A gauge-fixed action supplements a gauge-invariant action with a term imposing the gauge condition and with Faddeev-Popov ghost fields representing the Faddeev-Popov determinant. A local gauge condition gives local differential operators and a local action.

In gauge field theory its standard form is $A^{+}=n_{\mu }{A}^{\mu }=0$ for a fixed null direction $n$. A light-cone gauge uses light-cone coordinates to fix a longitudinal variable, often choosing $x^{+}$ as the evolution parameter for a relativistic particle phase-space action. Related field gauges set selected minus components to zero when a field has the required gauge invariance. A massive field without that gauge freedom must instead eliminate dependent components through its equations of motion.

The Jacobi identity for gauge covariant derivatives gives $D_{[\mu }{F}_{\nu \rho ]}=0$. In four dimensions this is $D_{\mu }{\tilde{F}}^{\mu \nu }=0$, with ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. It holds independently of the Yang-Mills equations.

In the standard Dirac representation of the gamma matrices, $({\gamma }^0{)}^{†}={\gamma }^0$ and $({\gamma }^i{)}^{†}=-{\gamma }^i$, so ${\gamma }^{a†}={\gamma }^0{\gamma }^a{\gamma }^0$. The transpose signs in that representation are positive for $a=0,2$ and negative for $a=1,3$. The charge-conjugation matrix expresses the basis-independent intertwining relation $C^{-1}{\gamma }^{a}C=-({\gamma }^a{)}^T$. Under an arbitrary nonunitary similarity transformation, the hermitizing matrix changes by inverse Hermitian congruence; it need not equal the transformed temporal gamma matrix.

A hermitizing matrix for a set of gamma matrices is an invertible Hermitian matrix $H$ satisfying ${\gamma }^{a†}=H{\gamma }^a{H}^{-1}$. Equivalently, each $H{\gamma }^a$ is Hermitian. It defines a covariant adjoint $\bar{\psi }={\psi }^{†}H$. If ${\gamma }^{\prime a}=M{\gamma }^a{M}^{-1}$, then $H^{\prime }=M^{-†}H{M}^{-1}$ has the same property. In a standard Lorentzian Dirac basis, $H={\gamma }^0$ is a conventional choice.

A generalized mass function with low-mass slope $-7/4$ and cutoff $e^{-\sqrt{M_{\ast }/M_{\ast ,\mathrm{ch}}}}$ differs from the ordinary exponential Schechter cutoff. Its total stellar-mass density is finite, even though its untruncated total galaxy count diverges. This form arises from a scale-free Press-Schechter mass function with a constant stellar-to-halo mass mapping.

The cosmic stellar mass density is the mass moment of a galaxy stellar mass function in a stated volume convention. With a comoving mass function it is mass per comoving volume, while proper density is larger by $(1+z{)}^3$. Comparing it to a critical or matter density requires keeping the epoch and volume convention explicit.

In a dilute, homogeneous population with number density $n$, an effective encounter cross section $\Sigma =\pi R_{\mathrm{enc}}^2$ gives the rate per target galaxyHere $R_{\mathrm{enc}}$ is the maximum encounter impact parameter, not automatically one galaxy's visible radius. If the cross section depends on relative speed, average the product instead: ${\Gamma }_{\mathrm{enc}}=n⟨\Sigma (v_{\mathrm{rel}})v_{\mathrm{rel}}⟩$. Gravitational focusing can create such speed dependence.

With independent encounters at a constant rate, a Poisson process givesThe approximation ${\Gamma }_{\mathrm{enc}}T$ is valid only for rare encounters. This is an encounter probability; it is not automatically a galaxy merger probability. Passing galaxies can remain unbound, and a bound pair can take additional time to coalesce.

As a simple additional model, suppose each encounter independently produces a merger within the relevant window with a fixed conditional probability $p$. The probability generating function of the Poisson count givesso the corresponding merger probability is $1-e^{-p{\Gamma }_{\mathrm{enc}}T}$. A realistic $p$ depends on encounter geometry, relative energy, angular momentum, gas and the allowed delay. It cannot be inferred from the geometric encounter rate alone.

For an entire volume $V$ of identical potential targets, the total distinct encounter-pair rate is $nV{\Gamma }_{\mathrm{enc}}/2$, because summing the per-target rate counts each unordered pair twice. This pair correction is separate from the conditional chance of merging.

A major galaxy merger joins galaxies with comparable masses. Rapid changes in the gravitational potential can redistribute stellar orbital energies through violent relaxation, substantially disturbing preexisting galactic disks. The outcome depends on gas content, orbit and subsequent accretion; a major merger does not universally prohibit disk rebuilding.

A gas-rich galaxy merger involves a substantial reservoir of gas capable of radiative cooling. Dissipation and gravitational torques permit gas inflow and enhanced star formation, unlike the nearly collisionless stellar evolution in a dry galaxy merger. A surviving or replenished gas reservoir can subsequently form a galactic disk.

For a bar rotating with constant pattern speed ${\Omega }_b$, use ${\psi }_{\mathrm{eff}}=\psi +{\Omega }_b^2{R}^2/2$ in the force convention $+\nabla \psi$. The centrifugal acceleration is its added gradient. This is the negative of the usual physical effective potential ${\Phi }_{\mathrm{eff}}$; Coriolis acceleration is separate.

For independent stage means $X,Y\sim N(\delta ,s^2)$ and combined mean $S=(X+Y)/2$, conditioning on the first-stage sample mean $X$ exceeding $c$ raises that mean. In this equal-stage setting, the estimator bias after continuation is $E(S\mid X\ge c)-\delta =(s/2)\lambda ((c-\delta )/s)$, where $\lambda$ is the upper-tail inverse Mills ratio. It is positive and tends to zero as continuation becomes virtually certain.

A functor $F:\mathcal{C}\to \mathbf{Set}$ is finite-limit-preserving if it carries each finite limit cone to a limiting cone. When $\mathcal{C}$ is a small category with finite limits, this is equivalent to $F$ being a filtered colimit in a category of covariant representable functors, and to the opposite of its category of elements being a filtered category. It is also equivalent to every comma category $(S\downarrow F)$ having finite limits, for all sets $S$.

An endofunctor is a functor from a category to itself. A monad adds a unit and multiplication to an endofunctor, satisfying the unit and associativity laws.

A functor discarding some structure, for example an action on a module. It need not be full or faithful in general.

Relative to a specified class of diagram shapes, a functor preserves limits when it takes each existing categorical limit cone of those shapes to a categorical limit cone. The usual completeness and general adjoint functor theorem hypotheses refer to small shapes; a possibly large diagram requires an explicit preservation hypothesis covering its shape.

Relative to a specified class of diagram shapes, a functor preserves colimits when it takes each existing colimit cocone of those shapes to a colimit cocone. A left adjoint preserves all existing colimits, including large ones when their universal properties are meaningful. Small-colimit preservation alone does not imply preservation of a class-indexed colimit.

A decomposition $H={\sum }_Z{h}_Z$ has frustration freeness when a global ground state minimizes every term simultaneously. For finite-dimensional Hermitian terms this implies $E_0(H)={\sum }_Z{E}_0(h_Z)$. Shifting each term by its minimum produces positive operators with a common null vector. The property depends on the decomposition, not only the total operator.

In a nearest-neighbour frustration-free Hamiltonian chain, multiply the bond orthogonal projections in two disjoint alternating layers. The resulting product $K$ fixes the common ground state space but contracts its orthogonal complement by a factor bounded away from one in terms of the spectral gap. A bound $\Vert K{|}_{\mathrm{supp}H}\Vert \le (1+\Delta /2{)}^{-1/3}$ implies exponentially convergent powers. The product is generally neither Hermitian nor normal.

A shallow gravity current is subcritical when its Froude number is less than one: a gravity disturbance can propagate upstream relative to the bed. This definition uses the characteristic speed $\sqrt{g^{\prime }h}$ appropriate to the stated model.