The Theory of the Universal Wave Function

You and Aage were starting to say some ridiculous things about the implications of Quantum Mechanics and I was having a little fun joshing you and telling you some of the outrageous implications of what you said, and, as we had a little more sherry and got a little more potted... in the conversation. Don't you remember, Charlie? You were there! You had too much sherry.

Hugh Everett was not happy with the mainstream interpretation of Quantum Mechanics�the Copenhagen Interpretation; so in his PhD thesis he set out to offer an alternative. One that didn't have "outrageous implications". Though perhaps what he proposed might also be considered outrageous to some.

Everett first describes the two processes central to the more orthodox interpretations of quantum mechanics. The popular Copenhagen Interpretation is a slightly more restrictive form of these two processes.

Process 1: The discontinuous change brought about by the observation of a quantity with eigenstates ??,??,..., in which the state ???? will be changed to the state ?j with probability |(????,?j)|�.

Process 1 is what happens, in orthodox quantum mechanics, when a quantum system is observed�the result is randomly determined according to a probability distribution defined by the state of the system.

Process 2: The continuous, deterministic change of state of the (isolated) system with time according to a wave equation ????????/????t = Ut, where U is a linear operator.

Process 2 is what happens when a system is not observed. The system behaves as a deterministic wave.

Everett then tells a story, to show the paradox that arises from orthodox interpretations:

Isolated somewhere out in space is a room containing an observer, A, who is about to perform a measurement upon a system S. After performing his measurement he will record the result in his notebook. We assume that he knows the state function of S (perhaps as a result of previous measurement), and that it is not an eigenstate of the measurement he is about to perform. A, being an orthodox quantum theorist, then believes that the outcome of his measurement is undetermined and that the process is correctly described by Process 1.

In the meantime, however, there is another observer, B, outside the room, who is in possession of the state function of the entire room, including S, the measuring apparatus, and A, just prior to the measurement. B is only interested in what will be found in the notebook one week hence, so he computes the state function of the room for one week in the future according to Process 2. One week passes, and we find B still in possession of the state function of the room, which this equally orthodox quantum theorist believes to be a complete description of the room and its contents. If B's state function calculation tells beforehand exactly what is going to be in the notebook, then A is incorrect in his belief about the indeterminacy of the outcome of his measurement. We therefore assume that B's state function contains non-zero amplitudes over several of the notebook entries.

At this point, B opens the door to the room and looks at the notebook (performs his observation). Having observed the notebook entry, he turns to A and informs him in a patronizing manner that since his (B's) wave function just prior to his entry into the room, which he knows to have been a complete description of the room and its contents, had non-zero amplitude over other than the present result of the measurement, the result must have been decided only when B entered the room, so that A, his notebook entry, and his memory about what occurred one week ago had no independent objective existence until the intervention by B. In short, B implies that A owes his present objective existence to B's generous nature which compelled him to intervene on his behalf. However, to B's consternation, A does not react with anything like the respect and gratitude he should exhibit towards B, and at the end of a somewhat heated reply, in which A conveys in a colorful manner his opinion of B and his beliefs, he rudely punctures B's ego by observing that if B's view is correct, then he has no reason to feel complacent, since the whole present situation may have no objective existence, but may depend upon the future actions of yet another observer.

Deciding that the orthodox interpretation is untenable, Everett goes on the suggest some alternative interpretations:

Alternative 1: To postulate the existence of only one observer in the universe. This is the solipsist position, in which each of us must hold the view that he alone is the only valid observer, with the rest of the universe and its inhabitants obeying at all times Process 2 except when under his observation.

This view is quite consistent, but one must feel uneasy when, for example, writing textbooks on quantum mechanics, describing Process 1, for the consumption of other persons to whom it does not apply.

His first suggestion is effectively a joke. Though it could be true, no good can come from believing it to be true.

Alternative 2: To limit the applicability of quantum mechanics by asserting that the quantum mechanical description fails when applied to observers, or to measuring. apparatus, or more generally to systems approaching macroscopic size.

This is effectively saying that, due to the fact that there is an observer in the room, the second observer outside of the room cannot apply the laws of quantum mechanics to the room. Everett dismisses this:

For what n might a group of n particles be construed as forming a measuring device so that the quantum description fails?

He argues that it is impossible to draw a line between observers and non-observers without the line being completely arbitrary. Laws of nature that rely on arbitrary values are often wrong, therefore this interpretation is probably wrong.

Alternative 3: To admit the validity of the state function description, but to deny the possibility that B could ever be in possession of the state function of A + S. Thus one might argue that a determination of the state of A would constitute such a drastic intervention that A would cease to function as an observer.

Everett has two objections to this. Firstly that there is no intrinsic restriction on the knowability of any state function, so you would need additional axioms regarding the state function know-ability that currently have no evidence to support them. His second objection is that it is not actually vital that the state function is known�the paradox arises anyway: is not particularly relevant whether or not B actually knows the precise state function of A + S. If he merely believes that the system is described by a state function, which he does not presume to know, then the difficulty still exists. He must then believe that this state function changed deterministically, and hence that there was nothing probabilistic in A's determination.

So alternative 3 is not particularly useful.

Alternative 4: To abandon the position that the state function is a complete description of a system. The state function is to be regarded not as a description of a single system, but of an ensemble of systems, so that the probabilistic assertions arise naturally from the incompleteness of the description.

This is effectively saying that the randomness in quantum mechanics is not actually as a result of the state function, but due to hidden variables that make an otherwise deterministic system seem random. Hidden variables, however, are a poor explanation without evidence.

Alternative 5: To assume the universal validity of the quantum description, by the complete abandonment of Process 1. The general validity of pure wave mechanics, without any statistical assertions, is assumed for all physical systems, including observers and measuring apparata. Observation processes are to be described completely by the state function of the composite system which includes the observer and his object-system, and which at all times obeys the wave equation (Process 2).

Here Everett is saying that Process 1, which describes observations, never happens. Instead the entire system�the entire universe�can be described as a deterministic wave function. The remainder of the thesis shows that Alernative 5, the theory of the universal wave function, is self consistent, and that it is consistent with our own experiences�that it can adequately explain the real world. Vital to this, is explaining why Process 1 appears to happen, even though it does not.

To be continued...