Theory of Everything: A geometric approach to the standard model

Here is Garrett’s conclusions paragraph giving a descriptive overview of what he has accomplished:

“This paper has progressed in small steps to construct a complete picture of gravity and the standard model from the bottom up using basic elements with as few mathematical abstractions as possible. It began and ended with the description of a Clifford algebra as a graded Lie algebra, which became the fiber over a four dimensional base manifold. The connection and curvature of this bundle, along with an appropriately restricted BF action, provided a complete description of General Relativity in terms of Lie algebra valued differential forms, without use of a metric. This “trick” is equivalent to the MacDowell-Mansouri method of getting GR from an so(5) valued connection. Hamiltonian dynamics were discussed, providing a possible connecting point with the canonical approach to quantum gravity. Further tools and mathematical elements were described just before they were needed. The matrix representation of Clifford algebras was developed, as well as how spinor fields fit in with these representations. The relevant BRST method produced spinor fields with gauge operators acting on the left and right. These pieces all came together, forming a complete picture of gravity and the standard model as a single BRST extended connection. If this final picture seems very simple, it has succeeded. As a coherent picture, this work does have weaknesses. Everything takes place purely on the level of “classical” fields – but with an eye towards their use in a QFT via the methods of quantum gravity, which must be applied in a truly complete model. The BRST approach to deriving fermions from gauge symmetries, although a straightforward application of standard techniques, may be hard to swallow. If this method is unpalatable, it is perfectly acceptable to begin instead with the picture of a fundamental fermionic field as a Clifford element with gauge fields acting from the left and right in an appropriate action. The model conjectured at the very end, based on the related u(4) GUT, is yet untested and should be treated with great skepticism until further investigated. In a somewhat ironic twist, after arguing in the beginning for the more natural description of the MM bivector so(5) model in terms of mixed grade Cl1,3 vectors and bivectors, this conjectured model is composed purely of bivector gauge fields. Although the model stands on its own as a straightforward Cl8 fiber bundle construction over four dimensional base, there are many other compatible geometric descriptions. One alternative is to interpret ⇁ ̃A as the connection for a Cartan geometry with Lie group G – with a Lie subgroup, H, formed by the generators of elements other than ⇁e, and the spacetime “base” formed by G/H. Another particularly appealing interpretation is the Kaluza-Klein construction, with four compact dimensions implied by the Higgs vector, φ = −φ ψΓ ψ, and a corresponding translation of the components of ⇁ ̃A into parts of a vielbein including this higher dimensional space. The model may also be extended to encompass more traditional unification schemes, such as using a ten dimensional Clifford algebra in a so(10) GUT. All of these geometric ideas should be developed further in the context of the model described here, as they may provide valuable insights. In conclusion, and in defense of its existence, this work has concentrated on producing a clear and coherent unified picture rather than introducing novel ideas in particular areas. The answer to the question of what here is really “new” is: “as little as possible.” Rather, several standard and non-standard pieces have been brought together to form a unified whole describing the conventional standard model and gravity as simply as possible.”

see pre-print on http://arxiv.org/abs/0711.0770

Doubly Special Relativity

 

Doubly-special relativity — also called deformed special relativity or, by some, extra-special relativity — is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but an observer-independent minimum length (the Plank length).

This was first proposed in a paper by Giovanni Amelino-Camelia, though it is at least implicit in a paper by Paul Merriam. An alternate approach to doubly-special relativity theory, inspired by that of Amelino-Camelia, was proposed later by João Magueijo and Lee Smolin. There exist proposals that these theories may be related to loop quantum gravity.

One of the motivations for this work is the observation of high-energy cosmic rays that appear to violate the Greisen-Zatsepin-Kuzmin limit: the so-called Oh-My-God particles.

The theory is highly speculative as of first publishing in 2002. The theory is built using a well-established approach in theoretical physics named invariance under transformation, which is colloquially (even in science) called relativistic. Nevertheless the theory is not considered a promising approach by a majority of members of the high-energy physics community.

DSR is based upon a generalization of symmetry to quantum groups. The Poincaré symmetry of ordinary special relativity is deformed into some noncommutative symmetry and Minkowski space is deformed into some noncommutative space. This theory is not a violation of Poincaré symmetry as much as a deformation of it and this symmetry is exact. This deformation is scale dependent in the sense that the deformation is huge at the Planck scale but negligible at much larger length scales. Many models which are significantly Lorentz violating at the Planck scale are also significantly Lorentz violating in the infrared limit because of nasty radiative corrections. Without any exact Lorentz symmetry to protect them, such Lorentz violating terms will be generated with abandon by quantum corrections. However, DSR models do not succumb to this difficulty since the deformed symmetry is exact and will protect the theory from unwanted radiative corrections — assuming the absence of quantum anomalies.

Jafari and Shariati have constructed canonical transformations that relate both the doubly-special relativity theories of Amelino-Camelia and of Magueijo and Smolin to ordinary special relativity. They claim that doubly-special relativity is therefore only a complicated set of coordinates for an old and simple theory. However, all theories are related to free theories by canonical transformations. Therefore supporters of doubly-special relativity may claim that while it is equivalent to ordinary relativity, the momentum and energy coordinates of doubly-special relativity are those that appear in the usual form of the standard model interactions. This implies that ordinary special relativity and doubly-special relativity make distinct physical predictions in high energy processes, and in particular the derivation of the Greisen-Zatsepin-Kuzmin limit is not valid if one asserts that quantum electrodynamics takes its usual Maxwell form only in the coordinate systems of doubly-special relativity.

Literature

• Fabio Cardone[1], Roberto Mignani , Energy and Geometry: An Introduction to Deformed Special Relativity, World Scientific 2004, ISBN 981-238-728-5

External references

• Giovanni Amelino-Camelia, “Doubly-Special Relativity: First Results and Key Open Problems.” Int. J. Mod. Phys. D11 (2002) 1643. Online copy at http://arxiv.org/abs/gr-qc/0210063
• Giovanni Amelino-Camelia, Doubly-Special Relativity.’ http://lanl.arxiv.org/abs/gr-qc/0207049, a non-technical review
• Nosratollah Jafari, Ahmad Shariati, Doubly Special Relativity: A New Relativity or Not? http://arxiv.org/abs/gr-qc/0602075

Supersymmetry for the masses

Introduction
Since the times of the Ancient Greece, Natural Philosophers (they had several names, from time to time) have often asked themselves what the “world” is really made of. The first answer was “atoms” (“impossible to divide further”), and, as today, we still can’t prove if it’s true or not. Nowadays, this word has the different meaning of “minimal entities of chemistry”, but chemical atoms are not impossibile to divide since now we know they are made of more fundamental constituents called quarks and electrons.

The best description we (I mean the western Natural Philosophers) have found so far is called “Standard Model”, and it’s based on a more deeper cookbook called “quantum field theory”. While nobody has yed proved the latter to be wrong, we know the former to be an incomplete description of the world we live in. For some aspects it’s even flawed, since it predicts that some elusive objects called neutrinos are weightless, but experiments show they are not. This is not bad news: after all, it suggests that there’s still a lot of work to do. In physics, greek atoms are called elementary particles.

Relativistic Quantum Field Theory
At the beginning of the XXth century, physicists (mainly Planck, Einstein, Heisemberg, Schroedinger) discovered that our world, when considered at very short length, behaves differently from, let me say, at our “scale of distances”. They discovered the “quantum mechanics”. A famous physicist once noticed that we are all well accustomed to quantization: for example, when we order a beer at the pub, we buy it in pints, even if the beer flows continuously from the cask.
Basically, in the atomic world, energy is sold in glasses of different size. But always in glasses.

What does it mean the word “field” in the name of this theory? A field is an entity defined everywhere in the space. For example, when we watch forecasts on tv, we actually observe the “temperature field” or the “wind field”. We define the temperature to be a “scalar” (it can be measured with some kind of scaler), while the wind is a “vector” (wind has a magnitude and a direction, like “from south to west”: this makes a vector). There can be more uncommon type of fields, and they turn out to be very important for our discussion.
Anyway, the whole thing is that when you mix up “quantum” and “field” you discover that the particles are fields of waves. When two bodies exchange energy they do this through waves of the field that we recognize as particles.

Let’s talk about the word “relativistic”. Again, at the dawn of the previous century, Albert Einstein proved that the laws describing electrical and magnetic phenomena imply that space and time must be considered in a particular combination as a single entity (the so-called spacetime). This is inavoidable if we believe that everyone, independently of their place and speed (and possibily religious beliefs).
What followed from this discover has been an incredible breakthrough in Natural Philosophy: the speed of light in vacuum is the fastest speed that everything can reach, and it is the same for everyone, independently of their place and speed. Experiments boldly confirmed this theory. It’s so difficult to understand how it can be, that I suggest you to accept it as a fact, until it will be proven false. But don’t get crazy, nobody can understand it: We can only describe this weird thing with math.

Now, relativity let us toss and turn object in 4 dimensions: time and space. When we rotate something in space we call this operation…well, rotation. When we rotate something in 2 spatial dimensions and time we obtain a “Lorentz boost”. It has no obvious meaning in everyday life, but scientists discovered that it can explain a property of particles called “spin”: when placed in a magnetic field, electrically charged particles behave like they’re rotating, but they are not.
Particles can be classified according to the numerical value of their spin, which is always the same for each one. In fact, if you put together relativity and quantum mechanics, you discover that the spin is quantized (“sold in glasses”). If you measure the spin in a unit called “Planck constant”, it turns out that all the particles can have a spin that is either an integer multiple of this unit (let’s say 1, 2, 3…), or a semi-integer multiple (1/2, 3/2…). The first kind is called “boson” after the indian physicist Satyendra Bose, while the second is called “fermion” after Enrico Fermi. From calculations, it turns out that spin-0 particles are scalars, spin-N/2 are “spinors”, spin-1 are “vector bosons” and spin-2 are “tensors”. Don’t be afraid of spinors and tensors: you don’t need to know exactly what they are.

The Standard Model
Ok, now we have all the ingredients. What can we do with them? Let’s assign a kind of field for each particle discovered so far.
Experiments show that bosons are responsible of forces (like electromagnetism and nuclear reactions), while fermions are the “building blocks” of the world called quarks and leptons, let’s say, they form atoms, molecules, human beings etc. It is still experimentally unproven, but very likely, that gravitation is due to a spin-2 boson called “graviton”, but remember: gravitation is not a part of the Standard Model!

Now, a main issue is: “how come that particles have mass?”. Despite the fact that we can fancy several ways to get this results, the more simple and elegant is through the so-called “Higgs mechanism”, after the physicist who proposed it. Basically, Standard Model states that there is a field (a scalar one) that interacts only with some particles, but not all. Heavy particles interact very strongly with the Higgs field, while photons do not at all (light is “weightless”). It act, more or less, like putting a sheet of paper on a wet table top: paper absorbs water and it gets heavy. If you try doing the same with a plastic card it doesn’t work.

The problem is that nobody has ever officially seen a Higgs particle (some rumors from Fermilab). So far. Moreover, this theory doesn’t predicts the value of the masses, and other parameters.

So we’ve got:

• Spin-0: scalar particles (Higgs)
• Spin-N/2: “spinor” fields (quarks, electrons, neutrinos..)
• Spin-1: vector boson fields (electroweak interaction)

Is this all?

Beyond the Standard Model: Merits and flaws

Despite some rumors, the SM is a magnificent theory. Of course, it’s not perfect. For example, a lot of parameters are put in “by hand” (like the value of the masses, which is, IMHO, the biggest flaw), but it let us explain why the sun is so hot and so long-lived, and a lot of other important stuffs. Moreover, gravity is not included. Nobody tells us that we will eventually find a “Theory of everything”, but it is so attracting that it’s worth trying.

In the past 30 years (or more) a lot of effort has been made in this direction. Einstein has tried, too, but without much success. Actually, Grand Unification is the Tree of Life of particle physics, and it seems to have a large number of limbs. But only one trunk. Supersymmetry is one branch that is very promising.

You can see SUSY from different points of view, anyway, it states that fermions and bosons are two faces of the same coin, if you knew how to see it properly. Of course, in our world, they are not the same thing: fermions are the bricks of the Cosmos, and bosons act like glue. So what has SUSY to do with out world?

In Field Theory there’s an important operation called “commutation”. Commutation is made of “operators”, which are mathematical objects describing some kind of transformation.

The point is that if you apply SUSY twice and then come back to the initial state, you get a shift in one direction.

String Theory for Thirsty Pub Customers

Rumors say that string theorists are vacuum cleaner door-to-door salesmen. Perhaps. So I thought it could have been interesting to shed some light on this topic.

Once upon a time (1968), Gabriele Veneziano, an italian physicist, was struggling to find a consistent model for nucelar dynamics. He found a model, but it wasn’t as consistent as he hoped. Anyway, his model treated nuclear constituents not as pointlike particles, but like “strings” (that was so because the force between quarks acts quite like a spring). Unfortunately, this model turned out to predict the existence of some weird particles which has been never observed. They are spin-2 massless bosons, and they look like not playing any role in nuclear physics…But two years later Nambu, Susskind (not the one who wrote “The perfume”) and Nielsen (not the one who sells soap) independently discovered that the same formulas describe relativistic quantum vibrating strings, and that orphan boson was the quantum of gravity. That’s only the beginning.

In the past, magnetism and electricity seemed to be completely different kind of phenomena. Now we call them electromagnetism. In the past, electromagnetism and nuclear decays seemed very different processes. Now we call them electroweak interactions. Today, electroweak and strong interactions on a side, and gravity on the other, seem to be separate forces. Are they, really? We don’t know. A lot of physicists now climb on the shoulders of Newton, Maxwell and Weinberg trying to see a new landscape, in which there’s only one bigbigbig force, still nameless. Some of them are called “string theorists”.

String theory is not really a theory of unification, but it can provide some clues of it. As a matter of fact, if you make the assumption that everything is made up of tiny vibrating strings, then you must have gravity and you can have fermions and bosons as side dishes through the famous supersymmetry (without supersymmetry you can have only bosons). Strings themselves can be open (like in a guitar) or closed. Depending on the way they vibrate, they can describe different particles. For examples, at the lowest energy closed strings have the same properties (mass and spin) of the quanta of the gravity: the gravitons. So, if you have a volume filled with gravitons, then you have a curved space. But there can be a lot of other esoteric vacuum states called for examples dilatons, which can explain what happened during the “inflation era”, just after the Big Bang (has inflation era really happened, after all?). Another really striking and startling feature of the strings is that they require more than the usual 3+1 dimensions to rest in peace. Gravity seems to be self-consistent only in 10+1 dimensions. Now, the question is: where are those extra-dimensions? Nobody really knows. Maybe they are packed, rolled and very tiny so we can’t see them, or better we do live in 11 dimensions, but our senses are “constrained” in a 3+1 world. In fact, open strings can be “attached” to vibrating multi-dimensional membranes (called Dirichelet-branes or D-branes for short) and closed strings can travel between two D-branes and then be “absorbed” by one of them, causing gravity. So our world is a D-3 brane.

String theorists introduced the metaphysical concept of “principle of elegance”. That is: they believe that Mother Nature is simple and elegant in her very building blocks. So a theory describing The Nature must be mathematically elegant, indeed. Everything should be derived as a “natural” consequence, with no phenomenological-driven assumptions. After all, it is an answer to the Einstein’s cosmic question ”Had God any choice in creating the Universe?” It was his fondest hope that the answer was no. So, if He had no choice, string theory can be a one-way mathematic pathway.

Today, string theory is still “redundant”: we have 6 theory, linked in pairs by a symmetry. None of them describes correctly our Universe. Some have no fermions, some have tachyons (particles that travel faster than light in vacuum). Theorists do believe that they are very close to the “central” theory that links all of them. They call it M-Theory (magical? mesmerizing? metaphysical? mysterious?). I’m waiting.