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March 16, 2000 \hfill LBNL-45229 \\
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{\large \bf Decoherence, Quantum Zeno Effect, and the Efficacy of Mental
Effort.}
\footnote{This work is supported in part by the Director, Office of Science,
Office of High Energy and Nuclear Physics, Division of High Energy Physics,
of the U.S. Department of Energy under Contract DE-AC03-76SF00098}
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Henry P. Stapp\\
{\em Lawrence Berkeley National Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
Recent theoretical and experimental papers support the prevailing
opinion that large warm systems will rapidly lose quantum coherence,
and that classical properties will emerge. This rapid loss of coherence
would naturally be expected to block any critical role for quantum theory
in explaining the interaction between our conscious experiences and the
physical activities of our brains. However, there is a quantum theory of
mind in which the efficacy of mental effort is not affected by decoherence
effects. In this theory the effects of mental action on brain activity is
achieved by a Quantum Zeno Effect that is not weakened by decoherence. The
theory is based on a relativistic version of von Neumann's quantum theory.
It encompasses all the predictions of Copenhagen quantum theory, which include
all the validated predictions of classical physical theory. In addition, it
forges two-way dynamical links between the physical and experiential aspects
of nature. The theory has significant explanatory power.
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\noindent {\bf 1. Introduction.}
The experimental work of the Paris group of S. Haroche [1] and of the
Boulder group of D. Wineland [2] demonstrate convincingly that the theoretical
ideas of quantum theory really do work in careful experiments
performed, in effect, on individual atoms interacting with
controlled electromagnetic probes and environments. It is
an impressive tribute to the power of human reason and logic that the
creators of quantum theory were able to accurately forecast
effects so far removed in scale and intricacy from the data
that they possessed.
The experiments of these groups both confirm the
emergence of decoherence effects whose strength and rapidity of onset
increase rapidly with the size of the system being disturbed
by interactions with its environment.
In recent theoretical paper [3] Max Tegmark computes, on the basis of the
thus-confirmed ideas, some expected time intervals for the disappearance of
quantum coherence in various brain structures that have been proposed as
the seat of the neural correlates of consciousness. He finds that quantum
coherence disappears on time scales of $10^{-13}$ to $10^{-20}$ seconds,
and concludes from this that classical concepts should provide a
completely adequate basis for understanding the dynamical
connection between mind and brain.
This conclusion depends on the idea that the quantum interaction between
mind and brain depends upon quantum coherence. It is indeed usually thought
that coherence is the essence of quantum theory, and that all quantum effects
depend upon it. But the development of the von Neumann-Wigner quantum theory
of mind pursued by this author was specifically designed so that the effect of
mental effort on brain process is not weakened by decoherence. Indeed,
quantum decoherence was {\it assumed} to decompose the state of the brain
into a mixture of essentially classical states. But the quantum effect of
mental effort on brain activity is not curtailed by this decomposition.
I shall now explain how this works.
{\bf 2. Overview of the Theory}
Before giving the specific computation I must first describe
the general form of the theory. It is based on objectively interpreted
von Neumann-Wigner quantum theory. I have argued elsewhere [4,5] that
the evolving state S(t) of von Neumann-Wigner quantum theory can be
construed to be our theoretical representation of an objectively existing
and evolving informational structure that can properly be called ``physical
reality''.
The theory has four basic equations. The first defines the
state of a subsystem. If S(t) is the operator that represents
the state of the universe and b is a subsystem of the
universe then the state of b is defined to be
$$
S(t)_b = Tr_b S(t), \eqno(2.1)
$$
where $Tr_b$ means the trace over all variable except
those that characterize b.
The second basic equation specifies von Neumann's process I.
This process ``poses a question''. If
$S(t-0)$ represents the limit of $S(t')$ as $t'$ approaches t from
below then at certain times t the following jump occurs:
$$
S(t)= P S(t-0) P + (1-P) S(t-0) (1-P). \eqno(2.2)
$$
Here P is a projection operator (i.e., $P^2 = P$) that acts as
the unit operator on all degrees of freedom except those
associated with the processor b.
The third basic equation specifies the (Dirac) reduction. This
reduction specifies nature's answer to the question:
$$
S(t+0)= P S(t) P \mbox{ with probability } Tr P S(t)/ Tr S(t) \eqno (2.3)\\
$$
or
$$
S(t+0)=(1-P) S(t) (1-P) \mbox{ with probability }Tr (1-P) S(t)/Tr S(t).
$$
Between jumps the state evolves according to:
$$
S(t+\Delta t)= \exp(-iH\Delta t) S(t) \exp(+iH\Delta t). \eqno (2.4)
$$
The projection operator P has two eigenvalues, 1 and 0, and is
therefore associated
with a Yes-No question: the two alternative possible
reductions specified
in (2.3) are associated with the two alternative possible
answers, Yes or No, to the question associated with P. Thus
the reduction (2.3) specifies
one bit of information, and implants that information in the
state S(t) of the physical universe. This state S(t) can be
regarded as just the evolving carrier of the bits of
information generated by these reduction events.
Information is normally conceived to be associated with an
interpreting system. In Copenhagen quantum theory each
reduction is associated with an increment in human knowledge,
and the interpreting system is the brain and body of the
observer. Generalizing from this one known
kind of example, I shall assume that each reduction (2.3) is
associated with a quantum information processor, call it b,
that both poses the question
---picks P---and, when nature responds by picking, say, the
answer P=1, `interprets' that bit of information by
evolving in a characteristic way.
The projection operator P cannot be local: any point-like
projection would inject infinite energy into the processor.
This jump of S(t) to P S(t)P, because it is basically a
nonlocal process, has no counterpart in classical dynamics:
it is a new kind of element, relative to classical physical
theory. Generalizing again from the one known example, I
assume that each reduction event is connected to some sort
of ``knowing'': each such event has a characteristic
experiential ``feel''.
Each thought involves an effort to attend to something---
i.e., to pose a question---followed by a registration of the
answer. This conforms exactly to the quantum dynamics.
Normally a sequence of thoughts consists of a string of
thoughts each of which differs just slightly from its
predecessor: the sequence becomes a `stream' of
consciousness. So the basic process is self-replication:
the thought T creates conditions that tend to create a
likeness of T.
This means that a key requirement for P is that PSP not
evolve rapidly out of the subspace defined by P, or at least
that PSP quickly evolve into a state nearly the same as PSP,
so that the sequence of thought is likely to be a sequence of
similar thoughts.
This suggests that the projection operator P may act in
the space of a set of conjugate variables that is undergoing
periodic motion, and that it projects onto a band of neighboring
orbits in phase space. For a simple harmonic oscillator in
a state of high energy one could take the projection operator
P to be the sum of the projection operators onto a large set
of neighboring energy eigenstates. This would effectively
project onto a band of neighboring orbits in phase space.
\noindent {\bf 3. The Quantum Zeno Effect}
In this theory the main effect of mind on brain is via the
quantum Zeno effect. Suppose the initial state is PS(t)P,
and that in that state the next question is again P, and that
this question repetitiously repeats.
If these questions are posed at intervals $\Delta t$ then
equations (2.4) and (2.2) give
$$
S(t+\Delta t) = P \exp (-iH\Delta t) PS(t)P \exp (+iH\Delta t) P
$$
$$
+ (1-P) \exp (-iH\Delta t) PS(t)P \exp (+iH\Delta t) (1-P).
$$
If $\Delta t$ is small on the scale of the leakage of PS(t)P
out of the subspace defined by P then the second term is
small and of second order in $\Delta t$. Thus as $\Delta t$
gets small, on the scale of the leakage of $PSP$ into the subspace
associated with $(1-P)$, the Hamiltonian $H$ gets effectively
replaced by $PHP$: evolution within the $P$ subspace proceeds
normally, but leakage out of that subspace is blocked.
The point here is that the linear-in-time leakage out of the
subspace defined by $P$ is killed by the reduction events.
Thus only the quadratic and higher terms survive, and these
are damped out if the reductions occurs fast on the time
scale of the relevant oscillations.
This replacement of the full Hamiltonian H by PHP is the usual
quantum Zeno effect. We see that it is just as effective
for a statistical mixture S(t) of quasi-classical states as
for a pure state: the decoherence generated by interaction
with the environment does not weaken this quantum effect.
\noindent {\bf Explanatory Power}
Von Neumann-Wigner quantum theory
encompasses all the valid predictions of classical physical
theory. So for any computation, or argumentation, for which
quantum effects are unimportant one can use classical physics.
Hence vN/W theory is at least as good as classical physical
theory: the two theories are effectively equivalent insofar as
quantum effects are unimportant. In the purely physical
domain the vN/W theory is certainly better, because it
predicts also all of the quantum effects, including all of
the ``nonlocal'' quantum effects. But our interest here
is on the nature of the dynamical link between mind and brain,
and the nature of the consequences of this connection.
The only power given to the mind by this theory is the power
to choose the questions P. And the only effects of these
choices that has thus far been identified are the consequences
achieved by the quantum Zeno effect. This effect is to keep
the brain activity focussed on a question for longer than
it would stay focussed in the classical theory.
To make the theory still more constrained, let me assume that
the quantum processor, in this case the human brain/body,
possesses a certain set off possible questions P, and that
at a prescribed sequence of instants the processor can either
consent, or not consent, to posing a certain possible question P.
Let this question P be the one that maximizes $Tr_b P S(T)/Tr_b S(t)$.
To accomodate our intuitive feeling that mental `effort' does effect
brain/body activity I add the postulate that the rapidity of
the sequence of instants can be increased by mental effort.
This is a simple theory. But the effect of mind on brain
is highly constrained. The only variables under mental control
are ``consent' and `effort'.
Does this theory explain anything?
Consider the following passage from ``Psychology:
The Briefer Course'' by William James [7]. In the final
section of the chapter on Attention he
writes:
``I have spoken as if our attention were wholly
determined by neural conditions. I believe that the array of {\it things}
we can attend to is so determined. No object can {\it catch} our attention
except by the neural machinery. But the {\it amount} of the attention which
an object receives after it has caught our attention is another question.
It often takes effort to keep mind upon it. We feel that we can make more
or less of the effort as we choose. If this feeling be not deceptive,
if our effort be a spiritual force, and an indeterminant one, then of
course it contributes coequally with the cerebral conditions to the result.
Though it introduce no new idea, it will deepen and prolong the stay in
consciousness of innumerable ideas which else would fade more quickly
away. The delay thus gained might not be more than a second in duration---
but that second may be critical; for in the rising and falling
considerations in the mind, where two associated systems of them are
nearly in equilibrium it is often a matter of but a second more or
less of attention at the outset, whether one system shall gain force to
occupy the field and develop itself and exclude the other, or be excluded
itself by the other. When developed it may make us act, and that act may
seal our doom. When we come to the chapter on the Will we shall see that
the whole drama of the voluntary life hinges on the attention, slightly
more or slightly less, which rival motor ideas may receive. ...''
Posing a question is the act of attending. In the chapter on Will, in the
section entitled ``Volitional effort is effort of attention'' [7]
James writes:
``Thus we find that {\it we reach the heart of our inquiry into volition
when we ask by what process is it that the thought of any given action
comes to prevail stably in the mind.}''
and later
``{\it The essential achievement of the will, in short, when it is most
`voluntary,' is to attend to a difficult object and hold it fast before
the mind. ... Effort of attention is thus the essential phenomenon
of will.''}
Still later, James says:
{\it ``Consent to the idea's undivided presence, this is effort's sole
achievement.''} ...``Everywhere, then, the function of effort is the same:
to keep affirming and adopting the thought which, if left to itself, would
slip away.''
The vN/W theory, with the quantum zeno effect incorporated,
explains naturally the features that are the basis of James's
conception of the action of human volition.
\noindent {\bf References}
1. M. Brune, et. al. Phys. Rev. Lett. {\bf 77}, 4887 (1996)
2. C.J. Myatt, et. al. Nature, {\bf 403}, 269 (2000)
3. Max Tegmark, ``The Importance of Quantum Decoherence in Brain
Process,'' Phys. Rev E, to appear.
4. H.P. Stapp, ``Nonlocality, Counterfactuals, and Consistent Histories,\\
http://xxx.lanl.gov/abs/quant-ph/9905055
5. H.P. Stapp, ``From Einstein Nonlocality to Von Neumann Reality,''\\
http://www-physics.lbl.gov/$\sim$stapp/stappfiles.html
6. H.P. Stapp, ``Attention, Intention, and Will in Quantum Physics,''\\
in J. Consc. Studies {\bf 6}, 143-64 (1999).
7. Wm. James, ``Psychology: The Briefer Course'', ed. Gordon Allport,
University of Notre Dame Press, Notre Dame, IN. Ch. 4 and Ch. 17
\end{document}
Pre Comments
2. Seife's article about Tegmark's paper quotes me as saying that my theory
is not upset by the decoherence effects that he computes.
My principal aim in this paper is to explain why this is true.
But in order to do this I must first explain the general picture of
the quantum theory of mind that seems to me to flow most naturally from
the general precepts of the von Neumann-Wigner formulation of quantum
theory. The specific form of the mind-brain connection emerges
from this description. Then the explanatory power of the theory
can be discussed.
3. The experimental papers of Haroche (ENS, Paris) and Wineland
(NIST, Boulder) groups both confirm that the decoherence effects
predicted by QT do indeed occur. These experiments examine situations
in which a mesoscopic coherent state of a quantum oscillator is the
analog of the state of a ``pointer'' or ``cat''. [In the Haroche-ENS
experiment the quantum coherent state is a cavity state of the EM field;
in the Wineland-NIST experiment the quantum coherent state is the state
of an atom in a simple harmonic oscillator potential]. In both cases
a quantum superposition of two different states is created, where each
of these two states is a product of a pointer/cat state times a
corresponding state of an atom, and certain interference effects are
studied. These interference effect drop off as the distance
$|\alpha_1 -\alpha_2|$ between the two pointer/cat states increases.
In the Wineland experiment the pointer/cat state is coupled to an
``engineered'' random environment, and the interference drops off
exponentially with $|\alpha_1 -\alpha_2|^2 $, where $ $
is the mean square voltage (random) noise. Thus the interference effects
drop off rapidly with the increasing ``size'' of the pointer/cat,
and ``temperature'' of the environment.
The drop-off of the interference effects is represented by the fact that
the density matrix of the pointer/cat changes from one corresponding to
a quantum superposition of the two states to one that represents a mixture.
This transformation to a mixture is sometimes interpreted as saying that
the pointer/cat is now in EITHER one state OR the other. However, in both
the ENS and NIST experiments there is no real reduction to the either-or
situation: further procedures can (at least partially) reverse the
separation of the pointer/cat states and recover corresponding
interference effects.
These experiments are wonderful confirmations of the predictions
of quantum theory, and in particular of the validity of the density
matrix formalism that is spelled in von Neuman's book. That formalism
is the basis of my development of his theory. Thus the experiments
are, basically, empirical validations of the principles that are the basis
of my theory.