Saturday, February 14, 2015

Men are classical, women are quantum

Man can be understood in the framework of classical physics, but for woman you'll need quantum physics.

Wednesday, November 5, 2014

50 Years of Misunderstanding Bell's Theorem

Precisely 50 years ago, Bell's paper "On the Einstein Podolsky Rosen Paradox", containing his famous theorem was received by the journal Physics. Today is John Bell Day.

Bell's theorem is one of the most influential result in physics, despite the fact that it is a negative result. Contrary to what many people believe, Bell was actually searching for a hidden variable theory, and he found instead some severe limitations of such theories. The limitation expressed by Bell's theorem celebrated today is that hidden variable theories have to be nonlocal. The outcome of measurements are correlated in a way which seems to ignore the separation in space. Some misunderstand this result as rejecting determinism, or as rejecting any kind of hidden variables, or at least as proving that any theory which describes the quantum world using hidden variables has to rely on instantaneous communication.

Maybe others searching for a hidden variables description of quantum phenomena hit the same wall Bell hit, but rather than having the same revelation as Bell, they ignored it and continued to search for a replacement or completion of quantum mechanics. For example, Einstein had all the data to find Bell's theorem almost 30 years before Bell. The paper coauthored by Einstein, Podolsky and Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" shown that entanglement allows nonlocal correlations. But Einstein disliked nonlocality because it seemed to violate special relativity. So he concluded that quantum mechanics was incomplete, by interpreting those correlations as revealing that Heisenberg's uncertainty principle can be trespassed. So Einstein and coauthors hit the same wall as Bell, only that they considered that the problem could be solved by completing quantum mechanics. Bell's theorem clarifies their findings in showing that no matter how you put it, the world is nonlocal (if Bell's inequality is violated, as it was confirmed by experiments).

Almost 30 years later, Bell understood nonlocality as the major consequence of the EPR "paradox", and expressed it in the form of his theorem. Today, at 50 years after Bell clarified the problem, there are so many who consider that Einstein was a crackpot in what concerns quantum mechanics, and Bell defeated him. Today it is easy for any student who took a class of quantum mechanics or philosophy of physics, to consider that he has a better understanding of quantum mechanics that Einstein, and to feel superior to him (true story, just search the physics blogs and forums and you will see many examples). Most often they believe (as they are taught) that quantum mechanics is so radically different because it is not deterministic, and that what Einstein searched was a deterministic theory. And that EPR suggested this, and Bell rejected it. This is so unfair for EPR, but also for Bell.

The truth is that despite the 10 hot years of discoveries in quantum mechanics, when nearly every aspect was understood, and the foundations were laid down, nobody before Einstein, Podolsky and Rosen found that "paradox", which is true and relevant. It is unfair to consider the EPR an attack against quantum mechanics, as it is seen by many since the beginning. Rather, it is a most important discovery, which could only be made because three rebels were not satisfied with Bohr's prescriptions. Moreover, in almost 30 years since the EPR paper, nobody solved their "paradox". Not even Bohr, who rushed to respond too quickly with an article bearing the same name as the EPR one. And the solution was found by Bell, who was a supporter of hidden variables, and maybe he wouldn't find it either, without the reformulation of the EPR argument due to the main exponent of hidden variable theories at that time, David Bohm.

Now, the reader may think that I am defending the hidden variables, by praising hidden variables theorists like Einstein, Bohm, and Bell. I actually don't defend hidden variables, and I don't say this just because of the witch hunt against "Bohmians". I just want to emphasize that without these "crackpots", we would not have today the understanding of entanglement and nonlocality which allows scientists to put at use the "magic" of quantum mechanics at work in quantum computing, quantum information, quantum cryptography, and other recent hot areas.

Actually, to be honest, among Einstein, Bohm, and Bell, only the first two are considered a lacking understanding of quantum mechanics, and Bell is considered as the one who defeat them, so he is celebrated, while the other two are not. But this is only because Bell is perceived as being, because of his theorem, against hidden variables, while in fact he was also searching for a hidden variable theory.

Moreover, for some reason, many consider that Bell's theorem is only about hidden variable theories, while in fact it is about any quantum theory or interpretation which describes quantum correlations as are observed in nature, and therefore violates Bell's inequality. Including therefore standard quantum mechanics. So, quantum mechanics is nonlocal too, and no Copenhagen Interpretation, no Many Worlds Interpretation, no Decoherence Interpretation can make it otherwise. Similarly, quantum mechanics is contextual too, despite the fact that the Bell-Kochen-Specker theorem is considered to apply to hidden variable theories only.

But why some tend to consider only hidden variable theories guilty of the sins of nonlocality and contextuality? Maybe because they just want to reject such theories? Or could it be because they believe that it makes no sense to think about what happens between measurements (as Bohr teaches us)? Or because nearly everyone, when first learning about quantum mechanics, has the instinct of finding a local realist explanation, and fails, of course, and then denies having this sin by throwing stones at those who seem to have it? I think this is fine, since this is what we should do, we should question everything, and that the persistence with which we should question a claim has to increase with the degree by which that claim contradicts what we learned before, as is the case of quantum mechanics.

For lack of time, for the rush of getting published, for the fear of getting rejected for having unorthodox views, we tend to eat much more than we can digest, and actually we cease digesting. This is why misunderstanding are propagated even at the top of the scientific community. Misunderstandings concerning quantum mechanics and Bell's theorem prevent us from seeing both the truth, and the amazing beauty of quantum mechanics, which is transformed into a mere tool to calculate probabilities, and any attempt at understanding it is regarded with disdain.

I find very fortunate the fact that Tim Maudlin wrote for the 50th anniversary of Bell's theorem a paper named "What Bell did", in which he explains that Bell's result is that indeed our world, hence quantum mechanics, is nonlocal. He makes a thorough and in my opinion probably the most down to earth analysis of the meaning of the EPR paper and of Bell's theorem, and how they are misunderstood. He identifies a cluster of misunderstandings that are propagated among physicists and philosophers of physics. This is one of the cases when a philosopher really can help physicists understand physics. I'll leave you the pleasure to read it.

Tuesday, November 4, 2014

Happy Birthday, Nature!

Exactly 145 years ago, on 4 November 1869, the first number of Nature appeared.

According to the current Romanian prime minister Victor Ponta, Nature is controlled (unofficially) by the current president of Romania, Traian Băsescu, with the main purpose to accuse Ponta of plagiarism. A possible explanation is that some collaborators of Băsescu traveled in time to create the journal. And then they also founded 145 years ago a secret society of scientists, who kept making great scientific discoveries. This also explains why most scientific discoveries were made in the last 1.5 centuries. The reason to make these scientific breakthroughs is not to advance the world, but to publish them in Nature, or in the other journals citing Nature, to make it world's most cited journal, so that, when the time comes, Nature's accusations against Ponta will have greater impact. And to invent rules that it is dishonest to copy text from other books and articles without attributing it explicitly when you write your PhD thesis. The second reason would be that the research made by this secret society of genii secretly led by the mastermind Băsescu will eventually lead to the discovery of time travel [1,2,3,4], which would allow him to send his people 145 years back in time...

Now, presidential elections are taking place in Romania. Two days ago was the first round, and now Ponta is the favorite to become the new president after the second round, in 12 days. Ponta has now the chance to forbid time travel, to change the history back to its track, which is a world without the Nature journal and all that scientific research in it, a world without physicists who can discover time travel. Or quite the opposite, he may actually make use of time travel, to set our history back to 25 years ago, when people overthrew Ceaușescu, and the communists were forced to disguise themselves as social-democrats to continue to keep the power.

Monday, October 6, 2014

Dots plus dots equal spheres

I took this photo in a bus in Pisa. We can see a pattern of spheres.

Here is how to obtain it. We overlap these dots

over a shrunk version of theirs

and we get the following pattern:

Wednesday, October 1, 2014

Living in a vector

Vectors are present in all domains of fundamental physics, so if you want to understand physics, you will need them. You may think you know them, but the truth is that they appear in so many guises, that nobody really knows everything about them. But vectors are a gate that allows you to enter the Cathedral of physics, and once you are inside, they can guide you in all places. That is, special and general relativity, quantum mechanics, particle physics, gauge theory... all these places need vectors, and once you master the vectors, they become much simpler (if you don't know them and are interested, read this post).

The Cathedral has many gates, and vectors are just one of them. You can enter through groups, sets and relations, functions, categories, through all sorts of objects or structures from algebra, geometry, even logic. I decided to show you now the way of vectors,  because I think is fast and deep in the same time, but remember, this is a matter of choice. And vectors will lead us, inevitably, to the other gates too.

I will explain some elementary and not so elementary things about vectors, but you have to read and practice, because here I just give some guidelines, a big picture. The reason I am doing this is that when you study, you may get lost in details and miss the essential.

Very basic things

A vector can be understood in many ways. One way is to see it as a way to specify how to move from one point to another. A vector is like an arrow, and if you place the arrow in that point, you find the destination point. To find the new position for any point, just place the vector in that point, and the tip of the vector will show you the new position. You can compose more such arrows, and what you'll get is another vector, their sum. You can also subtract them, just place their origins in the same point, and the difference is the vector obtained by joining their tips with another arrow.

Once you fix a reference position, an origin, you can specify any position, by the vector that tells you how to move from origin to that position. You can see that vector as being the difference between the destination, and the starting position.

You can add and subtract vectors. You can multiply them with numbers. Those numbers are from a field $\mathbb{K}$, and we can take for example $\mathbb{K}=\mathbb{R}$, or $\mathbb{K}=\mathbb{C}$, and are called scalars. A vector space is a set of vectors, so that no matter how you add them and scale them, the result is from the same set. The vector space is real (complex), if the scalars are real (complex) numbers. A sum of rescaled vectors is named linear combination. You can always pick a basis, or a frame, a set of vectors so that any vector can be written as a linear combination of the basis vectors, in a unique way.

Vectors and functions

Consider a vector $v$ in an $n$-dimensional space $V$, and suppose its components in a given basis are $(v^1,\ldots,v^n)$. You can represent any vector $v$ as a function $f:\{1,\ldots,n\}\to\mathbb{K}$ given by $f(i)=v^i$. Conversely, any such function defines a unique vector. In general, if $S$ is a set, then the set of the functions $f:S\to\mathbb{K}$ form a vector space, which we will denote by $\mathbb{K}^S$. The cardinal of $S$ gives the dimension of the vector space, so $\mathbb{K}^{\{1,\ldots,n\}}\cong\mathbb{K}^n$. So, if $S$ is an infinite set, we will have an infinite dimensional vector space. For example, the scalar fields on a three dimensional space, that is, the functions $f:\mathbb{R}^3\to \mathbb{R}$, form an infinite dimensional vector space. Not only the vector spaces are not limited to $2$ or $3$ dimensions, but infinite dimensional spaces are very natural too.

Dual vectors

If $V$ is a $\mathbb{K}$-vector space, a linear functions $f:V\to\mathbb{K}$ is a function satisfying $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u,v\in V,\alpha\in\mathbb{K}$. The linear functions $f:V\to\mathbb{K}$ form a vector space $V^*$ named the dual space of $V$.


Consider now two sets, $S$ and $S'$, and a field $\mathbb{K}$. The Cartesian product $S\times S'$ is defined as the set of pairs $(s,s')$, where $s\in S$ and $s'\in S'$.  The functions defined on the Cartesian product, $f:S\times S'\to\mathbb{K}$, form a vector space $\mathbb{K}^{S\times S'}$, named the tensor product of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, $\mathbb{K}^{S\times S'}=\mathbb{K}^{S}\otimes\mathbb{K}^{S'}$. If $(e_i)$ and $(e'_j)$ are bases of  $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, then $(e_ie'j)$, where $e_ie'_j(s,s')=e_i(s)e'_j(s')$, is a basis of $\mathbb{K}^{S\times S'}$. Any vector $v\in\mathbb{K}^{S_1\times S_2}$ can be uniquely written as $v=\sum_i\sum_j \alpha_{ij} e_ie'j$.

Also, the set of functions $f:S\to\mathbb{K}^{S'}$ is a vector space, which can be identified with the tensor product $\mathbb{K}^{S}\otimes(\mathbb{K}^{S'})^*$.

The vectors that belong to tensor products of vector spaces are named tensors. So, tensors are vectors with some extra structure.

The tensor product can be defined easily for any kind of vector spaces, because any vector space can be thought of as a space of functions. The tensor product is associative, so we can define it between multiple vector spaces. We denote the tensor product of $n>1$ copies of $V$ by $V^{\otimes n}$. We can check that for $m,n>1$, $V^{\otimes (m+n)}=V^{\otimes {m}}\otimes V^{\otimes {n}}$. This can work also for $m,n\geq 0$, if we define $V^1=V$, $V^0=\mathbb{K}$. So, vectors and scalars are just tensors.

Let $U$, $V$ be $\mathbb{K}$-vector spaces. A linear operator is a function $f:U\to V$ which satisfies $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u\in U,v\in V,\alpha\in\mathbb{K}$. The operator $f:U\to V$ is in fact a tensor from $U^*\otimes V$.

Inner products

Given a basis, any vector can be expressed as a set of numbers, the components of the vector. But the vector is independent of this numerical representation. The basis can be chosen in many ways, and in fact, any non-zero vector can have any components (provided not all are zero) in a well chosen basis. This shows that any two non-zero vectors play identical roles, which may be a surprise. This is a key point, since a common misconception when talking about vectors is that they have definite intrinsic sizes and orientations, or that they can make an angle. But in fact the sizes and orientations are relative to the frame, or to the other vectors. Moreover, you can say that from two vectors, one is larger than the other, only if they are collinear. Otherwise, no matter how small is one of them, we can easily find a basis in which it becomes larger than the other. It makes no sense to speak about the size, or magnitude, or length of a vector, as an intrinsic property.

But wait, one may say, there is a way to define the size of a vector! Consider a basis in a two-dimensional vector space, and a vector $v=(v^1,v^2)$. Then, the size of the vector is given by Pythagoras's theorem, by $\sqrt{(v^1)^2+(v^2)^2}$. The problem with this definition is that, if you change the basis, you will obtain different components, and different size of the vector. To make sure that you obtain the same size, you should allow only certain bases. To speak about the size of  a vector, and about the angle between two vectors, you need an additional object, which is called inner product, or scalar product. Sometimes, for example in geometry and in relativity, it is called metric.

Choosing a basis gives a default inner product. But the best way is to define the inner product, and not to pick a special basis. Once you have the inner product, you can define angles between vectors too. But size and angles are not intrinsic properties of vectors, they depend on the scalar product too.

The inner product between two vectors $u$ and $v$, defined by a basis, is $u\cdot v = u^1 v^1 + u^2 v^2 + \ldots + u^n v^n$. But in a different basis, it will have a general form $u\cdot v=\sum_i\sum_j g_{ij} u^i v^j$, where $g_{ij}=g_{ji}$ can be seen as the components of a symmetric matrix. These components change when we change the basis, they form the components of a tensor from $V^*\otimes V^*$. Einstein had the brilliant idea to omit the sum signs, so the inner product looks like $u\cdot v=g_{ij} u^i v^j$, where you know that since $i$ and $j$ appear both in upper and in lower positions, we make them run from $1$ to $n$ and sum. This is a thing that many geometers hate, but physicists find it very useful and compact in calculations, because the same summation convention appears in many different situations, which to geometers appear to be different, but in fact are very similar.

Given a basis, we can define the inner product by choosing the coefficients $g_{ij}$. And we can always find another basis, in which $g_{ij}$ is diagonal, that is, it vanishes unless $i=j$. And we can rescale the basis so that $g_{ii}$ are equal to $-1$, $1$, or $0$. Only if $g_{ii}$ are all $1$ in some basis, the size of the vector is given by the usual Pythagoras's theorem, otherwise, there will be some minus signs there, and even some terms will be omitted (corresponding to $g_{ii}=0$).

Quantum mechanics

Quantum particles are described by Schrödinger's equation. Its solutions are, for a single elementary particle, complex functions $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}$, or more general, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$, named wavefunctions. They describe completely the states of the quantum particle. They form a vector space $H$ which also has a hermitian product (a complex scalar product so that $h_{ij}=\overline{h_{ji}}$), and is named the Hilbert space (because in the infinite dimensional case also satisfies an additional property which we don't need here), or the state space. Linear transformations of $H$ which preserve the complex scalar product are named unitary transformations, and they are the complex analogous of rotations.

The wavefunctions are represented in a basis as functions of positions, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$. The element of the position basis represent point particles. But we can make a unitary transformation and obtain another basis, made of functions of the form $e^{i (k_x x + k_y y + k_z z)}$, which represent pure waves. Some observations use one of the bases, some the other, and here is why there is a duality between waves and point particles.

For more elementary particles, the state space is the tensor product of the state spaces of the individual particles. A tensor product of the form $|\psi\rangle\otimes|\psi'\rangle$ represents separable states, which can be observed independently. If the system can't be written like this, but only as a sum, the particles are entangled. When we measure them, the outcomes are correlated.

The evolution of a quantum system is described by Schrödinger's equation. Basically, the state rotates, by a unitary transformation. Only such transformations conserve the probabilities associated to the wavefunction.

When you measure the quantum systems, you need an observable. One can see an observable as defining a decomposition of the state space, in perpendicular subspaces. After the observation, the state is found to be in one of the subspaces. We can only know the subspace, but not the actual state vector. This is strange, because the system can, in principle, be in any possible state, but the measurement finds it to be only in one of these subspaces (we say it collapsed). This is the measurement problem. The things become even stranger, if we realize that if we measure another property, the corresponding decomposition of the state space is different. In other words, if you look for a point particle, you find a point particle, and if you look for a wave, you find a wave. This seems as if the unitary evolution given by the Schrödinger's equation is broken during observations. Perhaps the wavefunction remains intact, but to us, only one of the components continues to exist, corresponding to the subspace we obtained after the measurement. In the many worlds interpretation the universes splits, and all outcomes continue to exist, in new created universes. So, not only the state vector contains the universe, but it actually contains many universes.

I have a proposed explanation for some strange quantum features, in [1, 2, 3], and in these videos:

Special relativity

An example when there is a minus signs in the Pythagoras's theorem is given by the theory of relativity, where the squared size of a vector is $v\cdot v=-(v^t)^2+(v^x)^2+(v^y)^2+(v^z)^2$.

This inner product is named the Lorentz metric. Special relativity takes place in the Minkowski spacetime, which has four dimensions. A vector $v$ is named timelike if $v\cdot v < 0$, spacelike if $v\cdot v > 0$, and null or lightlike if $v\cdot v = 0$. A particle moving with the speed of light is described by a lightlike vector, and one moving with an inferior speed, by a timelike vector. Spacelike vectors would describe faster than light particles, if they exist. Points in spacetime are named events. Events can be simultaneous, but this depends on the frame. Anyway, to be simultaneous in a frame, two events have to be separated by a spacelike interval. If they are separated by a lightlike or timelike interval, they can be connected causally, or joined by a particle with a speed equal to, respectively smaller than the speed of light.

In Newtonian mechanics, the laws remain unchanged to translations and rotations in  space, translations in time, and inertial movements of the frame - together they form the Galilei transformations. However, electromagnetism disobeyed. In fact, this was the motivation of the research of Einstein, Poincaré, Lorentz, and FitzGerald. Their work led to the discovery of special relativity, according to which the correct transformations are not those of Galilei, but those of Poincaré, which preserve the distances given by the Lorentz metric.

Curvilinear coordinates

A basis or a frame of vectors in the Minkowski spacetime allows us to construct Cartesian coordinates. However, if the observer's motion is accelerated (hence the observer is non-inertial), her frame will rotate in time, so Cartesian coordinates will have to be replaced with curved coordinates. In curved coordinates, the coefficients $g_{ij}$ depend on the position. But in special relativity they have to satisfy a flatness condition, otherwise spacetime will be curved, and this didn't make much sense back in 1905, when special relativity was discovered.

General relativity

Einstein remarked that to a non-inertial observer, inertia looks similar to gravity. So he imagined that a proper choice of the metric $g_{ij}$ may generate gravity. This turned out indeed to be true, but the choice of $g_{ij}$ corresponds to a curved spacetime, and not a flat one.

One of the problems of general relativity is that it has singularities. Singularities are places where some of the components of $g_{ij}$ become infinite, or where $g_{ij}$ has, when diagonalized, some zero entries on the diagonal. For this reason, many physicist believe that this problem indicates that general relativity should be replaced with some other theory, to be discovered. Maybe it will be solved when we will replace it with a theory of quantum gravity, like string theory or loop quantum gravity. But until we will know what is the right theory of quantum gravity, general relativity can actually deal with its own singularities (while the ones mentioned above did not solve this problem). I will not describe this here, but you can read my articles about this, and also this essay, and these posts about the black hole information paradox [1, 2, 3]. And watch this video

Vector bundles and forces

We call fields the functions defined on the space or the spacetime. We have seen that fields valued in vector spaces are actually vector spaces. On a flat space $M$ which looks like a vector space, the fields valued in vector spaces can be thought of as being valued in the same vector space, for example $f:M\to V$. But if the space is curved, or if it has nontrivial topology, we are forced to consider that at each point there is another copy of $V$. So, such a field will be more like $f(x)\in V_x$, where $V_x$ is the copy of the vector space $V$ at the point $x$. Such fields still form a vector space. The union of all $V_x$ is called a vector bundle. The fields are also called sections, and $V_x$ is called the fiber at $x$.

Now, since $V_x$ are copies of $V$ at each point, there is no invariant way to identify each $V_x$ with $V$. In other words, $V_x$ and $V$ can be identified, for each $x$, up to a linear transformation of $V$. We need a way to move from $V_x$ to a neighboring $V_{x+d x}$. This can be done with a connection. Also, moving a vector from $V_x$ along a closed curve reveals that, when returning to $V_x$, the vector is rotated. This is explained by the presence of a curvature, which can be obtained easily from the connection.

Connections behave like potentials of force fields. And a force field corresponds to the curvature of the connection. This makes very natural to use vector bundles to describe forces, and this is what gauge theory does.

Forces in the standard model of particles are described as follows. We assume that there is a typical complex vector space $V$ of dimension $n$, endowed with a hermitian scalar product. The connection is required to preserve this hermitian product when moving among the copies $V_x$. The set of linear transformations that preserve the scalar product is named unitary group, and is denoted by $U(n)$. The subset of transformations having the determinant equal to $1$ is named the special unitary group, $SU(n)$. The electromagnetic force corresponds to $U(1)$, the weak force to $SU(2)$, and the strong force to $SU(3)$. Moreover, all particles turn out to correspond to vectors that appear in the representations of the gauge groups on vector spaces.

What's next?

Vectors are present everywhere in physics. We see that they help us understand quantum mechanics, special and general relativity, and the particles and forces. They seem to offer a unitary view of fundamental physics.

However, up to this point, we don't know how to unify
  • unitary evolution and the collapse of the wavefunction
  • the quantum level with the mundane classical level
  • quantum mechanics and general relativity
  • the electroweak and strong forces (we know though how to combine the electromagnetic and weak forces, in the unitary group $U(2)$)
  • the standard model forces and gravity