Friday, February 11, 2011

Heisenberg's Relations and Uncertainty

Quantum Mechanics, in particular the Uncertainty Relations, need indeed a good interpretation. Well, I think that it is more than a matter of interpretation. If its internal logic is self-consistent, then there would not be needed an interpretation. The long discussions about interpretations actually reveal the existence of internal inconsistencies in the formalism of Quantum Mechanics. The "no interpretation" alternative, the "operational interpretation", tries to ignore the inconsistencies by avoiding discussing about reality, focusing only on the operations we perform when making experiments of Quantum Mechanics. I think that what really is needed is to resolve the internal conflicts of Quantum Mechanics. Actually, I think that the expression "interpretation of Quantum Mechanics" is used in fact for alternative theories, which propose mechanisms by which QM is implemented. Because what we can observe is described already by QM, such mechanisms are usually hidden, practically impossible to observe. So, in my opinion, they are named "interpretations" and not "theories" because of the exigencies of modern science to name them "theories" only if they are testable. We may call them "hypotheses", because they are not interpretations - they actually propose new mechanisms, but they cannot be tested, so they don't qualify to the modern definition of the word "theory". Of course, it can be argued that the assumption (superstition?) that Nature really gave us access to all its mechanisms, as if She had the purpose to allow us to test every statement we can make about them, should be kept open to debate.

Seeing the Uncertainty Relations as fundamental is indeed problematic for several reasons. First, they are in fact the mix of two principles. The second of these principles is the Born rule, giving the probability to obtain a given state as outcome of an observation of a quantum state. The Born rule, by specifying the probability, provides the probabilistic interpretation of a wavefunction. If the Born rule already contains the probabilities, I think it would be better if we could see the Heisenberg Relations separated of the probabilities.

If we take the solutions of the Schrödinger's equation - that is, the wavefunctions - as fundamental, then the basic Heisenberg relations appear from their very properties. We just take the relations between the size of the interval of the time (position) and the size of the interval of the frequency (wave vector), known from Fourier analysis. These relations are much more general: if we represent the same wavefunction in two different bases in the space of all possible wavefunctions, there is always such a relation between the corresponding intervals. Of course, an observable (Hermitian operator) comes with its own set of eigenfunctions, which are orthogonal, so it is naturally to obtain similar relations if we refer only to the observables and their commutation relations.

Therefore, the Uncertainty Relations come directly from the wave nature of the solutions to Schrödinger's equation, combined with the Born rule. By "Heisenberg Relations", I will refer to the relations as they appear from the wave nature of the wavefunction, reserving the names "Heisenberg Uncertainty Relations" or "Uncertainty Relations" for their probabilistic interpretation.

In a similar way, the entanglement between two or more particles is in fact a property of the tensor products between wavefunctions representing single particles. When the total state cannot be represented as a pure tensor product (which can be a combination of symmetric and antisymmetric products), but only as a superposition, we have entanglement. When we appeal to the Born rule, the entanglement manifests as correlations between the possible outcomes of the observation of the particles.

The Born rule has been thus tested by all experiments in QM, involving entanglement or not. Being probabilistic, they are tested only statistical, but this doesn't mean that they reveal an intrinsic probabilistic reality.

One central problem of Quantum Mechanics is to accommodate the unitary evolution described by the Schrödinger's equation, and the apparent collapse of the wavefunction due to the observation. There is clearly a contradiction here. If we introduce an internal mechanism to explain this collapse, then we have to make this mechanism able to explain both the unitary evolution and the collapse. This is difficult, because both processes are very simple. In a vector space, what can be simpler than unitary transformations and projections? Any hidden mechanism would have to compete with them. This is why it is so difficult to explain QM in terms of hidden variables, of multiverse, of nonlinear collapse and spontaneous diagonalization of the density matrix caused by the environment.

On the other hand, there are already enough unknown factors even if we consider the wavefunction as the only real element. The Schrödinger's equation gives us the evolution, it doesn't give us the initial conditions. The initial conditions can be partially obtained from observation. Due to the particular nature of quantum observation, our choice of what to observe also is a choice of what the initial conditions were (yes, in the past). This is why the initial conditions are delayed until the measurement is taken. To this, let us add that we do not observe the initial conditions of just a particle, but of that particle and every system with which it interacted in the past - such as the preparation device, which ensures the state of that particle at a previous time. Since such a device is large and complex, we don't really know its initial conditions, so when we observe the particle, we also observe the preparation device, and everything with which they interacted. Therefore, there are much more factors to introduce in the Schrödinger's equation. These factors are complex enough to make the conclusion that the wavefunction collapse is discontinuous not so necessary as it initially seemed. It is possible to have a unitary evolution leading from the state before the preparation to that after the measurement, given that we need to account for the interaction with the preparation device, which also have much freedom in its initial conditions. I described these ideas here, and there is also a video. In this view, the wavefunctions are real, therefore the Heisenberg Relations are real too. By applying to them the Born rule, it follows their probabilistic meaning, the Heisenberg Uncertainty Relations. It would be nice to have an explanation for the Born rule as well, because it is very plausible that it just follows somehow from a measure defined over the space of all possible wavefunctions.

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