Mathematical Prerequisites

A smooth (or differentiable) manifold of dimension \(n\) is a topological space \(M\) with a smooth structure, meaning:

• \(M\) is a Hausdorff space.
• \(M\) is covered by open sets \(\{U_i\}\), each homeomorphic to an open set in \(\mathbb{R}^n\).
• Transition functions between overlapping charts are smooth (infinitely differentiable).

A smooth map \(f: M \to N\) between manifolds is continuous and smooth in local coordinates. A diffeomorphism is a smooth bijection with smooth inverse.

At each point \(p\) on a smooth manifold \(M\), the tangent space \(T_pM\) is the vector space of all tangent vectors at \(p\). A tangent vector can be represented as a directional derivative acting on smooth functions. The dimension of \(T_pM\) equals the dimension of \(M\). The tangent bundle \(TM\) is the union of all tangent spaces: \(TM = \bigcup_{p \in M} T_pM\).

Given a smooth map \(f: M \to N\) and a point \(p \in M\), the differential (or pushforward) \(df_p: T_pM \to T_{f(p)}N\) is the linear map that takes a tangent vector \(v \in T_pM\) to its image under \(f\). In local coordinates, \(df_p\) is the Jacobian matrix of \(f\) at \(p\).

A vector field \(X\) on a manifold \(M\) is a smooth assignment of a tangent vector \(X_p \in T_pM\) to each point \(p \in M\). Vector fields can be added and scaled, forming a module over the ring of smooth functions \(C^\infty(M)\).

A key example is the gradient of a function: if \(f: M \to \mathbb{R}\) is smooth, the gradient vector field \(\nabla f\) points in the direction of steepest increase of \(f\).

A \(k\)-form on a manifold \(M\) is a smooth assignment to each point \(p \in M\) of an alternating multilinear functional on \(T_pM\).

Key examples:

• A 0-form is a smooth function \(f: M \to \mathbb{R}\).
• A 1-form \(\alpha\) assigns to each point and tangent vector a real number: \(\alpha_p: T_pM \to \mathbb{R}\).
• A 2-form \(\omega\) assigns to each point an alternating bilinear functional on pairs of tangent vectors.
• The exterior derivative \(d\) increases the degree by one: \(d: \Omega^k(M) \to \Omega^{k+1}(M)\).

The exterior derivative satisfies \(d^2 = 0\) and is defined without reference to a metric.

A Riemannian metric \(g\) on a smooth manifold \(M\) assigns to each point \(p \in M\) an inner product \(g_p: T_pM \times T_pM \to \mathbb{R}\) that varies smoothly. In local coordinates \((x^1, \ldots, x^n)\), the metric is written:

An affine connection \(\nabla\) on a manifold \(M\) is a rule for differentiating vector fields. Given vector fields \(X\) and \(Y\), the covariant derivative \(\nabla_X Y\) is another vector field satisfying:

• Linearity: \(\nabla_{aX + bY} Z = a\nabla_X Z + b\nabla_Y Z\) for smooth functions \(a, b\).
• Product rule: \(\nabla_X(fY) = (Xf)Y + f\nabla_X Y\) for smooth \(f\).
• Torsion: The torsion tensor \(T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]\) vanishes for a torsion-free connection.

In local coordinates, the connection is specified by Christoffel symbols \(\Gamma^\lambda_{\mu\nu}\):

On a Riemannian manifold, the unique torsion-free connection compatible with the metric is called the Levi-Civita connection. Its Christoffel symbols are given by:

The Riemann curvature tensor \(R\) measures how much parallel transport around an infinitesimal loop fails to return a vector to its original value. In components, it is defined through the commutator of covariant derivatives:

The Ricci tensor is the trace of the Riemann tensor:

For a 2-dimensional Riemannian manifold, the Gaussian curvature \(K\) at a point is the product of the principal curvatures. It can be computed from the metric:

The Euler characteristic \(\chi(M)\) of a finite CW complex \(M\) is defined as:

Key values:

• \(\chi(S^2) = 2\) (2 vertices, 3 edges, 1 face in one triangulation).
• \(\chi(T^2) = 0\) (flat torus).
• \(\chi(\Sigma_g) = 2 - 2g\) (general surface of genus \(g\)).

A homeomorphism is a continuous bijection with continuous inverse. Two topological spaces are homeomorphic if there exists a homeomorphism between them.

A diffeomorphism is a smooth bijection with smooth inverse. Two smooth manifolds are diffeomorphic if there exists a diffeomorphism between them.

Every diffeomorphism is a homeomorphism, but not conversely. Diffeomorphism is a stronger equivalence.

A fiber bundle with total space \(E\), base space \(B\), and fiber \(F\) is a space \(E\) with a surjective map \(\pi: E \to B\) such that:

• For each \(b \in B\), the fiber \(\pi^{-1}(b)\) is homeomorphic to \(F\).
• Locally, \(E\) looks like \(B \times F\): for each \(b \in B\), there exists a neighborhood \(U\) of \(b\) and a homeomorphism \(\phi: \pi^{-1}(U) \to U \times F\) such that \(\pi\) becomes projection onto the first factor.

A principal \(G\)-bundle is a fiber bundle with structure group \(G\) acting on the fibers. The gauge fields of physics are connections on principal bundles.

A curve \(\gamma(t)\) on a Riemannian manifold \((M, g)\) is a geodesic if its acceleration vector is orthogonal to the manifold, i.e., if \(\nabla_{\dot{\gamma}} \dot{\gamma} = 0\). In local coordinates, the geodesic equation is:

Geodesics minimize length locally: a short enough geodesic segment between two points is the shortest curve connecting them.

A vector field \(V\) is parallel along a curve \(\gamma\) if \(\nabla_{\dot{\gamma}} V = 0\). Given an initial vector \(V_0\) at \(\gamma(0)\), parallel transport along \(\gamma\) defines a unique vector field \(V(t)\) satisfying the parallel transport equation.

The parallel transport map \(P_\gamma: T_{\gamma(0)}M \to T_{\gamma(1)}M\) is a linear isomorphism. When the curve is closed, parallel transport around the loop generally does not return vectors to their original position; the failure is measured by the holonomy.

The exponential map \(\exp_p: T_pM \to M\) at a point \(p\) is defined by:

For a Riemannian manifold, \(\exp_p\) is a diffeomorphism from a neighborhood of the origin in \(T_pM\) to a neighborhood of \(p\) in \(M\) (the normal neighborhood). This allows us to use Cartesian coordinates on \(T_pM\) to describe distances in \(M\).

A Lie group is a smooth manifold \(G\) with a group structure such that:

• Multiplication \(G \times G \to G: (g, h) \mapsto gh\) is smooth.
• Inversion \(G \to G: g \mapsto g^{-1}\) is smooth.

Examples:

• \(\mathbb{R}\) under addition.
• \(\mathbb{R}^+\) under multiplication.
• The circle \(\mathrm{U}(1) = \{z \in \mathbb{C} : |z| = 1\}\) under multiplication.
• The special unitary group \(\mathrm{SU}(n)\): \(n \times n\) unitary matrices with determinant 1.
• The special orthogonal group \(\mathrm{SO}(n)\): \(n \times n\) orthogonal matrices with determinant 1.

The Lie algebra \(\mathfrak{g}\) of a Lie group \(G\) is the tangent space at the identity element \(e\): \(\mathfrak{g} = T_e G\). It is equipped with the Lie bracket \([X, Y] = \lim_{t \to 0} t^{-2}(\exp(tX)\exp(tY)\exp(-tX)\exp(-tY))\) for \(X, Y \in \mathfrak{g}\).

The Lie bracket encodes the local structure of the group. For matrix groups, the Lie bracket is the commutator: \([A, B] = AB - BA\).

Key examples of Lie algebras:

• \(\mathfrak{u}(1)\): the real numbers \(\mathbb{R}\).
• \(\mathfrak{su}(2)\): \(2 \times 2\) traceless skew-Hermitian matrices (dimension 3).
• \(\mathfrak{su}(3)\): \(3 \times 3\) traceless skew-Hermitian matrices (dimension 8).
• \(\mathfrak{so}(3)\): \(3 \times 3\) skew-symmetric real matrices (dimension 3).

For a Lie group \(G\) with Lie algebra \(\mathfrak{g}\), the exponential map \(\exp: \mathfrak{g} \to G\) is defined by:

The exponential map is a local diffeomorphism from a neighborhood of zero in \(\mathfrak{g}\) to a neighborhood of the identity in \(G\).

A representation of a Lie group \(G\) is a smooth homomorphism \(\rho: G \to \mathrm{GL}(V)\), where \(\mathrm{GL}(V)\) is the group of invertible linear transformations of a vector space \(V\).

A representation of the Lie algebra \(\mathfrak{g}\) is a linear map \(\rho: \mathfrak{g} \to \mathfrak{gl}(V)\) (the Lie algebra of \(\mathrm{GL}(V)\)) satisfying \(\rho([X,Y]) = [\rho(X), \rho(Y)]\).

For each Lie group representation, there is an associated Lie algebra representation obtained by differentiating at the identity.

The isometry group \(\mathrm{Iso}(M)\) of a Riemannian manifold \((M, g)\) is the group of all diffeomorphisms that preserve the metric: \(\phi^* g = g\), where \(\phi^*\) denotes the pullback of the metric.

The special unitary group \(\mathrm{SU}(n)\) is the group of \(n \times n\) unitary matrices with determinant 1:

The Lie algebra \(\mathfrak{su}(n)\) consists of traceless skew-Hermitian \(n \times n\) matrices. Its dimension is \(n^2 - 1\).

Key dimensions:

• \(\dim \mathfrak{su}(2) = 3\). Generators: Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\) (up to normalization).
• \(\dim \mathfrak{su}(3) = 8\). Generators: Gell-Mann matrices \(\lambda_1, \ldots, \lambda_8\).

The adjoint representation of a Lie group \(G\) is the action of \(G\) on its own Lie algebra \(\mathfrak{g}\). For \(g \in G\) and \(X \in \mathfrak{g}\):

The corresponding Lie algebra representation (adjoint representation of \(\mathfrak{g}\)) is:

An inner product space is a vector space \(V\) over \(\mathbb{C}\) (or \(\mathbb{R}\)) equipped with an inner product \(\langle \cdot, \cdot \rangle: V \times V \to \mathbb{C}\) satisfying:

• Linearity in the second argument: \(\langle x, ay + bz \rangle = a\langle x, y \rangle + b\langle x, z \rangle\).
• Conjugate symmetry: \(\langle x, y \rangle = \overline{\langle y, x \rangle}\).
• Positive definiteness: \(\langle x, x \rangle \geq 0\), with equality iff \(x = 0\).

The inner product induces a norm \(\|x\| = \sqrt{\langle x, x \rangle}\) and a distance \(d(x,y) = \|x - y\|\).

A Hilbert space is a complete inner product space: an inner product space in which every Cauchy sequence converges. Completeness is essential for analysis, as it ensures that limits of sequences belong to the space.

Examples:

• \(\mathbb{C}^n\) with the standard inner product \(\langle x, y \rangle = \sum_i \bar{x}_i y_i\).
• \(\ell^2(\mathbb{N}) = \left\{(x_1, x_2, \ldots) : \sum_i |x_i|^2 < \infty\right\}\) with \(\langle x, y \rangle = \sum_i \bar{x}_i y_i\).
• \(L^2(\mathbb{R}) = \left\{f: \mathbb{R} \to \mathbb{C} : \int_\mathbb{R} |f(x)|^2\,dx < \infty\right\}\) with \(\langle f, g \rangle = \int_\mathbb{R} \bar{f}(x)g(x)\,dx\).

A linear operator \(A: H \to H\) on a Hilbert space \(H\) is bounded if there exists \(M > 0\) such that:

The operator norm is defined as:

Examples of bounded operators: matrices (acting on \(\mathbb{C}^n\)), multiplication operators, compact operators.

An unbounded operator \(A\) is defined not on all of \(H\), but on a dense subspace \(\mathrm{Dom}(A) \subset H\) (the domain). Many operators in quantum mechanics are unbounded: the position operator \(\hat{x}\), the momentum operator \(\hat{p}\), and the Hamiltonian \(\hat{H}\) are all unbounded.

For physical observables, we require self-adjoint operators (their own adjoints): \(A = A^\dagger\).

For a linear operator \(A\) on a Hilbert space \(H\), the resolvent is:

The spectrum \(\sigma(A)\) is the complement of the resolvent set. For a self-adjoint operator, the spectrum is real.

The spectrum is partitioned into:

• The point spectrum (eigenvalues): values \(\lambda\) such that \(Ax = \lambda x\) for some nonzero \(x\).
• The continuous spectrum: limit points of eigenvalues (not eigenvalues themselves).
• The residual spectrum: part of the spectrum that is neither point nor continuous.

A bounded linear operator \(K: H \to H\) on a Hilbert space is compact if it maps bounded sets to relatively compact sets (closure is compact).

Equivalently, \(K\) is compact if every bounded sequence has a convergent subsequence in the image.

Examples: finite-rank operators, Hilbert-Schmidt operators, integral operators with certain kernels.

A quantum observable is a self-adjoint operator \(\hat{A}\) on the Hilbert space of quantum states. Its eigenvalues are the possible measurement outcomes. An eigenstate \(|\psi\rangle\) corresponds to a definite value of the observable.

The spectral theorem ensures that any self-adjoint operator can be written in diagonal form (relative to appropriate basis), making its physical meaning clear.