The Dancing Wu Li Masters

by Gary Zukav

  If we consider light as a particle phenomenon, this paradox becomes graphic. Namely, how can one photon be broken down into a horizontally polarized component and a vertically polarized component? (By definition, it can’t.)

  This paradox is at the heart of the difference between quantum logic and classical logic. It is caused by our thought processes which are following the rules of classical logic. Our intellect tells us that what we are seeing is impossible (after all, a single photon must be polarized one way or the other). Nonetheless, whenever we insert a diagonal polarizer between a horizontal polarizer and a vertical polarizer, we see light where there was none before. Our eyes are ignorant of the fact that what they are seeing is “impossible.” That is because experience does not follow the rules of classical logic. It follows the rules of quantum logic.

  The thing-in-itselfness of diagonally polarized light reflects the true nature of experience. Our symbolic thought process imposes upon us the categories of “either-or.” It confronts us always with either this or that, or a mixture of this and that. It says that polarized light is either vertically polarized or horizontally polarized or a mixture of vertically and horizontally polarized light. Thus are the rules of classical logic, the rules of symbols. In the realm of experience, nothing is either this or that. There is always at least one more alternative, and often an unlimited number of them.

  Finkelstein put it this way in reference to quantum theory:

  There are no waves in the game. The equation that the game obeys is a wave equation, but there are no waves running around. (This is one of the mountains of quantum mechanics.) There are no particles running around either. What’s running around are quanta, the third alternative.10

  To be less abstract, imagine that we have two different pieces from a chess set, say a bishop and a pawn. If these macroscopic chess pieces followed the same rules as quantum phenomena, we would not be able to say that there is nothing between being either a bishop or a pawn. Between the extremes of “bishop” and “pawn” is a creature called a “bishawn.” A “bishawn” is neither a bishop nor a pawn nor is it half a bishop and half a pawn glued together. A “bishawn” is a separate thing-in-itself. It cannot be separated into its pawn component and its bishop component any more than a puppy which is half collie and half German shepherd can be separated into its collie “component” and its German shepherd “component.”

  There is more than one type of “bishawn” between the extremes of bishop and pawn. The bishawn that we have been describing is one-half part bishop and one-half part pawn. Another type of bishawn is one-third part bishop and two-thirds part pawn. Still another type of bishawn is three-fourths part bishop and one-fourth part pawn. In fact, for every possible proportion of parts bishop to parts pawn there exists a bishawn which is quite distinct from all the others.

  A “bishawn” is what physicists call a coherent superposition. A “superposition” is one thing (or more) imposed on another. A double exposure, the bane of careless photographers, is a superposition of one photograph on another. A coherent superposition, however, is not simply the superposition of one thing on another. A coherent superposition is a thing-in-itself which is as distinct from its components as its components are from each other.

  Diagonally polarized light is a coherent superposition of horizontally polarized light and vertically polarized light. Quantum physics abounds with coherent superpositions. In fact, coherent superpositions are at the heart of the mathematics of quantum mechanics. Wave functions are coherent superpositions.

  Every quantum mechanical experiment has an observed system. Every observed system has an associated wave function. The wave function of a particular observed system (like a photon) is the coherent superposition of all the possible results of an interaction between the observed system and a measuring system (like a photographic plate). The development in time of this coherent superposition of possibilities is described by Schrödinger’s wave equation. Using this equation, we can calculate the form of this thing-in-itself, this coherent superposition of possibilities which we call a wave function, for any given time. Having done that, we then can calculate the probability of each possibility contained in the wave function at that particular time. This gives us a probability function, which is not the same as a wave function, but is calculated from a wave function. In a nutshell, that is the mathematics of quantum physics.

  In other words, in the mathematical formulations of quantum theory nothing is either “this” or “that” with nothing in between. Graduate students in physics routinely learn the mathematical technique of superimposing every “this” on every “that” in such a way that the result is neither the original “this” nor the original “that,” but an entirely new thing called a coherent superposition of the two.

  According to Finkelstein, one of the major conceptual difficulties of quantum mechanics is the false idea that these wave functions (coherent superpositions) are real things which develop, collapse, etc. On the other hand, the idea that coherent superpositions are pure abstractions which represent nothing that we encounter in our daily lives also is incorrect. They reflect the nature of experience.

  How do coherent superpositions reflect experience? Pure experience is never restricted to merely two possibilities. Our conceptualization of a given situation may create the illusion that each dilemma has only two horns, but this illusion is caused by assuming that experience is bound by the same rules as symbols. In the world of symbols, everything is either this or that. In the world of experience there are more alternatives available.

  For example, consider the judge who must try his own son in a court of law. The law allows only two verdicts: “He is guilty” and “He is innocent.” For the judge, however, there is another possible verdict, namely, “He is my son.” The fact that we prohibit judges from trying cases in which they have a personal interest is a tacit admission that experience is not limited to the categorical alternatives of “guilty” and “innocent” (or “good” and “bad,” etc.). Only in the realm of symbols is the choice so clear.

  During the Lebanese civil war, a story goes, a visiting American was stopped by a group of masked gunmen. One wrong word could cost him his life.

  “Are you Christian or Moslem?” they asked.

  “I am a tourist!” he cried.

  The way that we pose our questions often illusorily limits our responses. In this case, the visitor’s fear for his life broke through this illusion. Similarly, the way that we think our thoughts illusorily limits us to a perspective of either/or. Experience itself is never so limited. There is always an alternative between every “this” and every “that.” The recognition of this quality of experience is an integral part of quantum logic.

  Physicists engage in a particular kind of dance which is foreign to most of us. Being around them for any length of time is like entering another culture. Within this culture every statement is subject to the challenge, “Prove it!”

  When we tell a friend, “I feel great this morning,” we do not expect him to say, “Prove it.” However, when a physicist says, “Experience is not bound by the same rules as symbols,” he invites a chorus of “Prove it’s. Until he can do this, he must preface his remarks with, “It is my opinion that…” Physicists are not very impressed with opinions. Unfortunately, this sometimes makes them narrow-minded to an extraordinary degree. If you are not willing to follow their dance, they won’t dance with you.

  Their dance requires a “proof” for every assertion. A “proof” does not verify that an assertion is “true” (that that is the way the world really is). A scientific “proof” is a mathematical demonstration that the assertion in question is logically consistent. In the realm of pure mathematics, an assertion may have no relevance to experience at all. Nonetheless, if it is accompanied by a self-consistent “proof” it is accepted. If it is not, it is rejected. The same is true of physics except that the science of physics imposes the additional requirement that the assertion relate to
physical reality.

  So much for the relationship between the “truth” of a scientific assertion and the nature of reality. There isn’t any. Scientific “truth” has nothing to do with “the way that reality really is.” A scientific theory is “true” if it is self-consistent and correctly correlates experience (predicts events). In short, when a scientist says that a theory is true, he means that it correctly correlates experience and, therefore, it is useful. If we substitute the word “useful” whenever we encounter the word “true,” physics appears in its proper perspective.

  Birkhoff and von Neumann created a “proof” that experience violates the rules of classical logic. This proof, of course, is embedded in experience. In particular, it is based upon what does and does not happen with various combinations of polarized light. Finkelstein uses a slightly modified version of Birkhoff and von Neumann’s original proof to demonstrate quantum logic.

  The first step of this proof is to experiment with all of the possible combinations of horizontally, vertically, and diagonally polarized light. In other words, the first step is to do what we have done already: discover which light emissions pass through which polarizers. Observe for yourself that light passes through two vertical polarizers, two horizontal polarizers, two diagonal polarizers, a diagonal and a horizontal polarizer, and a diagonal and a vertical polarizer. All of these combinations are called “allowed transitions” because they actually happen. Similarly, observe for yourself that light does not pass through a horizontal and a vertical polarizer, or any other combination of polarizers oriented at right angles to each other. These combinations are called “forbidden transitions” because they never happen.

  The second step of the proof is to make a table of this information called a transition table. A transition table looks like this:

  The row of letters on the left are emissions. An emission is just what it sounds like. In this case, an emission is a light wave that is emitted from a light bulb. The “)” sign to the right of a letter indicates an emission. For example, “H)” means horizontally polarized light emitting from a horizontal polarizer. The row of letters on the top are admissions. An admission is the reception of an emission. The “)” sign to the left of a letter indicates an admission. For example, “)H” means a horizontally polarized light wave reaching an eyeball.

  The zeros with the lines through them (pronounced “oh”) stand for the “null process.” The null process means that we decided to go to the movies today and not do the experiment. The null process stands for no emissions, nothing. The letter “I” stands for “identity process.” The identity process is a filter that passes everything. In other words, “I” tells us what kinds of polarized light pass through, say, an open window: namely, every kind.

  Two kinds of diagonally polarized light are included in the table to make it complete. The “D” represents light diagonally polarized to the left, and the “” represents light diagonally polarized to the right (or the other way round).

  To use the transition table we pick the type of emission in which we are interested and follow it across the table. For example, an emission of horizontally polarized light, H), will pass through another horizontal polarizer, so an “A,” for allowed, is placed in the square where the horizontally polarized emission line intersects the horizontally polarized admission column. Horizontally polarized light also passes through a diagonal polarizer tilted to the left,)D, a diagonal polarizer tilted to the right,), and an open window, I. An “A” is placed in each appropriate square.

  Notice that the square where the horizontally polarized emission line intersects the vertically polarized admission column is blank. This is because horizontally polarized light does not pass through a vertical polarizer. The blank squares show the forbidden transitions. All of the null process squares are blank because nothing happens if we don’t do the experiment. All of the “I” squares are marked “A” because every kind of light, polarized and otherwise, passes through an open window.

  The third step in the proof is to make a simple diagram of the information contained in the transition table. The diagram made from this particular transition table looks like this:

  This type of diagram is called a lattice. Mathematicians use lattices to show the ordering of events or elements. Lattices are similar to the genealogical trees that we construct when we research our family roots. The higher elements include the lower elements. The lines show who is connected to whom and through whom.

  A lattice is not exactly a family tree, but it shows the same kind of inclusive ordering. At the bottom is the null process. Nothing is below the null process since the null process represents no emissions of any kind. In the next level up are the various states of polarization. The elements at this level are called singlets. Singlets are the simplest statements that we can make about the polarization of a light wave. “This light is horizontally polarized,” is the most that we can say about the state of polarization, even though it doesn’t tell us anything else. It is a “maximal but incomplete description,” a limitation inherent in the use of language.

  The next level up contains the doublets. In this lattice there is only one doublet. Doublets comprise the next level of maximal but incomplete statements that we can make about the polarization of light in this simple experiment. Lattices representing more complex phenomena can have considerably more levels—triplets, quadruplets, etc. This lattice is the simplest of them all, but it graphically demonstrates the nature of quantum logic.

  First, notice that the doublet, I, contains four singlets. This is typical of quantum logic but an incomprehensible contradiction to classical logic wherein every doublet, by definition, contains only two singlets, no more and no less. Lattices are graphic demonstrations of the quantum postulate that there is always at least one alternative between every “this” and every “that.” In this case, two alternatives (“D” and “”) are represented. There are many more available alternatives that are not represented in this lattice. For example, the light in the lattice represented by the symbol is diagonally polarized at 45°, but we also can polarize light at 46°, 47°, 48½°, etc., and all of these states of polarization could be included in the doublet, I.

  In both classical logic and quantum logic a singlet can be represented by a dot. In classical logic a doublet is represented by two dots. In quantum logic, however, a doublet is represented by a line drawn between the two dots. All of the points on the line are included in the doublet—not only the two points that define it.

  Now let us return to the distributive law: “A, and B or C” equals “A and B, or A and C.” (The whole purpose of making a transition table was to make a lattice to use in disproving the distributive law.)

  Mathematicians use lattice diagrams to determine which elements in the lattice are connected and in what way.

  For example, to see how two elements in the lattice are connected by the word “and,” follow the lines leading from the elements in question down to a point where they both meet (which mathematicians call the “greatest lower bound”). If we are interested in “H and D,” we follow the lines downward from H and from D and find that they meet at ø. Therefore, the lattice tells us that “H and D” equals “ø.” If we are interested in “I and H,” we follow the line downward from the highest starting point on the lattice (I) and find that the lowest common point of I and H is H. Therefore, the lattice tells us that “I and H” equals “H.”

  To see how two elements in the lattice are connected by the word “or,” follow the lines leading from the elements in question up to a point where they both meet (which mathematicians call the “least upper bound”). For example, if we are interested in “H or V,” we follow the lines upward from H and from V and find that they meet at I. Therefore, the lattice tells us that “H or V” equals “I.” Similarly, to find “D or I,” we follow the lines upward to their highest common point, which is I. Therefore, the lattice tells us that “D or I” equals “I.”

  The rule i
s simple: “and” goes down, “or” goes up.

  Go down the lattice to find “and,” go up the lattice to find “or.”

  Now we come to the proof itself. The proof itself is considerably simpler than the preliminary explanations. The distributive law says that “A, and B or C” equals “A and B, or A and C.” To see whether this is true of experience or not we simply insert some of our actual states of polarization into the formula and solve it using the lattice method. For example, the distributive law says that “Horizontally polarized light and vertically polarized light or diagonally polarized light” equals “Horizontally polarized light and vertically polarized light, or horizontally polarized light and diagonally polarized light.” Using the abbreviations that we already have used, this is written: “H, and D or V” equals “H and D, or H and V.”

  Returning to the lattice, let us examine the left side of this statement first. Solving for “D or V,” we follow the lines on the lattice upward from D and from V to their highest common point (“or” goes up). They meet at I. Therefore, the lattice tells us that “D or V” equals “I.” Substituting “I” for the original “D or V,” we have left on this side of the statement “H and I.” Following the lines from H and from I downward on the lattice (“and” goes down), we find that their lowest common point is at H. Therefore, the lattice tells us that “H and I” equals “H.”

  In short:

  “H, and D or V” equals “H and D, or H and V”

  “H and I” equals “H and D, or H and V”

  “H” equals “H and D, or H and V”

  We solve the right side of this statement in the same way. Solving for “H and D,” we follow the lines on the lattice downward from H and from D to their lowest common point. They meet at ø. Therefore, the lattice tells us that “H and D” equals “ø.”