Is Classical Mechanics Really Time-reversible and Deterministic?

J. M. Domingos

Bernard d’Espagnat

The problem of retrodiction is associated with basic questions, but for this very reason it is, unfortunately, quite difficult, even in classical physics. In quantum physics it is even more subtle. In quantum mechanics, a consistent definition of what is called a "system" leads to problems that do not exist in classical physics, and correspondingly the answers to the questions raised are less straightforward. This chapter discusses the conditions under which it is possible to speak of an "independent time evolution" of a system S and to study what kinds of predictions can legitimately be made when these conditions are fulfilled. It describes retroactive computation to investigate whether or not the assumed knowledge in question is really consistent with the standard quantum mechanical formalism. The concept of retrodiction is related to the notions of time reversal and of irreversibility. Both these notions play a central role in physics.

Max Born

Hrvoje Nikolić

Nicolas Gisin

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers.However, almost all real numbers contain an infinite amount of information.I argue that a finite volume of space can't contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant.Moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random.I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers.This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory.Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic.Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.

Nicolas Gisin

Abstract It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.