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\title{The histories interpretation:
stability instead of consistency?}

\author{C J S Clarke \\
Faculty of Mathematical Studies,
University of Southampton,\\ 
Southampton, SO17 1BJ, UK\\
cjsc@maths.soton.ac.uk}

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\def\rr#1{${}^{(#1)}$}
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\maketitle


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\vskip 5mm
{\it
\vskip 5mm
\hrule
\vskip 0.5cm

Using a reformulation of conventional results in decoherence theory, a 
condition is proposed for singling out a distinguished class of
histories which includes those which use the ``pointer basis'' of Zurek.
\vskip 5mm
\hrule
}
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\section{What's good and bad about Consistent Histories}

The basic problem faced by any extension of quantum theory outside the
laboratory realm of Bohr's interpretation can be expressed by the
slogan\rr{5} ``how does the classical world emerge?''
Unpacking the meaning of ``classical world'' we can extract several
interlinked problems: 
\par

{\parindent=40pt

\begin{list}{}{}
\item[1.] How is it that the logic of propositions about macroscopic
objects is classical (that is, a Boolean lattice) whereas the logic of
quantum propositions is an orthocomplemented lattice\rr{1}? 
\item[2.] Where the classical behaviour is, as a consequence of
underlying quantum processes, indeterministic, then 

\begin{list}{}{}
\item[(a)] what determines the particular Boolean sub-lattice of
quantum states that can be expressed as corresponding classical states
(why does the Schr\"odinger's cat Gedankenexperiment have the result
that we believe it would have)?  
\item[(b)] how is the statistical mechanics of these
probabilistically chosen classical states derived from the dynamics of
quantum theory? 
\item[(c)] on each individual occurrence, what determines which one of the classically allowed states is in fact actualised?
\end{list}

\item[3.] When the classical behaviour is (at least to a high accuracy) deterministic, how is the classical dynamics derived from the quantum dynamics?
\end{list}

\par}

I would claim that, apart from Bohm's interpretation, which relies on
intrinsically unobservable hidden variables, and interpretations
involving essentially new and untested physics, existing
interpretations can solve 2b and 3 in many situations, but can only
partially solve the others. The consistent histories approach\rr{6,10}
has proceeded furthest with
resolving these problems, and has done so in a way that introduces the
minimum of controversional ontological scafolding (many worlds etc). I
want to propose a modification of the histories approach that solves 1
and 2a more satisfactorily. 2c remains a problem for me, and for
versions of the histories approach other than that of Omn\`es\rr{11}, who deconstructs the question on metaphysical grounds. 

Attempts to solve 1 and 2a rest (as does 2b) on the phenomenon of
decoherence which ensures that a statistical ensemble of macroscopic
systems linked to microscopic states, all initially prepared in the
same state, will as a result of environmental influences, evolve to an
ensemble described by a density matrix that is almost exactly diagonal
in a basis (the  pointer basis\rr{10,15})
adapted to the distinct classical states of the macroscopic
system. This leaves two unresolved difficulties, however, which I will
justify shortly: 
\par

{\parindent=40pt

\begin{list}{}{}

\item[A.] The consistency condition used by the histories programme
(see (\ref{e2}) below) in fact implies, tautologously without any physics,
that the histories obey classical logic (in the sense of satisfying
probability sum rules: see Dowker and Kent\rr{4} eqn (2.5)). Thus 1
is not being solved by a physical explanation, but is in effect just
being put in ``by hand.''  

\item[B.] Even if we accept the consistency condition, which forces a
Boolean lattice, how do we know that there is not some other Boolean
lattice, in addition to that defined by macroscopically
distinguishable states, which still satisfies the consistency
condition? If that were the case, then the consistency condition would
not determine the lattice of classical states and 2a would not be
solved. 

\end{list}

\par
}

Both these problems are of physical interest; for, if we could clearly
articulate physical conditions under which a unique Boolean lattice
emerged in the quantum limit, then it would become of great interest,
and would be theoretically grounded, to look for areas where there
were slight departures from a Boolean lattice. Some aspects of biology
might provide evidence of such areas\rr{8}. 
Point B rests (i) on the circumstances that no rigorous mathematical
proof exists of the uniqueness of the pointer basis in generating a
classical logic, and (ii) on the demonstration by Dowker and
Kent\rr{4}
that, both in specific examples and in general on
dimension-counting grounds, this basis is not unique. The latter
argument is not cast-iron, since it could be that special symmetries
invalidate the general dimension-counting arguments; but the onus is
now on those who claim uniqueness of the pointer basis to demonstrate
it rigorously. 
Point A, however, provides the main focus of this paper

\section{Consistent histories and the stability condition }

\subsection{The consistent histories formalism}

The idea of consistent (or decoherent) histories was intoduced by
Griffiths and others\rr{6} as a means of avoiding both an excessively
realistic approach to the wave function and a split between classical
and quantum realms. The version of the histories
formalism that I am using here is taken from Dowker and Kent\rr{4}
except that my notation interchanges their sub- and superscripts. For
more recent work see the references in Halliwell\rr{7}.  This
version is not explicitly relativistic, but I do not regard this as
essential for the point being made here; for relativistic developments
see Isham et al.\rr{9}. I assume here that we are dealing with
conventional quantum mechanics over a given Hilbert space ${\cal H}$.  

A {\it history set} is a pair $\hbox{\bf H}=(\rho, (\sigma_1,\sigma_2,\ldots,\sigma_n))$ for some $n$ where 
{\parindent=40pt

\begin{list}{}{}

\item[]$\rho$ is a density matrix (unit trace non-negative Hermitean operator on ${\cal H}$)

\item[]for each $i$, $\sigma_i=(P^{(i)}_1, \ldots, P^{(i)}_{k_i})$ with the $ P^{(i)}_j$ being projections (interpreted as Heisenberg picture operators) satisfying

\begin{list}{}{}
\item[] $ P^{(i)}_j P^{(i)}_k = \delta_{jk} P^{(i)}_j$
\item[] $\sum_{j=1}^{k_i} P^{(i)}_j = 1$

\end{list}
\end{list}
\par}

Note that through these conditions we {\it are} putting in by hand the classicality of the propositions for any one instant, but we are {\it not} demanding it overall in the way that the different $\sigma_i$ relate to each other.

A {\it history} belonging to \hbox{\bf H} is a sequence $H = (P_1,\ldots,P_n)$ with $P_i \in \sigma_i$ for all $i$ and the probability of $H$ in the initial state $\rho$ is given by
%
\begin{equation}
\hbox{\bf P}(\rho;H) := \hbox{Tr}(P_n\ldots P_1\rho P_1\ldots P_n).\label{e1}
\end{equation}

The {\it consistency condition} on {\bf H} in its strongest 
form\footnote{More recently\rr{7} this condition has been
termed {\it decoherence}, with {\it consistency} reserved for the
equality of the real parts of the sides of (\ref{e2})} 
(the arguments given above also apply to many of the weaker forms) is that 
%
\begin{equation}
\hbox{Tr}(P^{(n)}_{i_n}\ldots P^{(1)}_{i_1}\rho P^{(1)}_{j_1}\ldots P^{(n)}_{j_n}) = \delta_{i_1j_1}\ldots\delta_{i_nj_n}\hbox{\bf P}(\rho;( P^{(1)}_{i_1},\ldots P^{(n)}_{j_n})) .\label{e2}
\end{equation}

\subsection{The stability condition}

This condition is based on the well known distinction (see, for
example, the survey by Tegmark\rr{12}) between the dynamical timescale
$t_d$ and the decoherence timescale $t_{dc}$ for a system. The
dynamical time scale is determined by the {\it system} Hamiltonian,
independently of the environment; whereas both are involved in
decoherence. Dynamical timescales can vary widely: for human
experience, based on neuronal firing rates, this might be $10^{-3}$s,
for cosmology in the present era $10^{15}$s and for electon-positron
pair production (where the logic would be highly non-classical)
$10^{-20}$s. The principle of the stability condition is that the
probabilities for histories should not vary (as a function of the
timing of their propositions) on a timescale (the stability timescale)
that is very much less than the dynamical timescale $t_d$. We cannot,
for instance, claim to be talking about human experience and then
introduce a proposition whose probability changes on a timescale of
$10^{-10}$s. It is straight forward (see \S 3 below) to see that the
probabilities for ``unphysical'' propositions (referring, for example,
to superposed states of the human brain) vary on the decoherence
timescale, and so this condition does precisely what is required.  

This proposal still has an air of the ad hoc about it, and needs to be
related to a more fundamental theory. But it is, unlike the
consistency condition, non-trivial (in the sense of point A, that it
does not beg the question it is trying to solve) and is sufficiently
grounded physically to point the way to a correct fundamental
theory. The rest of the paper is devoted to spelling out in more
detail how this operates in practice. 

We consider a situation where we are examining the effect of a proposition $P$ posed after a subhistory $H^{(i)}=(P_1,\ldots,P_i)$ with an initial state of $\rho$. Thus we are concerned with $\hbox{\bf P}(\rho;H^{(i)},P)$. Now let $P_t$ denote the proposition obtained by evolving $P$ for time $t$, namely
%
\begin{equation}
P_t := \exp (iHt/\hbar)P \exp (-iHt/\hbar).\label{e3}
\end{equation}
%
Then we define the {\it repetition probability}  by
%
\begin{equation}
p(t):= \hbox{\bf P}(\rho;H^{(i)},P,P_t).\label{e4}
\end{equation}
%
Note that $p(0) > p(t)$ for $t>0$. 

We can now define the (repetition) {\it stability timescale} $t_s$ for
$P$ in this context. The idea is elementary but its formulation rather
tedious; again, an indication that the theory cannot be in any sense
fundamental. 

We want to define the stability timescale as the inverse of the
gradient of $p(t)$ near $t=0$ (more precisely: the slope of the chord
from $t=0$ to a suitable point). Unfortunately, $p(t)$ may be subject
to small fluctuations due to perturbations from background noise (of a
normal physical kind unconnected with decoherence) which could give
rise to large gradients on a very small timescale. We define the
magnitude of these possible fluctuations away from $t=0$ by setting 
%
\begin{equation}
V(t,c) := \sup_{{t\leq t_1 < t_2 \leq t_d} \atop {t_2 - t_1 \geq c}}
{|p(t_2) - p(t_1)|\over t_2 - t_1}   \label{e5}
\end{equation}

Then let $F$ be the ratio of the slope of the chord from $t=0$ to the slope of the following fluctuations:
%
\begin{equation}
F(t) := {p(0) - p(t)\over t }\Big/ V(t,t)  \label{e6}
\end{equation}
%
having a supremum of $F^*$, and let $t^*$ be the smallest point (it exists!) for which 
%
\begin{equation}
\limsup_{t\to t^*}F(t) = F^*.   \label{e7}
\end{equation}

The stability timescale is then the inverse slope of the chord to $t^*$:
%
\begin{equation}
t_s := {t^*\over p(0) - p(t^*)} \label{e8}
\end{equation}
%
and finally
the stability condition is then the requirement on each $\sigma_i$ that $t_s > \lambda t_d$ where $\lambda$ is some chosen small parameter.


\section{Stability and decoherence }


This section fills in the obvious connection between stability and
decoherence, showing that as a result of the latter, the stabiity
condition rules out superpositions of macroscopically distinct states,
and thus gives rise to a Boolean lattice. Decoherence involves the
setting where ${\cal H} = {\cal H}_E \otimes {\cal H}_S$ where $S$
refers to the system and $E$ to the environment. It is hard to give a
completely general formulation of the results concerning decoherence
and the pointer basis\rr{5}; but a model of the
idea sufficient for our purposes might be the proposition that each
$\sigma_i $ can be chosen so that 

{\parindent=40pt 
\begin{list}{}{}
\item[(a)] For all $k$, $\hbox{Range} P^{(i)}_k  = {\cal H}_E \otimes V^{(i)}_k $ for a subspace $ V^{(i)}_k $ of ${\cal H}_S $
\item[(b)] If $|e\rangle \in V^{(i)}_k $ and $|f\rangle \in V^{(i)}_l $ for $k\neq l$, then $\langle e|\rho_S|f\rangle \to 0$ in the decoherence timescale, where $\rho_S$ is the density matrix projected to ${\cal H}_S$ by tracing over environment variables.
\end{list}
\par}

Consider, then, the possibility of measuring a projection on a
superposition  $|k\rangle = a|e\rangle + b|f\rangle$ (i.e.\ a  ``live
cat $+$ dead cat'' situation) with $|a|^2 + |b|^2 = 1$. Thus let $P =
|k\rangle \langle k|$. Let $\rho' = H^{(i)T}\rho H^{(i)}$ where ${}^T$
denotes adjoint. Then from (\ref{e4})  
%
\begin{equation}
p(t) = \hbox{Tr} P_tP\rho' PP_t.  \label{e9}
\end{equation}

The effective density matrix following $P$, projected onto the system variables, is
%
\begin{equation}
\rho^* := \left({P\rho'P\over p(0)}\right)_{\! S}. \label{e10}
\end{equation}
%
This is a unit trace matrix proportional to $|k\rangle \langle k|$, and hence is equal to $|k\rangle \langle k|$ (${}=P$), the non-zero terms, in a basis containing $|e\rangle$ and $|f\rangle$ being
%
\begin{equation}
\rho^* \sim \left[
                 \begin{array}{cc}
                       |a|^2 & a\bar b\\
                       \bar a b & |b|^2
                 \end{array}
           \right]. 
\label{e11}
\end{equation}

The off-diagonal terms decay with the decoherence time $t_{dc}$, while
the diagonal terms are stable and so from (\ref{e9}) and (\ref{e10}), 
noting that tracing over the environment commutes with $P$ (but not with $H$)
%
\begin{equation}
p(t) =  p(0) [\begin{array}{cc}
              a & b
              \end{array}]
        \left[\begin{array}{cc}
              |a|^2 & \epsilon\\ 
              \epsilon & |b|^2
              \end{array} 
        \right]
        \left[\begin{array}{c}
              \bar a\\
              \bar b
              \end{array}
        \right] \label{e12}
\end{equation}
%
where $|\epsilon | \sim e^{-t/t_{dc}}$.
Thus 
$$
p(t) \to p(0)(|a|^4 + |b|^4) = p(0)(1 - 2|a|^2|b|^2).
$$
This will violate the stability condition unless either $|a|$ or $|b|$ is very small, that is, unless the superposition is very close to a pure macroscopic state, as required.



\section{Conclusion }
\smallskip
I have stressed that this is a provisional theory with ad hoc
elements. There seems to be a growing feeling among some workers (such
as Zeh\rr{14}) that the inadequacies of the current situation can
only be overcome by a theory of mind; a view that I would endorse,
though without thereby endorsing a many-minds metaphysics. There
remains, however, a considerable gap between approaches to a theory of
mind starting from the requirements of quantum theory (e.g.\
Donald\rr{3}) 
and those starting from psychology (e.g.\ Velmans\rr{13}). Work
under way to bridge this gap (e.g.\ Clarke\rr{2}) still needs to
develop an adequate dynamics; but the requirement of a stability
condition can now be used to provide a clear goal in this work. 


\section{REFERENCES }

{\begin{enumerate}
\item E G Beltrametti and G Cassinelli,  {\it The Logic of
   Quantum Mechanics}, (Addison Wesley, Reading, 1981)

\item C J S Clarke,  ``Consciousness and non-hierarchical physics'' in
   {\it The physical nature of consciousness,}  Ed.  Philip van Looke,
   (John Benjamins Publishing, Amsterdam, 2000)

\item M Donald, ``A mathematical characterisation of the physical
   structure of observers, {\it Foundations of Physics,} {\bf  25},
   529-571 (1995)

\item F Dowker and A Kent, ``On the consistent histories approach to
   quantum mechanics'' {\it J. Stat. Phys.} {\bf }82 1575  (1996)

\item D Giulini, E Joos, C Kiefer, J Kupsch, I-O Stamatescu, and
H D Zeh (Eds)  {\it Decoherence and the appearance of a classical
world}, (Springer-Verlag, Berlin and Heidelberg, 1996)

\item J J Halliwell, ``A review of the decoherent histories approach to
quantum mechanics'' {\it Ann.\ NY Acad.\ Sci.} {\bf 755}, 726-740 (1995) 

\item J J Halliwell,  ``Approximate Decoherence of Histories and 't
Hooft's Deterministic Quantum Theory'', quant-ph/0011103

\item M-W Ho,  {\it The Rainbow and the Worm} (2nd Edition)
(World Scientific, Singapore, 1998)

\item C J Isham, N Linden, K Savvida, and S Schreckenberg, 
``Continuous time and consistent histories'', {\it J.\ Math.\ Phys.}
{\bf 39} 1818-34 (1999)

\item R Omn\`es,  {\it Understanding Quantum Mechanics} (Princeton
   University Press, Princeton, 1999)

\item R Omn\`es, {\it Quantum philosophy} (Princeton University
Press, Princeton, 1999)

\item M Tegmark,  {\it Phys.\ Rev.\ E} {\bf 61} (4) 4194--4206 (2000)

\item M Velmans, {\it Understanding Consciousness} (Routledge. London, 2000)

\item H D Zeh,  ``The problem of conscious observation in quantum
mechanical description'', {\it Found.\ Phys.\ Lett.} {\bf 13} 221-233 (2000)

\item W H Zurek,  {\it Phys.\ Rev.\ D}, {\bf 24} 1516 (1981)
\end{enumerate}}
\end{document}