Group Theory and Gauge Symmetries in Physics

Demonstrates the invariance of the spacetime interval ds^2 = c^2 dt^2 - (dx^2+dy^2+dz^2) across inertial frames using three frames and light-ray conditions; concludes the invariant form requires a constant function.

Demonstrates how the invariant ds squared comes from the Minkowski metric eta_mu nu using Einstein summation, and shows Lorentz transformations preserve this metric.

Discover how a vector field transforms under infinitesimal and finite Lorentz transformations, using generators sigma beta alpha and the Lorentz algebra, and compare with spinor rotation properties.

Explore how gamma matrices define spinor transformations under the Lorentz algebra, using the Clifford algebra to show spinors acquire minus identity after a full 360-degree rotation.

Demonstrates that the vector transformation generators sigma mu nu satisfy the Lorentz Lie algebra, using matrix and index notation, commutators, and relabeling to reveal the minus i structure.

Show how the spinor transformation generators sigma mu nu satisfy the Lorentz lie algebra, using gamma matrices, their anti-commutators, and commutators to derive relations.

From the Lorentz lie algebra, define J_i = -1/2 epsilon_{i mn} M^{mn}, and show [J_i, J_j] = i hbar epsilon_{ijk} J_k.

Show how charge conservation arises from the Dirac equation through the current j_mu = psi_bar gamma_mu psi, with del_mu j_mu = 0, and the hermitian Dirac Lagrangian for local gauge invariance.

Derive the commutator of covariant derivatives to obtain the electromagnetic tensor f_mu nu. Relate the gauge covariant derivative to photons and electrons, and mention the Riemann tensor.

Explore the Dirac lagrangian for quarks, their flavors and color SU(3) structure, and introduce gluons and the covariant derivative to realize local gauge invariance in QCD.

derive the gluon field strength tensor in QCD from SU(3) covariant derivatives, compute the invariant trace, and establish the QCD Lagrangian with the gluon kinetic term.

Explore the orbital angular momentum l^2 in quantum mechanics, deriving its form with Levi-Civita and Kronecker deltas, and connecting it to r^2 laplacian and r d/dr in spherical coordinates.

Derive the Laplacian in spherical coordinates from the gradient using r hat and the theta and phi unit vectors, and relate it to the L squared operator.

This lecture derives separation of variables for angular momentum, showing common eigenstates of L^2 and Lz, and derives the Legendre differential equation using x = cos phi.

Solve the Legendre differential equation for m=0 using a power-series around x=1, derive a recurrence, and connect to Legendre polynomials P_l(x) via hypergeometric series.

The lecture explains that the eigenfunctions y_lm(omega) of L^2 and L_z satisfy L^2 y_lm(omega) = hbar^2 l(l+1) y_lm(omega) and L_z y_lm(omega) = m hbar y_lm(omega); a general L^2 eigenfunction is a linear combination over m.

Explore Legendre polynomials p_l(x), their derivative-related relatives, and spherical harmonics y_lm, proving orthogonality on [-1,1] and introducing the normalization constant h_l while deriving the Legendre differential equation.

Explore the Rodrigues formula for Legendre polynomials, prove p_l(1)=1 by induction, and demonstrate how derivatives of (1−x^2)^l yield a solution to Legendre's differential equation.

Apply Rodrigues formula to compute the Legendre normalization constant h_l, showing h_l = 2/(2l+1) from ∫_{-1}^{1} p_l(x)^2 dx. Normalize Legendre polynomials so ∫ p_l p_{l'}/√(h_l h_{l'}) dx = δ_{ll'}.

The lecture proves the completeness relation for Legendre polynomials and shows how normalized Legendre polynomials reproduce the Dirac delta through function expansion.

Explore the generalized Legendre polynomials p_ml(x), derive their Rodriguez-based form, and show a large-x proportionality p_ml(x) = (-1)^m (l+m)!/(l-m)! p_-m l(x), setting up orthogonality in the next lecture.

Demonstrate the orthogonality of generalized Legendre polynomials p_ml by evaluating ∫_{-1}^1 p_ml(x) p_l'm(x) dx via Rodrigues formula, giving zero for l ≠ l' and h_ml for l = l'.

Expand any wave function on the sphere into spherical harmonics Y_lm(omega) with coefficients c_lm. The squared magnitudes |c_lm|^2 give probabilities to measure L^2 and L_z eigenvalues through orthogonality relations.

Show the addition theorem of spherical harmonics by applying the completeness relation, expanding the delta function in Legendre polynomials, and expressing cos gamma as a dot product of normals.

Explore the addition theorem for spherical harmonics and its realness under complex conjugation. Verify L=0 and L=1 cases using Y_l^m and cos gamma from normals.

Define the Wigner D matrices as rotation representations of spherical harmonics. Rewrite y_{m' l} as a rotation acting on a solid angle omega and expand in terms of dl_{m' m}.

Derives the explicit Wigner matrices d^j_{m' m}(u) by expanding rotated spinor polynomials via the binomial theorem for co2 rotations, linking them to three-parameter rotations and Euler angles.

Master the gradient in spherical coordinates using index notation and the Jacobian. Transform from x, y, z to r, theta, phi and implement the calculation in Matlab.

Derive the laplacian in spherical coordinates by applying the nabla operator to the gradient, using index notation and Matlab to verify equivalence with standard spherical forms.

Define an invariant measure for SU(2) via the Euler-angle parameterization, deriving the invariant volume element from the metric determinant, and verify with a Matlab script.

Compute the normalized invariant volume element for SU(2) by evaluating a triple integral over Euler angles and dividing by 16 pi squared, ensuring the measure du integrates to one.

Compare the L2 space of square-integrable functions on U(1) and SU(2), using Fourier basis e^{inθ} and Wigner matrices, highlighting Peter–Weyl completeness and delta constructions.

Examine how rank-two symmetric traceless covariant tensors transform under rotations and correspond to spin two, via the rotation generator j and e^{i theta j}, with tracelessness ensuring proper spin-two states.

Learn how gauge symmetry in electromagnetism yields a constrained path integral and introduces Faddeev-Popov ghosts, with landau gauge fixing and determinants ensuring consistency in the quantum theory.

Explore how path integrals yield the canonical commutation relation between the phi field and its momentum pi for a real scalar field, via time slicing.

Derives the path integral for a single particle from the Schrödinger equation, introducing the kernel, completeness, and time slicing, and presents the sum over all paths weighted by e^{i S[q]}.

Apply path integral to evolve from initial to final configurations by summing over all paths with exp(i S/ħ) and extend to fields via a Lagrangian density and partition function Z.