Cours

Symmetry made computable. Hall works with matrix groups so you can start with linear algebra alone — and Appendix C is precisely the angular-momentum machinery quantum mechanics runs on.

Book: Hall, Lie Groups, Lie Algebras, and Representations (GTM 222)

Read: Parts I–II (Ch. 1–9) + Appendix C

Skip: Ch. 10 and Part III (Ch. 11–13) unless you want the compact-group theory for its own sake.

Prerequisites: None

Worked: Tsirelson’s bound in four lines

Let $S = A_1(B_1{+}B_2) + A_2(B_1{-}B_2)$ with $A_i^2 = B_j^2 = \mathbb{1}$ and $[A_i, B_j] = 0$. Square it: $S^2 = 4\cdot\mathbb{1} - [A_1, A_2]\,[B_1, B_2].$ Operator norms: $\|[A_1,A_2]\| \le 2\|A_1\|\|A_2\| = 2$, same for $B$, so $\|S^2\| \le 4 + 4 = 8$ and $|\langle S\rangle| \le \sqrt{\|S^2\|} = 2\sqrt2.$ Classical bound $2$ is the commuting case ($[A_1,A_2] = 0$). The entire quantum advantage lives in one product of commutators — and saturating it requires anticommuting observables (mutually unbiased measurements) on a maximally entangled state.

Worked: teleportation, the actual algebra

Alice holds unknown $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ and half of $|\Phi^+\rangle$. Rewrite the 3-qubit state in Alice’s Bell basis: $|\psi\rangle_1|\Phi^+\rangle_{23} = \tfrac12\sum_{k} |\beta_k\rangle_{12}\otimes \sigma_k|\psi\rangle_3,$ where $(\beta_k, \sigma_k)$ runs over $(\Phi^+, \mathbb{1}), (\Phi^-, \sigma_z), (\Psi^+, \sigma_x), (\Psi^-, \sigma_z\sigma_x)$ — an identity you verify by expanding both sides once in your life. Alice’s Bell measurement selects branch $k$ (2 classical bits); Bob applies $\sigma_k^{-1}$ and holds $|\psi\rangle$ exactly. Notice the audit: no cloning ($|\psi\rangle$ destroyed at Alice’s side), no signaling (Bob’s marginal is $\mathbb{1}/2$ until the bits arrive), one Bell pair consumed. Resource arithmetic: 1 ebit + 2 cbits $\ge$ 1 qubit.

Worked: reduced state of a Bell pair = 1 bit of entropy

$\rho_A = \operatorname{tr}_B|\Phi^+\rangle\langle\Phi^+| = \tfrac12\big(|0\rangle\langle0| + |1\rangle\langle1|\big) = \tfrac{\mathbb{1}}{2}$: the local view of a maximally entangled pure state is a perfect coin. Entanglement entropy $S(\rho_A) = \log 2 = 1$ bit — the unit “ebit.” The global state is pure ($S_{AB} = 0$) while the parts are maximally uncertain: $S_A + S_B > S_{AB}$, impossible classically. Information lives in the correlations, not in the pieces — the single cleanest statement of what entanglement is.

Worked: PPT catches the isotropic state at $p = 1/3$

Mix a Bell state with noise: $\rho = p\,|\Phi^+\rangle\langle\Phi^+| + (1-p)\tfrac{\mathbb{1}}{4}$. Partial transpose flips the singlet sector: $(|\Phi^+\rangle\langle\Phi^+|)^{T_B}$ has eigenvalues $+\tfrac12$ (triplet, $\times3$) and $-\tfrac12$ (singlet). So $\rho^{T_B}$’s least eigenvalue is $\tfrac{1-p}{4} - \tfrac{p}{2} = \tfrac{1 - 3p}{4}$, negative iff $p > \tfrac13.$ For two qubits PPT is exact (Horodecki): below $1/3$ the state is genuinely separable — a Bell state survives 66% white noise before its entanglement dies. This one computation is the daily bread of experimental entanglement verification.

The physics spine begins here: a complete quantum mechanics course written by people who knew quantum information was coming. Everything in Parts II–III, and all of QFT, stands on these ten chapters.

Book: Bertlmann & Friis, Modern Quantum Theory — Part I

Read: Ch. 1–10

Skip: If you have had a serious QM course: skim Ch. 1–7, read Ch. 8–10 carefully (spin, EM coupling, perturbation theory) — Williams leans on those.

Prerequisites: None

Why quantum: wave–particle duality

Planck resolves the black-body catastrophe only by quantizing exchange: $E = \hbar\omega$. Einstein makes light corpuscular (photoelectric effect); de Broglie retaliates by making matter wavy, $p = \hbar k$; the double slit shows single particles interfering with themselves. The experiments come first in B&F because the formalism is unbelievable without them.

The Schrödinger equation and the Born rule

$i\hbar\,\partial_t\,\psi = \hat H\psi$ — linear, hence superposition; first-order in time, hence the state is everything. $|\psi(x)|^2$ is a probability density (Born), conserved via the continuity equation. Determinism survives in the wavefunction; probability enters only at measurement — the split that Part II will interrogate.

Voir aussi : Entanglement & Nonlocality

The Hilbert-space formalism

States are rays in a Hilbert space; observables are self-adjoint operators; outcomes are eigenvalues; expectation is $\langle\psi|\hat A|\psi\rangle$. Noncommuting observables cannot be jointly sharp: $\Delta A\,\Delta B \ge \tfrac{1}{2}\left|\langle[\hat A,\hat B]\rangle\right|.$ Dirac notation makes the linear algebra frictionless — and it is secretly the string-diagram notation of the capstone.

Voir aussi : Categorical Quantum Mechanics

Bound states and tunneling

The time-independent equation $\hat H\psi = E\psi$ in one dimension: infinite well (discrete spectrum from boundary conditions alone), finite well, and barrier penetration — transmission through classically forbidden regions, decaying like $e^{-2\kappa L}$. Tunneling is not exotic: alpha decay, scanning tunneling microscopes, and flash memory run on it.

Ehrenfest and the classical limit

Expectation values obey classical-looking laws: $\tfrac{d}{dt}\langle \hat A\rangle = \tfrac{i}{\hbar}\langle[\hat H,\hat A]\rangle + \langle\partial_t \hat A\rangle$, giving $\tfrac{d}{dt}\langle x\rangle = \langle p\rangle/m$ and $\tfrac{d}{dt}\langle p\rangle = -\langle V’(x)\rangle$. Newton survives on average — exactly when $\langle V’(x)\rangle \approx V’(\langle x\rangle)$, i.e. for wavepackets narrow against the potential’s variation. The commutator-to-Poisson-bracket dictionary starts here.

Voir aussi : Smooth Manifolds

Probability current

$|\psi|^2$ is conserved locally: $\partial_t|\psi|^2 + \nabla\!\cdot\!\vec j = 0$ with $\vec j = \tfrac{\hbar}{m}\mathrm{Im}(\psi^*\nabla\psi)$. Transmission/reflection coefficients are ratios of currents, not amplitudes — the bookkeeping that makes scattering probabilities add to one. A continuity equation identical in form to charge or fluid conservation: the first hint that $\psi$ is a field.

Voir aussi : Quantum Field Theory, Acoustics

The harmonic oscillator

Factor $\hat H$ with ladder operators $[a, a^\dagger] = 1$: $\hat H = \hbar\omega\left(a^\dagger a + \tfrac12\right), \quad E_n = \hbar\omega\left(n+\tfrac12\right).$ The algebraic method matters more than the spectrum: quantum fields are infinite families of these oscillators, and $a^\dagger$ becomes particle creation. Master this and Williams Ch. 6 is half-familiar on arrival.

Voir aussi : Quantum Field Theory

Angular momentum and spin

$[\hat L_i, \hat L_j] = i\hbar\,\varepsilon_{ijk}\hat L_k$ — the $\mathfrak{su}(2)$ algebra — forces the ladder structure: $j = 0, \tfrac12, 1, \dots$ with $2j{+}1$ states each. Half-integer $j$ cannot come from orbital motion: spin is an internal degree of freedom, revealed by Stern–Gerlach, represented by Pauli matrices, and mathematically a representation of $SU(2)$ — the double cover, exactly Hall’s Ch. 4 and the covering-space story from topology.

Voir aussi : Lie Groups, Algebras & Representations, Topology & the Fundamental Group

The hydrogen atom

The Coulomb problem separates in spherical coordinates; quantization yields $E_n = -13.6\,\mathrm{eV}/n^2$ with quantum numbers $(n, \ell, m)$ and the $n^2$-fold degeneracy that hides an extra symmetry (the Runge–Lenz vector — hydrogen secretly has $SO(4)$). The one atom solved exactly, and the calibration standard for everything approximate.

The 3D Schrödinger equation and $Y_{\ell m}$

Central potentials separate: $\psi = R_{n\ell}(r)\,Y_{\ell m}(\theta,\phi)$, the angular factor forced by rotational symmetry alone — $Y_{\ell m}$ are the matrix elements of $SO(3)$ representations (Hall Part III in disguise). The radial equation acquires the centrifugal barrier $\hbar^2\ell(\ell{+}1)/2mr^2$. Every atom, nucleus, and quantum dot begins with this separation.

Voir aussi : Lie Groups, Algebras & Representations

Identical particles and exchange

Permuting identical particles must act trivially on physical states up to phase: $\psi \to \pm\psi$. Symmetric = bosons, antisymmetric = fermions; the Pauli exclusion principle is the antisymmetric case’s refusal to double-occupy. Consequences: the periodic table, stability of matter, lasers, superconductivity. QFT (spin–statistics) will explain why the sign is tied to spin.

Voir aussi : Quantum Field Theory

Heisenberg picture

Freeze states, evolve operators: $\hat A_H(t) = e^{i\hat Ht/\hbar}\hat A e^{-i\hat Ht/\hbar}$, $\tfrac{d\hat A_H}{dt} = \tfrac{i}{\hbar}[\hat H, \hat A_H]$. Physically identical to Schrödinger’s picture, but the natural language of QFT (where fields are operators at spacetime points) and the cleanest bridge to classical equations of motion. Fluency in switching pictures is a prerequisite for Williams Ch. 6.

Voir aussi : Quantum Field Theory

Charged particles in electromagnetic fields

Minimal coupling $\hat{\mathbf p} \to \hat{\mathbf p} - q\mathbf A$ makes the Schrödinger equation gauge covariant: physics is invariant, phases are not — and the Aharonov–Bohm effect proves the potential’s phase is physical where fields vanish. Landau levels and Zeeman splitting live here. This is the gauge principle in embryo; Williams Ch. 8–9 grows it into the Standard Model.

Voir aussi : Quantum Field Theory

Perturbation theory

Stationary: $E_n \approx E_n^{(0)} + \langle n|\hat V|n\rangle + \sum_{m\ne n}\frac{|\langle m|\hat V|n\rangle|^2}{E_n^{(0)}-E_m^{(0)}}$, with the degenerate case forcing diagonalization first (fine structure, Stark). Time-dependent: transition rates from Fermi’s golden rule, $\Gamma = \frac{2\pi}{\hbar}|\langle f|\hat V|i\rangle|^2 \rho(E_f)$. This is also the conceptual template for Feynman diagrams: QFT is time-dependent perturbation theory grown up.

Voir aussi : Quantum Field Theory

Symmetry in quantum mechanics

Wigner’s theorem: symmetries act as unitary (or antiunitary — time reversal) operators on states. Continuous symmetries have self-adjoint generators, and $[\hat H, \hat Q] = 0$ makes $\hat Q$ conserved: momentum generates translations, $\hat L$ generates rotations. Quantum numbers are irreducible-representation labels — Schur’s lemma wearing a lab coat.

Voir aussi : Lie Groups, Algebras & Representations

The delta potential: a one-line bound state

$V(x) = -\alpha\,\delta(x)$: integrating the Schrödinger equation across the spike gives a derivative jump $\psi’(0^+) - \psi’(0^-) = -\tfrac{2m\alpha}{\hbar^2}\psi(0)$, and matching decaying exponentials yields exactly one bound state, $E = -\tfrac{m\alpha^2}{2\hbar^2}$. The minimal model of binding — and of renormalization thinking: a zero-range interaction with one physical parameter.

Voir aussi : Quantum Field Theory

Runge–Lenz and hydrogen’s hidden $SO(4)$

The Coulomb problem conserves an extra vector $\hat{\vec A}$ (quantized Laplace–Runge–Lenz); together with $\hat{\vec L}$ it closes into $\mathfrak{so}(4)$. Pure algebra then delivers $E_n \propto -1/n^2$ and the $n^2$ degeneracy — no differential equation solved. Degeneracy is never an accident: it is always a symmetry’s signature, here a hidden one.

Voir aussi : Lie Groups, Algebras & Representations

Coherent states

Eigenstates of the annihilation operator, $a|\alpha\rangle = \alpha|\alpha\rangle$: minimum-uncertainty Gaussians whose centers follow the classical trajectory exactly. Overcomplete, non-orthogonal, and the natural basis for radiation — the laser’s state. They open B&F Ch. 25 and the whole phase-space formulation; in QFT they become the closest quantum approximation to a classical field.

Voir aussi : Entropy, Channels & Open Systems, Quantum Field Theory

Worked: the infinite square well from scratch

Inside $[0,L]$: $\psi’’ = -k^2\psi$ with $\psi(0) = \psi(L) = 0$. Sine solutions with $kL = n\pi$: $\psi_n = \sqrt{\tfrac{2}{L}}\sin\tfrac{n\pi x}{L}, \qquad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}.$ Read off the morals: quantization came from boundary conditions, not postulates; the ground state has nonzero energy (confinement costs momentum, by uncertainty: $\Delta p \sim \hbar/L$ gives $E_1 \sim \hbar^2/2mL^2$ — the scaling is right before you solve anything); nodes increase one per level. This $1/L^2$ is why quantum dots glow by size and why nuclear energies are MeV while atomic are eV.

Worked: the oscillator by ladder, in five lines

Define $a = \sqrt{\tfrac{m\omega}{2\hbar}}(\hat x + \tfrac{i\hat p}{m\omega})$; then $[a, a^\dagger] = 1$ and $\hat H = \hbar\omega(a^\dagger a + \tfrac12)$. If $\hat H|E\rangle = E|E\rangle$, the commutators give $\hat H(a|E\rangle) = (E - \hbar\omega)(a|E\rangle)$: $a$ descends the spectrum. Positivity of $\|a|\psi\rangle\|^2 = \langle a^\dagger a\rangle \ge 0$ forces a floor: $a|0\rangle = 0$, i.e. $E_0 = \tfrac{\hbar\omega}{2}$, and the ladder gives $E_n = \hbar\omega(n + \tfrac12)$. The ground state condition is first-order: $\psi_0 \propto e^{-m\omega x^2/2\hbar}$, a Gaussian. No Hermite polynomials were harmed — and in QFT, $a^\dagger$ will simply be renamed “create a particle.”

Worked: spin precession (the two-level workhorse)

Spin-$\tfrac12$ in a field along $z$: $\hat H = \tfrac{\hbar\omega}{2}\sigma_z$. Heisenberg equations: $\dot\sigma_x = -\omega\sigma_y$, $\dot\sigma_y = \omega\sigma_x$ — so $\langle\vec S\rangle$ precesses about $z$ at the Larmor frequency $\omega$. Equivalently, a state on the Bloch sphere rotates rigidly: $|\psi(t)\rangle = e^{-i\omega t\sigma_z/2}|\psi(0)\rangle$. Note the telltale $4\pi$: at $\omega t = 2\pi$ the state vector has acquired a factor $-1$ (the $SU(2)$ double cover, measurable in neutron interferometry). NMR, qubit control, and atomic clocks are this calculation with decorations.

Worked: first-order perturbation and a selection rule

Ground state of hydrogen in a uniform field $\hat V = eEz$: the first-order shift is $\langle 100|z|100\rangle = 0$ — the integrand is odd under parity, and $|100\rangle$ has definite parity. So the Stark effect starts at second order, $\Delta E = \sum_{m\ne0} \tfrac{|\langle m|\hat V|0\rangle|^2}{E_0 - E_m} < 0$ (every term negative: the ground state is always pushed down — level repulsion). Meanwhile excited hydrogen, with its $\ell$-degeneracy, shows a linear Stark effect: degenerate perturbation theory mixes $2s$–$2p$ first. Parity forbidding matrix elements is the prototype of every selection rule; Wigner–Eckart (Hall App. C) is its industrial form.

The load-bearing wall of the whole math track. Everything downstream — manifolds, Lie groups, algebraic topology — speaks this language of open sets, continuity, and loops.

Book: Lee, Introduction to Topological Manifolds (GTM 202)

Read: Ch. 1–12

Skip: Ch. 13 (homology) — Hatcher Ch. 2 does it better; read it there.

Prerequisites: None

Calculus rebuilt on curved spaces. This is the mathematical home of classical mechanics, gauge fields, and general relativity — and the manifold view of the Lie groups Hall treats by matrices.

Book: Lee, Introduction to Smooth Manifolds (GTM 218)

Read: Ch. 1–17 core · Ch. 19–22 selectively

Skip: Ch. 18 (de Rham theorem proof) on a first pass; ch. 6 (Sard) can be skimmed for statements.

Prerequisites: Topology & the Fundamental Group

Smooth structures and charts

A smooth manifold is a topological manifold with an atlas of charts whose transition maps are $C^\infty$. Smoothness is not intrinsic to the space — it is added structure (some topological manifolds carry none, some carry many). Once fixed, it makes sense to ask whether functions and maps are differentiable, and diffeomorphism becomes the notion of sameness.

Voir aussi : Topology & the Fundamental Group

Tangent vectors as derivations

With no ambient space, a tangent vector at $p$ is defined as a derivation: a linear map $v:C^\infty(M)\to\mathbb{R}$ obeying Leibniz, $v(fg) = f(p)\,v(g) + g(p)\,v(f)$. These form the tangent space $T_pM$ with basis $\partial/\partial x^i$. A smooth map $F$ gets a best linear approximation, the differential $dF_p : T_pM \to T_{F(p)}N$ — the chain rule, globalized.

Immersions, submersions, embeddings

Classify maps by the rank of $dF_p$: injective (immersion), surjective (submersion), or immersion + homeomorphism onto image (embedding). The regular level set theorem is the workhorse: if $c$ is a regular value of $F:M\to N$, then $F^{-1}(c)$ is an embedded submanifold of codimension $\dim N$. This is how $S^n$, $SO(n)$, and most manifolds you meet are actually exhibited.

Voir aussi : Lie Groups, Algebras & Representations

Vector fields, flows, and the Lie bracket

A vector field is a smooth section of $TM$; its integral curves stitch into a flow, a one-parameter group of diffeomorphisms — the geometric meaning of solving an ODE. Two fields fail to commute by exactly the Lie bracket $[X,Y] = XY - YX$, itself a vector field. The bracket measures whether coordinate grids can be built from flows, and it is the same bracket that rules Lie algebras.

Voir aussi : Lie Groups, Algebras & Representations, Quantum Field Theory

Partitions of unity

Smooth bump functions summing to $1$, subordinate to any open cover — the device that glues local constructions into global ones (metrics, integrals, extensions all exist because of it). This is the payoff of second countability + paracompactness, and the single biggest technical difference between smooth topology (soft, flexible) and complex/analytic geometry (rigid, no bumps).

Voir aussi : Topology & the Fundamental Group

Immersed vs. embedded submanifolds

Embedded: image carries the subspace topology (the regular level set theorem produces these). Immersed: locally embedded but possibly self-crossing or densely wrapped — the irrational line on the torus is the canonical warning. Lie subgroups are in general only immersed; integral manifolds of distributions likewise. Knowing which kind you hold determines which theorems apply.

Bundles, covectors, and tensors

The tangent spaces assemble into the tangent bundle $TM$; dually, covectors form $T^*M$, whose sections are 1-forms like $df$. Tensor fields are multilinear machines fed vectors and covectors. A Riemannian metric — a smooth positive-definite symmetric 2-tensor $g$ — equips each tangent space with an inner product: lengths, angles, and geometry proper begin here.

Differential forms and $d$

Alternating tensors with a wedge product and one miracle operator, the exterior derivative: $d^2 = 0, \qquad F^*(d\omega) = d(F^*\omega).$ Forms are the objects born to be integrated; $d$ unifies grad, curl, and div; and pullback-compatibility means the whole calculus is coordinate-free.

Voir aussi : Algebraic Topology

Orientation, integration, Stokes

An orientation is a consistent choice of ordered bases; on oriented $n$-manifolds, $n$-forms integrate. Then one theorem swallows the fundamental theorem of calculus, Green, Gauss, and classical Stokes whole: $\int_M d\omega = \int_{\partial M}\omega.$ Conservation laws in physics are Stokes in costume.

Voir aussi : Quantum Field Theory, Acoustics

Whitney embedding

Every smooth $n$-manifold embeds in $\mathbb{R}^{2n+1}$ (Lee proves the easy compact case; Whitney’s hard theorem sharpens to $\mathbb{R}^{2n}$). Moral: abstract manifolds are not more general than submanifolds of Euclidean space — the abstraction buys convenience, not new objects. Proof is a partitions-of-unity showcase.

Sard’s theorem and transversality

Critical values have measure zero: almost every value is regular. Consequence machine: generic level sets are submanifolds, generic intersections are transverse ($T_pA + T_pB = T_pM$, giving $\operatorname{codim}(A\cap B) = \operatorname{codim} A + \operatorname{codim} B$), and degree/intersection counts are well defined. “Wiggle until generic” becomes a proof technique.

Voir aussi : Algebraic Topology

The Lie derivative

Differentiate any tensor along a flow: $\mathcal{L}_X T = \frac{d}{dt}\big|_0 (\varphi_t^* T)$. On functions it is $Xf$; on vector fields, $\mathcal{L}_X Y = [X,Y]$ — the bracket is a derivative; on forms, Cartan’s magic formula $\mathcal{L}_X = d\,\iota_X + \iota_X\, d$ turns flow-invariance questions ($\mathcal{L}_X\omega = 0$) into algebra. Conservation laws in Hamiltonian mechanics are exactly such statements.

Voir aussi : Quantum Field Theory

Musical isomorphisms and the gradient

A metric converts vectors to covectors and back: $X^\flat = g(X,\cdot)$, $\omega^\sharp$. The gradient is properly $\operatorname{grad} f = (df)^\sharp$ — $df$ exists on any smooth manifold, but pointing “uphill” requires a metric. Divergence, Laplacian ($\Delta = \operatorname{div}\operatorname{grad}$), and all of vector calculus reassemble on Riemannian manifolds this way — the geometry under Pierce’s acoustic operators.

Voir aussi : Acoustics

De Rham cohomology

Closed forms ($d\omega=0$) modulo exact ones ($\omega = d\eta$): $H^k_{\mathrm{dR}}(M)$. Locally every closed form is exact (Poincaré lemma), so nonzero classes detect global holes — analysis discovering topology. The de Rham theorem says these groups agree with the singular cohomology of Hatcher: two roads, one invariant.

Voir aussi : Algebraic Topology

Lie groups as manifolds

A Lie group is a group that is a smooth manifold with smooth operations. Left-invariant vector fields form its Lie algebra, and the flow of such a field through $e$ gives the exponential map — recovering Hall’s $e^{tX}$ intrinsically, with no matrices in sight. Read this chapter as the coronation of Hall Part I.

Voir aussi : Lie Groups, Algebras & Representations

Frobenius and foliations

A $k$-plane distribution is integrable — tangent to a family of immersed submanifolds slicing $M$ like pages of a book — iff it is closed under the Lie bracket. Frobenius’ theorem is the integrability criterion behind constrained mechanics, control theory, and the geometry of gauge fixing.

Voir aussi : Quantum Field Theory

Symplectic manifolds

A closed nondegenerate 2-form $\omega$ makes $M$ a phase space: every Hamiltonian $H$ determines a vector field by $\iota_{X_H}\omega = dH$, whose flow is Hamiltonian mechanics; Darboux says all symplectic manifolds look locally like $\sum dp_i \wedge dq^i$. This closes the loop with Williams Ch. 2 — classical mechanics is symplectic geometry.

Voir aussi : Quantum Field Theory

Mayer–Vietoris for de Rham

An open cover $M = U \cup V$ yields a long exact sequence linking $H^*_{dR}$ of $M, U, V, U\cap V$ — the analyst’s copy of the homology tool, with the connecting map built from a partition of unity. Compute $H^*_{dR}(S^n)$ by induction exactly as Hatcher computes $H_*(S^n)$: the two theories rhyme because (de Rham) they are the same.

Voir aussi : Algebraic Topology

Degree via integration

For a smooth map $F: M \to N$ between compact oriented $n$-manifolds, $\int_M F^*\omega = (\deg F)\int_N \omega$ — the degree counts preimages with orientation signs, and it is an integer that survives homotopy. Analysis computing topology; this is the smooth face of Hatcher’s homological degree, and the ancestor of every index theorem.

Voir aussi : Algebraic Topology

Quotient manifolds

When does $M/G$ inherit a smooth structure? Quotient manifold theorem: for a free, proper smooth action, $M/G$ is a manifold of dimension $\dim M - \dim G$ with a submersion projection. Produces $\mathbb{RP}^n$, lens spaces, and — with $G$ a Lie subgroup — homogeneous spaces $G/H$: spheres as $SO(n{+}1)/SO(n)$, the arenas of symmetry physics.

Voir aussi : Lie Groups, Algebras & Representations

The summit of the physics track: special relativity and quantum mechanics forced into one framework. Particles become excitations of fields, forces become gauge symmetries, and the Standard Model is assembled.

Book: Williams, Introduction to Quantum Field Theory

Read: Ch. 1–5, then Ch. 6–9

Skip: Skim Williams’ QM recap where it repeats B&F Part I; boxed proofs are designed to be deferred on first reading. The Appendix is a formulary you will live in during Ch. 6–9.

Prerequisites: Quantum Mechanics, Lie Groups, Algebras & Representations

Lorentz and Poincaré groups

Boosts and rotations form $SO(3,1)$; adding translations gives Poincaré. The finite-dimensional representations are labeled by two spins $(A,B)$ via $\mathfrak{so}(3,1)_{\mathbb C} \cong \mathfrak{sl}(2,\mathbb C)\oplus\mathfrak{sl}(2,\mathbb C)$: scalars $(0,0)$, Weyl spinors $(\tfrac12,0)$, vectors $(\tfrac12,\tfrac12)$. Wigner’s little-group analysis then classifies particles as unitary irreps labeled by mass and spin (or helicity if massless). Hall is the rigor behind every line of this chapter.

Voir aussi : Lie Groups, Algebras & Representations

Classical fields and Noether’s theorem

Promote mechanics to fields: $S = \int d^4x\, \mathcal{L}(\phi, \partial_\mu\phi)$, Euler–Lagrange $\partial_\mu \frac{\partial\mathcal L}{\partial(\partial_\mu\phi)} = \frac{\partial\mathcal L}{\partial\phi}$. Noether: every continuous symmetry yields a conserved current, $\partial_\mu j^\mu = 0$ — translations give energy–momentum, phase rotations give charge. The single most consequential theorem in physics, and the reason symmetry is the organizing principle of this whole library.

Voir aussi : Smooth Manifolds

Canonical quantization and Fock space

Fourier-decompose the free Klein–Gordon field: each mode is a harmonic oscillator. Promote to operators, $[a_{\mathbf p}, a^\dagger_{\mathbf p’}] = (2\pi)^3\delta^3(\mathbf p - \mathbf p’)$: particles are quanta of fields. Fock space stacks them; the Feynman propagator $\langle 0|T\,\phi(x)\phi(y)|0\rangle$ encodes causal structure — commutators vanish at spacelike separation, so relativity survives quantization.

Voir aussi : Quantum Mechanics

The Dirac equation

Seeking a first-order relativistic equation forces anticommuting coefficients: $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}$, and $(i\gamma^\mu\partial_\mu - m)\psi = 0$ acting on 4-spinors built from $(\tfrac12,0)\oplus(0,\tfrac12)$. Out fall spin-$\tfrac12$, the magnetic moment $g\approx2$, and — reinterpreting negative energies — antimatter. Quantizing consistently demands anticommutators: the spin–statistics connection.

Voir aussi : Lie Groups, Algebras & Representations

Natural units and dimensional analysis

Set $\hbar = c = 1$: everything is a power of energy (mass $=$ energy, length $=$ time $=$ energy$^{-1}$). A field’s mass dimension follows from its kinetic term ($[\phi] = 1$, $[\psi] = \tfrac32$ in 4D); couplings with negative dimension flag non-renormalizable interactions. Half of QFT sanity-checking is this arithmetic — make it reflexive before Ch. 6.

Discrete symmetries: C, P, T and the CPT theorem

Charge conjugation, parity, time reversal (the antiunitary one, via Wigner). Nature breaks each — P maximally in weak interactions, CP subtly (kaons, B mesons) — but any local, Lorentz-invariant QFT must preserve the product: the CPT theorem. Consequences with no known exceptions: particle and antiparticle share mass and lifetime.

Voir aussi : Lie Groups, Algebras & Representations

Interactions, Wick, LSZ

With interactions, transition amplitudes come from the Dyson series for the S-matrix; Wick’s theorem reduces time-ordered products to sums over pairings — propagators — which is precisely what Feynman diagrams draw. The LSZ reduction formula makes it honest: S-matrix elements are amputated correlation functions on shell. Scattering theory becomes combinatorics.

Feynman diagrams and QED processes

Vertices from $\mathcal L_{\mathrm{int}} = -q\bar\psi\gamma^\mu\psi A_\mu$, lines from propagators, loops integrated over. Tree-level QED: $e^+e^- \to \mu^+\mu^-$, Compton scattering, with cross sections matching experiment to spectacular precision. Perturbation theory from B&F Ch. 10, now Lorentz-covariant and drawn rather than summed.

Voir aussi : Quantum Mechanics

Path integrals

Equivalently: sum over all field histories, $Z[J] = \int\!\mathcal D\phi\; e^{\,i\left(S[\phi] + \int J\phi\right)/\hbar},$ correlation functions from functional derivatives. Manifestly covariant, the natural home of gauge-fixing and the semiclassical limit ($\hbar\to0$ recovers stationary action), and the formulation that generalizes to everything from statistical mechanics to string theory.

The gauge principle

Demand invariance under local phase rotations $\psi \to e^{iq\alpha(x)}\psi$ and a compensating field is forced into existence: $D_\mu = \partial_\mu + iqA_\mu$, transforming as a connection. Electromagnetism is not added to the theory — it is deduced from symmetry. Constrained-Hamiltonian care (Dirac–Bergmann, Williams Ch. 2) is what makes quantizing such redundant descriptions honest.

Voir aussi : Smooth Manifolds, Quantum Mechanics

Cross sections, decay rates, Mandelstam

The dictionary from amplitudes to experiment: $d\sigma \propto |\mathcal M|^2 \times$ (flux)$^{-1}$ $\times$ (Lorentz-invariant phase space); lifetimes from $\Gamma$. Kinematics organized by Mandelstam $s, t, u$ with $s + t + u = \sum m_i^2$: $s$-channel resonances, $t$-channel exchange. Williams Ch. 7 grinds these carefully — it is where theory meets the detector.

Gauge fixing and ghosts

Gauge symmetry means redundant description: quantization must fix a gauge ($R_\xi$ families; physics is $\xi$-independent — a running consistency check). In nonabelian theories the Faddeev–Popov determinant materializes as anticommuting scalar ghosts circulating in loops; BRST symmetry is the bookkeeping that certifies unitarity of the physical sector. Williams’ Dirac–Bergmann groundwork (Ch. 2) is why this is honest rather than sleight of hand.

Effective field theory

The modern renormalization moral inverted into a method: at energy $E \ll \Lambda$, write all operators allowed by symmetry, organized by powers of $E/\Lambda$; heavy physics survives only as coefficients. Fermi’s four-fermion theory is the EFT of the W boson; the Standard Model itself is presumably an EFT. “Non-renormalizable” now means “predictive at low energy, with an expiry scale.”

Renormalization

Loop integrals diverge; regularize (dimensional regularization: work in $d = 4-\epsilon$), absorb infinities into redefined masses and couplings, and physics emerges finite — with couplings that run with energy scale, $\alpha(\mu)$. Renormalization is not sweeping infinity under a rug: it is the discovery that theories are effective descriptions whose parameters depend on the resolution at which you look.

Nonabelian gauge theory

Gauge the internal symmetry group $SU(N)$: $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + ig[A_\mu, A_\nu]$ — the field strength now carries the Lie bracket, so gluons interact with gluons. Quantization needs Faddeev–Popov ghosts (BRST symmetry organizing the bookkeeping). The payoff is asymptotic freedom: the $SU(3)$ coupling weakens at high energy — QCD. Hall’s structure constants are now coupling constants.

Voir aussi : Lie Groups, Algebras & Representations

Symmetry breaking, Higgs, Standard Model

A symmetric Lagrangian with an asymmetric vacuum: global breaking yields massless Goldstone bosons; gauging turns them into the longitudinal modes of now-massive vector bosons — the Higgs mechanism. Electroweak $SU(2)_L\times U(1)_Y$ breaks to electromagnetism, $W^\pm, Z$ acquire mass, fermions couple through Yukawas, and the Standard Model — Williams’ declared endpoint — stands assembled.

Anomalies

A classical symmetry the path-integral measure refuses to respect. The chiral anomaly $\partial_\mu j_5^\mu \propto F\tilde F$ correctly predicts $\pi^0 \to \gamma\gamma$; gauge anomalies would destroy consistency — and their cancellation in the Standard Model works only because quark and lepton charges conspire, generation by generation. A quantum effect that legislates the particle content of the universe.

Voir aussi : Smooth Manifolds

Flavor: CKM and neutrino mixing

Yukawa matrices diagonalize differently for up- and down-type quarks; the mismatch is the CKM matrix — three angles and one CP-violating phase, the Standard Model’s only source of matter–antimatter asymmetry (and, per Sakharov, not nearly enough). Neutrino oscillations replicate the story in the lepton sector (PMNS). Williams Ch. 5 + 9; B&F Ch. 26 tests Bell inequalities in exactly these systems.

Voir aussi : Entanglement & Nonlocality

Confinement and the mass of the world

QCD’s running coupling explodes near $\Lambda_{\mathrm{QCD}} \sim 200$ MeV: quarks and gluons never appear free; flux tubes make the potential grow linearly, and only color singlets escape. Consequence worth savoring: $\sim\!99\%$ of the proton’s mass is confined gluon and quark kinetic energy — your weight is mostly QCD binding, not Higgs.

Worked: the Noether current of a complex scalar

$\mathcal L = \partial_\mu\phi^*\partial^\mu\phi - m^2\phi^*\phi$ is invariant under $\phi \to e^{i\alpha}\phi$. Vary with local $\alpha(x)$ and collect the $\partial_\mu\alpha$ terms — or apply the Noether recipe directly: $j^\mu = i\big(\phi^*\partial^\mu\phi - (\partial^\mu\phi^*)\phi\big), \qquad \partial_\mu j^\mu = 0 \ \text{(on-shell)}.$ The conserved charge $Q = \int d^3x\, j^0$ becomes, after quantization, (particles $-$ antiparticles): electric charge conservation is literally the phase symmetry of the field. Gauge this global symmetry and $j^\mu$ is what couples to the photon — the seam where Noether meets the gauge principle. Compare the nonrelativistic limit: it reduces to B&F’s probability current.

Worked: quantizing the Klein–Gordon field

Expand $\phi(x) = \int\!\tfrac{d^3p}{(2\pi)^3}\tfrac{1}{\sqrt{2\omega_p}}\big(a_p e^{-ipx} + a_p^\dagger e^{ipx}\big)$ and impose the field–momentum commutator $[\phi(\mathbf x), \pi(\mathbf y)] = i\delta^3(\mathbf x - \mathbf y)$. This forces $[a_p, a_{p’}^\dagger] = (2\pi)^3\delta^3(\mathbf p - \mathbf p’)$ — one oscillator algebra per momentum, exactly the QM ladder wholesale. The Hamiltonian becomes $H = \int\!\tfrac{d^3p}{(2\pi)^3}\,\omega_p\big(a_p^\dagger a_p + \tfrac12\delta^3(0)\big)$: normal-order away the (infinite, unobservable-in-flat-space) zero-point sea, and states $a_{p_1}^\dagger\cdots|0\rangle$ carry energy $\sum\omega_{p_i}$ — particles, derived rather than postulated. Causality check: $[\phi(x), \phi(y)] = 0$ at spacelike separation, by an honest contour computation.

Worked: Yukawa — force from particle exchange

Two heavy sources exchanging a scalar of mass $m$: the tree amplitude carries the propagator $\tfrac{-ig^2}{q^2 - m^2}$ with spacelike $q^2 = -|\mathbf q|^2$. Match to Born-approximation scattering off a potential and invert the Fourier transform: $V(r) = -\frac{g^2}{4\pi}\frac{e^{-mr}}{r}.$ Massive mediator $\Rightarrow$ exponential range $1/m$ (Yukawa’s 1935 logic: nuclear range $\sim$ fm predicts the pion at $\sim\!100$ MeV — found); massless $\Rightarrow$ Coulomb $1/r$. Forces are exchanged quanta — the sentence that separates field theory from everything before it, verified in one Fourier integral. Sign fine print: scalar exchange attracts like charges; vector (photon) exchange makes them repel.

Worked: running couplings, both signs

One-loop QED: vacuum polarization by electron loops screens charge, $\alpha(\mu) = \frac{\alpha(m_e)}{1 - \tfrac{2\alpha}{3\pi}\ln(\mu/m_e)}:$ $\alpha$ grows with energy — $1/137$ at atomic scales, $\approx 1/127$ at the Z mass (measured!), with a formal Landau pole far beyond physical relevance. One-loop QCD flips the sign: $\beta \propto -\big(11 - \tfrac{2n_f}{3}\big)g^3/16\pi^2$, and with $n_f \le 16$ flavors the gluon self-interaction anti-screens: asymptotic freedom (Nobel 2004) at high energy, confinement at low. Two signs of one function organize hadron physics, deep-inelastic scattering, and grand-unification dreams ($\alpha_1, \alpha_2, \alpha_3$ nearly meeting near $10^{16}$ GeV).

Topological spaces and continuity

A topology on a set $X$ is a collection $\tau$ of subsets (the open sets) closed under arbitrary unions and finite intersections, with $\varnothing, X \in \tau$. A map $f:X\to Y$ is continuous iff preimages of open sets are open — no distances required. Homeomorphism (continuous bijection with continuous inverse) is the notion of sameness: topology studies what survives stretching but not tearing.

Building spaces: subspaces, products, quotients

New spaces come from old by four constructions. The crucial and subtlest one is the quotient: glue points of $X$ together via an equivalence relation and give the result the finest topology making the projection continuous. The torus is a square with opposite edges glued; $\mathbb{RP}^n$ is the sphere with antipodes identified. Quotients are how every interesting space in this library gets built.

Connectedness and compactness

Connected: not splittable into two disjoint nonempty open sets; path-connected implies it. Compact: every open cover has a finite subcover — in $\mathbb{R}^n$ this is closed + bounded (Heine–Borel). Both are preserved by continuous maps, which is why they prove theorems: a continuous real function on a compact space attains its extremes.

Hausdorff spaces and topological manifolds

Hausdorff: distinct points have disjoint neighborhoods, so limits are unique. A topological $n$-manifold is a Hausdorff, second-countable space that is locally homeomorphic to $\mathbb{R}^n$. Spheres $S^n$, tori $T^n$, projective spaces $\mathbb{RP}^n$. This definition is the entry ticket to both Lee volumes and, ultimately, to spacetime.

Voir aussi : Smooth Manifolds

Metric spaces vs. topology

Every metric gives a topology (balls generate open sets), but not conversely: topology remembers nearness, forgets distance. Different metrics can induce the same topology (equivalent metrics), and some topologies admit no metric at all. Knowing which properties are metric (completeness, boundedness) versus topological (compactness, connectedness) prevents a whole genre of beginner errors.

Bases and subbases

A base is a family of opens from which all others arise as unions — balls in a metric space, products of intervals in $\mathbb{R}^n$. Second countable = countable base, one of the two hygiene conditions in the manifold definition (it buys partitions of unity later). Checking continuity or openness on a base is enough — the standard labor-saving device.

Voir aussi : Smooth Manifolds

CW complexes and the classification of surfaces

Build spaces by attaching cells ($n$-disks) along their boundaries — the combinatorial skeleton of topology. Payoff theorem: every compact surface is homeomorphic to exactly one of $S^2$, a connected sum of $g$ tori, or a connected sum of $k$ projective planes. One of the great complete classifications in mathematics, and your first taste of what invariants can do.

Voir aussi : Algebraic Topology

Homotopy and homotopy equivalence

Two maps are homotopic if one deforms continuously into the other; two spaces are homotopy equivalent if maps compose to identity up to homotopy. A disk is homotopy equivalent to a point (contractible); an annulus to a circle. Algebraic topology is blind to homotopy equivalence — that blindness is its power.

Voir aussi : Algebraic Topology

The fundamental group $\pi_1$

Loops at a basepoint, up to homotopy, with concatenation as the product: $\pi_1(X,p)$. The first bridge from topology to algebra, and a functor: continuous maps induce group homomorphisms. The founding computation is $\pi_1(S^1) \cong \mathbb{Z},$ the winding number — proved via the lifting theory that becomes covering-space theory two chapters later.

Voir aussi : Algebraic Topology, Lie Groups, Algebras & Representations

Local compactness and one-point compactification

Locally compact Hausdorff: every point has a compact neighborhood ($\mathbb{R}^n$ yes, $\mathbb{Q}$ no). Such a space embeds in a compact one by adding a single point at infinity: $\mathbb{R}^n \cup \{\infty\} \cong S^n$. This is why spheres keep appearing as completed Euclidean spaces — and stereographic projection is the inverse of this construction.

Group actions and orbit spaces

A group $G$ acting on $X$ yields the quotient $X/G$ of orbits. If the action is free and properly discontinuous, the quotient map is a covering and $X/G$ is as nice as $X$: $\mathbb{R}^2/\mathbb{Z}^2 = T^2$, $S^n/\{\pm 1\} = \mathbb{RP}^n$. Most covering spaces you will ever meet are built exactly this way — and $\pi_1(X/G) \cong G$ when $X$ is simply connected.

Voir aussi : Lie Groups, Algebras & Representations

Free groups and free products

Lee’s Ch. 9 algebra interlude: the free group $F(a,b)$ is all reduced words in $a^{\pm1}, b^{\pm1}$ — no relations at all; free products $G * H$ interleave syllables. Universal property: a homomorphism out of $F(S)$ is exactly a choice of images for $S$. Needed because van Kampen answers arrive as free products with amalgamation, and $\pi_1(\text{figure eight}) = F(a,b)$.

Voir aussi : Category Theory

Seifert–van Kampen

If $X = U \cup V$ with everything nicely path-connected, $\pi_1(X)$ is the free product $\pi_1(U) * \pi_1(V)$ amalgamated over $\pi_1(U\cap V)$. This is the computational engine: wedges of circles give free groups, and the genus-$g$ surface gives the one-relator group $\langle a_1,b_1,\dots \mid \prod [a_i,b_i]\rangle$.

Covering spaces and the Galois correspondence

A covering $q:\tilde X \to X$ is a local homeomorphism with evenly-covered neighborhoods; paths and homotopies lift uniquely. The classification theorem is a Galois correspondence: connected coverings of $X$ $\leftrightarrow$ conjugacy classes of subgroups of $\pi_1(X)$, with the simply connected universal cover at the top and deck transformations as the Galois group. $\mathbb{R}\to S^1$, $S^n \to \mathbb{RP}^n$, and — crucially for physics — $SU(2)\to SO(3)$: spin is a covering-space phenomenon.

Voir aussi : Lie Groups, Algebras & Representations, Quantum Mechanics

Where homology takes over

$\pi_1$ sees dimension one and is nonabelian, hence hard to compute. Homology $H_n$ trades subtlety for computability: abelian groups in every dimension, with $H_1 \cong \pi_1^{\mathrm{ab}}$ (Hurewicz in degree 1). Stop Lee here and cross to Hatcher Ch. 2 — that is the designed hand-off in your curriculum.

Voir aussi : Algebraic Topology

The circle’s covers, classified

Connected covers of $S^1$: the $n$-fold circle covers ($z \mapsto z^n$) for each $n \ge 1$, and the universal cover $\mathbb{R}$. These correspond exactly to the subgroups $n\mathbb{Z} \le \mathbb{Z}$ and the trivial subgroup — the Galois correspondence in its smallest complete example. Internalize this one picture and the general theorem feels inevitable.

Deck transformations

Automorphisms of a covering $\tilde X \to X$ — the symmetries invisible downstairs. For the universal cover, $\mathrm{Deck}(\tilde X) \cong \pi_1(X)$: the fundamental group acts on the universal cover with quotient $X$. For $\mathbb{R} \to S^1$ the deck group is translation by integers; for $S^n \to \mathbb{RP}^n$ it is the antipodal flip.

Manifolds with boundary

Model on the half-space $\mathbb{H}^n$: boundary points have half-ball neighborhoods, and $\partial M$ is an $(n{-}1)$-manifold without boundary — $\partial(\partial M) = \varnothing$, the topological shadow of $d^2 = 0$. Indispensable for Stokes’ theorem and for cobordism ideas later; Lee sets the conventions everyone else borrows.

Voir aussi : Smooth Manifolds

Worked: $\pi_1(S^1) \cong \mathbb{Z}$, the actual argument

Take the covering $p:\mathbb{R} \to S^1$, $p(t) = e^{2\pi i t}$. Path lifting: any loop $\gamma$ at $1 \in S^1$ lifts uniquely to a path $\tilde\gamma$ in $\mathbb{R}$ starting at $0$; since $p(\tilde\gamma(1)) = 1$, the endpoint $\tilde\gamma(1) = n$ is an integer — the winding number. Homotopy lifting: homotopic loops lift to homotopic paths with the same endpoint, so $n$ depends only on $[\gamma]$. The map $[\gamma] \mapsto n$ is a homomorphism (concatenated loops lift end-to-start, endpoints add), surjective ($t \mapsto e^{2\pi i n t}$ hits every $n$), and injective (if $n = 0$, $\tilde\gamma$ is a loop in the contractible space $\mathbb{R}$, so $\gamma$ is null-homotopic downstairs). Every ingredient here — lifting, deck action, contractible total space — is the covering-space machine in miniature.

Worked: van Kampen on the torus and Klein bottle

Present $T^2$ as a square with word $aba^{-1}b^{-1}$. Let $U$ = square minus center (deformation retracts to the boundary wedge $S^1 \vee S^1$, so $\pi_1(U) = F(a,b)$), $V$ = an open disk (trivial $\pi_1$), $U \cap V \simeq S^1$ generated by a loop that reads the boundary word. Van Kampen kills exactly that word: $\pi_1(T^2) = \langle a, b \mid aba^{-1}b^{-1}\rangle \cong \mathbb{Z}^2.$ Same square, word $abab^{-1}$, gives the Klein bottle: $\langle a,b \mid abab^{-1}\rangle$ — nonabelian, with $\mathbb{Z}^2$ sitting inside as an index-2 subgroup: the torus double-covers the Klein bottle. One template, every surface group.

Worked: Euler characteristic from a polygon word

A $4g$-gon with word $\prod_{i=1}^g a_i b_i a_i^{-1} b_i^{-1}$ has, after gluing: $1$ vertex (all corners identify), $2g$ edges, $1$ face. So $\chi = V - E + F = 1 - 2g + 1 = 2 - 2g.$ Sphere: $\chi = 2$; torus: $0$; genus-2: $-2$. Since $\chi$ is a homotopy invariant (provable once you have homology), surfaces of different genus are genuinely different spaces — the classification theorem’s uniqueness half in one line.

Worked: the universal cover of the torus

$\mathbb{Z}^2$ acts on $\mathbb{R}^2$ by translations — freely and properly discontinuously — with quotient $T^2$. So $q:\mathbb{R}^2 \to T^2$ is a covering; $\mathbb{R}^2$ is simply connected, hence universal. Consequences read off instantly: $\pi_1(T^2) \cong \mathrm{Deck} \cong \mathbb{Z}^2$ (matching van Kampen), every loop on the torus is classified by two integers (how many times around each handle), and higher homotopy $\pi_n(T^2) = \pi_n(\mathbb{R}^2) = 0$ for $n \ge 2$ — tori are aspherical.

Matrix Lie groups

Closed subgroups of $GL(n;\mathbb{C})$: the rotation groups $SO(n)$, unitary groups $U(n), SU(n)$, symplectic groups, the Heisenberg group, and the Lorentz group $O(3,1)$. Hall’s wager: essentially all the Lie theory physics needs lives here, and matrices let you compute from day one, deferring manifolds entirely.

Voir aussi : Smooth Manifolds, Quantum Field Theory

The matrix exponential

$e^X = \sum_{k\ge 0} X^k/k!$ converges for every matrix and solves $\tfrac{d}{dt}e^{tX} = Xe^{tX}$. One-parameter subgroups of $G$ are exactly $t\mapsto e^{tX}$; the exponential is the bridge from linear data to group elements. Alongside it: the matrix logarithm and the polar decomposition.

The Lie algebra of a group

$\mathfrak{g} = \{X : e^{tX}\in G \text{ for all } t\in\mathbb{R}\}$ — the group’s tangent space at the identity, closed under the commutator $[X,Y]=XY-YX$. The algebra linearizes the group: $\mathfrak{su}(2)$ is spanned by $-\tfrac{i}{2}\sigma_j$ with $[\sigma_i/2, \sigma_j/2]$ giving the angular-momentum relations. Almost everything about $G$ near the identity is readable from $\mathfrak{g}$.

Voir aussi : Quantum Mechanics

Representations and Schur’s lemma

A representation is a homomorphism $\Pi: G \to GL(V)$ — the group acting as linear symmetry of a vector space. Irreducible: no invariant subspaces. Schur’s lemma: an operator commuting with an irrep is a scalar — the reason Casimirs like $J^2$ take fixed values on irreducible multiplets, and the deep source of quantum numbers.

Voir aussi : Quantum Mechanics, Quantum Field Theory

Connectedness, components, and $SO$ vs. $O$

$O(n)$ has two components ($\det = \pm 1$); $SO(n)$ is the identity component. The Lorentz group has four; the proper orthochronous piece $SO^+(3,1)$ is what “Lorentz group” means in QFT. Lie algebras only see the identity component — discrete quotients and components must be tracked by hand, which is where parity and time reversal live.

Voir aussi : Quantum Field Theory

One-parameter subgroups

Every continuous homomorphism $\mathbb{R} \to G$ is $t \mapsto e^{tX}$ for a unique $X \in \mathfrak{g}$ — continuity forces smoothness (a small miracle). This is the precise sense in which $\mathfrak{g}$ is the space of “infinitesimal motions,” and in quantum mechanics it is Stone’s theorem’s finite-dimensional shadow: unitary evolution $e^{-iHt/\hbar}$ has a self-adjoint generator.

Voir aussi : Quantum Mechanics

$\mathfrak{sl}(2;\mathbb{C})$: the atom of representation theory

Basis $H, X, Y$ with $[H,X]=2X$, $[H,Y]=-2Y$, $[X,Y]=H$. Ladder analysis: every irreducible representation is determined by a highest weight $m \in \{0,1,2,\dots\}$ and has dimension $m+1$. Physicists know this as the spin-$j$ classification with $m = 2j$; raising and lowering operators are $X$ and $Y$. Every semisimple algebra is glued from copies of this one.

Voir aussi : Quantum Mechanics

Baker–Campbell–Hausdorff and the group–algebra dictionary

$\log(e^X e^Y) = X + Y + \tfrac12[X,Y] + \tfrac{1}{12}[X,[X,Y]] - \cdots$ — the group law is encoded in brackets alone. Consequence: for simply connected groups, Lie algebra homomorphisms exponentiate to group homomorphisms. When simple connectivity fails, projective phases appear: exactly why quantum mechanics represents $SU(2)$, the double cover, rather than $SO(3)$.

Voir aussi : Topology & the Fundamental Group, Quantum Mechanics

$\mathfrak{sl}(3;\mathbb{C})$ and weights

Two commuting Cartan elements now grade representations by weights in a plane; six root vectors ladder between them. The hexagonal and triangular weight diagrams you draw here are, character for character, the meson octets and baryon decuplets of the eightfold way — Gell-Mann’s flavor $SU(3)$.

Voir aussi : Quantum Field Theory

The adjoint representation and the Killing form

$G$ acts on its own algebra by conjugation: $\mathrm{Ad}_g X = gXg^{-1}$, with derivative $\mathrm{ad}_X Y = [X,Y]$. The Killing form $B(X,Y) = \operatorname{tr}(\mathrm{ad}_X\,\mathrm{ad}_Y)$ is the canonical inner product; nondegeneracy of $B$ defines semisimplicity (Cartan). For compact groups $B$ is negative definite — the reason compact gauge groups give positive kinetic terms in Yang–Mills.

Voir aussi : Quantum Field Theory

Cartan subalgebras and weights

Fix a maximal commuting set $\mathfrak{h}$ (for $\mathfrak{su}(n)$: diagonal matrices). Representations diagonalize simultaneously over $\mathfrak{h}$; the joint eigenvalues are weights, living in $\mathfrak{h}^*$. Physics reading: $\mathfrak{h}$ = the complete set of commuting quantum numbers (e.g. $I_3$ and hypercharge $Y$), weights = the labels on the states in a multiplet.

Voir aussi : Quantum Mechanics

Simple, semisimple, reductive

Simple: no nontrivial ideals (irreducible symmetry). Semisimple: direct sum of simples. Reductive: semisimple $\oplus$ center — e.g. $\mathfrak{u}(n) = \mathfrak{su}(n)\oplus\mathfrak{u}(1)$. The Standard Model group $SU(3)\times SU(2)\times U(1)$ is reductive, not simple — which is exactly why it has three independent couplings and why grand unification (embedding in one simple group) is tempting.

Voir aussi : Quantum Field Theory

Root systems and Dynkin diagrams

Abstracting the pattern: roots are the nonzero weights of the adjoint representation, reflected into each other by the Weyl group. The axioms force the crystallographic classification $A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$ — all possible simple Lie algebras, hence in a sense all possible continuous symmetries, from a finite list of diagrams.

The theorem of the highest weight

Irreducible finite-dimensional representations of a semisimple Lie algebra $\leftrightarrow$ dominant integral weights, bijectively. Hall proves it three ways (Verma modules, unitarian trick, Peter–Weyl-style); the Verma-module proof — generate freely from a highest weight vector, then quotient — is the template for infinite-dimensional representation theory too.

Weyl character formula

$\mathrm{ch}\,V_\lambda = \dfrac{\sum_{w\in W} \varepsilon(w)\, e^{w(\lambda+\rho)}}{\sum_{w\in W} \varepsilon(w)\, e^{w(\rho)}}$ — every character, hence every multiplicity and dimension, from Weyl-group combinatorics. The dimension formula it implies ($\dim V_\lambda = \prod_{\alpha>0}\langle \lambda+\rho,\alpha\rangle / \langle\rho,\alpha\rangle$) is the standard tool for counting states in a multiplet.

Clebsch–Gordan and Wigner–Eckart (Appendix C)

Tensor products decompose: $V_j \otimes V_k \cong V_{j+k}\oplus V_{j+k-1} \oplus \cdots \oplus V_{|j-k|}$ — the addition of angular momenta, with Clebsch–Gordan coefficients as the change of basis. Wigner–Eckart then factors matrix elements of tensor operators into (geometry) $\times$ (one reduced element): the origin of atomic selection rules. This appendix is where the math track pays its debt to quantum mechanics in full.

Voir aussi : Quantum Mechanics, Entanglement & Nonlocality

The unitarian trick

Representations of a compact group are all unitarizable (average an inner product over Haar measure), hence completely reducible. Weyl’s trick transports this to noncompact/complex settings: $\mathfrak{sl}(n,\mathbb{C})$ shares representation theory with compact $\mathfrak{su}(n)$. This is why physicists compute with hermitian generators and get away with it.

Fundamental weights and Dynkin labels

Dominant integral weights are $\mathbb{Z}_{\ge0}$-combinations of fundamental weights $\varpi_i$ (one per node of the Dynkin diagram); an irrep is a tuple of Dynkin labels. $\mathfrak{su}(3)$: $(1,0) = \mathbf{3}$, $(0,1) = \bar{\mathbf 3}$, $(1,1) = \mathbf 8$, $(3,0) = \mathbf{10}$ — the quark, antiquark, meson octet, and baryon decuplet, straight off the diagram.

Voir aussi : Quantum Field Theory

Peter–Weyl in one breath (Part III)

For a compact group, matrix coefficients of irreps are dense in $L^2(G)$: harmonic analysis is representation theory. Fourier series = Peter–Weyl for $U(1)$; spherical harmonics = Peter–Weyl for $SO(3)$ acting on $S^2$. Optional in your plan, but it explains why $Y_{\ell m}$ keep appearing in B&F Ch. 6–7.

Voir aussi : Quantum Mechanics

Worked: exponentiating $\mathfrak{so}(2)$ and a nilpotent

Rotation generator $J = \begin{pmatrix}0&-1\\1&0\end{pmatrix}$: powers cycle ($J^2 = -\mathbb{1}$), so the series splits into sine and cosine: $e^{\theta J} = \cos\theta\,\mathbb{1} + \sin\theta\, J = \begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.$ Nilpotent $N = \begin{pmatrix}0&1\\0&0\end{pmatrix}$: series truncates, $e^{tN} = \mathbb{1} + tN$ — shears. Every matrix exponential you will ever need is a mixture of these two behaviors (plus real eigenvalue stretching), via the Jordan form.

Worked: $SU(2) \to SO(3)$, the double cover explicitly

Map $\mathbb{R}^3 \to$ traceless hermitian matrices by $\vec v \mapsto \vec v\cdot\vec\sigma$. For $g \in SU(2)$, conjugation $\vec v\cdot\vec\sigma \mapsto g(\vec v\cdot\vec\sigma)g^\dagger$ preserves trace, hermiticity, and $\det(\vec v \cdot \vec\sigma) = -|\vec v|^2$ — so it is a rotation $R(g)\vec v$. The map $g \mapsto R(g)$ is a homomorphism onto $SO(3)$ with kernel $g(\vec v\cdot\vec\sigma)g^\dagger = \vec v\cdot\vec\sigma\ \forall \vec v \Rightarrow g = \pm\mathbb{1}$. Hence $SO(3) \cong SU(2)/\{\pm\mathbb 1\}$: two-to-one, and since $SU(2) \cong S^3$ is simply connected, it is the universal cover. Check the famous consequence: $g = e^{-i\pi\sigma_z/2\cdot 2} = e^{-i\pi\sigma_z}= -\mathbb 1$ maps to a $2\pi$ rotation — spinors flip sign.

Worked: building the spin-1 irrep from the highest weight

In $\mathfrak{sl}(2)$ take $H, X, Y$ with $[H,X] = 2X$, $[X,Y] = H$. Start from $v_0$ with $Hv_0 = 2v_0$, $Xv_0 = 0$ (highest weight $m = 2$). Ladder down: $v_1 = Yv_0$, $v_2 = Yv_1$; the relations force $Hv_1 = 0$, $Hv_2 = -2v_2$, and $Yv_2 = 0$ (the series terminates because $XY^{k+1}v_0 = (k{+}1)(m-k)Y^k v_0$ vanishes at $k = m$). Three states of weight $2, 0, -2$: in physics units ($H = 2J_z$), that is $j = 1$ with $m_j = 1, 0, -1$. The Casimir $J^2$ acts as $j(j{+}1) = 2$ on all three — Schur in action. Every irrep of every semisimple algebra is built by exactly this descent.

Worked: $\tfrac12\otimes\tfrac12 = 1 \oplus 0$, and it is the Bell basis

Two spin-$\tfrac12$ systems: total $J_z$ eigenvalues are $1, 0, 0, -1$. Top state $|{\uparrow\uparrow}\rangle$ seeds the triplet; lowering gives $\tfrac{1}{\sqrt2}(|{\uparrow\downarrow}\rangle + |{\downarrow\uparrow}\rangle)$ and $|{\downarrow\downarrow}\rangle$. The orthogonal combination $|0,0\rangle = \tfrac{1}{\sqrt2}(|{\uparrow\downarrow}\rangle - |{\downarrow\uparrow}\rangle)$ is annihilated by all of $\vec J$: the singlet. Now notice: the singlet is the Bell state $|\Psi^-\rangle$, and the triplet’s $m=0$ member is $|\Psi^+\rangle$. Clebsch–Gordan coefficients are the entries of the change of basis product $\to$ Bell — representation theory and entanglement theory meet in a $4\times4$ matrix.