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\centerline{\bf SELFCONSISTENCY AND OBJECTIFICATION}
\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{\sit 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 Three mathematical
models of interaction of a ``small''
quantummechanical system with a ``large'' one are presented. It is discussed
their possible r\^ole in solving interpretation problems of quantum mechanics
(QM). The third model determines the (``unsharp'')
quantity measured in the described process as a consequence of the chosen
interaction of the ``object'' with a macroscopic ``apparatus''.
The fourth of the presented models introduces a natural nonlinearity
into dynamics of QM; the nonlinear evolution allows to distinguish between
the two kinds of mixed states: between the improper ones (called also
``elementary mixtures''), and the genuine mixtures (called also ``Gemenge'').
A r\^ole of ``selfconsistency'' in trials of interpretation
of QM as a universal theory is briefly discussed. The paper contains some
remarks connecting the ``objectification problem''
of quantum measurement theory with the problem of universality of quantum
mechanics.\par}
\baselineskip=14true pt
\vskip15true pt
\leftline{\bf 1. General Framework}
\vskip8true pt
\leftline{\it 1.1. Some Philosophical Notes}
\vskip2.2true pt
We shall restrict present discussion to technical questions mainly, leaving the
relevant but difficult philososphical considerations almost untouched. These
notions have to offer to readers just very rough orientation in some relevant
aspects of the philosophical
position accepted by the present author.
Let us first mention that the notions like ``object'', ``physical system'', or
``observer'' are considered here as just names for specified sets of
statistically
correlated ``events'' (considered as elements of {\it our} knowledge; however,
the concept {\it of ourselves} is also of this kind).
Hence, we do not pursue here any ``realistic'' point of view. A physical theory
is considered as a model (consisting of {\it formalism} and of its {\it
interpretation}) which is to be compared with ever anew empirically selected
phenomena. {\it Universal theory} contains (sub-) models for (potential)
description of arbitrary phenomena (at least from a specified ``Universe'').
Theories are human (although not arbitrary) ``constructs'' which allow us
increase probabilities of our realized empirical expectations. A key
problem of a ``working'' application of a theory is the problem of
{\it identification} of specific events with a certain classes (e.g. concepts)
dealt with (resp. contained in) the theory. Expressed briefly, the problem often
arises to decide ``what is the same as$\dots$'' in specified conditions.
If a universal theory includes a model of human mind (i.e. of its own
``constructor''), and if one wants maintain in such a theory the man's
``free will'', the theory should contain possibility of more than one of
different responses
of a man to each of most of conceived states of his mind. This consideration
alown (before taking into account any specific physical knowledge)
seems to be sufficient to require
an introduction of some irreducible statistical elements into that theory. This
in turn seems to indicate that any ``completely hidden variables theory''
(in which all situations could be described by classical probabilities) cannot
be universal: it makes conceivable fully deterministic behaviour of Universe.
\subhead 1.2. On Universality and Objectification~
In considerations on foundations of QM, we are faced with situation when
an interpretation of any quantummechanical (quantal) ``observable'' is
specified in terms of kinematics of classical mechanics (CM) reflecting our
reproducible experience with macroscopic bodies. Hence, any interpretation of
universality of QM requires a possibility of description of concepts of CM
in framework of QM. This leads to a kind of selfreference of QM: Quantum
theory should contain its ``classical background'' which itself is a
prerequisite of any interpretation of QM. If inclusion of an ``observer''
into description of the process of measurement in QM would be inevitable,
then universal QM should describe also a model of human mind in its framework.
As a decisive problem of interpretation of QM,
the ``objectification problem'' is considered$^1$ often, posing the question
whether and when a state of the system S
(e.g. an apparatus or even of a microsystem) can be unambiguously considered
as a ``Genuine mixture'' (resp. ``Gemenge''), i.e. considered as a
state seen ``in all
possible situations'' as the same statistical ensemble of systems each of which
``really possesses'' definite values of certain properties (in the measurement
process these properties should be in a certain way related to values of
the measured quantity). To endow with an empirical meaning this vague
specification of that concept, let us determine here the ``objectification''
as a physical process
with the system S extended (into a larger system) by {\it any such system
correlations of which with S could be measured in the considered framework}
(the mentioned ``considered framework'' has to be specified, cf. also Secs.1.3
and 1.4); the state
{\it of the whole extended system} reached at the end of this process should
behave as the corresponding mixture \wrt {\it all possible measurements}
(cf. Secs.1.3 and 1.4). In this way, any member of that statistical ensemble
corresponds to a distinct superselection sector.$^{1-4}$ Single members
of the corresponding convex decomposition of such a state of S need not
represent pure states, neither should correspond to orthogonal projections in
its Hilbert state-space. It could be seen from these considerations that a
successful solution to the ``objectification problem'' requires a possibility
of such a dynamical process in QM which brings into correlations (in course of
measurement, e.g.) classically distinguished states with the corresponding
``objectified'' states of the
considered system S (e.g. microscopic; S could be, for example,
that ``second'' $1\over2$-spin system which was not actually measured in
the Bohm version of the EPR-like process). ``State-vector collapse'' comes
from emergence of such correlations.
\vfil\break
\subhead 1.3. On $C^*$-algebraic Approach~
The $C^*$-algebraic (quasilocal) description of systems with infinite
number of degrees of freedom offers mathematically well formulated models of
QM-systems containing a CM-subsystem: The classical quantities ${\cal C}_{cl}$
form a coveniently chosen subalgebra of the commutative subalgebra ${\cal Z}\
(\equiv$``the center of the second topological dual $\mA^{**}$ of the
quasilocal $C^*$-algebra \A''). The choice of $\mCc\subset \mZ$ (\Z\ is too
large to be useful whole at once) can be done with a help of dynamics of the
system described (locally) by \A, as it is the case of the mean-field models
of Hepp and Lieb$^{6-8}$. These models lead also to an effectively nonlinear
evolution of each elementary subsystem of the infinite system in translationally
invariant states. Such a nonlinear dynamics can be
described, however, as an autonomous (formally classical Hamiltonian) dynamics
on the (infinite dimensional) quantum state space of an elementary constituent
subsystem without any reference to the ``mean-field background'' produced by
the infinite colection of such systems (cf.Sec.3).\ct{11}
The algebra \A\ can be faithfully represented as an algebra of bounded operators
on a (possibly nonseparable) Hilbert space \H\ which is strictly smaller than
the algebra \LH\ of all bounded operators. The whole algebra \LH\ is traditionally
considered, however, to be an ``algebra of observables'' of closed finite
quantal system (with separable \H, and without superselection rules).\ct{1,4}
Hence, in the \Ca ic approach to (infinitely) large systems, it is used an
(effective) {\it restriction} of the set of observables: It is
assumed that only observables of (arbitrarily large, but) finite subsystems
are accepted as measurable quantities. This approach leads to a natural
appearance of superselection rules determined by classical ``observables at
infinity'' belonging to \Z.
The mentioned addition of the global quantities \Cc\ from \Z\ is a
mathematically
independent theoretical procedure made without generally accepted principles.
Quasilocal \Ca s are used in Secs.2.1 and 2.3.
\subhead 1.4. Other Restrictions to the Sets of Observables~
If one wants to avoid a use of infinite systems in QM, and if one wants to have
a universal (version of) QM, then one needs some, at least effective
introduction of superselection rules.\ct1\ An idea how to do this consists in
proposal of
detailed consideration of what a quantal measurement is, to obtain a subsequent
restriction to those quantities which could be ``really measured''. We
expect that the empirically found restriction to the known
kinds of interactions will lead to restricted possibility of constructing
measuring apparatuses, hence, perhaps, also to some superselection
rules. Any general theory of this kind of selection of observable quantities
does not exist yet. It might be interesting in this context to follow the
effort of experimental physicists to measure some multiparticle observables.
It might be seen that it is very difficult to measure even three particle
quantities.\ct9 We shall discuss here briefly only a simple specific proposal
how to formulate a restriction of the set of observables to the set of at most
$k$-particle quantities in an $N$-particle system (with very large $N$).
Let the large system 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 all the
observables be of the form
$$A:=\sum_{p_1,p_2,\dots p_k} X^1_{p_1}\otimes X^2_{p_2}\otimes \dots\otimes
X^k_{p_k}\otimes I_{\{N\backslash \{p_1,p_2,\dots p_k\}\}}\eqno(1.1)$$
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\ct3
$$\vert\lb\Psi\vert A\Phi\rb\vert0$. This presents a kind of ``effective disjoitness'' of the
states described by $\Psi$ and $\Phi$ \wrt the chosen set of observables
of the form (1.1).
\subhead 1.5. On Relevance of Nonlinear Evolution~
An effectively nonlinear evolution of the constituent subsystem of an infinite
system appeares in models of large systems with mean-field type long-range
interaction.\ct{7,10} Such a time evolution can be described also as
autonomous evolution of the subsystems.\ct{11} Whether such an evolution
will be observed or not, we want to point out that nonlinear motions
only could distinguish between ``elementary mixtures'' and ``Gemenge''.
Any ``Gemenge'' evolves, from its definition, in such a way that its
density matrix \rh~ expressed as the convex combination
$$\mrh~:=\sum_j\lambda_j\mrh j~\eqno(1.4)$$
of density matrices \rh j~
of the single systems included into the statistical ensemble desribed
by this ``Gemenge'' evolves with time $t$ as the same convex combination
of independently time-evolved density operators \rh j~:
$$\mrh t~\equiv\sum_j\lambda_j\mrh{jt}~ .\eqno(1.5)$$
Such an evolution corresponds to linear QM. In the case of nonlinear
evolutions one cannot keep Eq.(1.5) to stay valid for any \rh~ and for each
of its decompositions of the form (1.4).
Nonlinear evolution can be considered as an evolution with Hamiltonian
depending on state of the system.\ct{14,11} Let us assume that the considered
microsystem evolves interacting with its ``purely macroscopic'' environment,
i.e. it has ``its own'' Hamiltonian depending on an actual state of
the environment. If, moreover,
the (macroscopic) state of the environment depends on the state of the
microsystem,\ct{15} we
arrive at nonlinear evolution. Such a situation will occur at
the process of quantal measurement, if there would be some additional
(up to now, perhaps, unknown) long range interaction between
the macrosystems and microsystems.\ct{7,11}
\vfil\break
\hhead 2. Models of Quantum Measurement~
In the first two models of this section the ``measured microsystem'' is a
$1\over 2$-spin with $n = 0$\ lying at the very
beginning of a linear chain consisting of such spins. In the third model
a particle is detected by such a chain.
\subhead 2.1. An Unstable Infinite Spin Chain~
Let $a_n^{(*)}\ (n\in{\bf Z}_+)$ be the spin creation - annihilation operators,
and let
$$H:=H_{\infty}:=\sum_{n\in{\bf Z}_+}a_n^*a_n(a_{n+1}+
a^*_{n+1})a_{n+2}a^*_{n+2}\eqno(2.1)$$
be formal expression for the sequence of local Hamiltonians defining time
evolution as a one-parameter group $\tau_t\ (t\in{\bf R})$
of automorphisms of the quasilocal
\Ca\ \A\ generated by the set $\{ a_n, a_n^*: n=0,1,2,\dots\}$. Let
the initial state $\omega_0$ (i.e. the expectation functional) of the
whole infinite chain (including the site $n=0$ representing the ``measured
system'') be given by
$$\omega_0(a_n^*a_n):=0\quad {\rm for}\ n>0,\quad
\omega_0(a_0^*a_0):=1.\eqno(2.2)$$
An explicit solution of this model$^5$ gives for any $n\ge 0$ (the state-vector
$\Omega$ corresponds to the state with all the spins of the
compound system pointing down):
$$\eqalignno{\omega_t(a_n^*a_n) &:=\omega_0(\tau_t(a^*_na_n))&(2.3a)\cr
{}&=1-\sum_{m=0}^{n-1}\lb a^*_0\Omega\vert e^{itH}{\textstyle\prod_{j=0}^m}a^*_j
\Omega\rb^2&(2.3b)\cr
{}&=1-\sum_{m=1}^n(-1)^{m+1}{m^2\over t^2}\left[J_m(2t)\right]^2&(2.3c)\cr
{}&=1-O(t^{-3}),\qquad for\quad t \to\infty.&(2.3d)\cr}$$
Here $J_m$ are the Bessel functions.
Hence, the (weak-$^*$) limit of \om t~ is the state \om1~ determined by the
relation: $$\mom 1~(a^*_na_n)=1\qquad {\rm for\ all}\ n\ge0.\eqno(2.4)$$
This shows that if the stationary state of the chain with all spins pointing
down is perturbed by turning up just the first spin (i.e. the measured
microsystem)
then the whole chain would evolve into a new stationary state with all spins
pointing up. But these two states, the initial and final ones, are
mutually disjoint. So, the state described by a (coherent) linear combination
of states differing initially just
in the state of the first spin of the chain (pointing down, or up)
evolves as $t \to \infty$ into a mixture of
two macroscopically distinguishable states.
\vfil\break
\subhead 2.2. Radiating Finite-Size Apparatus~
Let us take now the finite initial segment of the previously considered chain
of spins of the lenght N with Hamiltonian
$$H_N:= \sum_{n=1}^{N-2}a_n^*a_n(a_{n+1}+
a^*_{n+1})a_{n+2}a^*_{n+2}\eqno(2.5)$$
(We shal use previously introduced notation, except of redefinition of the
total system Hamiltonian $H$.)
Let this spin chain interacts with a scalar Fermi field described by linear
ascription of creation operators $b^*(\mph~)$ to one particle wave functions
$\mph~\in \mLq$. The annihilation operators are $b(\mph~)=
(b^*(\mph~))^*$. Let $d\Gamma (\hat p^2)$ be the second quantized\ct4
one particle
Hamiltonian $\hat p^2$, with $\hat p:=-i\partial_x$. The interaction is
defined by
$$V:=\kappa a^*_{N-1}a_{N-1}(a^*_Nb^*(\sigma)+a_Nb(\sigma)),\quad\kappa>0,
\ \sigma\in \mLq,\ \Vert\sigma\Vert=1.\eqno(2.6)$$
Let the total Hamiltonian of our system in the common Fock representation
(i.e. the Fermi field is in the finite particle number sector) be
$$H:=H_N-\bar\varepsilon a^*_Na_N+d\Gamma(\hat p^2)+V.\eqno(2.7)$$
One can show that the common vacuum (i.e. no particles, and all spins ``down'')
is stationary. Let us initially turn up
just the first spin from this common vacuum. Then one obtains an ``almost
exponential decay'' into the stationary state where all spins are pointing
up, and the Fermi field is again in its vacuum.\ct{12} For the chosen initial
state \om 0~ the dynamics given by Eq. (2.7) leads, e.g., to
$$\mom t~(a^*_Na_N)=1-{\rm o}(t^{-m})\qquad {\rm for}\
t\to\infty,\ {\rm for\ all}\ m\in {\bf Z}_+.\eqno(2.8)$$
\subhead 2.3 Unsharp Measurement~
In this subsection we introduce a model for nonideal measurement. The measured
quantity could be defined from the interaction of the apparatus (it is again
the infinite spin chain from Sec. 2.1) with the measured ``microsystem'' (this
is now a scalar nonrelativistic particle interacting with {\it first two}
spins in the chain). The microsystem (particle) is described by the Fermi
field from Sec. 2.2. The total Hamiltonian is now
$$H:=H_{\infty}+d\Gamma(\hat p^2)+W,\eqno(2.9)$$
where the interaction W is defined by
$$W:=\gamma(a^*_1+a_1)a_2a^*_2\otimes b^*(\sigma)b(\sigma),\eqno(2.10)$$
with $\gamma \in {\bf R}$ and $\sigma \in \mLq,\
\Vert\sigma\Vert=1$. Let the chain be again at $t=0$ in its ``vacuum''.
Let the initial state of the particle be described by an arbitrary normalized
wave function $\mph0~\in\mLq$. (The ``incoming'' asymptotics for $t\to-\infty$
of the scattering theory is not used here.) Denoting by \om t~ the state
of the total system in time t, and by $A$ an arbitrary observable of
that composed system, one can obtain\ct{13} that
$$\bar\omega (A) := \lim_{t\to\infty}\mom t~(A)\eqno(2.11)$$
exists, and one arrives for suitable parameters of W at:
$$\bar\omega =(w\mom1{chain}~+(1-w)\mom0{chain}~)\otimes\mom0{field}~,\eqno(2.12)$$
where \om j{chain}~ $(j=0,1)$ denotes the state of the chain with all spins
pointing down, resp. pointing up (the spin with $n=0$ is excluded now).
The field state \om0{field}~
is the Fermi vacuum. We again obtained a mixture of two disjoint states.
The probability $w$ of switching the apparatus can be expressed explicitly in
terms of dynamics of the chain, as well as of the particle-chain interaction
parameters.\ct{13} It can be written in the form
$$w=\lb\mph0~\vert\Lambda\mph0~\rb.\eqno(2.13)$$
Here $\Lambda$ is a positive operator of the norm $\Vert \Lambda\Vert \leq1$
acting on the Hilbert space \Lq. This operator represents the measured
(unsharp) observable of the particle. In engineer's terms,
$\Lambda$ can represent
also a ``characteristic'' of apparatus.
\hhead 3. Nonlinear Quantum Mechanics~
Let us now consider QM of a finite (closed, or in external field) system.
The sets of its mutually equivalent vector states are in bijective
correspondence with points
of the projective
Hilbert space \PH\ elements of which are one-dimensional projections in
\H\ (e.g., \H\ := \Lq). Similarly, each ``mixed state'' in QM is represented by
a density operator \rh~. As a quantal analogy with phase space of Hamiltonian
CM, we shall consider the set \Ss\ of all density matrices. (We do not use
the larger set of states \S\ containing, e.g., states with sharp values
of continuous quantities.) We shall define Poisson
brackets on \Ss. Let for all ``sufficiently nice'' real-valued functions
$f,\ h,\dots$ the symbols $d_{\mrh~}f,\ d_{\nu}h\dots$ denote the (Fr\`echet)
differentials at the points $\varrho,\ \nu,\dots$ of \Ss. We can consider
these differentials as unambigously defined bouned operators in \LH.
Then we define the Poisson bracket of $f$ and $h$ by\ct{11}
$$\{f,h\}(\nu):=i\,Tr(\nu[d_{\nu}f,d_{\nu}h]),\eqno(3.1)$$
where $[\cdot,\cdot]$ means the commutator of bounded operators in \LH,
and $Tr$ is the trace in the space $\cal T(H)$, ${\mLH\supset\cal
T(H)}\supset \mSs$, of trace class operators.
Time evolution given by the Hamiltonian function $h$ is ``essentially''
determined now
in the usual way (except of some complications with infinite
dimensionality of \H, and occurence of unbounded operators)\ct{11}
from Eq. (3.1). It can be shown\ct{11} that the
evolution of density matrices can be described by the `cocycle'
of unitary operators $u_h(t,\nu)$ (with $t\in{\bf R},\ \nu\in\mSs$)
satisfying Schr\"odinger-like (in general nonlinear)
equation with the `Hamilton operator' $d_{\mrh~}h$:
$$i\,{d\over dt}u_h(t,\nu)=d_{\nu(t)}h\ u_h(t,\nu).\eqno(3.2)$$
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 (3.1).
If $h(\nu)\equiv Tr(\nu H)$ for some $\nu$-independent operator $H$,
then the Eq. (3.2) 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.
Such a nonlinear quantum mechanics of a finite system arises naturally\ct7\
from an automorphism group of an algebra of infinite collection of the same
systems with usual kinematics in \LH, interacting mutually via a long range
very weak ({\it linear}) interactions introduced in Ref. 6; i.e., it is a
{\it reduced dynamics}. Its `Hamilton operator' expresses the influence of
the classical ``mean-field background''.\ct{7,8}
Let us note that this theory contains much more observables and
states than linear QM. Probability measures on \Ss\ could
describe ``Gemenge''. A consistent statistical interpretation of this
theory (different from the Weinberg's one)\ct{14}\ is also proposed.\ct{11}
\hhead 4. References~
\item{\ 1.}\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{\ 2.} R. Haag and D. Kastler, {\it J. Math. Phys.} {\bf 5}\ (1964) 848.
\item{\ 3.}K. Hepp, {\it Helv. Phys. Acta} {\bf 45}\ (1972) 237.
\item{\ 4.}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{\ 5.} P. B\'ona, {\it acta phys. slov.} {\bf 27} (1977) 101.
\item{\ 6.}K. Hepp and E. H. Lieb, {\it Helv. Phys. Acta} {\bf 46} (1973) 573.
\item{\ 7.}P. B\'ona, {\it J. Math. Phys.}\ {\bf 29} (1988) 2223.
\item{\ 8.}P. B\'ona, {\it Czech. J. Phys.} {\bf B37} (1987) 482.
\item{\ 9.}D. M. Greenberger, M. A. Horne and A. Zeilinger, {\it Physics Today} {\bf
46} (August 1993) 22.
\item{10.}{P. B\'ona}, {\it J. Math. Phys.} {\bf 30} {(1989)} {2994}.
\citb{11}{P. B\'ona}{Quantum Mechanics With Mean-Field Background}{Comenius
University, Bratislava 1991}{, preprint Ph10-91.}~
\citb{12}{P. B\'ona}{Unpublished Notes}{Bratislava 1993}~
\citj{13}{P. B\'ona}{ACTA F.R.N. Univ.Comen. PHYSICA,}{XX}{1980}{65}~
\citj{14}{S. Weinberg}{Ann. Phys.}{194}{1989}{336}~
\citj{15}{W. H. Zurek}{Phys. Rev.}{D 26}{1982}{1862}~
\bye