Unitary Flow

Wooden Alien

2012-02-11T05:54:00.000-08:00

Is this is an alien, hidden in the closet in my hotel room?

Halkidiki, Greece, June 2007.

We evolved to find pattern in everything, because we live in an environment, and we need to interact with it. Even if this is for survival only, we need to decode the environment - to find food, water, to identify friends and enemies and so on. This is why we are that good at finding patterns.

This doesn't mean that we really understand the reality, we didn't need to understand it to evolve. We just needed to identify patterns, the way they are organized, to be able to predict what's waiting for us around the corner. Any story we tell us as an explanation is good, so long as it helps us guessing what's next. The story doesn't have to be true, just practical.

Sometimes it's usefulness is not in predicting what's to come, but to give us confidence that we understand a bit of this world. We learn pattern recognition by experience, and the experience being individual, what we learn is subjective. What we see is not what it is. We may see in a piece of wood a human face, even if there's no face there, because we are trained to see human faces. Our experience provided us filters, which allow us to recognize patterns when we don't have full access to all information. But the drawback is that we fill the blanks and connect the dots, in a subjective way.

Our subjective filters may put us in conflict with the others, because we see the things differently than they do. I don't say that the truth is relative, or that there is no right or wrong. What I say is that we should be very careful when judging, when deciding what's right and wrong, because we are allowed by our filters to see different patterns than our peers do.

A person, scientist or not, should always remember that appearance is not the same as reality. The way we decode what is presented to our senses is not how reality is in fact. Or if it is, we don't know, because we only can access the patterns which the filters from our minds allow us to access. We only see the shadows of reality, as illustrated by Plato's allegory of the cave in The Repubic.

When our subjectivity filters the patterns in a way shared by others, they become organized in complex constructions like the paradigms described by Thomas Kuhn in his masterpiece The Structure of Scientific Revolutions.

Is semi-classical gravity wrong?

2012-01-22T07:44:00.000-08:00

Semi-classical gravity is not considered fundamental, yet it escaped to experimental falsification. Of course, maybe we don't have yet the technology, or at least ideas of experiments we can do, to falsify it. It would be nice to be able to differentiate it experimentally from various quantum gravity approaches. But theoretically, it stands pretty well: being the most straightforward union between general relativity and quantum mechanics, it inherits their successes.

Are we sure that the theoretical reasons to reject it are so good? The regularization works promising for the semi-classical Einstein equation. The main problem seems to be that of singularities, but is there any evidence that this will not be solved?

One possibility is to rewrite Einstein's equation in a different way, which is equivalent to the original, but works in singularities. A simple type of metric singularity is when the metric becomes degenerate. The metric can be smooth (hence its components in a chart remain finite), yet the Kretschmann scalar can diverge, sure sign of a singularity. In arXiv:1105.0201, arXiv:1105.3404, arXiv:1111.0646 is developed the mathematics of such metrics, and it is consistent and without infinities, if the proper variables are used (for example, we have to use $g_{ab}$ and $R_{abcd}$, but not $g^{ab}$ and $R^a{}_{bcd}$).

Once we have this extension of the semi-Riemannian geometry developed, we need to show that we can apply it to the singularities of the Schwarzschild, Reissner-Nordstrom and Kerr-Newman singularities. In the standard expressions of these solutions, some components of the metric diverge. But there are coordinates which make the metric smooth - similar to how the Eddington-Finkelstein coordinates removed the apparent singularity on the event horizon, only that in our cases the metric becomes degenerate at the singularities. So, we can now write an equation equivalent to Einstein's, valid even at the singularities of these black holes, or at more general black holes which change in time, for example by Hawking evaporation (arXiv:1111.4837, arXiv:1111.4332, arXiv:1111.7082, arXiv:1108.5099). We can write field equations on such spacetimes, and the information can now pass through these singularities. Similarly, we can write this extend version of Einstein's equation through a FLRW singularity, without having problems with the infinities (arXiv:1112.4508).

About the problem of the wavefunction collapse. If it is discontinuous, it will lead to violations of the energy conservation. It will also imply (never observed) violation of the conservation of other quantities like spin or electric charge. So, maybe the wavefunction remains all the way unitary. How can this be possible, when the projection postulate seems to tell that it is discontinuous? A possibility is described here.

Local Hidden Variables Correlations

2011-08-10T23:57:00.000-07:00

To generate random angular momentum variables, press "Generate Random Data". In this case the correlations will be the classical ones (along the blue line), as it is predicted.To test it on random generated orientations and plot the result, press "Plot Data".Alternatively, if anybody believes that a local hidden variable theory can provide a set of data which gives the correlation of Quantum Mechanics, he or she can paste the data instead of randomly generating it, and then plot it.For example, Joy Christian claims in arxiv:0806.3078v2, page 4, that an experiment he describes can provide a list of angular momenta which gives correlations = -cos of the angle between the two orientations chosen by Alice and Bob (the green curve). He claims by this that his local hidden variables can reproduce the outcomes of the EPR-Bohm experiment.More exactly, Joy saids that after an experiment involving balls which explode in halves which have total angular momentum 0, a list of angular momenta can be collected. The second part of his experiment is to randomly generate on a computer pairs of directions a and b in space, and calculate the result using equation (16) from his paper, page 4. He then claims that the result will be -cos of the angle between the two orientations (the green curve), rather than the linear function represented in blue.My application does exactly the second part of Joy's experiment. If Joy Christian or anybody else can produce this kind of data, they he can test it in this application. I already provided a mathematical proof that the only possible correlation depending only on the angle between a and b is the linear one, but there are people who don't trust the mathematical proof. Therefore, I challenge them to produce the data which will contradict my proof by counterexample.Given that the output of the first part of Joy's experiment is just a list of angular momenta, you can produce it by performing the first part of Joy's experiment. But I will not require anybody to get the data only by actually making the experiment. Joy can produce the list by any means he wants, I will not constrain him to make the experiment. Just to provide a list of angular momenta which give his prediction.I used JavaScript, so that anyone can easily verify the source code.P.S. For the moment, there is a problem viewing the results in Internet Explorer, so please use Mozilla, Chrome or Opera instead.P.P.S. Thanks to Florin Moldoveanu for the uniformization of the random generator.

On Singular Semi-Riemannian Manifolds

2011-05-19T06:22:00.000-07:00

Abstract
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this article we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.
On Singular Semi-Riemannian Manifolds on arXiv

"Bit from It" vs. "It from Bit"

2011-02-16T01:16:00.000-08:00

Julian Barbour presented his essay "Bit from It" at the FQXi essay contest Is Reality Digital or Analog?.

The essay is beautiful and I agree with the conclusion "Bit from It", in a way I will try to make clear. But I disagree with the way the conclusion was reached - it seems to me that the central part of Wheeler's 'ontology' "It from Bit" was overlooked, and this makes it look naive, while it is in fact very profound.

In a classical world, Wheeler's "It from Bit" would be obviously silly. When we measure something, we can write down the outcome as a string of digits, and by collecting all these digits we can determine the state. In such a world, "bit" would indeed originate from "it".

But Wheeler is discussing the quantum world. And for Wheeler, the quantum world is not just "classical world" plus "probability". Julian Barbour said: "Crucially, even if individual quantum outcomes are unpredictable, the probabilities for them are beautifully determined by a theory based on 'its'", but this is not the whole story. If this would be all, then he would indeed be right to say "I see nothing in Wheeler's arguments to suggest that we should reverse the mode of explanation that has so far served science so well". Julian Barbour tries to understand how Wheeler could do so trivial mistakes: "Wheeler's thesis mistakes abstraction for reality", and "A 'bit' has no meaning except in the context of the universe". Yet, there is no such a gross mistake.

Wheeler's "It from Bit" can be understood in the context of the "delayed choice experiment". He realizes that it is not enough to specify the outcome, but also what we measure - for example "which way" or "both ways" in the Mach-Zehnder experiment. But he realizes that our choice of what to measure determines how the state was (yes, in the past). This is the key problem of quantum mechanics, and this is the fundamental obstacle of all realistic interpretations of quantum mechanics: we choose "now" what to measure, and our present choice dictates how the state was, long time before we made our choice. We can think that there is an ontology behind the outcomes of our measurements, as in the classical world. But the "delayed choice experiment" shows that the "elements of reality" depend of the future choice of our measurements. And the outcomes depend of these choices too. So, it is in fact "the choice of what to measure" (Hermitian operator) plus "the outcome" (eigenvalue) that forms the "Bit" from Wheeler's "It from Bit". And the "It" is in fact the eigenstate corresponding to the obtained eigenvalue, given that the observable was that particular Hermitian operator. Wheeler was not that naive to think that eigenvalues determine eigenstates by themselves, without considering the Hermitian operator, so he accounted well for the prescription "A 'bit' has no meaning except in the context of the universe".

The central point of Wheeler's "It from Bit" is that the reality of today depends on the choices we make tomorrow, when we decide what to observe, and of the outcomes of the observations. He compares this with the game of 20 questions, when we try to guess a word by asking 20 yes/no questions, under the prescription that the choice of the word is not done at the beginning. The person who "knows" the word changes it by wish, so long as it remains consistent with the answer she already gave to our question. Wheeler wants to emphasize by this the similarity with the quantum state we try to determine, but which depends on what we choose to observe. This is why he was led to the idea that the state of the universe (it) results from the observations (bit).

I give more credit than Julian Barbour to the "It from Bit" philosophy - I view it as a way to present a central problem of quantum mechanics. I think, nevertheless, that it is exaggerated to conclude from this, as many do, that the world is digital. It may be or it may be not, but we should not force the conclusion. After all, the "It from Bit" philosophy is intended to clarify some points of a theory based on continuum - Quantum Mechanics.

My viewpoint on "It from Bit" is that we should regard the outcomes of measurements as "delayed initial conditions" for the Schrödinger's equation. I presented my view in this article, and this video. A solution of a partial differential equation like Schrödinger's is determined by a set of initial conditions. Classically, the initial conditions can be determined from future observations. In Quantum Mechanics, the future observations determine the state in the two meanings of the word "determine": passive - "find out what it is" (by the selection of an eigenvalue of the observable), and active - "choose what it is" (by the choice of that observable). Another central problem is that two consecutive observations of the same quantum system are incompatible, if the observables do not commute. That is, they impose incompatible initial conditions to the wavefunction. But, the second measurement is not, in fact, a measurement of the same system. The system interacted with the first measurement device, and this measurement device has many degrees of freedom which are not determined yet. So, the second observation measures in fact the composed system - the observed system plus the apparatuses used for the previous observations, and all the past interactions of the observed system. This may offer enough degrees of freedom to maintain the unitary evolution and to avoid a discontinuous collapse of the wavefunction.

My interpretation comes with a realistic wavefunction, which is not yet determined among the possible wavefunctions, but whose "delayed initial conditions" are determined by all future and past observations. I think that we cannot avoid the idea of "delayed initial conditions", no matter what "It" we choose to consider as the underlying ontology.

My view is therefore that "It from Bit" and "Bit from It" are reciprocal: a set of possible "It"s (solutions to the Schrödinger's equation), a set of possible "Bit"s (observations, delayed initial conditions) and the Universe is a pair (It, Bit), so that the "It" and the "Bit"s are compatible.

On the other hand, the "Bit" itself is part of the solution of the Schrödinger's equation, that is, of the "It". This is why I said at the beginning that I agree with "Bit from It". But if we have some "delayed initial conditions" - the "Bit"s - the "It" that satisfies to them is not necessarily unique. So, in fact, what we have is not a pair (It, Bit), but a pair ("It"s that satisfy to the observed "Bit", the observed "Bit"). There is a relation "one-to-many" between the "Bit" and the "It"s. The "Bit" appears to be discrete, but the "It" may very well be continuous. So, although "It from Bit" reflects an important aspect of Quantum Mechanics, it should not be taken too far.

Heisenberg's Relations and Uncertainty

2011-02-11T05:09:00.000-08:00

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.

"Explanation" between concrete and abstract

2011-02-11T05:04:00.000-08:00

I realized that an apparently well-understood word, "explanation", may lead to controversies in discussions about the foundations of physics. The foundations are already controversial enough, but this adds even more to the confusion. It gives you a double featured feeling: on the one hand, of being misunderstood, and on the other hand, that you don't understand where the interlocutor is going on.

What is an "explanation"? Probably the most usual meaning is that explanation is to reduce the unknown to the known, the unfamiliar to the familiar. When this happens, we get the sense of understanding.

Even since childhood, we had so many questions, and the grown ups explained them - reduced the unfamiliar to more familiar notions. In school, the teachers continued to provide us explanations, and we appreciated most the teachers who managed to make the unclear things more intuitive for us. When reading about the foundations of physics, we usually start with popular physics books. The most recommended such books are those providing the feeling of understanding, appealing to our intuition. When we try to read something more advanced, even if it is recommended by our favorite pop-sci books, we find ourselves in a totally different situation. Instead of finding the deeper explanations we are looking for, we find ourselves thrown in the turbulent torrents of the abstract mathematics, drifting without an apparent purpose. And what is most annoying, these textbooks and articles full of equations actually claim to explain things!

Why is this happening? I think that they are guided by another meaning of the term "explanation": "to give an explanation to a phenomenon is to deduce the existence of that phenomenon from hypotheses considered more fundamental. For example, when from the principles of General Relativity was deduced the correct value from the perihelion precession of Mercury, it was considered that GR explained this precession. On the other hand, the deflection of light by the Sun was considered a prediction. After the full experimental confirmation, it became an explanation. I consider that "prediction" is just a temporary status of a scientific explanation, and that the fact that many explanations are first predictions is a historical accident.

There seem to be a similarity between principles/phenomena and axioms/theorems. This similarity suggests the reason why mathematics plays such an important role in the explanation of phenomena. To deduce more from less, complicated from simple, diverse from universal, this means to use logic and mathematics. And there is no limit of the difficulty of the needed mathematics, even if the principles are not that difficult.

This notion of explanation, I understand now, it is not shared by all of us. The reason is simple: because "explanation" usually means to reduce the unfamiliar to familiar. When somebody claimed to explain a phenomenon, we expect him to show how this strange phenomenon can be described in more familiar, concrete terms. Instead, we find that he or she starts describing it in more abstract terms. How come that such more and more abstract terms are shamelessly named "more basic principles", "more elementary principles" and so on? Isn't this a lie?

Maybe the explanation by "reducing to concrete things" has pedagogical reasons, and the explanation by "reducing to universal principles" is in fact foundational research. But does this means that the gap between pedagogical and scientific explanation should grow as it does nowadays? Wouldn't be much, much better to have a mechanistic explanation? After all, Maxwell sought for such an explanation of the electromagnetic waves, even though he had the equations! The ether theorists of the XIXth century tried to reduce electromagnetism to vibrations in a medium. This tradition still continues, and we encounter on a daily basis renowned scientists trying to explain things which other renowned scientists consider to be already explained: electromagnetism, wave-particle duality, gravity, entropy, the Unruh effect, spacetime, time, black holes and so on.

Probably it would be better to have a mechanistic explanation of everything. This would definitely help the public outreach of physics, and will help physics to advance faster. This may have a huge impact on technology, and on our lives. But who can bet that God, when created the world, bothered about our need to reduce the things to what we know? Why would the universe care about our limited understanding, when decided what principles to follow? Who are we, why would we be so important? I think that, although it would be desirable to find concrete, familiar universal principles behind this complex and diverse world, we have no guarantee that this will ever happen. "You shall not make for yourself a carved image, or any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth."

The definition of "explanation" as a reduction to universal principles has its own advantages, given that we do not take these principles as ultimate truths, but just as hypotheses. One of these advantages is that it allows us to equally appreciate theories which seem to contradict each other. We can appreciate its explanatory power in the sense stated above: as its efficient encapsulation of a wide variety of phenomena in fewer, simpler, and more general principles. This doesn't mean that we should consider these principles as being "true". It is not about being "true", just about encapsulating as much phenomena as possible in as few principles as possible, even if these principles are more abstract. If we insist to become fans of one theory or another as the ultimate "truth", we may reduce our capacity to grasp other explanations. This would not be a problem, if we could prove our theories beyond any doubt, but the truth is that we cannot, no matter how convincing they may look to us.

The Essence of Quantum Theory

2010-04-25T02:20:00.000-07:00

The purpose of this short post is to provide a very brief presentation of Quantum Theory.

Short:

In quantum theory, particles are waves of various shapes. You cannot directly observe the waves, only some of their properties. Each property is well defined only for some of the possible shapes. There is no shape for which the properties "position" and "momentum" are simultaneously well defined (Heisenberg's principle). When you observe a property, you find the wave in a shape corresponding to that property (like magic!), without regard of its previous shape. Entanglement: n particles are a single wave in a space with n x 3 dimensions, they don't have individual shapes.

Details:

In classical physics, particles are points moving on well-defined trajectories. This picture turned out to be an approximation: a particle is in fact a wave (although there is no waving medium for this wave). We know it is a wave, because it interferes, it can be diffracted, its allowed states in an atom are those corresponding to an integral number of wavelengths, and it is governed by a wave equation. As a wave, it has no definite trajectory, and insisting in discussing in terms of position and momentum as for point particles leads to problems.

But you can't observe the wave directly, only classical properties, like position or momentum. Each property you observe is well defined only for a particular set of possible shapes of the wave. When you observe its position, the wave appears to be concentrated at a point, but it has an undefined momentum. Conversely, the possible shapes that have well defined momentum have no well defined position – they are spread in all the space. Similar things happen when you want to observe any other classical property.

The first strangest thing about quanta is that when you look at them, they take precisely one of those shapes corresponding to the property you observe, without regard of their previously known shape. If further you try to observe another property, which is not well defined for the previously observed shape, you will find the new kind of shape, allowed by the new property. Knowing its shape before an observation, you can not predict which of the allowed shapes you obtain, but only the probability for each allowed shape.

The second strangest thing is the entanglement. When dealing with more particles, say n, they are not described by individual waves, but by a single wave on a space obtained by multiplying the usual three-dimensional space with itself n times. This means that after two particles interact, they have no individual shape, but a common shape on this space with 6 dimensions. We can still observe one of the particles, and obtain a particular 3-dimensional shape for it, but if we try to observe both particles, the shape of one is dependent on the shape of the other. The strangest part is that their shapes are correlated even if the particles are separated by very large distances.

Are vector bundles fundamental in Physics?

2010-04-22T02:29:00.001-07:00

Vector bundles and gauge theory

The idea in Gauge Theory is that the fields of the known forces can be expressed starting with some principal bundles and their associated vector bundles. To be more precise, let's consider Maxwell's electromagnetic field $F_{ab}$. It can be represented with the help of a principal bundle of group $U(1)$, and a connection on this bundle. The connection corresponds to the electromagnetic potential, and the curvature to the electromagnetic field. It is known that we can modify the potential to $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$, and obtain the same $F_{ab}$. In terms of bundles, this transformation corresponds to a gauge transformation of $\mathbb C$ by the action of $e^{i\theta(t,x)}$. The connection will appear to depend on the gauge, but the curvature is gauge invariant.

A bundle is just another manifold

Both principal bundles and vector bundles are differential manifolds (that is, topological spaces which looks locally, from topological viewpoint, like a vector space with with a fixed number of dimensions, and on which we can define partial derivatives). A fiber bundle over spacetime looks locally like the cartesian product between the spacetime and a fixed manifold named fiber. For the vector bundles the fiber is a vector space, for the principal bundle it is a Lie group. The $U(1)$ bundle looks locally like a cartesian product between the spacetime and a circle. This space is 5-dimensional, and it was used by Kaluza and Klein in their attempt to unify electromagnetism with gravity by using a 5-dimensional version of general relativity.

After the electromagnetic force was understood as a gauge field, Yang and Mills provided a generalization which allowed us to see as gauge fields also the strong and electroweak forces. It seemed as easy as replacing the $U(1)$ group with a non-abelian group like $U(2)$ for the electroweak force, and $SU(3)$ for the color force. New bundles resulted, and they can be viewed as well as spacetimes with more dimensions, from which some are compactified.

The obvious problem with these extra dimensions is that we cannot "see" them. What explanation is that we cannot test? To avoid this questions, these dimensions are referred as corresponding to "internal spaces", and the Kaluza-Klein interpretation is in general avoided, being preferred that in terms of bundles.

What is more fundamental, the field or the connection?

It was believed that the potential is only a mathematical trick to simplify Maxwell's equations, and that it has no correspondent in reality. There are some reasons to change this view.

One is, as I explained here, in chapter III., the following. Maxwell's equations contain constraints imposed on the field for equal time, that is, between the values of the field at spacelike separated points (Gauss' law). This may seem a little bit acausal, because requires the initial conditions at two spacelike separated points to be related. Of course, the separation between the two points is infinitesimal, but it still exists, and has non-local consequences. In terms of the potential, these constraints are no longer needed. If we consider the connection as fundamental, then the curvature will be a derived field. It will still obey Gauss' law, but this time just as a consequence of being associated to the connection, which is the true fundamental field. And the connection is not constrained.

Taking a charged field, such as the Dirac electron field, under a gauge transformation it is multiplied by $e^{i\theta(t,x)}$. The Dirac-Maxwell equations maintain their form, if we apply the corresponding gauge transformation to the potential. This allows us to perform an experiment to see whether the potential is a real field, or just a mathematical trick. This experiment was imagined by Werner Ehrenberg and Raymond E. Siday, and Aharonov and Bohm, a decade later. It was verified experimentally by S. Olariu and I. Iovitzu Popescu, and confirmed two years later by Osakabe et. al..

Basically, this effect shows that the electromagnetic potential has a fundamental nature. But how can a potential be the fundamental quantity? Which potential, considering that there can be an infinity such choices, related by $A_a(t,x)\mapsto A_a(t,x) + \partial_a \theta(t,x)$? The only way known for this is if it represents a connection on a $U(1)$-bundle. This way, the potential is just the expression of the connection, in a particular frame on the bundle. Gauge transformations are just changes of that frame.

The Aharonov-Bohm effect is interpreted topologically as an effect of the holonomy of a connection on this bundle (which is the electromagnetic potential). These properties are captured by Wilson’s loops.

Are those "internal spaces" real?

It is easy to check the number of dimensions of our space: it is the number of coordinates required to indicate the position of a point, that is, 3. The number of numbers needed to express a rotation, 3(3-1)/2=3, indicates also that we live in a 3-dimensional space. How can we check the extra, "internal" dimensions? We just count the numbers needed to represent them. Since the electromagnetic potential can be changed in a way indicating a rotation of a circle, we conclude that the internal space has one dimension. It is the same as in the case of the 3-dimensional space. The only difference is that we can actually move in this space, and this is why we consider it real. We cannot move in the internal dimensions. But can we, at least, send particles to move in those dimensions?

In fact we can. The Aharonov-Bohm effect shows that we can rotate the wavefunction of an electron. We can compare the rotation of a part of the wavefunction of an electron with that of another part. To do this, we just make them interfere, and see the relative rotation between them. Isn't this remind us of comparing the speed of light in two arms of the Michaelson-Morley interferometer? Only that the Aharonov-Bohm effect succeded, and showed that there is an "internal rotation".

Now, it is time to remember the notion of existence as it is used by mathematicians. Something exists from a mathematical viewpoint if it is logically consistent. The 5-dimensional spacetime (3+1+1) of the electromagnetism exists, in this respect. Did the rotation verified by the Aharonov-Bohm effect confirm its physical existence? In fact, we can take for the internal space, instead of a circle, the complex space $\mathbb C$. The group $U(1)$ acts as well on this space, and we can think that the physical spacetime is in fact 6-dimensional (3+1+2). What is the true number of dimensions? I would say that this number is given by the number of dimensions of the $U(1)$-bundle, that is, 3+1+1. And the internal space happens to be a circle because the $U(1)$ group itself is, topologically, a circle. It has one dimension too. And both the circle bundle and the $\mathbb C$ bundle are associated to this principal bundle, that is, they are obtained from representations of the $U(1)$ group.

OK, so the space dimensions are more real for us, because we can move almost freely in these dimensions. Time is the fourth, at least mathematically, and some people can accept that it is the fourth physically too. They think that this is true, because of the great beauty and symmetry of the Lorentz group. But the internal dimensions, have they more than a mathematical existence? We can ask as well whether the three space dimensions are true or not. What if the real number of space dimensions is two, as the holographic principle suggests?

Do we have a criterion to distinguish between real dimensions and simple mathematical constructions in physics? Can this criterion be the experiment?

Why are vector bundles natural in Physics?

2010-04-18T14:09:00.000-07:00

Aren't usual vector fields enough?

When we work with a space $M$ (a differentiable manifold in fact), we may need to consider fields on that space. The fields can be scalar, vector, tensor, spinor fields, depending on the possible values they can take - scalars, vectors, tensors, spinors. But all these can be considered vectors in some spaces, so in general the fields will be considered to be vector fields.

We can think that, considering vector fields on a space, it is as simple as considering functions on that space $M$, valued in a vector space $V$. Unfortunately, this is not the case, and there is a very good reason for this. I will explain it here.

When working with a function $f:M\to V$, we can represent it by its graph, which is in fact a subset of the cartesian product, $\{(x,f(x))|x\in M\}\subset M\times V$. Therefore, we may hope that all the vector fields on $M$ valued in $V$ are subsets of $M\times V$. If the base manifold is the sphere $S^2\subset \mathbb{R}^3$, its tangent vector fields cannot be, in general, represented as subsets of the cartesian product $S^2\times\mathbb{R}^2$ (we say that $S^2$ is not parallelizable). This and other simple examples force us to consider a more general definition of vector fields.

On the other hand, there are spaces with which this representation works always. For example, we can take $M=\mathbb{R}^n$ or a simply connected open subset of it, $M\subset\mathbb{R}^n$. All possible vector fields of such an $M$ can be represented as subsets of $M\times V$.

The idea behind the vector bundles was to consider the base space $M$ as being covered by open sets like above. The restriction of a vector fields to such an open set $U\subset M$ can be represented as a subset of $U\times V$. But the way they are glued together can vary very much, because when they are glued together, the vector space $V$ can be transformed relatively to $V$ on another open set. Take for example a circle as the basis manifold, and consider as a vector space the Euclidean one-dimensional space. We can glue it to each point o the circle in two ways: as a cylinder, and as the Mobius strip. The idea is that we can cover the base manifold with opens such that the way we associate vector spaces to its points is trivial on each open set from the covering.

The basic point is that, in order to have fields of any kind on a manifold, you need bundles. The fields are "sections" in the bundles. Now, these fields can be combined as we do with the vectors. In fact, what we can do with the vector spaces, we can do with vector bundles as well. We can construct direct sums, duals, tensor products.

Vector bundles and quantum entanglement

There is an important difference between two types of tensor products. The fields which are sections of a given vector bundle $E\to M$ form themselves a vector space $\Gamma(E\to M)$. Two such vector spaces of sections can as well be tensored. The tensor product $\Gamma(E_1)\otimes \Gamma(E_2)$ of two vector spaces of sections of two bundles over the same base manifold $M$ is larger than the vector space defined by the sections of the tensor product of the two bundles, $\Gamma(E_1\otimes E_2)$. The first contains nonlocal fields of the form $\phi(x,y)$, while the second contains only local fields, of the form $\varphi(x)=\phi(x,x)$. $\Gamma(E_1)\otimes \Gamma(E_2)$ are no longer sections of a vector bundle. The entangled states in Quantum Mechanics are represented by such nonlocal fields.

One of the most important applications of vector bundles in Physics is related to the Gauge Theory. We will discuss more about this other time.

Polyhedra and Groups

2009-06-06T14:47:00.000-07:00

November 21, 2004

The central idea of this article is a direct, geometric method to multiply/compose permutations, by using polygons and regular polyhedra.

I begin with a short review of the properties of regular polyhedra and permutation groups, as well as of the relations between these two areas. These relations led me to the geometric method of multiplying permutations.

Introduction
Regular polyhedra

The regular polyhedra, also known as platonic solids, are represented below:

Inscribing one regular polyhedron in another

There are many ways to inscribe one regular polyhedron in another. One useful case is that when each vertex of the first polyhedron lies at the center of a face of the second, and each face of the second polyhedron contains one vertex of the first.

The cube can be inscribed in this way in the octahedron, but also reciprocally. The same reciprocity holds between the icosahedron and the dodecahedron. Such polyhedra are said to be dual. The tetrahedron is self-dual.

The following ways to inscribe regular polyhedra will be useful too: one or two tetrahedra in a cube, and cube in a dodecahedron.

Permutations

You can permute the set {1, 2} in two ways: (1, 2) and (2, 1). What about the set {1, 2, 3}? It is easy to see that we have six possibilities: (1, 2, 3), (2, 3, 1), (3, 1, 2), (2, 1, 3), (3, 2, 1), (1, 3, 2). The result is the same, for any sets of three objects, not only numbers. The number of permutations depends only on the number of elements of the set, and not on their nature. For a set with n elements, this number is 1·2·...·n, and is named the factorial n! of the number n.

An ordered set of the first n positive integers can also be used to express the permutation, as a specific operation to be applied to another ordered set. For example, (2, 1) shows that the elements of an ordered set of two elements are reverted. (2, 3, 1) shows that an ordered set of three, for example (1, 2, 3), changes its order to (2, 3, 1). Such transforms are named permutations of the set {1, 2, ... , n}.

We can multiply two permutations, which means that we apply them successively to the ordered set. All the permutations of a set with n elements forms a grup, because the operation of multiplication of permutations is associative, has a neutral element (which is the identical permutation (1, 2, ... , n), leaving the order unchanged), and every permutation has an inverse which cancels its effect. The group of permutations of the set {1, 2, ... , n} is named the symmetric group of degree n, denoted by $S_n$, and having n! elements.

A transposition is a permutation which interchanges only two elements. For example, (2, 1, 3), (3, 2, 1), (1, 3, 2) are transpositions of the group $S_3$. Any permutation can be decomposed as a product of transpositions in a non unique way. Yet, there is something independent of the way we decompose the permutation as a product of transpositions. If the number of transpositions in such a decomposition is even, it will be even for any other decomposition of the same permutation, and we name such a permutation even. Otherwise, we call it odd. The set of even permutations of a set with n elements forms a subgroup of $S_n$, named the alternating group, denoted by $A_n$, and having n!/2 elements.

The symmetries of a regular polyhedron

By rotating the regular tetrahedron around one of its heights with 120º or 240º, this one remains unchanged. We say that the regular tetrahedron is unchanged by these transformations. We can rotate a regular polyhedron so that, after this transformation, it occupies exactly the same position, but having the faces not necessarily in the same positions. Also, they admit symmetry planes. In fact, this is the idea about the regular polyhedra – their rich symmetry.

The transformations leaving unchanged a polyhedron are named symmetry transformations. One can multiply the transformations. Because each symmetry transformation interchanges the faces of the polyhedron, we can associate to the transformation a permutation from $S_n$, n being the number of faces of the polyhedron. The symmetry transformations leaving invariant a polyhedron form a subgroup of $S_n$, named the automorphism group of the polyhedron.

How many symmetry transformations have each of the regular polyhedra? One easy method to count them is the following. Choose a face a. After a transformation it will take the place of another face a’. Since the polyhedron has n faces, we have n possibilities. Let’s choose a face b, neighbor to the first face before transformation, a. After the transformation, b goes into a face b’, neighbor to the face a’. Each face has the same number of edges k. The face b’ can be one of the k faces neighbor to a’. Therefore, we have nk possibilities. These transforms are the rotations of the polyhedron. But the face a is a polygon, therefore it has two sides. Consequently, when we move the face a in a’, this can flip. In this case, the transformation is no longer a rotation. It can no longer be obtain simply by moving the polyhedron, but by taking its mirror image.

Thus, among all transformations of the polyhedron, a special subgroup is formed by the nk rotations, but the total number of transformations is 2nk. Both these groups are subgroups of $S_n$. For the regular tetrahedron we obtain 2·4·3 = 24 automorphisms, from which the rotations are 4·3 = 12. The cube’s automorphisms group contains 2·6·4 = 48 automorfisms, from which 6·4 = 24 are rotations. The regular octahedron has 2·8·3 = 48 automorfisms, from which 8·3 = 24 rotations, like the cube. Both the icosahedron and the dodecahedron have 2·4·3 = 24 automorfisms, 4·3 = 12 rotations.

We see that the dual polyhedra have the same symmetries.

The symmetries of regular polyhedra and the permutations

Let’s label the vertices of a regular tetrahedron with the numbers {1, 2, 3, 4}:

At a symmetry transformation, the vertex 1 can go in any of the four vertices. The vertices 2, 3 and 4 are all neighbor with the vertex 1, and they will remain so after the transformation too. Their order around the vertex 1 is preserved in the case of rotations, otherwise it is reverted. Therefore, the vertices’ permutation is even if and only if the transformation is a rotation. The regular tetrahedron having 24 automorphisms, they coincide with the elements of the group $S_4$. The rotations correspond to the elements of the alternating group $A_4$.

Let us now label the cube’s vertices such that the ends of each large diagonal have identical labels from the set {1, 2, 3, 4}, like below:

Let’s choose one of the two regular tetrahedra inscribed in the cube. Its vertices are labeled with the numbers {1, 2, 3, 4}. A rotation of the cube rotates also the tetrahedron. Each cube rotation which let the tetrahedron in place corresponds to a rotation of the tetrahedron, and it is an even permutation. The even permutations of the cube’s labels correspond to transformations which interchange the two tetrahedra. Thus, the cube’s rotations correspond to the symmetric group $S_4$.

By labeling the faces of the regular octahedron, we obtain that it has the same symmetry groups as the cube (being its dual).

To obtain the symmetry group of the regular dodecahedron, let’s label its edges with the numbers {1, 2, 3, 4, 5}, like in the image:

For each edge of the dodecahedron, we take the four adjacent edges, and the other vertices of these edges form a square (hint: the four edges are diagonals in identical regular pentagons). These edges form five cubes. Each of the five cubes highlights six of the dodecahedron’s edges. It is easy to see that for each face we obtain a different ordering for the labels. Each rotation of the dodecahedron will take the face labeled by the ordered set (1, 2, 3, 4, 5) (we count starting with the vertex 1) in another face, also labeled with a permutation of the five numbers, so that 1 goes to one number, 2 to another etc. Thus, to the permutation (1, 2, 3, 4, 5) we can associate, as a result of the rotation, another permutation. Because we limit ourselves to rotations, we can choose one orientation (for example clockwise). The rotations will keep this orientation. Because we can start from any vertex of a face when we read the permutation, we will have five permutations for each face. This way, each corner of a face represents a permutation. We can check that these permutations are always even. Each dodecahedron’s rotation corresponds to an even permutation of the face labeled by (1, 2, 3, 4, 5). The dodecahedron’s rotations group is isomorphic with the alternating group $A_5$.

The regular icosahedron being dual to the regular dodecahedron, its 30 edges can be labeled like the ones of the dodecahedron. In this case, we will use cubes with the vertices in the centers of the icosahedron’s faces.

Group operations with polygons

"Calculator" for the Klein group

The Klein’s group has four elements {E, A, B, C}, and its multiplication is given by the label:

This group is isomorphic with the automorphism group of a rectangle. We can construct a „calculator” for multiplying elements of the Klein group by using to identical cards:

The first card will be used as witness card. For obtaining all the multiplications with the element B, hence the „multiplication with B table”. For doing this, we rotate the second card (the result card) so that the element B is moved in the position corresponding to the neutral element E of the witness card:

Now, to obtain the product of any element X from the group with B, we just read from the result card the element from the position corresponding to the position of X in the witness card. For example, to see the result of the operation C·B, we look for the element C in the witness card. It is in the lower right corner. We look in the same position in the result card. The corresponding element is A. We can check in the multiplicative table that indeed C·B = A.

Calculator for the groups $A_3$ and $S_3$

We start with the group $A_3$ of the even permutations of a set with 3 elements. Its elements are the permutations (1, 2, 3), (2, 3, 1), (3, 1, 2). We construct a card shaped as an equilateral triangle and we label its vertices with the numbers 1, 2 and 3:

The rule is: we label the vertices with the numbers {1, 2, 3} clockwise, and the edges with the permutations given by the order in which we met the vertices by starting from that edge and go clockwise.

We see that the rotations with 120º, 240º or 360º preserve this triangle. To find the multiplications of the elements of the $A_3$ with one element of the group, say (3, 1, 2), we keep the witness card in the normal position, and rotate the result card such that the permutation (3, 1, 2) corresponds to the position of the identical permutation (1, 2, 3) from the witness card. In order to find the result of the multiplication of a permutation with the permutation (3, 1, 2), we search on the witness card the position of the desired permutation, and on the result card we just read the permutation on the corresponding position. For example, to find the result of multiplying the permutations (3, 1, 2) and (2, 3, 1), we look on the result card for the permutation corresponding to the position occupied by the permutation (2, 3, 1) on the witness card. We see that the result is the permutation (1, 2, 3):

The group $S_3$ contains, in addition to the even permutations (1, 2, 3), (2, 3, 1), (3, 1, 2) from the group $A_3$, the odd permutations (2, 1, 3), (3, 2, 1), (1, 3, 2). This is why we will allow, besides the rotations maintaining the triangle in plane, transformations obtainable by lifting the triangle from the table and flipping it. On its back face we will mark the odd permutations:

Calculator for the groups $C_n$ and $D_n$

A finite group such that all its elements can be obtained by multiplying one element (named generator of the group) with itself, is named cyclic group. We denote the cyclic group with n elements by $C_n$. The cyclic group $C_n$ is isomorphic with the group of integers modulo n, $\mathbb{Z}_n$. The cyclic group C3 is isomorphic with the alternating group $A_3$, but this doesn’t hold for n > 3. We can consider the group $C_n$ as representing the plane rotations of a regular polygon with n edges. If we allow mirror symmetries, obtained by taking the polygon outside the plane and flipping, the number of the possible symmetries doubles, and their group is named the dihedral group, $D_n$. We observe from the definition that for n = 3 the dihedral group is isomorphic to the permutation group of a set of three elements: $D_3$ ~ $S_3$, but, as for the cyclic group, we can’t generalize for n > 3.

The calculators for the cyclic and dihedral groups can be made from regular polyhedra. They can be used similarly to those described for the groups $A_3$ and $S_3$, with a witness polygon and a result polygon.

Polyhedral calculators of permutations

Calculator for the group $S_4$

Let’s consider again the cube with vertices labeled like this:

Each face has four edges, and we label each of them, on that face, with the permutations obtained by reading the vertices’ labels, starting from that edge and walking clockwise. We obtain a cube labeled on each side of each edge with permutations. The reader can construct her own cube by printing this image:

After assembling it in 3D, it will look like this:

For finding all the multiplications of the permutations from $S_4$ with a particular permutation, say (4132), we put the witness cube with the face containing the identical permutation (1234) in front, such that the identical permutation is below:

Then we put the resulting cube in a similar position, only that on the position of the identical permutation we put the permutation (4132):

Let’s suppose we want the result of the multiplication of the permutation (4132) with another permutation from $S_4$, for example (3124). We look in the witness cube the position of the permutation (3124), and in the result cube we read the permutation from the corresponding position. Because the permutation (3124) occurs in the witness cube on the left edge of the right face, we read the permutation from the left edge of the right face of the resulting cube. This is (3412).

Once we position the cubes, we can simply read all the results of the multiplications with the chosen permutation (4132) with any element of $S_4$ on the result cube.

The reader is invited to study the symmetries of the permutations written on the cube.

For example, two permutations associated to the same edge differ by a transposition between the elements labeling the edge’s ends. The permutation (4123) is neighbor of the permutation (3124) – they are on the same edge of the cube. Their common edge has the vertices labeled with the numbers 4 and 3. These numbers are transposed in one of the two permutations to obtain the other. In fact, any two permutations associated to the same edge differ by inverting the positions 1 and 4.

Another interesting property of this cube is that, applying a rotation, we obtain on the initial position of the identical permutation a permutation showing how the four large diagonals of the cube were permuted.

Calculator for the group $A_5$

As we remember, the group of rotations of the dodecahedron is isomorphic with the alternating group $A_5$. We can label the dodecahedron by using even permutations of the set {1, 2, 3, 4, 5}, like this:

By printing it and gluing the edges, you can construct a dodecahedron. By constructing two identical such dodecahedra, we can use them to multiply permutations from the alternating group $A_5$, in a similar way we used two cubes to multiply elements of $S_4$.

Smooth Quantum Mechanics: 0. Main Article

2009-02-16T01:48:00.000-08:00

Smooth Quantum Mechanics

Quantum Mechanics without discontinuities in time evolution

Abstract

I show that the apparent wave function collapse can take place smoothly, without discontinuities. The projections on the observable's eigenspaces can be obtained by delayed initial conditions, imposed to the smooth time evolution of the observed system entangled with the measurement device used for the preparation. Since the quantum state of this device is not entirely available to the observer, its unknown degrees of freedom inject, by the means of entanglement, an apparent randomness in the observed system, leading to a probabilistic behavior. By using this mechanism, we can construct a Smooth Quantum Mechanics (SQM), without the need of discontinuities in time evolution. Therefore, the probabilities occur because not all the involved systems have determined quantum states. The evolution is deterministic, but for an observer who has access only to an incomplete set of initial conditions, it appears to be indeterministic.

The problem of discontinuities in Quantum Mechanics

The time evolution of a Quantum System

A quantum system which is in a pure state, and not entangled with another system, evolves according to the Schrödinger equation:

$$\Bigg\{
\begin{array}{ll}
i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=H(t)|\psi(t)\rangle \\
|\psi(t_0)\rangle=|\psi_0\rangle,
\end{array}
$$

where $|\psi(t)\rangle\in\mathcal{S}$ is a state vector from the state space $\mathcal{S}$, and $H(t)$ is the Hamiltonian, usually a hermitian operator on $\mathcal{S}$. If the quantum system is closed, then $H$ is time independent, but in general, it is time dependent, because of the interactions with other systems. For more general interactions, the observed system can become entangled with other systems, and its state will no longer be pure. In this case, as well as in the case when we don't know the initial data, but rather a probability distribution, we represent the state by a density operator $\rho$ on $\mathcal{S}$. For these situations we will employ, instead of the Schrödinger's equation, the Liouville - von Neumann equation:

$$\Bigg\{
\begin{array}{ll}
i\hbar\frac{\partial\rho(t)}{\partial t}=[H(t),\rho(t)] \\
\rho(t_0)=\rho_0.
\end{array}
$$

The measurement problem

When a measurement is performed to a quantum system, the system is found to be in an eigenstate of the observable. There are two main problems raised by this fact. First: why only eigenstates of the observables are obtained as outcomes of the observations? Why don't we observe superpositions of such eigenstates, like a $|\textrm{dead}\rangle+|\textrm{alive}\rangle$ Schrödinger cat? This is the main problem of the measurement, which, in this article, will be accepted as it is, without offering an explanation. The second problem is the following. Knowing the state of a quantum system, and assuming that a measurement will find the system in an eigenstate of the observable, it seems like a discontinuous jump happens. How is this happening? Is this jump really discontinuous? What is its nature? In this article, I propose a solution to this discontinuity problem in Quantum Mechanics.

One-measurement

The Schrödinger's equation, as well as the Liouville - von Neumann equation, are PDE equations. Each solution can be uniquely specified by an initial condition. The initial condition is obtained by performing a measurement at an initial moment $t_0$.

Let's consider that we measure the polarization of a photon, at the instant $t_0$. The obtained result will determine the photon's polarization both for moments $t$ with $t$<$t_0$, and with $t>t_0$. The last statement needs some argumentation. A measurement is in general required to produce a minimal disturbance of the system; it is required to determine the system's state, and to let the system in the state it was found. At the quantum level, this is sometimes difficult. The outcome of a measurement depends on the observable we choose to measure. For example, by choosing a direction along which we measure the spin of an electron, we determine only two possible eigenstates, out of an infinity. In the case of the photon, we can impose the left or right circular polarization with the help of a wave plate. This is an initial condition imposed to the photon. Consider a pair of entangled photons, being obtained from a parametric down conversion, such that they have linear polarizations along directions which are orthogonal to each other. A polarization imposed to one of them, applies also to the other one. This mechanism is used in the delayed choice quantum eraser experiment [SMKYKS00]. In this experiment, the polarization of one photon determines if we have interference or not. We can choose the polarization by imposing to the other photon a polarization, and we can do this after the other photon hit the screen. The polarization imposed to one of the two photons, is revealed to apply in past, until the pair of the two photons has been emitted, so that the second photon has the correct polarization. This shows that we can impose a polarization at $t_0$, which applies for instants of time before and after $t_0$.

The classical view is that the measurement only revealed the state of the system, and the solution of the evolution equation preexisted to the measurement. On the other hand, in Quantum Mechanics, we can choose what observable to measure, thus, we can choose the set of admitted eigenstates. So, if the solution we detected by measurement preexisted, it did this in a way that anticipated our choice of the observables. This choice can be performed with a delay, to make sure that it doesn't affect in a causal way the observed system. This was emphasized by Wheeler [Whe77,Whe78,WZ83] when he revived (It seems that similar suggestions were made before by Weiszäcker [Wei31] and Bohr [Sch49].) the idea of delayed-choice experiments. In the case of the electron spin, when we choose the direction to measure the spin, we let available only two possible eigenstates for the spin. Had we choose a different direction, the eigenstates would be different. So, our choice limited the possible outcomes. And when we measure the spin, we determine not only what spin the electron has at $t_0$, but also for previous $t$, as we can see from entanglement situations like the one pointed by Einstein, Podolsky, and Rosen [EPR35] (in Bohm's version [Bohm51]).

We can conclude that one measurement determines uniquely the state at $t_0$, hence the solution, and this determination seems to affect the past in a weird way. We cannot say that it can change it, rather it is only an initial condition, established with a (very large) delay. Two measurements of the same system may be incompatible, and require something that looks like a wavefunction collapse. In the case of only one measurement (or when the measurement is the first one), the state prior to the measurement is not sustained by any data, so we can not talk about collapse. The one-measurement situation makes apparent that the eigenstate can be selected without involving the discontinuous wave function collapse. Two initial conditions may be incompatible, but only one cannot be, since it is the only condition. If after the first measurement we perform another, the things become more difficult, but the solution is similar, as we will see next.

Two-measurements case and the wave function collapse

Let's consider a system whose evolution is described by the Schrödinger's equation. Suppose that, after an observation at $t_0$ finds the system in an eigenstate, we perform a second observation, at the time $t_1$. If the state predicted by the evolution equation is an eigenstate of the second observable, then it will be obtained at the second measurement. If not, then the second observation cannot impose an initial condition at $t_1$, compatible with the unitary evolution governed by the Schrödinger equation.

We can see that one observation imposes an initial condition to the Schrödinger equation, choosing a solution, but a second observation either confirms the solution, or it is incompatible with it. In this case, it should not be possible to perform more than one observation of a system.
A quantum system has a wave-like behavior, described by the Schrödinger equation, or by the Liouville - von Neumann equation, and a quantum behavior, expressed by the condition to be found in an eigenstate of the observable. These two behaviors seem to be incompatible.

But we know from experience that we can perform more observations to the same quantum system. This has the appearance of a jumping from one solution of the Schrödinger equation to another one, in a discontinuous fashion.

The analysis of delayed-choice experiments suggests that, if a collapse happened, it took place in advance, during the previous interaction, possible even at $t_0$ (please refer to figure 1).

Figure 1 In a delayed-choice experiment, the reduction seems to take place in advance, anticipating the experimenter's choice of the observable.

Why discontinuities cause problems?

The acceptance that a quantum system is subject of a discontinuous wave function collapse can raise several problems. On the one hand, if we consider the observed system as being a part of a larger one (perhaps the Universe), containing the measurement device too, as a quantum subsystem, the measurement can be described by the evolution equation, and we expect that no discontinuous collapse appears. But, when we refer to the observed system only, we cannot see how the discontinuity can be avoided. We seem to have a paradox: a system evolving with discontinuities, being in the same time a subsystem of one evolving smoothly.

Another problem is that the discontinuous collapse has been postulated, but never observed directly. We don't know of any mechanism which can produce it, and we don't know when exactly it takes place. An explanation is required, since we cannot accept that it simply happens.

In Quantum Mechanics, an observable that commutes with the Hamiltonian of the system is conserved during the evolution.
But the conservation holds only as long as the system evolves governed by the Hamiltonian (Schrödinger equation or Liouville - von Neumann equation).
Since performing a measurement makes the system jump in a totally different state, it is expected that the conservation laws are broken. For example, if we measure the momentum of the system, and then measure its position, then the initial momentum is lost. If we measure again the momentum, we should expect to obtain a totally different value than the first time. We can expect that, after several measurements, the conserved quantities of the system be totally blown up.

The discontinuities are incompatible with the conservation laws, but the conservation laws don't break down as a result of measurements. Something happens always to restore them.
To make them compatible, we need to appeal to a "magical postulate":

During the state vector reductions, the conservation laws can no longer be deduced from the Hamiltonian, but they must be restored in some way or another.

The problem is that we don't know any explanation for the conservation laws, other than the time evolution described by the Schrödinger's equation and Liouville - von Neumann equation. Breaking this evolution should break the conservation laws, contrary to our experimental observations.

The quantum world is like a great illusionist, who has in his sleeves a lot of tricks that make us believing that the quantum system jumps discontinuously from time to time. But we have to remember that, in the end, there must be a logical explanation for the illusion number presented in the show, and to look for the strings.

Can discontinuities be avoided?

In the following, I will show that the apparent wave function collapse can be explained by the standard Quantum Mechanics, minus the discontinuity, as a smooth and natural phenomenon. The first ingredient comes from the discussion above, concerning a system undergoing only one measurement. A measurement fixes the initial data for a quantum system; going to a larger system, makes those initial conditions insufficient, therefore, a new measurement is allowed. The second ingredient is the entanglement with the device performing the previous measurement (which will be named preparation device).

Quantum Mechanics without discontinuities

We begin by considering the measurement from a semi-classical viewpoint: the observed system is quantum, and the preparation device is classical.

The semi-classical interaction approach

Let's consider a quantum system evolving according to the Schrödinger's equation, subject to a first measurement (the preparation) starting at the instant $t_0$ and ending at $t_0+\varepsilon$, and a second measurement at $t_1>t_0+\varepsilon$. If we consider the preparation device as being classical, its influence can be described by an interaction Hamiltonian $H_{\textrm{int}}(t)$. Thus, in the Dirac picture, the Hamiltonian is:
$$H(t)=H_0+H_{\textrm{int}}(t).
$$

The preparation device is considered classical, this meaning that its true state, which is quantum, is unknown. There will be a large set of quantum states which, at the classical level, will look identical. This set of equivalent quantum states can be parameterized, with both discrete and continuous parameters. Let's take a smooth parameterization $u(t)$ of its continuous degrees of freedom. The interaction Hamiltonian $H_{\textrm{int}}(t)$ will depend on $u(t)$, such that $H_{\textrm{int}}(t)=H_{\textrm{int}}(t,u)$.

Each choice of the parameters $u(t)$ will lead to a state of the system at $t$ given by
$$|\psi(t,u)\rangle=U(t,t_0,u)|\psi(t_0)\rangle.
$$

Before the introduction of the degrees of freedom parameterized by $u(t)$, there was only one possible state at $t_1$ for the observed system. Now, by varying $u$, $|\psi(t_1,u)\rangle$ also changes.

We ask the following question:

What condition should the parameters $u$ satisfy, such that all possible outcomes of any possible observation taking place at $t_1$ are reached by $|\psi(t_1,u)\rangle$?

This is a problem of Quantum Control Theory. Under some general assumptions on $u(t)$, the condition is that the Lie group associated to the Lie algebra generated by the matrices of the form $iH(t,u)$ should contain all the possible unitary transformations. If the dimension of $\mathcal{S}$ is $n<\infty$, then it is enough that the rank of the Lie algebra generated in this way to be identical to the rank of the unitary Lie algebra $\mathfrak u(n)$. This holds when there is no time limit, but in our case, the time is limited to $t_0+\varepsilon$, bringing a new restriction. On the other hand, the Born rule assigns zero probability to projections on an orthogonal state. Therefore, we don't need to obtain at $t_1$ states orthogonal to $|\psi_0\rangle$.The parameters $u(t)$ can be determined by appropriate initial conditions. Similarly to the one-measurement case, the initial conditions are determined such that the system evolves to be the appropriate eigenstate of the observable, at $t_1$.In the figure 2 we can see how the Hamiltonian can prepare the observed system to be in an eigenstate of the observable.

Figure 2 The disturbance in the evolution of the quantum system, introduced by the measurement device performing the preparation, needs to be taken into account by modifying the Hamiltonian from $H_0$ to $H(t,u)=H_0+H_{\textrm{int}}(t,u)$ for the time interval $(t_0, t_0+\varepsilon)$. This will "repair" the discontinuity presented in the figure 1.

Assuming that the observable corresponding to the measurement at $t_1$ is $O_1$, for each outcome $|\psi_{O_1,\lambda}\rangle$ of the measurement, corresponding to an eigenvalue $\lambda$, there must exist a choice $u_{O_1,\lambda}$ of the parameters $u(t)$ such that the interaction send the observed system in $|\psi_{O_1,\lambda}\rangle$. The corresponding unitary operator is $U_{O_1,\lambda}(t_1,t_0)$, so that $|\psi_{O_1,\lambda}\rangle =U_{O_1,\lambda}(t_1,t_0)|\psi_0\rangle$.

Let us consider the following example, raised by Einstein to Bohr, at the Fifth Solvay Conference (Brussels 1927). Einstein said that, in a two-slit experiment, if we measure the recoil of the wall containing the two slits, when the light passes through it, one should be able to deduce whether the photon passed through one slit or the other. As Bohr replied to him, if we detect a significant recoil, the interference pattern is destroyed.
Let's reverse a bit the reasoning, and apply it to the delayed-choice [Whe77, Whe78, WZ83, Wei31, Sch49] version of the two-slit experiment. We can decide after the photon has passed through the slit(s) whether to observe the "which way" or the "both ways" aspects. If we decide to observe the "which way" behavior, we cause the wall with the two slits to undergo a significant change of momentum (corresponding to the cases when the photon has passed through one slit or the other). If we choose to observe the interference, the change in momentum will be undefined. The wall with the two slits will get in a superposition of eigenstates of momenta.
This example shows that, indeed, the interaction with the wall with the two slits, happening in the interval $(t_0,t_0+\varepsilon)$, takes place in such a manner that the outcome of the measurement is one of those expected by the choice of the observable.

The entanglement approach

The previous analysis simplified the interaction between the preparation device and the observed system. A more general description will consider that the preparation device is quantum, not classical. In this case, its interaction with the observed system leads to an entanglement between the two. The evolution of the observed system can no longer be considered unitary: its state may go from being pure, at $t_0$, to being mixed at $t_0+\varepsilon$. Of course, the combined system made of the observed system and the preparation device, may be isolated, and undergo unitary evolution, but the observed system's state will be obtained by partial tracing the density operator of the larger system, and it will not necessarily be pure.
A correct description will use density operators to represent the state, and the Liouville - von Neumann equation, for its evolution.

Let us consider that the state of the observed quantum system is described by the density operator $\rho_q$, on the state space $\mathcal{S}_q$, and the one of the preparation device is described by a density operator $\rho_p$ on the state space $\mathcal{S}_p$. We consider that the combined system, represented by a density operator $\rho_{q,p}$ on $\mathcal{S}_q\otimes \mathcal{S}_p$, is isolated. If it is not isolated, then we complete the system with remaining systems $\rho_r$, so that we obtain an isolated system. We can consider, without loosing the generality, that the preparation device incorporates all these systems, so it will be enough to work in the state space $\mathcal{S}_q\otimes \mathcal{S}_p$. The combined system will have a unitary evolution between $t_0$ and $t_0+\varepsilon$, given by the unitary operator $U_{q,p}=U_{q,p}(t_0+\varepsilon,t_0)$:
$$\rho_{q,p}(t_0+\varepsilon)=U_{q,p}\rho_{q,p}(t_0)U_{q,p}^\dagger
$$

The initial and the final density operators for the observed system can be obtained by partial trace:
$$\begin{array}{ll}
\rho_q(t_0)&=\textrm{tr}_p\rho_{q,p}(t_0)\\
\rho_q(t_0+\varepsilon)&=\textrm{tr}_p\rho_{q,p}(t_0+\varepsilon),
\end{array}
$$

and we have
$$\rho_q(t_0+\varepsilon)=tr_p(U_{q,p}\rho_{q,p}(t_0)U_{q,p}^\dagger).
$$

In general, the transformation from $\rho_q(t_0)$ to $\rho_q(t_0+\varepsilon)$ is not unitary.

We perform now another simplification, again without loosing generality, by purifying the state. We can purify the state $\rho_{q,p}$ by expanding the state space from $\mathcal{S}_q\otimes \mathcal{S}_p$ to
$$\mathcal{S}:=\mathcal{S}_q\otimes \mathcal{S}_p \otimes \mathcal{S}'_q\otimes \mathcal{S}'_p,
$$

with $\mathcal{S}_q\cong \mathcal{S}'_q$ and $\mathcal{S}_p\cong \mathcal{S}'_p$. The two extra state spaces $\mathcal{S}'_q$ and $\mathcal{S}'_p$ does not necessarily represent physical systems, but they allow us to consider $\rho_{q,p}$ as the partial trace of a pure state on $\mathcal{S}$. The composed system's evolution can be considered to be described by Schrödinger's equation on $\mathcal{S}$, although the state $\rho_{q,p}$ still needs to obey Liouville - von Neumann equation.
We denote the state space which is external to our observed system by
$$\mathcal{S}_e:=\mathcal{S}_p \otimes \mathcal{S}'_q\otimes \mathcal{S}'_p,
$$

and the density operator describing the evolution on this space by $\rho_e$.

The conditions imposed by the observations to the system described by $\rho_q$ at $t_0$ and $t_1$ imply that $\rho_q(t_0)$ and $\rho_q(t_1)$ represent pure states:
$$\begin{array}{l}
\rho_q(t_0)=|\psi_0\rangle\langle\psi_0|\textrm{ and} \\
\rho_q(t_1)=|\psi_1\rangle \langle\psi_1|.
\end{array}
$$

This imposes restrictions also on the combined system $\rho_{q,p}$. After $t_0$ the systems $\rho_q$ and $\rho_p$ become entangled, and the second observation disentangles them, and also imposes to $\rho_e$ a purity condition
$$\rho_e(t_1)=|\eta_1\rangle \langle\eta_1|,
$$

with $\eta_1\in\mathcal{S}_e$.

Since at $t_0$ the preparation device and the observed system were being separated, the preparation device was in a state $\rho_p(t_0)$, which can be obtained by partial tracing from a pure state $|\eta_0\rangle\in \mathcal{S}_e$. Although the state vector $|\eta_1\rangle$ is uniquely determined by the observation at $t_1$, it depends on $|\eta_0\rangle$. Because we don't know the value of $|\eta_0\rangle$, to each possible $|\eta_0\rangle$, and to each possible outcomes $|\psi_0\rangle$ and $|\psi_1\rangle$ of the two measurements, will correspond a unique $|\eta_1\rangle$. In order to clarify this correspondence, we need to study some properties of linear operators acting between tensor products of vector spaces.

Let $\mathcal{V}_A$, $\mathcal{V}_B$, $\mathcal{V}_C$ and $\mathcal{V}_D$ be four vector spaces over a field $\mathbb{K}$, $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, and let

$$T:\mathcal{V}_A\otimes\mathcal{V}_B\to\mathcal{V}_C\otimes\mathcal{V}_D
$$

be a $\mathbb{K}$-linear morphism of vector spaces. We are interested in identifying the possible separable vectors $|A\rangle \otimes|B\rangle \in\mathcal{V}_A\otimes\mathcal{V}_B$ and $|C\rangle \otimes|D\rangle \in\mathcal{V}_C\otimes\mathcal{V}_D$ such that:

$$T(|A\rangle \otimes|B\rangle )=|C\rangle \otimes|D\rangle .
$$

Proposition 1.

Let us consider $|A\rangle$ and $|C\rangle$ fixed. The set of vectors $|B\rangle \in\mathcal{V}_B$, and the set of vectors $|D\rangle \in\mathcal{V}_D$, satisfying the equation above, form vector subspaces $\mathcal{V}_B^{AC}\leq\mathcal{V}_B$, respectively $\mathcal{V}_D^{AC}\leq\mathcal{V}_D$.

Proof:
If $|B'\rangle$ and $|B''\rangle$ are such that $T(|A\rangle \otimes|B'\rangle )=|C\rangle \otimes|D'\rangle$ and $T(|A\rangle \otimes|B''\rangle )=|C\rangle \otimes|D''\rangle$ for some $|D'\rangle$ and $|D''\rangle \in\mathcal{S}_D$, then for any $z',z''\in\mathbb{C}$,
$$T(|A\rangle \otimes(z'|B'\rangle +z''|B''\rangle ))=z'T(|A\rangle \otimes|B'\rangle )+z''T(|A\rangle \otimes|B''\rangle )=\\
z'|C\rangle \otimes|D'\rangle +z''|C\rangle \otimes|D''\rangle =|C\rangle \otimes(z'|D'\rangle +z''|D''\rangle ),
$$

therefore the solutions $|B\rangle \in\mathcal{V}_B$ form a vector subspace $\mathcal{V}_B^{AC}\leq\mathcal{V}_B$. Consequently, the solutions $|D\rangle \in\mathcal{V}_D$ form a vector subspace $\mathcal{V}_D^{AC}=T(\mathcal{V}_B^{AC})\leq\mathcal{V}_B$.

Remark 1.
Since we have $\mathcal{V}_C\otimes\mathcal{V}_D\cong\mathcal{V}_D\otimes\mathcal{V}_C$, it follows from Proposition 1 that a similar result holds for $|A\rangle$ and $|D\rangle$ fixed.

Proposition 2.
If the space $\mathcal{V}_C$ has a scalar product $\langle \;|\;\rangle$, the linear operator
$$T_B^{AC}:=T|_{\mathcal{V}_B^{AC}}
$$

is given, for $|C\rangle \neq 0$, by
$$T_B^{AC}(|B\rangle )=\frac{\textrm{tr}_C(T(|A\rangle \otimes|B\rangle )\otimes\langle C|)}{\langle C|C\rangle}.
$$

Proof:
To remove the $|C\rangle$ part from $|C\rangle \otimes|D\rangle$, we tensor $|C\rangle \otimes|D\rangle$ with $\langle C|\in\mathcal{V}_C^*$, partial trace over $|C\rangle \langle C|$, and then divide by $||C\rangle |^2$.
\end{proof}

Remark 2.
If $T$ defined above is isomorphism, then $T_B^{AC}$ is isomorphism onto its image.

We can now apply the previous results to a unitary operator $U$ acting on our space $\mathcal{S}_q\otimes\mathcal{S}_e$:
$$U:\mathcal{S}_q\otimes\mathcal{S}_e\to\mathcal{S}_q\otimes\mathcal{S}_e,
$$

and to the equation
$$U(|\psi_0\rangle\otimes|\eta_0\rangle)=|\psi_1\rangle \otimes|\eta_1\rangle,
$$

obtaining the following corollaries.

Corollary 1.
Let us consider $|\psi_0\rangle$ and $|\psi_1\rangle$ fixed. The set of vectors $|\eta_0\rangle\in\mathcal{S}_e$, and the set of vectors $|\eta_1\rangle \in\mathcal{S}_e$, satisfying the equation above, form isomorphic vector subspaces $\mathcal{S}_{e0}^{\psi_0\psi_1}\leq\mathcal{S}_e$, respectively $\mathcal{S}_{e1}^{\psi_0\psi_1}\leq\mathcal{S}_e$.
Proof:
Follows immediately from Proposition 1. and Remark 2.

A measurement at $t_1$, although determines the observed state to be in $|\psi_1\rangle$, it does not necessarily determine completely the state $|\eta_1\rangle$ of the preparation device.

Corollary 2.
Let us consider $|\psi_0\rangle$ and $|\eta_1\rangle$ fixed. The set of vectors $|\eta_0\rangle\in\mathcal{S}_e$, and the set of vectors $|\psi_1\rangle \in\mathcal{S}_q$, satisfying the equation from Remark 2, form isomorphic vector subspaces $\mathcal{S}_{e0}^{\psi_0\eta_1}\leq\mathcal{S}_e$, respectively $\mathcal{S}_{q1}^{\psi_0\eta_1}\leq\mathcal{S}_q$.
Proof:
Follows from the Remark 1.

We denote the isomorphism obtained by restricting the unitary operator $U$ to $\mathcal{S}_{e0}^{\psi_0\eta_1}$ by $K_{\psi_0\eta_1}$.

From Corrolary 2 we obtain:

Theorem.
The set of all states $|\psi_1\rangle \in\mathcal{S}_q$ that can appear in the right side of the equation in Remark 2, for a fixed $|\psi_0\rangle\in\mathcal{S}_q$ is given by the following union of subspaces:
$$\mathcal{S}_{q1}^{\psi_0}:=\bigcup_{|\eta_1\rangle \in\mathcal{S}_e}\mathcal{S}_{q1}^{\psi_0\eta_1},
$$

obtained by varying the state vector $|\eta_0\rangle$ in the set
$$\mathcal{S}_{e0}^{\psi_0}:=\bigcup_{|\psi_1\rangle \in\mathcal{S}_q}\mathcal{S}_{e0}^{\psi_0\psi_1}.
$$

Remark:
A good preparation must satisfies the condition
$$\mathcal{S}_{q1}^{\psi_0} \geq \{|\psi_1\rangle \in\mathcal{S}_q|\langle\psi_1|U(t_1,t_0)|\psi_0\rangle\neq 0\},
$$

where $U(t_1,t_0)$ is the evolution operator of the observed system, if the system is undisturbed.

We recall that the state space $\mathcal{S}_e$ is an extension of a $\mathcal{S}_q$, made for working with purified states, but this is not a problem, since we can always recover the density operators of the subsystems by partial tracing.

The mechanism proposed here is represented in the figure 3. The preparation should consist in an interaction with the property that any possible outcome $|\psi_1\rangle$ of the second measurement can be fitted by an appropriate choice of the initial conditions for the preparation device, represented by the state vector $|\eta_0\rangle$.

Figure 3 Each possible outcome $|\psi_1^i\rangle$ can be obtained by choosing the appropriate states $|\eta_0^i\rangle$ representing the preparation device.

The smooth projection mechanism

Because the first measurement can find the observed system in the state $|\psi_0\rangle$, while the second one in $|\psi_1\rangle \neq U(t_1,t_0)|\psi_0\rangle$, it is easy to understand why it seemed that the state vector suffers a discontinuous jump, somewhere between $t_0$ and $t_1$. But we can now explain the wave function collapse as taking place smoothly, restoring the continuity in its evolution.

In order to do this, we had to go to the level of a larger system, composed by the observed system and the preparation device. At that level, the unitary evolution has been restored, and we have seen that the observed system (although its evolution may no longer be unitary, being entangled with the preparation device) can undergo a "smooth collapse".

The price to be paid was the acceptance that the observed system acts, somehow, anticipating the set of possible eigenstates. This feature may seem acausal, but it is present also in the standard Quantum Mechanics, as we have learned from the "delayed-choice experiments". In this article, the collapse was only pushed to the "beginning of times", and the initial conditions remained at the time $t_1$, being thus "delayed initial conditions".

Each measurement specifies the initial conditions of a system. When a system is measured a second time, the initial conditions need to be restated. To be restated without contradicting the previously observed initial conditions, they should be lost somehow. I hypothesized here that they are lost because of the interaction with the preparation device, which, although determines the previous set of conditions, transfers from its own indeterminacy of initial conditions to the observed system. Any interaction of a system with another system which has some freedom in the choice of its initial conditions, will make the former system loose its specification of the initial conditions. The observation only shows what the state was, and not what it will be at the next measurement.

Our mechanism allows us to see the projection, usually being associated to the wave function collapse, as taking place continuously, smoothly, and not discontinuously. The projector operator is not present explicitly in the evolution equation, but it is "embedded" in a set of operators parameterized by $|\eta_0\rangle$ - it can be reconstructed, for each pair $(|\psi_0\rangle,|\psi_1\rangle )$, by choosing an appropriate $|\eta_0\rangle\in\mathcal{S}_e$.

Discussion

Smooth Quantum Mechanics

Smoothness

This article provides a scenario of how the wave function collapse can take place without discontinuities, in a smooth way. We can reconstruct the Quantum Mechanics into a Smooth version, but we have to remember that this is not the only place where discontinuities occur. For example, the eigenstates of the position are distributions, and the eigenstates of the momentum have infinite norm. If we consider the state space as being a Hilbert space, then we have to accept such problems. Yet, we can avoid this kind of problems by renouncing at the completeness - the idea that the state space should contain limits for any Cauchy sequence. We can instead use a rigged Hilbert space, $\mathcal{S}\subset \mathcal{H}\cong\mathcal{H}^*\subseteq\mathcal{S}^*$. The state vectors will then be elements of a space $\mathcal{S}$ of smooth functions of finite norms, but the (ideal) eigenstates of various operators will belong to $\mathcal{H}$.

Probabilities

The evolution equations are deterministic, and since we have eliminated the discontinuities, the only source of randomness is in the initial conditions. Therefore, both the Born rule and the Heisenberg relations have to be reinterpreted. The Born rule doesn't expresses the probabilities of collapse, but those of the initial conditions to lead to each outcome.

We can derive the original Heisenberg relations [WZ83] by multiplying the relations $\Delta\omega\Delta t\geq 2\pi$ and $\Delta k_x\Delta x\geq 2\pi$, from the Fourier analysis, with the reduced Plank constant $\hbar$.
To obtain similar Heisenberg relations for other pairs of conjugated operators, we do the same for the corresponding eigenbases.
These relations refer to how large the support of a state vector can be, when expressed in two different bases, and have nothing intrinsically probabilistic built in.
For example, the relation $\Delta k_x\Delta x\geq 2\pi$ shows that if the wave packet is too located in space, then in the momenta space it will be more spread.
We can obtain also the Heisenberg's relations from the commutation relations of the operators.
A version of Heisenberg's relations, which is used frequently, is $\sigma(p_x)\sigma(x)\geq \frac 1 2\hbar$, expressed in terms of the standard deviation, defined for an operator $A$ by $\sigma(A):=\sqrt{\langle A^2\rangle-\langle A\rangle^2}$. Again, the probabilities have not yet entered into the play, because the standard deviations, in this case, refer to the components of the wave packet, expressed in two conjugate bases.

It is only when the state vector is disturbed by a preparation, and we apply the Born rules in relation to an eigenbasis of an observable, when Heisenberg's relations become the uncertainty relations. It follows that the probabilistic meaning of the Heisenberg's relations also reflects our ignorance of the initial conditions.

The observers don't have access to the full set of initial conditions. The observations allows them collect only a set (which we will name registry) of partial initial conditions. Therefore, although the evolution is deterministic, they perceive the time evolution as being indeterministic.

Causality and delayed initial conditions

The mechanism in Smooth Quantum Mechanics resumes to fixing the initial conditions at a moment of time, even for events prior to that moment. In fact, from mathematical point of view, the same solution of a PDE can be obtained from appropriate initial conditions imposed at any instant of time.

The main motivation for choosing the solution of delayed initial conditions resides in the avoidance of discontinuities. The discontinuities are source of problems. They imply that there are two sets of physical laws at quantum level. The discontinuities have never been observed directly, only the incompatibility between the outcomes of the measurements. They require additional explanations for the conservation laws, which are respected always, although only the smooth evolution leads to them, as they are obtained from commutation with the Hamiltonian (or by Noether's theorem). The price for avoiding the discontinuous collapse is to allow the initial data to be specified with a delay.

But we should clarify whether or not the mechanism of delayed initial conditions breaks down the causality. As already mentioned, each new condition should be chosen such that it is compatible with the already chosen conditions. It is even possible to choose two or more conditions at positions and moments which can be related only with signals traveling with velocity greater than that of the light. In this case, if we account for relativity, we can add the two conditions to the registry simultaneously, taking care not to violate the compatibility. In a deterministic world, with the initial data completely specified, there is no room for phenomena which contradict the initial data. Similarly, in the Smooth Quantum Mechanics, all the initial data is required to be compatible with the initial data already acquired. A similar apparent acausality, manifested by the anticipation of future initial conditions, is also present in experiments with photons having negative group velocity [GSBKB06]. Consequently, the causal loops and breaking of causality are avoided.

What remains to be done

This article only shows that it is possible to have a smooth, instead of a discontinuous, wave function collapse, and shows that it is possible a smooth reconstruction of Quantum Mechanics. Not any interaction is able to provide the freedom in initial conditions required to solving this problem. Perhaps, this is why not any interaction is a measurement, but this point needs to be developed better. Ideally, we would have a precise mathematical description of a measurement, and a theorem showing that from this description, we obtain precisely the required range of outcomes at a second observation. Having a good definition of the measurement apparatus will allow us to predict, for example, which interactions qualify as measurements. Maybe, for this understanding, we will have to wait until more challenging parts of the Quantum Mechanics - the reconstruction of the classical world from the quantum world, and the explanation of why a measurement can obtain only eigenstates of the observable as outcomes - will receive better explanations. Another important progress would be a deduction of the Born rule. At the current moment, it seems that this rule is independent on the Smooth Quantum Mechanics' principles, but it would be desirable to have at least a good definition of a measurement which will lead easier to a smooth version of the Projection Postulate, including the Born rule.

Relations with other interpretations of Quantum Mechanics

After about eight decades of progresses in Quantum Mechanics, the discussions between Einstein and Bohr remain of actuality.
Although their views seemed incompatible one another, the Smooth Quantum Mechanics presented here is friendly with both of them. I don't say that, had they living today, they would say that they had in mind this solution, but I hope that this is at least a small step towards a reconciliation between their viewpoints. In a way, Bohr was right to say that "a phenomenon does not exist, until is observed" [Bohr28, WZ83], and Einstein was right to hope for a better, more complete, explanation of the quantum phenomena.

Perhaps, Schrödinger's ideas are most compatible with the SQM, since he disliked the discontinuous collapse, and believed in the physical reality of the wave functions. For example, he took the charge density in the electron's wave function literally, not as a probability distribution, and, according to SQM, he may be right.
The electron is the electron's wave function, since it is not a point, it is a wave, having different "shapes", depending on the measured observables.

The determinism is regained, since the evolution is again deterministic. The efforts of de Broglie, Vigier, culminating with David Bohm's causal or ontological interpretation of Quantum Mechanics [Bohm52, BH93], are theories whose purpose is to restore the determinism, the causality, and the reality and independence of the world. The price, as we now know, was the necessity to admit the nonlocality [Bel64, CHSH69, CS78, ADR82, Asp99]. SQM provides a deterministic theory without extra "hidden variables", rather, it is based on undetermined variables, or undetermined initial data. Here, "to determine" has a passive meaning - "to measure/observe", and an active one - "to choose". The initial data is determined by measurements, but we can choose what to measure. We can look at the indeterministic QM as applying to open systems only, whose description can be completed to a deterministic image by accounting for the parameters "hidden" in the environment.

The indeterministic view is not completely lost, since what the observer has is the registry, which is never a complete set of initial data. Each new measurement can bring new information, and the registry can be extended in different ways. We can interpret this in two ways. The first way is that the past is not established, and it is progressively created by each new choice of the observables, and, consequently, by each new outcome of the measurement. The second way to see the things is that all possible worlds exist, like a sheaf, and when we choose the observable we reduce the sheaf of worlds compatible with our registry. Each extension of the registry reduces this sheaf. Therefore, SQM is compatible with the Many Worlds Interpretation, with the amendment that each world is deterministic, and the only split is in the observer's registry, which can be completed in many ways. We can call this version of MWI the Many Registry Interpretation.

Perhaps one reason in the acceptance of a fundamentally indeterministic behavior in Quantum Mechanics was the belief that this is the only way to allow the existence of free-will [CK06, CK08]. The Smooth Quantum Mechanics offers an alternative, a deterministic view, which is still compatible with the free-will, at the same extent as the standard QM. We have the same freedom in choosing what observable to measure, influencing by this the initial conditions [tH07], but in a smooth and deterministic manner.

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Smooth Quantum Mechanics: 4. The Video

2009-02-02T00:24:00.001-08:00

Here is a five minutes video explaining the main idea of the Smooth Quantum Mechanics.

What is the collapse of the wave function? Is it necessary? The Quantum Mechanics can be built without making use of discontinuities in the time evolution. The appearance of the collapse is due to the entanglement between the preparation device and the observed system, in combination with the delayed choice initial conditions.

Smooth Quantum Mechanics: 3. Registry and Evolving Block Universe

2009-01-06T05:06:00.000-08:00

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

2009-01-06T05:04:00.001-08:00

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.

Smooth Quantum Mechanics: 1. The Smooth Particle

2009-01-04T12:56:00.000-08:00

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.

The Counterintuitive Time: 5. Quantum Time

2009-01-02T06:36:00.000-08:00

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”.

The Counterintuitive Time: 4. Time and General Relativity

2008-12-30T22:43:00.000-08:00

This is a series of posts about the counterintuitive nature of time in Physics. This post presents some difficult aspects of time in General Relativity, such as the notion of curved spacetime, the looping time, and the idea of the beginning of time.

After the Special Relativity, Einstein tried to express various physical laws in this formalism. Basically, their mathematical form is required to be invariant at Lorentz transformations. The main difficulty he encountered was to express Newton’s gravitation field in this way. After many years of research, Einstein obtained the movement of bodies under the effect of gravity as simply an inertial movement in a curved spacetime. The inertia and the gravity become unified. The spacetime itself is curved by the masses, and the universal attraction was just an effect of this curvature. General Relativity was born. The experimental consequences eventually confirmed the theory, which become widely accepted. Among its surprising features is that that the time flow changes in the presence of massive bodies.

The main equation of General Relativity, Einstein’s equation, relates the curvature to the distribution of energy. One very important difference between this equation, and the previously known equations in mathematical Physics, is the following: finding the solution, means also finding the background (meaning the spacetime itself). In the Newtonian and special relativistic cases, the spacetime was fixed, but in General Relativity, it is part of the solution itself. Perhaps, this is the most striking difference.

The main counterintuitive aspect of the curved spacetime is caused by our tendency to consider it as a subspace of a space with more dimensions. Many persons, when learn for the first time that the spacetime is curved, tend to interpret this as being curved in a fifth dimension. As a simpler but historic example, when we think at a curved surface, we tend to consider it a subspace of the Euclidean space. Gauss realized that the intrinsic geometry of every surface can be expressed independently on the Euclidean space in which this is embedded. The main ingredient is the metric tensor, which provides a point-dependant measure of the lengths of the curves embedded in the surface. Riemann generalized the surfaces to curved spaces with any number of dimensions. Their work helps understanding that the curved spaces in Riemannian geometry do not rely on a Euclidean space in which they may be embedded. Einstein found the four-dimensional Riemannian geometry as the ideal tool for General Relativity, provided that we replace the Euclidean metric tensor with the Lorentz metric.

The Einstein’s equation may have solutions that contain closed timelike curves. Spacetime may be curved in such a manner, that the future of an event becomes also its past. This looping time highly contradicts our intuition. Yet, unlike the other counterintuitive aspects of time, this one may not even exist, as Hawking’s Chronology Protection Conjecture states.

Another hard to grasp aspect of time is the beginning. Our experience teaches us to consider the time as being linear, infinitely continued in past and future. Why do we have this intuition, considering that our lives are finite? Perhaps because the daily events succeed linearly, at our scales, and because the History of our countries, and of our planet, and solar systems, appear to be linear. But when we hear about the Big Bang, two questions we may find natural is “what was before the Big Bang?”, and “when happened the Big Bang?”. We find difficult to accept that even the time may have a beginning.

Other difficult aspects of time in General Relativity are related to special situations, like the time in the presence of a Black Hole, in a Worm Hole, time traveling using Worm Holes, Hawking’s imaginary time, the time near/at the initial singularity. I will not detail these problems.

The Counterintuitive Time: 3. The Time's Arrows

2008-12-30T22:42:00.000-08:00

This is a series of posts about the counterintuitive nature of time in Physics. In this post it is analyzed the difference we perceive between past and future, as it appears in irreversible phenomena.

Seeing that the equations are symmetric at time reversal, we may legitimately wonder why the time has a direction. Boltzmann answered this question long time ago, when he explained the entropy, but since then, many felt that the things are not clear yet.

If at microscopic level the laws are symmetric to time reversal, why are they irreversible at larger scales? At larger scales, two systems which differ at small scale, may look identical. For example, to spheres made of the same material, and of the same radius, having the same density, may be considered identical, although their microscopic structures are far from being identical. Two glass balloons of identical shapes, filled with the same quantity of the same gas, will look identical at macroscopic level, but very different at atomic scale. Boltzmann defined the entropy of a macroscopic state of a system as minus the logarithm of the number of distinct microscopic states that macroscopically look identical to the macroscopic state. This definition fit well the entropy as it was known in Physics, and also has an analog in Shannon’s information theory, which led to an informational interpretation of the entropy. For our discussion, we will deal with its probabilistic meaning. A system tends to evolve to a more probable state, and a state with larger entropy is more probable. This is the key to understanding phenomena which are thermodynamically irreversible, like boiling an egg or breaking a cup.

The entropy will increase only to a maximum value corresponding to the most probable state, after that it will just fluctuate around that value. But then, it seems to follow that the present state is most likely to be one of the most probable, with the maximum of entropy, therefore we should not observe an increase of entropy, and no special arrow of time. The answer is that our present state is one of the most improbable, and therefore the entropy has enough room to increase. Moreover, it appears that the entropy increases since the Big Bang, and at that initial moment the entropy was very low. Very low entropy means very improbable, so the matter distribution at the Big Bang was very improbable. The permanent increase of entropy is explained not by a universal law of Physics, like the fundamental laws, but by a special property of the initial conditions. It is a “historical law”, and not a “universal law”.

The Big Bang itself seems to provide initial conditions improbable enough to activate the Second Law of Thermodynamics, by the simple fact that the matter was all concentrated in a very small region, most probably a singularity. But not all scientists consider this concentration enough. For example, Roger Penrose proposed an explanation of the thermodynamic arrow of time based on the condition that the Weyl tensor canceled. The tensor describing the curvature of the spacetime in General Relativity contains a part corresponding to the energy-momentum tensor, the other part is the Weyl tensor. But the Weyl tensor can be viewed, by the mean of the Bianchi identity, as corresponding to the gravitational field generated by the energy-momentum tensor of the matter. I interpret Penrose’s condition Weyl=0 as simply stating that in gravitation, only the “retarded gravitational potential” should be considered (similar to the retarded potential in Electrodynamics). Therefore, it seems that Penrose’s condition refers to a “radiative arrow of time”. It seems that the Big Bang, the cosmological arrow, is tied with the thermodynamic and radiative arrows.

The psychological arrow of time, corresponding to our minds remembering only the past, is perhaps the most difficult to grasp. It is habitually to be explained by comparing the brain with a computer who, in order to use its memory, needs to heat the environment, increasing the entropy.

I believe that the explanations of the arrows of time are very counterintuitive, and one reason is that they are based on symmetry breaking. The PDE expressing the fundamental physical laws are time-symmetric, but the solutions are not necessarily so. The time asymmetry is related very well with the existence of a special time, of minimum entropy, and that time is, naturally, the origin of time’s arrows. Because of the difficulty in accepting the arrow of time in a world governed by time-symmetric fundamental laws, some physicists try to find fundamental laws which exhibit time-asymmetry. In most cases, the asymmetry is searched in quantum phenomena, especially in the measurement process. But many consider the time arrows explained well enough, not requiring supplemental assumptions.

Yet, if one of the time’s arrows is less understood, I think that this is a psychological one, not necessarily restrained to memory, but to the whole psychological meaning of the words “time flows”. Perhaps the central point of the flow of time is the subject experiencing it, the “I” of each one of us.

The Counterintuitive Time: 2. The Geometric Time

2008-12-30T22:39:00.000-08:00

This is a series of posts about the counterintuitive nature of time in Physics. This post tries to identify the problem in accepting the geometric nature of time implied by Special Relativity, as well as the differences between space and time in relativistic spacetime.

In Newtonian Mechanics, the world evolves deterministically. The time is a parameter similar to the space coordinates. The PDE describing the evolution respect symmetries at orthogonal transformations of space, and at time translation and time reversal. Another symmetry is related to the speed of the inertial reference frames: the laws do not depend on the speed of an inertial frame.

A challenge of the Galilean relativity is provided by the Maxwell’s equations. The Electrodynamics suggested another group of symmetries, the Poincaré group, and its Lorentz subgroup, which are associated to the Special Relativity.

The introduction of the Lorentz transformations shed a new light on the nature of time. The time is no longer a parameter, but it gains a geometric meaning, which brings new counterintuitive aspects. The geometric meaning of time comes from the Lorentz invariance. The Lorentz transforms can “mix” space and time dimensions, like a spatial rotation can mix two directions of space. The relativity of simultaneity, which is a central point of Einstein’s theory, provides a physical interpretation of this character. This challenges our intuition, because it suggests that spacetime is a single geometric and timeless entity. Each direction in the Minkowski spacetime corresponds to a speed. The relative speed between two such directions can be obtained by applying the hyperbolic arctangent to the angle between them. This shows that two inertial frames moving with a relative speed, have different time direction in spacetime.

When somebody hears about the Minkowski spacetime as a symmetric space, may think why we couldn’t move through time like we are moving through space. The usual answer involves the idea of lightcone, but this explanation is not enough. But let us first discuss the lightcone and the causal structure of Relativity.

The lightcone is the set of all spacetime directions which corresponds to light speed. The 4-vectors from inside the cone, represents time directions, and the ones from outside, spatial directions. The squared norm of a spacelike vector has opposite sign than the squared norm of a timelike vector, and the lightlike vectors have zero norm, being also named null vectors. The Lorentz transformations preserve the norms, therefore they cannot be used to turn a timelike vector into a spacelike vector.

It seems impossible for an object having a velocity smaller than the speed of light to change smoothly the direction in spacetime and go back in time. The main reason is that its velocity will need to become the speed of light, and then larger (to go out of the light cone). But what is the problem with a body being accelerated to the speed of light? The answer is that we would need an infinite amount of energy for doing this. When the body increases its speed, its mass also increases, and the energy required for increasing further its speed becomes larger. For going to the speed of light, we will need to give it an infinite amount of energy.

Although we understand that the Relativity explains well our limitations in moving through time like we are moving through space, this difference between space and time are still so deep rooted in our intuition, that we find very difficult to accept the geometric nature of time.

A second counterintuitive aspect is the difference between the spacelike and the timelike vectors. If they can be rotated one into another by Lorentz transforms, this rotation is only partial, because we cannot transform a spacelike vector into a timelike vector. This asymmetry is not that annoying from mathematical viewpoint, and, as we saw from the previous argument, it is even useful. But many find it disturbing, and they feel like there is a need to replace the Lorentz metric with a Euclidean one (by some mathematical trickery). In general, these attempts ended up by complicating the things, and the mainstream physicists remained with the Lorentz metric. But, I cannot deny that there may be persons who consider simpler the Euclidean approach, because the price of accepting an indefinite metric seems too high for them. Maybe it is a matter of taste.

The Counterintuitive Time: 1. Time and Determinism

2008-12-30T12:35:00.000-08:00

This is a series of posts about the counterintuitive nature of time in Physics. This first post tries to identify the reasons why people may not agree with the deterministic view, and with the block spacetime view, in our sense of free-will and of flowing time.

The paradigmatic example of a world governed by determinism is provided by a system of differential equations (DE), or, more generally, of partial differential equations (PDE). In Physics, the PDE are required to have as solutions functions of position and time. In Newtonian Mechanics, the position is a point in the 3-dimensional Euclidean space, and the time is a real number. The positions and the instants form a 4-dimensional real vector space. The solutions of the PDE are functions $f(x, t)$ defined on this space. The state of the system at a moment of time $t$ is given by $f_t(x) = f(x, t)$.

The PDE systems appearing in mathematical physics have the nice property that, by knowing the state at a moment of time $t_0$, and the values of some additional partial derivatives of $f$ (in general the first order ones are enough), we can determine uniquely the states for another time $t$. For the solution to exist at $t$, the initial states and the derivatives appearing in the initial conditions are required to be well-posed, but I will not detail here. What is important is that we can extend the state at a time $t$ to contain not only $f_t$, but also the partial derivatives involved in the initial condition. This way, all the information about the system and its time evolution is contained in the extended state at each instant.

The fundamental equations of Physics are PDE, and they satisfy these conditions. One important exception seems to be provided by the Quantum Mechanics, where the indeterminism seems to be fundamental, but for the moment I will concentrate on the deterministic situation.

In such a deterministic world, the extended states contain all the information about the system. There is no physical property which is not contained in the extended state. Is the world we live in, of this type? It may be, or it may be not. If the world is like this, then it is a block world. The solution of the PDE is defined on the spacetime, and together they form a timeless, frozen entity, the function $f(x, t)$.

Many biologists and neurobiologists believe that, at least in principle, life and consciousness can be explained by making use of the deterministic properties of atoms and molecules, and perhaps more complex systems only, and not appealing to the indeterminism. Many persons understand what a deterministic world is, and even believe that our world may be of this type. Yet, they hardly accept the block world. A deterministic world contains all the information regarding all moments of time, at the extended state at each instant. There is no need to “play” this world, like playing a pick-up disk. If our world is deterministic, and if the minds are reducible to configurations of matter, then the extended state contains also the mind state of a possible observer. Are the observers just states depending on a real number (which is interpreted as time)? If it would be so, then there will be no change, in the sense that, at any instant, the 3d-observer at that instant will contain in its state the impression that he or she perceive a time flow, and a dynamic evolution of his/hers state. There will be only timeless 3d observers containing in their states the impression of time evolution. For a person, there will be one such 3d timeless observer, associated to each instant (which is just a real number). Considering all the instants making a “lifetime”, there will be an infinity of such 3d timeless observers, connected.

It is easy to understand why such view is rejected by many. Determinism leads to a block world view. Some may accept determinism, and reject block view. When they understand the relation between them, they may continue rejecting the block view, and therefore reject also the determinism. The main problem seems to be the block view. If our world is such, then we are also reduced to parts of a set of timeless states.

Perhaps the main parts of our intuition contradicted by this view are the following. First, the feeling of subjectivity, the sense of “I”. Second, the feeling that we have free-will. A block view seems to make everything frozen, predetermined.

The Spinning Dancer's Mistery

2008-10-11T17:45:00.001-07:00

I explain why the "spinning dancer" is perceived by the most as spinning clockwise. I will then show why the laws of perspective indicate that in fact she spins counter-clockwise.

What is "The Spinning Dancer"?

The spinning dancer is an interesting optical illusion created by Nobuyuki Kayahara. The ambivalence of the image makes some observers seeing that the dancer is spinning clockwise, while others have the impression that she is spinning counter-clockwise. This is an example of bistable optical illusion.

Some tests presented at Cognitive Daily show that the ones perceiving the dancer spinning clockwise are twice as many as the others.

Why two people out of three see the dancer spinning clockwise?

I will explain now why. The answer is based on our experience. We are accustomed to see other people from one specific angle - from about the level of their eyes. For example, in the case of the dancer, the rotating leg’s toe describes a (almost) horizontal circle. Well, at least the circle would be horizontal, if the dancer would not go up and down. Our brains tend to make the supposition that we are looking at that circle from the above, that the circle is at a lower level than our eyes. The figure below shows this:

This is also the reason why, when you look at the dancer’s reflection, you switch the to counter-clockwise mode. The reflected circle, seen from above, corresponds to the counter-clockwise spin.

Why the correct answer is that the dancer spins counter-clockwise?

Well, although the illusion is usually presented as lacking any hint for the depth, I will show you that this is not the case. If we look at the dancer’s reflection in the floor, we can see that, assuming that the circle is more or less horizontal, we obtain a clue. If we assume that the circle described by the dancer’s toe is horizontal, then its plane should be parallel to the plane containing the circle described by the reflection of the toe in the floor. But the laws of the perspective tell us that a closer object look larger than a distant one. Accordingly, the distance between the toe and its reflection should look larger when the toe is closer to the observer. In our case, this happens if the dancer is spinning counter-clockwise. Of course, this means that the observer’s point of view is somewhere near the floor, because the circle described by the toe is now perceived from below.

To give a chance to the clockwise interpretation, we can assume that the floor is very inclined, so that the circle is far from being parallel to its reflection. Even in this case, the point of view should be under the floor’s level, because the closest side of the reflected circle appears to be above the other one. But in this case, the reflection makes no sense. If it is shadow and not reflection, it should not appear upside down, when we see it from below the floor. Therefore, the only interpretation compatible with the laws of perspective is the counter-clockwise one.

It is therefore safe to affirm that the original 3d animation (perhaps created with a 3d computer graphics software like 3ds Max) from which Nobuyuki Kayahara exported the animation rotates indeed counter-clockwise.

Breaking the illusion

How many of our perceptions, how many of our beliefs, are illusions? Is there a possibility to understand that we are living in an illusory world? I not necessarily refer to the objective world as being an illusion. By definition, the illusions are subjective. Between us and the objective world, is a thick layer of more or less illusory beliefs.

To realize whether a belief is based on illusion, we can use the logic. We can check the consistency; this is the best reality test. If the laws of perspective tell us that the reflection of the dancer is not consistent with her spinning clockwise, then we can try the opposite interpretation. We can try the opposite interpretation anyway, as a mean to wonder whether our point of view is the only possible, or as a way to understand other perspectives.

The opinion of the majority cannot be taken as a reality test, or as a proof of objectiveness. Their opinion matters in politics, it matters in their personal lives, or in marketing. The opinion of the others matter for us, because we care about them. But we should not consider the opinion of the majority as our reference frame. The majority believes that the dancer is spinning clockwise, but the logic and the geometry showed us that she is spinning counter-clockwise.

Even if you are among the few that saw the dancer spinning counter-clockwise, this may be an illusion too. You can dream that you are in your bed, and when you wake up you find that you were right, but it was only an illusion. You can consider that you were not fooled by the illusion only when you not only were right, but you understood, or experienced, the reason why you are right.

An example makes the things clearer. If you simplify in the fraction 16/64 the figure 6, you obtain the correct answer 1/4. But you obtained it by pure chance; the same solution will not apply to other fractions. The way to solve the problem was completely wrong, but this does not guarantee you that the answer will be wrong too.

Let me summarize: don’t let you be fooled by the illusions, question every preconception you are tempted to adopt. Whether the preconception belongs to the majority, to a minority you trust, to a person you cherish… Even better: question every preconception of yourself. A preconception is not necessarily false, but it is based on false arguments. The logical consistency should be your first guide out of the illusion.

The Illusion of Center

2008-09-25T08:31:00.000-07:00

There really exists a center, or it is an illusion?

The O’Reilly’s rotating grid is an example of an illusion of center, as I will explain. Then, some math skills will allow us to find a grid rotating around two exact centers. The next part will be about the self and the illusion of center.

O'Reilly's rotating grid

Let's take a look at an interesting effect, discovered by David O’Reilly:

As its discoverer noticed, it is an effect of temporal aliasing, but I will add an ingredient - the illusion of center. The temporal aliasing is an effect which usually appears when the moving images are presented as a sequence of frames. Presenting the image by frames creates the illusion of motion, but sometimes the information contained in the frames is ambiguous. As we can see below, if we highlight the parts that seem to rotate, the region delimited by the yellow square is rotating around a central point. The regions delimited by the red squares, although may appear to have their own centers, at a second examination are found not to have a center.

If we look at a real rotating grid, the effect cannot appear. It happens only because the frames present us only some lines. Two rotating orthogonal lines pass through the center. Together with them, other parallel lines rotate around the center. Near every point on the image there are lines, which are parallel to the lines passing through the center. The "approximate centers" are near more lines than the others, but it is only the center that contains lines for all possible directions. We can see this by overlapping all the frames:

Rotation around two centers

If we rotate the grid at each step with an angle of 360/n, the center is the only point containing all the time vertices of the grid (except, of course, the case when n=1, 2 or 4, when no rotation is viewed). But if we are good enough at math, we can modify the animation such that we obtain more than one fixed point (please click to see the full image):

The animation above is based on some properties of the number 65. This number plays the role of the hypotenuse in eight Pythagorean triples:

65²=16²+63²=25²+60²=33²+56²=39²+52²=52²+39²=56²+33²=60²+25²=63²+16².

If we allow the hypotenuse not necessarily be integer (for example to square to 325), then the obtained image can be smaller:

and a tile version is here.

Here is a checker board pattern:

and you can see some dots here.

The self and the illusion of center

Why are our minds tricking us into believing that there is a center where in fact it is not? Perhaps this happens because, in a perpetually moving world, we need to believe that there is a fixed point of the Universe. We need to believe that we can see where the phenomena converge, so that we can understand the laws of Nature, so that we can survive and have benefits. In order to want to survive and evolve, we need to believe that we are important, that we have a central self.

At different stages in our lives, we may be so different, that we hardly can say that we are the same person. Our interests and ideals as children may differ very much from those we have as adults. Our priorities and even values at the office or school may be very different than the ones we manifest in family, or with our friends. If we are changing that much during our lifetimes, is there a convergence point of all our facets? Do we really have a self? Or this is just another trick of our mind?

In everything happens to us, sometimes we may feel the need to believe that there is a reason, other than blind chance. We may be tempted to hope that there is a reason behind everything, a central point towards everything converges. Is there such a Center of the World, like the one marked by cross? If there is, how are we, the other beings, like the areas marked with red squares, with no real center? Or are we all centers in the same time, like the vertices marked with red dots? Or there is nothing else but the emptiness of the grid?