For an odd prime $p$, the even residues are the negatives of the odd residues. If $O=1\cdot 3\cdots (p-2)$, then $(p-1)!\equiv (-1{)}^{(p-1)/2}{O}^2$. Apply Wilson theorem to obtain the displayed congruence. The empty product for $p=3$ is one.

For an odd prime number $p$, put $m=(p-1)/2$. Both $(p-1)!$ and $p-1$ are divisible by $m$. By Wilson's theorem, they are also congruent modulo $p$, because each is $-1$ modulo $p$. The coprime integers $m,p$ therefore both divide their difference, giving the displayed integer congruence. This combines the prime-modulus factorial identity with an elementary divisibility constraint.

For $i{\Psi }_t=-\Delta \Psi /2$, differentiate the two-point product $\Psi (x+y/2)\overline{\Psi }(x-y/2)$. The difference of its two Laplacians is $2{\nabla }_x\cdot {\nabla }_y$, so Fourier transformation in $y$ gives the displayed free-transport equation for the Wigner transform. Its solution is $w(x,\xi ,t)=w(x-t\xi ,\xi ,0)$.

The analytic Wiener algebra consists of the power series $f(z)={\sum }_{n\ge 0}{a}_n{z}^n$ with ${\sum }_{n\ge 0}|a_n|<\infty$, identified with their values on the unit circle. The norm is $\Vert f\Vert =\sum |a_n|$. Discrete convolution gives multiplication and a submultiplicative norm; completeness follows from the absolutely summable sequence space. Each series converges uniformly on the closed unit disk and is holomorphic in its interior.

Every algebra character of the Wiener algebra is evaluation at a unique point of the unit circle. Writing $u(w)=w$, both $u$ and $u^{-1}$ have norm one. Thus $z=\varphi (u)$ satisfies $|z|\le 1$ and $|z^{-1}|\le 1$, hence $|z|=1$. Continuity then gives $\varphi ({\sum }_{n\in \mathbb{Z}}{a}_n{u}^n)=\sum a_n{z}^n$. Conversely, evaluation defines a continuous algebra character. The resulting bijection is a homeomorphism for the Gelfand topology.

Split each free real scalar field as $\varphi ={\varphi }^{(-)}+{\varphi }^{(+)}$, into creation and annihilation parts. For chronologically ordered arguments, the first field has the latest time. Multiplying it into a normal-ordered product of the remaining fields givesThis follows by commuting the annihilation part through each creation part; each commutator is a scalar, and all creation parts commute among themselves, as do all annihilation parts. The scalar commutator is the Feynman propagator for the chronological pair. Induction on the number of fields therefore lists every partial pairing exactly once: either field 1 remains unpaired or it pairs with one $j$. This proves the Wick theorem for free bosonic fields, including the empty pairing. Vacuum expectation retains only complete pairings. The identities are understood for regulated or smeared operator-valued distributions.

For a free real scalar field, split $\varphi ={\varphi }^{(+)}+{\varphi }^{(-)}$ into annihilation and creation parts. Moving the creation operators left gives $\varphi (x)\varphi (y)=:\varphi (x)\varphi (y):+[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$. The commutator is the Wightman function. Choosing either time order gives the displayed identity with the Feynman propagator. It is an identity of operator-valued distributions; coincident-point products need a suitable prescription.

Split a free scalar field into its annihilation and creation parts. For $x^0>y^0$, changing the product into normal ordering adds $[{\varphi }^{(+)}(x),{\varphi }^{(-)}(y)]$, the positive-frequency two-point function. For the opposite ordering interchange the two points. Their combination is the Feynman propagator, equal to the vacuum expectation of the time-ordered product. Iterating this identity gives the scalar Wick theorem.

In characteristic $p>0$, multiplication by $x_i$ and formal differentiation by $x_i$ act on the displayed vector space, because differentiation preserves the ideal generated by the $p$th powers. They satisfy the Weyl algebra relations and yield a nonzero module of dimension $p^n$. The central elements $x_i^p$ and ${\partial }_i^p$ also show why the characteristic-zero simplicity argument cannot apply. In particular the characteristic-zero Bernstein inequality for Weyl algebra modules fails for this finite-dimensional module.

// Destination: module-theory.bigb

The defining commutators $[{\partial }_i,x_j]={\delta }_{ij}$ reorder every word into the displayed monomials. Over a characteristic-zero field, the polynomial representation proves independence: iterated commutation with the coordinate multiplications extracts a highest derivative coefficient, multiplied by the nonzero factorial $\beta !$. Hence the representation is faithful and every polynomial-coefficient operator has a unique ordered expression.

In a nonzero two-sided ideal choose an element of least Bernstein filtration degree. Commuting it with any position or differentiation generator lowers degree, so all such commutators vanish. The ordered monomial basis of a Weyl algebra and characteristic zero then force it to be a nonzero scalar. The ideal is the whole ring. The argument fails in positive characteristic, when powers of the generators can be central.

For the relative differential operators over $\mathbb{C}$ on a polynomial ring, coordinates have order zero and partial derivatives have order one. The associated graded ring is $\mathbb{C}[x,\xi ]$ with degrees zero for $x_i$ and one for ${\xi }_i$. This differs from the Bernstein filtration, in which both sets of generators have degree one and every filtration piece is finite-dimensional.

The line where the interfaces of three phases meet, for example where a liquid film meets a solid and the surrounding gas. Continuum assumptions at this line can differ from the bulk lubrication approximation.

A nitrocellulose Western blot membrane is a Western blot membrane made from nitrated cellulose. Its retention capacity is finite; the original Western blot transfer experiment tested breakthrough with a second receiving sheet.

For canonical Kähler potential and $W=m{\Phi }^2/2+g{\Phi }^3/3$, eliminating the auxiliary field gives $V=|m\varphi +g{\varphi }^2{|}^2$. The cubic interaction Lagrangian is $-m^{\ast }g{\varphi }^{\ast }{\varphi }^2-m{g}^{\ast }\varphi ({\varphi }^{\ast }{)}^2$. Differentiating with respect to the three external fields gives the all-incoming Feynman rules $V_{\varphi \varphi {\varphi }^{\ast }}=-2i{m}^{\ast }g$ and $V_{{\varphi }^{\ast }{\varphi }^{\ast }\varphi }=-2im{g}^{\ast }$; the factor two counts identical scalar legs. For real $m,g$ and $\varphi =(A+iB)/\sqrt{2}$, the cubic Lagrangian is $-mgA(A^2+B^2)/\sqrt{2}$, giving $V_{AAA}=-3\sqrt{2}img$ and $V_{ABB}=-\sqrt{2}img$.

Localizing the rigid supersymmetry of a free Wess–Zumino model produces a derivative of the supersymmetry parameter multiplying the supercurrent. The Noether gauging procedure introduces a gravitino coupled to this current. Closure also requires a vierbein, the Einstein-Hilbert action, covariant matter kinetic terms and higher-order interactions. The leading gravitino-matter coupling contains the derivative of the scalar, not its undifferentiated value.

In a canonical Wess–Zumino model, complete the static transverse energy into $|{\varphi }^{\prime }-e^{i\alpha }\overline{W^{\prime }}{|}^2+2\mathrm{Re}(e^{-i\alpha }dW/dz)$. Choosing the phase of $\Delta W$ gives the tension bound. A solution of the displayed first-order equation attains it and preserves two of four real supercharges. For $W={\mu }^2\Phi /g-g{\Phi }^3/3$, $\varphi (z)=(\mu /g)\tanh (\mu z)$ interpolates between the two vacua and has tension $8{\mu }^3/(3g^2)$.

For $F=dA$ and an embedding variation $V$, the pullback Lie derivative and Cartan's magic formula give the displayed variation. Consequently the interior force from the Wess-Zumino brane coupling depends only on the gauge-invariant field strength. A change $A\mapsto A+d\Lambda$ changes the action only on the boundary, by the Generalized Stokes theorem; boundaries require the appropriate fixed boundary data or additional boundary coupling.

For an oriented brane worldvolume $W$, a gauge transformation $C\mapsto C+d\Lambda$ changes the Wess-Zumino brane coupling by $q{\int }_{\partial W}{X}^{\ast }\Lambda$, by the Generalized Stokes theorem. If the brane boundary lies on a manifold carrying a form $B$ of one lower degree, adding the displayed boundary term and transforming $B\mapsto B-\Lambda$ cancels this variation. The sign uses the induced boundary orientation. A closed worldvolume needs no boundary term; noncompact cases require appropriate support or decay conditions.

A $(p+1)$-form gauge potential couples electrically through its pullback to a $(p+1)$-dimensional worldvolume, whose object has $p$ spatial dimensions. Its field strength has degree $p+2$ and its magnetic dual potential has degree $d-p-3$, hence magnetic spatial dimension $d-p-4$. Electric and magnetic fluxes are integrated over linking spheres of dimensions $d-p-2$ and $p+2$, respectively. This dimensional dictionary applies when the indicated objects and dual descriptions exist; top forms need separate treatment and carry no ordinary propagating polarization.