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%%% Discontinuity and Continuity of Definite Properties
%%% in the Modal Interpretation.
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%%% From ``The Modal Interpretation of Quantum Mechanics'',
%%% pages 213--222, edited by D. Diecks and P.E. Vermaas,
%%% Kluwer (1998).
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%%% Plain TeX, 10 pages
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%%% Matthew J. Donald
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%%% web site: http://people.bss.phy.cam.ac.uk/~mjd1014
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%%% e-mail : mjd1014@cam.ac.uk
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{ /Title (Discontinuity and Continuity of Definite Properties in the Modal Interpretation..)
/Author (Matthew J. Donald)
/CreationDate (June 1996)
/ModDate (\number\year/\number\month/\number\day)
/Subject (The Modal Interpretation of Quantum Theory.)
/Keywords (modal interpretation, quantum theory, mixed state eigendecompositions)}
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\vbox to 2cm{}
\centerline{\bf Discontinuity and Continuity of Definite Properties}
\centerline{\bf in the Modal Interpretation.}
\vbox to 1.5cm{}
\centerline{\bf Matthew J. Donald}
\vbox to 1.5cm{}
{\bf \hfill The Cavendish Laboratory, JJ Thomson Avenue,
\hfill Cambridge CB3 0HE, Great Britain.}
\vbox to 0.75cm{}
{\bf \hfill e-mail:\quad mjd1014@cam.ac.uk}
\bigskip
{\bf \hfill web site:\quad \catcode`\~=12
\outlink{http://people.bss.phy.cam.ac.uk/~mjd1014}
{http://people.bss.phy.cam.ac.uk/\tilbf mjd1014}}
\medskip
\vbox to 2cm{}
\proclaim{abstract}{} Some technical results about discontinuity and
continuity of eigenprojections of reduced density operators are discussed in
an elementary context. It is argued that these results suggest serious
obstacles both to the goal of applying the modal interpretation to
measurement theory in the context of the quantum statistical mechanics of
macroscopic objects and to the goal of extending the modal interpretation
to be compatible with relativistic quantum field theory. The paper is based
on joint work with Guido Bacciagaluppi and Pieter Vermaas.
\endproclaim
\vbox to 1cm{}
{\bf From ``The Modal Interpretation of Quantum Mechanics'',
pages 213--222, edited by D. Diecks and P.E. Vermaas, Kluwer (1998).}
\vfill \eject
\proclaim{1 \quad Technical Results in an Elementary Context.}
\endproclaim
In this paper, we shall consider the Vermaas-Dieks version of the modal
interpretation [1]. Suppose that the Hilbert space $\H$ of the universe
takes the form of a tensor product $\H \cong \H_s \otimes \H_e$ where
$\H_s$ represents the Hilbert space of states of a system of interest, and
$\H_e$ represents the Hilbert space of the environment. Suppose also that
the quantum state of the universe is some pure state
$|\Psi\>\<\Psi|$. Then the state of system $s$ is the reduced state
$(|\Psi\>\<\Psi|)_s$ defined by taking the partial trace of $|\Psi\>\<\Psi|$
over
$\H_e$.
$(|\Psi\>\<\Psi|)_s$ is a self-adjoint trace class operator on $\H_s$ and so
has a unique spectral resolution of the form
$$(|\Psi\>\<\Psi|)_s = \sum_m p_m P_m \eqno(1)$$
where the $P_m$ are orthogonal projections such that $\sum_m P_m = 1$,
and the $p_m$ are distinct and $\sum_m p_m \dim P_m = 1$. According to
[1], the $P_m$ represent the definite properties of the system
$s$.
$(|\Psi\>\<\Psi|)_s$ also possesses eigendecompositions of the form
$$(|\Psi\>\<\Psi|)_s = \tsize \sum_n r_n |\psi_n\>\<\psi_n|$$ where
$(\psi_n)_n$ is an orthonormal basis for $\H_s$, and $\sum_n r_n = 1$. If
all the $r_n$ are distinct, then the eigendecomposition is the spectral
resolution, and so is unique -- apart from phase factors. More generally,
however, any sequence of bases for the subspaces $P_m \H_s$ gives rise to
an eigendecomposition. This means that the eigendecomposition is
non-trivially non-unique whenever any of these subspaces has dimension
greater than unity.
In [2], Bacciagaluppi, Vermaas, and I analyse the properties of the
$P_m$ and $\psi_n$ and consider how they change with time, under the
assumption of a global Hamiltonian
$H$ acting on the total Hilbert space $\H$, so that the reduced density
matrix has time dependence
$$\rho(t) = (e^{-itH} |\Psi\>\<\Psi| e^{itH})_s.$$ The first part of this paper
reviews results from [2]. The results and examples in this part are quoted
from [2] and complete technical details, proofs, and references, may be
found there.
An elementary example shows how problems may arise:
\medskip
\noindent{\bf example} \quad Suppose $\H_s$ is
two-dimensional. Consider, for $0
\leq \varepsilon \leq {1\over 2}$, reduced density matrices $\rho
_\varepsilon$ and $\sigma _\varepsilon$ given by
$\rho _\varepsilon = \pmatrix{{1\over 2}+\varepsilon & 0 \cr 0 & {1\over
2} - \varepsilon \cr}$ and $\sigma _\varepsilon =
\pmatrix{ {1\over 2} & \varepsilon \cr \varepsilon & {1\over 2}
\cr}$. As long as
$\varepsilon > 0$, $\rho _\varepsilon$ and
$\sigma _\varepsilon$ each have unique pairs of one-dimensional
eigenprojections; $\pmatrix{1 & 0 \cr 0 & 0 \cr}$ and $ \pmatrix{ 0 & 0
\cr 0 & 1 \cr}$ for $\rho _\varepsilon$ and $
\pmatrix{{1\over2} & {1\over2} \cr {1\over2} & {1\over2}
\cr}$ and $\pmatrix{ {1\over2} & -{1\over2} \cr -{1\over2} & {1\over2}
\cr}$ for $\sigma_\varepsilon$. Continuity and stability problems arise
because, although these pairs are independent of
$\varepsilon$, $\rho _\varepsilon$ is arbitrarily close to
$\sigma _\varepsilon$ for $\varepsilon$ sufficiently small. $\varepsilon =
0$ is the degeneracy point, where $\rho _\varepsilon = \sigma
_\varepsilon$, any normalized vector is an eigenvector, and the spectral
resolution contains the two-dimensional eigenprojection
$\pmatrix{ 1 & 0 \cr 0 & 1 \cr}$.
\medskip
Choosing $\H_s$ to be two-dimensional is sufficient to exhibit most of the
technical results from [2]. In this case, the ``Bloch sphere'' construction
allows us to represent the states on $\H_s$ by the points of the unit ball in
three-dimensional real space
$\Real^3$.
\medskip
\noindent{\bf The Bloch Sphere.}
\medskip
Let $\Sigma$ be the set of $2\times2$ density matrices $\rho = \pmatrix{
\rho_{11} & \rho_{12} \cr \rho_{21} & \rho_{22} \cr}$.
\newline $\rho$ is a self-adjoint positive matrix with trace unity, so that
$$\displaylines{ \tr(\rho) = \rho_{11} + \rho_{22} = 1, \quad \rho_{12} =
\overline{\rho_{21}}, \quad 0
\leq \rho_{11} \leq 1, \cr \hbox{ and} \quad 0 \leq \det(\rho) = \rho_{11}
\rho_{22} - \rho_{12} \rho_{21} \leq \tsize{1\over 4}. }$$
A mapping $\chi: \Sigma \rightarrow \Real^3$ is defined by
$$\chi(\rho)^1 = \rho_{12} + \rho_{21},\quad
\chi(\rho)^2 = i(\rho_{12} - \rho_{21}),\quad \chi(\rho)^3 = \rho_{11} -
\rho_{22}.$$
$\chi$ maps $\Sigma$ into the unit ball $B^3 \subset \Real^3$:
$$|\chi(\rho)|^2 = 4 \rho_{12}
\rho_{21} + \rho_{11}^2 - 2 \rho_{11} \rho_{22} + \rho_{22}^2 =
(\rho_{11} + \rho_{22})^2 - 4 \det \rho = 1 - 4 \det \rho \leq 1.$$
$\chi$ is a bijection onto $B^3$ with inverse $\varphi$ defined by
$$\varphi({\bf x}) = {\textstyle{1\over 2}} \pmatrix{ 1 + x^3 & x^1-ix^2
\cr x^1 +i x^2 & 1-x^3
\cr}.$$
$\det \varphi({\bf x}) = {1\over4}(1 - (x^1)^2 - (x^2)^2 - (x^3)^2)$.
$\rho$ is pure $\iff \det \rho = 0 \iff |\chi(\rho)|^2 = 1$. This means that
the pure states are mapped onto the surface of the ball.
For any ${\bf x}$, $\varphi({\bf x}) + \varphi(-{\bf x}) = 1$ so that
$\varphi({\bf x})$ and $\varphi(-{\bf x})$ commute. In particular, for
$|{\bf x}|^2 = 1$,
$\varphi({\bf x})$ and $\varphi(-{\bf x})$ are orthogonal pure states.
$\chi$ is an affine isomorphism because, for $0 \leq \lambda \leq 1$,
$$\chi(\lambda \rho + (1-\lambda) \sigma) = \lambda \chi(\rho) +
(1-\lambda) \chi(\sigma)$$
$$\varphi(\lambda {\bf x} + (1-\lambda) {\bf y}) = \lambda \varphi({\bf
x}) + (1-\lambda) \varphi({\bf y}).$$
From this it follows that the state represented by a point ${\bf x}$ inside
the ball can be decomposed into the orthogonal pure states represented by
the end points of the line passing through ${\bf x}$ and the centre of the
ball. This line is unique unless
${\bf x} = \bf 0$. Thus ${\bf x}$ is non-degenerate unless $\bf x = 0$.
As $B^3$ is a manifold with three (real) dimensions and $\bf 0$ is a
manifold of zero dimensions, we have an example of the first of the results
which I shall take from [2]:
\medskip
\noindent{\bf theorem} {\sl The space of degenerate density
operators on a finite-dimensional Hilbert space $\H_s$ has $3$ dimensions
fewer than the space of all density operators on $\H_s$.}
\medskip
Our discussion in [2] of the continuity of eigenvectors of reduced density
matrices is based on work by Rellich on the perturbation theory of linear
operators. Rellich's main theorem states, when applied to our case, that if
the time-dependence of a density matrix $\rho(t)$ is sufficiently smooth
-- more precisely, if $\rho(t)$ is an analytic function of $t$ -- then it is
possible to find eigenvectors of $\rho(t)$ which are themselves
analytic functions of $t$.
By trying to construct a counter-example in the Bloch sphere, it is fairly
straightforward to see that some such result must hold in the
two-dimensional case. As noted above, the eigenvectors of a non-zero point
in $B^3$ can be found by projecting from the point out to the surface, along
the line through the centre. This projection is clearly continuous if we
avoid the centre, and, indeed, a continuous choice can be made even if we
do go through the centre, unless we ``turn a sharp corner'' there. Smooth
paths do not turn sharp corners.
If the global Hilbert space $\H$ is finite-dimensional, then any Hamiltonian
$H$ is bounded, and all reduced states of the form $\rho(t) =
(e^{-itH}|\Psi\>\<\Psi|e^{itH})_s$ are analytic functions of $t$. Rellich's
theorem on the existence of analytic eigenvectors can also be applied in the
modal interpretation if $\H$ is infinite-dimensional because of the
following result:
\medskip
\noindent{\bf lemma}\quad {\sl If $H$ is a Hamiltonian on a tensor product
Hilbert space $\H = \H_s \otimes \H_e$ then there is a dense set of vectors
$\Psi \in \H$ such that the reduced density operator $\rho(t) =
(e^{-itH}|\Psi\>\<\Psi|e^{itH})_s$ is analytic in $t$.}
\medskip
It should be noted that Rellich's result is slightly complicated in the
infinite-dimensional case. In finite dimensions, there is a time-dependent
basis of analytic eigenvectors for $\H_s$. In infinite dimensions however,
even with the best analyticity properties for $\rho(t)$, Rellich's theorem
only applies to eigenvectors $\psi(t)$ for which the corresponding
eigenvalue $r(t)$ is greater than zero. If $r(t) \rightarrow 0$, then it is
possible for $\psi(t)$ to disappear.
\medskip
\noindent{\bf example}\quad {\sl Let $(t_n)_{n\geq1}$ be any sequence of
real numbers (for example, some counting of the rational numbers). Then
there is a vector $\Phi $ in a Hilbert space $\H= \H_s \otimes \H_e$ and a
bounded Hamiltonian $H$ on $\H$ such that the density operator $\rho (t) =
(e^{-itH}|\Phi \>\<\Phi |e^{itH})_s$ has an eigenvector disappearing at each
point of the sequence $(t_n)_{n\geq1}$.}
\medskip
Equating eigenvalue $r(t)$ with probability, suggests that such
disappearing eigenvectors may not be a problem of great physical
significance. Much more important, in my opinion, is the problem of
instability.
We have seen that for a given Hamiltonian $H$, smooth eigenvectors of the
reduced state can be chosen. However, these eigenvectors may change
uncontrollably under arbitrarily small changes in $H$.
This problem also is easily exemplified in the Bloch sphere. Ima\-gine
that $H$ depends on a parameter $\eta$, and that the reduced state
$$\rho(t, \eta) = (e^{-itH(\eta)}|\Psi\>\<\Psi|e^{itH(\eta)})_s$$ sweeps
through the degeneracy point at $t = t_0$ and $\eta = \eta_0$. By
considering how the projection from the centre of the Bloch sphere to the
surface changes as a reduced state moves close to the centre of the sphere,
it is easy to see that, if a suitable choice of parameter dependence can be
found, then the eigenvectors of $\rho$ can be made to move, for example,
from equator to pole for arbitrarily small change in $t$. This is the basic
idea behind the following example.
\medskip
\noindent{\bf example}\quad {\sl There exists a Hamiltonian $H(\eta)$ on a
Hilbert space $\H = \H_s \otimes \H_e$ and a vector $\Psi \in \H$ such that
$H(\eta)$ is bounded and depends analytically on the parameter $\eta$ and
$(e^{-itH(\eta)}|\Psi\>\<\Psi|e^{itH(\eta)})_s$ is jointly analytic in $t$ and
$\eta$. However, there exist $t_0$ and $\eta_0$ such that, for any
$\varepsilon > 0$ there exist $t_1$, $t_2$, $\eta_1$, and $\eta_2$, with $|t_1
- t_0| + |t_2-t_0| + |\eta_1 - \eta_0| + |\eta_2-\eta_0| <
\varepsilon $ and $||\xi - \xi '|| > {1\over 2}$ for any pair
$(\xi , \xi ')$ consisting of an eigenvector $\xi $ of
$(e^{-it_1H(\eta_1)}|\Psi\>\<\Psi|e^{it_1H(\eta_1)})_s$ and an eigenvector
$\xi '$ of $(e^{-it_2H(\eta_2)}|\Psi\>\<\Psi|e^{it_2H(\eta_2)})_s$.}
\medskip
This suggests that what the modal interpretation takes to be the ``real''
properties of a physical subsystem may fluctuate uncontrollably under
environmental perturbations. A similar problem arises in the very
identification of subsystems. According to the modal interpretation, a
subsystem is given as a factor space in the Hilbert space of the universe.
However arbitrarily small changes in the identification of such factors may
give rise to large changes in the properties of the corresponding systems.
A Hilbert space $\H$ of dimension $N_s N_e$ can be expressed as a tensor
product $\H = \H_s \otimes \H_e$ by giving an indexed basis $(\chi
_{mn})_{m = 1}^{N_s}{}_{n= 1}^{N_e}$ for $\H$ and equating $\chi_{mn}$
with $\varphi_m \otimes \psi_n$ where $(\varphi_{m})_{m = 1}^{N_s}$ is a
basis for $\H_s$ and $(\psi_{n})_{n = 1}^{N_e}$ is a basis for $\H_e$.
Two such expressions corresponding to bases
$(\chi_{mn})_{m = 1}^{N_s}{}_{n= 1}^{N_e}$ and
$(\chi'_{mn})_{m = 1}^{N_s}{}_{n= 1}^{N_e}$ for $\H$ may be considered to
be close if the basis vectors $\chi_{mn}$ and $\chi'_{mn}$ are sufficiently
close, for all $m$ and $n$.
Let $\Psi \in \H$ with $||\Psi|| =1$, and let $(|\Psi\>\<\Psi|)_s$ (resp.
$(|\Psi\>\<\Psi|)_{s'}$) denote the density operator on $\H_s$ defined by
$$\eqalignno{
\<\varphi_m|(|\Psi\>\<\Psi|)_s|\varphi_{m'}\>
&=
\sum_n \<\chi_{mn}|\Psi\>\<\Psi|\chi_{m'n}\> \cr
\hbox{(resp.)} \quad \<\varphi_m|(|\Psi\>\<\Psi|)_{s'}|\varphi_{m'}\> &=
\sum_n \<\chi'_{mn}|\Psi\>\<\Psi|\chi'_{m'n}\>. \cr}$$
If for some $\delta > 0$,
$$\sum_{m=1}^{N_s}\sum_{n= 1}^{N_e} ||\chi _{mn} - \chi '_{mn}|| <
\delta \eqno(2)$$ then, for any $\Psi \in \H$,
$$||(|\Psi\>\<\Psi|)_s - (|\Psi\>\<\Psi|)_{s'}||_1 < 2\delta. \eqno(3)$$
\medskip
\noindent{\bf example}\quad {\sl Choose $\delta > 0$. There exists a
Hilbert space $\H$ which can be expressed as a tensor product $\H = \H_s
\otimes \H_e$ in two possible ways, corresponding to bases $(\chi
_{mn})_{m = 1}^{N_s}{}_{n= 1}^{N_e}$ and $(\chi '_{mn})_{m=1}^{N_s}{}_{n=
1}^{N_e}$ which are close by criteria (2) and (3). There is a vector $\Psi
\in \H$ such that $||\xi - \xi '|| > {1\over 2}$ for any pair $(\xi , \xi ')$
consisting of an eigenvector $\xi $ of $(|\Psi\>\<\Psi|)_s$ and an
eigenvector $\xi '$ of $(|\Psi\>\<\Psi|)_{s'}$.}
\bigskip
\noindent{\bf 2 \ The problem of instability.}
\medskip
In my opinion, instability near degeneracy points is a fundamental problem
for the modal interpretation. Theoretical physics is based on a long chain of
approximations. Instability means that eigendecompositions do not behave
well under approximation. This makes it impossible for the modal
interpretation to claim that it is dealing with any sort of approximation to
the truth, while it continues to rely on non-relativistic and
few-dimensional models of quantum mechanics. Indeed, near-degeneracies
and degeneracies are inevitable for real physical systems according both to
the many-dimensional models suggested by quantum statistical mechanics
and to the infinite-dimensional models of local relativistic quantum field
theory.
Macroscopic systems have high entropy and, therefore, according to
quantum statistical mechanics, they should be assigned highly mixed, nearly
degenerate states with correspondingly uncontrollable eigenfunctions.
These states appear to be quantum mechanical ana\-logs of the ensembles
of classical statistical mechanics. However, this does not mean that the
interpretation of these states is entirely straightforward. Indeed, the
modal interpretation in general can be seen as an attempt to grapple with
the problem of extending an ensemble picture to quantum states
by providing an exact and unambiguous definition of the ensemble
which corresponds to such a state. This problem is just the same for states
of macroscopic quantum systems as it is for microscopic quantum systems.
There is also a question as to whether the high entropy states of quantum
statistical mechanics are the states which we should assume that the
corresponding systems occupy. Consider, for example, a hot cup of coffee.
According to a direct application of the modal interpretation to macroscopic
objects, this coffee has a quantum state $(|\Psi\>\<\Psi|)_c$, which is the
reduction of the state of the entire universe to the Hilbert space $\H_c$
defined by the particles making up the coffee. More precisely,
$(|\Psi\>\<\Psi|)_c$ is the reduction to $\H_c$ of the state of the entire
universe, given a considerable amount of prior information; for example,
given that the cup of coffee exists, or given the observations of the coffee
drinker. Statistical mechanical arguments then suggest that the most
plausible guess we can make for $(|\Psi\>\<\Psi|)_c$ is that it is the state on
$\H_c$ with highest entropy given our prior information. If the modal
interpretation is really the complete and universal interpretation of
quantum mechanics which it purports to be, then appropriate prior
information would correspond to fixed definite properties on some
suitable systems. If we assume that the prior information is limited by our
observations of the coffee, then $(|\Psi\>\<\Psi|)_c$ will be a high-entropy,
near-degenerate, quasi-equilibrium, thermal state.
Absolute entropies $S$ at $25^\circ$C are 2.4 J K$^{-1}$
mol$^{-1}$ for diamond and 205 J K$^{-1}$ mol$^{-1}$ for oxygen. If we
equate $S$ with $k_B \log N$, then $N$ is a measure of the minimum
number of orthogonal wave-functions into which the equilibrium
quantum state decomposes with significant probability. $N \sim 10^{6.3
\times 10^{21}}$ for one gram of diamond and $N \sim 10^{2.6 \times
10^{23}}$ for one litre of oxygen.
Also relevant in this context, is work by Lubkin [3] and Page [4], which has
recently been turned into a theorem by Foong and Kanno [5]. These authors
have shown that most pure states on a Hilbert space of sufficiently large
dimension give rise to nearly maximally degenerate states on restriction to
a subspace of appropriate dimension. More precisely, they have shown
that if $|\Psi\>\<\Psi|$ is a randomly chosen pure state on a Hilbert space
$\H = \H_s \otimes \H_e$ of dimension $N_s N_e$ with $N_e \gg N_s \gg 1$
then
$$S((|\Psi\>\<\Psi|)_s) = \tr(-(|\Psi\>\<\Psi|)_s \log (|\Psi\>\<\Psi|)_s) \sim
\log N_s.$$
In the Vermaas-Dieks form of the modal interpretation, the possessed
properties are fixed by an algorithm; the possessed properties correspond
to the eigenprojections of the density matrix of the subsystem. Having such
an algorithm is an enormous advance compared to the conventional
interpretation of quantum theory, in which we are supposed to look for the
eigenfunctions of some ``measured'' operator, but no precise means is
provided by which we can identify that operator. Nevertheless, in my
opinion, the motivation behind the modal interpretation remains the
suggestion that the mysterious nature of quantum mechanical states can be
resolved if a subsystem ``possesses'' properties, which are (somehow)
``quasi-classical''. Thus, in the modal interpretation also, it is implied that
the possessed properties should, in some sense, correspond to definite
values of what is being measured. Unfortunately, the algorithm supplied
by the Vermaas-Dieks modal interpretation does not necessarily yield
``quasi-classical'' properties. The eigenfunctions of a reduced density
matrix near an $N$-fold degeneracy vary over an $N$-dimensional space.
It has been suggested that decoherence theory solves this problem. This
suggestion is incorrect. Decoherence theory does tell us that the reduced
state is close to a state which has a decomposition into pure states with
physically desirable properties, but this is NOT equivalent to saying that the
eigendecomposition of the reduced state is into pure states with close to
physically desirable properties. Once again, the algorithmic nature of the
modal interpretation, which is its greatest strength, makes it impossible for
the interpretation to hide behind the usual ``for all practical purposes''
(FAPP) arguments. The following examples demonstrate this point. Similar
examples will be presented by Bacciagaluppi [6] in a forthcoming paper.
\medskip
\noindent{\bf example}\quad The one-particle reduced density matrix for
a one-dimen\-sional ideal gas of particles of mass $m$ confined to an
interval $[0, L]$ in the classical regime at temperature $T$ is given by
$$\displaylines{ \rho_{\beta, L}(x, y) = {1\over Z} e^{-\beta H}(x, y) =
{2\over Z L}\sum_{n = 1}^\infty e^{-\alpha n^2} \sin {n\pi x \over L}\sin
{n\pi y \over L} \cr \quad
\hbox{where } \beta = 1/kT, \alpha = {\hbar^2 \pi^2 \over 2m L^2 kT},
\hbox{ and }Z = \tr(e^{-\beta H}) = \sum_{n = 1}^\infty e^{-\alpha n^2}. }$$
The eigenfunctions of $\rho_{\beta, L}$ given by $\sqrt{2\over L} \sin
{n\pi x \over L}$ are unique and non-local\-ized, but, by expanding
${\tsize{1\over2}} {\tsize\sqrt{\pi\over \alpha}} \sum_{s = -\infty}^\infty
e^{-\pi^2(x+2s)^2/4\alpha}$ in co\-sines, it is possible to show that, for $|x
- y| \ll x, y \ll L$,
$$ \rho_{\beta, L}(x, y) \sim {1 \over L} e^{-{m \over2 \hbar^2 \beta}(x
- y)^2}.$$
Thus $\rho_{\beta, L}$ has a decoherence length $\dsize\sqrt{{2
\hbar^2 \beta \over m}}$, corresponding to the de Broglie thermal
wavelength, which is of order $10^{-11}$ m for atoms at room
temperature. This means that $\rho_{\beta, L}$ is exactly the type of
density operator which decoherence theory claims is typical for a
macroscopic system. Nevertheless, the eigenfunctions of $\rho$ are utterly
``quantum mechanical'' in nature.
\medskip
\noindent{\bf example}\quad Consider a particle of mass
$m$ in a harmonic potential in one dimension. The Schr\"odinger equation
is:
$$i \hbar {\partial \Psi \over \partial t} = -{\hbar^2 \over 2m} {\partial^2
\Psi \over \partial x^2} + {1\over 2} m \omega^2 x^2 \Psi.$$
At temperature $T$, the density matrix of the particle is
$$\rho_\beta =(1 - e^{- \beta \hbar \omega})\sum_{n=0}^\infty e^{- \beta
n \hbar
\omega} |\psi_n\>\<\psi_n|$$ where $\beta = 1/kT$.
The harmonic oscillator eigenfunctions $\psi_n$ are independent of
temperature and have length scale
$\sqrt{{(n+{1\over2}) \hbar \over m \omega}}$. Again, these are utterly
``quantum mechanical'' eigenfunctions.
Using the generating function for the $\psi_n$, it can be shown that
$$\rho_\beta(x, y) = {\tsize\sqrt{ m \omega \over \pi \hbar \lambda}}
e^{ - {\lambda m \omega \over 4 \hbar} (x- y)^2 - { m \omega \over 4
\hbar \lambda}(x + y)^2}$$ where $\lambda = {1 + e^{-\beta \hbar
\omega} \over 1 - e^{-\beta \hbar \omega}}$.
In the high temperature limit, $\beta \rightarrow 0$, and $\lambda \sim
2/(\beta \hbar \omega)$, so that
$$ \rho_\beta(x, y) \sim {\tsize\sqrt{ m \omega^2 \beta \over 2 \pi }}
e^{ - { m \over 2 \hbar^2 \beta} (x- y)^2 - { m \omega^2 \beta \over 8
}(x + y)^2}. $$ This has the same form as the exact result. The correlation
length at high temperature $\sqrt{{2\hbar^2 \beta \over m}}$ is the same
as that for the free particle. The term $\dsize e^{ - { m \omega^2 \beta
\over 8 }(x + y)^2}$ provides a long length-scale decrease, so that
$\rho_\beta$ is normalized. Once again, this is a ``decoherent'' density
operator.
Other decompositions of $\rho_\beta$ are possible. For example, following
Glauber [7], but using an explicit width parameter $\xi$,
$\rho_\beta$ can also be represented as a Gaussian distribution of coherent
states
$$\psi_{u, v, \xi}(x) = {\tsize{1 \over \sqrt{\sqrt{\pi}\xi}}} e^{-(x -
u)^2/2\xi^2 + i v x/\hbar}.$$
For ${\hbar \lambda \over m \omega} > \xi^2 > {\hbar \over \lambda m
\omega}$,
$$\displaylines{ \rho_\beta = {\tsize\sqrt{{ m \omega \xi^2 \over
\pi^2 (\hbar \lambda - \xi^2 m \omega)( \lambda m \omega\hbar \xi^2 -
\hbar^2) }}} \hfill \cr
\int |\psi_{u, v, \xi}\>\<\psi_{u, v,
\xi}| e^{- { m \omega \over \hbar \lambda - \xi^2 m \omega} u^2 - {\xi^2
\over \lambda m \omega\hbar \xi^2 - \hbar^2} v^2} du dv. }$$
These decompositions decompose $\rho_\beta$ into a range of
well-localized quasi-classical particle states. Such states are satisfactory
``for all practical purposes''. It would be splendid if an algorithmic
interpretation could be used to break the non-uniqueness represented
by $\xi$ and pick out exactly one of these decompositions. The modal
interpretation algorithm, of course, does not do this.
\medskip
The problems raised for the modal interpretation by quantum statistical
mechanics are serious. At the very least, instability near degeneracy points
implies that the analysis of prior information in the modal interpretation is
not a problem which can be ignored; the modal interpretation cannot take
advantage of anything analogous to the free choice of Heisenberg cut
between measuring apparatus and measured system. However, the
problems raised by relativistic quantum field theory are perhaps even
more fundamental. The modal interpretation is supposed to be a universal
``no collapse'' theory. This means that our ultimate goal should be to
analyse a universal wave-function $\Psi \in \H$ which would be an
uncollapsed state arising from the big bang. $\Psi$ would be a
superposition of all possibilities. In any regime of space-time, $\Psi$ would
be close to a thermal equilibrium state. Even stars would be superposed in
$\Psi$. Until we began the process of assigning definite properties, there
would be no definite macroscopic objects; no measuring devices in ready
states. In this context, it seems to me that the only natural subsystems
with which we can start our analysis of the universal wave-function, are
the ``local algebras'' of the Haag-Schroer-Kastler axioms [8]. However,
these local algebras are type III von Neumann algebras and, as such, they
have {\it no} pure normal states. It is possible to define reduced states on
such algebras, but these states have {\it no} eigendecompositions which
correspond in any relevant way to analogs of (1). In a very real sense, in
relativistic quantum field theory, local systems are irreducibly degenerate.
\medskip
\noindent{\bf 3 \ Where does the modal interpretation go from here?}
\medskip
One possibility for modifying the Vermaas-Dieks modal interpretation
might be to consider an alternative algorithm yielding alternative
decompositions. Some sort of maximum entropy decomposition might well
be desirable, but there would be problems with continuous distributions,
and with infinite dimensional systems.
A second possibility would be to use decoherence theory to say that there
are suitable decompositions. This, of course, is mere FAPP.
The path I favour, involves going back to Everett, who was the first to use
the Schmidt decomposition as the technical foundation of an interpretation,
and starting again. My own version of the many-minds interpretation [9] is
also an algorithmic interpretation. However, unlike the modal
interpretation, it is fully compatible with relativistic quantum field theory
and it is mathematically stable under approximation. Working in the context
of a universal ``no collapse'' theory, it involves a detailed, mathematical,
analysis of the structure of observers, which takes into account the
macroscopic, localized, and thermal nature of observers. It does not
attempt to associate an individual wavefunction at each moment to an
observer. In my opinion, all such attempts are essentially unphysical,
ignoring as they do, not only the mathematics of relativity, but also the
continuous and unavoidable interactions between a warm, breathing
observer and his environment. Instead, in my theory, observers are taken
to occupy suitable mixed thermal states.
\bigskip
\noindent{\bf References.}
\frenchspacing
\noindent [1]\quad P.E. Vermaas and D. Dieks: `The modal interpretation of
quantum mechanics and its generalization to density operators', {\it Found.
Phys.\/} {\bf 25}, 145--158 (1995).
\noindent [2]\quad G. Bacciagaluppi, M.J. Donald, and P.E. Vermaas:
`Continuity and discontinuity of definite properties in the modal
interpretation', {\it Helv. Phys. Acta\/} {\bf 68}, 679--704 (1995).
\noindent [3]\quad E. Lubkin: `Entropy of an $n$-system from its
correlation with a $k$-reser\-voir', {\it J. Math. Phys.\/} {\bf 19},
1028--1031 (1978).
\noindent [4]\quad D.N. Page: `Average entropy of a subsystem',{\it Phys.
Rev. Lett.\/} {\bf 71}, 1291--1294 (1993).
\noindent [5]\quad S.K. Foong and S. Kanno: `Proof of Page's conjecture on
the average entropy of a subsystem', {\it Phys. Rev. Lett.\/} {\bf 72},
1148--1151 (1994).
\noindent [6]\quad G. Bacciagaluppi: `Delocalized properties in the modal
interpretation of a continuous model of decoherence', preprint, June 1996.
\noindent [7]\quad R.J. Glauber: `Coherent and incoherent states of the
radiation field', {\it Phys. Rev.\/} {\bf 131}, 2766--2788 (1963).
\noindent [8]\quad R. Haag: {\it Local Quantum Physics: Fields, Particles,
Algebras} (Berlin: Springer, 1992).
\noindent [9]\quad M.J. Donald: `A mathematical characterization of the
physical structure of observers', {\it Found. Phys.\/} {\bf 25}, 529--571
(1995).
\end