Why is the fundamental theoretical framework of contemporary physics a probability calculus, and why are the events to which this assigns probabilities measurement outcomes?
It seems to me that all previous attempts to arrive at satisfactory answers have foundered on two assumptions that are at odds with the theory’s ontological implications. The first is the notion that physics can be neatly divided into kinematics, which concerns the description of a system at an instant of time, and dynamics, which concerns the evolution of a system from earlier to later times. We may call this notion “the principle (or paradigm) of evolution.”
By itself, this principle implies that time is completely differentiated — or else that the world is completely differentiated with respect to time. It permits us to think of time as a set of durationless instants. When combined with the special theory of relativity — specifically, its frame-dependent stratification of spacetime into hyperplanes of constant time — it implies that space, too, is completely differentiated — or else that the world is completely differentiated with respect to space. This permits us to think of it as a set of extensionless points. We may call these notions “the principle (or paradigm) of complete spatiotemporal differentiation.”
In keeping with the principle of evolution, the wave function ψ(x,t) is usually awarded primary status, while the propagator <B,tB|A,tA> is seen as playing a secondary role, notwithstanding that both encapsulate the same information. At the bottom of this partiality lies the erroneous notion that wave functions — and quantum states in general — are constructs that are meaningful even in the absence of measurements. If this were the case, measurements would merely contribute (or even merely appear to contribute) to determine quantum states, and if that were the case, quantum states would determine probabilities that are absolute. In reality, the probabilities determined by quantum states are always conditional on measurement outcomes.
The prevalence of these mistaken ideas can be traced back to two fortuitous cases of historical precedence: that of Schrödinger’s “wave mechanics” over Feynman’s propagator-based approach, and that of Kolmogorov’s formulation of probability theory over an axiomatic alternative developed by Rényi.[2,3] (Every result of Kolmogorov’s theory has a translation into Rényi’s, but whereas in Kolmogorov’s theory absolute probabilities have primacy over conditional ones, Rényi’s theory is based entirely on conditional probabilities.)
The principle of evolution is moreover at odds with the time-symmetry of quantum mechanics. For Born’s rule can be used not only to predict the probabilities of the possible outcomes of a later measurement on the basis of the actual outcome of an earlier measurement but also to retrodict the probabilities of the possible outcomes of an earlier measurement on the basis of the actual outcome of a later measurement. Quantum mechanics even allows us to assign probabilities symmetrically with respect to time, on the basis of both earlier and later outcomes. Positing an interpolating quantum state evolving from later to earlier times would therefore be just as legitimate — or, rather, illegitimate — as positing an interpolating quantum state evolving from earlier to later times.
The difference between the two uses of Born’s rule diminishes if we think in terms of the ensembles needed to (approximately) measure Born probabilities. Such ensembles can be postselected as well as preselected. To preselect an ensemble is to take into account only those instances of a measurement performed at the earlier time t1 that yield a particular outcome v. The preselected ensemble — an ensemble of identically prepared systems — serves to measure, as relative frequencies, the probabilities of the possible outcomes of a measurement performed at the later time t2. To postselect an ensemble is to take into account only those instances of a measurement performed at t2 that yield a particular outcome w. The postselected ensemble — an ensemble of identically retropared systems — serves to measure, as relative frequencies, the probabilities of the possible outcomes of a measurement performed at t1.
To obtain the rule to be used for assigning probabilities on the basis of both earlier and later outcomes, we assume that three measurements are performed at the respective times t1<t2<t3, that the measurement at t1 yields the outcome u, and that the measurement at t3 yields the outcome w. We then calculate the probability p(w,v|u) of obtaining v at t2 and w at t3, given that u has been obtained at t1. This is the product of two probabilities: the probability |<v|u>|2 of obtaining v given u and the probability |<w|v>|2 of obtaining w given v:
p(w,v|u) = |<w|v> <v|u>|2.
Next we calculate the probability pV(w|u) of obtaining w given u, and given that a measurement M with the possible outcome v is made at t2 (whose actual outcome is not taken into account). This probability is obtained by adding the probabilities p(w,vk|u), k = 1,2,3…, for all possible outcomes of M:
pV(w|u) = |<w|v1> <v1|u>|2 + |<w|v2> <v2|u>|2 + |<w|v3> <v3|u>|2 + ···
pV(w|u) = Σk |<w|vk> <vk|u>|2.
If we want to measure the probability p(v|w,u) of a particular outcome v of M, given both the initial outcome u and the final outcome w, we use an ensemble that is both pre- and postselected — in other words, an ensemble of physical systems that are both identically prepared and identically retropared. To create this ensemble, we take into account only those instances in which the initial measurement yields u and the final measurement yields w. We discard all runs in which either the initial measurement yields an outcome different from u and/or the final measurement yields an outcome different from w.
We gather from this that in order to calculate p(v|w,u), all we have to do is divide p(w,v|u) by pV(w|u). The result is known as the ABL rule:
p(v|w,u) = |<w|v> <v|u>|2 / Σk |<w|vk> <vk|u>|2.
An experiment with three holes
It will be instructive to apply the ABL rule to the following setup. Instead of a plate with two slits we have a plate we three holes, labeled A, B, and C. In front of the plate and equidistant from the holes there is a particle source (say, an electron gun G), and behind the plate, again equidistant from the holes, there is a particle detector D. Finally, somewhere between C and D there is a device that causes a phase shift by 180°, which means that the amplitude of the corresponding alternative gets multiplied by the complex number [1:180°] = −1.
An electron emerging from the holes is thus prepared “in” the state
u = (A + B + C)/√3
and retropared “in” the state
w = (A + B − C)/√3.
We now introduce an apparatus MA that can indicate whether or not a particle went through A. The possible outcomes of this measurement are represented by two projectors, one projecting into the 1-dimensional subspace containing the vector A, and one projecting into the 2-dimensional subspace containing the vectors B and C. If we now calculate the probability p(A|w,u) of finding that the particle has gone through A, given that it was prepared as described by u and retropared as described by w, we obtain the result p(A|w,u) = 1. Given the manner in which the particle is pre- and retropared, we are sure to find that it went through A.
Instead of MA, we next introduce an apparatus MB that can indicate whether or not a particle went through B. The possible outcomes of this measurement are represented by two different projectors, one projecting into the 1-dimensional subspace containing B, and one projecting into the 2-dimensional subspace containing A and C. If we now calculate the probability p(B|w,u) of finding that the particle has gone through B, given that it was prepared as described by u and retropared as described by w, we obtain the result p(B|w,u) = 1. Given the manner in which the particle is pre- and retropared, we are sure to find that it went through B.
Finally we use an apparatus that can indicate through which of the three holes a particle went. The possible outcomes of this measurement are represented by three projectors, one projecting into the 1-dimensional subspace containing A, one projecting into the 1-dimensional subspace containing B, and one projecting into the 1-dimensional subspace containing C. If we now calculate the probabilities p(A|w,u), p(B|w,u), and p(C|w,u), we obtain the results
p(A|w,u) = p(B|w,u) = p(C|w,u) = 1/3.
Time-symmetric probability assignments thus are (in general) contextual. If a particle launched at (or by) G and detected at (or by) D, the probability with which one finds that it went through A depends on the possible outcomes of the intermediate measurement. If the only possible outcomes are “through A” and “not through A,” then p(A|w,u) = 1. If the only possible outcomes are “through B” and “not through B,” then p(B|w,u) = 1. And if the possible outcomes are “through A,” “through B,” and “through C”, then the three outcomes are equally likely.
1. [↑] Kolmogorov, A.N. (1950). Foundations of the Theory of Probability, Chelsea Publishing Company.
2. [↑] Rényi, A. (1955). A new axiomatic theory of probability. Acta Mathematica Academia Scientiarum Hungaricae 6, 285–335.
3. [↑] Rényi, A. (1970). Foundations of Probability, Holden–Day.
4. [↑] Aharonov, Y. Bergmann, P.G., and Lebowitz, J.L. (1964). Time symmetry in the quantum process of measurement. Physical Review B 134, 1410–1416.