Is consciousness grounded in matter or the other way around?
We will ask physics about physicalism, zombies, and consciousness. This video is an as-simple-as-I-could explanation of this article https://philsci-archive.pitt.edu/23108/. We will use a bit of physics and math, but I'll introduce these smoothly, to make the trip easy and interesting. You can verify each step of the proof, and if you can find an error, please let me know. Many of the implications of this result remain open to be discussed later.
Abstract of the article
If the mind of a sentient being would be reducible to its structure, any system with identical structure should be equally sentient. Based on the structural symmetries of Physics, I prove that this thesis has two unexpected consequences:
1) There would be an inflation of minds, living in apparently different worlds. 2) The content of these minds would be independent of the properties of the external world. That is, these minds would be unable to know anything about the world.
Since this contradicts empirical observations, structure alone is insufficient for sentient experience.
This excludes the purely physicalistic approaches to physics and consciousness. For physics to be as we know it, all physical properties have to be grounded in something sentiential.
Kerr (yes, who found the well-known Kerr black hole solutions) disagrees with Penrose's singularity theorem and its variations. Namely these theorems prove the existence of geodesics that can't be extended beyond a finite affine length, but Kerr finds numerous examples of inextensible light rays that don't contain singularities. These geodesics go all the way to the null infinity, and yet the affine parameter remains finite. And there are such light rays through every point of the Kerr spacetime. Only some geodesics hit the ring singularity, but this region can be replaced by a nonsingular one, perhaps matter can do this. Kerr thinks that his perfectly symmetric vacuum solution doesn't happen in reality (despite the "no-hair theorem", which is in fact a conjecture improperly called "theorem", stating that all black holes evolve into a Kerr solution), even though he thinks that black holes exist.
Now, how is this possible? I mean the singularity theorems, now sealed forever by a Nobel prize, prove that there are conditions that necessarily lead to singularities. That if there's a black hole, there must be a singularity beyond its horizon. Or do they?
This is a bit of a word play. There are more meanings of the word "singularity". Normally singularity means a place where the metric blows up. Or its inverse. Or the curvature, or any field that we think it's physical. But then we can think of excluding these points from spacetime. If these points are "in the way" of the physical fields, if the evolution equations can't go beyond such a place but they should, this would be a problem even if we exclude them from spacetime. But if these singularities are somewhere at the "edge" of spacetime, and the spacetime admits a nice foliation so that the evolution equations work fine across the entire spacetime, why would this be a problem? And yet, the other definition of singularity, the one that is actually the object of the singularity theorems, includes such cases as well. That is, as a diagnostic method, it gives numerous false positives.
Here's what happened. And I don't say it's a plot against General Relativity, rather an accident, perhaps welcomed by many. If your spacetime contains singularities, we can think of excluding them from spacetime. But this, as I said, doesn't solve the problem. So maybe there is a way to detect this pathology even with the singularities removed, and talk about such a spacetime as being singular anyway. And here comes into play the redefinition of singular spacetime in terms of geodesic incompleteness. And it is said in the Hawking & Ellis bible, on page 258:
I don't want to single out this great book, it explains well the adoption of this diagnosis, and others said similar things. But here I think lies the problem. Because this definition can be misunderstood (unintentionally I think) in a way that makes the singularity theorems seem about singularities even if there are no singularities in the interior of spacetime, even if the spacetime can be nicely foliated, offering a nice home to the evolution equations.
The singularity theorems prove (and they indeed prove this) that there are incomplete geodesics, where incomplete means they can't be extended beyond a finite affine length. Whether all of them deserve to be called "incomplete" is also questionable. If the affine length (which is not the same as geometric length anyway) of a timelike or null geodesic is finite, but it goes to the "real edge" of spacetime, as in Kerr's paper, why should it be called incomplete? This already seeds in our minds the idea that there's something wrong with them.
So, one on top of another, the meaning of words shifted so that now it's widely believed that General Relativity breaks down, due to the singularities. And Kerr gives nice rich counterexamples, all in the same spacetime of a Kerr black hole. I mean, his spacetime has a singularity, but the singularity theorem doesn't even predict that singularity. It predicts some singularities, but they are false positives, they don't occur on the geodesics up to the boundary of spacetime. It doesn't predict the ring singularity, because, as Kerr says, there is no trapped surface inside the inner horizon of the Kerr black hole. So, if we cut out the spacetime around that ring, and replace that region (and the "other universe" beyond the ring) with one without singularities, we get a spacetime without singularities (and from what we know matter may do this), and yet the singularity theorems as usually cited say it has singularities (outside that region)!
I'd like to add that I was convinced as well, for a long time, that the singularity
theorems imply the kind of metric singularities that are problematic. They were the reason why I worked
to save General Relativity by reformulating it in a way that doesn't
have infinities at the singularities. And I repeated numerous times the
claim that the singularity theorems prove that the metric tensor has
singularities, assuming that they are of this kind. And I might have regarded people who didn't believe in
singularities as, let's say, not very serious. Despite being aware that
there was a step in the proof of the singularity theorems that I never understood,
namely exactly the step where from inextensibility we conclude the
existence of such singularities. Despite never being able to find a place where this step is proved for a limited person like me. And that while knowing that I didn't
understand that step, and being limited, I considered that I should trust the experts about
it, or maybe just my limited understanding of what experts say. And now, after seeing Roy Kerr's counterexamples, I think I was wrong.
So yes, Kerr is right, to be able to say that General Relativity breaks down because of singularities we need a proof for exactly such singularities, and the singularity theorems alone don't do the job, and there are counterexamples showing this. But of course counterexamples are a no go mainly for the more mathematically inclined (and some of them noticed this at some times, but somehow the most spread interpretation of the singularity theorems remained unaffected). Many physicists may still use the confusion between the two notions of singular spacetimes (assuming they're aware of them) to reject classical General Relativity, and at the same time they would claim that quantum gravity doesn't have this problem, again without proof, without even a theory of quantum gravity! (The only argument is that quantum fields may violate a condition in the singularity theorems, but this doesn't prove that this avoids the alleged singularity)
But what if somebody takes notice now of Kerr's paper, and of the disambiguation of the term "singular spacetime", and finds a singularity theorem, with different conditions evidently, that is actually about such singularities? Even so, General Relativity can be formulated in terms of finite geometric objects, which can evolve beyond the singularities, as I showed some time ago https://arxiv.org/abs/1301.2231. This formulation is equivalent with the usual one outside the singularities, but it extends at the singularities too, at least in the usual cases.
So I see no reason why General Relativity is so often pronounced dead. I mean, sure, we need a quantum theory of gravity, but let's stop throwing the baby with the bathwater. There's no reason to treat like a stepchild one of the two babies, General Relativity and Quantum Theory, and favor the other one. The really naughty one ;)
Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.
One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.
I illustrate this with this animated gif:
I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:
A. The standard Penrose diagram of an evaporating black hole.
B The diagram from the analytic solution I proposed.
___________________________
* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.
A simple geometric algebra is shown to contain automatically the leptons and
quarks of a generation of the Standard Model, and the electroweak and color
gauge symmetries. The algebra is just the Clifford algebra of a complex
six-dimensional vector space endowed with a preferred Witt decomposition, and
it is already implicitly present in the mathematical structure of the Standard
Model. The minimal left ideals determined by the Witt decomposition correspond
naturally pairs of leptons or quarks whose left chiral components interact
weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals
representing leptons and quarks. The resulting representations on the ideals
are invariant to the electromagnetic and color symmetries, which are generated
by the bivectors of the algebra. The electroweak symmetry is also present, and
it is already broken by the geometry of the algebra. The model predicts a bare
Weinberg angle θW given by sin2(θW)=0.25.
I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."
My paper in cause about Big-Bang singularities is arXiv:1112.4508 (The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in arxiv:1105.0201. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis arxiv:1301.2231. In the paper arXiv:1112.4508, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.
The paper attempting to refute my result is arxiv:1603.02837 (Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities, by L. Fernández-Jambrina). Both my paper and this one appeared this year in International Journal of Theoretical Physics. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was explicitly not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.
Once you can send information faster than the speed of light, you can
also send it back in time. If you can send information back in time, you
can create inconsistent histories, that is, you can create various
different pasts, a problem commonly known as “grandfather paradox:” What
happens if you travel back in time and kill your grandpa? Will Marty
McFly be born if he doesn’t get his mom to dance with his dad? Exactly
this problem.
This is correct. Special relativity implies that, if faster than light signaling would be true, you would be able to signal to your own past, and this can lead to paradoxes. Here I will explain how exactly this can happen. This is rather elementary special relativity stuff, but I realized there is much confusion around it. First, I never saw a precise scenario in which faster than light (FTL) signaling can be used to signal back to your own past, so I will give one. Second, I have the feeling that when people make statements like this,
they either refer to the fact that, if an observer A sends FTL signals in her own future, for another observer B it may look like sending in back in time, in B's reference frame, as in this figure:
Orange lines represent light cones, blue represent timelike curves
(observers), red represents the proper space of an observer, and green
represents FTL signals. While the picture represents the proper space of A as
a horizontal red line, the proper space of B is oblique, due to
the Lorentz transformation (relativity of simultaneity).
The first scenario is not that
paradoxical, because observer B can always reinterpret the signal from A
to B as a signal going in his own future, from B to A. But even in this
case, we will have the problem of who actually created the message in the first place.
or they refer to examples where the observer sends an FTL signal toward her own past, as in this figure:
The second scenario is the usual example of causality violation due to FTL you will find, but is refutable on the grounds that you are not allowed to send signals directly to your own past, or to receive signals directly coming from your own future.
Here is how FTL signaling would imply that one can signal back in time, using only signals sent in the future and received from the past, with respect to the proper reference frame:
The inertial observer A accelerates away from B, then sends an FTL signal at t₀. Observer B receives it at t'₀ in his proper time, then accelerates away from observer A, then sends it back, at t'₁. Observer A receives the signal at t₋₁, where t₋₁< t₀.
So indeed FTL implies signaling back in your own past, even if FTL signals are sent only to the proper future and received only from the proper past.
Let us see how this allows paradoxes. Suppose that earlier A and B agreed on the following: if A receives the message "Yes", she sends the message "No", and if she receives "No", she sends "Yes". If B receives a signal, he just resends it without changing. Then, we have a paradox: does A send the message "Yes", or "No"? It is similar to the liar paradox, since if she sends "Yes", then she receives "Yes", so she sends "No", and so on. But it is also like grandfather's paradox, because B can send instead of a message, a killing FTL ray, to kill A or her grandfather before she was born.
So far there is no evidence of FTL signaling, except for some misunderstandings of the EPR "paradox". I don't know either of a fundamental physical law which prevents it, given that tachyonic solutions are mathematically consistent, both in special relativity, and in quantum field theory. But as we have seen, FTL would lead to time travel paradoxes.
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 $V$.
Tensors
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 $U(2)$)
At first, math seemed to show that anything that enters a black hole, is lost forever. Later, it seemed that black holes evaporate, but the secrets remain lost. But maybe it is not so.
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I made a 7 minutes video introducing some puzzling aspects of quantum mechanics to a general audience. At the end it contains a proposed view which, at least to me, makes the things clearer, so I hope it can help others too.
You can check and rate other videos, ranging from fun to informative, at FQXi Video Contest - Spring, 2014. You can submit your own video until August 22.
Last week, on December 6, 2013, I defended my PhD Thesis.
The Thesis is named Singular General Relativity, and can be found at arXiv:1301.2231.
Thesis Abstract:
This work presents the foundations of Singular Semi-Riemannian Geometry and
Singular General Relativity, based on the author's research. An extension of
differential geometry and of Einstein's equation to singularities is reported.
Singularities of the form studied here allow a smooth extension of the Einstein
field equations, including matter. This applies to the Big-Bang singularity of
the FLRW solution. It applies to stationary black holes, in appropriate
coordinates (since the standard coordinates are singular at singularity, hiding
the smoothness of the metric). In these coordinates, charged black holes have
the electromagnetic potential regular everywhere. Implications on Penrose's
Weyl curvature hypothesis are presented. In addition, these singularities
exhibit a (geo)metric dimensional reduction, which might act as a regulator for
the quantum fields, including for quantum gravity, in the UV regime. This opens
the perspective of perturbative renormalizability of quantum gravity without
modifying General Relativity.
The Thesis is based on a series of papers, from which the following are published or accepted:
[1] C. Stoica On Singular Semi-Riemannian Manifolds, Int. J. Geom. Methods Mod. Phys., 0(0):1450041, March 2014, arXiv:1105.0201.
[2] C. Stoica Schwarzschild Singularity is Semi-Regularizable, Eur. Phys. J. Plus, 127(83):1–
8, 2012, arXiv:1111.4837.
[3] C. Stoica Analytic Reissner-Nordstrom Singularity, Phys. Scr., 85(5):055004, 2012, arXiv:1111.4332.
[4] C. Stoica Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves, To appear in U.P.B. Sci. Bull., Series A, arXiv:1111.7082.
[5] C. Stoica Spacetimes with Singularities, An. St. Univ. Ovidius Constanta, 20(2):213–238, July 2012, arXiv:1108.5099.
[6] C. Stoica Einstein Equation at Singularities, Cent. Eur. J. Phys., 12 (2014), 123-131, arXiv:1203.2140.
[7] C. Stoica Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity, Commun. Theor. Phys., 58(4):613–616, March 2012, arXiv:1203.1819.
[8] C. Stoica On the Weyl Curvature Hypothesis, Annals of Physics, 338:186–194, November 2013, arXiv:1203.3382.
[9] C. Stoica The Geometry of Black Hole Singularities, Advances in High Energy Physics, Volume 2014 (2014), Article ID 907518 arXiv:1401.6283.
[10] C. Stoica Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91,arXiv:1205.2586.
Others are not yet published:
[11] C. Stoica Warped Products of Singular Semi-Riemannian Manifolds, arXiv:1105.3404.
[12] C. Stoica Cartan's Structural Equations for Degenerate Metric, Balkan J. Geom. Appl., Vol. 19, No. 2, (2014), p. 118-126, arXiv:1111.0646.
[13] C. Stoica Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime, arXiv:1112.4508.
Quantum mechanics forces us to reconsider certain aspects of classical
causality. The 'central mystery' of quantum mechanics manifests in different
ways, depending on the interpretation. This mystery can be formulated as the
possibility of selecting part of the initial conditions of the Universe
'retroactively'. This talk aims to show that there is a global, timeless,
'bird's view' of the spacetime, which makes this mystery more reasonable. We
will review some well-known quantum effects from the perspective of global
consistency.
This picture (which I made for the slides) represents the directions used in the proof to the Kochen–Specker theorem, simplified by A. Peres, and arranged by R. Penrose in a pattern inspired by M. C. Escher's Waterfall.
The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity.
In previous papers it has been proposed an equivalent form of Einstein's equation, which extends it and remains valid at an important class of singularities (including in particular the Schwarzschild, FLRW, and isotropic singularities). Here it is shown that if the Big-Bang singularity is from this class, it also satisfies the Weyl curvature hypothesis.
As an application, we study a very general example of cosmological models, which generalizes the FLRW model by dropping the isotropy and homogeneity constraints. This model also generalizes isotropic singularities, and a class of singularities occurring in Bianchi cosmologies. We show that the Big-Bang singularity of this model is of the type under consideration, and satisfies therefore the Weyl curvature hypothesis.
Recently, a new paper by Maldacena and Susskind appears, named Cool horizons for entangled black holes (arxiv:1306.0533). In the paper, the two authors propose that two entangled particles are connected by an Einsten-Rosen bridge, a wormhole. Their stake is in fact related to the black hole information paradox, the Maldacena correspondence, and the recent idea of black hole firewalls. It was covered, among others, by Sean Carroll.
AN EXPLICIT LOCAL VARIABLES TOPOLOGICAL MECHANISM FOR THE EPR CORRELATIONS
It is based on a non-trivial topology (wormholes).
Cut
two spheres out of our space, and glue the two boundaries of the space
together. This wormhole can be traversed by a source free electric
field, and used to model a pair of electrically charged particles of
opposite charges as its mouths (Einstein-Risen 1935, Misner-Wheeler's
charge-without-charge 1957, Rainich 1925).
For EPR we need a
wormhole which connects two electrons instead of an electron-positron
pair. A wormhole having as mouths two equal charges can be obtained as
follows: instead of just gluing together the two spherical boundaries,
we first flip the orientation of one of them. Since the electric field
is a bivector, the change in orientation changes the sign of the
electric field, and the two topological charges have the same sign.
Now
associate to the two electrons your favorite local classical
description. The communication required to obtain the correlation can be
done through the wormhole.
----------------
This may be
the basis of a mathematically correct local hidden variable theory.
Also, it seems to disprove, or rather circumvent, Bell's theorem. For
Bohm's hidden variable theory, it provides a mechanism to get the
correlation without faster than light signals. I proposed it here for
theoretical purposes only, as an example. My favorite interpretation is another one.
Cristi
I did not want to spend more time on this, only to break my neck proposing local models of entanglement, especially since I did not find the idea of hidden variables relevant.
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, here, and at arxiv:1309.2309.
I don't want by this to claim that General Relativity, and semi-classical gravity, have no problems, or that they are all solved by the solutions presented here. What I want to say is that, to justify other more radical theories, people frequently make claims about how General Relativity fails in one place or another. GR will most likely be replaced by a more complete theory, but it is important to know how much we can keep from it, and how much we should change.
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?
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.
