TUTORIAL IN QUANTUM MECHANICS AND THE MIND-BRAIN CONNECTION

Henry P. Stapp

I have written extensively of the topic of this tutorial. But in order
to reach a broad audience I have in many of my more recent works
refrained from using equations. That approach makes those works
accessible in principle both to readers who are repelled by equations,
and also to quantum physicists who are sufficiently familiar with the
details of the quantum theory of measurement to be able to fill in for
themselves the omitted equations. However, that approach means also that
an important class of potential readers is not being optimally reached.
This class consists of persons having an engineering or mathematical
background, and possessing perhaps even some familiarity with classical
physics, but who have little knowledge of quantum mechanics.

I recently encountered two such readers.I gave them private tutorials,
which they found very helpful. This tutorial is essentially the one that
I gave to those two engineers.

Coordinate and Momentum Variables.

Consider first a physical system that consists of a single (rigid) bead
confined to move on a wire, so that the position of the center of the
bead at the instant of time t would be given by x(t), where x is a real
number. Suppose that there is no rotational motion, so that the function
x(t) specifies completely the positional state of the system at the
instant t.

But the bead can be moving, so the full state at time t is given by a
pair (x(t), v(t)), where v(t) is a real number that specifies the
velocity of the bead at time t..

For a simple (non-relativistic) case the product of the velocity v(t) of
the bead times the mass m of the bead is called the "momentum":
p(t)=mv(t). It is useful to characterize the state of the system at time
t by the pair (p(t), q(t)), where p is the "momentum", and in this case
the "coordinate" is q(t) = x(t).

The full physically described state of this system at the instant of
time t corresponds to a point (p(t), q(t)) in (p, q) space. This space
is called "phase space". The set of points {p(t), q(t)} for all t in the
range 0≤t≤1 constitutes the phase-space trajectory of the system during
this interval of time.

Diagram 1 A trajectory in phase space for a system described by one
coordinate variable q and one momentum variable p. (Vertical p axis:
Horizontal q axis: And a parabola facing right centered on the
horizontal axis)

In this simplest example the variables p and q are one-dimensional
variables, In general,

a classical system has many degrees of freedom, and p and q will be
variables in the corresponding many-dimensional space.

Classical Statistical Mechanics.

Quantum mechanics is closely connected, mathematically, to classical
\*statistical\* mechanics. In classical statistical mechanics "our
knowledge" of the system is represented by a probability density
function ρ(p.q).

Representation of an increment in knowledge.

Suppose our initial knowledge of the system is represented by one
particular probability density ρ(p.q). Suppose we then acquire the added
information that the system lies in region R of phase space, but acquire
no other information. Then the new state of knowledge is represented by
P(R) ρ(p.q), where the action of the operator P(R) on ρ(p.q) is to
"project" the function ρ(p.q) onto the region R. That "projection"
operation P(R) transforms to zero the value of ρ(p.q) at all points
outside region R, but leave unchanged the value of ρ(p.q) at all points
inside R.

Note that P(R) P(R) = P(R). This property entails the "eigenvalue"
property that if any ρ(p.q) is taken into some constant multiple of
itself by the action of P(R), then that constant C must be such that
C^2^ = C. But the only such numbers are zero and one.Thus the
eigenvalues of any operator P that satisfies P^2^ =P are zero and one.
Operators that satisfy this property P^2^ =P are called projection
operators.

The set of points (p,q) represent the set of possible physically defined
states of our system. Physically definable properties of the system are
properties that are defined as functions of p and q. The subset of
systems that possess some specific physically defined property X can be
characterized as set of point (p,q) that lie in some region (or subset)
R(X) of phase space. Thus a density function ρ(p.q) that represents a
statistical ensemble that possesses a physically definable property X is
an eigenfunction with eigenvalue one of a projection operator P(R(X))
corresponding to property X. If ρ(p.q) is an eigenfunction of P(R(X))
with eigenvalues is zero then property X is definitely false in the
ensemble specified by ρ(p.q).

Thus the two eigenvalues, one and zero, correspond to the two answers,
'Yes' and 'No' to the query "Does property X hold?" If ρ(p.q) represents
our knowledge of the system just before an observation/measurement that
reveals only that property X holds then the ρ(p.q) that represents our
knowledge just after the observation is P(R(X)) ρ(p.q), where P(R(X)) is
the projection operator that corresponds to our finding out that
property X holds.

The probability that we will find that property X holds is the integral
of P(R(X)) ρ(p.q)

over all (p, q) space divided by the integral of ρ(p.q) over all (p, q)
space, in case this latter ρ(p.q) has not already been normalized to
unity.

If we know that a certain measurement/observation on a system will give
either the answer that the system definitely does have property X or
definitely does not the property X, then our knowledge of the state of
the system after we know that the measurement has been performed, but
before we have observed the outcome is

ρ'(p.q) = P(R(X)) ρ(p.q) + P(R'(X)) ρ(p.q).,

where R'(X) is the complement (I-R(X)) of R(X), with I representing the
whole (p, q) space. Notice that ρ'(p.q) is the same as ρ(p.q). This
identity does not generally hold true in quantum mechanics.

Transition to Quantum Mechanics.

In quantum mechanics the states ρ and the projection operators P are
represented by matrices.

In classical mechanics the density function was a function in (p, q)
space and the projection operators were projections onto subsets.
Consequently, the order of application of two projection operators did
not matter: the projection operators all commuted with one another. But
if the projection operators are represented by matrices then the order
of applying projection operators can matter.

The law of multiplication of matrices asserts that if M is a matrix that
has matrix element

M~ij~ in the ith row and the jth column, and N is a matrix with
similarly labeled elements N~ij~ , then the matrix called MN has
elements (MN)~ij~ = ∑ M~ik~ N~kj~.where the sum is over the repeated
(inner) index k.

As the simplest example consider the case of 2x2 matrices. It is
convenient to introduce

the four Pauli matrices.

σ~0~ = 1 0 σ~1~ =1 0 σ~2~ = 0 1 σ~3~ = 0 - i

0 1 0 -1 1 0 i 0

The square of each Pauli matrix is the unit matrix 1= σ~0~ , and σ~1~
σ~2~ = iσ~3~ = - σ~2~ σ~3~ , together with its analogs under cyclic
permutations (1231).

Note that ½(1-σ~i~ ) is a projection operator for all i = 1, 2, or 3,
but these three operator do not commute.

If ρ represents our knowledge of system before a measurement that will
inform us, after our observation of the results, either that property X
definitely holds, or that property X definitely does not hold, then the
implementation of this measurement is represented by the action

ρ  ρ' = P ρ P + P' ρ P',

where P' =(1-P).

If the answer to the query "Does property X hold?" turns out to be
'Yes'.then

ρ' P ρ P.

If the answer to the query is 'No', then

Ρ'  P' ρ P'

That is, in quantum mechanics, the reduction to the 'Yes' outcome places
the 'Yes' projection operator P on both sides of the matrix ρ that
represents our prior knowledge, and similarly for 'No' and P'= (1-P).

By comparing ρ' to the identity

ρ = (1-P)ρ(1-P) + (1-P)ρP + Pρ(1-P) + PρP.

we see that, in contrast to the corresponding classical situation, ρ' is
not generally equal to ρ. The parts of the prior ρ that have a P on one
side but a P' on the other are absent.

The quantum counterpart of the integral over all (p, q) space is the
trace operation.

The trace of any matrix M is the sum of the diagonal elements M~ii~ of
M.

Notice that that, for any M and N,

Trace MN =∑~i~∑~k~ M~ik~ N~ki~ = Trace NM.

The probability the answer to the query is 'Yes' is

Prob (Yes) = Trace P ρ P/Trace ρ = Trace P ρ /Trace ρ

The probability that the answer is 'No' is

Prob (No) = Trace P' ρ P'/Trace ρ = Trace P' ρ / Trace ρ

The transformation from the first to the second forms of these two
equations

(i.e., the elimination of a P or a P') is achieve by using the identity

Trace MN = Trace NM to move that P or P' around to the front, where it
can be eliminated by using PP=P or P'P'=P'. The second form is similar
to the classical formula mentioned above, and ensures that Prob (Yes) +
Prob (No) = 1.

.

Measurement and Environment.

Suppose we put behind one slit of a double-slit experiment an apparatus
that has a movable element that is initially in a location close to the
beam that passes through this slit. Suppose that this element is very
light, and is such that if the beam particle passes through this slit,
then the element is moved far away from its initial location. We can
suppose that the beam particle is very heavy and that its motion will
not be significantly affected by the presence of the light element.

In the absence of the apparatus just described let the probability of
finding the beam particle in some small region of the screen is give by

Prob = P ρ P = (ψ~1+~ ψ~2~)^\*^ (ψ~1+~ ψ~2~),

where ψ~1~ represents the amplitude associated with the path that passes
through the first slit, and ψ~2~ the amplitude associated with the path
that passes through the second slit.

The apparatus will now be placed behind the second slit. Now we have a
system of two particles, the heavy beam particle and the light apparatus
particle. The ψ~1~ will now be multiplied by initial wave function φ~1~
of the apparatus particle, whereas the ψ~2~ will be multiplied by the
greatly displaced wave function φ~2~ of the apparatus particle. If no
measurement is performed on the apparatus particle then the associated
projection operator in φ space is unity. Then

Prob = Trace~φ~ (ψ~1~φ~1~ + ψ~2~φ~2~)^\*^(ψ~1~φ~1~ + ψ~2~φ~2~).

The Trace~φ~ instructs us to form the inner products of the φ^\*^
factors with the factors φ

These two functions are both normalized (because we have normalized the
initial Trace~φ~ to unity), and the assumed large displacement means
that their inner product is essentially zero. But then the effect of
taking this trace is to eliminate the interference term.

Prob = ψ~1~^\*^ ψ~1\ +~ ψ~2~^\*^ ψ~2.~

Thus the effect of allowing the beam to interact with the light movable
element of the apparatus, but *not* observing the effects of the beam on
that movable element, destroys the interference between the two beams.
If we later do observe the light moveable to be in one or the other of
these two far apart locations then we can say which slit the heavy
particle went. Still, without observing the apparatus particle we cannot
conclude that the beam particle went through only one slit. Because if
we bring the two parts φ~1~ and φ~2~

back together and perform a measurement whose observed outcome shows
them to in phase, then the interference between the two beams will again
be observable

Discussion.

The simple calculations given above are the basis of the quantum theory
of measurements. The way the theory is formulated in order to get
predictions about future possible increments of knowledge on the basis
of information provided by past knowledge, revolves around the idea of
human knowledge and specific human actions that are intended to produce
new increments of "our knowledge". Key parts of the theory involve the
reduction events described above, which are specified by human choices
of specific questions put to nature.

The theory described above of the interactions of systems with their
environments raises the question of whether these
measurement/observation reduction events ever really actually occur. Of
course, we seem to experience "facts": the definite resolutions one way
or the other of certain 'Yes-No' questions. .But scientists are
generally reluctant to put human beings in a really privileged place in
the general scheme of things, and the results exhibited above raise some
doubts about whether such reductions ever really occur.

But the direct interpretation of the non-emergence of facts in that
treatment is that facts do not emerge from a consideration of the
physical aspects of nature alone. In the quantum formalism facts emerge
only in association with the actions of a knowing agent!

Quantum theory provides a framework, tested in actual practice, for
introducing facts into our understanding of reality, and it does so only
in association with the physical actions of knowing agents

But I believe that no rationally coherent explanation of the occurrence
of facts concordant with quantum mechanics has been found that removes
from the dynamics knowledge carrying agents such as ourselves. I believe
a useful, and very possible the only useful, approach should build upon
the essential components of quantum theory as it is used successfully in
actual practice. This involves an acceptance of the key element of non
Neumann-Heisenberg quantum formalism, namely that the generation of
facts involves actions by agents that pose questions that nature answers
in accordance with the probability rules described above. I follow von
Neumann's idea of pushing the mind-matter interaction up to the level of
mind-brain interaction. This does not mean that the only allowed
mind-matter interactions in involve human beings, but it means that a
main topic of scientific interest is the human-brain case, where the
empirical data of science of is directly available on both sides of the
mind-brain dualistic description..