Tuesday, January 6, 2009

Smooth Quantum Mechanics: 3. Registry and Evolving Block Universe

The time evolution appears to humans as a flow, subject to our free-will. The past appears to be frozen forever, and the future seems open to various possibilities. This post compares this view, of Evolving Block Universe, due to Professor George Ellis, with the registry approach of Smooth Quantum Mechanics.

Registry and Evolving Block Universe

The block view and the time evolution view are not as incompatible as they may look at a first sight. We can recover the time evolution by watching the entropy distribution between the events of the block world, and the causal co-relations between them. I would like to compare the registry time evolution with the Evolving Block Universe of George Ellis. Professor Ellis proposes an evolving block universe, perhaps the most credible proposed so far. He explains that the quantum phenomena (in the standard indeterministic interpretation of QM) should have gravitational effects. Consequently, they must change the spacetime. I agree with this argument. Further, he details a theory in which the time flows, evolves, in a sort of presentist way, and the past, which already happened, is “archived” in a block universe. The block universe increases with time, as new “presents” adds to it. The future is not decided yet, and as it happens, it becomes present, and then it is archived. This view is well elaborated, and reflects well our feelings of time flow, free-will, carved in stone past, and open future. On the other hand, I do not agree with Professor Ellis that QM proves the indeterminism. Even so, assuming the indeterminism valid, this doesn’t eliminate the possibility of the standard block view.

Perhaps the most important difficulty of such an Evolving Block Universe is the possibility, offered by QM, of deciding the past events chronologically after they took place. This implies that we have to wait to archive the passed times. Moreover, it is possible to never be able to determine the past completely. Consider Wheeler’s delayed choice experiment, with the photon emitted by a distant star. The observer watching the star will decide whether to measure the “both ways”, or the “which way”. Her decision affects the past history of the observed photon, hence of the observed star. Of course, it is unlikely that she affected the star’s state in a significant way, but she affected it at least in a small way. Until the observation, the photon, hence the star (by entanglement), was in an undefined state. Assume now that the photon is never observed, and escapes far from any planet and any possible observer. The Universe will remain in an undetermined state. So, we cannot say that the past block will be ever created. On the other hand, my proposal of a “registry” of incomplete initial data which increases with each observation, relying on Smooth Quantum Mechanics, allows the possibility that the state of the Universe remains undetermined. Professor’s Ellis idea of foliating the spacetime so that the spacelike surfaces contains the wavefunction collapses may be unreachable, because the entanglement makes the collapses impossible to be ordered temporally. I am afraid that the entanglement can be complicated enough. The measurements of the spins of the two electrons in the EPR-B experiment can be in any spacetime relation. We cannot consider that the wavefunction collapse takes place necessarily along such preferred spacelike surfaces, which are compatible with a spacetime foliation. It is easy to see that, if we associate spacelike surfaces to the collapse, it is possible that these intersect in complicated ways. Moreover, collapse can take place also between events that cannot belong to the same spacelike surface, being for example one in the other’s future.

The standard BU attempts to express the temporal structures in terms of timeless structures. We can consider it, in a way, as a research program of explaining the time itself in terms of timeless structures. But, by adhering to a presentist view, and by reducing the BU functionality to a purely archiving role, there is the danger of explaining the time by appealing to time in a circular way: the EBU includes the passed time in the archived BU, but the evolution happens in a metatime. Another interesting feature the BU has is that it contains all the physical fields in its description. By giving a special role to the present, we introduce a feature which has no correspondent in the matter fields. The BU accounts for the physical fields, but it cannot include an intrinsic present, and maybe doesn’t even need. Yet, if it would need to mark the present, a “BU with a bookmark” would solve the problem.

The registry view is compatible with both time evolution, and with the standard block universe view. And it shares with the EBU picture the compatibility with our feelings of time flow, free-will, open future, but not the carved in stone past.

Smooth Quantum Mechanics: 2. Registry and Time Evolution

In the Smooth Quantum Mechanics, the evolution is deterministic, but the initial data is not determined. The registry is a collection of partial initial data, which is in general incomplete. This leads in particular to a version of the Many Worlds Interpretation: the Many Registries Interpretation.

Registry and Time Evolution

In the Smooth Quantum Mechanics, the evolution is, for the complete system, deterministic and unitary. The randomness appears because of the incomplete knowledge of the initial conditions. Not only that we don’t know these initial conditions, but they are even not defined, until we perform the measurement. This is because of Bell’s theorem. We can choose what observable to measure, and each observable limits the possible outcomes in a different way. We can have two observables so that it is clear that their outcomes are relatively incompatible. When we choose one of them, we chose what already happened. We determine the past in the active sense of the word “to determine”.

Each observation we make increases the constrains of the solution of the evolution equation. The set of (delayed) initial conditions known at a time is named “the registry”. This registry increases as new measurements are related causally to the registry. For the observer, the time evolution correlates with the registry expansion. Some measurements are independent, and others are correlated, since they measure entangled particles or degrees of freedom. The registry is a network of such events, and the way it expands is correlated with the time arrow. The physical laws being time symmetric at a fundamental level, the relations between the measurements are probably correlated with the thermodynamic time arrow. Therefore, the registry can be viewed in a timeless way, providing a block world (or block universe, or BU) view. On the other hand, it can be viewed as being correlated with the time’s arrow, providing an evolving world view.

Many Registries Interpretation of Quantum Mechanics

The way the registry expands by adding new observations is not determined by the present state of the registry (except when the registry is complete, providing a full description of initial data). This means that we can expand the registry in many ways. These multiple possibilities remind us the Many Worlds Interpretation of Quantum Mechanics, in which the world splits with each new measurement, according to each outcome. There is a difference, in that the split in the MWI is due to the indeterministic character of the wavefunction collapse (although the “total” wavefunction evolves deterministically). In the Many Registries Interpretation, each world is deterministic, but the observer has not identified/chosen yet her world, so she perceived the evolution as indeterministic. By adding new observations to the registry, she can increase the information about the world. Thus, she selects the world, and she even seems to have a small possibility of choosing the world, by choosing the observable.
By adhering to Smooth Quantum Mechanics, we can keep the idea of MWI, in the form of Many Registries Interpretation. The idea of “registry” reconciles the indeterminism perceived by the observer, with the fundamental determinism of the unitary evolution.
Even if we don’t know if we will ever be able to provide experimental support for one or another version of the MWI, I think that the MWI has at least a great pedagogical value. Also, it allows a better reconciliation of the block world with the apparent openness of the future.

Sunday, January 4, 2009

Smooth Quantum Mechanics: 1. The Smooth Particle

Smooth Quantum Mechanics provides a way to avoid the discontinuity in the wavefunction collapse. This post presents the wave-like nature of the particles in Smooth QM, as compared to the standard QM. The probabilities also have a different character, being rooted in the initial conditions, instead of discontinuities.

In Smooth Quantum Mechanics, there are only (entangled) waves.

In standard QM, the particle behavior is manifest when the observable is a position operator acting on the Hilbert space. In this case, the system is found in an eigenstate of the position operator, which is a Dirac distribution (a generalized function which is zero everywhere, except in one point, where is infinite, such that its integral is 1). The wave and particles are, in standard QM, all vectors in the Hilbert space, being therefore wave functions or distributions. When expressed as wavefunction (in a basis of positions), the “pure waves” are eigenstates of momenta, and the “pure Dirac distributions” are eigenstates of positions - these are extreme situations, in which infinities occur.

One interpretation of the waves in QM is that they underlie probabilities of finding the particles at a given point in space. Of course, this is true, grace to the Born rule, but this does not necessarily mean that the point particles are the fundamental ones, and the wave represents only probabilities. Yet, many like to think at particles as being fundamentally point-like, therefore the waves need to be interpreted as underlying probabilities.

In Smooth QM, it is considered that the only real physical states of the system are smooth: it is employed a space of smooth wavefunctions of finite norm. By completing the space, we obtain a rigged Hilbert space, which contains state vectors that are abstract, being distributions or having infinite norm. By observing the position, we get only a smooth wave concentrated around that position, but not a Dirac distribution. The position is never determined to be in a point, but in an open subset of the space. I think that the two extremes: eigenstates of pure momenta and pure positions, are non-physical abstractions, because they bring in infinities, but they are useful to express and explain the wavefunction duality. In Smooth QM, by not allowing the distributions, by eliminating the infinities, the unity between the two complementary aspects is more manifest and more physical. All particles are waves, in various shapes, depending on the observable. Of course, the Hilbert space is very useful, and it is easy to solve the equations in this space, and many operations become simpler. But in Smooth QM it is considered abstract. The wave is physical, not a probability wave.

Therefore, we return to the old idea of particles as wave packets. One main problem of this view was the dispersion of the wave packet. In Smooth QM, the laws are the same, so the dispersion is not eliminated, but the mechanism is such that the dispersion is no longer a problem. A measurement of position, finding the wave-like particle localized very well, will imply dispersion both in the future, and in the past. The waves converge, the wave packet is “con-persed” until the small sized packet is obtained and determined by measurement, and then it disperses again. The “delayed initial conditions” mechanism allows this.

The probabilistic behavior occurs only when we measure an observable which does not contain the observed state among its eigenstates. In this case, the Born rule expresses the probabilistic character, but the probability in Smooth QM resides only in the undetermined initial conditions. Here “determined” has two meanings, active and passive: to cause, and to measure. The probabilistic behavior occurs when a collapse occurs, but in this approach the collapse is smooth and deterministic, and the probabilities reside in the initial conditions. Even the Heisenberg relations, which are often thought as representing irreducible probabilities, are not necessarily such. They can be obtained from Fourier analysis, and apply for deterministic waves. The probabilistic character occurs because of measurements, which entails the (smooth) collapse, which brings the probabilities of the unknown initial conditions.

Although I said that the waves are physical, and real, we should not forget that they are entangled. For a particle, the wave can be a field depending on space and position, but for more particles, we need to consider the tensor products, therefore the entanglement.

Friday, January 2, 2009

The Counterintuitive Time: 5. Quantum Time

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”.