## Thursday, June 23, 2011

### Unification as a source of certainty

People have observed billions of objects and experienced trillions of events and processes.

When a monkey was born somewhere on the Earth a million of years ago and inherited almost no serious knowledge from her ancestors, and instead, she was just asked to behave as a homo sapiens, she had to be overwhelmed! ;-)

No surprise that she had to see dragons and gods behind every corner.

It's just so confusing. And the math class is tough!

However, she has tried hard - and especially he has tried hard - and many things started to make sense. Those trillions of events and processes suddenly looked like manifestations of the same underlying logic. In many cases, pairs of very different objects or pairs of very different processes turned out to be manifestations of the same underlying physics and the same underlying logic and maths.

Whenever this occurs, the hypothesis about the common ancestry usually implies some predictions that a priori didn't have to be satisfied - but they can often be verified and confirmed. Those predictions are sometimes qualitative; sometimes they are quantitative or even very accurate. Such confirmed predictions always increase the probability that the unified description is valid.

By Bayesian inference, the hypotheses trying to explain these phenomena that avoid the "unifying assumption" become much less likely because for those theories, the observed prediction confirming the unity is an unlikely coincidence with probability "p" which is smaller or much smaller than 1.

So the probability that the non-unifying theories are valid also decreases - it gets multiplied by "p". On the other hand, the unifying explanation predicted the right outcome with a 100% certainty, so its probability doesn't drop. Because the competing non-unified hypotheses are punished in this way, the unifying theory gains, relatively speaking. This is true for all cases of unification of concepts and I won't emphasize this point every minute.

In this text, I will mostly focus on some "ancient" scientific discoveries that many of us consider obvious today but there used be times when they were very far from obvious. The list will surely not be complete but it is a reminder of the immense increase of the elegance of our description of the laws of Nature that has occurred in the recent 1 million years or so. Well, most of it has occurred in the recent 110 years. ;-)

Unification of water, ice, and vapor

People - at least those who lived far away from the equator - have always experienced snow, ice, and everyone knows water. A priori, these are different substances. However, the people ultimately had to notice that ice can melt and water may freeze again. And both of them may evaporate or sublimate and the vapor may condense.

These transformations imply predictions about the amount of water, ice, snow, and vapor that you get by a transformation of others. Phase transitions may be viewed as a primitive example of people's suspicion that seemingly different things in Nature have a common origin.

Solids like to contain regular lattices, liquids have chaotic molecules in contact, and gases contain molecules separated by gaps. But the molecules may be the same ones.

Unification of the Sun and stars

When you look at the skies at night, you may see stars. Even today, many people don't know why stars shine - and I wouldn't have to go too far to offer you examples. ;-) I expect 99.9% of the TRF readers to know the answer. Well, no: they don't reflect the light rays from the Sun. Instead, they are emitting theirs.

Not only they're "similar" sources of light as the Sun which looks so big and different. The Sun is actually a pretty average star. The apparent huge difference between the Sun and the stars may be explained by their vastly different distances.

If you measure the apparent size of the Sun and the nearby stars and their distance (either by the parallax or by the luminosity), the ratios will be pretty much fixed. A priori, this would be a very unlikely insight because these distance scales span over many orders of magnitude. But those things are true. The Sun is just another star and the stars are just other distant cousins of the Sun.

Well, some stars may be much larger than the Sun but there are still many stars that are comparable.

Unification of compounds of common elements

People would experience diamonds as well as graphite. Again, they look so different. Nevertheless, it can be seen that if you burn them, you get the same result - the gas we call life (CO2). This "shared outcome" suggests that the solid materials were related to start with. And indeed, they were just differently organized atoms of carbon.

Diamonds are not forever...

This is not quite "chemistry" but I want to include millions of insights of chemistry under this umbrella. Clever people have always had some guesses that everything was composed of the same "elements". However, only when alchemy got advanced enough and especially when chemistry was born, they were able to figure out what the elements were and what the composition of various compounds was.

Not only the composition but also the detailed structure of the bonds in the molecules mattered. But that's already a different story. The punch line is that the "materials" we encounter - and we really know many of them - are ultimately made out of a limited number of elements.

Many of the hypotheses in chemistry - and in science - were wrong and were refuted by the evidence. But those that survived were often verified by so many tests that the degree of certainty that they're right could have converged "arbitrarily close" to 100 percent.

Chemistry and biochemistry contains lots of partial "unification insights". For example, there is a unification of animals' eating and metabolism and steam engines. Our lungs and bodies etc. are burning carbon just like the steam engines or coal power plants (i.e. in fire)! We get some energy in the form of materials with carbon (or other things), it's being burned, under some elevated temperatures, usually, and some gas goes out. I remember that I was very happy and intellectually thrilled when I reached that conclusion at the age of 6 or so. :-) Well, it would be easy to offer more sophisticated insights of this type, too.

Unification of celestial and terrestrial gravity

Isaac Newton realized that what keeps the celestial bodies on their orbits around the Sun - or the Earth (I mean the Moon, not the planets) - is the same force that attracts us to the Earth's surface. Once he determined that gravity had to decrease with the inverse squared distance (from Kepler's laws that Kepler found by a creative analysis of high-quality data from Tycho Brahe), he could predict relationships between the gravitational acceleration at different places.

Needless to say, the predictions worked; at least those that were not instantly eliminated i.e. those that we consider correct today. Again, it's amazing to realize that a priori, the motion of the celestial bodies has nothing to do with our free fall on Earth. Ultimately, it's the same thing. All of them are governed by a simple formula. The parameters - such as mass - are associated with the sources of gravity (the centers around which things revolve).

Unification of people, animals, and plants

Darwin's theory of evolution shed completely new light on the origin of species and has eliminated the certainty that animal species (and the human race) had to be created one by one by the creator. Our composition was so complicated that there couldn't have been a better solution than a hard-working creator, people thought, but in practice, they were only building on their traditions they heard from their ancestors and a lack of imagination rather than solid arguments. There was an alternative and Darwin and Wallace found it. As Weinberg said, it didn't make religion impossible - but it did make irreligion possible.

Evolution is an example of unification, too. It's just a fact that despite the similarities, there also seem to be substantial "qualitative" differences between the humans on one side and  animals and plants - especially the less advanced ones - on the other side. Darwin has effectively shown that these differences are not qualitative. They can be explained by a rather continuous evolution of some parameters - the appearance of the organisms and/or their DNA code, as we would put it today.

Showing that previous "huge qualitative differences" are just "minor quantitative differences in the details" is tautologically the same thing as finding a more universal perspective on the world. If the distance between humans and animals used to look large but it looks small today, it obviously means that we can suddenly see longer distances in the virtual "space of objects and ideas" which means that we can look further.

There is also one characteristic feature of the Darwinian example that is far more general. The common ancestry may be realized not only in the "formal evolution tree" of the species. This tree has actually "materialized" in the real world in such a way that the branches closer to the "root" lived at earlier times. The common ancestors lived before us and their descendants evolved out of them. This basic idea is superficially linked to "historical sciences" such as the "evolution of species" or geology or cosmology but it is actually much more general.

Quite generally, the more unified forms of existence "existed" in the past and they split by subsequent evolution. This theme is omnipresent in particle physics, too. The early Universe had a high temperature which unified many things that look different today. The different types of objects "crystallized" once the Universe cooled down. Cooling down - going to low energies per particle (it's the same thing!) - is usually related to the breaking of the fundamental symmetries. The electroweak symmetry or supersymmetry are obvious examples. But there are also more mundane example - e.g. crystals (think of ice) are only symmetric under certain rotations while the warmer liquids (when crystals melt) are symmetric under all rotations.

The breaking of the fundamental symmetries obscures these symmetries - but fundamentally speaking, they're still there. The whole quest for unification may be interpreted as the search for the unity and symmetries that began to look obscure for various dynamical and historical reasons - and because of vastly different values that various parameters may take.

It's important that unification isn't and cannot be just about our "loss of resolution". Someone could try to "unify" things just by being vague, sloppy, drunk, and fuzzy. This is not a genuine unification, however. A and B are only unified if the differences between A and B may be parameterized by features and parameters of a full-fledged theory that can predict what these differences may be and mustn't be - and if none of the allowed values of the difference can be falsified.

Unification of heat and energy

James Prescott Joule showed that the heat - what you need to melt ice or cook a soup - may be converted to energy - what you need to bring a heavy piano to the 7th floor - and vice versa. Ultimately, the temperature of warm objects was unified with the motion of the atoms.

The "hot surface" and a "moving object" sound as very different things but at the fundamental level, as people found out when thermodynamics was swallowed by statistical physics, they're the same thing. A high temperature simply means lots of kinetic energy of each molecule or atom in the material. The difference between heat and other motion is only in the size of the objects that are moving.

We consider this interpretation of heat obvious today. But just try to think how nontrivial it has been. People surely have a different feeling when they burn their hand or when they have to run from a tiger. But it's almost the same thing. Note how many predictions we can make once we know this "kinetic" origin of the thermal phenomena - predictions that the primitive people could never do.

Unification of electricity and magnetism

Electricity - including thunderstorms etc. - and magnetism - especially ferromagnets - were similar and yet independent phenomena. James Clerk Maxwell, building on the work of others, ultimately unified them under the umbrella of electromagnetism.

The same theory also predicted electromagnetic waves and Maxwell conjectured that light was an electromagnetic wave, too!

Unification of colors of light and other radiation

This was an amazing finding but let me discuss it at a somewhat more primitive level because this insight includes much more unification than what is normally appreciated. For example, people could have seen many colors for quite some time. How is light of one color related to the light of another color?

The electromagnetic explanation of light of course answers all such questions. Light is ultimately the same thing - it's a stream of electromagnetic waves or photons, once we go quantum - and the only difference between the colors such as blue, green, yellow, and red is the frequency or, equivalently, the wavelength.

This is cool because this actually quantifies and "explains" where colors come from. The explanation of light as electromagnetic waves obviously promotes the frequency to a free parameter and this new parameter is predicting that we must see many kinds of light. These kinds of light may be identified with colors. Again, this predicts many other things which may be verified and confirmed.

Moreover, once we represent the color of a monochromatic light by the frequency which is a real number, there are many values of this real number that become possible. By thinking how electromagnetic waves of other frequencies would look like, we find out that the infrared radiation - the invisible "heat" that is coming from a fire, even if there is some vacuum separated by transparent materials in between - is an electromagnetic wave, too. It only has a slightly lower frequency than visible light.

Even lower frequencies are predicted and used in cell phones, television signals, microwave ovens, and so forth. On the other hand, higher frequencies exist as well. People ultimately learn of the ultraviolet rays that are coming from the Sun and that may be unhealthy as well as the X-rays and gamma rays that are even more unhealthy but that can be used to look inside objects - including humans - to find out perhaps other unhealthy things which may ultimately turn out to be pretty healthy for humans.

(The previous contrived sentence says that doctors may sometimes help you with X-ray images haha.)

The periodic functions encoding the color of sound from several musical instruments

A whole analogous discussion would apply to the explanations of the origin of sound. You might call it "unification of sound and wind". The sound is just a vibration of the air - that may also spread via vibrations of all other materials such as railway tracks. The pressure is oscillating. The difference between music and the wind is that the pressure oscillates more quickly if you listen to the music than if there is just some wind. The whole music boils down to the Fourier analysis of the frequencies of the sound. This is another amazing class of seemingly trivial insights. An octave means a doubling of the frequencies. Pleasant chords are those that link frequencies whose ratios are p/q, i.e. rational with small enough values of "p" and "q", or that are close to it.

Unification of space and time, matter and energy, energy and momentum, and so on

Relativity is an excellent example of unification in physics. It reduced the number of independent concepts in physics considerably. Space and time used to be viewed as totally different things. Time was "only" an independent variable while positions of things in space were "only" functions of this independent variable. There used to be no similarity.

Einstein showed that space and time were two different examples of the same thing - a spacetime dimension - and a sign in the metric tensor was the only thing in which they differed. Fundamentally, the qualitative difference was rephrased as a quantitative difference once again. Eigenvalues of the metric tensor may be either positive or negative (let's omit zero here which gives the non-relativistic limits) and the sign is the only thing that distinguishes space and time. Of course, the different sign in the metric tensor has lots of consequences - but they're just consequences of a "technical detail": they're no longer assumptions about a qualitatively different origin.

This amazing unification of space and time in special relativity has also made the unification of electricity and magnetism inevitable. Also, it has led to a unification of the energy and the mass via E=mc^2, and the incorporation of energy and the momentum to the same 4-vector. Many other things were unified into representations of the Lorentz group. I won't discuss is in detail but any previously unknown symmetry between previously known concepts and objects inevitably brings new insights resulting from the unification of those objects.

(One could discuss tons of example, including the obvious ones such as the unification of the spectrum of hadrons by the approximate flavor SU(3) symmetry.)

Unification of gravity and acceleration

General relativity is based on the equivalence principle which of course states that fictitious forces - inertia, centrifugal force, Coriolis force etc. - are locally and fundamentally indistinguishable from gravity. This is, once again, a great advance towards the unification of physics which has had - and still has - many consequences. Because general relativity explains many things, it also qualitatively unifies extreme conditions at the center of the black hole with those of the Big Bang, and so on, and so on.

General relativity was the first really "canonical" example of a theory that disagreed with the previous ones but the difference was tiny. So there were various deviations or anomalies - the precession of Mercury's perihelion and the (later found) bending of the starlight during solar eclipse - which were explained from the same equations.

Quite generally, new theories predicting many new phenomena may be viewed as frameworks to "unify the anomalies". Of course, "anomaly" is just another, different effect that is named by this derogatory word only because people don't find it "important" enough, usually because it's small and it doesn't change things "qualitatively". In the fully scientific approach, however, the explanation and unification of "anomalies" is as important as the explanation of any other effects that may be (equally reliably) observed. Whether something is important for the human lives or is just an irrelevant and useless deviation - an "anomaly" - is a subjective difference that has no difference for our purely scientific understanding of the real world.

Unification of particles and waves

Quantum mechanics itself is usually not presented as a step towards unification - in a general sense - but it may be described in this way, too. The world as imagined at the beginning of the 20th century contained lots of particles which were essentially point-like. But as we wrote in the previous section, it also contained waves that were inevitably spread in space.

Quantum mechanics showed that these two forms of existence - localized particles and delocalized waves - are actually two extreme faces of the same thing, too. It's because the waves may be thought of as a stream of particles and the profile of the wave just mimics the probabilistic waves of the particles; or, using the opposite perspective, the particles may be viewed as excitations of the quantum fields describing the waves whose energies have to be quantized for the same reason why the harmonic oscillators have quantized energy in quantum mechanics: the number of zeros (roots) of the wave function of a bound state has to be an integer.

The quantum mechanical calculations explain all of atomic physics, chemistry, biology, and material science, among other things, so there's of course a lot of room for details that have been unified.

Unification of bosons and fermions, particles and forces

In the same way, the basic structure of quantum field theory itself is almost never presented as a unification framework. But it is a unification framework, too. In the early days of non-relativistic quantum mechanics - and even in the 1930s - physicists would still misunderstand that photons and the electrons really follow the "same" logic. They didn't know that the second quantization had to be applied both to the electromagnetic and Dirac field and both of them resulted in analogous "multiparticle Hilbert spaces" where particles always have "wave properties", too.

The new unifying framework was most clearly articulated by Wolfgang Pauli. He has figured out that the only difference between particles that obey the exclusion principle - e.g. electrons - and those that don't - e.g. photons - is a sign in the wave function after you permute a pair of particles.

We say that the bosons have symmetric wave functions, psi(x2,x1)=+psi(x1,x2), while the fermions have antisymmetric wave functions, psi(x2,x1)=-psi(x1,x2). The latter implies that psi(x1,x1)=0, the exclusion, while the former doesn't. Moreover, these two outcomes that used to look like "qualitatively distinct" phenomena (electrons make up "impenetrable matter" while light is "freely penetrable") may be produced from the same quantum fields that either commute or anticommute.

Wolfgang Pauli was the key person who made everyone think about quantum fields all the time. Before its replacement by string theory, quantum field theory was the "state-of-the-art" description of physics. It could describe waves as well as particles (any number of them) and for fermions, it guaranteed the Pauli exclusion principle.

Electroweak unification, GUT, strings

I don't want to discuss these typical examples too much because they're the standard examples of what we call "unification" all the time. Nevertheless, the electromagnetic interaction - previously made concise by Maxwell - was unified with the weak nuclear force that causes beta-decay.

A priori, the beta-decay looks as different from an electromagnetic force as you can get. Electromagnetic forces just change the speeds and positions of objects; the weak nuclear force changes a neutron to a proton, electron, and an antineutrino - it totally changes the particle composition (so the laymen wouldn't even call it a "force"). Electromagnetism operates at long distances while the weak force only applies inside the neutron, i.e. it is a short-range force, and so on.

However, the first difference is explained by the fact that the photon - the messenger of electromagnetism - is neutral while the W-bosons - the messengers of the weak nuclear force causing beta decay - are charged in various ways. Because they're charged, they inevitably change particles into other particles. However, charged and uncharged messenger particles may be unified into the concept of gauge bosons. The electromagnetic U(1) generator doesn't have to commute with other generators - which means that the other generators may be connected with charged particles.

The second difference (long-range vs short-range) results from the fact that the electromagnetic U(1) symmetry is unbroken - which keeps the photon massless and the range of the electromagnetic force stays infinite - while the SU(2) x U(1) symmetry underlying W-bosons and Z-bosons is broken to the electromagnetic U(1) - which makes W-bosons and Z-bosons massive and the forces mediated by them are short-range forces.

Again, the presence of the Higgs mechanism that breaks the symmetry is just a "quantitative difference". If the Higgs potential is "H^4+A.H^2", then the potential has a minimum at "H=0" for a positive "A", but a minimum at a nonzero "H" for a negative "A". The sign of "A" decides about the qualitative issue - whether the symmetry remains unbroken - and once again, broken and unbroken symmetries are unified into the same equations and only differ by "numerical details" such as this sign of "A". The adjustment of "A" is fundamentally an innocent continuous change but it has consequences that look "qualitative" and far-reaching.

The electroweak theory makes a huge number of predictions because it's very constrained at this level - much more constrained than possible life forms in Darwin's evolution tree which allows the animals to evolve in much more creative ways. ;-)

The strong nuclear force keeps protons and neutrons together in the nuclei - despite the electromagnetic repulsion between the protons. As we know today, it really keeps quarks together inside the protons and neutrons, too. Again, the strong force looks very different both from the electromagnetism and the weak force. And once again, this difference may be explained by just "minor quantitative" differences.

The SU(3) color symmetry remains unbroken but it is confining so only color-neutral objects (e.g. protons composed of 1 red, 1 green, 1 blue quark) may exist in isolation. In between these "separately allowed" color-neutral objects, only residual forces from the color force may exist and those residual forces are therefore short-range forces once again, despite the massless gluon. The confining character of the color force may be explained by another sign analogous to the sign of "A" in the Higgs discussion above - in this case, it is the sign of the beta-function. Moreover, this beta-function is not quite a free and adjustable parameter; it is calculable and QCD has a negative beta-function because SU(3) is a pretty large group and there are not too many SU(3)-charged particle species.

At the level of formalism, there is a unification because electroweak and strong forces may be described by the same language of quantum field theory - in fact, by gauge theory which is a special type of quantum field theories. However, this unification could be much tighter if the grand unified theories are right (the groups could be a part of a bigger, more unified group), and there are good reasons to think that they could be although it's not an established fact and there arguably exist alternatives that have a different but comparably convincing explanatory logic (type II intersecting brane worlds).

Lots of symmetries that were used to unify the spectrum of elementary particles should be added here. Some of the symmetries are approximate - the SU(3) flavor symmetry that was used by Gell-Mann and others to explain the spectrum of the hadrons. Others are accurate - they have to be gauge symmetries, at the end. Others are not yet established, and so on.

Black holes and elementary particles

String/M-theory is the modern science's ultimate layer of the unification of science - at least, at this moment it seems that there can't be any "distinctly higher" or "deeper" layer anymore. The theory incorporates all matter species and all forces - whether they're caused by spin-1 gauge bosons or spin-2 gravitons - and interprets them as dynamical state of the same fundamental objects.

Most famously, in the weak coupling stringy limit, all of them are vibrations of fundamental strings found in different mass eigenstates. At stronger coupling, the strings are replaced by some democratic hybrids of strings and branes and other objects which are as constrained as the strings but are more "bootstrapy" in character - in general, they can't be described by equally "constructive" and readable theories that we use for the weakly coupled strings.

But I want to stress some other kinds of unification that string theory brings us. One of them is the unification of black holes and elementary particles.

In the previous sketch of the history of science, I mentioned that particles were unified with waves by quantum mechanics and bosons were unified with fermions, and so on. But there still existed other mostly localized - but not point-like - objects that didn't seem to be composed out of point-like particles. They include things like monopoles and other "solitons" (topologically nontrivial solutions of classical equations of motion that behave as semi-localized but extended objects) and especially black holes.

Progress in field theory and string theory has shown that monopoles (and, more generally, other solitons) are actually just a different face of point-like particles. In particular, S-duality interchanges electricity with magnetism in non-Abelian gauge theories in such a way that the point-like "electric" excitations such as electrons are exchanged with their dual "magnetic" solitons such as the magnetic monopoles.

Quantum mechanics has unified the completely localized particles with the completely delocalized waves but S-duality and related insights have also shown that the "intermediate", semi-localized objects such as solitons ("large particles") whose solutions have a typical nonzero radius are manifestations of the same objects, too.

In fact, all the dualities in string theory are obviously examples of unification. The electric charge is unified with the momentum of a particle; this insight was already known to Kaluza or at least Klein. That's true because the electric charge is the generator of an isometry in a new 5th circular dimension - which is the momentum in this direction. More generally, T-dualities unify the winding (how many times a string is wound around or a brane is wrapped around a circle or a cycle in a manifold) with the momentum and other charges, and so on. The details of the K3 geometry are remarkably unified with Wilson lines in the E8 x E8 or SO(32) gauge group, and so on, and so on. Some of these "unifications" are beautiful for reasons that look "purely mathematical" to some people but they're really physical relationships - just ones that affect very unfamiliar and "faraway" physical environments.

So every object that didn't seem to fit into the "quantum field theory"-like description based on quantum fields describing strictly point-like particles has been shown to be just a physical manifestation of the same objects, too. In fact, the unification may often be presented in many ways.

AdS/CFT is a unification of field theory and string theory - a kind of spectacular meta-unification, too.

But I want to dedicate a few paragraphs to the unification of the black holes and elementary particles. In general relativity, one may derive the existence of black holes that may have a given size = mass, charge, and angular momentum but that are not composed of normal elementary particles, either. So why do they exist?

String theory, once again, allows one to show that in practice, the black holes are composed of the very same "building blocks" as other objects except that in the case of black holes, the elementary objects - strings and branes - are extremely strongly bound so you don't learn much if you just naively try to deconstruct the black hole into pieces.

Moreover, black holes have been shown to carry a nonzero (and very large, in some "holographic" sense, maximally large) entropy - S=A/4G if the horizon area A is large. So there exists a large density, exp(A/4G), of microstates describing a particular macroscopic shape of a black hole. This large number of microstates may be thought of as a large number of arrangements of the fundamental building blocks. In some situations, they may be counted using dual descriptions and a very nontrivial agreement with the Bekenstein-Hawking entropy may be reached - an industry initiated by the revolutionary Strominger-Vafa insights. It's also plausible that each of the microstates has a quasi-local description in terms of "fuzzball geometries".

However, the individual microstates of the black holes may also be viewed as "elementary particles", something that can be localized just like the electron. So string theory i.e. quantum gravity tells you that there are many species of elementary particles. The light ones are well-described by the quantum field theories - an approximate, effective description. However, you may continuously try to increase the mass of the particles, look for ever heavier species, and when you get to very high particle species, they may be viewed as higher excited, "large" strings or branes and, when they're even heavier, as black hole microstates!

Some of the previous steps in the unification - such as S-duality - have found rigorous maps between seemingly elementary objects (such as electrically charged "electrons") - and the seemingly composite objects (such as "solitons", e.g. magnetic monopoles). So it's not quite a new insight that elementary and composite objects can't be strictly separated - an aesthetically appealing intuition that was promoted by people such as Werner Heisenberg when he was older and the S-matrix school but that didn't lead to any solid insights prior to the birth of string theory.

UV/IR mixing

However, there's still some difference between the light elementary particle species - and the numerous heavy elementary particle species such as the black hole. The latter may still be described as some kind of "composites" of the former ones. However, string/M-theory has made the notion of compositeness more subtle. Compositeness has been de facto replaced by the UV/IR-mixing.

What is that? It's a rule that in a theory that is as powerful as string theory and/or quantum gravity, the spectrum of elementary particles is far from arbitrary. In fact, the statistical details of the high-mass spectrum (and the properties of high-mass particles) is fully encoded in the properties of the low-mass excitations and vice versa!

In perturbative string theory, this insight boils down to the "modular invariance". One-loop partition sums may be represented as the path integral over histories that resemble a toroidal world sheet. A torus is a rectangle with the periodic identification of the left/right boundaries and the upper/lower boundaries. However, if you rotate this rectangle or torus by 90 degrees, you get the same shape back. So the path integral has to be invariant under this transformation. But you may read both forms of it differently and it is damn useful to read it in both ways!

A "thin" and "thick" torus are ultimately the same thing but they may be linked to the properties of the high-mass excitations or the low-mass excitations. This implies an infinite number of relationships between the high-mass behavior of the spectrum and the properties of the low-mass particles in the spectrum. The theory constrains itself!

This modular invariance is very explicit in perturbative string theory. More generally, in quantum gravity, we qualitatively know that "something similar" is operating but we don't quite know the same "exact counterpart" of the modular invariance that would be valid in the generic quantum gravity regime.

However, one may mention some trivial examples why we're sure that such UV/IR (ultraviolet/infrared i.e. short-distance/long-distance) links do exist. For example, if your low-energy theory has 2011 types of photons, the spectrum of black hole microstates will inevitably know about the fact that the black holes may carry 2011 different types of electric charges and 2011 different types of magnetic charges. This is translated to some details in the density of states as a function of the mass, and other things.

This is just a trivial example but there surely exist much more nontrivial examples that pretty much constrain "everything".

The modular invariance in the case of perturbative string theory ultimately boils down to a rotation of a toroidal world sheet. The world sheet may be viewed as an "auxiliary construct" of string theory because we don't "directly" observe it in spacetime. Still, even auxiliary constructs have symmetries acting on them and these symmetries lead to insights and unifying ideas that tell us new things about the phenomena in the spacetime, too.

In some sense, all the principles of quantum field theory and general relativity - such as the Yang-Mills symmetries and the equivalence principle etc. - may be derived from a more fundamental starting point in perturbative string theory, namely the diff/Weyl symmetry on the world sheet - that may be gauge-fixed to the conformal symmetry which includes large conformal transformations such as the modular transformations of the torus.

Since the mid 1990s, we no longer say that the world sheet and its symmetries are "fundamental" because they only apply in a particular perturbative description. But in this limit, they surely do apply, and at least morally, it seems true that a "similarly powerful set of mathematical structures and symmetries" applies outside the perturbative approximations, too. In Matrix theory or the AdS/CFT correspondence, all of the gravitational bulk spacetime physics may be derived from some ordinary Yang-Mills principles of the dual model, and so on.

String theory which is the state-of-the-art culmination of all the human knowledge about Nature provides us with very many "dimensions" of unification of previously unrelated phenomena, types of phenomena, objects, types of objects, and even unification of methods how to unify things, whether those things are objects, phenomena and principles that play a key role in the previous advances in physics or whether they are just notions in mathematics or whole subdisciplines of maths.

They can't be covered here because I didn't want this text to be a hardcore text about string theory but I may return to them sometime in the future.

Instead, I hope that I have convinced the dear reader that despite the constantly expanding realm of our knowledge and the number of objects and phenomena in Nature we have observed or inventions and technologies that we have made up, science is keeping a unified perspective on the world that depends on a number of independent fundamental concepts and the number of these independent fundamental concepts is actually not increasing - it is arguably decreasing.

At the end of this journey, science is de facto going to understand "everything from nothing" - more precisely, a very sophisticated version of "nothing". ;-) We're not there yet but we're much closer to this point than 1 million years ago, 110 years ago, and even then 20 years ago.

And that's the memo.

Bonus: Cameron's unification in Prague

British PM David Cameron was on his visit to Prague today. He met his "friend and ally" Czech PM Petr Nečas as well as our president. Of course, both sides agreed about all kinds of things about the EU, avoiding of the bailouts for Greece, competitiveness, collaboration in investigation of corruption, and so on. Smooth.

But he also achieved another kind of unification - of the history. Cameron, who was born in 1966, "remembers how he went to Prague during the Prague Spring [1968] and he couldn't find too many hotels but he was feeling the immense excitement of the country that was suddenly free and democratic."

Well, maybe instead of hotels, he should have been looking for a daycare center - with diapers.

Cameron also said that the Prague Spring, which ended with the Soviet Occupation, is similar to the recent events in Libya. Well, this guy knows how to entertain! :-) Just to be sure, Cameron confused or "unified" the Prague Spring - an era of socialism with the human face in 1968 - with the Velvet Revolution in 1989. :-) Just to be sure, this is exactly a kind of unification I didn't have in mind in the essay above.

Cameron came with a standard daily commercial flight of the British Airways. But together with Nečas, they took a Czech government aircraft to leave for the Brussels.

Bonus II: Baby boxes

The only link between this second off-topic addition and unification is that it is about a complaint by the United Nations against the Czech baby boxes which were introduced here by Mr Ludvík Hess.

About 0.6% of Czech newborns are placed in institutional care - a percentage much higher than the EU average of 0.1%. I guess that the de facto absence of religion in Czechia may be correlated with this particular huge discrepancy - just to assure everyone that not everything will look "positive" if you get a nation of infidels.

The baby boxes, in which mothers can anonymously deposit their newborns, just saved the 50th Czech child since 2005 when they were introduced. Children from baby boxes may be instantly adopted so "not knowing your real parents" is the only temporary drawback they bring - but the right to know your parents is less important than the right to live, the baby box pioneer points out. Many of those children would suffer under parents who don't want them but don't want to openly give them up for adoption - or could be killed (or preemptively aborted).

You could think that baby boxes and the recommendation by the United Nations is a controversial question in the Czech Republic. Well, you would be wrong. 98.5 percent of the Czech population is against the cancellation of the baby boxes. In other words, 98.5% of the Czechs say "screw you, United Nations".

1. Careful, such holistic thinking flies in the face of Climatology and Environmentalism.

2. Dear Aaron, thanks for your warning but could you please offer some details how it works?

As far as I can say - and what I meant - this text is one big promotion of reductionism which is exactly the opposite thing than holism. ;-)

Even if one made this sign flip, I don't quite see what it has to do with environmentalism or climatology...

Environmentalism and climatology - if they can be interpreted as sciences at all - have exactly the opposite desire then unification. They're all about the fragmentation and "interdisciplinarity". It's the opposite tendency than unification because unification always wants to incorporate insights under the same umbrellas so in a sufficiently unified frameowork of science, there's really no room for interdisciplinary disciplines.

3. Everything from nothing?
Isn’t this in contrast with Everything from the Modular transformation of the torus?
See: 3-Dimensional String Based Alternative Particle Model.
http://vixra.org/abs/1103.0002

4. With unification, even reductionism holism can get confused. :)

5. Hi Molt,

I'm from Brazil and I'm working on a didactic approach of String Theory for undergraduate public. This post is very interesting in the sense of providing metaphors and analogies that could be used on my plan.

Very nice!

Diogo