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If $\gamma \ne 0$ is algebraic of degree at most $D$, then every embedding satisfiesIndeed, $H({\gamma }^{-1})=H(\gamma )$ and the contribution of one Archimedean place to the height bounds $|\gamma {|}^{-1/D}$.

Let $P,Q\in \mathbb{Z}[X_1,\dots ,X_k]$ have degree at most $n_j$ in $X_j$. Thenwhenever the quotient is defined. At each non-Archimedean place, the integral coefficients contribute at most one; at the Archimedean places, the triangle inequality contributes the polynomial length. Multiplying these local estimates and using the product formula gives the claim.

If the primitive minimal polynomial of an algebraic number $\alpha$ has degree $d$, then

For a nonzero vector $\mathbf{a}=(a_0,\dots ,a_n)$ over a number field $K$, its absolute projective height isThe product formula makes this unchanged by multiplying all coordinates by the same nonzero scalar. The projective height of a linear form is the projective height of its coefficient vector.

The Absolute logarithmic Weil height is the logarithm $h(\alpha )=\log H(\alpha )$ of the absolute multiplicative height of an algebraic number. If its primitive integer minimal polynomial is $f(X)=a_d{\prod }_{i=1}^d(X-{\alpha }_i)$, thenThus it is the logarithm of the Mahler measure divided by the degree.

For a reduced rational $a/b$, $h(a/b)=\log \max (|a|,|b|)$. It satisfies $h({\alpha }^m)=|m|h(\alpha )$ for nonzero $\alpha$ and integral $m$. Unlike a single absolute value, it records denominator complexity and the behavior of all conjugates, making it suitable for arithmetic bounds that should not depend on a chosen embedding.

For coprime integers $p,q$ with $q>0$,This follows directly from the real and p-adic absolute values, or from the height-Mahler measure formula for the primitive polynomial $qX-p$.

For an invertible K-matrix, an Abelian Chern--Simons theory hastopological ground states on a spatial torus. On a closed orientable surface of genus of a surface $g$, the corresponding count is $|\det K{|}^g$. ^[ref]

The determinant counts the anyon classes in ${\mathbb{Z}}^N/K{\mathbb{Z}}^N$: the fundamental cell of the sublattice $K{\mathbb{Z}}^N$ has volume $|\det K|$ relative to the unit integer lattice. Transport around intersecting torus cycles gives the noncommuting Wilson-loop algebra of an Abelian Chern--Simons theory, which distinguishes these states.

For the toric code matrixthe quotient has four classes, represented by $(0,0),(1,0),(0,1),(1,1)$, so the torus count is four. These represent different global sectors rather than four locally distinguishable arrangements. In a finite microscopic realization, local perturbations can produce small finite-size splitting; the topological count describes the protected low-energy sector in the large-system limit.

The K-matrix is the symmetric integer matrix multiplying the coupled Abelian gauge fields in an Abelian Chern--Simons theory. For invertible $K$, the anyon lattice of an Abelian Chern--Simons theory isAdding $Kn$ to a charge vector changes it by a local particle. Its full braiding with any integer charge $q^{\prime }$ changes by $2\pi n^T{q}^{\prime }$, leaving the phase unchanged.

For example, $K=(2)$ has two classes, represented by $q=0,1$. Encircling two copies of class $1$ gives $\exp (2\pi i/2)=-1$; fusing them gives class $0$. This illustrates how an integer coupling produces fractional anyon statistics.

An integer unimodular matrix $W$ changes the basis asBecause $W^{-1}$ is also integral, it maps charge classes bijectively and preserves $q^T{K}^{-1}\tilde{q}$. It changes the coordinates of the theory rather than its braiding data. If electromagnetic coupling is specified by a charge vector $t$, that vector must be transformed consistently as $t^{\prime }=Wt$. ^[ref]

For $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$ with spatial $Q^{\mu }$, current conservation gives $n^{\mu }{\nabla }_{\mu }Q=KQ-D_{\mu }{Q}^{\mu }-Q^{\mu }{D}_{\mu }\log \alpha$ under the stated extrinsic-curvature convention.

With future unit normal $n^{\mu }$, the convention $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ measures how the hypersurface is embedded in spacetime.

The shift vector describes how spatial coordinates move tangentially between neighboring hypersurfaces.

The lapse function measures proper-time separation between neighboring spatial hypersurfaces along their unit normal.

Writing $v=e^S$ in $v^{\prime \prime }=q(x)v$ produces the Riccati equationExpanding $S^{\prime }$ in inverse powers of $x$ determines the exponential phase and algebraic prefactor recursively.

To classify infinity, set $z=1/x$ and classify $z=0$. For a second-order equation in normal form, a transformed coefficient with a pole of order greater than two makes infinity an irregular singular point.

Forthe substitutiongives the normal form

A WKB quantization condition selects approximate bound-state energies by requiring an oscillatory wavefunction to match both endpoint conditions. For a smooth one-dimensional well bounded by two separated simple classical turning points $a(E),b(E)$,Each soft turning point contributes an Airy turning-point connection formula phase. A hard Dirichlet wall and one soft turning point instead give $(n+3/4)\pi$, as in hard-wall and soft-turning-point WKB quantization. Two regular Dirichlet endpoints with no turning point give an integer multiple of $\pi$. The endpoint type is part of the formula.

For the quantum harmonic oscillator, the two-turning-point action is $\pi E/\omega$. The condition gives $E_n=\hslash \omega (n+1/2)$, which happens to agree with the exact spectrum. In a general potential it is a leading semiclassical estimate, and nearby or degenerate turning points can invalidate independent connection formulas.

For a one-dimensional energy $E$ and potential $V(x)$, a classical turning point satisfies $V(x)=E$. The classically allowed region has $E>V(x)$ and supports an oscillatory leading-order WKB solution.

Near a simple turning point, rescaling the independent variable reduces a second-order equation to the Airy equation. The decaying Airy solution on the forbidden side matches an oscillatory solution with phase shift $\pi /4$ on the allowed side.

A Gaussian wave packet is a wavefunction whose spatial amplitude is a Gaussian function. Free quantum evolution preserves its Gaussian form while changing its complex width and spreading its probability density.

The probability current is the flux associated with a quantum state's probability density: its outward surface integral gives the rate at which probability leaves a region. Subtracting the Schrödinger equation times the conjugate wavefunction from its complex conjugate gives the probability continuity equation, identifying the current below. The displayed expression assumes a real scalar potential and no electromagnetic vector potential. In an electromagnetic field, the gauge-invariant current instead uses the kinetic momentum operator, including its vector-potential contribution. A changing spatial phase can therefore produce flow even where the probability density is constant.

For a nonrelativistic particle of mass $m$, the probability current is