Pure quantum evolution is deterministic, ψ → exp (—iHt/ħ)ψ , but as for classical mechanics probability enters because a given macroscopic initial condition contains microscopic states that lead to different outcomes; the relative probability of those outcomes equals the relative abundance of the microscopic states for each outcome. This is the postulated basis for the recovery of the usual quantum probabilities, as discussed in Chapter 6. In this chapter we take up the question of whether the allowable microstates (the ‘special’ states) do indeed come with the correct abundance. To recap: ‘special’ states are microstates not leading to superpositions of macroscopically different states (‘grotesque’ states). For a given experiment and for each macroscopically distinct outcome of that experiment these states form a subspace. We wish to show that the dimension of that subspace is the relative probability of that outcome. This is an ambitious goal, especially considering the effort needed to establish that there are any special states—the subject of Chapter 7. As remarked there, the special states exhibited are likely to be only a small and atypical fraction of all special states in the physical apparatus being modeled (e.g., the cloud chamber). In one example (the decay model) there is a remarkable matching of dimension and conventional probability, but I would not make too much of that. What is especially challenging about the present task is that we seek a universal distribution.
Pure quantum evolution is deterministic, ψ → exp (—iHt/ħ)ψ , but as for classical mechanics probability enters because a given macroscopic initial condition contains microscopic states that lead to different outcomes; the relative probability of those outcomes equals the relative abundance of the microscopic states for each outcome. This is the postulated basis for the recovery of the usual quantum probabilities, as discussed in Chapter 6. In this chapter we take up the question of whether the allowable microstates (the ‘special’ states) do indeed come with the correct abundance. To recap: ‘special’ states are microstates not leading to superpositions of macroscopically different states (‘grotesque’ states). For a given experiment and for each macroscopically distinct outcome of that experiment these states form a subspace. We wish to show that the dimension of that subspace is the relative probability of that outcome. This is an ambitious goal, especially considering the effort needed to establish that there are any special states—the subject of Chapter 7. As remarked there, the special states exhibited are likely to be only a small and atypical fraction of all special states in the physical apparatus being modeled (e.g., the cloud chamber). In one example (the decay model) there is a remarkable matching of dimension and conventional probability, but I would not make too much of that. What is especially challenging about the present task is that we seek a universal distribution.
The existence of ‘special’ states can be established with ordinary quantum mechanics. We seek particular microscopic states of large systems that have the property that they evolve to only one or another macroscopic outcome, when other microstates of a more common sort (having the same initial macrostate) would have given grotesque states. Justifying the hypothesis that Nature chooses these special states as initial conditions is another matter. In this chapter we stick to the narrower issue of whether there exist states that can do the job, irrespective of whether they occur in practice. In Section 7.1 we give several explicit examples. In Section 6.2 we exhibited an apparatus model and its special states. Here we look at the decay of an unstable quantum state, not as a single degree of freedom in a potential, but with elements of the environment taken into account as well. We also study another popular many-body system, the spin boson model. This has extensive physical applications and especially with respect to Josephson junctions has been used to address quantum measurement questions. A single degree of freedom in a potential, by the way, does not generally lend itself to ‘specializing,’ and an example is shown below. In recent years, exotic non-local effects of quantum mechanics have been exhibited experimentally. Behind many of these lies entanglement , the property of a wave function of several variables that it does not factor into a product of functions of these variables separately.
Why should special states occur as initial conditions in every physical situation in which they are needed? Half this book has been devoted to making the points that initial conditions may not be as controllable as they seem; that there may be constraints on microscopic initial conditions; that this would not have been noticed; that such constraints can arise from two-time or future conditioning; that in our universe such future conditioning may well be present, although, as remarked, cryptic . In this chapter I will take up more detailed questions: what future conditions could give rise to the need for our ‘special’ states, and why should those particular future conditions be imposed. Before going into this there is a point that needs to be made. Everything in the present chapter could be wrong and the thesis of Chapter 6 nevertheless correct. It is one thing to avoid grotesque states (and solve the quantum measurement problem) by means of special states and it is another provide a rationale for their occurrence. I say this not only to highlight the conceptual dependencies of the theses in this book, but also because there is a good deal of hand waving in the coming chapter and I don't want it to reflect unfavorably on the basic proposal. As pointed out earlier, the usual thermodynamic arrow of time can be phrased as follows: initial states are arbitrary, final states special.
In Chapter 6 I presented a proposal for how and why grotesque states do not occur in Nature. In subsequent chapters I explored consequences and found subsidiary requirements, such as Cauchy distributed kicks. To find out whether all or part of our scheme is the way Nature works, we turn to experiment. How to turn to experiment is not so obvious, since the basic dynamical law, ψ → exp(− iHt/ħ )ψ, is the same as for most other theories. Our basic assertion concerns not the dynamical law but the selection of states. Therefore it is that assertion that must be tested. For example, one way is to set up a situation where the states we demand, the ‘special’ states, cannot occur. Then what happens? Another part of our theory is the probability postulate, and this deals not only with the existence of special states but with their abundance. It enters in the recovery of standard probabilities but has far reaching consequences that may well lead to the best tests of the theory. Such tests arise in the context of EPR situations. The experimental tests fall into the following categories. Precluding a class of special states . This should prevent a class of outcomes. If the changes in the system (due to precluding the class of special states) do not change the predictions of the Copenhagen interpretation, then this provides a test. In particular, with a class of special states precluded, our theory forbids the associated outcome.
Less than meets the eye If even some of the ideas presented here are correct, the world is different from what it seems. The major theses of the book, on time's arrows and on quantum measurement theory, are unified by the notion of cryptic constraints . We see, sense, specify, macroscopic states, but what we predict about these states depends on an important assumption concerning their microscopic situation. The assumption is that the actual microstate is equally likely to be any of those consistent with the macrostate. And I say, not so. For various reasons, both classical and quantum, many otherwise-possible microstates are eliminated. As presented in detail in previous chapters, such elimination impacts many areas of physics, from the cosmos to the atom. But we also make the point that this elimination can be difficult to notice and in particular there is no experimental evidence that confirms the usual assumption. By an explicit example on a model (Fig. 4.3.2), we show that a future constraint eliminating 98% of the microstates can go completely unnoticed. My expectation is that this fundamental change in the foundations of statistical mechanics is needed. Whether or not it takes the forms I've proposed will be determined by future investigations. In the next section I will review open problems for the program implicit in this book. The tone will be that used in speaking with colleagues: an attempt to be frank about difficulties and a willingness to be wildly speculative.
Quantum mechanics is an immensely successful theory, occupying a unique position in the history of science. It has solved mysteries ranging from macroscopic superconductivity to the microscopic theory of elementary particles and has provided deep insights into the nature of vacuum on the one hand and the description of the nucleon on the other. Whole new fields such as quantum optics and quantum electronics owe their very existence to this body of knowledge. However, despite the stunning successes of quantum mechanics, there is no general agreement on the conceptual foundations and interpretation of the subject. The theory provides unambiguous information about the outcome of a measurement of a physical object. However, many feel that it does not provide a satisfactory answer to the nature of the “reality” we should attribute to the physical objects between the acts of measurement. The conceptual difficulty comes about because the wave function |ψ〉 is usually given by a coherent superposition of various distinguishable experimental outcomes. If we denote the collection of states that represent the possible outcomes of an experiment by |ψ j 〉, then |ψ〉 = ∑ j c j |ψ j 〉 where c j = 〈ψ j |ψ〉. The probability of the outcome |ψ j 〉 is P j =|c j | 2 . In the process of measurement, the so called collapse of the wave function takes place and a single, definite state |ψ i 〉 of the physical object is chosen. The difficulty comes about in the interpretation of the mechanism by which this definite state is chosen from amongst all the possible outcomes.
There are of course many different ways of generating probabilities. In many classical statistical experiments we can assume all outcomes are equally likely to occur. This chapter introduces different ways to produce the required probability distribution. Generally the probability of each of measured outcome would depend on the state of the system as well as that particular observable in question. A simple counting of number of outcomes without reference to the state cannot possibly lead to the correct probability distribution. The theory presented can be extended to lower as well as higher dimensional real vector spaces. For a model theory to have any chance of success it is necessary for the model to satisfy the following additional requirements: There must be enough unit vectors and selfadjoint operators to correspond to all the states and all the observables of the system; and the probability distribution for every observable in every state must agree with experimental results.