Symmetries

From Susskind's Theoretical Minimum Book

Classical Mechanics: Symmetries & Conservation Laws

1. General Conservation (Noether's Theorem)

Symmetry Transformation: $\delta q_i = \epsilon f_i(q)$
Conserved Quantity: $Q = \sum_i p_i f_i(q)$
Conjugate Momentum: $p_i = \frac{\partial L}{\partial \dot{q}_i}$
Conjugate (Generalized) Force: $F_i = \frac{\partial L}{\partial q_i}$ (connected to momentum via the Euler-Lagrange equation: $\dot{p}_i = F_i$)

2. Cyclic Coordinates & Explicit Dependence

If a Lagrangian $L(q, \dot{q}, t)$ or Hamiltonian $H(q, p, t)$ does not explicitly depend on a specific variable, the corresponding conjugate quantity is conserved.

Cyclic Coordinate ($q_i$): $\frac{\partial L}{\partial q_i} = 0 \implies F_i = 0 \implies \dot{p}_i = 0 \implies p_i$ is conserved.
Linear Coordinate: $\frac{\partial L}{\partial x} = 0 \implies p_x$ (linear momentum) is conserved.
Angular Coordinate: $\frac{\partial L}{\partial \theta} = 0 \implies p_\theta$ (angular momentum) is conserved.
Time Independence ($t$): $\frac{\partial L}{\partial t} = 0 \implies \frac{dH}{dt} = 0 \implies H$ (Energy) is conserved.
Hamiltonian Equivalence: The rules hold identically in phase space.
$\frac{\partial H}{\partial q_i} = 0 \implies p_i$ is conserved (since $\dot{p}_i = -\frac{\partial H}{\partial q_i}$).
$\frac{\partial H}{\partial t} = 0 \implies H$ is conserved.

3. Lagrangian Symmetries Table

Condition	Symmetry Transformation	Coordinate Shift	Function $f$	Conserved Quantity ($Q$)
$\frac{\partial L}{\partial q_i} = 0$	Spatial Translation (1D)	$\delta q_i = \epsilon$	$f_i = 1$	Linear Momentum: $p_i$
Uniform displacement	Spatial Translation (3D)	$\delta \vec{r} = \epsilon \hat{n}$	$\vec{f} = \hat{n}$	Total Momentum: $\vec{P} \cdot \hat{n}$
Axis $\hat{n}$ rotation	Spatial Rotation	$\delta \vec{r} = \epsilon (\hat{n} \times \vec{r})$	$\vec{f} = \hat{n} \times \vec{r}$	Angular Momentum: $\hat{n} \cdot \vec{L}$

4. Hamiltonian Evolution & Conservation

Equation of Motion: $\frac{dA}{dt} = {A, H} + \frac{\partial A}{\partial t}$
Conservation Condition (for $\frac{\partial A}{\partial t} = 0$): ${A, H} = 0$
Generator of Symmetry ($G$): $\delta A = \epsilon {A, G}$
Translation Generator Example: $\delta q_i = \epsilon {q_i, p_i} = \epsilon$

5. Poisson Brackets

General Definition: For any two observables $A(q,p,t)$ and $B(q,p,t)$:

$${A, B} = \sum_i \left( \frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial B}{\partial q_i} \right)$$

Fundamental Brackets:
Position-Position: ${q_i, q_j} = 0$
Momentum-Momentum: ${p_i, p_j} = 0$
Position-Momentum: ${q_i, p_j} = \delta_{ij}$

6. Poisson Bracket Operations & Checks

Linearity Rule: ${aA + bB, C} = a{A, C} + b{B, C}$
Product Rule (Leibniz): ${AB, C} = A{B, C} + {A, C}B$
Linear Momentum Conservation Check: ${p_i, H} = -\frac{\partial H}{\partial q_i} = 0$
Angular Momentum Conservation Check: ${L_z, H} = {x p_y - y p_x, H} = 0$
Expansion for $L_z$ Check: $x{p_y, H} + {x, H}p_y - y{p_x, H} - {y, H}p_x = 0$

7. Hamiltonian Formulation of Noether's Theorem

The equations below define an Infinitesimal Canonical Transformation generated by the phase-space function $g$.

$$q_i \to \bar{q}_i = q_i + \varepsilon \frac{\partial g}{\partial p_i} \equiv q_i + \delta q_i$$

$$p_i \to \bar{p}_i = p_i - \varepsilon \frac{\partial g}{\partial q_i} \equiv p_i + \delta p_i$$

$g$ is conserved if the Hamiltonian is invariant under this transformation of coordinates.

Dirac Notation

From R Shankar's Book, Ch 01

Vector Spaces and Inner Products

Inner Product Symmetry: $\langle V \vert{} W \rangle = \langle W \vert{} V \rangle^*$
Linearity (Ket): $\langle V \vert{} aW_1 + bW_2 \rangle = a\langle V \vert{} W_1 \rangle + b\langle V \vert{} W_2 \rangle$
Anti-linearity (Bra): $\langle aV_1 + bV_2 \vert{} W \rangle = a^\langle V_1 \vert{} W \rangle + b^\langle V_2 \vert{} W \rangle$
Norm (Length): $\vert{}V\vert{} = \sqrt{\langle V \vert{} V \rangle}$
Cauchy-Schwarz Inequality: $\vert{}\langle V \vert{} W \rangle\vert{} \le \vert{}V\vert{} \cdot \vert{}W\vert{}$
Triangle Inequality: $\vert{}V + W\vert{} \le \vert{}V\vert{} + \vert{}W\vert{}$
Orthonormality Condition: $\langle i \vert{} j \rangle = \delta_{ij}$
Completeness Relation (Discrete): $\sum_i \vert{}i\rangle\langle i\vert{} = I$
Expansion of a Ket: $\vert{}V\rangle = \sum_i v_i \vert{}i\rangle$ where $v_i = \langle i \vert{} V \rangle$
Gram-Schmidt Orthogonalization: Converts linearly independent vectors $\vert{}v_i\rangle$ into an orthonormal basis $\vert{}i\rangle$.
$\vert{}1\rangle = \frac{\vert{}v_1\rangle}{\vert{}v_1\vert{}}$
$\vert{}2'\rangle = \vert{}v_2\rangle - \vert{}1\rangle\langle 1 \vert{} v_2 \rangle$, then normalize: $\vert{}2\rangle = \frac{\vert{}2'\rangle}{\vert{}2'\vert{}}$

Operators and Matrices

Matrix Elements: $\Omega_{ij} = \langle i \vert{} \Omega \vert{} j \rangle$
Action of an Outer Product: $(\vert{}V\rangle\langle W\vert{})\vert{}X\rangle = \langle W \vert{} X \rangle\vert{}V\rangle$
Adjoint (Hermitian Conjugate): $\langle V \vert{} \Omega \vert{} W \rangle = \langle \Omega^\dagger V \vert{} W \rangle$
Matrix Elements of Adjoint: $(\Omega^\dagger){ij} = \Omega{ji}^*$
Adjoint of a Product: $(\Omega \Lambda)^\dagger = \Lambda^\dagger \Omega^\dagger$
Trace Definition: $\text{Tr}(\Omega) = \sum_i \Omega_{ii} = \sum_i \langle i \vert{} \Omega \vert{} i \rangle$
Cyclic Property of Trace: $\text{Tr}(\Omega\Lambda) = \text{Tr}(\Lambda\Omega)$
Basis Independence: Both $\text{Tr}(\Omega)$ and $\det(\Omega)$ are invariant under a change of basis.

Key Operator Types and Theorems

Hermitian Operator ($\Omega^\dagger = \Omega$): Represents physical observables.
Eigenvalues are strictly real ($\omega_i = \omega_i^*$).
Eigenvectors corresponding to distinct eigenvalues are orthogonal ($\langle \omega_i \vert{} \omega_j \rangle = 0$ for $\omega_i \neq \omega_j$).
Anti-Hermitian Operator: $\Omega^\dagger = -\Omega$
Unitary Operator ($U^\dagger U = U U^\dagger = I$): Represents physical transformations.
Preserves the inner product (norm): $\langle U V \vert{} U W \rangle = \langle V \vert{} W \rangle$.
Eigenvalues are unimodular phase factors ($u_i = e^{i\theta}$).
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Projection Operator: $\mathbb{P}_i = \vert{}i\rangle\langle i\vert{}$. Satisfies idempotence: $\mathbb{P}_i^2 = \mathbb{P}_i$.

Commutators and Functions of Operators

Commutator: $[\Omega, \Lambda] = \Omega\Lambda - \Lambda\Omega$
Anti-commutator: ${\Omega, \Lambda} = \Omega\Lambda + \Lambda\Omega$
Commutator Identities:
$[\Omega, \Lambda\Theta] = \Lambda[\Omega, \Theta] + [\Omega, \Lambda]\Theta$
$[\Lambda\Omega, \Theta] = \Lambda[\Omega, \Theta] + [\Lambda, \Theta]\Omega$
Taylor Series Expansion: $f(\Omega) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} \Omega^n$
Action on Eigenkets: $f(\Omega)\vert{}\omega_i\rangle = f(\omega_i)\vert{}\omega_i\rangle$
Function Matrix (Spectral Decomposition): $f(\Omega) = \sum_i f(\omega_i) \vert{}i\rangle\langle i\vert{}$

The Eigenvalue Problem

Characteristic Equation: $\det(\Omega - \omega I) = 0$
Spectral Decomposition: $\Omega = \sum_i \omega_i \vert{}i\rangle\langle i\vert{}$
Trace in terms of Eigenvalues: $\text{Tr}(\Omega) = \sum_i \omega_i$
Determinant in terms of Eigenvalues: $\det(\Omega) = \prod_i \omega_i$
Degeneracy: Occurs when multiple linearly independent eigenvectors share the same eigenvalue. The Gram-Schmidt procedure is used within this degenerate subspace to enforce orthogonality.
Simultaneous Diagonalization: Two Hermitian operators $\Omega$ and $\Lambda$ share a complete basis of simultaneous eigenvectors if and only if they commute ($[\Omega, \Lambda] = 0$).

Infinite Dimensions and Continuous Bases

Dirac Delta Function (Orthonormality): $\langle x \vert{} x' \rangle = \delta(x-x')$
Completeness Relation (Continuous): $\int \vert{}x\rangle\langle x\vert{} dx = I$
Inner Product of Functions: $\langle f \vert{} g \rangle = \int f^*(x)g(x) dx$
Expansion of a Ket: $\vert{}f\rangle = \int f(x)\vert{}x\rangle dx$, where $f(x) = \langle x \vert{} f \rangle$
Continuous Matrix Elements: $\langle x \vert{} \Omega \vert{} x' \rangle = \Omega(x, x')$
Derivative of Delta Function: $\int f(x)\delta'(x-a) dx = -f'(a)$ (Integrating by parts shifts the derivative to the function $f(x)$).

regstuff/symmetries_dirac_notation.md

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