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gauge invariant In field theory, a Lagran- geodesic In Riemannian geometry a curve γ
gian which admits a gauge transformation. whose velocity ˙γ is autoparallel to γ , i.e., satis-
fies the geodesic equations ∇ ˙ γ(t) = 0, or, in
˙ γ(t)
gauge theory A quantum field theory with a local coordinates
gauge invariant Lagrangian.
j
k
i
i
¨ γ + N ˙γ ˙γ = 0,
jk
gauge transformation In field theory a
i
where N are the Christoffel symbols. For exam-
transformation of the fields which leaves the jk
n
ple, in R geodesics are straight lines.
equations of motion invariant. For example, in
More generally, given a manifold M with a
electrodynamics if we add to the electromagnetic
connection N, a curve whose tangent vector ˙γ
potential A the gradient of a function ∇φ, then
is parallel transported along the curve itself, i.e.,
the Maxwell equations are invariant under this
∇ ˙γ = 0.
gauge transformation A → A +∇φ. ˙ γ
In terms of principal fiber bundles a gauge
geodesic equations See geodesic.
transformation is an automorphism of the total
space that covers the identity of the base space,
geodesic motion For the kinetic energy
i.e.: If π : P → M is a principal G bundle,
Lagrangian, with metric tensor g ij
then a diffeomorphism φ : P → P is called
a gauge transformation if φ(p · g) = φ(p) · g 1
n
i i i j
for all p ∈ P, g ∈ G and π(f (p)) = π(p). L(q , ˙q ) = g (q)˙q ˙q
ij
2
Gauge transformations form an infinite dimen- i,j=1
sional Lie group (under composition), called the
the Euler-Lagrange equations are called
group of gauge transformations or gauge group.
geodesic flow or geodesic motion; they are
Elements of its Lie algebra are called infinitesi-
equivalent to the geodesic equations.
mal gauge transformations.
Gibbs energy diagram A diagram showing
Gaussian chain The simplest mathematical
the relative standard Gibbs energies of reactants,
model for a polymer chain molecule. The model
transition states, reaction intermediates and
treatsthemoleculeasarandomwalkandneglects
products, in the same sequence as they occur in
theexcludedvolumeeffect. Eachstepinthewalk a chemical reaction. These points are often con-
isassumedtobearandomvariable. Inthelimitof nected by a smooth curve (a “Gibbs energy pro-
a large number of steps, the distribution for the file,” commonly still referred to as a “free energy
end-to-end distance is approximately Gaussian profile”) but experimental observation can pro-
distributed. (cf. P.J. Flory, Statistical Mechanics vide information on relative standard Gibbs ener-
of Chain Molecules, John Wiley & Sons, New gies only at the maxima and minima and not at
York, 1969.) the configurations between them. The abscissa
expresses the sequence of reactants, products,
2
general linear group The n dimensional reaction intermediates, and transition states and
n
n
Lie group of linear isomorphisms form R to R , is usually undefined or only vaguely defined by
denoted by GL(n, R). The group of linear iso- the reaction coordinate (extent of bond break-
n
n
morphisms form C to C is called the (complex) ing or bond making). In some adaptations the
general linear group, denoted by GL(n, C). A ∈ abscissasare, however, explicitlydefinedas bond
GL(n) if A is an invertible real (complex) n × n orders, Bronsted exponents, etc.
matrix, i.e., det A = 0. Contrary to statements in many textbooks, the
highest point on a Gibbs energy diagram does not
n
generalized function Let ; ⊂ R be open necessarily correspond to the transition state of
and C (;) the Frechet space of infinitely dif- the rate-limiting step. For example, in a stepwise
∞
0
ferentiable functions with compact support in reaction consisting of two reaction steps:
;.A generalized function or distribution is →
a continuous linear functional on the space (i.) A + B ← C;
C (;). (ii.) C + D → E.
∞
0
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