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Chance, Calculation and Life


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of randomness was given by Poincaré. By his “negative result” (his words) on the Three Body Problem (1892, relatively simple deterministic dynamics, see below), he proved that minor fluctuations or perturbations below the best possible measurement may manifest in a measurable, yet unpredictable consequence: “we then have a random phenomenon” (Poincaré 1902). This started the analysis of deterministic chaos, as his description of the phase-space trajectory derived from a nonlinear system is the first description of chaotic dynamics (Bros and Iagolnitzer 1973).

      In this classical framework, a random event has a cause, yet this cause is below measurement. Thus, Curie’s principle3 is preserved: “the asymmetries of the consequences are already present in the causes” or “symmetries are preserved” – the asymmetries in the causes are just hidden.

      For decades, Poincaré’s approach was quoted and developed by only a few, that is, until Kolmogorov’s work in the late 1950s and Lorentz in the 1960s. Turing is one of these few: he based his seminal paper on morphogenesis (Turing 1952) on the nonlinear dynamics of forms generated by chemical reactants. His “action/reaction/diffusion system” produced different forms by spontaneous symmetry breaking. An early hint of these ideas is given by him in Turing (1950, p. 440): “The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping”. This Poincarian remark by Turing preceded by the famous “Lorentz butterfly effect” (proposed in 1972) by 20 years on the grounds of Lorentz’s work from 1961.

      By “theory” we mean the equational or functional determination, possibly by a nonlinear system of equations or evolution functions.

      Quantum randomness is hailed to be more than “epistemic”, that is, “intrinsic” (to the theory). However, quantum randomness is not part of the standard mathematical model of the quantum which talks about probabilities, but is about the measurement of individual observables. So, to give more sense to the first statement we need to answer (at least) the following questions: (1) What is the source of quantum randomness? (2) What is the quality of quantum randomness? (3) Is quantum randomness different from classical randomness?

      A naive answer to (1) is to say that quantum mechanics has shown “without doubt” that microscopic phenomena are intrinsically random. For example, we cannot predict with certainty how long it will take for a single unstable atom in a controlled environment to decay, even if one has complete knowledge of the “laws of physics” and the atom’s initial conditions. One can only calculate the probability of decay in a given time, nothing more! This is intrinsic randomness guaranteed.

      Following Einstein’s approach (Einstein et al. 1935), quantum indeterminism corresponds to the absence of physical reality, if reality is what is made accessible by measurement: if no unique element of physical reality corresponding to a particular physical observable (thus, measurable) quantity exists, this is reflected by the physical quantity being indeterminate. This approach needs to be more precisely formalized. The notion of value indefiniteness, as it appears in the theorems of Bell (Bell 1966) and, particularly, Kochen and Specker (1967), has been used as a formal model of quantum indeterminism (Abbott et al. 2012). The model also has empirical support as these theorems have been experimentally tested via the violation of various inequalities (Weihs et al. 1998). We have to be aware that, going along this path, the “belief” in quantum indeterminism rests on the assumptions used by these theorems.

      An observable is value definite for a given quantum system in a particular state if the measurement of that observable is pre-determined to take a (potentially hidden) value. If no such pre-determined value exists, the observable is value indefinite. Formally, this notion can be represented by a (partial) value assignment function (see Abbott et al. (2012) for the complete formalism).

      When should we conclude that a physical quantity is value definite? Einstein, Podolsky and Rosen (EPR) defined physical reality in terms of certainty of predictability in Einstein et al. (1935, p. 777):

      If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity.

      EPR principle: If, without disturbing a system in any way, we can predict with certainty the value of a physical quantity, then there exists a definite value prior to the observation corresponding to this