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Schroedinger dynamics of microsystem-cum-measuring instrument; macroscopic phase cells as pointer positions; macroscopic decoherence; reduction of wave packet of microsystem
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\magnification 1200
\centerline {\bf On the Mathematical Structure of Quantum Measurement Theory}
\vskip 1cm
\centerline {{\bf by Geoffrey Sewell}\footnote*{e-mail: g.l.sewell@qmul.ac.uk}}
\vskip 0.5cm
\centerline {\bf Department of Physics, Queen Mary, University of London}
\vskip 0.5cm
\centerline {\bf Mile End Road, London E1 4NS, UK}
\vskip 1cm
\centerline {\bf Abstract}
\vskip 0.5cm
We show that the key problems of quantum measurement theory, namely the reduction of
the wave-packet of a microsystem and the specification of its quantum state by a
macroscopic measuring instrument, may be rigorously resolved within the traditional
framework of the quantum mechanics of finite conservative systems. The argument is
centred on the generic model of a microsystem, $S$, coupled to a finite macroscopic
measuring instrument ${\cal I}$, which itself is an $N$-particle quantum system. The
pointer positions of ${\cal I}$ correspond to the macrostates of this instrument, as
represented by orthogonal subspaces of the Hilbert space of its pure states. These
subspaces, or \lq phase cells\rq , are the simultaneous eigenspaces of a set of coarse
grained intercommuting macro-observables, $M$, and, crucially, are of astronomically
large dimensionalities, which increase exponentially with $N$. We formulate conditions
on the conservative dynamics of the composite $(S+{\cal I})$ under which it yields both
a reduction of the wave packet describing the state of $S$ and a one-to-one
correspondence, following a measurement, between the observed value of $M$ and the
resultant eigenstate of $S$; and we show that these conditions are fulfilled by the finite
version of the Coleman-Hepp model.
\vskip 1cm
{\bf Key Words:} Schroedinger dynamics of microsystem-cum-measuring instrument;
macroscopic phase cells as pointer positions; macroscopic decoherence; reduction of
wave packet of microsystem.
\vfill\eject
\centerline {\bf 1. Introductory Discussion.}
\vskip 0.3cm
The quantum theory of measurement is concerned with the determination of the state of a
microsystem, $S$, such as an atom, by a macroscopic measuring instrument, ${\cal I}$.
In Von Neumann\rq s [1] {\it phenomenological} picture, the $S-{\cal I}$ coupling
leads to two essential effects. Firstly it converts a pure state of $S$, as given by a linear
combination ${\sum}_{r=1}^{n}c_{r}u_{r}$ of its orthonormal eigenstates $u_{r}$,
into a statistical mixture of these states for which ${\vert}c_{r}{\vert}^{2}$ is the
probability of finding this system in the state $u_{r}$: this is the phenomenon often
termed the \lq reduction of the wave packet\rq. Secondly, it sends a certain set of
classical, i.e. intercommuting, macroscopic variables $M$ of ${\cal I}$ to values,
indicated by pointers, that specify the state $u_{r}$ of $S$ that is actually realised. The
problem of the quantum theory of this process is to characterise the properties of the
macroscopic observables $M$ and the dynamics of the composite $S_{c}=(S+{\cal I})$
that lead to these two effects. Our objective here is to treat this problem on the basis of
the model for which $S_{c}$ is a strictly conservative finite quantum system, whose
dynamics is governed by its many-particle Schroedinger equation; and our main result is
that this model does indeed contain the structures required for the resolution of this
problem. This result provides mathematical justification for the heuristic arguments of
Van Kampen [2], which led to essentially the same conclusion. It also establishes that
there is no need to base quantum measurement theory on the model, advocated by some
authors [3-7], in which $S_{c}$ is a dissipative system, as a result of {\it either} its
interaction with the \lq rest of the Universe\rq\ [3-6] {\it or} a certain postulated
nonlinear modification of its Schroedinger equation that leads to a classical deterministic
evolution of its macroscopic observables [7].
\vskip 0.2cm
As regards the main requirements of a satisfactory theory of the measurement process, it
is clear from the works of Bohr [8], Jauch [9] and Van Kampen [2] that such a theory
demands both a characterisation of the macroscopicality of the observables $M$ and an
amplification property of the $S-{\cal I}$ coupling whereby different microstates of $S$
give rise to macroscopically different states of ${\cal I}$. Evidently, this implies that the
initial state in which ${\cal I}$ is prepared must be unstable against microscopic changes
in the state of $S$. On the other hand, as emphasised by Whitten-Wolfe and Emch [10,
11], the correspondence between the microstate of $S$ and the eventual observed
macrostate of ${\cal I}$ must be stable against macroscopically small changes in the
initial state of this instrument, of the kind that are inevitable in experimental procedures.
Thus, the initial state of ${\cal I}$ must be {\it metastable} by virtue of this combination
of stability and instability properties.
\vskip 0.2cm
There are basically two ways of characterising the macroscopicality of the observables
$M$ of ${\cal I}$. The first is to represent this instrument as a large but finite $N$-
particle system for which $N$ is extremely large, e.g. of the order of $10^{24}$. $M$ is
then represented according to the scheme of Van Kampen [12] and Emch [13] as an
intercommuting set of observables, which typically are coarse-grained extensive
conserved variables of parts or of the whole of this instrument. The simultaneous
eigenspaces of these observables then correspond to classical \lq phase cells\rq\ , which
represent the possible positions of the pointers of ${\cal I}$. Moreover, for suitably
coarse-grained macroscopic observables $M$, the dimensionality of each of these cells is
astronomically large [2, 12] since, by Boltzmann’s formula, it is just the exponential of
the entropy of the macrostate that it represents, and thus it increases exponentially with
$N$.
\vskip 0.2cm
The second way of characterising the macroscopicality of $M$ is to idealise the
instrument ${\cal I}$ as an infinitely extended system of particles, with finite number
density, and to take $M$ to be a set of global intensive observables, which necessarily
intercommute [14, 10, 11]. This corresponds to the picture employed in the statistical
mechanical description of large systems in the thermodynamic limit [15-17], and it has
the merit of sharply distinguishing between macroscopically different states, since
different values of $M$ correspond to disjoint primary representations of the observables.
Moreover, in the treatments of the measurement problem based on this idealisation, the
models of Hepp [14] and Whitten-Wolfe and Emch [10, 11] do indeed exhibit the
required reduction of the wave-packet and the one-to-one correspondence between the
pointer position of ${\cal I}$ and the resultant state of $S$; and these results are stable
against all localised perturbations of the initial state of ${\cal I}$. On the debit side,
however, Hepp\rq s model requires an infinite time for the measurement to be effected
(cf. Bell [18]), while although that of Whitten-Wolfe and Emch achieves its
measurements in finite times, it does so only by dint of a physically unnatural, globally
extended $S-{\cal I}$ interaction. In view of these observations, it appears to be
worthwhile to explore the mathematical structure of the measuring process on the basis of
the model for which ${\cal I}$ is large but finite, with the aim of obtaining conditions
under which it yields the essential results obtained for the infinite model instrument, but
with a finite realistic observational time. Evidently, this requires rigorous control of any
approximations that arise as a result of the finiteness of $N$.
\vskip 0.2cm
The object of this article is to investigate the mathematical structure of the measurement
process by means of a dynamical treatment of the generic model of the composite,
$S_{c}$, of a microsystem, $S$, and a macroscopic, but finite, measuring instrument
${\cal I}$. Our treatment of this model is designed to obtain conditions on the $S-{\cal
I}$ coupling that lead to both the reduction of the wave-packet and the required
correspondence between the reading of the instrument’s pointer and the resultant state of
$S$. As in the works [2] , [8-11] and [14], we avoid the assumption of Von Neumann
[1] and Wigner [19] that the observation of the pointer of ${\cal I}$ requires another
measuring instrument, ${\cal I}_{2}$, which in turn requires yet another instrument, and
so on, in such a way that the whole process involves an infinite regression ending up in
the observer\rq s brain! Instead, we assume that the measurement process ends with the
reading of the pointers that evaluate the macrovariables $M$ of ${\cal I}$. This carries
the implicit assumption that the dynamics of these variables is sufficiently robust to
ensure that the act of reading the pointers has negligible effect on their positions. In this
sense, the macroscopic variables of ${\cal I}$ behaves radically differently from the
observables of $S$, since the states of \lq small\rq\ quantum microsystems are
susceptible to drastic changes as a result of microscopic disturbances. A further crucial
property of ${\cal I}$ is that, as pointed out above, the dimensionality of each of its
macroscopic phase cells is of astronomically large dimensionality, which increases
exponentially with $N$; and, by a similar argument, the same is true for its density of
energy eigenstates We shall see that the enormous phase cells of the finite instrument
${\cal I}$ play the essential role of the disjoint representation spaces of the infinite one
and consequently that the finite model possesses all the positive properties of the infinite
one, with the bonus that it achieves its measurements in finite, realistic times.
Furthermore, the enormity of its density of energy eigenstates ensures that the periods of
its Poincare recurrences are astronomically long. We can therefore discount these
recurrences by restricting our treatment of the dynamics to finite intervals of much
shorter duration.
\vskip 0.2cm
Turning now to the formulation of the measurement process, we assume, in a standard
way, that the observables of $S$ and the macroscopic ones, $M$, of ${\cal I}$, on which
measurements are performed, generate $W^{\star}$-algebras ${\cal A}$ and ${\cal M}$,
respectively, the latter being abelian. The process is expressed in terms of the state on the
algebra ${\cal A}{\otimes}{\cal M}$ that results from the evolution of $S_{c}$ from an
initial state obtained by independent preparations of $S$ and ${\cal I}$. The resultant
evolved state of $S_{c}$ then determines the expectation values of the observables, $A$,
of $S$ and their conditional expectation values, $E(A{\vert}{\cal M})$, given the
$M$\rq s. Thus it determines the probabilistic state, ${\rho}$, of $S$ {\it before} those
macroscopic variables are measured and its {\it subsequent} state, as given by the form
of $E(.{\vert}{\cal M})$, following the measurement.
\vskip 0.2cm
On relating the state ${\rho}$ and the conditional expectation functional $E(.{\vert}{\cal
M})$ to the $S-{\cal I}$ interaction, we find that the mathematical model yields two
classes of effective instruments ${\cal I}$, though from the empirical standpoint these
classes are essentially equivalent. The first class of instruments comprises those for
which the wave packet of $S$ is reduced according to Von Neumann\rq s prescription
and the correspondence between the observed value of $M$ and the microstate of $S$ is
strictly one-to-one. The second class of instruments comprises those for which this result
arises with overwhelming probability, for large $N$, rather than with absolute certainty.
Thus, in this case, if the result of a measurement is interpreted on the basis of an
assumption of a perfect correspondence between the microstate of $S$ and the macrostate
of ${\cal I}$, then there is a miniscule possibility that the pointer position will
correspond to a state of $S$ quite different from (in general orthogonal to) the indicated
one. We term the instruments of the first class {\it ideal} and those of the second class
{\it normal}. As support for this classification of instruments, we show that, in the case of
a finite version of the Coleman-Hepp model [14], the instrument is generically normal,
though it is ideal for certain special values of its parameters. Furthermore, in the former
case, the odds against the pointer indicating the \lq wrong\rq\ state of $S$ increase
exponentially with $N$.
\vskip 0.2cm
We present our mathematical treatment of the measurement process as follows. In
Section 2, we formulate the generic model of the composite quantum system $S_{c}$,
employing the phase cell representation of Van Kampen [12] and Emch [13] for the
description of the macroscopic observables of ${\cal I}$. In particular, we formulate the
time-dependent expectation values of the observables, $A$, of $S$ and of the
macroscopic ones, $M$, of ${\cal I}$, as well as the conditional expectation value of
$A$, given ${\cal M}$, subject to the assumption that $S$ and ${\cal I}$ are
independently prepared and then coupled together at time $t=0$. In Section 3, we
formulate the conditions on the dynamics of the model under which the measuring
instrument ${\cal I}$ is ideal or normal, in the sense described above. In Section 4, we
show that the general scheme of Sections 2 and 3 is fully realized by the finite version of
the Coleman-Hepp model [14]. There we show that the instrument ${\cal I}$ for this
model is generically normal, though for certain special values of its parameters it is ideal.
Moreover, we show that these results are stable under localised perturbations of the initial
state of ${\cal I}$, and even under global ones that correspond to small changes in the
values of intensive thermodynamical variables (e.g. temperature, polarisation) of that
state. Here we take the generic prevalence of normality of ${\cal I}$, for this model, to be
an indication that real quantum measuring instruments are generally normal rather than
ideal. We conclude, in Section 5, with a brief resume of the picture presented here.and a
suggestion about a possible further development in the physics of the quantum
measutrement process.
\vskip 0.5cm
\centerline {\bf 2. The Generic Model.}
\vskip 0.3cm
We assume that the algebras of observables, ${\cal A}$ and ${\cal B}$, of the
microsystem $S$ and the instrument ${\cal I}$, are those of the bounded operators in
separable Hilbert spaces ${\cal H}$ and ${\cal K}$, respectively. Correspondingly, the
states of these systems are represented by the density matrices in the respective spaces.
The density matrices for the pure states are then the one-dimensional projectors. For
simplicity we assume that ${\cal H}$ is of finite dimensionality $n$.
\vskip 0.2cm
We assume that the coupled composite $S_{c}:=(S+{\cal I})$ is a conservative system,
whose Hamiltonian operator $H_{c}$, in ${\cal H}{\otimes}{\cal K}$, takes the form
$$H_{c}=H{\otimes}I_{\cal K}+I_{\cal H}{\otimes}K+V,\eqno(2.1)$$
where $H$ and $K$ are the Hamiltonians of $S$ and ${\cal I}$, respectively, and $V$ is
the $S-{\cal I}$ interaction. Thus, the dynamics of $S_{c}$ is given by the one-
parameter group $U_{c}$ of unitary transformations of ${\cal H}{\otimes}{\cal K}$
generated by $iH_{c}$, i.e.
$$U_{c}(t)={\rm exp}(iH_{c}t) \ {\forall} \ t{\in}{\bf R}.\eqno(2.2)$$
We assume that the the systems $S$ and ${\cal I}$ are prepared, independently of one
another, in their initial states represented by density matrices ${\omega}$ and
${\Omega}$, respectively, and then coupled together at time $t=0$. Thus the initial state
of the composite $S_{c}$ is given by the density matrix ${\omega}{\otimes}{\Omega}$
in ${\cal H}_{c}:={\cal H}{\otimes}{\cal K}$. Further, we assume that the initial state
of $S$ is pure, and thus that ${\omega}$ is the projection operator $P({\psi})$ for a
vector ${\psi}$ in ${\cal H}$. The initial state of $S_{c}$ is then
$${\Phi}=P({\psi}){\otimes}{\Omega}.\eqno(2.3)$$
Since ${\cal H}$ is $n$-dimensional, we may take as its basis a complete orthonormal set
of eigenvectors, $(u_{1},. \ .,u_{n})$, of $H$ . Hence, the initial state vector ${\psi}$ of
$S$ is given by a linear combination of these vectors, i.e.
$${\psi}={\sum}_{r=1}^{n}c_{r}u_{r},\eqno(2.4)$$
where
$${\sum}_{r=1}^{n}{\vert}c_{r}{\vert}^{2}=1;\eqno(2.5)$$
while the action of $H$ on $u_{r}$ is given by the equation
$$Hu_{r}={\epsilon}_{r}u_{r},\eqno(2.6)$$
where ${\epsilon}_{r}$ is the corresponding eigenvalue of this operator.
\vskip 0.2cm
We assume that the instrument ${\cal I}$ is designed to perform measurements of the
first kind (cf. Jauch [9]), whereby the $S-{\cal I}$ coupling does not induce transitions
between the eigenstates ${\lbrace}u_{r}{\rbrace}$ of $S$. This signifies that the
interaction $V$ takes the form
$$V={\sum}_{r=1}^{n}P(u_{r}){\otimes}V_{r},$$
where $P(u_{r})$ is the projection operator for $u_{r}$ and the $V_{r}$\rq s are
observables of ${\cal I}$. Hence, by Eq. (2.1), the Hamiltonian of the composite system
$S_{c}$ is
$$H_{c}={\sum}_{r=1}^{n}P(u_{r}){\otimes}K_{r},\eqno(2.7)$$
where
$$K_{r}=K+V_{r}+{\epsilon}_{r}I_{\cal K}.\eqno(2.8)$$
Consequently, by Eqs. (2.2) and (2.7), the dynamical group $U_{c}$ is given by the
formula
$$U_{c}(t)={\rm exp}(iH_{c}t)={\sum}_{r=1}^{n}P(u_{r}){\otimes}U_{r}(t),
\eqno(2.9)$$
where
$$U_{r}(t)={\rm exp}(iK_{r}t).\eqno(2.10)$$
Consequently, since the evolute at time $t \ ({\geq}0)$ of the initial state ${\Phi}$ of
$S_{c}$ is $U_{c}^{\star}(t){\Phi}U_{c}(t):={\Phi}(t)$, it follows from Eqs.(2.3), (2.4)
and (2.10) that
$${\Phi}(t)={\sum}_{r,s=1}^{n}{\overline c}_{r}c_{s}P_{r,s}
{\otimes}{\Omega}_{r,s}(t),\eqno(2.11)$$
where $P_{r,s}$ is the operator in ${\cal H}$ defined by the equation
$$P_{r,s}f=(u_{s},f)u_{r} \ {\forall} \ f{\in}{\cal H}\eqno(2.12)$$
and
$${\Omega}_{r,s}(t)=U_{r}^{\star}(t){\Omega}U_{s}(t).\eqno(2.13)$$
\vskip 0.3cm
{\bf 2.1. The Macroscopic Observables of ${\cal I}$.} We assume that these conform to
the following scheme, due to Van Kampen [12] and Emch [13].
\vskip 0.2cm\noindent
(1) They are intercommuting observables, which typically are coarse grained extensive
conserved variables of parts or of the whole of the system ${\cal I}$. The algebra, ${\cal
M}$, of these observables is therefore an abelian subalgebra of the full algebra, ${\cal
B}$, of bounded observables of ${\cal I}$. For simplicity, we assume that ${\cal M}$ is
finitely generated and thus that it consists of the linear combinations of a finite set of
orthogonal projectors ${\lbrace}{\Pi}_{\alpha}{\vert}{\alpha}=1,2. \ .,{\nu}{\rbrace}$
that span the space ${\cal K}$. It follows from these specifications that
$${\Pi}_{\alpha}{\Pi}_{\beta}={\Pi}_{\alpha}{\delta}_{{\alpha}{\beta}},\eqno(2.14)$$
$${\sum}_{{\alpha}=1}^{\nu}{\Pi}_{\alpha}=I_{\cal K}\eqno(2.15)$$
and that any element, $M$, of ${\cal M}$ takes the form
$$M={\sum}_{{\alpha}=1}^{\nu}M_{\alpha}{\Pi}_{\alpha},\eqno(2.16)$$
where the $M_{\alpha}$\rq s are constants.The subspaces ${\lbrace}{\cal K}_{\alpha}:=
{\Pi}_{\alpha}{\cal K}{\rbrace}$ of ${\cal K}$ correspond to classical phase cells.
Each such cell then represents a macrostate of ${\cal I}$, and is identified by the position
of a pointer (or set of pointers) in a measurement process.
\vskip 0.2cm\noindent
(2) As noted in Section 1, the dimensionality of each cell ${\cal K}_{\alpha}$ is
astronomically large, since it is given essentially by the exponential of the entropy
function of the macro-observables and thus grows exponentially with $N$. The largeness
of the phase cells is closely connected to the robustness of the macroscopic measurement.
\vskip 0.2cm
Note here that these properties of ${\cal M}$ are just general ones of macroscopic
observables and do not depend on ${\cal I}$ being a measuring instrument for the system
$S$. The coordination of these properties with those of $S$ that are pertinent to the
measuring process will be treated in Section 3.
\vskip 0.3cm
{\bf 2.2. Expectation and Conditional Expectation Values of Observables.} The
observables of $S_{c}$ with which we shall be concerned are just the self-adjoint
elements of ${\cal A}{\otimes}{\cal M}$. Their expectation values for the time-
dependent state ${\Phi}(t)$ are given by the formula
$$E\bigl(A{\otimes}M\bigr)={\rm Tr}\bigl({\Phi}(t)[A{\otimes}M]\bigr) \ {\forall} \
A{\in}
{\cal A}, \ M{\in}{\cal M},\eqno(2.17)$$
In particular, the expectation values of the observables of $S$ and the macroscopic ones
of ${\cal I}$ are given by the equations
$$E(A)=E(A{\otimes}I_{\cal K})\eqno(2.18)$$
and
$$ E(M)=E(I_{\cal H}{\otimes}M) ,\eqno(2.19)$$
respectively. Further, in view of the abelian character of ${\cal M}$, the expectation
functional $E$ is compatible with a unique conditional expectation functional on ${\cal
A}$ with respect to ${\cal M}$, as the following argument shows. Such a conditional
expectation is a linear mapping $E(.{\vert}{\cal M})$ of ${\cal A}$ into ${\cal M}$ that
preserves positivity and normalisation and satisfies the condition
$$E\bigl(E(A{\vert}{\cal M})M\bigr)=E(A{\otimes}M) \ {\forall} \ A{\in}{\cal A}, \
M{\in}{\cal M}.\eqno(2.20)$$
Therefore since, by linearity and Eq. (2.16), $E(.{\vert}{\cal M})$ must take the form
$$E(A{\vert}{\cal M})={\sum}_{\alpha}{\omega}_{\alpha}(A){\Pi}_{\alpha},
\eqno(2.21)$$
where the ${\omega}_{\alpha}$\rq s are linear functionals on ${\cal A}$, it follows from
Eq. (2.20) that
$${\omega}_{\alpha}(A)E({\Pi}_{\alpha})=E(A{\otimes}{\Pi}_{\alpha})$$
and consequently, by Eq. (2.21),
$$E(A{\vert}{\cal M})={\sum}_{\alpha}^{\prime}E(A{\otimes}{\Pi}_{\alpha})
{\Pi}_{\alpha}/ E({\Pi}_{\alpha}) \ {\forall} \ A{\in}{\cal A},\eqno(2.22)$$
where the prime over ${\Sigma}$ indicates that summation is confined to the
${\alpha}$\rq s for which $E({\Pi}_{\alpha})$ does not vanish. In view of Eq. (2.15),
this formula for $E(.{\vert}{\cal M})$ meets the requirements of positivity and
normalisation.
\vskip 0.3cm
{\bf 2.3. Properties of the Expectation Functional $E$.} By Eqs. (2.11)-(2.13), (2.16)
and (2.17),
$$E\bigl(A{\otimes}M\bigr)={\sum}_{r,s=1}^{n}{\sum}_{{\alpha}=1}^{\nu}
{\overline c}_{r}c_{s}(u_{r},Au_{s})M_{\alpha}F_{r,s;{\alpha}},\eqno(2.23)$$
where
$$F_{r,s;{\alpha}}={\rm Tr}\bigl({\Omega}_{r,s}(t){\Pi}_{\alpha}\bigr).\eqno(2.24)$$
Key properties of $F_{r,s:{\alpha}}$, which follows from Eqns. (2.13), (2.15) and (2.24)
are that
$${\sum}_{{\alpha}=1}^{\nu}F_{r,r;{\alpha}}=1,\eqno(2.25)$$
$$1{\geq}F_{r,r;{\alpha}}{\geq}0\eqno(2.26)$$
and
$$F_{r,s:{\alpha}}={\overline F}_{s,r:{\alpha}},\eqno(2.27)$$
where the bar over $F$ on the r.h.s. indicates complex conjugation. It also follows from
those formulae that, for
$z_{1},. \ .,z_{n}{\in}{\bf C}$, the sesquilinear form ${\sum}_{r,s=1}^{n}
{\overline z}_{r}z_{s}F_{r,s;{\alpha}}$ is positive, and hence
$$F_{r,r;{\alpha}}F_{s,s;{\alpha}}{\geq}{\vert}F_{r,s;{\alpha}}{\vert}^{2}.
\eqno(2.28)$$
\vskip 0.5cm
\centerline {\bf 3. The Measurement Process}
\vskip 0.3cm
As noted in Section 2, a pointer reading of ${\cal I}$ serves to identify the phase cells
${\cal K}_{\alpha}$ that represents its macrostate. Eq. (2.22) therefore signifies that the
expectation values of the observables of $S$ following that measurement is
$$E(A{\vert}{\cal K}_{\alpha}):= E(A{\otimes}{\Pi}_{\alpha})/E({\Pi}_{\alpha}).
\eqno(3.1)$$
Now, in order that the pointer reading specifies the eigenstate of $S$, we require a one-
to-one correspondence between the phase cells ${\cal K}_{\alpha}$ and the eigenstates
$u_{r}$ of $S$. Accordingly, we assume that, for an instrument designed to identify the
microstate of $S$, the number of these phase cells is just the number of the eigenstates
$u_{r}$ of $S$, namely $n$.
\vskip 0.3cm
{\bf 3.1. The Ideal Instruments.} We term the instrument ${\cal I}$ {\it ideal} if there is
a one-to-one correspondence between the pointer reading ${\alpha}$ and the eigenstate
$u_{r}$ of $S$, on a realistic observational time scale. Thus ${\cal I}$ is ideal if, for
times $t$ greater than some critical value, ${\tau}$, and less, in order of magnitude, than
the Poincare’ recurrence times,
\vskip 0.2cm\noindent
(I.1) the time-dependent state ${\Omega}_{r,r}(t)$ of ${\cal I}$, that arises in
conjunction with the state $u_{r}$ of $S$ in the formula Eqs. (2.13), lies in one of the
subspaces ${\cal K}_{\alpha}$ of ${\cal K}$;
\vskip 0.2cm\noindent
(I.2) the correspondence between $r$ and ${\alpha}$ here is one-to-one, i.e.
${\alpha}=a(r)$, where $a$ is an invertible transformation of the point set ${\lbrace}1,2,.
\ .,n{\rbrace}$; and
\vskip 0.2cm\noindent
(I.3) this correspondence is stable with respect to perturbations of the initial state
${\Omega}$ of ${\cal I}$ that are localised, in the sense that each of them leaves this
state unchanged outside some region contained in a ball of volume $O(1)$ with respect to
$N$.
\vskip 0.2cm\noindent
The conditions (I.1) and (I.2) signify that, for times $t$ in the range specified there,
$${\rm Tr}\bigl({\Omega}_{r,r}(t){\Pi}_{\alpha}\bigr)={\delta}_{a(r),{\alpha}},$$
i.e., by Eq. (2.24),
$$F_{r,r:{\alpha}}={\delta}_{a(r),{\alpha}}.\eqno(3.2)$$
Moreover, it follows from Eqs. (2.25) and (2.26), together with the invertibility of the
function $a$ that Eq. (3.2) not only implies but is actually equivalent to the condition
$$F_{r,r;a(r)}=1.\eqno(3.2)^{\prime}$$
Further, by Eqs. (2.28) and (3.2) and the invertibility of $a$,
$$F_{r,s;{\alpha}} =0 \ {\rm for} \ r{\neq}s.\eqno(3.3)$$
Consequently, by Eqs. (2.23), (3.2) and (3.3),
$$E(A{\otimes}M)={\sum}_{r=1}^{n}w_{a(r))}M_{a(r)}
(u_{r},Au_{r}),\eqno(3.4)$$
where
$$w_{a(r)}={\vert}c_{r}{\vert}^{2}.\eqno(3.5)$$
Hence, by Eqs. (3.1) and (3.4) and the invertibility of $a$,
$$E({\Pi}_{\alpha})=w_{\alpha},\eqno(3.6)$$
$$E(A)={\sum}_{r=1}^{n}w_{a(r)}
(u_{r},Au_{r})\eqno(3.7)$$
and
$$E(A{\vert}{\cal K}_{a(r)})= (u_{r},Au_{r}).\eqno(3.8 )$$
\vskip 0.2cm
Eqs. (3.6) and (3.7) signify that, {\it before} the pointer position is read, $w_{\alpha}$ is
the probability that the reading is ${\alpha}$ and the state of $S$ is given by the density
matrix
$${\rho}={\sum}_{r=1}^{n}w_{a(r)}P(u_{r}),$$
i.e., by Eq. (3.5),
$${\rho}={\sum}_{r=1}^{n}{\vert}c_{r}{\vert}^{2}P(u_{r}).\eqno(3.9)$$
Thus we have a reduction of the wave packet, as given by the transition from the pure
state ${\psi} \ (={\sum}_{r=1}^{n})$ to this mixed state ${\rho}$.
\vskip 0.2cm
According to the standard probabilisitic interpretation of quantum mechanics, Eq. (3.9)
specifies the state of $S$ just prior to the reading of the pointers, whereas Eq. (3.8) serves
to specify its state following that reading. Thus, by Eq. (3.9), ${\vert}c_{r}{\vert}^{2}$
is the probability that the pointer reading will yield the result that $u_{r}$ is the state of
$S$; while Eq. (3.8) signifies that, following a reading that yields the result that
${\alpha}=a(r)$, the state of $S$ is $u_{r}$. In the standard picture of quantum theory,
there is no causality principle that determines which of the states $u_{r}$ will be found.
\vskip 0.3cm
{\bf Comments.} (1) As shown above, the property (3.2) ensures that ${\cal I}$ enjoys
all the essential properties of a measuring instrument since it implies both the reduction
of the wave-packet and the one-to-one correspondence between the pointer position and
the microstate of $S$. On the other hand, the property (3.3), which ensures the reduction
of the wave-packet, does not imply Eq. (3.2) and therefore does not, of itself, imply that
${\cal I}$ serves as a measuring instrument
\vskip 0.2cm
(2) The property (3.3) signifies that the $S-{\cal I}$ coupling removes the interference
between the different components $u_{r}$ of the pure state ${\psi}$ and thus represents
a {\it complete decoherence} effect. To see how this is related to the structure of a typical
phase cell ${\cal K}_{\alpha}$, we introduce a complete orthonormal basis
${\lbrace}{\theta}_{{\alpha},{\lambda}}{\rbrace}$ of this cell, where the index
${\lambda}$ runs from $1$ to ${\rm dim}({\cal K}_{\alpha})$, the dimensionality of
${\cal K}_{\alpha}$. We then infer from Eqs. (2.13) and (2.23) that
$$F_{r,s;{\alpha}}={\sum}_{{\lambda}=1}^{{\rm dim}({\cal K}_{\alpha})}
\bigl(U_{r}(t){\theta}_{{\alpha},{\lambda}},{\Omega}
U_{s}(t){\theta}_{{\alpha},{\lambda}}\bigr),$$
Hence, as $iK_{r}$ is the generator of $U_{r}$, this equation signifies that the
decoherence arises from the aggregated destructive interference of the evolutes of the
vectors ${\theta}_{{\alpha},{\lambda}}$ generated by the different Hamiltonians
$K_{r}$ and $K_{s}$. This picture of decoherence corresponds to that assumed by Van
Kampen [2].
\vskip 0.3cm
{\bf 3.2. Normal Measuring Instruments.} We term the instrument ${\cal I}$
{\it normal}\footnote*{We conjecture that the behaviour of real instruments is generally
normal in the sense specified here and thus that the use of this adjective is approriate.
Some support for this conjecture is provided by the results of Section 4 for the Coleman-
Hepp model.} if the following conditions are fulfilled.
\vskip 0.2cm\noindent
(N.1)) A weaker form of the ideality condition (3.2), or equivalently (3.2)$^{\prime}$,
prevails, to the effect that the difference between the two sides of the latter formula is
negligibly small, i.e., noting Eq. (2.25), that
$$0<1-F_{r,r;a(r)}<{\eta}(N),\eqno(3.10)$$
where, for large $N, \ {\eta}(N)$ is miniscule by comparison with unity: in the case of
the finite version of the Coleman-Hepp model treated in Section 4, it is ${\exp}(-cN)$,
where $c$ is a fixed positive constant of the order of unity. We note that, by Eq. (2.25)
and the positivity of ${\Pi}_{\alpha}$, the condition (3.10) is equivalent to the inequality
$$0<{\sum}_{r{\neq}a^{-1}({\alpha})}F_{r,r;{\alpha}}<{\eta}(N).
\eqno(3.10)^{\prime}$$
Further, it follows from Eqs. (2.28), (3.10) and (3.10)$^{\prime}$ that
$${\vert}F_{r,s;{\alpha}}{\vert}<{\eta}(N)^{1/2} \
{\rm for} \ r{\neq}s,\eqno(3.11)$$
which is evidently a {\it decoherence condition}, being a slightly weakened version of
the complete one given by Eq. (3.3).
\vskip 0.2cm\noindent
(N.2) This condition (N.1) is stable under localised modifications of the initial state
${\Omega}$ of ${\cal I}$. This stability condition may even be strengthened to include
global perturbations of ${\Omega}$ corresponding to small changes in its intensive
thermodynamic parameters (cf. the treatment of the Coleman-Hepp model in Section 4).
\vskip 0.2cm
It follows now from Eq. (3.11) that the replacement of the ideal condition (3.2) by the
normal one (3.10) leads to modifications of the order ${\eta}(N)^{1/2}$ to the formula
(3.4) and its consequences. In particular, it implies that a pointer reading ${\alpha}$
signifies that it is overwhelmingly probable, but not absolutely certain, that the state of
$S$ is
$u_{a^{-1}({\alpha})}$, as the following argument shows. Suppose that the initial state
of $S$ is $u_{r}$. Then, by Eq. (2.11), the state of $S_{c}$ at time $t$ is
$P(u_{r}){\otimes}{\Omega}_{r,r}(t)$; and by Eqs. (3.10)$^{\prime}$, there is a
probability of the order of ${\eta}(N)$ that the pointer reading is given by a value
${\alpha}$, different from $a(r)$, of the indicator parameter of ${\cal I}$. In the freak
case that this possibility is realised, this would mean that the state $u_{r}$ of $S$ led to a
pointer reading ${\alpha}{\neq}a(r)$. Hence, in this case, any inference to the effect that
a pointer reading ${\alpha}$ signified that the state of $S$ was $u_{a^{-1}({\alpha})}$
would be invalid.
\vskip 0.3cm
{\bf Comments.} The scheme proposed here admits two kinds of measuring instruments,
namely the ideal and the normal ones. The former fulfill perfectly the demands for the
reduction of the wave-packet and the one-to-one correspondence between the pointer
reading of the measuring instrument and the eigenstate of the observed microsystem. On
the other hand, in the case of a normal instrument, there is just a minuscule possibility
that the pointer reading might correspond to the \lq wrong\rq\ eigenstate of the
microsystem. However, as the odds against such an eventuality are overwhelming, the
distinction between the two kinds of instruments is essentially mathematical rather than
observational.
\vskip 0.5cm
\centerline {\bf 4. The Finite Coleman-Hepp Model.}
\vskip 0.3cm
This model is a caricature of an electron that interacts with a finite spin chain that serves
to measure the electronic spin [14]. In order to fit this model into the scheme of the
previous Sections, we regard the electron, ${\cal P}$, as the composite of two entities,
namely its spin, ${\cal P}_{1}$, and its orbital component, ${\cal P}_{2}$. We then take
the system $S$ to be just ${\cal P}_{1}$ and the instrument ${\cal I}$ to be the
composite of ${\cal P}_{2}$ and the chain ${\cal C}$. Thus, we build the model of
$S_{c}=(S+{\cal I})$ from its components in the following way.
\vskip 0.3cm
{\bf 4.1. The System $S={\cal P}_{1}$.} This is just a single Pauli spin. Thus, its state
space is ${\cal H}={\bf C}^{2}$ and its three-component spin observable is given by the
Pauli matrices $(s_{x},s_{y},s_{z})$. We denote by $u_{\pm}$ the eigenvectors of
$s_{z}$ whose eigenvalues are ${\pm}1$, respectively. These vectors then form a basis
in ${\cal H}$. We denote their projection operators by $P_{\pm}$, respectively.
\vskip 0.3cm
{\bf 4.2. The System ${\cal I}=({\cal P}_{2}+{\cal C})$.} We assume that ${\cal P}$
moves along, or parallel to, the axis $Ox$ and thus that the state space of ${\cal P}_{2}$
is ${\tilde {\cal K}}:=L^{2}({\bf R})$. We assume that ${\cal C}$ is a chain of Pauli
spins located at the sites $(1,2,. \ .,2L+1)$, of $Ox$, where $L$ is a positive integer.
Thus, the state space of ${\cal C}$ is ${\hat {\cal K}}:=({\bf C}^{2})^{(2L+1)}$, and
therefore that of ${\cal I}$ is ${\cal K}={\tilde {\cal K}}{\otimes}{\hat {\cal K}}$.
\vskip 0.2cm
The spin at the site $n$ of ${\cal C}$ are represented by Pauli matrices
$({\sigma}_{n,x},{\sigma}_{n,y},{\sigma}_{n,z})$ that act on the $n$\rq th
${\bf C}^{2}$ component of ${\hat {\cal K}}$. Thus, they may be canonically identified
with operators in ${\hat {\cal K}}$ that satisfy the standard Pauli relations
$${\sigma}_{n,x}^{2}={\sigma}_{n,y}^{2}={\sigma}_{n,z}^{2}={\hat I}; \
{\sigma}_{n,x}{\sigma}_{n,y}=i{\sigma}_{n,z}, \ {\rm etc},\eqno(4.1)$$
together with the condition that the spins at different sites intercommute.
\vskip 0.2cm
We assume that ${\cal P}_{1}, \ {\cal P}_{2}$ and ${\cal C}$ are independently
prepared before being coupled together at time $t=0$. Further, we assume that the initial
states of ${\cal P}_{1}$ and ${\cal P}_{2}$ are the pure ones, represented by vectors
${\psi}$ and ${\phi}$ in ${\cal H}$ and ${\tilde {\cal K}}$, respectively, while that of
${\cal C}$ is given by a density matrix ${\hat {\Omega}}$, in ${\hat {\cal K}}$, whose
form will be specified below, by Eqs. (4.3) and (4.4).Thus, the initial state of ${\cal I}$
is
$${\Omega}=P({\phi}){\otimes}{\hat {\Omega}},\eqno(4.2)$$
where $P({\phi})$ is the projection operator for ${\phi}$. We assume that ${\phi}$ has
support in a finite interval $[c,d]$ and that ${\hat {\Omega}}$ takes the form
$${\hat {\Omega}}={\otimes}_{n=1}^{2L+1}{\hat {\omega}}_{n},\eqno(4.3)$$
where ${\hat {\omega}}_{n}$, the initial state of the $n$\rq th spin of ${\cal C}$, is give
by the formula
$${\hat {\omega}}_{n}={1\over 2}(I_{n}+m{\sigma}_{n,z}),\eqno(4.4)$$
where $0