# Knowledge of Quantum Hidden Variables Enables Backwards-In-Time Signaling
## 4 Bohmian Mechanics  
In the de [PERSON] interpretation of quantum mechanics [28], spins do not have hidden variables: the only hidden variables are the particles' positions, but our argument employs spin hidden variables. Nonetheless, we can easily translate the spin degrees of freedom into trajectories. Consider a \"full loop Stern-Gerlach apparatus\" (FULO) where we do not determine whether the particle goes up or down, but where we just subject the particle to a magnetic field gradient and then we use the opposite magnetic field gradient to re-merge the two arms of the wave function (e.g., see [29]). Such an apparatus does not measure anything: it is described by an identity transformation in quantum mechanics, as it is a unitary transformation followed by its inverse unitary. Nonetheless, the spin information is transferred into the trajectory information inside the FULO. To determine the \(x,y,z\) spin hidden variables, [PERSON] can perform one FULO along the \(x\) axis followed by one FULO along the \(y\) axis and one FULO along the \(z\) axis and track (but not measure!) the internal trajectories. These trajectories encode the spin \"hidden variables\" because the trajectories in the FULOs are consistent when repeated: if an \(x\)-directed FULO is followed by a \(y\)-directed one and then by a second \(x\)-directed one, the trajectories along the two \(x\)-directed ones coincide. Namely, any time dependence of the trajectories is irrelevant in the encoding of the spin hidden values into FULOs trajectories. Moreover, the FULOs trajectory encodes the outcome of future spin measurements (as requested by the hidden variable definition we use in this paper), since the measurement trivially consists of measuring in which of the two arms the particle is present, and since the trajectories in successive FULOs match. Then, we can conclude that the trajectory in the \(w\)-directed FULO encodes the \(w\) hidden value of the spin for \(w=x,y,z\). Namely, the \"non-existent\" hidden variables for spin are translated into hidden variables for position. Then our argument _seemingly_ applies: by knowing (not measuring!) the trajectories of the particles through the three successive FULOs, [PERSON] seemingly would be able to signal back in time. Indeed, we have shown how [PERSON] can map the spin quantum degree of freedom into position degrees of freedom of a particle which are the hidden variables of Bohmian mechanics. However, in reality, our argument is not applicable to Bohmian mechanics because of the time evolution of the hidden variables. Since only position is a hidden variable in Bohmian mechanics, one must also postulate that positions _at different times_ may be connected to different (incompatible) hidden variables. In our protocol we must postulate that the successive FULOs refer to the spin hidden variables _at the same time_. While this is a reasonable postulate, it is false. It is reasonable because the FULOs are just identity transformations for quantum mechanics and they do not change the quantum state at all, so one would expect that they cannot change the hidden variables that encode the \"true\" value of the spin (as confirmed by the fact that the trajectories match in successive FULOs oriented along the same directions). However, it is false because if one looks at the trajectory across two FULOs that are rotated by 180 degrees, one finds a trajectory inconsistent with the hidden variable (see Figure 2). Namely, if one looks at the trajectory along a FULO oriented along the \(x\) direction and then at the trajectory along a flipped FULO oriented along \(-x\), one must conclude that the hidden variable for \(\sigma_{x}\) is flipped by the second FULO. In other words, Bohmian mechanics cannot be used for signaling back in time by our protocol because Bohmian mechanics is an explicit realization of the second way to bypass the argument that we discussed in the previous section. The time dependence of the hidden variables is nontrivial: even unitary evolutions may change the hidden variables of spins.