Abstract
At the onset of quantum mechanics, it was argued that the new theory would entail a rejection of classical logic. The main arguments to support this claim come from the non-commutativity of quantum observables, which allegedly would generate a non-distributive lattice of propositions, and from quantum superpositions, which would entail new rules for quantum disjunctions. While the quantum logic program is not as popular as it once was, a crucial question remains unsettled: what is the relationship between the logical structures of classical and quantum mechanics? In this essay we answer this question by showing that the original arguments promoting quantum logic contain serious flaws, and that quantum theory does satisfy the classical distributivity law once the full meaning of quantum propositions is properly taken into account. Moreover, we show that quantum mechanics can generate a distributive lattice of propositions, which, unlike the one of quantum logic, includes statements about expectation values which are of undoubtable physical interest. Lastly, we show that the lattice of statistical propositions in classical mechanics follows the same structure, yielding an analogue non-commutative sublattice of classical propositions. This fact entails that the purported difference between classical and quantum logic stems from a misconstructed parallel between the two theories.
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Notes
For historical details on quantum logic see Jammer (1974).
It is worth noting that von Neumann in his treatise Mathematische Grundlagen der Quantenmechanik published in 1932 anticipated that from the algebraic structure of quantum theory it would have been possible to formulate a new propositional calculus. These ideas would be fully expressed in his successive collaboration with Garrett Birkhoff.
This conclusion is now accepted among experts; for details see e.g. Bacciagaluppi (2009) and Dalla Chiara and Giuntini (2002). Referring to this, the latter authors stated explicitly that “quantum logics are not to be regarded as a kind of “clue”, capable of solving the main physical and epistemological difficulties of QT [quantum theory]. This was perhaps an illusion of some pioneering workers in quantum logic” (Dalla Chiara & Giuntini, 2002, p. 225).
In the present essay we will be concerned only with propositional calculus in the context of non-relativistic QM. First-order and higher-order logics will not be discussed here, nor the logic of relativistic quantum theories.
It is worth noting that in what follows we are not going to discuss whether it is useful or possible to describe quantum phenomena with a non-classical type of logic.
In QM the wave function of a system provides the maximal information available about it.
In the context of quantum logic, it is usually believed that distributivity must be replaced by a weaker law: \((p\wedge q)\vee (p\wedge r)\longrightarrow (p\wedge (q\vee r)).\)
Cf. also David (2015, p. 78), where we read that “An orthogonal projector \({\textbf {P}}\) onto a linear subspace \(P\subset \mathcal {H}\) is indeed the operator associated to an observable that can take only the values 1 (and always 1 if the state \(\psi \in P\) is in the subspace P) or 0 (and always 0 if the state \(\psi \in P^{\perp }\) belongs to the orthogonal subspace to P). Thus we can consider that measuring the observable \({\textbf {P}}\) is equivalent to performing a test on the system, or to checking the validity of a logical proposition p on the system”.
More on this below.
Clearly, a quantum disjunction is false when both disjuncts are false; however, the interesting case for our discussion is the one in which quantum propositions have undetermined truth values.
Alternatively stated, in classical mechanics it is usually claimed that the sentence “The value of the quantity y for the system s is x” can be considered equivalent to the statement “After a measurement M of the quantity y on the system s, the value x is obtained”. This equivalence is due to the fact that measurements in the classical context do not alter the observed system, or in less idealized scenarios, that the disturbance caused by the interaction between observed system and measuring device is negligible. A more elaborate treatment of measurements in classical mechanics goes beyond the scope of the present paper.
In operational quantum theory the preparations are a set of instructions that an agent has to follow in order to prepare a quantum system in a certain state. We are aware that preparations can be considered special kinds of measurements. For the sake of the argument, however, in what follows we take preparations to be the instructions to arrange a physical system in a given state before the measurement of a certain quantity is actually performed.
In fact, note that \(q_f\) is true if and only if \(q_o\) is true, and, similarly, \(p_f\) is true if and only if \(r_o\) is true.
We thank one anonymous reviewer for having pointed out Heelan’s work to us.
Heelan exemplifies such a distinction as follows: “If the event, for example, is a particle-location event, the event-language is position-language, and the physical context is a standardized instrumental set-up plus whatever other physical conditions are necessary and sufficient for the measurement of a given range of possible particle-position events” (Heelan, 1970, p. 96).
We should stress, however, that \(\sigma \)-algebras are defined independently of what the elements of X actually are. For a simpler example, X could correspond to the set of the real numbers for a continuous quantity.
In the quantum case this is exemplified also in Sect. 2.
Note that the interpretation used here is exactly the same used in probability theory. In fact, a probability space is defined by three elements: a sample space \(\Omega \) which represents the set of all possible outcomes; a \(\sigma \)-algebra over \(\Omega \) which represents all events, all propositions; and a measure that assigns a probability to each event.
Here the topology is important for us only as a means to construct the Borel algebra.
We can now see precisely that by generating a lattice of statements we mean starting with a set of propositions that are experimentally accessible (i.e. the basis of the topology) and closing under negation (i.e. complement) and countable disjunction (i.e. union).
The choice of the square root of the density rather than the density itself is merely a technical choice.
To give more context, the set \(U = \{ v \}\) can be written as \(U = \{w \; | \; |w - v|^2 = 0\}\) the set of all elements with zero distance from v. This can be understood as the limit (i.e. countable intersection) of a sequence of propositions “the object is within \(\epsilon \) of v” where \(\epsilon \) tends to zero.
Obviously, the inclusion map from the lattice of closed subspaces is not an order isomorphism as the join and complement are not preserved. However, these are still expressible as subspace closure and subspace complement.
Technically, position and momentum are not bounded operators. However, one can argue that measured position and momentum are indeed bounded as the acceptance of a detector is always limited. Note that all operators for which the spectral theorem applies can be written as the limit of a sequence of bounded operators.
In most cases, one assumes states to be normalized, which is what we have done here for simplicity. Renormalization would lead to the set \(\{ \psi \in \mathcal {H} \, | \, \langle \psi , A \psi \rangle / \langle \psi , \psi \rangle \in (x_0, x_1) \}\), which is still a Borel set since renormalization is a continuous function over its domain \(\mathcal {H} \setminus \{ 0 \}\).
Note that we are not claiming that all Borel sets can be constructed in this way or that all Borel sets correspond to physically interesting propositions. First of all, it is not true: \(\{ \psi \}\) and \(\{ e^{\imath \theta } \psi \}\) are different Borel sets, yet they are not physically distinguishable since a difference in absolute phase is not physically relevant. This can be fixed by considering the Borel sets of the projective space. Second, to really examine the physicality of all Borel sets we need a “general theory of experimental logic”, of the type provided by Kelly (1996) or Carcassi and Aidala (2021), which would go well beyond the present discussion. With appropriate caveats (e.g. the Hilbert space must be separable), one can say that the Borel sets correspond to propositions that can be associated to an experimental test (what Carcassi and Aidala (2021) call “theoretical statements”) which may or may not terminate in any or all cases. We leave this discussion for another work.
Since probabilities are assigned to Borel sets and only Borel sets, the Borel algebra will contain all physically interesting statements. Further expansions (e.g. to the power set) would only include unphysical statements.
To be clear, we are not arguing that this construction is the most appropriate or useful. Simply that it can be done.
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Acknowledgements
Andrea Oldofredi is grateful to the Fundacão para a Ciência e a Tecnologia (FCT) for financial support (Grant no. 2020.02858.CEECIND). This work is part of a larger project, Assumptions of Physics (Carcassi and Aidala (2021)), which aims to identify a handful of physical principles from which the basic laws can be rigorously derived.
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Oldofredi, A., Carcassi, G. & Aidala, C.A. On the Common Logical Structure of Classical and Quantum Mechanics. Erkenn 89, 1507–1533 (2024). https://doi.org/10.1007/s10670-022-00593-z
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DOI: https://doi.org/10.1007/s10670-022-00593-z