String theory is what?

[ChildInAZoo]

HUP is a good guide. But look at http://en.wikipedia.org/wiki/Uncertainty_principle and see where it says

"In quantum physics, a particle is described by a wave packet..."

Good stuff. But then look at what comes next:

"..Consider the measurement of the absolute position of a particle. It could be anywhere the particle's wave packet has non-zero amplitude, meaning the position is uncertain – it could be almost anywhere along the wave packet."

Aaargh! Something like a photon is just a wave in the bulk of space. It's an extended entity, a smeared out thing, there is no absolute position, and it has no surface.

This is a circumstance where knowing the actual scientific details would aid your commentary, Farsight. If you look at the passage you cited, it refers to "the measurement of the absolute position of a particle." Such a measurement would involve the interaction of that particle, probably through absorption. This does happen at a specific place, as the experimental evidence demonstrates. In the details of quantum mechanics, the measurement of the absolute position indicates a position where the particle actually has an interaction, thus it is by definition a specific place or region.

[lpetrich]

I don't understand Farsight's difficulties with the Dirac equation - it can't be much more horrible than Maxwell's equations.

Furthermore, the Dirac equation is much less arbitrary than one might at first think, and this comes about from the quantum-mechanical theory of angular momentum.

To get QM angular momentum, one can take the classical expression and go from classical to quantum mechanics in the usual fashion. One can find wavefunction solutions, and one finds them to be spherical harmonics. But one can go a more abstract route, avoiding spherical harmonics entirely, and one finds in general:
Square of angular momentum: J2 = j(j+1)
Angular momentum along some direction: J3 = m
where m = -j, -j+1, ..., j-1, j

One gets the spherical-harmonic solutions for integer j, and one also finds that half-odd j is possible. The simplest such solution has
j = 1/2, m = -1/2 or +1/2

A vector can be interpreted as a j = 1 angular-momentum solution, and a symmetric trace-free n-tensor can be interpreted as a j = n solution, with angular-momentum operators being related to the vector product. So one needs a special sort of entity for a j = 1/2 solution. In the late 1920's, Wolfgang Pauli came up with a solution: a 2-component "spin vector" or "spinor". One component is for m = 1/2, and the other for m = -1/2. He also introduced spinor angular-momentum operators: the Pauli matrices.


One can extend this angular-momentum analysis to more dimensions. For the 4 space-time dimensions, one adds boosts (Lorentz transformations), and the resulting Lorentz-group states are designated with a pair of angular momenta: (j1,j2). The multiplicity of each state is the product of the individual multiplicies or (2j1+1)*(2j2+1).

A space-time position or gradient vector is thus Lorentz state (1/2,1/2), with multiplicity 2*2 = 4. Let's see what various sorts of fields turn out to be:

Spin 0
The Klein-Gordon equation describes the trivial case of a scalar field: (0,0)

Spin 1/2
A Dirac field is (1/2,0) + (0,1/2) - separate left-handed and right-handed spinors.

Spin 1
Photons and gauge particles in general are vector fields.
Potential: (1/2,1/2)
Field (Faraday tensor): (1,0) + (0,1)
The parts correspond to E+i*B and E-i*B.

Spin 3/2
The Rarita-Schwinger equation describes a vector-spinor field.
R-S: (1,1/2) + (1/2,1)

Spin 2
In general relativity, the metric is described by a symmetric tensor: (1,1) + (0,0)
From the metric:
Riemann tensor: (2,0) + (0,2) + (1,1) + (0,0)
From the Riemann tensor:
Weyl tensor: (2,0) + (0,2)
Ricci tensor: (1,1) + (0,0)
Ricci scalar: (0,0)

So the space-time component structures are more-or-less fixed by various symmetries.

[Farsight]

Sorry I haven't responded to that post, lpetrich. I've been tied up, and have to go away this weekend. I'll get back to you and ChildInAZoo and everybody else next week.

[lpetrich]

I'll now give an indication of some of what string theory is supposed to explain, by going into detail about the Standard Model. But first something about symmetry groups.

The group of all orthogonal matrices with dimension n is called O(n). Orthogonal: inverse = transpose. Its elements have determinant values of either +1 (proper rotations) or -1 (rotations + reflections). The proper rotations form a group called SO(n), the special orthogonal matrices.

The group of all unitary matrices with dimension n is called U(n). Unitary: inverse = transpose complex conjugate. Those with determinant = 1 are the special unitary matrices, or SU(n). In effect, U(n) = SU(n) * U(1)

Groups SO(n), U(n), and SU(n) have the property that every element has a continuous path of elements from the identity element. This makes them "Lie groups", and they are generated by "Lie algebras" ("Lee" ones). For elements close to the identity element, their departure from that element is proportional to some Lie-algebra members. Lie algebras are often easier to study than the groups they generate. The smaller Lie algebras are related as follows:

SO(1) ~ SU(1) ~ identity group
SO(2) ~ U(1) ~ circle group
SO(3) ~ SU(2) ~ quantum-mechanics angular-momentum group
SO(4) ~ SU(2) * SU(2) ~ related to the Lorentz group
SO(6) ~ SU(4)

Now for the Standard Model's particle spectrum. At low energies, less than electroweak-unification energies, the gauge symmetry of the SM is
SU(3)C * U(1)em
Their associated quantum numbers are the QCD multiplet number and the electric charge: (QCD #, E.C.)

Higgs: spin 0, superpartner spin 1/2
(1, 0), (1, +/-1)

Gauge: spin 1, superpartner spin 1/2
Photon: (1,0)
W: (1, +/-1)
Z: (1,0)
Gluons: (8, 0)

Elementary fermions: spin 1/2, superpartner spin 0
Quarks and their antiparticles
Up-like: (3, 2/3), (3*, -2/3)
Down-like: (3, -1/3), (3*, 1/3)
Leptons and their antiparticles
Neutrinos: (1,0), (1,0)
Electron-like (1, -1), (1,1)

Gravitons (spin 2) and their superpartners, gravitinos (spin 3/2) are gauge singlets in every theory. Here, they are (1,0).

[lpetrich]

It must be pointed out that unbroken electroweak symmetry looks even more horrible. It is
SU(3)C * SU(2)L * U(1)Y
C - QCD interaction - carried by gluons
L - weak isospin, part of electroweak - carried by charged and neutral W's. Has a mathematical analogy to angular momentum because of its being SU(2)
Y - weak hypercharge, part of electroweak - carried by B's. The B works like the photon

Due to electroweak symmetry breaking, the charged W gets a mass of about 80 GeV, while the neutral W and the B mix to make the Z, with mass of 91 GeV, and the photon, with zero mass and symmetry U(1)em. The Higgs particles also lose some of their modes to the W and the Z.

The particles:

Gauge: spin 1 (1/2)
gluon: (8,1,0)
W: (1,3,0)
B: (1,1,0)

Higgs: spin 0 (1/2)
Particle / antiparticle
Hu (1,2,1/2), Hu* (1,2,-1/2)
Hd (1,2,-1/2), Hd* (1,2,1/2)
The plain Standard Model needs only one Higgs multiplet; supersymmetric extensions of it need two Higgs multiplets to work properly.

Elementary fermions: spin 1/2 (0)
Left-handed particle / right-handed antiparticle
Quark (up + down): Q (3,2,1/6), Q* (3*,2,-1/6)
Anti-Up: U* (3*,1,-2/3), U (3,1,2/3)
Anti-Down: D* (3*,1,1/3), D (3,1,-1/3)
Lepton (neutrino + electron): L (1,2,-1/2), L* (1,2,1/2)
Anti-Neutrino: N* (1,1,0), N (1,1,0)
Anti-Electron: E* (1,1,1), E (1,1,-1)

Note that the elementary fermions are split into left- and right-handed parts. Charged weak interactions turn left-handed parts of different-charged flavors into each other. The particles get their masses by way of these interactions with the Higgses:

yu Hu Q U* + yd Hd Q D* + yn Hu L N* + ye Hd L E* + mnr N* N* + (conjugates; particle <-> antparticle)

where the y's are dimensionless parameters that vary from about 10-4 for electrons and up and down quarks to 0.1 - 1 for the tau lepton and the bottom and top quarks. The mnr is for the "seesaw model" of neutrino masses. It is about 1012 GeV.

This sort of gawdawful mess has appeared before: in chemical elements, nuclei, and hadrons. But each previous time, it was possible to recognize various interrelationships, which in turn were explained by the discovery of underlying simplicity.

That's expected to happen for the Standard Model in the form of a Grand Unified Theory, but GUT's are still rather poorly constrained. The Large Hadron Collider should be able to make particles and effects that will supply some additional constraints.

[lpetrich]

The simplest Grand Unified Theory that units the gauge fields is the Georgi-Glashow SU(5) model. Its gauge symmetry is, of course, SU(5), and its multiplet structure is:

Gauge: spin 1, 1/2
G(24) = gluons + W + B + XY + XY*
It has some new particles: XY (3*,2,5/6), XY* (3,2,-5/6), which contain X (3*,4/3) and Y (3*,1/3). These particles can make protons and bound neutrons decay.

Higgs: spin 0, 1/2
Hu(5) = Hu + Hql / Hu*(5*) = Hu* + Hql*
Hd(5*) = Hd + Hqr* / Hd*(5) = Hd* + Hqr
A "Higgs quark" or Higgs triplet appears: Hql and Hqr form a particle with SM quantum numbers (3,1,-1/3), thus resembling a down quark. It also can make protons and bound neutrons decay.

Both the XY and the Higgs quark must have their masses forced up to 1015 - 1016 GeV, in order to be within the experimental bounds of the proton lifetime, about 2*1029 years. This is close to where one gets gauge unification with the Minimal Supersymmetric Standard Model (MSSM): about 2*1016 GeV. In fact, the best fit is with the supersymmetry-breaking energy around 1 TeV, something which can be tested with the LHC!

In the MSSM, one may also get gaugino mass unification at GUT energy scales; gauginos are the supersymmetry partners of the gauge particles. The LHC ought to be able to produce all three gauginos, the gluino, the wino, and the bino, though the latter two will be mixed with the higgsinos to make 2 "charginos" and 4 "neutralinos".

The elementary-fermion multiplets (spin 0, 1/2) are:
F(1) = N* / F*(1*) = N
F(10) = Q + U* + E* / F*(10*) = Q* + U + E
F(5*) = L + D* / F*(5) = L* + D

yu Hu(5) F(10) F(10) + yn Hu(5) F(5*) F(1) + yd Hd(5*) F(10) F(5*) + mnr F(1) F(1) + (conjugates; particle <-> antparticle)
where ye = transpose(yd).

One gets bottom-tau mass unification at GUT energy scales!

So the LHC, by probing the masses where we expect to see supersymmetry partners, will help us test GUT possibilities.

[lpetrich]

In summary, we ought to be able to test:

• Gauge-coupling unification
• Gaugino-mass unification
• Bottom-tau mass unification

by extrapolating to GUT energies.

One can go further than SU(5) to bigger gauge groups. The smallest one that unifies the elementary fermions into one multiplet per generation is SO(10), and it breaks down:
SO(10) -> SU(5) * U(1)Y'
where Y' is a quantum number related to (baryon number) - (lepton number):
B - L = (Y' + 4Y)/5
I'll do (SO(10) multiplet number), then (SU(5) multiplet number, Y')

Gauge: G(45) -> G(24,0) + G(1,0) + G(10,-4) + G(10*,4)
The first one contains the familiar Standard-Model gauge fields, while the other three are forced up to GUT energies. However, SO(10) contains no new particles for the others:

Higgs: H(10) -> Hu(5,-2) + Hd(5*,2) / H*(10) -> Hu*(5*,2) + Hd*(5,-2)
One Higgs multiplet!

Elementary fermions: F(16) -> F(1,5) + F(10,1) + F(5*,-3) / F*(16*) -> F*(1*,-5) + F*(10*,-1) + F*(5,3)
One EF multiplet per generation!

Mass terms: y H(10) F(16) F(16) + conjugate
where y = yu = yd = yn = ye = symmetric
No cross-generation quark decays, no neutrino oscillations.
No right-handed neutrino mass, however, and no seesaw effect.

So SO(10) mass unification is too successful. Cross-generation quark decay, neutrino oscillations, and neutrino masses all require breaking of SO(10).

[Twiglet]

Ipetrich, I'd be interested to hear your take on this, because to me it all seemed rather like cookery (and was described as such by the lecturer who taught it). A few basic ingredients cooking up different particles to explain phenomenon, with a great deal of complexity arising simply from the different ways of cooking basic ingredients together.

Science (or the explanations) felt...very frail...in a way. I know solid predictions arose from it (like the muons etc), and it has all kinds of pretty maths....but it feels so...I don't know - inelegant? Or like we are scratching at the surface of something, which has something else much deeper beyond it.

At a certain level, the idea that particles could be some kind of resonance pattern, with stabilities occuring because they correspond to favourable geometries (think Hamiltonian least energy configurations) is incredibly appealing, though I would have no idea where to start with such ideas. I love the way Hamiltonian mechanics is such a natural fit to QM, describing possible modes of behaviour based on the energy of a system. In a way, it might be possible to come up with something that models quark/gluon interactions but I'd have no idea even really how to start, or what to start with.

I think at that level, I quite sympathise with farsights aims (but not methods), there's a yearning for some underlying principle to make sense of the mess on a deeper level than what we have - does. I suppose string theory is an attempt as well....

Anyway... just thoughts in the dark really.

[lpetrich]

Ipetrich, I'd be interested to hear your take on this, because to me it all seemed rather like cookery (and was described as such by the lecturer who taught it). A few basic ingredients cooking up different particles to explain phenomenon, with a great deal of complexity arising simply from the different ways of cooking basic ingredients together.

That's an interesting way of putting it.

Science (or the explanations) felt...very frail...in a way. I know solid predictions arose from it (like the muons etc), and it has all kinds of pretty maths....but it feels so...I don't know - inelegant? Or like we are scratching at the surface of something, which has something else much deeper beyond it.

But whatever it is must include known physics as a special case. Complete with the numbers that one can find for it. That's the problem I have with such self-styled physicists as Farsight; they ignore the necessity of being able to get good numerical agreement with present theories.

Consider relativity vs. Newtonian mechanics, quantum mechanics vs. classical mechanics, etc.

At a certain level, the idea that particles could be some kind of resonance pattern, with stabilities occuring because they correspond to favourable geometries (think Hamiltonian least energy configurations) is incredibly appealing, though I would have no idea where to start with such ideas.

Except that "particles" aren't classical billiard balls or whatever -- they are quantized fields, and rather simple quantized fields at that. Look at Maxwell's equations or the Dirac equation or the Klein-Gordon equation.

I love the way Hamiltonian mechanics is such a natural fit to QM, describing possible modes of behaviour based on the energy of a system. In a way, it might be possible to come up with something that models quark/gluon interactions but I'd have no idea even really how to start, or what to start with.

We already have a successful model, Quantum Chromodynamics. That's a nonabelian gauge field theory. Think 8 photonlike quantum fields, all interacting with each other in a certain way.

I think at that level, I quite sympathise with farsights aims (but not methods), there's a yearning for some underlying principle to make sense of the mess on a deeper level than what we have - does. I suppose string theory is an attempt as well....

And GUT's in general.

[Twiglet]

Indulge me in a thought experiment Ipetrich...

Imagine that we had an accelerator with 10-20 times the collision energies of the LHC, and we found not just the Higss boson but some other new stuff as well.

I guess we would just keep adding quarks and colours or whatever to explain.. i.e. a few more basic ingredients.

What I wonder is if particles (not hung up about whether they are wavefunctions, localised wavepackets or whatever) are possible normal modes of behaviour dependent on the energy of the system, i.e. they are stabilities. The more energy is available, the more normal modes are possible - the greater the number of possible variations (particles)

This isn't a radical rethink as far as existing explanations go, in fact it is entirely consistent with them. QCD could just "fall out" of such a system. It's not a reversion to classical ideas either, it's more focused on the idea that a particle (wavicle etc) isn't something intrinsic or basic, it's rather a stable "normal mode of oscillation" made stable by the energy density it inhabits.

Superficially an idea like this would predict that more and more stuff happens the higher the energy density is ad infinitum perhaps. Every elementary particle isn't elementary at all, just a stable configuration within an energy density. Perhaps by examining the energy densities that create stabilities for particular particles it could be possible to make predictions about where the next islands of stability would be, but that would require coming up with something concrete in the first place. Which of course, I haven't.

[lpetrich]

I love the way Hamiltonian mechanics is such a natural fit to QM, describing possible modes of behaviour based on the energy of a system.

"Hamiltonian mechanics" is essentially the Hamiltonian formulation of Newtonian mechanics, much like the Lagrangian and Hamilton-Jacobi formulations.

The Schroedinger and Heisenberg formulations of quantum mechanics both utilize the Hamiltonian formulation of classical mechanics, which was likely from the helpfulness of its mathematical elegance. One even gets Heisenberg -> Hamilton and Schroedinger -> Hamilton-Jacobi in the classical limit. However, it's difficult to make Schroedinger/Heisenberg relativity-friendly, and that difficulty has been important for developing quantum field theory. The problem is that there is no "universal time" in relativity. A Lorentz-transformed time has some space admixture, just like the time admixture of space in both relativity and its Newtonian limit.

So in QFT, it has been easier to start from a Lagrangian than a Hamiltonian; a Lagrangian will be "manifestly covariant", a quantity with directly-evident Lorentz invariance. Strictly speaking, a QFT Lagrangian is a Lagrangian density, integrated over all of space-time, like the Lagrangian of classical-limit electromagnetism. The space-time volume integral is, of course, Lorentz-invariant.

There is even an approach to quantizing fields that uses a Lagrangian directly: the path-integral approach.

Anyway... just thoughts in the dark really.

I appreciate such humility.

I also have very little sympathy for Farsight's approach -- it features a neglect of the successes of QFT-based theories.

[lpetrich]

For SO(10), we have:

• One gauge multiplet: 45
• One Higgs multiplet: 10, anti: 10
• One multiplet per elementary-fermion generation: 16, anti: 16*

Can we go further? Let's consider the "exceptional Lie algebra" E6. It breaks into SO(10) in this fashion:
E6 -> SO(10) * U(1)Y''
Using the same notation as before, we find:

Gauge: G(78) -> G(45,0) + G(1,0) + G(16,-3) + G(16*,3)
The first one contains the Standard Model gauge fields, the others must be forced up to GUT energies.

Elementary fermions: F(27) -> F(16,1) + H(10,-2) + X(1,4), F*(27*) -> F*(16*,-1) + H*(10*,2) + X*(1*,-4)
The first one contains all the Standard Model elementary-fermion multiplets. But one generation of the second one becomes the Higgs; the other ones get forced up to GUT energies. This may explain the mass hierarchy of the elementary fermions; the Higgs makes one generation much more massive than the others because it had come from that generation. Its associated X may also be present at low energies, and it may be the additional particle in the Next-to-Minimum Supersymmetric Standard Model. It is a Standard-Model scalar that interacts only with the Higgs particles, helping to give them their masses.

The mass terms are:
y F(27) F(27) F(27) + conjugate

A very simple-looking expression.

[JimC]

I cruise this thread from time to time, because of past flamewars...

It makes my brain hurt, but I am most impressed with the lack of not-niceness...

(excuse my double negative, I blame the gin...)

[lpetrich]

Thanx.

Here's a summary of these complicated multiplets.

Low-Energy Standard Model (electroweak, supersymmetry broken)
Gauge: 5
Higgs: 4
Neutralinos, charginos: 4, 2
Elementary (s)fermions / generation: 5

Unbroken Standard Model
Gauge: 3
Higgs: 2
EF's /gen: 6

GUT's:

SU(5) (Georgi-Glashow)
Gauge: 1
Higgs: 2
EF's/gen: 3

SO(10)
Gauge: 1
Higgs: 1
EF's/gen: 1

E6
Gauge: 1
EF's/gen with Higgs: 1

Why E6? Let's now go the reverse direction from string theory.

There are 5 kinds of supersymmetric string or superstring. In the low-energy limit, they all have some supersymmetric gravity or supergravity multiplet in them, and some of them have supersymmetric gauge multiplets. The most interesting starting point for trying to get the Standard Model is the E8*E8 Heterotic Superstring. The E8*E8 is its gauge group; all the gauge particles are in two multiplets, each with size 248.

It seems like there are no Higgses or elementary fermions in sight, but a certain feature of superstrings comes to the rescue. They prefer to live in 10-dimensional space-time, which is 6 more space dimensions than what we have. A common solution is to suppose those extra dimensions to be "compactified", curled up into a tiny ball with GUT-scale to Planck-scale sizes. This ball then interacts with one of the gauge multiplets to break its symmetry down to something like E6*SU(3), where the SU(3) part interacts with the details of that ball. The multiplet itself breaks down into

G(248) -> G(78,1) + G(1,8) + G(27,3) + G(27*,3*)

The first of them makes the E6 gauge fields, the second one is likely pushed up to GUT energies, and the third and fourth become the elementary fermions and Higgses with their antiparticles. The first one's low-energy modes are the vector components along the large dimensions, and the third and fourth ones' low-energy modes are likely the vector components along the small dimensions, because those ones are scalars in the large dimensions.

The topology of the six small dimensions causes a lot of symmetry breaking, by forcing many of the field modes up into GUT-scale energies. What pattern of low-energy particles depends on what topology, and string theory itself does not appear to offer much constraint of possible topologies. Thus, the Standard Model is a far-from-unique solution of string theory, as far as anyone can tell.

However, the string's supergravity multiple produces the graviton, the gravitino, and possibly some additional low-energy particles with spins 0, 1/2, and 1. Those additional ones, like the graviton and gravitino, are all gauge singlets, meaning that Standard-Model particles essentially look alike to them.

[Farsight]

Sorry I didn't get back to you as promised, lpetrich. I don't want to derail the thread, but can I just say this:

That's the problem I have with such self-styled physicists as Farsight; they ignore the necessity of being able to get good numerical agreement with present theories.

I don't want to ignore it, it's just that it's so much work, and I'm not equipped to do it. That's why I talk to guys like you. I have this big picture that's only an outline, covering the deep fundamentals* but lacking in detail. You're close-in, with lots of detail, but with respect, you don't seem to understand those deep fundamentals of have any kind of big picture. I'd hope there's mileage in talking things through, but there seems to be a kind of professional pride that prevents people like yourself from accepting any input from people like me. I think it's a pity, because progress in physics has been stalled for many years. IMHO there's an impasse, it has to be broken, and I'm doing what I can to help.

* See Time Explained as an example.

I think at that level, I quite sympathise with farsights aims (but not methods), there's a yearning for some underlying principle to make sense of the mess on a deeper level than what we have - does. I suppose string theory is an attempt as well....

I'd say you should take a good look at Topological Quantum Field Theory. I have to use circuituous method because it's very difficult to get a journal to accept a paper that delivers a new idea. I'm not just talking about my own experiences here. For the new idea to be right, the old idea has to be wrong. And the sort of people who came up with the old idea, or adhere to it, are the sort of people on the editorial committee.