The rest mass of a stable soliton is the lowest energy in its topological sector minus the vacuum energy, in the infinite-volume limit. If Euclidean kernels $K_Q(\tau )$ have nonzero overlap with the sector's lowest-energy states, then $M=-{\lim }_{\tau \to \infty }{\tau }^{-1}\log [K_1(\tau )/K_0(\tau )]$, followed by the infinite-volume limit. A Euclidean path integral represents these kernels, with boundary wavefunctionals included if necessary.

A weak-coupling expansion of a vacuum-subtracted soliton mass begins with the classical energy and adds the Gaussian fluctuation approximation. For Sine-Gordon theory, $M=8m/{\beta }^2+\Delta M_{1\mathrm{loop}}+O(m{\beta }^2)$. The term of order $m$ includes both the vacuum-subtracted fluctuation frequencies and the appropriate mass counterterm.

A breather is a spatially localized nonlinear-wave solution periodic in time. It does not separate into independently traveling lumps while it breathes. A Sine-Gordon breather arises from a complex-conjugate soliton pair. In the quantum sine-Gordon model, the associated neutral kink-antikink bound particles are also called breathers.

A classical field theory is a gauge theory when its description has a local redundancy represented by gauge transformations. A gauge field defines the covariant derivative that compares matter fields at different points, and its gauge field strength measures the curvature of that comparison. Electromagnetism is the familiar Abelian example; Yang-Mills theory generalizes it to a non-Abelian internal group. Both classical equations and their quantum field theory can have this structure. Physically equivalent gauge descriptions must give the same observables, even though the gauge potential changes.

A Hamiltonian density is a local integrand for the field-theory Hamiltonian. For canonical fields with invertible velocity-momentum relation it is the Legendre transform in mechanics of the Lagrangian density, summing over all conjugate field components. Gauge theories require constraints and multiplier terms as well.

An antisymmetric tensor $A_{mn}$ with $F=dA$ obeys ${\partial }^m{F}_{mnp}={\mu }^2{A}_{np}$ for $\mu \ne 0$. Taking a divergence yields ${\partial }^m{A}_{mn}=0$, and the remaining equation is the Klein-Gordon equation for each component. The physical particle polarizations form the exterior square of the massive little group vector representation, of dimension $(D-1)(D-2)/2$. Unlike a two-form gauge field, it has physical longitudinal polarizations.

For uniform circular motion of radius $R$ and angular speed $\omega$, the acceleration is directed toward the centre and has magnitude ${\omega }^{2}R=v^2/R$. Writing $\mathbf{r}$ for the position relative to the centre gives $\ddot{\mathbf{r}}=-{\omega }^2\mathbf{r}$. The required resultant force is $m\ddot{\mathbf{r}}$; it may be supplied by a normal force, gravity, or another physical interaction.

A variation of the connection one-form has $\dot{F}=D_A\dot{A}$. The Bianchi identity and graded cyclic matrix trace imply ${\partial }_t\mathrm{Tr}(F\wedge F)=2d\mathrm{Tr}(\dot{A}\wedge F)$. Thus the Chern-Weil theory four-form changes by an exact differential form, and its integral on a closed four-cycle is connection-independent. For a general unitary bundle, include the corresponding trace-product term in the Second Chern form.

A Keller--Segel model couples the density of moving cells to a chemical field that biases their motion. A common cell conservation law is $b_t=D{b}_{xx}-{\partial }_x(\chi (c)b{c}_x)$. Chemical dynamics depend on the application: the attractant may be produced by cells, or a nutrient may instead be consumed. The nutrient-consumption model $c_t=-kb$ with no nutrient diffusion gives travelling bacterial bands rather than prescribing the chemical externally.

For a particle obeying $m\dot{\mathbf{v}}=q\mathbf{v}\times \mathbf{B}$ in a constant uniform magnetic field, the displayed combination is conserved. The derivative of mechanical angular momentum projected onto the field is canceled by the quadratic position term. This is the rotational analogue of the distinction between mechanical and canonical momentum in an electromagnetic field.

For an Inhomogeneous Poisson process whose known positive intensity changes from ${\lambda }_1$ to ${\lambda }_2$ at an unknown time $\theta \in (0,T)$, the ordered arrival-time likelihood function isUnder a uniform prior, each interval between successive arrivals has an exponential posterior density with rate in the exponent ${\lambda }_2-{\lambda }_1$. Its integrated weights allow exact interval selection followed by a truncated exponential draw; when the two rates coincide, the Bayesian posterior is uniform and the data contain no change-time information.

Suppose a body of mass $M_0$ has centre of mass ${\mathbf{R}}_0$, and material of mass $M_c<M_0$ with centre of mass ${\mathbf{R}}_c$ is removed. Additivity of the first mass moment gives the remaining centre of massWhen the original centre of mass is at the origin, the remaining centre of mass points opposite the removed material's first mass moment. For a constrained rigid body rotating about a fixed axis, this displaced centre of mass undergoes centripetal acceleration even if the applied axial torque is zero.

The celestial equator is the great circle perpendicular to the rotation axis of Earth; declination measures angular displacement from it.

The celestial poles are the two intersections of the rotation axis of Earth with the celestial sphere. The northern pole has declination $+90^{\circ }$.

An observer’s celestial meridian is the great circle containing the zenith and the celestial poles. Its upper half passes through the local north and south directions.

For a central-force radial energy equation, choose a monotone radius $q=q(r)$ and clock $\tau$ such that $d\tau /dt=(q/r)dq/dr$. Then $q^{\prime }=(r/q)\dot{r}$, and multiplying the old energy equation by $r^2/q^2$ preserves the centrifugal form $L^2/(2q^2)$. A split of the remaining terms into constant energy and a new Newtonian gravitational potential produces a transformed radial orbit.

The causal future of $S$ consists of points reached from $S$ by future-directed causal curves, including points of $S$ itself. In a globally hyperbolic spacetime, the causal future of a compact set is closed.

The causal past is the time-reversed causal future. Relative to a specified component ${\mathcal{I}}^{+}$ of future null infinity, the black hole region is the complement of its causal past: $\mathcal{B}=\mathcal{M}\setminus J^{-}({\mathcal{I}}^{+})$.

A causal curve has an everywhere nonspacelike tangent of consistent time orientation; its smooth segments are timelike or null. More generally, curves with local Lipschitz continuity and future causal tangent almost everywhere are allowed. The causal future and causal past are defined by reachability along these curves.

A time orientation is a continuous choice of future cone throughout a time-orientable spacetime. A continuous nonzero timelike vector field supplies such a choice. For metric signature $(-,+,+,+)$, a nonzero causal vector $Z$ is future-directed relative to a future timelike $W$ exactly when $g(W,Z)<0$.