**4. THE OBSERVER.**

From the time of Isaac Newton to the beginning of the twentieth century
science relegated consciousness to the role of passive viewer: our
thoughts, ideas, and feelings were treated as impotent bystanders to a
march of events controlled wholly by contact interactions between tiny
mechanical elements. Conscious experiences, insofar as they had any
influences at all on what happens in the world, were believed to be
completely determined by the motions of miniscule entities, and the
behaviors of these minute parts were assumed to be fixed by laws that
acted exclusively at the microscopic level. Hence the *idea-like* and
*felt* realities that make up our streams of conscious thoughts were
regarded as redundant, and were denied fundamental status in the basic
theory of nature.

The revolutionary act of the founders of quantum mechanics was to bring
conscious human experiences into the basic theory of physics in a
fundamental way. In the words of Niels Bohr the key innovation was to
recognize that \"in the drama of existence we ourselves are both actors
and spectators.\" \[Bohr, Essays 1958/1962 on Atomic Physics and Human
Knowledge\]. After two hundred years of neglect, our thoughts were
suddenly thrust into the limelight. This was an astonishing reversal of
precedent because the enormous successes of the prior physics was due in
large measure to the policy of keeping idea-like qualities out.

What sort of crises could have forced scientists to this wholesale
revision of their idea of the role of mind in their description of
Nature? The answer is the discovery and integration into physics of the
\"quantum of action.\'\' This property of matter was discovered and
measured in 1900 by Max Planck, and its measured value is called
\"Planck\'s Constant.\" It is one of three absolute numbers that are
built into the fundamental fabric of the physical universe. The other
two are the gravitational constant, which fixes the strength of the
force that pulls every bit of matter in the universe toward every other
bit, and the speed of light, which controls the response of every
particle to this force, and to every other force. The integration into
physics of each of these three basic quantities generated monumental
shifts in our conception of nature.

Isaac Newton discovered the gravitational constant, which linked our
understanding of celestial and terrestrial dynamics. It connected the
motions of the planets and their moons to the trajectories of cannon
balls here on earth, and to the rising and falling of the tides. Insofar
as his laws are complete the *entire physical universe* is governed by
mathematical equations that link every bit of matter to every other bit,
and that moreover fix the complete course history for all times from
conditions prevailing in the primordial past.

Einstein recognized that the \"speed of light\" is not just the rate of
propagation of some special kind of wave-like disturbance, namely
\"light\". It is rather a fundamental number that enters into the
equations of motion of every kind of material substance, and that, among
other things, prevents any piece of matter from traveling faster than
this universal limiting value. Like Newton\'s gravitational constant it
is a number that enters ubiquitously into the basic structure of Nature.
But important as the effects of two quantities are, they are, in terms
of profundity, like child\'s play compare to the consequences of
Planck\'s discovery.

Planck\'s \"quantum of action\" revealed itself first in the study of
light, or electromagnetic radiation. The radiant energy emerging from a
tiny hole in a heated hollow container can be decomposed into its
various frequency components. Classical nineteenth century physics gave
a clean prediction about how that energy is should be distributed among
the frequencies, but the empirical facts did not fit that theory.
Eventually, Planck discovered that the correct formula could be obtained
by assuming that the energy was concentrated in finite packets, with the
amount of energy in each such unit being directly proportional to the
frequency of the radiation that was carrying it. The ratio of energy to
frequency is called \"Planck\'s constant\". Its value is extremely small
on the scale of normal human activity, but becomes significant when we
come to the behavior of the atomic particles and fields out of which our
bodies and brains and all large physical objects are made.

It took twenty-five years for Planck\'s \"quantum of action\" to be
integrated coherently into physics. During that interval from 1900 to
1925 many experiments were performed on atomic particles and it was
repeatedly found that classical laws did not work. They gave well
defined predictions that were contradicted by the empirical facts. But
it was clear that that all of these departures of fact from theory were
linked to Planck\'s constant.

Heisenberg finally discovered in 1925 the completely amazing and wholly
unsuspected and unprecedented solution to the puzzle of the failure of
the classical laws: the quantities that classical physical theory was
based upon, and which were thought to be numbers, *are not numbers at
all*. Ordinary numbers, such as 2 and 3, have the property that the
product of any two of them does not depend on the order of the factors:
2 times 3 is the same as 3 times 2. But Heisenberg discovered that one
could get the correct answers out of the old classical laws if one
decreed that the *order in which one multiplies* certain quantities
matters!

This \"solution\" may sound absurd or insane. But mathematicians had
already discovered that completely coherent and logically consistent
mathematical structures exists in which the order in which one
multiplies quantities together matters. Ordinary numbers are just a very
special case in which A times B happens to be the same as B times A.
There is no logical reason why Nature should not exploit the more
general case, and there is no compelling reason why our physical
theories must be based exclusively on ordinary numbers. Quantum theory
exploits the more general logical possibility.

An example may be helpful. In classical physics the center-point of each
object has, at each instant, a well defined location, which can be
specified by giving its three coordinates *(x, y, z)* relative to some
coordinate system. For example, the location of a spider dangling in a
room can be specified by letting *z* be its distance from the floor, and
letting *x* and y be its distances from two intersecting walls.
Similarly, the *velocity* of that dangling spider as drops to the floor,
blown by gust of wind, can be specified by giving *the rate of change*
of these three coordinates (x, y, z). If each of the three numbers that
together specify the velocity are multiplied by the weight (=mass) of
the spider, then one gets three numbers, say (p, q, r), that define the
\"momentum\" of the spider.

Now in classical mechanics the symbols x and p described above both
represent *numbers*: the symbol *x* represents the distance of the
spider from the first wall, measured in some appropriate units, say
inches; and the symbol p likewise represents some *number* connected to
the velocity and weight of the spider. Because *x* and *p* both
represent just ordinary *numbers*, the product x times p is the same as
p times x, as we all learned in school. But Heisenberg\'s analysis
showed that in order to make the formulas of classical physics describe
quantum phenomena, x times p must be different from p times x. Moreover,
he found that the difference between x times p and p times x must be
Planck\'s constant. \[Actually, the difference is Planck\'s constant
multiplied by the imaginary unit *i,* which is a number such that *i*
times *i* is minus one.\] Thus quantum theory was born by recognizing,
or declaring, that the symbols used in classical physical theory to
represent ordinary numbers actually represented mathematical objects
such that their ordering in a product was important. The procedure of
creating the mathematical structure of quantum mechanics from classical
physics by replacing the ordinary numbers by these more complex objects
is called \"quantization.\" That process is an essentially
straightforward mathematical operation, but it needed to tied in some
well-defined way to empirical data before becoming part of science.
Establishing this link involves not just mathematics, in a narrow sense,
but involves also philosophy, in a broad sense.

By the year 1900 scientists believed that they had certainly discovered
the nature of the fabric of reality into which our experiences are
woven. External physical reality was understood to be composed of moving
atomic particles and changing physical fields. The classical laws
governing the behavior of these physical realities had been proposed by
Isaac Newton, James Clerk Maxwell, and Albert Einstein. What Heisenberg
found out was that in order to accommodate phenomena in which the value
of Planck\'s quantum of action is important the symbols, such as *x* and
*p,* that appear in the earlier theory have to be replaced by
mathematical objects such that their ordering in a series of factors
matters. What this mathematical change does*, conceptually*, is to
convert the conception of a particle as a minute entity into the
conception of a \"particle\" as an extended cloud-like structure.

Physicists had, for more than two hundred years, imagined Nature to be
composed, at least in part, out of entities resembling miniature
planets. This idea of minute physical entities had become so deeply
entrenched the psyches of scientists that it had acquired almost the
status of an article of faith: if you do not belief it your not a
scientist. But Nature, as she now reveals herself to us through our
observations and our mathematics, appears to be made out of a very
different kind of stuff. Careful analysis shows that atomic particles
can never reveal themselves to be the tiny moving objects that they had
been imagined to be since the time of Isaac Newton. Nor is there any
reason to believe that such tiny objects exist at all. Each
\"particle\", insofar as we can ever know it, may be associated with a
particular mass (e.g., the mass of an electron) and a particular charge
(e.g., the charge of the electron), but there is no evidence that it has
a particular location. All the empirical evidence is most parsimoniously
represented by taking each atomic particle to be a cloud-like structure
that has a strong proclivity to spread out over ever-larger regions.

However, this diffusing tendency of the \"clouds\" does not proceed
unchecked forever. There is a *counter process*, which consists of a
sequence of \"quantum jumps.\" These events are sudden collapses of the
cloud. At one moment the form may extend over miles, but an instant
later it is reduced to the size of a speck. Yet how can we make good
scientific sense out of such a crazy idea of how the world behaves?

Einstein described a simple situation that illustrates the puzzling
character of these quantum jumps. Suppose a radioactive atom is placed
in a detecting device that responds to the decay of this atom by sending
an electrical pulse to a recording instrument that draws a line on a
moving scroll. A blip in this line will indicate the time at which the
electrical signal arrives. Next, suppose some scientists are observing
the instrument and reporting to each other where the blip is located on
the scroll. What we know is that these observers will more or less agree
amongst themselves as to the position of the blip. But quantum theory
has stringent laws that govern in principle the behavior of all physical
systems. If one applies these rules to the entire system under
consideration here, which consists of the radioactive atom, the
detecting device, the electrical pulse, the recording instrument, the
bodies and brains of the human observers, and all other physical systems
that interact with them, then one arrives at a contradiction. What we
know is that the blip seen by the observers occurs at a fairly definite
location. But according to the quantum laws the full physical system
will be a smeared out continuum of possible worlds of the kind that
occurred in classical physics. In each of these classical-type worlds
the blip will occur at some particular location on the scroll, and all
of the observers will be reporting to each other that they see the blip
at that location. However, the full cloud-like quantum state will
include, for each of the infinity of possible locations of the blip on
the scroll, possible worlds in which the blip occurs at that location,
and in which all of the observers report seeing the blip occurring at
that particular location.

In short, the quantum law, or rule, that governs the behavior of matter
generates a whole continuum of possible worlds of the kind that appear
in our streams of conscious experiences. The world that you experience
is just one tiny slice of the full world generated by the quantum laws
obtained by incorporating the correct measured value of Planck\'s
constant into the otherwise incorrect laws of classical physics.

This mismatch, which lies at the central core of quantum theory, is a
discord between the two distinct parts of science, the theoretical and
the empirical: it is a disparity between theory and (experienced) fact.
These two interrelated aspects of science are extremely different in
character. Each *fact* comes as a chunk of somebody\'s experience. But
these disjoint chunks are related to each other. At one moment you see a
chair, then look away. Upon looking back you see a chair that resembles
the one you saw before. You were alone in the room, hence no continuous
human experience bridges the gap between these experiential moments. Yet
the two experiences are obviously linked together by something.

How do we human beings, scientist and nonscientist alike, deal with the
manifest linkages between the disconnected perceptual facts? The answer
is this: we concoct *theories*! We create ideas about persisting
realities that exist even when no one is watching them, and that bind
the disjoint facts together. Our physical theories are conceptual
frameworks that we create for the purpose not only of organizing our
perceptual experiences, but also of permitting us even to have
understandable and describable experiences. We need at least a
rudimentary \"theory of reality\" even to grasp and describe the idea
that some piece of apparatus has been placed in a certain location and
is, itself, behaving in a certain way. As Niels Bohr succinctly puts it:
\"The task of science is both to extend the range of our experience and
reduce it to order.\'\' \[N. Bohr. Atomic Physics and Human knowledge,
p.1 \].

The unique quantum laws produced by the quantization procedure make
predictions about empirical data that are accurate to as much as one
part in a hundred million, and they correctly describe various features
of the behavior of systems of billions of particles. But Einstein\'s
example shows that these quantum laws of motion lead also to cloud-like
physical states that are grossly discordant with the more narrowly
defined character of our actual experiences.

You might think that this huge conflict between the mathematical theory
and the empirical facts would render the theory false and useless. But
the amazing thing is that the creators of quantum theory found that *all
of the successes of classical physics* and a great deal more could be
explained, without any contradiction ever arising, by adopting the
following dictum: assume that the quantum generalization of the old
classical laws do indeed hold, but if they lead to a physical state that
disagrees with your empirical observation then simply discard the part
of that (mathematically computed) state that disagrees with your
observations, and keep the rest. This sudden resetting of the physical
state is the \"quantum jump.\" By itself it would yield nothing. But it
is accompanied by a natural statistical law, which will be described
later, that produces all of the wondrous results just described.

How can a theory of this kind make sense? Well, notice that you,
yourself, like all of us, are continually creating, on the basis of the
best information and ideas available to you a theoretical image of the
physical world around you: you have an idea about the status of all
sorts of things that you are not currently experiencing. Each time you
gain information you revise that theoretical picture to fit the newly
experienced facts.

Quantum theory instructs the scientist to do the same. That simple
dictum (revise your theoretical picture of the world to fit the
empirical facts), together with its statistical partner, produces not
only incredibly accurate predictions, but every successful result of the
earlier classical physics, and all of the thousands of successes of
quantum theory where classical physics fails. These impressive results
are achieved by simply allowing the beautiful, internally consistent,
and unique quantized version of the old classical laws to hold whenever
we are not actually acquiring knowledge about a physical system, but
incorporating promptly any knowledge we acquire. The close connection
maintained in this way between what the mathematical description
represents and what we empirically know underlies Heisenberg\'s
assertion that the quantum mathematics \`\`represents no longer the
behavior of particles but rather our knowledge of this behavior.\"

The shocker, however, is that Bohr and the other founders have argued
persuasively that *no other description of nature in terms of its atomic
constituents can be more complete than this one,* in the scientific
sense of telling us more about relationships between human experiences.
That is, this theory, constructed by incorporating the measured value of
Planck\'s constant into the old and incorrect classical laws, appears to
tell us everything that a basic physical theory could ever tell us about
connections between empirical facts.

This claim of scientific completeness, made by Heisenberg, Bohr, and the
other founders of quantum theory, was disputed by Einstein, who tried
repeatedly to devise a counter-example. But the quantum theorists shot
down every try. Thus it does indeed seem to be true that this
fantastically coherent quantization of the older laws generates
*everything that is knowable about reality*: this quantum description of
nature, crazy jumps and all, appears to be, from a narrow scientific
point of view, our best picture of nature. Any attempt to add something
more may please some philosophers, but carries us outside of science,
regarded simply as a tool for \"expanding our experience and reducing it
to order.\"

This apparent scientific completeness of quantum theory, together with
the fact that the \"quantization\" procedure totally *eliminates* the
classically conceived tiny entities, and replaces them by cloud-like
forms, make plausible the conclusion that there simply are no
*classical-type or quasi-classical-type* realities lying behind our
thoughts, and that searching for them is a futile endeavor. The
presumption that such realities exist is therefore a gross philosophical
blunder. There is absolutely no empirical evidence that rationally
supports the notion that there is a physical reality out there that is
better defined than what quantum theory allows. The assumption that such
a quasi-classical type reality exists is not justified by the scientific
evidence, and is thus likely to produce a conception of both nature and
human beings that is fundamentally incorrect.

But let us be clear about one thing: although quantum theory is an
endeavor to rationally order the empirical facts, and is therefore
erected upon human experience, it does not assert that our thoughts are
the *only* realities, and matter naught but an invention of the mind.
The founders did not espouse the philosophy of idealism. (Everything is
made out of ideas alone: \"To be is to be perceived.\") Their position
was the more conservative one that science is about what we can know,
and that a physical theory must be judged not by concordance with
intuition, but rather by its rational coherence, its capacity to
accommodate the known facts, and its power to make reliable predictions
about future experiences. This view liberates theoretical creativity: it
allows science, unfettered by ancient prejudice and fallible intuition,
to construct a practically useful and *empirically based new idea of the
nature of reality.*

Revamping the physics and the philosophy in this radical way did not
satisfy everyone, Einstein and Schroedinger being the most notable
hold-outs. But it did allow the scientists who accepted it to get on
with the business of developing, testing, and using this immensely
successful theory.

I have stressed that the founders of quantum theory brought human
consciousness into basic theory of nature. But the really essential
point is that your mental life inserted in two distinctly different
ways. The first is as a passive stream of conscious thoughts that
constitutes a growing reservoir of knowledge: each waking moment adds
something new to what you knew before. The second is as an active agent
endowed with a free will that can influence both how your body moves and
how your thoughts unfold. Explaining and exploring that active aspect is
the purpose of this book. But to understand yourself as participant you
must also appreciate yourself as the intertwined expanding collection of
knowings upon which your actions are based.

As might be expected, this radical restructuring of physics is not
achieved without some rearrangements of old ideas. For example, \"The
Observer\" as understood in the original \"Copenhagen\" formulation of
quantum theory differs from what one would normally mean by this term.
In particular, it involves an extension of the human observer outside
his physical body. Bohr mentioned several times the example of a man
with a cane: if he holds the cane loosely he feels himself to extend
only to his hand. But if he holds the cane firmly then the outer world
seems to begin at the tip of his probing cane.

In analogy, \"The Quantum Observer\" is considered to include not only
the body and mind of the experimenter himself, but also the measuring
devices that he uses to probe what is outside that extended \"self\".
Thus the world is imagined to be cleaved into two parts, which are
described in different ways. The outer \"observed system\" is described
in terms of the quantum mathematics, whereas the inner \"observing
system\'\' is described as a collection of empirical (i.e., phenomenal
or experiential) facts. This way of dividing the world implements the
point, stressed already above, that quantum physics \-\-- like all of
science \-\-- rests on two disparate kinds of descriptions, the first
being of conscious experiences that we can record, remember, and
communicate to our colleagues, and which form the empirical database,
and the second being of a theoretical structure that we have invented
for the purpose of comprehending the structure of our experience.

Copenhagen quantum theory regards the measuring instruments as part of
the observer because these devices are described not in terms of their
atomic constituents but rather in terms of our conscious knowings. Bohr
repeatedly points to this key feature of quantum theory, in statements
such as:

\"The decisive point is that the description of the experimental
arrangement and the recording of the observations must be given in plain
language, suitably refined by the usual terminology. This is a simple
logical demand, since by the word \`experiment\' we can only mean a
procedure regarding which we are able to communicate to others what we
have done and what we have learnt.\" (Essays 1958/1962....p.3)

I have described here the basic ideas of the Copenhagen approach to
physics, and explained its essential reliance upon the experiences of
the observers. This philosophical framework is conceptual container for
a rigid mathematical structure, and this receptacle, or mold, was shaped
in large part by the rigid form that it was designed to hold. That
uncompromising structure, which was generated by incorporating Planck\'s
quantum of action into the old pre-quantum laws of physics, captures and
reveals aspects of nature that supercede and negate all the scientific
ideas about the fundamental nature of reality that existed at the dawn
of the twentieth century. So having now described the philosophical
environment that was created to cradle the new mathematics, and to
connect it to empirical data, we are in a good position to see what that
new mathematical structure is like. The ideas are basically simple, once
old prejudices and obscuring jargon are stripped away. With the correct
tools then in hand, we shall be able to describe the mathematical rules
that allow your thoughts, feelings, and efforts to influence your mental
and physical behavior.

**5. THE UNSEEN.**

Quantum theory represents our knowledge about the unseen system being
probed by means of a mathematical structure called the quantum state.
This state normally evolves continuously in accordance with a
deterministic law that is closely connected to the \"laws of nature\"
used in classical physics. However, at certain instants this orderly
progression is suddenly interrupted by an abrupt \"quantum jump\". Such
a jump occurs each time one of the observers gains new knowledge: the
jump brings the quantum state into concordance with the new state of our
knowledge. Thus the quantum state of the system being examined
represents always the evolving knowledge of the community of
communicating observers.

But how can a mathematically described state represent human knowledge?.
Our knowledge seems to be an ephemeral and ineffable vagary, whereas the
mathematically described states of quantum theory are precise structures
that allow empirically observed numbers to be computed to an accuracy of
one part in a hundred million.

To explain this connection I need to introduce two mathematical ideas:
\"Hilbert space\", and \"projection operator\". These names may sound
intimidating, but the ideas are basically simple, and understanding them
will allow you to grasp the essence of quantum theory.

A Hilbert space is a collection of *vectors*, and a vector is a
displacement by a specified amount in a specified direction.

Two vectors, A and B, can be added together to give a vector C, which is
formed by adding together the displacements A and B.

Consider, for example, the displacement from the corner of a room where
two walls and the floor meet to a point on one of these two walls. That
displacement is the sum of a vertical displacement up from the corner
plus a horizontal displacement along the wall that contains the point.

If the two vectors A and B that add to give C are perpendicular to each
other, as in this example, then the theorem of Pythagoras asserts that
the square of the length of A plus the square of the length of B equals
the square of the length of C.

This celebrated theorem is tied to the probability rules of quantum
theory: If C is a vector of length one (i.e., unity) and A and B are two
perpendicular vectors that sum to C, then the square of the length of A
plus the square of the length of B is unity (i.e., one). The two
perpendicular vectors will correspond to two *alternative possible
outcomes of a probing action*, and the square of the length of A will be
the probability for the event associated with A to occur, and the square
of the length of B will be the probability for the event associated with
B to occur. The sum of these two probabilities is unity by virtue of the
theorem. This accords with the fact that the probabilities associated
with alternative possibilities must sum to unity.

In the example of the point on the wall, the space of vectors is
two-dimensional: any point on the wall can be reached from the corner by
a sum of just two displacements, one in each of the two pre-specified
perpendicular directions, vertical and horizontal. We can also easily
visualize displacements in a three-dimensional space. But it is possible
to consider mathematically an N-dimensional vector space in which there
are exactly N mutually perpendicular directions, and each vector in the
space is a sum of N vectors, one directed along each of these N
directions. We allow null displacements and also negative displacements,
which are the same as positive displacements in the reverse direction.

A set of N vectors, each perpendicular to every other one, is not easy
to visualize, geometrically, for large N. But if one uses an algebraic
approach in terms of sets of numbers, then the examples of vector spaces
in one, two, and three dimensions are easily generalized to spaces of
arbitrarily large but finite dimension N. With a little more effort one
can even go to the case where N is infinite. Hilbert spaces include the
infinite-N cases, but that is a technical matter that need not concern
us here. It will be enough to think of simple cases where N in finite.

If a vector V is composed of a sum of N perpendicular vectors then a
generalization of the theorem of Pythagoras shows that the square of the
length of V is the sum of the squares of the lengths of these N mutually
perpendicular vectors that add up to form V.

The second important concept is the idea of a projection operator. A
*projection operator* P acts on a vector V to give a new vector PV. The
action of P eliminates a specified subset of a set of perpendicular
vectors that add up to give the vector V upon which it acts, but leaves
unaffected the remaining vectors in the sum. Thus, for example, the
vector V from the corner of a room to any point in the interior of the
room would be converted by a certain projection operator P to the vector
PV that is the displacement from the corner to the point on the floor
that lies directly under that point in the room: the vertical vector is
eliminated by the action of this particular projection operator P.

That example is a very special case. For one thing the three
perpendicular vectors were very special, involving one vertical vector
and two particular horizontal ones. But one can imagine replacing the
room by a cubic box, and consider the infinity of ways that this box
could be oriented relative to the room. For each of these orientations
the three edges that meet at a corner define three perpendicular
directions. Then one can go from N = 3 to arbitrary N, and select any
subset of the set of N perpendicular directions to be the set that is
not set to zero. This obviously gives a huge set of logically possible
projection operators P.

For each projection operator P there is a unique complementary
projection operator P\' that does *not* set to zero exactly the subset
of the N perpendicular vectors that *is* set to zero by P. Thus for any
vector V, it is true that PV + P\'V = V. The vectors PV and P\'V are two
perpendicular vectors that sum to V.

Given this simple idea of a vector, and how a vector in an N dimensional
space can be considered to be a sum of a set of N vectors, each of which
is perpendicular to every other one, we can now state the basic idea of
quantum theory: Our knowledge about the unseen system, gleaned from
earlier experience about things we *can* see, is represented, under
certain ideal conditions, by a vector V of unit length. This vector
evolves under the action of a rule called \"the Schroedinger equation\",
which alters the direction that V points, but leaves its length
unchanged.

When the outcome of a probing action appears the vector V suddenly jumps
to the vector PV or to P\'V, where P is the projection operator
associated with the probing action, and P\' is the complementary
projection operator. The probability of V jumping to PV is the square of
the length of PV and the probability that the jump will be to P\'V is
the square of the length of P\'V. These two probabilities add to unity,
by virtue of the theorem of Pythagoras. This property matches the
property of probabilities that their sum over any set of alternative
possibilities must be unity.

The essential point here is that our knowledge of the unseen system
being probed can, according to quantum theory, be associated with a
vector V in a Hilbert space, and this association gives simple rules for
the probabilities for the alternative possible outcomes of our probing
action to appear, once the form of the projection operator P is known.

With this general picture in mind we can now return to the question of
how our knowledge is represented mathematically.

According to quantum theory the polarization of a photon is represented
by a vector V in the two dimensional space that is perpendicular to the
photon\'s line of flight. Suppose a photon is allowed to fall on a
crystal that splits the beam so that the part polarized along direction
A1 is deflected to a photon detector D1 and the part polarized in the
direction A2, perpendicular to A1, is deflected to a photon detector D2.
If the detectors are 100% efficient then one or the other of the two
detectors will fire, but not both.

In this example the probing action is associated with the projection
operator P such that the vector PV is directed along A1 and P\'V is
directed along A2. The vectors PV and P\'V represent the *alternative*
possible outcomes of the probing action. If the observer sees detector
D1 fire that he knows that the system being probed is in state PV; if he
sees detector D2 fire then he knows that the system being probed is in
the state P\'V. Thus quite accurate information about the new state of
the unseen system can be gleaned from the empirically discernible fact
of whether D1 fires or D2. Discarding the part of the state V=PV +P\'V
that is incompatible with the empirical fact that D1 fires and D2 does
not, or vice versa, is non problematic.

The successive action upon a vector V, first of the projection operator
P1 and then of the projection operator P2 is represented by V\'=P2 P1 V.
In general this vector is different from V\'\'=P1 P2 V: the order of in
which the two operators P1 and P2 appear matters! This can be easily
verified in simple cases, and is not mysterious.

For example, consider a space of two dimensions. Each nontrivial
projection operator P will be the projection onto some line through
\"the origin\" in space, which is the point corresponding to the unique
vector of length zero, the null vector. Suppose P1 is the projection
unto the horizontal line through the origin and P2 is the projection
onto the line that makes an angle of 45 degrees with the horizontal
line. Suppose V is a vector of length one (i.e., unity) that lies in the
*vertical* line through the origin. The P1 V is the null vector:
projecting a vertical vector onto a horizontal line gives the vector of
length zero. Then P2 P1 V is also the null vector. But a moments
reflection shows that P1 P2 V is not zero: it is a horizontal vector of
length 1/2. Thus the action of the product P2 P1 is different from the
action of P1 P2: the order in which the two factors P1 and P2 occur
matters.

All dependence in quantum theory of products on the order in which the
factors occur can be traced to this completely understandable dependence
of the product of projection operators upon the order in which they
occur.

I specified at the beginning that V represented of *our knowledge* of
the system being probed. Thus there is no problem with the fact that V
suddenly changes when an observer acquires new knowledge by seeing one
of the two the detectors, D1 or D2, fire and the other one not fire.
However, this facile way of speaking glosses over some deep problems.
This vector V seems to be connected more closely to the state of the
photon itself than to human consciousness. The very fact that the photon
could be represented by a vector, and that this vector should evolve
normally in accordance with the Schroedinger equation was a consequence
of incorporating Planck\'s constant into the equations of classical
physics. That quantization procedure converted the old classical-type of
reality into the new cloud-like (or vector-type) replacement. This
transformation seems to be an objective change, not related specifically
to human consciousness. Moreover, the stability of matter itself, and
the formation of the elements, are all understood in terms of these
quantum equations of motion. All of this structure and process predates
human existence. The original Copenhagen pragmatic way of understanding
the quantum mathematics, while tremendously useful as a stepping stone,
closes the door to any real understanding of the reality that replaces
the one that empirical phenomena has ruled out.

**6. THE PARTICIPANT**

Niels Bohr, the principal architect of quantum philosophy, wrote that we
cannot forget \"that in the drama of existence we ourselves are both
actors and spectators.\" \[Essays 1958/62 on Atomic Physics and Human
knowledge, p.15\] This comment succinctly captures a key point of
quantum theory: the human observers are no longer passive witnesses to a
flow of physical events that they cannot influence. They are essential
players in the action: their

\"free\" choices can influence the flow of physical events.

Quantum theory, in spite of its idealistic content, is formulated in a
completely realistic and practical way. It is structured around the
activities of human agents, who can probe nature in any of many possible
ways. Bohr emphasized this freedom of the experimenters in passages such
as:

\"The freedom of experimentation, presupposed in

classical physics, is of course retained and corresponds

to the free choice of experimental arrangement for which

the mathematical structure of the quantum mechanical

formalism offers the appropriate latitude.\"

The point here is that quantum theory, in its original form, is set up
in terms of an interaction between conscious human agents and an
invisible quantum system. As already discussed, that system is
represented in the theory by a vector in Hilbert space. This vector
usually evolves according to the quantum law of motion, the Schroedinger
equation. But to get information about that system the experimenter must
ask specific questions by setting up corresponding probes. For example,
the experimenter can orient the crystal in the photon experiment
described earlier in any way he chooses. Different choices correspond to
different choices of the two perpendicular directions A1 and A2.
Moreover, the human agent can choose to do or not do the experiment, or
to do it sooner or later. These choices are, according to our basic
physical theory, quantum theory, not fixed by any yet-known laws of
nature. Hence these choices are, in that specific sense, \"free
choices\".

We have now arrived at the crux of the matter! Quantum theory, in its
orthodox formulation, involves the human observer not merely as a
passive recipient of data, but also as an active agent that enters into
natural process in ways that are not controlled by the known physical
laws. His specific role, *as it appears in the world of experience*, is
to select which experiment is to be performed: i.e., to choose which
aspect of nature will be probed. This role, *as it appears in the
mathematics*, is to select some one single projection operator P. This P
is the projection operator such that the initial vector V will become PV
if nature delivers the affirmative answer \'yes\' to the posed question,
but will become the vector P\'V if nature delivers the negative answer
\'no\'. Here P\' is the projection operator that keeps the vectors that
P eliminates, and hence satisfies the condition PV+P\`V=V for all V.

\[Sometimes a complex question involving a combination of several
projection operators can be considered. But these can be regarded as a
sequence of individual P\'s, and I shall adopt this simpler way of
speaking.\]

The essential point here is that, according to orthodox ideas, nature\'s
process of generating the experiences that appear in our streams of
consciousness cannot proceed without particular questions being posed.
But then the two key questions become:

If the known physical laws do not determine the choices that need to be
made by the human agents in order for nature\'s process of generating
human experiences to proceed, then what sort of considerations do
influence or determine these choices?

and

What effect do these \"free\" choices have on the course of physical
events?

The evolution of the unseen system involves a sequence of questions with
Yes or No answers. Hence the interaction between nature and agent is
like a game of twenty-questions: the agent is free to choose a sequence
of Yes-No questions, and nature delivers an answer, Yes or No, to each,
in the form of an experience that answers the posed question. The choice
of question is represented within the mathematical description as the
selection of one projection operator P from among an infinite continuum
of possibilities. Each possible P corresponds to *particular* way of
orienting a set of N mutually perpendicular vectors, and, then, a
*particular* choice of which of these N vectors will be eliminated to
form PV. The process of making these *particular choices* of P\'s is not
specified by the Schroedinger equation, which controls the *continuous*
evolution of the physical state between observations, but neither the
agent\'s choices of the questions nor nature\'s choices of the answer.
In particular, a complete account of the dynamics needs to explain an
as-yet-unexplained process that picks a sequence of particular P\'s from
a continuum of possibilities. This selection process is associated,
according to orthodox quantum theory, with consciously made free choices
on the part of human beings. And, according to the quantum rules
themselves, these \"free choices\" can influence the course of physical
events, as we shall presently see.