The counterintuitive nature of time in Physics series continues with Quantum Mechanics, with entanglement and delayed choice experiments. It is presented the Smooth Quantum Mechanics, which eliminates the discontinuity from the wavefunction collapse. It happens to be deterministic, but the compatibility with free-will is maintained.

Nonrelativistic Quantum Mechanics describes a system by a vector, named state vector, from a complex Hilbert space (a special type of complex vector space, endowed with a special type of scalar product). To the classical quantities, we associate selfadjoint operators on the Hilbert space. The space differs from the time, because there are position operators, while the time is only a parameter.

Schrödinger proposed an equation, describing the evolution of the state of a system. Schrödinger’s equation is of PDE type, and it is deterministic, linear, even unitary (it preserves the scalar product). What we can observe or measure is an operator, representing the observable we want to measure. What we can get as outcome, is that the state vector of the system is one of the observable’s eigenvectors (special vectors associated to each operator). This means that we can never know what the system’s state is, without disturbing it, because there are few chances that the system is already in an eigenstate of the observable.

In the standard interpretation, the system jumps into one of the eigenstates of the observable. We cannot know before in which, but we can know the probability for each possible outcome, due to Born’s rule. This introduces the indeterminism at the very fundamental level of reality. The time gains a strange feature, because it appears that, at any moment, a system can jump in a state without an apparent cause. The Classical Mechanics paradigm identifying the causality with the deterministic evolution lasted for centuries. QM introduced the possibility that a system jump out of the blue, and opened a totally different perspective. To resolve some problems of QM, Hugh Everett III proposed an interpretation of QM which states that each possible jump takes in fact place, but the world splits in many worlds, each of them containing one of the possible jumps. In this interpretation, time itself looks like it is branching, or forking, although the observers cannot check the existence of the other alternative histories. Despites the fact that for each observer, “prisoner” of one of these worlds, the wavefunction collapse and other strange quantum phenomena remain unexplained as before, this interpretation offers a intuitive and unitary view of what happens.

Some of the founders of QM, Einstein, de Broglie, Schrödinger, felt that accepting the indeterminism means to give up the search for a better explanation. Nowadays, when the indeterministic view in QM is well established, they are sometimes presented like conservators, with little understanding of quantum phenomena. This is unfair, because not only they co-initiated the quantum revolution, together with Bohr, Born and Heisenberg, but they also expressed the problems which this new born theory encountered, this leading to a refinement of the theory and its interpretations. Schrödinger explained the idea of entanglement, which springs from the very fundamental principles of Quantum Mechanics. Einstein, Podolsky, and Rosen, proposed an experiment which showed a paradoxical behavior of quantum mechanics, which is in fact the entanglement between two particles that previously interacted. This brings a weird aspect of time: they interacted in the past, and now, by measuring one of them, we can limit the possible outcomes of a measurement performed to the other one. It appears that the wavefunction has a nonlocal character over space and time.

One strange quantum effect is visible by the “delayed choice experiments”, made popular by Wheeler. Wheeler provides the example of a photon emitted by a very distant star. He considers the case when between us and that star there is a galaxy, which bend the light ray, according to General Relativity. According to QM, among the possible experiments we can make with the incoming photon, there are two mutually exclusive. First, we can observe whether it passes through the left, or through the right of that galaxy - the “which way” measurement. The second possibility is to put the two possible ways to interfere one another, like the photon was traveled “both ways”. The problem is that we can make our choice now, long time after the photon was emitted by the distant star, and long time after it was bent by that galaxy. We can choose now what kind of behavior had the photon thousands of years ago. This is really something that bends our intuition on time very much. We tend to believe that the past determines, or at least influences the future, but future influencing the past?

It is usually believed that the wavefunction, when measured, suffers a collapse. The corresponding state vector becomes suddenly projected on one of the observable’s eigenstates. This is a little strange, because it entails a discontinuity in evolution, which we never observed. This discontinuity makes more difficult the preservation of conserved quantities, because usually the conservation laws are effects of the unitary evolution, but a discontinuous jump may break them down. Yet, we haven’t observed such breaking of the conservation laws, nor we had observed other direct evidence of the jump, except our knowledge that we prepared the system to be in one state, and we detect it in another state. In the Smooth Quantum Mechanics eprint, I show that we can avoid the discontinuity of the wavefunction collapse. I use the entanglement between the observed system, and the measurement device that performed the previous measurement (the preparation device), and the possibility of choosing with a delay the initial conditions. What appears to be a jump, is described in a continuous, even smooth way (which is even unitary at a higher level). The past interaction with the preparation device happens in such a way, that it anticipates the outcome of the measurement. This interaction takes place during a finite time, and changes smoothly the state, such that, when it is measured, to be an eigenstate of the observable. I use a mechanism similar to the delayed choice experiment, but which, because of the smoothness, extends indefinitely in the past.

Because the smooth QM provides a smooth description of what was believed to be a discontinuous collapse, it appears that Einstein, de Broglie, and Schrödinger weren’t that wrong. The determinism was also brought back by Bohm, by using nonlocal hidden variables. In the smooth QM, the hidden variables are replaced naturally by the yet to be determined initial conditions. The nonlocality remains in all versions, but the determinism becomes possible just by unitary evolution (Schrödinger’s equation being replaced with the von Neumann’s, because we deal with entangled states). So, we can say that both sides in the Einstein-Bohr debate were simultaneously right, at an unexpected degree.

If the standard QM allows the free-will, so does the smooth version, because the freedom of choosing the observable is exactly the same. The smooth version is deterministic, but the initial conditions are not determined yet completely, and each new experiment adds new information about them. This is why they can be named “delayed initial conditions”.

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