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 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 the
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.
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.