### 2 On probability

With regard to the role of probability, three types of physical theories may be distinguished.

• Theories of type 1 are deterministic. Single events are completely described by their known initial values and deterministic laws (differential equations). Classical mechanics is obviously such a theory. We include this type of theory, where probability does not play any role, in our classification scheme because it provides a basis for the following two types of theories.
• Theories of type 2 have deterministic laws but the initial values are unknown. Therefore, no predictions on individual events are possible, despite the fact that deterministic laws describing individual events are valid. In order to verify a prediction of a type 2 theory a large number of identically prepared experiments must be performed. We have no problems to understand or to interpret such a theory because we know its just our lack of knowledge which causes the uncertainty. An example is given by classical statistical mechanics. Of course, in order to construct a type 2 theory one needs a type 1 theory providing the deterministic laws.
• It is possible to go one step further in this direction increasing the relative importance of probability even more. We may not only work with unknown initial values but with unknown laws as well. In these type 3 theories there are no deterministic laws describing individual events, only probabilities can be assigned. There is no need to mention initial values for particle trajectories any more (initial values for probabilistic dynamical variables are still required).

Type 2 theories could also be referred to as classical (statistical) theories. Type 3 theories are most interesting because we recognize here characteristic features of quantum mechanics. In what follows we shall try to make this last statement more definite.

Comparing type 2 and type 3 theories, one finds two remarkable aspects. The first is a subtle kind of “inconsistency” of type 2 theories: If we are unable to know the initial values of our observables (at a particular time), why should we be able to know these values during the following time interval (given we know them at a fixed time). In other words, in type 2 theories the two factors determining the final outcome of a theoretical prediction - namely initial values and laws - are not placed on the same (realistic) footing. This hybrid situation has been recognized before; the term ’crypto-deterministic’ has been used by Moyal [32] to characterize classical statistical mechanics (note that the same term is also used in a very different sense to characterize hidden variable theories [35]). Type 3 theories do not show this kind of inconsistency.

The second observation is simply that type 2 and type 3 theories have a number of important properties in common. Both are unable to predict the outcome of single events with certainty; only probabilities are provided in both cases. In both theories the quantities which may be actually observed - whose time dependence may be formulated in terms of a differential equation - are averaged observables, obtained with the help of a large number of single experiments. These common features lead us to suspect that a general structure might exist which comprises both types of theories.

Such a general structure should consist of a set of (statistical) conditions, which have to be obeyed by any statistical theory. In theories of this kind observables in the conventional sense do not exist. Their role is taken over by random variables. Likewise, conventional physical laws - differential equations for time-dependent observables - do not exist. They are replaced by differential equations for statistical averages. These averages of the (former) observables become the new observables, with the time playing again the role of the independent variable. In order to construct such general conditions one needs again (as with type 2 theories) a deterministic (type I) theory as a “parent” theory. Given such a type 1 theory, we realize that a simple recipe to construct a reasonable set of statistical conditions is the following: Replace all observables (of the type 1 theory) by averaged values using appropriate probability densities. In this way the dynamics of the problem is completely transferred from the observables to the probability distributions. This program will be carried through in the next sections, using a model system of classical mechanics as parent theory.

The above construction principle describes an unusual situation, because we are used to considering determinism (concerning single events) as a very condition for doing science. Nevertheless, the physical context, which is referred to is quite simple and clear, namely that nature forbids for some reason deterministic description of single events but allows it at least “on the average”. It is certainly true that we are not accustomed to such a kind of thinking. But to believe or not to believe in such mechanisms of nature is basically a matter of intellectual habit. Also, the fact that quantum mechanics is incomplete does not necessarily imply that a complete theory exists; the opposite possibility, that no deterministic description of nature will ever be found, should also be taken into account.