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Stable Facts, Relative Facts Stable Facts, Relative Facts Andrea Di Biagio1, ∗ and Carlo Rovelli2, 3, 4, y 1Dipartimento di Fisica, Universit`aLa Sapienza, I-00185 Roma, EU 2AMU Universit´e,Universit´ede Toulon, CNRS, CPT, F-13288 Marseille, EU 3Perimeter Institute, 31 Caroline Street N, Waterloo ON, N2L2Y5, Canada 4The Rotman Institute of Philosophy, 1151 Richmond St. N London N6A5B7, Canada (Dated: June 28, 2020) We define and characterise two kinds of facts that play a role in quantum theory: stable and relative. We describe how stable facts emerge in a world of relative facts and then discuss the respective roles they play in connecting the theory and the world. The distinction between relative and stable facts resolves the difficulties pointed out by the no-go theorems of Frauchiger and Renner, Brukner, and Bong et. al.. Basing the ontology of the theory on relative facts clarifies the role of decoherence in bringing about the classical world and solves the apparent incompatibility between the `linear evolution' and `projection' postulates. I. FACTS IN QUANTUM THEORY or cannot be extended to systems with an infinite num- ber of degrees of freedom [3], or else. But the universal The common textbook presentation of quantum theory success of the theory and all current empirical evidence assumes the existence of a classical world. A measure- strongly suggest that real physical objects are `classical', ment involves an interaction between the classical world meaning they do not display quantum properties, only and a quantum system. The measurement produces a approximatively. There are no exactly classical objects, definite result, for instance a dot on a screen. The result strictly speaking, as everything we interact with is made is a fact by itself, but also establishes a fact about a quan- of atoms and photons, which obey quantum theory. tum system. For instance, a certain measurement result- In formulating the fundamental theory of nature, the ing in a definite record establishes that at some time the use of effective concepts valid only within an approxi- ~ mation is unconvincing. Therefore the attempts to in- z-component of the spin of an electron is Lz = 2 . This is a fact. terpret quantum theory as a universal theory, such as Quantum probabilities are probabilities for facts, given Many Worlds, Hidden Variables, and others do not rely other facts. The facts are therefore entries of which the on postulating classical objects. A possibility to interpret probability amplitudes are function. In particular, facts quantum theory as a universal theory neither postulating are used as conditionals for computing probabilities of classical objects, nor postulating unobservable worlds, other facts. For instance, if the spin of the electron men- unobservable variables, or unobserved physics, is Rela- tioned above is then immediately measured in a direction tional Quantum Mechanics (RQM) [4,5]. RQM bases at an angle θ from the z-axis, the probability to find the the interpretation of the theory on a larger ensemble of ~ ~ facts, of which stable facts are only a subset. These are value Lθ = 2 (a fact), given the fact that Lz = 2 , is ~ ~ 2 called relative facts. P Lθ = 2 j Lz = 2 = cos (θ=2). Hence, facts are what quantum mechanics is about. Facts ascertained in a conventional measurement are A. Relative facts stable in the following sense. If we know that one of N mutually exclusive facts ai (i = 1 :::N) has happened, the probability P (b) for a fact b to happen is given by Relative facts are defined to happen whenever a phys- ical system interacts with another physical system. Con- X P (b) = P (bjai)P (ai); (1) sider two systems S and F. If an interaction affects F i in a manner that depends on the value of a certain vari- 1 able LS of S, then the value of LS is a fact relative to where P (ai) is the probability that ai has happened and F. This is true by definition irrespectively of whether F P (bjai) is the probability for b given ai. We take equation is a classical system or not. That is, whenever the two (1) as a characterisation of stable facts. systems interact, the value of the variable LS becomes a This textbook presentation of quantum mechanics is fact relative to F. incomplete because it assumes the existence of a classi- The interaction with F is the context in which LS takes cal world. An exactly classical world might exist because a specific value; we call the system F, in this role, a current quantum theory has limited validity|for in- stance is violated by physical collapse mechanisms [1,2], 1 We use `variable' to denote any quantity that in the classical the- ory is a function on phase pace. We prefer to avoid the expression ∗ [email protected] `observable', which is loaded with irrelevant extra baggage: the y [email protected] ideas of observation and a complex observer. 2 `context'.2 The interaction with the context determines B. Decoherence the fact that a certain variable, LS , has a certain value. Stable facts are only a subset of relative facts, as Since stability is a characteristic feature of the classi- there are many relative facts that are not stable facts. cal world (whose facts invariably satisfy (1)), answering Quantum theory provides probabilities that relate rela- the questions above amounts to explaining in terms of tive facts, but these satisfy (1) only if b and the ai are relative facts what it takes for a system to be classical. facts relative to the same system. That is, if we label Various characterisations of a classical or semiclassical facts with the system they refer to, writing a(F) for a fact situation can be found in the literature: large quantum relative to the system F, it is always true that numbers, semiclassical wave packets or coherent states, macroscopic systems, large or infinite number of degrees (F) X (F) (F) (F) P (b ) = P (b jai )P (ai ); (2) of freedom... All these features play a role in charac- i terising classical systems in specific situations. But the while it is in general false that key phenomenon that makes facts stable is decoherence [9{11]: the suppression of quantum interference that hap- (W) X (W) (F) (F) pens when some information becomes inaccessible. P (b ) = P (b jai )P (ai ) (3) i Consider two systems F and E (E for `Environment'), and a variable LF of the system F. Let F ai be the (F) if W 6= F. When (3) holds, we say that the facts ai are eigenvalues of LF . A generic state of the coupled system stable with respect to W. F − E can be written in the form The failure of (3) is easily understood in terms of the X standard language of quantum theory. If F is sufficiently j i = ci jF aii ⊗ j ii ; (5) isolated it may be possible to maintain the quantum co- i herence of the coupled system S−F formed by S and F where j ii are normalised states of E. Define together. The interaction entangles the two systems and 2 interference effects between different values of the vari- = max j h ij ji j : (6) i;j able LS can later be detected in the measurements by an observer W. The probabilities for facts of the S −F Now, suppose that: (a) is vanishing or very small and system relative to W, indeed, can be computed from an (b) a system W does not interact with E. Then the prob- entangled state of the form ability P (b) of any possible fact relative to W resulting from an interaction between F and W can be computed c1ja1i ⊗ jF a1i + c2ja2i ⊗ jF a2i (4) from the density matrix obtained tracing over E, that is 0 where ai are values of LS and F ai are values of F s X 2 ρ = TrE j ih j = jcij jF aiihF aij + O(): (7) `pointer variable' LF . Probabilities computed from this state feature interference terms, and this violates (3) be- i cause what sums is amplitudes, not probabilities. The (E) 2 By posing P (F ai ) = jcij , we can then write value of LS , therefore, is not a stable fact. Hence facts relative to a system F cannot in general be (W) X (W) (E) (E) P (b ) = P (b jF ai )P (F ai ) + O(): (8) taken as conditionals for computing probabilities of facts i relative to a different system W. Equation (1) holds only Thus, probabilities for facts b relative to W calculated if b and ai are facts relative to the same system, but fails in general if used for facts relative to different systems. in terms of the possible values of LF satisfy (3) up to a The notation S for `system', F for `Friend' and W small deviation. Hence the fact LF = ai relative to E for `Wigner' is meant to to evoke the famous Wigner's is stable with respect to W to the extent to which one friend ideal experiment [8]; there is no assumption, how- ignores effects of order . In the limit ! 0, the variable ever, about the system F being quantum or classical, LF of the system F is exactly stable with respect to W. microscopic or macroscopic. The extensive theoretical work on decoherence [12] has Relative facts play a central role in the Relational In- shown that decoherence is practically unavoidable and terpretation of Quantum Mechanics (RQM) [4,5]. We extremely effective as soon as large numbers of degrees shall discuss this role in detail later on.
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