This minipaper presents an early formulation of the quantum measurement problem in terms of a conflict between subjective and objective probabilities. Everett ultimately provided a much clearer formulation of the measurement problem in terms of a conflict between different state attributions in the long thesis. The structure of the experimental setup, however, is similar. There are two versions of this minipaper. One, handwritten, contains the drawing (pg. 15) of the Wigner’s Friend experiment. The second version (presented here) was typed after being edited and condensed from the handwritten draft.a
Everett
Objective vs Subjective probability.b
Since the root of the controversy over the interpretation of the formalism of quantum mechanics lies in the interpretation of the probabilities given by the formalism, we must devote some time to discussing these interpretations. There are basically two types of probabilities, which may be called subjective and objective probabilities, respectively. A subjective probability refers to an estimate by a particular observer which is based upon incomplete information, and as such is not a property of the system being observed, but only of the state of information of the observer. An objective probability on the other hand is regarded as an intrinsic property of a system, i.e., to what might be called “really” random processes. To illustrate, we consider the following experiment: A deck of cards is shuffled, and one card is selected and placed face down upon the top of a table. An observer, A, is asked whether or not that card is the ace of spades, whereupon he would probably reply that the “probability” that it is the ace of spaces is 1/52. This probability would be a subjective probability, because it clearly refers to the state of information of the observer, and not to the system, namely the card, which is in actuality either the ace, or not the ace, and not a probability mixture. Nevertheless, A’s statement that the probability is 1/52 has meaning, since if the experiment were to be repeated a large number of times, and he were to state each time that the card on the table is the ace of spades, then he would be correct roughly 1/52 of the time. However, if there were another observer, B, present, who is informed each time of the color of the card, he would answer differently, either 0 or 1/26 depending upon the color, and he would also be correct in the same sense that A was. Still another observer, who caught a glimpse of the card each time it was placed on the table would have yet another answer, in which no probability was present at all. This illustrates that subjective probabilities are not invariant from observer to observer, due to differing states of information.
Since an objective probability is conceived to be a property of a system, and hence independent of states of information, it must be invariant from one observer to the next. That is if two observersc ascribe different probabilities to some aspect of the same system,d then at least one of these probabilities is subjective! One possible method of arriving at a probability which satisfies this criterion of invariance is as a limit of subjective probabilities as information becomes more and more perfect. This obviously satisfies the criterion, but can never be verified in practice. We cannot go further towards a positive definition of objective probabilities, but we can investigate the consistency of ascribing objective probabilities to events in certain situations, with regard to the above negative criterion. In particular we wish to investigate the limitations of interpreting the probabilities of quantum mechanics as objective probabilities, which are imposed by this criterion. We shall refer to this view as the objective interpretation of quantum mechanics.
As a starting point we might investigate the consequences of the following set of postulates:e
We shall now show that the above postulates are inconsistent by consideration of the following situation: We suppose that we have a system, S1, and a measuring device M1, for measuring some property of S1, which is connected to a recording device which will record the results of the measurement at a classical level, such as the position of a relay arm, and we assume that the measuring device is arranged to make the measurement automatically at some time. We further assume that the entire system, S2, consisting of S1, M1, and the recording device, is isolated from any external interactions.
Now, the system S1 possesses a state function ψ1 which gives the objective probability of the results of the measurement. (We shall also assume that ψ1 is not an eigenstate of the measurement, so that we shall have a probability which is neither 0 nor 1. But if we now consider the total system S2, before the measurement, it also possesses a state function ψ2. Furthermore, ψ2 at any later time is strictly determined by our initial ψ2 so long as there are no outside interactions, so that in particular ψ2 for some time after the measurement has taken place is strictly determined by its value before the measurement. We now consider what this later ψ2 may say about the configuration of the recording device. If it gives a probability mixture over the configurations, then clearly these probabilities are of the subjectiveg type, since they refer to something which actually exists in a pure state, because in reality the configuration of the recording device has already been determined. (Just as in the case of the man and the card). On the other hand if this later ψ2 gives the exact configuration of the recording device, then clearly the outcome of the measurement was determined before it took place, since the later ψ2 was strictly determined by the earlier, in which case the probability given by ψ1 was not objective. So that we see that in any case at least one of the probabilities was subjective, and the postulates are untenable.
There are several ways of removing this inconsistency, such as the following modifications:
a See also the discussion of the minipapers in the biographical introduction (chapter 2, pg. 17). A scan of the earlier handwritten version can be found online at UCIspace (pg. xi).
b Everett’s copy of this document had handwritten marginal notes from his advisor John Wheeler. These are included here as notes with minimal editing in their approximate location.
c Wheeler boxes the word “observers” and writes in the margin: “distinguish observation and observer”.
d Wheeler writes in the margin: “either/or taken too strongly”.
e Postulates 1 and 2 roughly correspond to von Neumann’s (1955) Processes 1 and 2, respectively. See the conceptual introduction (chapter 3, pgs. 28–29) for a discussion of von Neumann’s postulates.
f Wheeler writes in the margin: “by whom”.
g Wheeler writes in the margin: “Orders of mag of time Amplif’n means not pure”.
h Although Everett’s language of pure and mixed states is at least somewhat nonstandard, the thought behind this option is clear enough. He is describing a theory that provides conditions under which a stochastic collapse of the quantum mechanical state would occur. The standard collapse theory might qualify if it were completed in an appropriate manner.