Quantum Mechanics - Some Math

[lpetrich]

I'll close with discussing in more detail how new theories relate to old theories, especially major theories like quantum mechanics and Newtonian mechanics.

Does QM mean that NM is wrong? That's what one might at first think, and that's the sort of thing that a lot of simplistic science journalism would make one think ("New results overthrow long-held theory!!!"). But NM had been extensively tested over the previous centuries, and for the most part, it had been a great success. So it must not be completely wrong.

Asimov - The Relativity of Wrong has a good discussion of this issue.

So let us look more closely at the relationship between QM and NM. In the limit of action-like quantities much greater than Planck's constant h, QM becomes NM. That is evident from these three formulations:

• Schroedinger -> Hamilton-Jacobi
• Heisenberg -> Hamilton
• Feynman path-integral -> Euler-Lagrange

Action-like quantities have dimensions (mass)*(length)²/(time), and include (position)*(momentum), (time)*(energy), and angular momentum.

So NM is an approximation that is only good for (action-like) >> h, what's often called the classical limit.

This explains why NM had been so successful before the development of QM. It had been successful because it was applied to systems where action-like quantities are much greater than h, and those systems are just about everything much larger than an atom.

The same is also true of relativity. NM is also a limiting case of it, and one can quantify how far off it is rather easily. Approximately (1/2)*(v/c)², where v is typical speeds, and c is the speed of light in a vacuum, a speed related to the geometry of space-time. Special relativity and quantum mechanics make relativistic quantum field theory, and that reduces to NM in two ways, through nonrelativistic QM and through classical-limit special relativity:

RQFT -> NRQM -> NM
RQFT -> SR -> NM

So if some theory is going to succeed in superseding QM, or more precisely, RQFT, it must have the same relationship to QM that QM has to NM. It must be a superset theory that reduces to QM in some appropriate limit, a limit where QM has been successfully tested.

Pseudoscientists and crackpots often get into trouble over this. They often make little effort to show that their theories account for what existing theories successfully account for.

-

One last thing. The classical limit of QM is a bit more complicated than what my discussion of it here may suggest. In a way, it has not one but two classical limits: the particle limit and the wave limit. Here is how they related to QM:

Particle limit. The familiar sort of physical objects are all in this limit. Their wavefunctions' extents are much, much, much smaller than their sizes.

QM statistics: Non-issue. The wavefunctions are too small to overlap, and macroscopic objects are almost always non-identical at the elementary-particle level.

Wave limit. Or more generally, the field limit. Electromagnetism and gravity are well-known fields in this limit, and this limit also includes collective effects like sound waves and Bose-Einstein condensates.

QM statistics: always Bose-Einstein.

[lpetrich]

Online resources

Wikipedia is often a good source, though I will concede that it can be hit-and-miss. A good Wikipedia article will contain references to its sources, and you can use it as a starting point. Like Quantum mechanics.

About physics, HyperPhysics is a very good resource. It's usually nonmathematical, and the mathematics it does have is much like the mathematics I've posted here.

One can find more good references at Subject hysical sciences - Wikibooks, open books for an open world, like This Quantum World - Wikibooks, open books for an open world, though they can get mathematical.

Here are some collections of quantum-mechanics visualizations:
Quantum Mechanics Animations
Physics Flash Animations
Both of them use Flash.


For the chemical elements, there are numerous online resources, like:
Dynamic Periodic Table - nice interactive one
Periodic Table of the Elements by WebElements
RSC Visual Elements Periodic Table
The Photographic Periodic Table of the Elements - has pictures of elements or their namesakes
Periodic table
Periodic Table of the Elements - from Hyperphysics

For chemical bonds, I've found
Chemical bond
Chemical Bonds - Types of Chemical Bonds - has some realistic depictions of atoms: fuzzy blobs
Chemical Bonds - from Hyperphysics

Some stuff on atomic energy levels:
Hydrogen energies and spectrum - the easiest case
Atomic Energy Levels
Atomic Energy Level Diagrams


Turning to atomic nuclei, I've found
Livechart - Table of Nuclides - Nuclear structure and decay data - pure webpage
Live Chart of Nuclides - Table of Nuclides - Java applet in webpage
Interactive Chart of Nuclides - pure webpage
which work something like the interactive Periodic-Table pages. The Q values in it are decay energies, and the FY's are nuclear-fission yields. The magic numbers of protons (Z) and neutrons (N) are marked out; they are 2, 8, 20, 28, 50, 82, 126.

Also,
Table of nuclides - links to several versions: a single table (very wide), that table split into several tables, and some lists.
Nuclear reaction


Some big collections of data:
Standard Reference Data at the NIST -- a huge collection
National Nuclear Data Center


A note: this is intended to be a resources thread, like the threads on evolution here. If you wish to comment on this thread, there's Quantum Mechanics - Some Math - Peanut Gallery - Secular Café

[lpetrich]

The elementary particles of the Standard Model?

First, let us discuss gauge symmetry in more detail, since that is an important part of the SM and extensions of it.

As I'd mentioned earlier, electromagnetism has a gauge symmetry, one related to the conservation of electric charge. Let us generalize it to what happens when one interchanges kinds of particles.

Let's start with the 2-component case, since it illustrates some important features of the general case. It is inspired by the 1-component case, what electromagnetism has:

For wavefunction X,
X -> U*X

where |U| is 1 and U = exp(i*q*L) ~ 1 + i*q*L in for some function L when it is small.

Now do X = {X1,X2}. U becomes {{U11,U12},{U21,U22}}. We need to put in a matrix multiply:
X -> U.X

Where U satisfies inverse(U) = Hermitian conjugate of U, its transpose of its complex conjugate. U is thus "unitary". If U is a real matrix, then U is "orthogonal".

As with the 1-dimensional case,
X -> exp(i*T_kL_k).X ~ (1 + i*T_kL_k).X

for generators T_k and functions L_k. One can take the T's to be
{{1,0},{0,0}}, {{0,1},{0,0}}, {{0,0},{1,0}}, {{0,0},{0,1}}.

One quickly notices that the T's do not commute, that different changes interact with each other in nontrivial ways. It turns out that one can find a linear combination of them that does not interact with certain other linear combinations of them.

Set of 1: {{1,0},{0,1}} - the identity matrix. It changes the overall phase of the system.
Set of 3: {{0,1},{1,0}}, {{0,-i},{i,0}}, {{1,0},{0,-1}} - it can interchange components and give them opposite phases.

In effect, we have decomposed the group of dimension-2 unitary matrices, U(2) into two others: the group of dimension-2 unitary matrices U(1) (the set of 1) and a group with determinant 1, the special dimension-2 unitary matrices SU(2) (the set of 3). More generally,
U(n) -> U(1) * SU(2)
U(1) - generated by the set of 1
SU(2) - generated by the set of 3

The orthogonal matrices decompose
O(n) -> {I,R} * SO(n)
since they are either pure rotations or rotation-reflections (rotoreflections). R is one of the latter.


The set of 3 is the Pauli matrices, which were introduced by Wolfgang Pauli in the late 1920's for handling the angular-momentum generators of spin-1/2 particles like electrons. Since angular momentum is related to rotation, one expects angular momentum to have the operator algebra of 3D space rotations, or SO(3). But the Pauli matrices have the same operator algebra, making SU(2) ~ SO(3) for generators. Thus angular momentum has symmetry group SU(2) ~ SO(3).

There are some other generator isomorphisms:
SO(2) ~ U(1)
SO(3) ~ SU(2)
SO(4) ~ SU(2) * SU(2)
SO(6) ~ SU(4)


Can we apply this sort of symmetry more broadly?

In the early 1930's, Werner Heisenberg noticed that protons and neutrons in atomic nuclei act almost like they are two versions of the same particle. He proposed that they are related by a symmetry that can interchange protons and neutrons, a symmetry that acts much like angular momentum for spin-1/2 particle. It got called "isotopic spin" or "isospin" for short. This symmetry is not exact, of course. Electromagnetism breaks it, as one can tell from the charges of the proton and the neutron.

When lots of hadrons were discovered in the 1940's and 1950's, they were also found to obey isospin symmetry, electromagnetic breaking and all. A property called "strangeness" seemed related to isospin, and isospin + strangeness were found to approximately obey a somewhat larger symmetry. Where were we in the early 1960's?
Electromagnetism: U(1) -- exact
Isospin: SU(2) -- approximate
Isospin + strangeness: SU(3) -- less approximate

With the discovery of quarks, it became evident that isospin and isospin+strangeness symmetries were the result of the lightest quarks being lighter than the hadron mass scale. The up and down quarks have masses around a few MeV, less than 1% of the nucleons' masses, thus isospin SU(2) symmetry is close to exact if one ignores electromagnetic and weak interactions. The strange quark has a mass around 100 MeV, about 1/9 the nucleons' masses, and the isospin+strangeness symmetry is thus not as good.

One can extend these symmetries to heavy quarks, but their high masses rather grossly break the symmetry.


Multiplets of isospin-related particles have average electric charges, of course, and that average is called the hypercharge.
Q = I3 + Y
(Electric charge) = (3rd component of isospin) + (hypercharge)

Worked examples for several elementary-particle multiplets:

Nucleons: isospin = 1/2, hypercharge = 1/2, strangeness = 0
Proton (uud): +1/2 + 1/2 = 1
Neutron (udd): -1/2 + 1/2 = 0

Delta baryons: isospin = 3/2, hypercharge = 1/2, strangeness = 0
Positive 2 (uuu): 3/2 + 1/2 = 2
Positive (uud): 1/2 + 1/2 = 1
Neutral (udd): -1/2 + 1/2 = 0
Negative (ddd): -3/2 + 1/2 = -1

Lambda baryon: isospin = 0, hypercharge = 0, strangeness = -1
Neutral (uds): 0 + 0 = 1

Sigma baryons: isospin = 1, hypercharge = 0, strangeness = -1
Positive (uus): 1 + 0 = 1
Neutral (uds): 0 + 0 = 0
Negative (dds): -1 + 0 = -1

Xi baryons: isospin = 1/2, hypercharge = -1/2, , strangeness = -2
Neutral (uss): 1/2 - 1/2 = 0
Negative (dss): -1/2 - 1/2 = -1

Omega baryon: isospin = 0, hypercharge = -1, strangeness = -3
Negative (sss): 0 - 1 = -1

Pions: isospin = 1, hypercharge = 0, strangeness = 0
Positive (ud*): 1 + 0 = 1
Neutral (uu*-dd*): 0 + 0 = 0
Negative (du*): -1 + 0 = -1

Kaons: isospin = 1/2, hypercharge = 1/2, strangeness = 1
Positive (us*): 1/2 + 1/2 = 1
Neutral (ds*): -1/2 + 1/2 = 0

Up and down quarks: isospin = 1/2, hypercharge = 1/6, strangeness = 0
Up: 1/2 + 1/6 = 2/3
Down: -1/2 + 1/6 = -1/3
Strange quark: isospin = 0, hypercharge = -1/3, strangeness = -1
0 - 1/3 = -1/3

Antiparticles have the same isospin and reversed hypercharge, electric charge.

For the SU(3) flavor symmetry,
Up, down, strange quarks: a 3 multiplet
Their antiparticles: a 3* multiplet

The spin-1/2 light baryons: nucleons, lambda, sigma, xi: a 8 multiplet
The spin-3/2 light baryons: delta, sigma-x, xi-x, omega: a 10 multiplet

The omega baryon was predicted using this approximate flavor symmetry by Murray Gell-Mann and Yuval Ne'eman in 1962, and it was discovered in 1964.


There is a curiosity that arises when one combines the spin and flavor symmetries of the light baryons. The spins of the baryons imply that quarks must have spin 1/2. They can combine to make spin 3/2 (symmetric) or spin 1/2 (mixed symmetry). Likewise, the quark flavors combine to make a 10 multiplet (symmetric) and a 8 multiplet (mixed). The symmetric spin and flavor combination is, of course, symmetric, and the mixed ones have a symmetric combination here. But quarks are supposed to follow Fermi-Dirac statistics, and have an antisymmetric wavefunction, and not Bose-Einstein statistics with a symmetric wavefunction. It took the development of Quantum Chromodynamics to resolve that paradox.


Most of this discussion has been about flavor symmetries for quarks, but that should be good for illustrating how elementary-particle symmetries.

[lpetrich]

The previous post had featured mainly flavor symmetries. These ones are global or rigid, being applied the same to every place in space-time.

The electromagnetic gauge symmetry is, however, local. One can change it at any point, while keeping its overall behavior the same. Can it be extended to multicomponent fields? Yes, it can.

As with electromagnetism, one constructs a "gauge-covariant" derivative of the particle fields that will cancel out the more troublesome gauge variation.

D_i(X) -> D_i(X) - i*g*T_kA_k,i.X

where A is the gauge field's potential and g is a charge-like value. Like the electromagnetic potential, it is a 4-vector, but it has one component for each symmetry generator.

As with electromagnetism, one can construct a field value that cancels out the more troublesome gauge variation:
T_kF_k,i1,i2 = D_i2(T_kA_k,i1) - D_i1(T_kA_k,i2) - i*g*[T_k2,T_k1]A_k2,i2A_k1,i1

One can easily find it by finding the commutator of a gauge-covariant derivative: D_i1(D_i2(X)) - D_i2(D_i1(X))

So if some symmetry generators do not commute with each other, they force their corresponding gauge fields to interact with each other. A gauge theory with this property is called nonabelian, after the usual term for a group whose members do not necessarily commute with each other.


It took a lot of trial and error for physicists to work out the Standard Model, but by the late 1970's and early 1980's, they had succeeded. The term "Standard Model" itself dates from the early 1980's.

Its gauge symmetry is symmetry group SU(3) * SU(2) * U(1)

SU(3) = Quantum Chromodynamics, QCD
SU(2) = Weak Isospin, WIS
U(1) = Weak Hypercharge, WHC

WIS and WHC are named in analogy with their quark-flavor-symmetry versions.


In QCD, quarks as having an additional quantum number, one that comes in threes. It was named color, after the three primary colors of our visual perception. However, the "colors" are outwardly indistinguishable -- QCD symmetry is unbroken.

Quarks: red, green, blue -- r, g, b
Antiquarks: cyan, magenta, yellow -- c, m, y

There are 9 ways to change quark fields by color, but one can get an overall phase change out of them, and that one is thus irrelevant. That leaves 8. In group-theory terms,
U(3) -> U(1) * SU(2)

The QCD gauge field is called the gluon, from its rather gluey properties in the light hadrons. Its has 8 color-anticolor combinations, with the 9th being the colorless one:

Gluon: rm, ry, gc, gy, bc, bm, (rc - gm)/sqrt(2), (rc + gm - 2by)/sqrt(6)
Colorless: (rc + gm + by)/sqrt(3)

The known hadrons are either mesons (quark-antiquark) or baryons (3 quarks). They are all outwardly colorless, and here are their color states:
Mesons: (rc + gm + by)/sqrt(3)
Baryons: (rgb - rbg + gbr - grb + brg - bgr)/sqrt(6)

Note that the quarks in baryons are asymmetric by color. Their spin-flavor symmetry combines with this asymmetry to make an overall asymmetry, thus rescuing Fermi-Dirac statistics for quarks.


Elementary particles' interactions makes their effective masses and interaction parameters charge with energy, usually in a slow logarithmic fashion. But this change will be faster for parameters near 1 in dimensionless units, and QCD illustrates it very well. Gluon self-interaction makes its effective "charge" go down with increasing interaction energy. At interaction energies of around 100 GeV, its effective "charge" is about 0.3, while it goes past 1 for light-hadron energies of about 1 GeV. The QCD interaction getting superstrong likely explains color confinement, an effect where one colored particle cannot get more than about one fermi or 10^(-15) m from another one, with overall states being colorless.

In fact, one can observe this "asymptotic freedom" in the more energetic collisions. Quarks and gluons behave like approximately free particles, until they get about 1 fermi from each other. If they are traveling with high energies relative to each other, they will pull quarks and antiquarks out of the vacuum as they go, making jets of hadrons. Jets that are often observed.

[lpetrich]

Turning to the weak-isospin and the weak-hypercharge symmetries, they are the symmetries of the electroweak interaction, the unification of the electromagnetic and the weak interactions.

The weak-hypercharge particle, the B, behaves much like the photon, while the weak-isospin particle, the W, behaves a bit like the gluon, but interchanging 2 instead of 3 states of other particles.

To see how it works, consider the Higgs doublet, two complex fields that the weak-isospin and weak-hypercharge symmetries interchange and change phases of: {Hc, Hn}. The WIS and WHC symmetries can make this {Hc+Hn,Hc-Hn}/sqrt(2), {Hn,Hc}, {Hn,-Hc}, {i*Hc,Hn}, {Hc,-Hn}, etc. The magnitude,
|H|² = |Hc|² + |Hn|²
stays constant.

From its self-interaction, the Higgs particle gets a nonzero Vacuum Expectation Value, and this can be arbitrarily distributed among Hc and Hn, with their phases arbitrarily varying.

This breaks the electroweak symmetry, and let's see what happens. For convenience, one does some "gauge fixing", imposing some condition that is satisfied by some gauge transformation from some arbitrary state. This fixing makes the Higgs vacuum state {0,1}*|H|, which is more convenient to work with.

This nonzero value gives masses to two of the W's, and a mass to a mixture of the third W and the B. The remaining one stays massless.

W⁺, W^-: low-energy W particles, mass 80 GeV
W⁰*cos(a) + B*sin(a): Z particle, mass 91 GeV
-W⁰*sin(a) + B*cos(a): photon, massless
a = Weinberg angle ~ 29d

Thus, the photon is related to the W and the Z.


Now the elementary fermions.

In electroweak unification, the left-handed parts of the neutrino and the electron are part of one WIS doublet (WIS = 1/2), and their right-handed parts are separate WIS singlets (WIS = 0). Likewise for the up and down quarks.

Here, "electron" is shorthand for electron, mu, tau leptons, "down quark" is shorthand for down, strange, bottom quarks, and "up quark" is shorthand for up, charm, top quarks.

The low-energy W particle can turn up quarks into down quarks and vice versa, and likewise with neutrinos and electrons. A neutron's decay works like this:

Neutron -> proton + W^-
(udd) -- d -> u + W^- -- (uud) + W^-
The W is too massive to be materialized, so it stays virtual. That gives the neutron a half-life of about 10 minutes instead of about 10^-24 seconds.

W^- -> electron + antineutrino
This is related to electron <-> neutrino flips, because creating an antiparticle is equivalent to destroying an ordinary particle, and vice versa.


The details the Higgs particle makes mass.

Since the Higgs particle interacts with the W and the Z, its kinetic terms have a derivative term that looks like
D_W,B(H) = D(H) - i*g*(T.W).H - i*g'*B*H

For the Higgs particle's vacuum field value, the kinetic term |D_W,B(H)|² yields terms
(g*|H|)² * |W^+-|²
and |H|² * (g*W⁰ + g'*B)²
Thus giving the low-energy W and the Z their masses.

For the elementary fermions, we have interaction terms

h.(RH electron).(Higgs).{LH neutrino, LH electron}
h.(RH neutrino).(flipped Higgs).{LH neutrino, LH electron}
h.(RH down).(Higgs).{LH up, LH down}
h.(RH up).(flipped Higgs).{LH up, LH down}
RH = right-handed, LH = left-handed

With the Higgs's vacuum field, we get
(h*|H|).(RH electron).(LH electron)
(h*|H|).(RH neutrino).(LH neutrino)
(h*|H|).(RH down).(LH down)
(h*|H|).(RH up).(LH up)

The h's are coupling constants, and they are really matrices over all 3 generations of elementary fermions. These matrices are not quite orthogonal, and that's what makes the quarks do cross-generation decays.

Neutrinos, however, get produced in a mixture of mass states that oscillate separately, making the weak-state content oscillate. These oscillations have been observed, though the observations continue to be difficult. To see how they work, consider a 2-neutrino system with weak states at angle a relative to the mass states. They are produced in the first weak state, with mass-state content
{cos(a), sin(a)}
They oscillate with phases p1 and p2:
{exp(i*p1)*cos(a), exp(i*p2)*sin(a)}

Amplitude for getting the first neutrino weak state back:
cos(a)²*exp(i*p1) + sin(a)²*exp(i*p2)

Strength = absolute square of amplitude =
cos(a)⁴ + 2*cos(a)²*sin(a)²*cos(p1-p2) + sin(a)⁴ = 1 - 2*cos(a)²*sin(a)²*(1-cos(p1-p2))

Amplitude for getting the second neutrino weak state {-sin(a), cos(a)}:
cos(a)*sin(a)*(exp(i*p2) - exp(i*p1))

Strength =
2*cos(a)²*sin(a)²*(1-cos(p1-p2))

[lpetrich]

Here is the Standard Model's zoo of particles.

First, the low-energy particles, with broken electroweak symmetry.

Q = electric charge
QCD = QCD multiplet (1 = colorless, 3 = quark-like, 3* = antiquark-like, 8 = gluon-like)

[table]Particle | Spin | QCD | Q | Mass
Z | 1 | 1 | 0 | 91 GeV
W | 1 | 1| +-1 | 80 GeV
photon | 1 | 1 | 0 | 0
Higgs | 0 | 1 | 0 | 125 GeV
gluon | 1 | 8 | 0 | 0
up, charm, top | 1/2 | 3 | 2/3 | 0.0024, 1.25, 171 GeV
(anti) | 1/2 | 3* | -2/3 | (same)
down, strange, bottom | 1/2 | 3 | -1/3 | 0.0048, 0.10, 4.2 GeV
(anti) | 1/2 | 3* | 1/3 | (same)
neutrinos | 1/2 | 1 | 0 | ~10^(-11) GeV
(anti) | 1/2 | 1 | 0 | (same)
electron, mu, tau | 1/2 | 1 | -1 | 0.000511, 0.106, 1.78 GeV
(anti) | 1/2 | 1 | 0 | (same)[/table]

A much-discussed theoretical possibility is an additional form of symmetry, supersymmetry. It relates particles with different spins. Although no known particle is a supersymmetry partner of another known particle, supersymmetry has some nice theoretical features.

The simplest supersymmetric extension of the Standard Model is, of course, the Minimal Supersymmetric Standard Model, and it features these additional particles in the low-energy limit:
[table]Particle | Spin | QCD | Q
charged Higgs | 0 | 1 | +-1
neutral Higgs (2) | 0 | 1 | 0
charginos (2) | 1/2 | 1 | +-1
neutralinos (4) | 1/2 | 1 | 0
gluino | 1/2 | 8 | 0
up squarks (6) | 0 | 3 | 2/3
(anti) | 0 | 3* | -2/3
down squarks (6) | 0 | 3 | -1/3
(anti) | 0 | 3* | 1/3
sneutrinos (3 or 6) | 0 | 1 | 0
(anti) | 0 | 1 | 0
selectrons (6) | 0 | 1 | -1
(anti) | 0 | 1 | 1[/table]

Neutralinos and charginos are mixtures of the wino, bino, and Higgsinos, superpartners of the W, B, and Higgs particles. They are mixed as a result of electroweak symmetry breaking.

Now for the unbroken-electroweak (Minimal Supersymmetric) Standard Model.

Hand = chirality or handedness, (L)eft or (R)ight
SuSp = superpartner spin
WIS = weak-isospin multiplicity = 2*(weak-isospin value) + 1
WHC = weak hypercharge
Masses not given because most unbroken-electroweak particles are massless.

[table]Particle | Hand | Spin | SuSp | QCD | WIS | WHC
up Higgs Hu | L | 0 | 1/2 | 1 | 2 | 1/2
(anti) | R | 0 | 1/2 | 1 | 2 | -1/2
down Higgs Hd | L | 0 | 1/2 | 1 | 2 | -1/2
(anti) | R | 0 | 1/2 | 1 | 2 | 1/2
B | - | 1 | 1/2 | 1 | 1 | 0
W | - | 1 | 1/2 | 1 | 3 | 0
gluon g | - | 1 | 1/2 | 8 | 1 | 0
quark Q | L | 1/2 | 0 | 3 | 2 | 1/6
(anti) | R | 1/2 | 0 | 3* | 2 | -1/6
up U | R | 1/2 | 0 | 3 | 1 | 2/3
(anti) | L | 1/2 | 0 | 3* | 1 | -2/3
down D | R | 1/2 | 0 | 3 | 1 | -1/3
(anti) | L | 1/2 | 0 | 3* | 1 | 1/3
lepton L | L | 1/2 | 0 | 1 | 2 | -1/2
(anti) | R | 1/2 | 0 | 1 | 2 | 1/2
neutrino N | R | 1/2 | 0 | 1 | 1 | 0
(anti) | L | 1/2 | 0 | 1 | 1 | 0
electron E | R | 1/2 | 0 | 1 | 1 | -1
(anti) | L | 1/2 | 0 | 1 | 1 | 1[/table]

There are two Higgs doublets listed here, Hu and Hd, because that's what the MSSM requires. However, the SM Higgs can be interpreted as either.

Note how the Higgs particles and the elementary fermions both have spins 0 and 1/2 here. They are all "Wess-Zumino" multiplets. Gauge particles have spins 1 and 1/2.

Higgs interactions:
(U*).Hu.Q, (D*).Hd.Q, (N*).Hu.L, (E*).Hd.L, Hu.Hd (self-interaction) + corresponding interactions of antiparticles
The * denotes an antiparticle here.

[lpetrich]

Research into elementary entities had produced some zoos before the Standard-Model zoo: the chemical elements, the atomic nuclei, and the hadrons. As they were discovered, some patterns became evident in them, and those patterns were eventually explained as results of relatively simple underlying causes.

Sure enough, one can find some patterns in the Standard-Model zoo. A notable one is this expression for the weak hypercharge:

WHC = (integer) + WIS - (1/3)*(QCD-multiplet triality)

WIS is 0 for WI singlets, 1/2 for WI doublets, and 1 for WI triplets like the W particle (unbroken SM).

QCD-multiplet triality is something that adds modulo 3 for combined multiplet, and for QCD multiplets, it has these values:
Colorless (1) -- 0
Quarks (3) -- 1
Antiquarks (3*) -- 2 or -1
Gluons (8) -- 0

Hadrons have total triality 0, the colorless value.


This is one of the inspirations of particle physicists' quest for Grand Unified Theories, theories in which all the elementary particles fit into a few large multiplets. Another one is what happens when one extrapolates the gauge particles' "charges" to very high energies. One finds that they converge at interaction energies around 10^(16) GeV.

SOFTSUSY Homepage – Hepforge: MSSM gauge unification: gauge unification (EPS file), zoom into convergence area (EPS file).

This suggests that the thee gauge particles, the gluon, W, and B, are parts of a single gauge particle's multiplet, and that that multiplet had gotten split up by symmetry breaking. Particle physicists have come up with several possibilities, and I will list some of them here.

The smallest one is the Georgi-Glashow theory, with gauge-symmetry group SU(5). Here is its particle content

Mult = multiplet type

[table]Particle | Hand | Spin | Mult | Composition
Gauge | - | 1, 1/2 | 24 | gluon + W + B + (3,2,-5/6) + (3*,2,5/6)
up Higgs | L | 0, 1/2 | 5 | Hu + (3,1,-1/3)
(anti) | R | 0, 1/2 | 5* | Hu* + (3*,1,1/3)
down Higgs | L | 0, 1/2 | 5* | Hd + (3*,1,1/3)
(anti) | R | 0, 1/2 | 5 | Hd* + (3,1,-1/3)
elem ferm | L | 1/2, 0 | 1 | N*
(anti) | R | 1/2, 0 | 1 | N
elem ferm | L | 1/2, 0 | 10 | Q + U* + E*
(anti) | R | 1/2, 0 | 10* | Q* + U + E
elem ferm | L | 1/2, 0 | 5* | L + D*
(anti) | R | 1/2, 0 | 5 | L* + D[/table]

Elementary fermion = elem ferm
Additional particles: (QCD multiplet, WIS multiplicity, WHC)

Higgs interactions: F(10).H(5).F(10) ... F(5*).H(5*).F(10) ... F(1).H(5).F(5*) ... F(10).H(5*).F(5*) ... H(5).H(5*)
F = elementary fermion, H = Higgs particle
Gives mass unification for the bottom quark and the tau lepton, at least.

The additional (3,2,-5/6) gauge multiplet contains particles with electric charges -4/3 and -1/3, particles that can turn quarks into antiquarks and leptons. The additional down-quark-like (3,1,-1/3) Higgs multiplet can also do that. However, their masses must be forced up to GUT energy levels by GUT symmetry breaking, or else protons would decay too fast.

Protons decaying? That's an outcome of the Georgi-Glashow GUT and most other GUT's. Not only isolated protons can decay in this fashion, but also isolated neutrons, and protons and neutrons in nuclei. Proton decay has been searched for in several experiments, with lower limits on the proton's half-life now greater than 10^(33) years. That is close to what one would expect for a GUT energy scale of about 10^(16) GeV, so we are getting close to detecting a GUT effect, if it happens.

[lpetrich]

Physicists have used several other gauge-symmetry groups to construct GUT's with, and some of them go further than Georgi-Glashow in unifying the partiles of the Standard Model. The simplest that puts each generation of elementary fermions into one multiplet uses the symmetry group SO(10).

It breaks down into SU(5) * U(1), where the U(1) is for an additional hypercharge-like factor that's related to B - L: (baryon number) - (lepton number).

[table]Particle | Hand | Spin | Mult | Decomposition
gauge | - | 1, 1/2 | 45 | (24,0) + (10,-1) + (10*,1) + (1,0)
Higgs | L | 0, 1/2 | 10 | (5,-1/2) + (5*,1/2)
(anti) | R | 0, 1/2 | 10 | (5,-1/2) + (5*,1/2)
elem ferm | L | 1/2, 0 | 16 | (1,5/4) + (10,1/4) + (5*,-3/4)
(anti) | R | 1/2, 0 | 16* | (1,-5/4) + (10*,-1/4) + (5,3/4)[/table]

Higgs interactions: F(16).H(10).F(16) (only one kind of EF-Higgs term!) ... H(10).H(10)
Thus, SO(10) successfully unifies the masses of the elementary fermions, but in a way, it is too successful. It has no generation mixing, and thus, no cross-generation decays. So generation mixing must be the result of something that breaks the SO(10) symmetry.


Another feature of SO(10) is right-handed neutrinos. These are not much in evidence at lab-accessible energies, like in neutrino oscillations. In fact, the neutrinos that we see likely have "Majorana" masses. They'd be purely left-handed, and flipping their spins would turn them into their antiparticles.

Charged elementary fermions, however, have "Dirac" masses due to their Higgs-particle interactions, masses that connect their left-handed and right-handed parts.

If neutrinos get their masses the way the charged EF's do, then there is a curious problem. Their Higgs interactions must be smaller than the others' Higgs interactions by a factor of 10^7 to 10^11, which seems odd.


There is an ingenious solution called the "seesaw mechanism". In it, the neutrinos get "normal" Dirac masses like the other EF's, but the right-handed neutrinos get very large Majorana masses. These effects combine to produce Majorana-ish neutrinos with very large and very small masses. Here is how it works out mathematically for one generation. Here's the appropriate wave equation for a stationary particle:

i*(d{nuL,nuR}/dt) = {m*nuR, m*nuL + M*nuR}

hbar = 1, m is the "normal" Dirac mass, and M is the much larger right-handed Majorana mass. It can be separated out:

Putting in variation nuX ~ exp(-i*E*t), we get

E*{nuL, nuR} = {m*nuR, m*nuL + M*nuR}

It has two solutions:

E*(nuR + (m/M)*nuL) = (M + m²/M) * (nuR + (m/M)*nuL)
E*(nuL - (m/M)*nuR) = - (m²/M) *(nuL - (m/M)*nuR)

For three generations, the masses m and M become 3*3 matrices, but the solution has a similar form:

E*{nuL, nuR} = {m⁺.nuR, m.nuL + M.nuR}

E*(nuR + M^-1.m.nuL) = (M + M^-1.m⁺.m) . (nuR + M^-1.m.nuL)
E*(nuL - m⁺.M^-1.nuR) = (- m⁺.M^-1.m) . (nuL - m⁺.M^-1.nuR)

m⁺ is the Hermitian conjugate of m (transpose complex conjugate), and M = M⁺. The mass matrices can have complex values because of the possibility of CP violation.

We find mostly-right-handed and mostly-left-handed neutrinos, with the mostly-right-handed ones having close to the right-handed ones' Majorana masses, and the mostly-left-handed one having tiny masses.

Taking m ~ 10 GeV, plausible for the most massive neutrino with a "normal" mass, and taking observed masses around 0.01 eV or 10^(-11) GeV, we get M ~ 10^(13) GeV. That is rather close to GUT energies of about 10^(16) GeV.


So neutrino masses and neutrino mixing can offer additional access to GUT-scale physics.

[lpetrich]

Some words on abstract-algebra groups, something that one sees a lot of in quantum mechanics. They are composed of a set of entities and a binary operation, an operation that takes two of them and returns one of them. It has these properties:

Associativity: a*(b*c) = (a*b)*c
Identity e: a*e = e*a = e
Inverse: a*inv(a) = inv(a)*a = e

If the operation is commutative, a*b = b*a, then the group is called abelian, after Norwegian mathematician Niels Henrik Abel. Otherwise, it's nonabelian.

All the finite abelian groups are known, and also all the finite "simple" groups, those without subgroups with certain nice properties. Among infinite groups, some are continuous, and those that are one-piece continuous and differentiable are called "Lie groups" ("Lee"), after Norwegian mathematician Sophus Lie.


Lie groups are generated by "Lie algebras", and it's often easier to work with Lie algebras than the groups that they generate. We've already run into some Lie algebras here, like the angular-momentum / rotation operators. Lie algebras are defined as a set of operators L and commutation on them:
[L_i,L_j] = L_i/L_j - L_j.L_i = f_ij,kL_k

where the f's are the "structure constants" of the algebra. If the commutation results span the entire algebra, then the algebra is called "semisimple". If a semisimple algebra cannot be expressed as the sum of semisimple algebras that commute with each other, then it is "simple".


First consider O(n), the group of rotations and reflections in n dimensions. It is not a Lie group because one cannot bridge the gap between pure rotations and rotation-reflections in it. That is evident from the matrices' determinants, which is 1 for pure rotations and -1 for rotation-reflections.

However, the pure rotations form SO(n), which is a Lie group.

Their complex counterparts, U(n) and SU(n), are both Lie groups. For U(n), the matrix determinants are complex numbers with magnitude 1, and are thus continuous with 1.


The algebras are often given the same designation as the groups, which can cause confusion. However, some authors use different typography for the names of the groups and the algebras.

Let's see what algebras are simple and what are not.
SO(1) ~ SU(1) -- empty algebra
SO(2) ~ U(1) -- not semisimple. One generator L: [L,L] = 0
SO(3) ~ SU(2) -- simple
SO(4) ~ SU(2) * SU(2) -- semisimple but not simple
SO(5) -- simple
SO(6) ~ SU(4) -- simple

Simple ones: SO(3), SO(n) for n >= 5, SU(n) for n >= 2

In addition to SU(n) and SO(n), there is another infinite family of algebras, Sp(2n), the "symplectic" ones. But they don't get much use.

There are 5 additional simple Lie algebras, the "exceptional" ones: G2, F4, E6, E7, E8.

Of these, E6 and E8 appear in some GUT's.

[lpetrich]

Where can we find Lie groups and Lie algebras? Rotation groups and gauge symmetry groups are all Lie groups -- they are continuous, differentiable, and connected. So to be a good theoretical particle physicist, one must understand Lie groups and Lie algebras.

Someone pointed out that the rotation and reflection group O(n) ought to be called a Lie group. But one has to relax the connectedness condition to do so.


How does this discussion of Lie algebras relate to the multiplet states that we've seen here? Like angular-momentum direction states and QCD states.

The connection is group representations. These are sets of matrices that act according to the group's operation. A representation or rep is reducible if one can turn it into block-matrix form, where each block is a smaller-sized rep. If it is not possible, the rep is an irreducible rep or irrep.

Finite groups have a finite number of irreps, and infinite groups an infinite number. Lie groups have reps and irreps, of course, and the idea of them carries over into Lie algebras.

Irreps may be designated in various ways, like the size of the matrices in it, a common particle-physicist approach. However, some irreps have the same size of matrices, so one may have to note additional info. Also, one may use various shortcuts to avoid constructing the irreps' matrices. For Lie algebras, one can construct the irreps' basis vectors, generalizing the construction of angular-momentum states X(j,m) for m = -j to +1 by integer steps.

Some irrep conventions for angular momentum j:
Size: 2j+1
Highest-weight vector: {2j} (unambiguous though arcane)

If one uses the algebra generators as a basis, one gets the "adjoint representation". Gauge particles are always in the adjoint rep, since a gauge multiplet has one member per generator.


The reps of SU(n) can be expressed in a nice form.

Its "fundamental rep" is a vector with n components, and the other reps are tensors, like vectors, but with multiple indices. A m-tensor, one with m indices, has n^m components. A scalar (indexless quantity) is a 0-tensor and a vector is a 1-tensor.

Let's see what the next ones are. If one combines two vectors with an "outer product", one where their indices are separate, one gets a 2-tensor. One must use a general 2-tensor, because quantum states often mix when they can.

A 2-tensor is indeed a rep of SU(n), but unlike the vector one, it is not an irreducible one. That's also true of all larger tensors. But the 2-tensor can be broken down into two irreducible ones, a symmetric one and an antisymmetric one, with sizes n(n+1)/2 and n(n-1)/2. The symmetry is over interchange of the indices. Having a symmetry giving irreducibility is true in general.
Symmetric: T(k2,k1) = T(k1,k2)
Aantisymmetric: T(k2,k1) = - T(k1,k2)

One can combine three vectors to get a 3-tensor, One gets symmetric and antisymmetric parts, and also two mixed-symmetry ones, with sizes n(n+1)(n+2)/6, n(n-1)(n-2)/6, and two n(n^2-1)/3. That is true of all larger tensors, though they have more mixed-symmetry ones.

The adjoint rep is more complicated, but it always has size (n^2-1) -- it's (vector * (conjugate vector)) - scalar.

One can combine higher-order tensors as well as vectors, but it's more complicated. One can use a nice graphical procedure with "Young diagrams", the "Littlewood-Richardson rule".


Let's put this to work.

First on particle spins: SU(2). The fundamental rep corresponds to a spin-1/2 particle, size 2 and all. Combining 2 of them yields a symmetric state with spin 1, size 3, and an antisymmetric state with spin 0, size 1. Combining 3 of them yields a symmetric state with spin 3/2, size 4, two mixed ones with spin 1/2, size 2, and no antisymmetric ones. All this agrees with what one gets with the rules of adding quantum-mechanical angular momentum.

Now for quark colors: SU(3). A single quark has 3 colors, from vector size 3. Two quarks together can have symmetric size 6 or antisymmetric size 3. The latter one is a conjugate of the vector one, as is evident by multiplying its 2-tensor by the antisymmetric symbol eps(i1,i2,i3) (n indices for n dimensions). It's the inner product, with indices matched and summed over. Three quarks together can have symmetric size 10, two mixed sizes 8, or antisymmetric size 1. Thus, three quarks that are antisymmetric in color are colorless, which is what baryons are.

That also works for quarks' 3 light flavors, getting the number of flavor states of the baryon multiplets. Symmetric: 10 (spin 3/2), mixed: 8 (spin 1/2).

For SU(5), the elementary fermions are in states 1, 5, 10, 10*, 5*, 1. These are all antisymmetric tensors, with numbers of indices 0, 1, 2, 3, 4, 5. The 0 one is a scalar, while the 1 one is a vector. The antisymmetric symbol has 5 indices, and it turns a 3-tensor into a conjugate of the 2-tensor, a 4-tensor into a conjugate of the vector, and a 5-tensor into a scalar.

The Higgs particles, however, are in 5 and 5*, the vector and its conjugate.


One can do SO(n) in a similar fashion, but one has to subtract out the identity matrix and separate out the subtractions as additional states. Thus, the symmetric 2-tensor T splits into
tensor T - (1/n)*I*(T.I), size (n-1)(n+2)/2
and a scalar multiplied by I.

Also, multiplying by the antisymmetric symbol gets an equivalent state, and for SO(2n), an antisymmetric n-tensor gets split into 2 equal-sized parts.

The adjoint rep of SO(n) is easy: an antisymmetric 2-tensor. For SO(3), it's equivalent to a vector. That's related to angular momentum being defined with a cross product.

SO(n) also has "spinor" ("spin vector") reps, where a spinor generalizes spin-1/2 spin states. For SO(2n) and SO(2n+1), one needs an outer product of n spin-1/2 states, getting size 2^n. In SO(2n), the spinor gets split into 2 irreducible ones, both with size 2^(n-1), while in SO(2n+1), the spinor is irreducible.

Thus, for SO(10), the Higgs particles are in its vector rep, size 10, and the elementary fermions in its spinor reps, size 16. Spinors in both spin and gauge symmetry.


The symplectic one, Sp(2n), works much like SO(n), but with no spinors, vector size 2n, and subtracting out an antisymmetric tensor instead: {{0,-I},{I,0}} where I has size n*n.


I don't know of any simple way to work with the exceptional Lie algebras' reps.

[lpetrich]

For handling symmetry breaking, it's necessary to consider a Lie algebra's subalgebras and how the algebra's reps decompose in them. But I can't think of a simple way to explain how that works, even in special cases like SU(n).

Back to GUT's.

So far, we have
SU(5) -> Standard Model, SU(3)*SU(2)*U(1)
Gauge: adjoint: 24
Higgs: vectors: 5, 5*
Elementary fermions: antisymmetric tensors: 1, 5, 10, 10*, 5*, 1

SO(10) -> SU(5)*U(1)
Gauge: adjoint: 45
Higgs: vector: 10
Elementary fermions: spinor parts: 16, 16*

Another breakdown of SO(10) gives the Pati-Salam model:
SO(10) -> SO(6)*SO(4) ~ SU(4)*SU(2)*SU(2)
It's notable for treating the leptons as a fourth color of quark, one that got separated by symmetry breaking.

The Salam is Abdus Salam, co-developer of the theory of electroweak unification.

The SU(4) breaks down into SU(3)*U(1), one of the SU(2)'s into U(1), and the two U(1)'s mix to get weak hypercharge and (baryon)-(lepton).


Can we go further? Yes, with the exceptional Lie algebra E6.
It breaks down E6 -> SO(10)*U(1)
It has yet another hypercharge-like factor.
[table]Particle | Hand | Spin | Mult | Decomposition
Gauge | - | 1, 1/2 | 78 | (45,0) + (16,-1) + (16*,1) + (1,0)
EF, Higgs | L | 0, 1/2 | 27 | (16,1/3) + (10,-2/3) + (1,4/3)
(anti) | R | 0, 1/2 | 27* | (16*,-1/3) + (10,2/3) + (1,-4/3)[/table]
27 = fundamental, 27* = its conjugate, 78 = adjoint

The Higgs particles can live alongside a generation of elementary fermions! That may explain why some of them are much massive than the others. However, the other generations' Higgs particles have to have GUT-scale masses, and this unification requires supersymmetry to work.

But the interactions are at least partially correct. From a 27 multiplet X = F + H + S (elementary fermions, Higgs particles, extra scalar), one gets
X³ ~ F.F.H + H.H.S

The first term is the EF-Higgs interaction term that we've been getting for the Standard Model and its other supersets here. The second one is a modification of the Higgs self-interaction H.H, since one can't get H.H alone from E6. The S appears in some (relatively) low-energy theories, like the Next-to-Minimal Supersymmetric Standard Model or NMSSM. The S makes 2 additional neutral Higgs particles and 1 additional neutralino, making 5 each.


E6 can break down to the Standard Model by another route, "trinification":
E6 -> SU(3)*SU(3)*SU(3) -> Standard Model

[lpetrich]

Despite all the successes of the Standard Model and plausible extrapolations to Grand Unified Theories, there is still one big difficulty for quantum mechanics.

Gravity.

So far, it has been VERY difficult to construct a quantum theory of gravity, despite many valiant efforts.

As with quantum mechanics in general, one must get the appropriate classical-mechanical theory in the classical limit, the limit of Planck's constant -> 0.

That theory is general relativity and possible extensions like the Generalized Brans-Dicke theory. GBD includes an additional field, a scalar field that makes the gravitational constant vary. However, one can make the amount of variation arbitrarily small, making GBD arbitrarily close to GR.

The Confrontation between General Relativity and Experiment describes several GR tests -- and GR passes to within experimental limits. The delay of the Cassini spacecraft's radio signals near the Sun agrees with GR to within 1 part in 10^5. That's 1 part in 100 thousand.


GR states that gravity is due to the distortion of space-time by the matter / energy / momentum / pressure in it. How does one quantify distortion of space-time? Here is how it works; I'll try to make it simple by doing the 2-dimensional case.

The distance between points 1 and 2 in rectangular coordinates {x,y} is, from Pythagoras's theorem,
s₁₂² = (x₂ - x₁)² + (y₂ - y₁)²

Going to polar coordinates {r,a} gives
s₁₂² = r₁² + r₂² - 2*r₁*r₂*cos(a₂ - a₁)

Other coordinate systems can give even more complicated results. But if one does only small coordinate differences dx, dy, dr, da, one finds for the distance ds:
ds² = dx² + dy²
ds² = dr² + r²*da²

Much simpler expressions, and easy to generalize:
ds² = g_xx*dx² + 2*g_xy*dx*dy + g_yy*dy²

where the g's are functions of x and y. The g's form the "metric tensor" of this space. GR treats the metric tensor of space-time as the gravitational potential.

Note that one can do arbitrary coordinate changes and still get a metric tensor. However, gradients of vectors and tensors have a problem. They do not transform very nicely with coordinate changes. They get extra terms, much like what one sees for position-dependent gauge transformations in gauge-field theories. So one has to introduce "connection coefficients" to cancel them out:
(Covariant derivative) = (ordinary derivative) + (connection coefficients) * (original quantity)

The connection coefficients are set by setting the covariant derivative of the metric tensor to zero -- one uses the metric tensor, because, as we've seen, it gives the shape of space-time.

How does one get a well-defined field strength? By doing what one does with gauge fields and finding the commutator of covariant derivatives. The result is the "Riemann tensor", which gives the amount of curvature. For 2D, it reduces to a single value, but for more dimensions, one gets more values.


Note that GR behaves much like like a gauge theory, much like QCD and electroweak unification. In fact, it can be interpreted as a gauge theory of space-time, where the gauge transformations are the coordinate transformations.

[lpetrich]

This gauge nature is not as helpful as one might hope, because there are serious difficulties with quantizing GR.

If one takes some space-time and finds a perturbation of it, and one ignores that perturbation's self-interactions, there is no big trouble with quantization. That perturbation is a field with spin 2, the "graviton".

It's those self-interactions that cause the trouble.

They makes the field equations nonlinear, and nonlinear systems are MUCH more difficult to solve than linear ones.


Particle physicists have had similar problems with Standard-Model interactions, but they have been able to solve those problems there and in many extensions of the SM. They have even learned how to recognize where their solutions work and where those solutions don't.

The first success was by Richard Feynman and his colleagues in the 1950's with the quantum field theory of electromagnetism, quantum electrodynamics or QED, a success which won RF and two of his colleagues a Nobel Prize in physics. As a photon travels, it can turn into and electron and a positron, and they can recombine to make that photon again. Likewise, as an electron travels, it can emit a photon and then re-absorb it. The "Feynman diagrams" that represent these events have loops in them; the lowest-order, loopless ones, are called "tree-level". These particles' comings and goings get multiplied ad infinitum, loops and all.

To get quantities like masses, interaction strengths, and field strengths, one has to integrate over those loops, and those integrals diverge at high energies. That was a very serious problem until RF and his colleagues found a solution. They noticed that one does not observe "bare" quantities, but instead those quantities with loop effects added on:
g(observed) = g(bare) + g(loop effects)

They decided to define g(bare) using values at some energy scale E:
g(bare) = g(observed at E) - g(loop effects at E)

Thus,
g(observed) = g(observed at E) + (g(loop effects) - g(loop effects at E))

A procedure called "renormalization".

When they did the calculations, they discovered that the difference terms are well-behaved, with divergences subtracted out. Thus, QED is renormalizable.

Physicists then discovered that several other sorts of field theories are renormalizable, including QCD and electroweak -- the entire Standard Model is renormalizable, and many of its extensions are.


However, some sorts of interactions cannot be renormalized, and a traditional example is Enrico Fermi's original theory of the weak interaction. It features a contact interaction between all the particles involved. But when one tries to do loop integrals with it, those integrals diverge uncontrollably at energy scales more than a few hundred GeV, and that is why this theory cannot be renormalized.

That was one of the motivations behind electroweak unification, and that theory can be renormalized. It also has the interesting consequence that the weak interaction is nonlocal, with a size scale of about the Compton wavelength of the W particle, about 10^(-17) m. Not zero, but small even when compared to a nucleon's size of about 10^(-15) m.


Returning to gravity, when one tries to do calculations of graviton loops in GR, one gets uncontrollable divergences past the quantum-gravity "Planck energy", about 10^(19) GeV. This means that straightforwardly quantized GR is not renormalizable.



Quantum gravity has an additional problem: time. In Newtonian space-time, time is well-defined and universal, and one can construct quantum mechanics in it without much trouble. But in special relativity, time is relative in the way that space is relative. That causes serious problems in constructing quantum mechanics, and that's what's led to quantum field theory. Because of its curved space-time, GR is even worse. While individual particles have well-defined times, time is not very well-defined globally.

[lpetrich]

Given the problems with quantizing GR in isolation, let us see if we can add something to GR that would make our life easier. Ideally, one would want some particles with a special relationship with the graviton. What might result in such a relationship? Being associated with some suitable symmetry.

The graviton is associated with space-time symmetry, and some extra particles ought to have some symmetry that interacts with space-time symmetry.

However, no other known elementary-particle symmetry can do so, and according to the "Coleman-Mandula theorem", no similar ones can. Gauge symmetries and space-time symmetries do not interact, gauge-symmetry generators are space-time scalars, and space-time symmetry generators are chargeless / colorless.

But there is a way to get around that theorem. It holds for symmetry generators that act like bosonic operators, like the generators of space-time and gauge symmetries. But what about symmetry generators that act like fermionic operators?

These generators generate "supersymmetry" or SUSY. They interact with space-time symmetry generators, but not with any other sort of symmetry generators, like gauge-symmetry ones. Thus, the Coleman-Mandula theorem is partially rescued as the "Haag–Lopuszanski–Sohnius theorem".

SUSY generators relate particles / fields with different spins, turning a particle into one with a spin differing by 1/2. In fact, from SUSY, the number of bosonic particle modes (integer spin) must equal the number of fermionic ones (half-odd spin). So could SUSY account for the spin-1/2 particles that are known to exist?

-

There are some problems with that. All the members of a SUSY multiplet must interact in the same way with other particles, they must all have the same quantum numbers in relation to those particles, and they must all be in the non-space-time symmetry multiplets.

For the Standard Model, it is not difficult to show that no known particle has another known particle as a SUSY partner. That's what's motivated SUSY extensions of it like the MSSM. But no non-SM MSSM particle has been detected, and if SUSY is real, then it must be broken at energy scales of at least 1 TeV (1000 GeV).

Turning to gravity, GR makes space-time symmetries local, that is, varying by position. This means that if SUSY is real, it must also be local. The result is supersymmetric gravity or supergravity or SUGRA.

The simplest possible SUGRA theory has these particles:
Graviton: spin 2
Gravitino: spin 3/2

More fancy ones include additional particles, with spins 1, 1/2, and 0.

The gravitino interacts with gravitational strength, the strength of the graviton's interactions. That makes both particles *very* hard to detect in particle-accelerator experiments. Gravity shapes everything large-scale in the Universe because the graviton is massless and because the gravitational "charge" is always positive, but the gravitino are another story. SUSY breaking will give it a nonzero mass, though estimates of that mass vary widely. However, most estimates of its mass are too large to make it have much effect. If the gravitino's mass is 1 eV (10^(-9) GeV), then the gravitino's range will be about a micron. Even worse, if a particle emits or absorbs a gravitino, it will change into its SUSY partner, and since known particles' SUSY partners have masses at least a few hundred GeV, that makes it difficult for gravitinos to interact with other particles. So gravitino effects will be insignificant at macroscopic scales.

[lpetrich]

When supergravity was worked out in the 1970's, it seemed like a route to a Theory of Everything. All one needs is a large-enough multiplet, and one can get the Standard Model out of the lower-spin particles.

But more work on this question revealed it to be a vain hope. It was not possible for supergravity in our 4 space-time dimensions. But what about more space-time dimensions? Even though we don't observe more than 4 of them.

A hint was discovered by Theodor Kaluza in 1921 and Oskar Klein in 1926, working with general relativity. They imagined that space-time has 5 dimensions, 1 time and 4 space, and that one of the space dimensions is curled up in a very tiny circle. If a field's behavior is to be unaffected by this "compactification", then it must be constant over that small dimension. But despite that constancy, if a field has built-in directions (vector, tensor indices), then it will get split into large-direction and small-direction fields.

The gravitational field's metric g_ij gets split up into 3 parts:
All large dimensions: g_ij -- familiar metric tensor
Large + small: g_iS = g_Si -- a 4D vector
All the small dimension: g_SS -- a scalar
S = the small dimension

The vector behaves something like the electromagnetic potential, also a 4-vector, and that seemed like the first step in unifying gravity and electromagnetism.

It did not work out in this form, but the idea could easily be generalized.


So could supergravity in higher dimensions do it? One can go up to 11 dimensions without running into certain awkward features, and 11-dimensional supergravity seems like plenty. One can compactify over 7 of the dimensions, turning them into a very tiny 7-dimensional ball, with 4 dimensions remaining large.

But that did not work either. There was no way to get the Standard Model out of it.

So a Theory of Everything would have to be supergravity + some superset of the Standard Model.

[lpetrich]

Now for one of the more contentious aspects of particle physics over the last few decades, string theory.

The story started in the late 1960's, when Gabriele Veneziano took a whack at puzzling out the strong interactions. He noted that hadron excited states often have masses that fit onto a straight line:

m² = M₀ + M₁*n

for nonnegative integer n and constants M₀ and M₁

He looked for a simple formula for an interaction amplitude that could produce those mass values as spikes or "poles" in it, and after a lot of experimenting, he found one in 1968. It used the Euler beta function, a function formed from the Euler gamma function, a generalization of the factorial function. His "Veneziano amplitude" is

B(-a(s),-a(t)) = G(-a(s)) * G(-a(t)) / G(-a(s)-a(t))

B = Euler beta function, G = Euler gamma function, a is some approx. linear function, s and t are more-or-less squares of combinations of incoming and outgoing particles' energy-momentum 4-vectors.


There was already a familiar system that produces modes like that: an elastic string. One with ends fixed has vibration modes with wavenumbers (reciprocal wavelengths) n*pi/L, where L is the length and n is a positive integer.

Although it is rather easy to construct Newtonian equations of motion, for relativistic strings, it is more difficult. The first effort, the "Nambu-Goto action" was rather awkward, but the "Polyakov action" was easier, though at the price of introducing extra internal parameters. These actions essentially minimize the area of a 2D sheet in space-time. One internal dimension is timelike and the other is spacelike.

For the simplest sheet topologies, one finds these masses/energies:
m² = m₀² * ( sum over positive k, transverse dimensions i of k*n_k,i )

where m₀ is the string mass scale ~ square root of the string tension. The transverse space-time dimensions are those other than time and one space dimension - the dimensions where the string oscillates.


Looking back to the theory of the strong interactions, an excited hadron may get a stringy shape, with the gluons forming a string between (anti)quarks on each end. Thus, that stringy spectrum of energy states.


A string can certainly have more complicated topologies. They correspond to interacting strings and the string equivalent of loop diagrams - strings with holes in their space-time sheets.

A remarkable result of string theory is that loop diagrams are well-behaved at high energies, not getting awkward infinities. This suggests that string theory could be a good Theory of Everything.


There are serious problems, however. This "bosonic string" is only quantum-mechanically self-consistent in 26 space-time dimensions, and its energies get pushed down by quantum-mechanical effects to make the lowest-energy mode a particle with imaginary mass, a tachyon, a faster-than-light particle. A field theory of FTL particles has certain awkward properties, like instability at low wavelengths.

Being bosonic is an additional difficulty; how does one add fermions? The solution is to add supersymmetry to the string, making it a superstring. It turns out that there are 5 superstring possibilities, and that they are all quantum-mechanically self-consistent in 10 dimensions. However, they have no tachyons.

It was later discovered that these 5 theories are related by "dualities", and that their low-energy limits could be derived from 11-dimensional supergravity by shrinking one of the dimensions. That's suggested something called M-theory, something that has those superstring theories and 11-D SUGRA as limits, but something whose nature continues to be obscure.

[lpetrich]

Let's see what particle content the 5 superstring theories have. Since the massive modes likely have masses at least the Planck mass, it's the massless modes that we need to consider.

Type I: Type I supergravity:

Graviton, gravitino, antisymmetric (AS) Kalb-Ramond 2-tensor, dilatino (spinor; like spin-1/2), dilaton (scalar)

Can have a gauge field, but the gauge-symmetry group is restricted to SO(32).

Type IIA: Type II supergravity:

Type I, with additional gravitino and dilatino modes, and a vector and AS 3-tensor field.

Type IIA: Type II supergravity:

Type I, with additional gravitino and dilatino modes, and a scalar, an AS 2-tensor, and 1/2 of an AS 4-tensor field.

HO Heterotic: Type I supergravity and a SO(32) gauge field

HE Heterotic: Type I supergravity and a E8*E8 gauge field

E8 is an "exceptional Lie algebra", and the largest one. It also has the property that its fundamental representation, the rep that one can derive every other rep from, is its adjoint rep. This rep has dimension 248.


Let's count particle modes.

For massive particles, their directions can vary over D - 1 dimensions, out of D space-time dimensions. That's because time cancels one of them out. For massless ones, the number of transverse dimensions is D - 2, because both the time and a space dimension cancel out directions.

For the graviton, one has to find a symmetric tensor in the transverse directions and subtract out an identity mode. For n-D, that's
n(n+1)/2 - 1 = (n-1)(n+2)/2

The gravitino is a vector-spinor, and
(vector) * (spinor) = (vector-spinor) + (spinor)
Thus, (n-1)*(spinor dimension)

Spinor sizes are complicated.
Even n: two, with 2^(n/2-1)
Odd n: one, with 2^((n-1)/2)

Antisymmetric m-tensors are easy: n(n-1)(n-m+1)/m!


For massless particles in 4D space-time, the spins reduce to helicities, projections along the direction of motion.
B = boson, F = fermion
B: Graviton (spin 2): +- 2
F: Gravitino (spin 3/2): +- 3/2
B: Gauge field (spin 1): +- 1
F: Elementary fermion (spin 1/2): +- 1/2
B: Scalar (spin 0): 0

Simple supergravity:
B: gravition (spin 2) = 2
F: gravitino (spin 3/2) = 2

SUSY gauge theory:
B: gauge particle (spin 1) = 2
F: gaugino (spin 1/2) = 2

SUSY Wess-Zumino multiplet (elementary fermions, Higgs particles):
B: scalar (spin 0) = 2
F: EF (spin 1/2) = 2
For WZ, one needs a complex scalar field, and thus 2 scalar modes.


Now to 10D space-time, with 8 transverse dimensions.

Type I supergravity:
Graviton = 35
Kalb-Ramond AS 2-tensor = 28
Dilaton scalar = 1
-- Bosons = 64
Gravitino (vector-spinor) = 56
Dilatino (spinor) = 8
-- Fermions = 64
(one of the 2 8D spinor modes)

Type IIA supergravity additions to Type I:
Vector = 8
AS 3-tensor = 56
-- Bosons = 64
Gravitino (vector-spinor) = 56
Dilatino (spinor) = 8
-- Fermions = 64
(the other of the 2 8D spinor modes)

Type IIB supergravity additions to Type I:
Scalar = 1
AS 2-tensor = 28
Half of AS 4-tensor = 35 (split by antisymmetric 8-symbol)
-- Bosons = 64
Gravitino (vector-spinor) = 56
Dilatino (spinor) = 8
-- Fermions = 64
(the other of the 2 8D spinor modes)

Gauge theory:
B: Gauge particle (vector) = 8
F: Gaugino (spinor) = 8
(one of the 2 8D spinor modes)


Finally, let's look at 11D supergravity, with 9 transverse dimensions. It has a graviton, a gravitino, and an antisymmetric 3-tensor.
Graviton = 44
AS 3-tensor = 84
-- Bosons = 128
Gravitino = 128
-- Fermions = 128

[lpetrich]

So from string theory, we have supergravity and gauge theories. Supergravity won't get us the Standard Model, so can the superstring gauge theories do so?

We find two possible gauge groups that can contain the Standard Model and various GUT's, SO(32), and E8*E8. SO(32) is the group of rotations of a 32-D space, but E8 has no such simple interpretation.

Types IIA and IIB superstrings have no gauge fields, so they are out.

SO(32) contains SO(10) as a subgroup, so it seems like a possible candidate. But since the only particles subject to the gauge group are the gauge fields, we must consider what we can get out of the adjoint rep. It's an antisymmetric 2-tensor, so one can only get vectors and AS 2-tensors out of it. No spinors. So Type I and HO superstrings are out.

However, E8 is much more promising. Let's break it down to E6*SU(3):

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

Fundamental and adjoint of E8 ->
Adjoint of E6
Fund of E6 * fund of SU(3)
Fund conj of E6 * fund conj of SU(3)
Adjoint of SU(3)


So one gets not only the adjoint of E6, but also its fundamental rep, 27, and that rep's conjugate, 27*. Just what one needs to be a superset of the Standard Model's symmetry group and reps.


But these fields live in 10 space-time dimensions, and not 4, so 6 space dimensions must be compactified into a Planck/GUT-sized tiny ball. This compactification interacts with one of the gauge E8's, breaking it to E6 * SU(3) or something similar, and even further to the Standard Model's symmetry. Which particles are present at low energies depends on the zero modes of this tiny ball, and that's in turn dependent on its topology. Physicists have investigated numerous possible topologies, notably various "Calabi-Yau" spaces and "orbifolds". It's possible to get much of the Standard Model in this way, like its multiplet structure, but I have not seen as much success in getting its parameter values.


Let's see how the various 10D space-time multiplets split up to 4D and 6D ones. Each one splits into sets of (4D space-time multiplet, 6D space multiplet)

Bosonic:

• Scalar -> (scalar,scalar)
• Vector -> (vector,scalar) + (scalar,vector)
• Graviton -> (graviton,scalar) + (scalar,graviton) + (vector,vector) + (scalar,scalar)
• Adjoint -> (adjoint,scalar) + (scalar,adjoint) + (vector,vector)

In 4D,

• Scalar = spin 0 -- dilaton, part of 10D gauge particle, 10D graviton, 10D KR field
• Vector = spin 1 -- gauge particle, part of 10D graviton, 10D KR field
• Graviton = spin 2
• Adjoint = spin 1 -- part of 10D KR field

KR = Kalb-Ramond antisymmetric 2-tensor field in 10D supergravity.

Fermionic:

• Spinor1 -> (spinor1,spinor1) + (spinor2,spinor2)
• Spinor2 -> (spinor1,spinor2) + (spinor2,spinor1)
• Gravitino1 -> (gravitino1,spinor1) + (gravitino2,spinor2) + (spinor1,gravitino1) + (spinor2,gravitino2) + (spinor1,spinor1) + (spinor2,spinor2)
• Gravitino2 -> (gravitino1,spinor2) + (gravitino1,spinor2) + (spinor1,gravitino2) + (spinor1,gravitino2) + (spinor1,spinor2) + (spinor2,spinor1)

Spinor1,spinor2 and gravitino1,gravitino2 are the splittings of the spinor and the gravitino fields, because 4, 6, and 10 are even numbers of space-time dimensions.
In 4D,

• Spinor = spin 1/2 -- gaugino, dilatino, part of 10D gravitino
• Gravitino = spin 3/2

So the 10D supergravity and gauge fields split up into several particles, with the 4D parts having spins 0, 1/2, 1, 3/2, and 2.

Which of all these particles survive to low energies depends on what zero modes their 6D parts have, and I can't find anything even halfway simple on that.


But one can get *all* the Standard-Model particles from one of the E8 gauge particles, by splitting the 10 space-time dimensions into 4 large and 6 small ones with some suitable topology. This multiplet not only produces all the gauge particles, but also the Higgs particle and all the generations of elementary fermions.

This raises the question of what happens to the other E8 gauge particle. I've seen a few speculations, but that's about it.