In the standard Weyl representation of the gamma matrices and mostly-minus metric signature, the left chirality component obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. A plane wave with $p^0>0$ exists precisely on the massless mass shell $p^0=|\mathbf{p}|$. Its coefficient is a negative helicity eigenspinor. For direction angles $\vartheta ,\phi$, one normalized choice isThe nonzero solution is unique up to overall normalization and phase. Different patches are required for a continuous phase choice over all directions; the direction of zero momentum is undefined.

A left-handed massless Weyl spinor has a positive-energy plane wave transforming by $e^{i\alpha /2}$ under a rotation about its momentum. Its particle has helicity $-1/2$. The negative-frequency term multiplies an antiparticle fermionic creation operator, so its corresponding one-particle state carries the conjugate rotation phase and helicity $+1/2$. Thus a left-handed field has one particle and one antiparticle helicity; the right-handed field interchanges the signs. A full massless Dirac field contains both chiral fields and hence both particle helicities.

For mostly-minus metric signature, the Weyl sigma matrices are ${\sigma }^{\mu }=(I,{\sigma }_1,{\sigma }_2,{\sigma }_3)$ and ${\bar{\sigma }}^{\mu }=(I,-{\sigma }_1,-{\sigma }_2,-{\sigma }_3)$, where ${\sigma }_i$ are the Pauli matrices. The Pauli matrix multiplication law provesA real Lorentz four-vector is equivalently a Hermitian matrix $X=x^{0}I+x^i{\sigma }_i$ with $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. The transformation $X\mapsto MX{M}^{†}$ for $M\in \mathrm{SL}(2,\mathbb{C})$ preserves this determinant and gives the Lorentz spinor double cover. The indices ${\sigma }_{\alpha \dot{\beta }}^{\mu }$ therefore convert a vector index into one undotted and one dotted Weyl spinor index. They are the invariant tensors in the mixed supercharge anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$.

A spacetime field valued in one irreducible chiral spinor representation. A free massless left-handed Weyl field obeys $i({\partial }_t-\mathbf{\sigma }\cdot \nabla ){\chi }_L=0$. The left-handed massless plane-wave spinor and helicity content of a quantized Weyl field describe its modes and particle states. Field chirality fixes one spinor representation; it does not give the same particle and antiparticle helicity.

The integrated Gaussian in any nonzero regular representation of the Weyl relations is an orthogonal projection and obeys $PW(a)P=e^{-|a{|}^2/2}P$. Gaussian integration with the Weyl phase proves both identities. It cannot be zero: conjugating it by all Weyl operators would otherwise make the Fourier transform of every Gaussian-weighted matrix coefficient vanish, including its nonzero value at the identity. In an irreducible representation its range has dimension one, and the orbit of a unit vector there gives a dense cyclic space with universal Gaussian inner products.

Every irreducible strongly continuous representation of the Weyl relations with fixed nonzero central character is unitarily equivalent to the translation-modulation representation on $L^2(\mathbb{R})$. The Gaussian projection in a Weyl representation supplies a cyclic vector whose Weyl-orbit inner products agree with those of a Gaussian in the standard representation. Matching those orbit vectors gives the unitary intertwiner, with no appeal to a classification assertion in place of the proof.

For $1\le j\le k$ and every $\eta >0$, the displayed estimate bounds a mean value of a polynomial exponential sum. For cubes its eighth-moment bound is $O_{\eta }(n^{5+\eta })$. It complements pointwise Weyl differencing estimates: a ninth moment over a region is bounded by its supremum there times this eighth moment. Leiden lecture notes, Section 8.1 give the cubic formulation.

If $S(\theta )={\sum }_{x\le n}e(\theta x^3)$ and $|\theta -a/q|\le q^{-2}$ with $(a,q)=1$, the displayed estimate holds. Twice applying Weyl differencing bounds $|S{|}^4$ by $O(n^3+n{\sum }_{0<|h|,|l|<n}\min (n,\Vert 6\theta hl{\Vert }^{-1}))$. Group by $6hl$ and use the subpower bound for the divisor function. The reciprocal fractional-part sum near a rational bounds the resulting sum by $O(\log (2q)(n^3/q+n^2+n+q))$. Take fourth roots, absorbing logarithms into $n^{\eta }$ when $q\le n^3$; for larger $q$ the trivial bound suffices. Leiden lecture notes, Section 7.2 develop this cubic differencing estimate.

For a real degree-$k$ polynomial with leading coefficient $\theta$, $k\ge 2$, and coprime integers $a,q$ satisfying $|\theta -a/q|\le q^{-2}$, the displayed bound applies to its exponential sum over $1\le n\le N$. Repeated Weyl differencing reduces the phase to slope $k!\theta h_1\cdots h_{k-1}$. A geometric-sum estimate, the subpower bound for the divisor function and the reciprocal fractional-part sum near a rational complete the estimate. The arbitrary positive exponent $\eta$ absorbs divisor bounds and logarithms.

A map $f:a\to \mathcal{P}(a)$ is recursive if every map $g:\mathcal{P}(b)\to b$ admits a unique $h:a\to b$ with $h(y)=g(\{h(x):x\in f(y)\})$. This holds exactly when the predecessor relation $x\in f(y)$ is a well-founded relation. For the forward construction, take the union of all consistent partial solutions on downward-closed domains; well-founded induction proves compatibility, and a minimal missing point allows extension. Conversely, a nonempty subset with no minimal point has an upward reachability closure $T$ satisfying $y\in T$ exactly when some predecessor is in $T$. With $b=\{0,1\}$ and $g(S)=1$ exactly when $1\in S$, both the zero map and the indicator of $T$ solve the recursion, contradicting uniqueness.

A symmetric positive-definite matrix $M$ defines an inner product on real coordinate vectors byFor diagonal $M$, positive diagonal entries weight the squared coordinate errors. This is the finite-coordinate version of a weighted inner product; the positive-definite matrix example appears in Francis Narcowich, Inner Products and Norms, standard inner products, example 8.

For $M=\mathrm{diag}(4,1)$, the squared norm of $(a,b{)}^T$ is $4a^2+b^2$. A first-coordinate error of one costs as much as a second-coordinate error of two. The unit sphere is the ellipse $4a^2+b^2=1$, and orthogonality means $4u_1{v}_1+u_2{v}_2=0$. Thus “closest” and “perpendicular” depend on the chosen inner product. The orthogonal projection residual example shows how this changes a best approximation.

Starting with $p_0=0$ on $[-1,1]$, this polynomial recurrence gives $0\le p_r(x)\le |x|$ and $p_{r+1}(x)\ge p_r(x)$. Indeed the error satisfies $|x|-p_{r+1}=(|x|-p_r)[1-(|x|+p_r)/2]$. Its pointwise limit solves $p^2=x^2$, so it is $|x|$. Dini's theorem gives uniform convergence. Each continuous piecewise linear function is an affine function plus a finite linear combination of $(x-t{)}_{+}=(x-t+|x-t|)/2$. Scaling and translating the recurrence therefore gives polynomial approximation of every such function, and their density gives the Weierstrass approximation theorem.

On $[0,1]$, the polynomials $p_n(x)=x^n/n$ converge in supremum norm to the zero function, since $\Vert p_n{\Vert }_{\infty }=1/n$. But $p_n^{\prime }(1)=1$ for every $n$, whereas the zero function has left derivative zero at that endpoint. Thus uniform convergence of function values does not imply uniform convergence of derivatives. This does not contradict uniform convergence of Bernstein polynomial derivatives, which has a particular approximation construction and additional differentiability hypotheses.

A discrete memoryless channel on the additive group $\mathbb{Z}/3\mathbb{Z}$ maps $x$ to $x$ with probability $1-q$ and to $x+1$ with probability $q$. Every input gives the same conditional entropy $h_2(q)$, whereas the output Shannon entropy is at most ${\log }_{2}3$. A uniform input produces a uniform output since every column of the transition matrix sums to one. Hence the channel capacity is ${\log }_{2}3-h_2(q)$ bits per use. This cyclic-error channel differs from a channel distributing its error probability equally over both other output symbols.

For a nonnegative sublinear operator, weak type $(1,1)$ is the displayed level-set bound for all integrable inputs and positive thresholds. It supplies an exceptional-set estimate for approximation errors; the uncentered maximal weak-type inequality is a central example.

For a normed space $E$ and $0<q<\infty$, an operator $T:E\to L^0$ is of weak type $(E,q)$ when the displayed distribution bound holds uniformly over $f\in E$ and $a>0$. No completeness of $E$ is needed. A sublinear operator has pointwise triangle domination and absolute homogeneity; its outputs may be understood modulo almost everywhere equality.

Let $S$ be a sublinear operator with positive constants and $\mu \{Sf>a\}\le C_1\Vert f{\Vert }_1/a$ and $\Vert Sf{\Vert }_{\infty }\le C_{\infty }\Vert f{\Vert }_{\infty }$. Split $f$ into its part above $a/(2C_{\infty })$ and its bounded remainder. The latter contributes at most $a/2$, so the weak bound controls the superlevel measure by $2C_1{a}^{-1}{\int }_{|f|>a/(2C_{\infty })}|f|$. Integrating this estimate with layer cake representation proves the displayed bound for $1<p<\infty$ and $q=p/(p-1)$. In particular it gives the strong $L^p$ estimate for the Hardy-Littlewood maximal function.

If $T=S^{\ast }$ for a bounded operator on a separable Banach space, and $C$ is a nonempty weak-star closed convex subset of the dual unit ball with $T(C)\subseteq C$, then $T$ has a fixed point in $C$. The Cesàro averages of the orbit remain in $C$, and $T$ minus the identity sends those averages to vectors of norm at most $2/n$. Weak-star compactness supplies a cluster point, and the adjoint action is weak-star continuous. The corresponding assertion with only weak closedness is false for the right shift on the probability simplex in ${\ell }^1$, the dual of $c_0$.

For a partial differential equation $F(x,u,Du,D^{2}u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\varphi (x_0),D^2\varphi (x_0))\le 0$ whenever a smooth test function $\varphi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\mathrm{tr}{D}^{2}u$, these conditions are respectively $\Delta \varphi \ge 0$ and $\Delta \varphi \le 0$. They make sense even when $u$ has no classical second derivatives.

If a field $L$ is finitely generated as an algebra over a subfield $k$, then $L$ is a finite algebraic extension of $k$. Applying this to a residue field of a polynomial ring proves the Weak Hilbert Nullstellensatz.