific very weak, and of very long range (``mean--field'', cf.[19], [20])
interactions. This describes in fact nonlinear situations in QM, and a
theory of such situations is sketched in the next section. We assume
that such nonlinear situation could occur in measurement -- like
situations, as it was mentioned also in [3].
%%
\hhead 5. On Nonlinear Extension of QM~
Let us consider a complex Hilbert space \H\ and the set \Ss\ of all density
matrices on it. The set \Ss\ is embedded into the Banach space ${\cal
T(H)}$ of all trace--class operators on \H. Let ${\cal U(H)}$ be the
unitary group of \LH. \Ss\ can be decomposed into orbits of \UH\ \wrt the
coadjoint representation:
$$Ad^*(u)\mrh~:=u\mrh~u^*$$
for all $u\in \mUH$, and all $\mrh~\in\mSs$. Each orbit consists of all
density matrices with the same spectral characteristics (i.e. spectrum and
spectral multiplicities). Orbits ${\cal O}_{\mrh~}$ corresponding to \rh~
with zero eigenvalue subspaces in \H\ of finite codimension form embedded
submanifolds of \TH, other orbits are not submanifolds of \TH, cf. [21].
Hence, the projective
Hilbert space \PH\ elements of which are one-dimensional projections in
\H\ (e.g., \H\ := \Lq) forms an orbit of the first mentioned kind.
The set \Ss\ will be considered here as the quantal analogy of the phase
space of Hamiltonian CM. Let us consider the set ${\cal F}$ of real valued
differentiable (for simplicity; for more general -- also unbounded -- cases
see [21]) functions on \TH\ representing {\sl generators} of
transformations of the generalized phase space \Ss. We shall define Poisson
brackets on \Ss. Let for all ``sufficiently nice'' real-valued functions
$f,\ h,\dots$ the symbols $d_{\mrh~}f,\ d_{\nu}h\dots$ denote the (Fr\`echet)
differentials at the points $\varrho,\ \nu,\dots$ of \Ss. We can consider
these differentials as unambigously defined bouned operators in \LH.
Then we define the Poisson bracket of $f$ and $h$ by, [21],
$$\{f,h\}(\nu):=i\,Tr(\nu[d_{\nu}f,d_{\nu}h]),\eqno(16)$$
where $[\cdot,\cdot]$ means the commutator of bounded operators in \LH,
and $Tr$ is the trace in the space $\cal T(H)$.
Time evolution given by the Hamiltonian function $h$ is determined now
in the usual way from Eq. (16). It can be shown, [21], that the
evolution of density matrices can be described by the `cocycle'
of unitary operators $u_h(t,\nu)$ (with $t\in{\bf R},\ \nu\in\mSs$)
satisfying Schr\"odinger-like (in general nonlinear)
equation with the `Hamilton operator' $d_{\mrh~}h$:
$$i\,{d\over dt}u_h(t,\nu)=d_{\nu(t)}h\ u_h(t,\nu).\eqno(17)$$
Here $\nu(t):=u_h(t,\nu)\,\nu\,u^*_h(t,\nu)$, if $\nu(0):=\nu$; this
determines time evolution of any $\nu\in\mSs$ under the Hamiltonian dynamics
corresponding to Poisson brackets (16) (there is an ambiguity in
formulation of eq. (17) for $u_h(t,\nu)$, cf.[21]).
We see that the evolution in this
NLQM satisfies conditions (a) and (b) of the preceding section,
with $U(t_1,t_2;\mrh~)\equiv u_h(t_1-t_2,\mrh~)$.
If $h(\nu)\equiv Tr(\nu H)$ for some $\nu$-independent operator $H$,
then the Eq. (17) converts into the usual Schr\"odinger equation
with the Hamiltonian $H$; the dynamics is linear in the usual
sense exactly in this last mentioned case.
A nonllinear Schr\"odinger equation of the form (17) (for vector states
$\rho^2\equiv\rho$ only) was published in [22]; the
corresponding NLQM
contained, however, interpretation, work with mixtures, and other aspects
different from those of the present one.
Let us now define (formal) {\sl observables} as differentiable functions
{\bf f,\ h} of two variables $(\nu;\rho)(\in \mSs\times\mSs)\mapsto {\bf R}$
depending on the first variable $\nu$ affinely:
$${\bf f}(\lambda_1\nu_1+\lambda_2\nu_2; \rho)= \lambda_1{\bf f}
(\nu_1;\rho)+\lambda_2{\bf f}(\nu_2;\rho).\eqno(18)$$
Such functions can be represented by operator--valued functions
${\bf F}: \rho(\in\mSs) \mapsto {\bf F}(\rho)\in \mLH$ in the following way:
$${\bf f}(\nu;\rho)=Tr(\nu{\bf F}(\rho)).\eqno(19)$$
The proposed interpretation of these ``observables'' is a natural
extension of the one from QM: $Tr(\rho{\bf F}(\rho)^n)$ equals to n--th moment
of the probability distribution of the observable {\bf f} (resp.
equivalently, of {\bf F}) in the state $\rho$ (corresponding
modifications for POV -- measures, [8], can be formulated easily).
Classical observables are described by {\bf R} -- valued functions
$f(\rho)$ on \Ss\ representing the operator -- valued function
$${\bf F}(\rho)\equiv f(\rho){\bf I}\eqno(20)$$ with {\bf I} denoting
the identity operator on \H.
Elements of \Ss\ can be, in accordance with the preceding section,
interpreted as Elementary (hence ``improper'') mixtures. The Genuine
mixtures (i.e. Gemenge) are proposed here to be described by probability
measures on \Ss.
We claim, [21], that the presented scheme is a mathematically consistent model
containing conventional QM as its subtheory: If we choose only those
``generators'', resp. ``observables'' which are of the form $f(\rho)\equiv
Tr(\rho F),\ (F\in\mLH)$, resp. ${\bf f}(\nu;\rho)\equiv Tr(\nu{\bf F}),\
{\bf F}\in\mLH$, we obtain QM (with restriction to bounded operators, cf.,
however, [21]). CT can be formulated in this framework with a help of the
``observables'' (19) with {\bf F} from (20). The Hamilton functions $h$ in
(17) can be chosen such that the corresponding time evolution leaves some
arbitrarily chosen
manifolds of generalized coherent states (in the sense of Perelomov, [23])
invariant. In this way, approximative theories to QT, like the time
dependent Hartree--Fock theory (cf. [24]), together with finite dimensional
Hamiltonian CT are included in our NLQM as
exact (i.e. obtained without any approximations) subtheories.
%\vfil\break
\hhead 6. References~
\item{\ 1.}P. Busch, P. J. Lahti, and P. Mittelstaedt (Eds.), {\it
Symposium on the Foundations of Modern Physics 1993 -- Quantum Measurement,
Irreversibility and The Physics of Information}\/, Cologne, Germany 1 -- 5
June 1993 (World Scientific, London, 1993).
\item{\ 2.}P. Mittelstaedt in [1].
\item{\ 3.}P. B\'ona in [1].
\item{\ 4.} According to some estimates the size of the corresponding
apparatus would be of the order (or even larger) of the size of our Universe.
\item{\ 5.}V. S. Varadarajan, {\it Geometry of Quantum Theory}, Vols. I and
II, (Van Nostrand Reinhold Co., New York, 1968, 1970).
\item{\ 6.}G. W. Mackey, {\it The Mathematical Foundations of Quantum
Mechanics}, (W. A. Benjamin, New York, 1963).
\item{\ 7.}P. B\'ona, {\sl ACTA F.R.N. Univ. Comen. PHYSICA} {\bf XX}
(1980) 65.
\item{\ 8.}E. B. Davies, {\it Quantum Theory of Open Systems}, (Academic
Press, London, 1976).
\item{\ 9.}P. Busch, in {\it Classical and Quantum Systems -- Foundations
and Symmetries}, (Eds. H. D. Doebner, W. Scherer and F. Schroeck, Jr.),
(World Scientific, Singapore, 1993).
\item{10.}\leftskip=0true pt \rightskip=0true pt \noindent P. Busch,
P. J. Lahti and P. Mittelstaedt, {\it The Quantum Theory of
Measurement}, (Springer, Berlin, 1992).
\item{11.}G. Ludwig, {\it An Axiomatic Basis for Quantum Mechanics}, Vols.
I and II (Springer, Berlin, 1985 and 1987).
\item{12.}G. Ludwig in [1].
\item{13.}K. Hepp, {\it Helv. Phys. Acta} {\bf 45}\ (1972) 237.
\item{14.}O. Bratteli and D. W. Robinson, {\it Operator Algebras and
Quantum Statistical
Mechanics} (Springer, New York - Berlin, 1979 and 1981), Vols. I and II.
\item{} R. Haag and D. Kastler, {\it J. Math. Phys.} {\bf 5}\ (1964) 848.
\item{15.}P. B\'ona: {\it On Nonlinear Quantum Mechanics}, in {\sl
Differential Geometry and Applications}, {\it Proceedings of 5 -- th
International Conference on Differential Geometry and Its Applications,
August 24 -- 28, 1992, Opava, Czechoslovakia}, Eds. O. Kowalski, and D.
Krupka (Silesian University Press, Opava, 1993).
\item{16.}P. Pt\'ak, S. Pulmannov\'a, {\it Orthomodular Structures as Quantum
Logics}, (Kluwer, Dordrecht, 1991).
\item{17.}P. B\'ona, {\it acta phys. slov.} {\bf 27} (1977) 101.
\item{18.} The present author apologizes to readers for this way of
expression where from
time to time into the formalism of QM are incorporated intrinsically
foreign elements as ``superselection rules'' in a trial to express
classical features of observed empirical systems in a modified version of
QT.
\item{19.}K. Hepp and E. H. Lieb, {\it Helv. Phys. Acta} {\bf 46} (1973) 573.
\item{20.}P. B\'ona, {\it Czech. J. Phys.} {\bf B37} (1987) 482;
{\it J. Math. Phys.}\ {\bf 29} (1988) 2223; {\sl ibid}
{\bf 30} {(1989)} {2994}.
\item{21.}P. B\'ona, {\it Quantum Mechanics With Mean-Field Background},
(Comenius University, Bratislava 1991), preprint Ph10-91.
\item{22.}S. Weinberg, {\it Ann. Phys.}, {\bf 194} (1989), 336.
\item{23.}A. M. Perelomov, {\it Commun. Math. Phys.}, {\bf 26} (1972) 222.
\item{24.}D. J. Rowe, A. Ryman, and G. Rosensteel, {\it Phys. Rev.}
{\bf A 22} (1980) 2362.
\bye