Standard Model

We have to go up the following steps:

• Lagrangian Mechanics for particles. A Lagrangian function determines the dynamics of the system.
• Lagrangian field theory (classical field theory). The same, but we deal with infinite degrees of freedom, a continuum of particles. They correspond to matter fields, and the "forces" affecting the matter are encoded into the Lagrangian itself.
• Gauge theory. The matter fields are recognized as sections of a certain bundle, instead of functions. We modify the notion of differentiation (connections), so we reorganize the Lagrangians: some ugly terms of the Lagrangian of the previous step are included inside the connection. Moreover, the curvatures of the connections encode the forces". They are also called fields (gauge strength tensor or gauge field or something like that). We have two Lagrangians, one for the matter field and other for the connection. See also the note quantum particle.
• Quantization. The matter fields (sections of a bundle) are promoted to operator-valued fields (fermions). Also, the gauge potentials $A_{\mu }$ are promoted to a operator (bosons).

Understanding the Standard Model: $U(1)\times SU(2)\times SU(3)$

(To be reviewed)
This note traces a path from a single particle to the Standard Model of particle physics. Each section cites the notes where the details can be found. Content not present in the vault is flagged with [NOT IN VAULT].

1. A single particle

Everything begins with a particle — a system with finitely many degrees of freedom. Its state is a point in a configuration space, and its evolution is a curve $\alpha :\mathbb{R}\to M$. Physics is introduced through a Lagrangian $L:TM\to \mathbb{R}$, and the physical trajectories are those that extremize the action:

This leads to the Euler-Lagrange equations. The key insight: the Lagrangian encodes everything — the "toll to pay" for passing through a point with a given velocity, as described in Lagrangian Mechanics.

2. From particles to fields

A classical field is the continuum generalization. Instead of a finite number of particles, we imagine a continuous family of "oscillators," one per point of space. The collective state is no longer a curve but a function

where $M$ is spacetime and $V$ is the space of internal states of each oscillator. See the motivating example in Classical Field Theory: a single oscillating particle gives ${\varphi }_1(t)$ (a $(0+1)$-dimensional field); infinitely many give $\varphi (x,t)$ (a string, a $(1+1)$-dimensional field).

The dynamics are governed by a Lagrangian density $L(\varphi ,{\partial }_{\mu }\varphi )$, and the physical fields extremize the action $S[\varphi ]=\int L{d}^{n}x$. This is Lagrangian field theory. And we do also have Hamiltonian field theory.

The internal space $V$ depends on what the "oscillator" carries. A vibrating string has $V=\mathbb{R}$. A charged field has $V=\mathbb{C}$ (the phase matters). A field with more complex internal structure may have $V={\mathbb{C}}^2$ or $V={\mathbb{C}}^3$.

3. The internal symmetry group

Now comes the crucial observation. The internal space $V$ has symmetries: transformations $g:V\to V$ that leave the physics unchanged. These form a group $G\subset GL(V)$, the structure group.
This is already present at the level of the classical field, before any connection or quantization. As developed in detail in on theories, symmetries and gauge: we start with a variational principle $\mathcal{E}$, and the gauge symmetry group is the group of fiber transformations leaving $\mathcal{E}$ invariant. The group $G$ comes from the physics — from what we observe in experiments.

What determines $G$? This is where the gauge principle enters, as discussed in on theories, symmetries and gauge#The gauge principle. The logic is not that we first know $V$ and then deduce $G$ from it. Rather, we observe patterns in nature — conserved quantities, selection rules, multiplet structures — and from them we infer a symmetry group $G$. Then we postulate $G$ as a foundational ingredient and demand that the Lagrangian be invariant under it. The group $G$ is an empirical input, not a mathematical derivation.

Concretely, here is how experiments led physicists to each factor of $G$:

• Electric charge and $U(1)$. In every reaction ever observed — scattering, decay, annihilation — the total electric charge before equals the total charge after. No exception has ever been found. Now, charge is a single real number attached to each particle ($-1$ for the electron, $+\frac{2}{3}$ for the up quark, etc.), and it is additive: the charge of a composite system is the sum of its parts. By Noether's theorem, an additive conserved quantity corresponds to a continuous one-parameter symmetry. The only continuous group acting on a complex field by multiplication with modulus 1 (so that probabilities are preserved) is $U(1)$: the phase rotations $\varphi \mapsto e^{in\theta }\varphi$, where the integer $n$ is the charge. So the mere bookkeeping fact that charge is an exactly conserved integer already forces $G=U(1)$, $V=\mathbb{C}$ (or a higher-dimensional space carrying a representation of $U(1)$).

• Weak isospin and $SU(2)$. In the 1930s, Heisenberg noticed that the proton and neutron have almost identical masses and behave nearly identically under the strong nuclear force — as if they were two states of a single entity, an "isospin doublet." Later, the weak force made this pairing exact: in beta decay, a neutron converts into a proton ($n\to p+e^{-}+{\overline{\nu }}_e$), and vice versa. The weak interaction rotates one member of the doublet into the other. Similarly, the electron and its neutrino form another doublet $({\nu }_e,e^{-})$: the weak force can convert one into the other. A doublet — two states that mix into each other — is the fundamental (2-dimensional) representation of $SU(2)$. The pattern of allowed and forbidden weak decays is precisely what you get from $SU(2)$ selection rules: transitions that change isospin by $\frac{1}{2}$ are allowed, others are suppressed. So the experimental observation of doublet structure and weak decay patterns forces $G=SU(2)$, $V={\mathbb{C}}^2$. The irreducible representations of SU(2), labelled by spin $j=0,\frac{1}{2},1,\frac{3}{2},\dots$, classify how each field transforms.

• [NOT IN VAULT] Color charge and $SU(3)$. Several independent experimental facts converge on the number 3. (1) The ${\mathrm{\Omega }}^{-}$ baryon and the "eightfold way": in the 1960s, Gell-Mann and Ne'eman noticed that hadrons (strongly interacting particles) organize into multiplets — octets, decuplets — that are exactly the representations of $SU(3{)}_{\text{flavor}}$. Gell-Mann predicted the ${\mathrm{\Omega }}^{-}$ particle from a gap in the decuplet; it was found in 1964. (2) But $SU(3{)}_{\text{flavor}}$ is only an approximate symmetry. The deeper $SU(3)$ is color: the hypothesis that each quark flavor comes in 3 invisible "colors." The evidence is indirect but compelling: the rate of $e^{+}{e}^{-}\to \text{hadrons}$ compared to $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ (the "$R$-ratio") is exactly 3 times what you'd expect for colorless quarks; the ${\pi }^0\to \gamma \gamma$ decay rate requires a factor of $N_c^2=9$ from the triangle anomaly; and baryons (proton, neutron) are bound states of 3 quarks, which can only form a color-singlet if the color group has a 3-dimensional fundamental representation. All of this points to $G=SU(3)$ with $V={\mathbb{C}}^3$. The adjoint representation is 8-dimensional, corresponding to 8 gluons. The group $SU(3)$, its Lie algebra $\mathfrak{su}(3)$, and the Gell-Mann matrices are not in the vault.

The Standard Model says that nature has all three kinds of internal structure simultaneously:

Each particle species is characterized by a representation of this product group — a triplet of labels $(Y,I,C)$ specifying how it transforms under hypercharge, weak isospin, and color, respectively. [NOT IN VAULT]: the specific particle assignments (which representations correspond to quarks, leptons, etc.) are not yet in the vault.

4. From global to local symmetry: the connection

So far, the symmetry is global: the same transformation $g\in G$ is applied at every point of spacetime. The Lagrangian $L(\varphi ,{\partial }_{\mu }\varphi )$ is invariant under such global transformations.

But nature tells us something stronger: the symmetry should be local — the transformation $g(x)\in G$ can vary from point to point. The problem is that ${\partial }_{\mu }\varphi$ compares the values of $\varphi$ at nearby points, and if the internal frame is different at each point, this comparison becomes meaningless.

This is the core insight of gauge theory: we must recognize that $\varphi$ is not really a function $M\to V$ but a section of a G-bundle $E\to M$. The fiber $E_x$ over each point $x$ is a copy of $V$, but there is no canonical way to identify different fibers. A principal bundle $P\to M$ encodes all possible $G$-frames.

To compare fibers, we introduce a connection on $P$: a rule for "parallel transporting" the internal frame along spacetime. This connection replaces the ordinary derivative ${\partial }_{\mu }$ with a covariant derivative

where $A_{\mu }$ is a $\mathfrak{g}$-valued 1-form called the gauge potential (or Yang-Mills field). As explained in gauge theory#Coming from Classical field theories, this reorganizes the Lagrangian: ugly interaction terms get absorbed into $D_{\mu }$, making the Lagrangian simpler and manifestly gauge-invariant.

The connection is not arbitrary decoration — it is forced on us by demanding local $G$-invariance. Different connections $A_{\mu }$ describe different physical situations: different external fields acting on the matter.

5. Curvature = forces

The connection $A_{\mu }$ has intrinsic information: its curvature. For a connection $\omega$ on a principal $G$-bundle, the curvature is

In terms of the local gauge potential $A_{\mu }$, the field strength tensor is

The curvature measures the failure of parallel transport around a closed loop to return to the starting point (see holonomy). This is the force field.

The dynamics of the connection are governed by the Yang-Mills Lagrangian:

[NOT IN VAULT]: The Yang-Mills Lagrangian as a standalone topic is not in the vault, although the ingredients (curvature, field strength) are well-covered.

6. The prototype: electromagnetism as $U(1)$ gauge theory

Electromagnetism is the simplest and historically first gauge theory. Your vault covers it in great detail:

• The gauge group is $G=U(1)$.
• The Lie algebra $\mathfrak{u}(1)\cong \mathbb{R}$ is abelian, so $[A_{\mu },A_{\nu }]=0$ and the curvature simplifies to $F=dA$, as in electromagnetic field#The vector potential.
• The field strength tensor $F_{\mu \nu }$ encodes both $\mathbf{E}$ and $\mathbf{B}$, as shown in electromagnetic field#Differential forms approach.
• Maxwell's equations become $dF=0$ and $\star d\star F=J^{\mathrm{♭}}$.
• A charged field with charge $q$ lives in the associated bundle $E_q$ corresponding to the representation ${\rho }_q(e^{i\theta })=e^{iq\theta }$, and its covariant derivative is $D_{\mu }\varphi =({\partial }_{\mu }+iq{A}_{\mu })\varphi$, as detailed in the dictionary at electromagnetic field#Gauge theory formulation.
• The full Lagrangian for the system (gauge field + matter) is written in gauge theory#Coming from Classical field theories:

This is the model for everything that follows. Each factor in $U(1)\times SU(2)\times SU(3)$ works the same way, but with a non-abelian group.

7. The non-abelian generalization

When $G$ is non-abelian (like $SU(2)$ or $SU(3)$), the commutator $[A_{\mu },A_{\nu }]\ne 0$, and the theory becomes richer:

• The field strength tensor picks up the extra term: $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$. This is the content of curvature of a connection#Principal $G$-bundles.
• The gauge field itself carries charge (unlike the photon, which is uncharged). This means the gauge bosons interact with each other.
• The gauge potential $A_{\mu }$ takes values in the Lie algebra $\mathfrak{g}$, and a gauge transformation acts by the adjoint representation: $A_{\mu }\mapsto g{A}_{\mu }{g}^{-1}+g{\partial }_{\mu }{g}^{-1}$. This is the transformation law (!) in principal connection on a principal bundle#Another characterization.

8. The three factors of the Standard Model

The Standard Model declares: the total gauge group is

Each factor corresponds to a different kind of "internal charge" and a different force:

$U(1{)}_Y$ — Hypercharge

This is the easiest factor. It works exactly like electromagnetism, but the quantum number is called hypercharge $Y$ (instead of electric charge $q$). The irreducible representations are labelled by integers (see group representation#Representations of $U(1)$). The gauge boson is called $B_{\mu }$.

[NOT IN VAULT]: The concept of hypercharge and its relation to electric charge via $Q=I_3+\frac{Y}{2}$ is not in the vault.

$SU(2{)}_L$ — Weak isospin

The structure group SU(2) acts on fields that carry weak isospin. The irreducible representations are classified in group representation#Representations of $SU(2)$: one for each dimension $n$, with spin $j=(n-1)/2$. The three gauge bosons are $W_{\mu }^1,W_{\mu }^2,W_{\mu }^3$ (corresponding to the three generators of $\mathfrak{su}(2)$, which can be taken to be the Pauli matrices).

The subscript $L$ stands for "left" — this force only affects left-handed particles. Left-handed fermions form doublets ($j=1/2$ representation), while right-handed fermions are singlets ($j=0$, invisible to $SU(2{)}_L$).

[NOT IN VAULT]: The chiral structure of the Standard Model (left-handed doublets vs right-handed singlets), weak isospin assignments, and the Weinberg angle are not in the vault.

$SU(3{)}_c$ — Color

[NOT IN VAULT]: This entire factor is absent from the vault. Here is a summary of what would be needed:

• $SU(3)$ is the group of $3\times 3$ unitary matrices with determinant 1. Its Lie algebra $\mathfrak{su}(3)$ is 8-dimensional (the traceless anti-Hermitian $3\times 3$ matrices), with a standard basis given by the Gell-Mann matrices ${\lambda }_1,\dots ,{\lambda }_8$.
• Quarks live in the fundamental representation ($\mathbf{3}$): each quark comes in three "colors" (red, green, blue).
• Gluons, the gauge bosons of $SU(3{)}_c$, live in the adjoint representation ($\mathbf{8}$): there are 8 gluons, and they carry color charge themselves (unlike the photon, which carries no electric charge).
• Leptons (electrons, neutrinos) are color singlets ($\mathbf{1}$): they don't feel the strong force.
• The theory of the strong interaction based on $SU(3{)}_c$ is called Quantum Chromodynamics (QCD).

9. Particles = representations

In the language of gauge theory, every particle species corresponds to a choice of representation of the full gauge group $G=U(1{)}_Y\times SU(2{)}_L\times SU(3{)}_c$. The general picture is:

• Fermions (matter particles) are sections of associated vector bundles, constructed from the principal bundle $P$ via a representation of $G$. This is stated explicitly in gauge theory#Coming from Classical field theories: "Fermions (particles) are related to sections of a complex vector bundle which is associated to a principal bundle over space-time."
• Bosons (force carriers) are connections on the principal bundle $P$: "Bosons (forces) are related to connections on the principal bundle." There is one set of bosons for each simple factor of $G$.
• The spinorial nature of fermions (spin $\frac{1}{2}$) comes from the spacetime part: they are sections of a spinor bundle associated to the spin group $Spin(1,3)$, as described in quantum particle.

[NOT IN VAULT]: The full particle content table. Here is a sketch:

(There are three generations of fermions, each with the same quantum numbers.)

10. The Lagrangian of the Standard Model

Combining everything, the Standard Model Lagrangian has the schematic form:

where:

• ${\mathcal{L}}_{YM}=-\frac{1}{4}{B}_{\mu \nu }{B}^{\mu \nu }-\frac{1}{4}{W}_{\mu \nu }^a{W}^{a\mu \nu }-\frac{1}{4}{G}_{\mu \nu }^A{G}^{A\mu \nu }$ is the Yang-Mills part — one field strength for each factor $U(1{)}_Y$, $SU(2{)}_L$, $SU(3{)}_c$. This generalizes what is shown in electromagnetic field#The Lagrangian and curvature of a connection.

• ${\mathcal{L}}_{matter}=\overline{\psi }i{\gamma }^{\mu }{D}_{\mu }\psi$ is the matter Lagrangian, where $D_{\mu }$ is the covariant derivative containing all three gauge potentials, acting on the appropriate representations. This is the generalization of the structure in gauge theory#Coming from Classical field theories.

• [NOT IN VAULT] ${\mathcal{L}}_{Higgs}$ describes the Higgs field and the mechanism of spontaneous symmetry breaking: the Higgs field acquires a nonzero vacuum expectation value, breaking $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{EM}$. This gives mass to $W^{\pm }$ and $Z^0$ while keeping the photon massless. The electric charge emerges as a combination of hypercharge and weak isospin: $Q=I_3+Y/2$.

• [NOT IN VAULT] ${\mathcal{L}}_{Yukawa}$ describes the coupling between fermions and the Higgs field, which is how fermions acquire mass.

11. Quantization

Finally, all these classical fields are quantized, as outlined in Standard Model:

• The matter fields (sections of the associated bundles) are promoted to operator-valued fields — these are the fermions. See quantum field for a detailed construction starting from the analogy with finitely many particles.
• The gauge potentials $A_{\mu }$ (connections) are also promoted to operator-valued fields — these are the bosons.

The detailed mechanism is explained in quantum field#Fields promoted to operators: the classical field $\varphi (x)$ becomes an operator $\hat{\varphi }(x)$ acting on a Hilbert space of field configurations. The path integral provides an alternative approach to the same quantization. The framework of effective field theories provides a way to handle the theory at different energy scales.

See also quantum particle for the summary of how a specific particle (e.g. an electron with charge $q$) is modeled: a principal bundle with group $S{O}^{+}(1,n)\times U(1)$, an associated bundle via a representation, and quantization of the sections.

12. Summary: the logic of the Standard Model

Sources (vault notes)

• Lagrangian Mechanics — particle dynamics, variational principle
• Classical Field Theory — particles to fields transition
• Lagrangian field theory — field Lagrangians and Euler-Lagrange
• classical field — definition and types of classical fields
• on theories, symmetries and gauge — theories, symmetries, gauge principle
• gauge theory — full gauge theory framework, matter fields as sections, bosons as connections
• Standard Model — the four-step ladder
• Lie group, Lie algebra — group and algebra foundations
• SU(2) — the weak isospin group
• group representation — representations of $U(1)$ and $SU(2)$
• adjoint representation — gauge field transformations
• spin representation — spinor representations
• principal bundle — principal bundles, $G$-frames, trivializations
• G-bundle — fiber bundles with structure group
• associated bundle — constructing matter bundles from representations
• principal connection on a principal bundle — connections, Yang-Mills fields
• curvature of a connection — curvature, field strength tensors
• exterior covariant derivative — covariant differentiation
• Maurer-Cartan form — Lie-algebra valued forms on groups
• parallel transport, holonomy — geometric meaning of curvature
• electromagnetic field — the $U(1)$ prototype in full detail
• spinor — spinor fields on spacetime
• spin group — double covers of $SO(n)$
• Clifford algebra, Pauli matrices — algebraic foundations for spinors
• action, Euler-Lagrange — variational machinery
• Noether's theorem — symmetries and conservation laws
• quantum field — quantization of fields, operator-valued fields
• quantum particle — modeling a specific particle
• Quantum Field Theory — overview of QFT
• Feynman's path integral formulation — path integral quantization
• effective field theories — multi-scale physics

Gaps (not in the vault)

The following topics are needed for a self-contained account and are currently absent:

1. $SU(3)$ — the Lie group, $\mathfrak{su}(3)$, Gell-Mann matrices
2. Representations of $SU(3)$ — fundamental $\mathbf{3}$, antifundamental $\overline{\mathbf{3}}$, adjoint $\mathbf{8}$
3. Yang-Mills Lagrangian — $\mathcal{L}=-\frac{1}{4}\text{tr}(F_{\mu \nu }{F}^{\mu \nu })$ for general compact groups
4. Electroweak theory — $SU(2{)}_L\times U(1{)}_Y$, hypercharge, weak isospin, Weinberg angle, chirality
5. Spontaneous symmetry breaking / Higgs mechanism — how $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{EM}$, mass generation for $W^{\pm }$ and $Z^0$
6. QCD (Quantum Chromodynamics) — color charge, gluon self-interaction, confinement, asymptotic freedom
7. Standard Model particle content — the explicit table of representations for all known particles