One can draw a revealing parallel between quantum mechanics and thermo-dynamics: the latter discipline is formulated in terms of macroscopic variables, such as pressure, temperature, volume, etc., that undergo fluctuations. Statistical mechanics, on the other hand, provides a more precise description in terms of microscopic states where the positions and velocities of particles are well-de-fined. Thermodynamic fluctuations are explained within this framework through averages over these microscopic states, accompanied by their dispersions. In this context, microscopic states play the role of hidden variables.

Similarly, current quantum mechanics could be seen, just like thermodynamics, as an “approximate” version of the theory, awaiting a more exhaustive, more fundamental formulation, which remains unknown at the moment. Thus, in this framework, Heisenberg’s inequalities could take the form of uncertainty relations, as a quantum system could indeed be characterized by a well-defined pair of observables (x, p). However, a set of such systems, even when prepared identically, would present—due to a reason still unknown—statistical dispersions for these quantities, precisely formalized by Heisenberg’s uncertainty relations. Of course, this theory would need to explain specifically quantum phenomena, which excludes the possibility that such systems possess exactly the same type of objective prop-erties as in the framework of Newtonian theory.

Two arguments will highlight the presumed inadequacies of the current formu-lation of the theory.

The EPR Argument

Einstein, along with his colleagues Boris Podolski and Nathan Rosen35, in what remains one of the most brilliant moments in the history of physics, formulated an argument of incompleteness by highlighting a paradox—known as the EPR paradox, after the initials of its authors—aiming to demonstrate that quantum theory, as currently formulated, leads to a contradiction. We will focus here on the version of this paradox proposed by David Bohm, which stands out for its simplicity. The fundamental idea is to consider a vector quantity, with no clas-sical equivalent, called spin S⃗, an intrinsic property of the particle, similar to its mass or charge. One peculiar feature of this quantity is that its projection along any direction n⃗, denoted Sn = S⃗.n⃗, can only take discrete values. In the case of a particle with spin , these values are . Moreover, and this is a fundamen-tal point for the following discussion, the components Sx, Sy, Sz of the spin obey Heisenberg’s inequalities:

± 1 2

(8)

which prohibit the existence of a state in which two—and by extension, three—components of the spin, such as Sx, Sy, Sz are simultaneously well-defined.

Bohm, following Einstein, Podolski, and Rosen, considers a situation in which two particles have interacted at some point or were emitted simultaneously from a common source, such that they are described, with regard to the observable spin36, by the following state:

(9)

In this expression, | ↑i,n⃗⟩, with i = {1, 2}, denotes the eigenstate of the spin projec-tion of particle i = {1, 2} along the direction n⃗, with the eigenvalue while | ↓i,n⃗⟩ denotes the eigenstate of the projection with the eigenvalue . The state |Ψ� thus consists of a superposition of two states, each formed by a product of individual states of the form |A1⟩ ⊗ |B2⟩, where the indices 1 and 2 always refer to the indices of the particles37. Thus, in the first state constituting the global state (9), particle 1

− 1 2
Δ S x Δ S y ≥ | ⟨ S z ⟩ | Δ S y Δ S z ≥ | ⟨ S x ⟩ | Δ S z Δ S x ≥ | ⟨ S y ⟩ |
+ 1 2
| Ψ ⟩ = 1 2 ( | ↑ 1 , n → ⟩ ⊗ | ↓ 2 , n → ⟩ − | ↓ 1 , n → ⟩ ⊗ | ↑ 2 , n → ⟩ ) .
S = 1 2

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