%REPORTMASTER (revised 8/24/88)
\documentstyle[12pt]{article}
%\input math_macros.tex
\def\baselinestretch{1.2}
\def\thefootnote{\fnsymbol{footnote}}
\begin{document}
\begin{titlepage}
\begin{center}
May 12, 2000 \hfill LBNL-45229 \\
\vskip .5in
{\large \bf A Quantum Theory of Mind.}
\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}
\vskip .20in
Henry P. Stapp\\
{\em Lawrence Berkeley National Laboratory\\
University of California\\
Berkeley, California 94720}
\end{center}
\vskip .10in
\begin{abstract}
A theory of consciousness is erected on von Neumann's formulation of
quantum theory. More specifically, a theory of attention and mental
effort is developed. This presentation is aimed at psychologists:
it assumes no familiarity with quantum theory. The theory is highly
structured, compared to theories based on classical concepts, because
it rests heavily on restrictive quantum conditions on the form of the
interaction between mind and brain. These restrictions invest the theory
with significant explanatory power. Supporting empirical evidence comes
both from the general observations of William James and from recent
experimental findings pertaining to attention and mental effort.
\end{abstract}
\medskip
\end{titlepage}
\newpage
\renewcommand{\thepage}{\arabic{page}}
\setcounter{page}{1}
\noindent {\bf 1. Introduction.}
Classical physical theory provides an unsatisfactory foundation for a
theory of consciousness. The precepts of classical physics require
all brain activity to be in principle completely determined by
bottom-up local atomic processes alone, with the experiential events
that form a person's stream of consciousness making no difference at
all either to his bodily behaviour or to what is happening in his brain.
This has two unsatisfactory consequences. It means that there is no reason
for the incredibly elaborate, complex, and sensitive system of human
consciousness ever to have evolved, and hence no way to understand the causes
of its development. And it means that conscious experiences must be added
to the theory ad hoc, simply because we know that they exist, rather than
because they are logically necessary parts of a unified dynamical theory
of the natural world. Consequently, the explanatory power of any theory of
consciousness based on classical physical theory is inherently limited
by this fact that in classical theory consciousness is a free-floating
supernumerary having no fixed logical connections to the physical world.
The situation is completely reversed if quantum theory is used. Within the
quantum framework a rational approach leads naturally to theory in which
mind enters into brain process as a integral and logically necessary
component of the basic dynamics. Thus the fundamental physical principles
imposes stringent conditions both on the way consciousness arises from
brain action, and on the way it acts back on the brain. The theory of
attention and mental effort that emerges from this tightknit approach
has impressive explanatory power. It explains both the general features
of attention and efficacious mental effort described by William James
at the end of the nineteenth century, as well as certain empirical findings
that have remained mysterious and unexplained since that time, and also
many hitherto unexplained features uncovered by recent empirical work.
It is well known that the founders of quantum theory introduced the
observer and his observations into physical theory. This innovation
constituted a sharp break from the ideas of classical physical theory,
which placed the conscious experiences of human observers outside the part
of nature dealt with by physical theory. But according
to quantum philosophy, as articulated by Niels Bohr:
``In our description of nature the purpose is not to
disclose the real essence of phenomena but only to track down as far as
possible relations between the multifold aspects of our experience.''[1].
In the words of Heisenberg: ``The conception of objective reality of the
elementary particles has evaporated not into the cloud of some new reality
concept but into the transparent clarity of a mathematics that represents
no longer the behaviour of the elementary particles but rather our
knowledge of this behaviour.''[2]. Human experience and knowledge thus
became identified as precisely the thing that basic physical theory is about.
Moreover, the human experimenter plays a key dynamical role:
a human intelligence standing outside the physical system, as it is
represented by the mathematical machinery of quantum theory, chooses
which aspect of nature will be probed. In Bohr's words:
``The freedom of experimentation, presupposed in classical physics,
is of course retained and corresponds to the free choice of experimental
arrangements for which the mathematical structure of the quantum
mechanical formalism offers the appropriate latitude.''[3]
This Copenhagen approach is ideally adapted to the
practical applications of quantum theory to laboratory experiments.
It enables quantum physicists to avoid getting entangled in metaphysical
issues that are, for all practical purposes, irrelevant to these
important applications. This Copenhagen quantum theory rests on the
idea of dividing nature into two parts: ``an observed system'',
which is described by the mathematical machinery of quantum theory, and
``an observing system'', which includes the human observers and their
measuring devices. The latter are described in everyday language, refined
by the concepts of classical physical theory, without, however, suggesting
that classical physical theory really holds exactly for this part of the
world: ordinary language is simply the language that we humans do in fact use
to communicate to others what we have done, in setting up the experiments,
and what we have learned from our observations.
The practical virtues of the Copenhagen approach stem directly from
this division of the dynamically unified world into these two
differently described parts. Though useful in practice, this tearing
asunder of the unified physical world is, from the standpoint of principle,
a disaster. However, this problem at the level of principle was solved
by John von Neumann, who showed that the predictions of Copenhagen quantum
theory could be obtained in principle by treating the entire physical
world, including the bodies and brains of the human observers, in the
mathematical language of quantum theory, and allowing the experiences of
each observer to interact dynamically, according to quantum precepts, with
the brain of that observer. This gives a highly constrained theory
that is, at the same time, both a logically coherent way of formulating
quantum theory and also a suitable physical basis for a theory of
consciousness.
There is, however, one effect, well known to physicists, that
seems at first to block any important contribution of quantum effects
to the efficacy of consciousness. This effect, called environmental
decoherence, is the strong tendency of a warm wet system, such as a human
brain, to be effectively decomposed by interaction with its environment
into a statistical mixture of quasi-classical states. This decomposition
would appear, superficially, to eradicate all macroscopic quantum effects
that might contribute to the efficacy of mental effort in the control
of brain activity [4].
Consideration of this decoherence effect played, in fact, a crucial
role in the development of the theory of consciousness described here.
It forced the theory to rely, in order to achieve strong effects of mental
effort upon behaviour, on the Quantum Zeno Effect. This is a well-known
and well-studied quantum effect that is not weakened by environmental
decoherence, and it is the basis of the theory of efficacious mental effort
described here. The essential point, which will be explained in detail,
is that focus of attention is maintained by mental effort via the physical
mechanism of the Quantum Zeno Effect, and this connection explains many
empirical features of the mind-brain connection.
This article is aimed at mainly psychologists. No knowledge of
physics is assumed. Indeed, it may be beneficial not to know any physics,
because quantum theory alters or reinterprets everything that classical
physical theory says. Hence understanding classical physical physics
creates prejudices that have to be overcome. I intend here to
be both precise enough to satisfy quantum physicists yet clear and simple
enough to reach readers with no knowledge of either classical or
quantum physics. Once I have introduced from scratch a few simple ideas,
I shall actually derive the Quantum Zeno Effect, and
show how it works to make mental effort causally efficacious in
the control of physical processes in a human brain. Then I shall
compare the resulting properties to both to the observations of William James,
and empirical results of the last half century, with emphasis on
results about attention and effort obtained in the last two decades.
{\bf 2. The Form of the Interaction Between Mind and Brain}
This work is based on objectively interpreted
von Neumann/Wigner quantum theory. I have argued elsewhere [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''. von Neumann/Wigner quantum theory is essentially a theory
of the interaction between this objective physical reality and
subjective human experiences. The earlier Copenhagen version was essentially
subjective, being, fundamentally, merely a set of rules that allow
scientists to make predictions pertaining to connections between possible
human experiences. It renounced the effort to understand the reality
lying behind these rules. The immediate objective of the von Neumann
theory was to show how these empirically validated Copenhagen predictions
about connections between human experiences can be deduced from a
rationally coherent conception of an objectively existing physical reality
that interacts in specified ways with the subjective elements described in
the Copenhagen formulation. Thus von Neumann and Wigner supplied, first,
a putative description of the underlying physical reality that the
Copenhagen approach tries to ignore, and, second, the form of the
interaction between this objective physical reality and the experiential
realities that are the basis of the Copenhagen approach.
I shall now describe the essentials of quantum theory, and of the
connection between Copenhagen quantum theory and von Neumann/Wigner
quantum theory.
Suppose there were just one particle in the universe. And suppose
that, according to classical ideas, it must lie at one of N possible
locations. Then one could represent the universe theoretically by
imagining a long row of N boxes, each capable of holding a number,
and then placing the number ``one'' in the box corresponding to the location
where the particle lies, and a ``zero'' in every other box. If one wanted to
represent statistical knowledge or information, then one could put
into each box a number representing the ``probability'', or
``statistical weight'' associated with that possible
location of the particle.
To pass to quantum theory, for this simple case, one would represent
the state S(t) of the universe at time t, not by a `row' of N boxes,
each containing some number, but by an N-by-N `square array' of boxes,
each containing some number.
The ``diagonal'' elements of this matrix play a special role.
The ``diagonal'' elements of an N-by-N matrix are the N numbers that lie
in the boxes that along the diagonal, i.e., in a box such that, for some
number n, the box lies in row n and column n of the square array.
The diagonal element that lies in row n and column n represents, in our
simple example, the probability that the particle is in location n. The
off-diagonal elements do not have this simple statistical interpretation,
but they enter into the dynamics, and hence cannot be ignored. A matrix
in which all of the off-diagonal entries are zero is called a
`diagonal matrix'.
For any matrix M, the `number' in row i and column j is denoted by $M_{ij}$.
These individual numbers are called the matrix elements of the matrix M.
The `diagonal' elements of M are the numbers $M_{ii}$.
Matrices are generalizations of numbers, and share many of their
properties. One can multiply a matrix a number: the rule is simply to
multiply each matrix element of the matrix by that number. One can add two
N-by-N matrices: the rule is simply to add together the two matrix elements
that occupy the same position in the two arrays. Two
N-by-N matrices, A and B, can be multiplied together to form the N-by-N
matrix called AB. The rule for constructing the matrix AB is this: the
matrix element of AB that lies in row i and column j is defined to be a
sum of N numbers each of which is a product of two numbers. The specific
rule is this:
$AB_{ij}$ = The sum over all N possible values k of $A_{ik} B_{kj}$.
Notice that the values i and j that define the row and column of the
matrix element $AB_{ij}$ appear also on the right-hand side of
this equation. The right-hand side contains also a variable k that runs,
in successive terms, over all the integers from 1 to N.
This multiplication rule may seem arbitrary, but from a certain point of view
it is the natural rule. Multiplication of matrices is important in both
mathematics and classical mechanics, as well as in quantum mechanics.
This multiplication rule does not ensure that AB=BA. This dependence of
the product of matrices on the order of the factors is important.
But apart from this dependence on the {\it order} in which they appear,
matrices can be treated like ordinary numbers.
If one merely preserves the ``order'' in which the matrix factors are
written down, one can do all of the usual operations of addition,
subtraction, and multiplication. This will make it easy
actually to derive the Quantum Zeno Effect from elementary
``arithmetic'', without writing down any numbers.
The term `operator' is often used in place of `matrix'. An `operator'
represents, in fact, a generalization that covers, for example, the case
in which. the discrete integer indices i and j are replaced by continuous
variables. I often use the word `operator', rather than `matrix', but the
difference is not important here.
Quantum dynamics consists of a deterministic evolution of the state
of the universe, punctuated by a sequence of sudden quantum events,
or `jumps'. The deterministic evolution between events is controlled by the
Schroedinger equation, which is the deterministic quantum counterpart of
the Newton/Maxwell equations of motion.
A key feature of Copenhagen quantum theory is that for each event
an experimenter, who stands outside the system that is represented
by the mathematical machinery, must choose a specific question:
he decides which aspect of the quantum system will be probed by the
observing system. This question is either itself a Yes-No question, or it
can be decomposed into a finite set of Yes-No questions.
Each such Yes-No question is represented within the mathematical machinery
by an associated `projection operator' P. A projection operator is an
operator (or matrix) that satisfies PP=P.
In our simple idealized example, where the universe consists of a single
particle that is located at one of N possible locations, a possible question
might be: Is the particle located at one of the first n locations?.
The projection operator P corresponding to this question is a diagonal
matrix that has a ``one'' in each of the first n boxes on the diagonal
and a zero in every other box. One can easily verify, from the
multiplication rule defined above, that PP=P, as required.
A basic rule of quantum mechanics asserts that if a question represented
by a projection operator P is asked at time t, and the state of the
physical system (or universe) just prior to time t is called $S(t-)$, and
the state just after time t is called $S(t+)$, then
$$
S(t+)= PS(t-)P
$$
or
$$
S(t+)=(1-P)S(t-)(1-P).
$$
That is, the state will make a sudden ``quantum jump'' from
$S(t-)$ to either $PS(t-)P$ or $(1-P)S(t-)(1-P)$.
What is the effect of the change from $S(t-)$ to $PS(t-)P$ ?
To find the answer one must perform the indicated matrix
multiplications.
Suppose P is the matrix described above that
corresponds to the question: ``Is the particle located
in one of the first n of the possible locations?'' Then the product
$PS(t-)P$ has a very simple form. It is the matrix obtained from
$S(t-)$ by leaving undisturbed the numbers in those boxes that lie
{\it both} in one of the first n rows {\it and} in one of the first
n columns, but setting to zero the numbers in all other boxes.
Thus $PS(t-)P$ is the `projection' of $S(t-)$ into the n-by-n part of itself
that corresponds to the answer `Yes' to the question associated with P:
all the numbers $S(t-)_{ij}$ with either i or j bigger than n are set to zero,
but those both i and j less than or equal to n are left unchanged.
The symbol $1$ in $(1-P)$ stands for the matrix with all `ones' on the
diagonal. Thus $(1-P)$ is the diagonal matrix with `ones' in the
{\it final} $N-n$ diagonal locations and zero elsewhere. Thus $(1-P)S(t-)(1-P)$
is the same as $S(t-)$ in the boxes that lie {\it both} in one of the
final $(N-n)$ rows {\it and} one of the final $(N-n)$ columns, but has zeros
in all other positions.
We thus see that the reduction of $S(t-)$ to $PS(t-)P$
reduces the state to the part of itself that is completely consistent
with the answer `Yes' to the question associated with P.
The alternative reduction of $S(t-)$ to $(1-P)S(t-)(1-P)$
reduces the state to the part of itself that is completely consistent
with the answer `No' to the question associated with P.
To compare the theory to experiment one needs to extract real numbers from
the theory. The basic operation that achieves this is the `trace'
operation. The trace of a matrix is just the sum of its diagonal elements:
Tr M = The sum over all values i of $M_{ii}$.
We are now in a position to state the basic interpretive law
of quantum theory: If the state of the universe at the time
immediately prior to time t is represented by $S(t-)$, and the
question associated with P is asked at time t, then the state
$S(t+)$ just after time t will be either $PS(t-)P$ or $(1-P)S(t-)(1-P)$,
and:
1) The probability that $S(t+)$ will be $PS(t-)P$ is
$Tr PS(t-)P/Tr S(t-)$, and
2) The probability that $S(t+)$ will be $(1-P)S(t-)(1-P)$ is
$Tr (1-P)S(t-)(1-P)/Tr S(t-)$.
To understand these probability formulas recall that the diagonal
elements of $S(t-)$ were the probabilities associated with the
corresponding possible locations of the particle.
That means that $Tr S(t-)$ should be `one': the sum of the probabilities
should add up to `one'. In this case the denominators in the above
expression would be `one'. However, it is advantageous to let the diagonal
elements represent only the {\it relative} probabilities. Then the
denominator is needed to get the overall normalization of the
probabilities correct.
Recalling the form of $PS(t-)P$ we see that $Tr PS(t-)P/Tr S(t-)$ is just
the sum, immediately prior to time t, of the probabilities associated
with first n diagonal possible locations of the particle. It is,
therefore, indeed, just the probability that the answer will be `Yes'
to the question, asked at time t, ``Is the particle in one of
the first n of its possible locations?'' The alternative form
$(1-P)S(t-)(1-P)$ corresponds to the answers `No'.
I have been speaking, for simplicity, about the example
of a single particle with N possible locations.
But the general rules apply equally well if one replaces the single particle
by entire physical universe, and allows the N possibilities to be
N possible states of the universe.
To get an adequate relativistic theory of the mind-brain connection
one needs to consider relativistic quantum field theory, and, in particular,
quantum electro-dynamics, as discussed in reference 5. But the general
logical structure is unaltered.
The general theory has four basic equations. The first defines the
state of a subsystem. If $S$ 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
$$
S_b = Tr_{-b} S, \eqno(2.1)
$$
where $Tr_{-b}$ means the trace over all variables {\it except}
those that characterize b. In quantum electrodynamics the variables that
characterize b are the local fields located in the body/brain of the
person. The variables i and j can each be written as a {\it pair} of
variables, where the first member of each pair refers to b and the second
member refers to the rest of the universe. Then $Tr_{-b}$ is constructed
by setting the {\it second} members of the two pairs equal to each other,
and then summing the equated variables over all N possible values. The
first members of the two pairs are retained as variables, and hence
$S_b=Tr_{-b}S$ is a matrix whose indices refer to the system b alone.
The second basic equation is von Neumann's process I.
This process specifies the question that is asked, but not the answer. If
a question is asked at time t, and $S(t-)$ represents the limit of $S(t')$
as $t'$ approaches t from values less than t, then the state at time t is
$$
S(t)= P S(t-) P + (1-P) S(t-) (1-P). \eqno(2.2)
$$
Here P is a projection operator (i.e., $P^2 = P$) that acts as
the unit operator on all degrees of freedom {\it except} those
associated with the body-brain b.
The third basic equation specifies the (Dirac) reduction. This
reduction specifies the {\it answer} that nature gives:
$$
S(t+)=PS(t)=PS(t-)P\mbox{ with probability } Tr PS(t)/Tr S(t), \eqno (2.3a)
$$
or,
$$
S(t+)=(1-P)S(t) \mbox{ with probability }Tr(1-P)S(t)/Tr S(t). \eqno (2.3b)
$$
Between jumps the state evolves according to:
$$
S(t+\Delta t)= \exp(-iH\Delta t) S(t) \exp(+iH\Delta t). \eqno (2.4)
$$
Here H is the energy (or Hamiltonian) operator of quantum electrodynamics.
The reduction (2.3) corresponds to a certain increase in information
It specifies one bit of information, Yes or No, and implants that
information in the objective state $S(t)$ of the physical universe,
by means of the reduction of $S(t-)$ to $PS(t-)P$ or to $(1-P)S(t-)(1-P)$.
This objective state $S(t)$, with variable $t$, can thus be regarded as
the evolving objective carrier of all of the bits of information generated
by all of the reduction events that have occurred up until time t, minus
the information that has been destroyed by the various reductions.
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. Thus in Copenhagen quantum theory the question
in not ``Is the particle in one of the first n
of the possible locations?'' It is rather: ``Will I find
the particle to be in one of the first n of the possible
locations?'' A probing question must be asked, and this
question must be such that the two answers, `Yes' or `No', are
experientially distinguishable.
In Copenhagen quantum theory
the projection operator P associated with the question acts
on the physical system that is being probed by the devices that
the human observer is observing. But in von Neumann-Wigner quantum
theory the projection operator P, which is---as in the Copenhagen
approach---associated with a question about what the observer's
experiences will be, acts on the degrees of freedom of the brain
(and possibly the body) of that observer. The occurrence of the
experience that corresponds to the answer `Yes' is associated with the
reduction of $S(t-)$ to $PS(t-)P$. This change reduces the state
of the brain of the observer to a state that is compatible with this
experience. This change is the interaction between mind and brain
specified by von Neumann/Wigner quantum theory. It is this
interaction that will be discussed and exploited in what follows.
One main point is that in order for the quantum dynamical process to
proceed, a sequence of questions must be asked.
Each question is associated with a quantum processor, which in the
original von Neumann/Wigner formulation is a human being.
This anthropocentric bias of the theory is due to the fact that
the immediate goal of the von Neumann/Wigner theory was to
reproduce the predictions of Copenhagen quantum theory, and in
the Copenhagen approach science is a strictly human endeavour
based on human knowledge. The von Neumann approach allows us
to relax this condition, and consider more general quantum
processors and address the problem of the evolution of
consciousness. But here I shall focus on the narrower issue
of the connection between human minds and human brains.
\noindent {\bf 3. The Macrolocal Process of Asking the Question}
The projection operator P cannot be local: any point-like
projection would inject infinite energy into the processor.
Because the reduction of S(t) to P S(t)P is a nonlocal physical
process it has no counterpart in classical
dynamics: it is a new kind of element, relative to classical
physical theory. But it is a key element
in both Copenhagen and von Neumann/Wigner dynamics.
The projection operator is associated with a question asked by
a person, or, more generally, by a quantum `processor' that plays
a dynamical role similar to a person.
Which question the person (or
processor) will ask is not specified by the known
rules of quantum theory. In Copenhagen quantum theory the choice of
which question to ask---of which aspect of nature to probe---
is left to the discretion of the experimenter, who is considered
to stand outside the quantum-mechanically described universe.
In von Neumann/Wigner quantum theory the body-brain
of the human experimenter is brought into the quantum-mechanically
described system, but the choice of the question he asks
remains unspecified by the known quantum mechanical rules. Hence
orthodox quantum theories do not specify how `the question' is fixed
or determined. This indeterminacy constitutes a causal
gap that opens the door to the possibility that mind can influence
brain in ways not determined by brain alone. The
fact that neither Copenhagen nor von Neumann quantum theory
specifies how these questions are determined means that orthodox
quantum theories leave the choices of the question to be asked
essentially free.
Because objective physical reality is informational in
character, rather than material, it is not clear that our
subjective knowings could not be, in some sense, composed out of
the objective physical stuff, or be completely determined by the
objective physical aspects of nature by rules not yet known to man.
Maybe future developments will shed some light on such questions. But
present-day physical theory, though it makes many sound predictions,
does not answer these ontological and causality questions: the
objectively-described and subjectively-described aspects
aspects of nature appear in the theory as differently described elements
that interact in accordance with certain dynamical rules.
The process of `Asking the Question' has an important property.
It is neither microlocal, nor globally nonlocal: it is
{\it macro }local. It is a process that is, in a certain
important sense, localized in the body-brain of the person
who is asking the question.
To see this, suppose there are two participants (i.e., persons)
located at time t in two well separated regions.
Suppose at time t the first one asks the question represented by $P_1$,
and the second one asks the question represented by $P_2$.
Then, because these two operators act at the same time on degrees of freedom
that are localized in two well separated regions of space, the rules of
relativistic quantum field theory demand that
$$
P_1 P_2 = P_2 P_1:
$$
the two projection operators commute.
The following result is then easy to derive:
The probability $$ for outcome `Yes' to the question
associated with by $P_1$ is not affected by just asking the question
associated with $P_2$.
PROOF:
According to (2.2), just asking the question associated with $P_2$
reduces S to the form
$$
[P_2 S P_2 +(1-P_2)S(1-P_2)].
$$
But then, according to (2.3a), the probability that the question
associated with $P_1$ has the answer `Yes' is
$$
= Tr P_1[P_2 S P_2 +(1-P_2)S(1-P_2)]/ Tr [P_2 S P_2 + (1-P_2)S(1-P_2)]
$$
$$
= Tr P_1[P_2 S + (1-P_2)S]/Tr[P_2 S + (1-P_2)S]
$$
$$
= Tr P_1 S/Tr S.
$$
To derive the second line one proceeds as follows. The numerator in the
first line breaks into two terms, the first of which is $Tr P_1 P_2 S P_2$.
The general identity $Tr AB = Tr BA$ is the used to move the factor $B = P_2$
to the left of $P_1$, giving $Tr P_2 P_1 P_2 S$. Then the equation
$P_2 P_1 = P_1 P_2$, is used to move this factor $P_2$ to the right of
$P_1$. giving $Tr P_1 P_2 P_2 S$. Then the property $P_2 P_2=P_2$
is used to obtain the first term in the numerator of the second line.
The second term in the numerator of the first line gives, by means of a
similar treatment, the second term in the numerator of the second line.
The two terms in the denominator are treated in the same way. This give line
two. Then a simple addition in the numerator, another one in the denominator,
gives the third line.
The first line represents the probability of getting the answer `Yes'
to the question associated with $P_1$ under the condition
that $P_2$ is asked, but no information about the answer (to that question
in region 2) is included.
The third line represents what the probability of $P_1$ would have been
if no question had been asked in region 2. Thus the computation
shows that {\it just asking} the question in region 2 (i.e., asking the
question, and adding the properly weighted contributions of the two possible
answers) does not affect the probability of obtaining
the answer `Yes' to question $P_1$ in region 1.
In this sense, the process of `just asking the question' associated with
$P_2$ is {\it macro}local; it has no instantaneous effects outside region
occupied by the body-brain b of participant 2.
However,{\it Answering} this question means keeping one of the two terms and
leaving out the other. But in that case the cancellations leading to the
macrolocality property derived above no longer occur. Thus {\it Answering} the
question is not macrolocal! It is globally nonlocal.
\noindent {\bf 4. The Quantum Zeno Effect}
In this theory the main effect of mind on brain is via the
Quantum Zeno Effect, which I shall now describe, and prove.
Suppose the initial state is PS(t)P, and that the question associated with P
is repetitiously repeated. 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). \eqno (4.1)
$$
If $\Delta t$ is small on the scale of the characteristic times for
$P\leftrightarrow (1-P)$ transitions to occur then in the terms where
$\exp (\pm iH\Delta t)$ is sandwiched
between a $P$ and a $(1-P)$ one can use the approximation
to it, $1\pm iH\Delta t$. But then the contribution from the unit term,
`one', drops out, by virtue of the equation $P(1-P)=(1-P)P=0$. Thus only
the single term $\pm iH\Delta t$ survives, Hence one sees that
every term that contributes to the $P \leftrightarrow (1-P)$ transition
occurs multiplied by at least {\it two} powers of $\Delta t$. But then these
$ P\leftrightarrow (1-P)$ transition terms drop out faster than the
other terms when one decreases $\Delta t$ by increasing the rapidity at
which questions are asked.
This is the Quantum Zeno Effect: if rate of asking the questions is
increased then the state tends to get trapped in the subspace that is
already in,
This effect is sometimes facetiously called the ``watched pot'' effect.
According to the old adage ``A watched pot never boils'', attending
intently to whether the water is boiling prevents it from boiling.
Similarly here, rapid-fire questioning effectively replaces the
Hamiltonian $H$ by $PHP+ (1-P)H(1-P)$: the other two terms,
which give the $P\leftrightarrow (1-P)$ transitions, tend to be
suppressed by rapid-fire `Askings'
This derivation shows that the Quantum Zeno Effect (QZE) is just as
effective for a statistical mixture S(t) of quasi-classical states as
for any other state: environmental decoherence does not eradicate
or weaken this particular quantum effect, which arises basically
from the reduction mechanism.
The only power given to the mind by this theory is the power to choose
the questions P, and the times at which they are asked. And the only
macroscopic dynamical effects of these choices thus far identified
are the consequences produced by the Quantum Zeno Effect. This effect
is merely to keep the brain activity focussed on a question for longer than
it would stay focussed if quantum effects were not present. This focussing
of attention is increased by increasing the rate at which the questions are
asked. It is reasonable to suppose, or propose, that this rate can be
increased by mental effort. Then mental effort can, by virtue of the
Quantum Zeno Effect, keep attention focussed.
\noindent {\bf 5. Explanatory Power}
Does the theory described above 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 indeterminate 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.''
von Neumann/Wigner quantum theory, with the quantum Zeno effect
incorporated, seems to explains naturally the way in which,
according to William James, human volition acts in our lives.
I shall tie down this connection by proposing a specific
model.
In the present context the questions that a person can ask are
supposed to be such that an affirmative answer corresponds to a
possible conscious experience. Let it be assumed that a person
with brain-body b has at each instant t a certain set of possible
questions that he can ask. Let $P_b$ mean $Tr_{-b}P$, and let
$\{P_b\}(t)$ be the set of $P_b$'s corresponding to the set of
possible conscious experiences at time t. Let $P_b(t)$ be the $P_b$ in
$\{P_b\}(t)$ the maximizes $Tr_b P_b S_b(T)/Tr_b S_b(t)$. If the brain
is a system whose job is to construct options for quantum events
that can actualize control sequences that can produce actions suited
to the circumstance in which the body-brain finds itself, then this
$P_b(t)$ should be the $P_b$ corresponding to the `best' question:
it is the $P_b$ that is given maximal statistical
weight by the evolving brain.
Let it further be supposed that at each time t the processor/person
can either consent, or not consent, to asking the question
associated with $P_b(t)$. To accommodate our intuitive feeling that
mental `effort' seems to focus attention I add the postulate that
the rate of `askings' can be increased by mental effort.
By virtue of QZE, increasing the rate of `askings'
should tend to keep attention focussed.
This is a simple theory in which the effect of mind on brain
is highly constrained: the only variables under mental control
are ``consent' and `effort': everything else contributing to fixing
the question asked has been ascribed to the brain. This brain computation
is macrolocal. Because this computation involves P it can access and
evaluate as a unit activities in different parts of the brain.
This simple quantum model fits James's descriptions very well.
\noindent {\bf 6. Explanatory Power: More Recent Data}
Much has happened in psychology since the time of William James.
So one must ask: How well does the close correspondence
between this quantum theory of consciousness and observations about
attention and effort by William James hold up in the light
of recent empirical work?
In this section I give an account of the data on attention and effort
described and analyzed in Harold Pashler's 1998 Book,
"The Psychology of Attention"[8]. Pashler treats James
with great respect, but explains how more recent empirical work
addresses issues that James was unable to deal with.
Pashler ties most of the empirical recent research on
attention that he considers to its impact on various
models of attention. He focusses on four different models:
1. Early selection theory (Broadbent's 1958 Filter theory.)
All stimuli are processed to the point where certain physical
attributes are represented, and then a filtering
occurs that determines which single stimulus is processed
for "identification", one at a time. (p.14, 219)
2. Late selection theories. "Identification processing" proceeds
unselectively without capacity limitation, then a later
process selects which of the "identified" possibilities is
selected for awareness.(p.16, 221)
3. Controlled Parallel Processing. The person chooses
which stimuli are attended to, and the attended-to
ones are processed, but the unattended-to ones are not
fully processed. (p.224)
Pashler finds that evidence rules out possibilities 1. and
2. and favors 3. , which, however, "provides only
the roughest scaffolding for a general account of perceptual
attention."
Further evidence shows that when several stimuli are selected for
attention they compete for a limited resource of
identifying capacity.
Pashler claims that "attention" must, in scheme 3, be identified either
with the gating part that selects what is to be attended to,
or with the capacity part that controls the allocation of the
identification-process resource. So one has two subpossibilities:
3.A. Attention controls which stimuli are selected for identification.
3.B. Attention controls allocation of limited identification-capacity.
Pashler asserts: "one thing we cannot choose is to maintain the
presupposition inherent in commonsense talk and early and late
selection theories that selectivity and capacity limitation
are inherent features of the very same process or mechanism."
(p.229)
Quantum theory makes one process both select stimuli
and allocate a finite resource. This process is the
ongoing process consisting of a sequence of reductions.
Each reduction associated with an answer `Yes' is
associated with a conscious event. Hence the process in
question is connected to conscious processing.
The reduction process effectively selects which stimuli will be
actualized by picking out, via the reduction of $S_b$ to $P_bS_bP_b$
(in case the answer is `Yes'), from the morass of prior superposed
partially processed stimuli, those stimuli that, by contributing
to $Tr_b P_bS_b(t)$, caused the question to be asked. The reduction also
controls, by actualizing the high-level control mechanism selected
by $P_bS_bP_b$, the further brain processing produced by the high-level
process actualized by the reduction event.
However, these reduction events are a limited resource that can be augmented
by mental effort, to produce more of them per second, or divided
by directing different events to different tasks.
This quantum theory appears to fit the "rough scaffolding"
suggested by recent empirical studies, but seems closer to intuition
that what Pashler deemed possible: the same mechanism of reduction
both selects the stimuli that are actualized and fully processed, from
a morass of stimuli that are processed in parallel superposed
branches, and it also allocates the limited resource of the
actualized higher-level processing.
I do not pretend that this sketch constitutes a full theory of
attention and effort. But it arises naturally from the basic
principles of physics, and already injects a fair amount of
specificity into the mind-brain connection.
I shall now examine in more detail the various empirical findings
described by Pashler that seem pertinent to this theory.
They all seem to be in line with the theory.
I do not claim that these features could not be explained by
a theory based on classical physical theory. But classical
physical theory does not lead in any natural way to the idea
that conscious experiences should even exist at all.
Hence it would, on the one hand, appear to be intrinsically
less constrained, hence less explanatory. On the other hand, it may
be unable to explain, in any natural way, features that arise naturally in
quantum theory from the macrolocal character of the reduction
process, or from the fact that the quantum processing involves
the whole state $S_b$, which is the quantum analog of the {\it whole
statistical ensemble} of quasi-classical states, or from
the natural quantum link between effort, rapidity of conscious events,
and confinement of attention.
Pashler has a section "Mental Effort" in the chapter "Automaticity,
Effort, and Control". He cites there first the 1968 experiments of Collyer
that show that "incentives to perform especially well led subjects to
improve both speed and accuracy." In the quantum model incentives should
lead to mental effort, which would increase the rate of processing, and
also increase accuracy insofar as the task is such that high-level processing
improves accuracy.
Pashler says that it is interesting that the speedup was additive
with variables affecting different processing stages, and that this
additivity is not easy to interpret. But it would be expected if effort
increases the overall rate of the conscious processing
events, as it does in the quantum model. [I need to examine more closely
the details of the experimental findings, but have not yet done so]
Pashler then considers a 1995 experiment by N.E. Johnson, Saccuzzo, and
Larson that shows that incentives improve performance more for complex process
that seem to require more conscious attention than for simple, more
automatic tasks. This is in line with the QTOC
(Quantum Theory of Consciousness), since effort merely speeds up
the event frequency, which keeps attention focussed: this improves
performance only on tasks where focussed attention is important.
Pashler then cites a 1983 study by Sanders which seems to be directly
interpretable as evidence that effort tends to sustain focus of attention.
A main basis of the QTOC is the physical explanation, on the basis
of the fundamental principles of quantum physics, of how effort
tends to sustain focus of attention.
Pashler then cites a 1986 work of Vidulich and Wickens,
which shows that increasing the rate at which events occur in
experimenter-paced tasks often increase effort ratings without affecting
performance. This is to be expected from QTOC by virtue of the
fact speeding up the task requires speeding up the mental processing,
and effort is needed to do that: the non effect on
performance should hold provided the task is simple enough so that good
performance does not require an extended period of focussed attention.
A 1988 work of Yeh and Wickens shows that if the task
difficulty is increased to a point that causes poor performance then
effortfulness ratings generally increases. QTOC suggest that poor performance
would lead to the need for a greater focus of conscious attention
which would require more effort.
Vidulich and Wickens (1986) find
that increasing incentives often increases workload ratings and
performance. This would be expected from QTOC if the task is
such that performance is enhanced by conscious focus of attention:
incentives would then lead to effort to perform better by increasing focus of
attention. [Again, a more detailed examination of the experimental
findings is needed here]
The next section of Pashler's book is entitled
"Subjective Effort: Causes and Consequences?"
He asks why people find effort taxing, and considers that people's aversion
to exerting effort may be that it consumes energy, which from an evolutionary
perspective, is better not squandered. However, the biophysical fact seems
to be that mental effort does not markedly increase energy consumption.
Increasing the rapidity of mental events should not increase
energy consumption significantly.
Pashler cites a puzzling phenomena discovered and confirmed at the end
of the nineteenth century: "Almost all mental tasks reduce the maximum
[physical] force [that one can exert], often by as much as 50 percent.
This puzzling phenomena remains unexplained."
But according to QTOC the maximum effort increases to a maximum
the number of brain events per second. This has the effect of
maximally focussing attention on the attended task of producing force
which has the effect of maximally controlling brain process
in such a way as to tend to produce the maximum physical force. If
attention is divided, then the action on the brain produced by the mental
effort to produce the physical force is correspondingly reduced, and this
reduced controlling action upon the brain will reduce the amount of physical
force applied. This heretofore unexplained phenomena is thus an automatic
consequence of QTOC.
This brief survey of the work on attention and effort described by
Pashler, while still a work in progress, suggests that the natural
quantum explanation of the features of conscious phenomena observed
by Wm. James may extend to the more subtle features revealed by new data.
\noindent {\bf References}
1. N. Bohr, ``Atomic Theory and the Description of Nature'',
Cambridge, CUP. (1934)
2. W. Heisenberg, ``The Representation of Nature in Contemporary
Physics'', Deadalus, {\bf 87}, 95-108 (1958)
3. N. Bohr, ``Atomic Physics and Human Knowledge'', Wiley, New York,
(1958) p.72
4. Max Tegmark, ``The Importance of Quantum Decoherence in Brain
Process,'' Phys. Rev E, {\bf 61}, 4194-4206 (2000).
5. H.P. Stapp, ``From Einstein Nonlocality to Von Neumann Reality,''\\
http://www-physics.lbl.gov/$\sim$stapp/stappfiles.html\\
quant-ph/0003064
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
8. Harold Pashler, ``The Psychology of Attention'',\\
MIT Press, Cambridge MA (1998)
\end{document}