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.