3 Strategy of interpretation

For those who adhere to the prin­ciple of evo­lu­tion, the piv­otal role played by mea­sure­ments in stan­dard for­mu­la­tions of quantum mechanics is an embar­rass­ment known as the “mea­sure­ment problem.” As an anony­mous ref­eree for a phi­los­ophy of sci­ence journal once put it to me, “to solve this problem means to design an inter­pre­ta­tion in which mea­sure­ment processes are not dif­ferent in prin­ciple from ordi­nary phys­ical inter­ac­tions.” How can reducing mea­sure­ments to “ordi­nary phys­ical inter­ac­tions” solve this problem, con­sid­ering that quantum mechanics describes “ordi­nary phys­ical inter­ac­tions” in terms of cor­re­la­tions between the prob­a­bil­i­ties of the pos­sible out­comes of mea­sure­ments per­formed on the inter­acting sys­tems? This kind of “solu­tion” merely sweeps the problem under the rug.

In actual fact, to solve the mea­sure­ment problem means to design an inter­pre­ta­tion in which the cen­tral role played by mea­sure­ments in stan­dard axiom­a­ti­za­tions of quantum mechanics is under­stood. But before there is any hope of under­standing it, the obvious must be acknowl­edged — namely, that the for­malism of quantum mechanics is a prob­a­bility cal­culus, and that the events to which this assigns prob­a­bil­i­ties are mea­sure­ment outcomes.

An algo­rithm for assigning prob­a­bil­i­ties to pos­sible mea­sure­ment out­comes on the basis of actual out­comes has two per­fectly normal depen­den­cies. It depends con­tin­u­ously on the time of mea­sure­ment: if this changes by a small amount, the assigned prob­a­bil­i­ties change by small amounts. And it depends dis­con­tin­u­ously on the out­comes that con­sti­tute the assign­ment basis: if this changes by the inclu­sion of an out­come not pre­vi­ously taken into account, so do the assigned probabilities.

But think of a quantum state’s depen­dence on time as the con­tin­uous time-​​dependence of an evolving state (rather than as a depen­dence on the time of a mea­sure­ment), and you have two modes of evo­lu­tion for the price of one:

  1. between mea­sure­ments, a quantum state evolves according to a uni­tary trans­for­ma­tion and thus con­tin­u­ously and predictably;
  2. at the time of a mea­sure­ment, a quantum state gen­er­ally “col­lapses” (or appears to “col­lapse”): it changes (or appears to change) dis­con­tin­u­ously and unpre­dictably into a state that depends on the measurement’s outcome.

Hence the mother of all quantum-​​theoretical pseudo-​​questions: why does a quantum state have (or appear to have) two modes of evo­lu­tion? And hence the embar­rass­ment. Get­ting rid of the pseudo-​​question is easy: we only have to rec­og­nize that the true number of modes of evo­lu­tion is nei­ther two nor one but zero. Get­ting rid of the embar­rass­ment requires more work, for we still have two Rules. Why two? What dis­tin­guishes this ques­tion from the afore­men­tioned pseudo-​​question is that it has a straight­for­ward answer:

When­ever quantum mechanics instructs us to add the ampli­tudes of alter­na­tives rather than their prob­a­bil­i­ties (that is, when­ever we are required to use Rule B rather than Rule A), the dis­tinc­tions we make between the alter­na­tives cor­re­spond to nothing in the actual world. They don’t exist in the actual world. They exist solely in our minds.

This answer lies at the heart of the inter­pre­ta­tional strategy we shall adopted. It does raise fur­ther ques­tions, but it also makes it pos­sible to answer them. What is more, it does not appeal to untestable meta­phys­ical assump­tions about what hap­pens between mea­sure­ments but pro­ceeds directly from the testable com­pu­ta­tional rules of quantum mechanics.