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An active scalar is a transported scalar field that feeds back on the velocity field, for example a density anomaly producing buoyancy. In an adjoint equations for Boussinesq scalar mixing calculation its dynamical coupling must also be transposed; a passive-scalar sensitivity equation generally omits that feedback.

A scalar field transforming in the Adjoint representation of a Lie algebra transforms by a Lie bracket with the infinitesimal gauge parameter.

For a free scalar field of mass $m$ in Anti-de Sitter spacetime of dimension $d+1$ and radius $L$, use coordinates withThe conformal boundary is at $z=0$. The Klein-Gordon equation gives the radial leading balance $z^2{\varphi }_{zz}-(d-1)z{\varphi }_z-m^2{L}^2\varphi \simeq 0$. A power law $z^{\Delta }$ therefore obeys the AdS scalar mass--dimension relationFor distinct roots and generic nonresonant parameters,In standard quantization, $J$ is the source of the dual local field operator; $A$ determines its expectation value after holographic renormalization, with a normalization fixed by the bulk action. ^[ref]

The Breitenlohner-Freedman bound $m^2{L}^2\ge -d^2/4$ makes $\nu$ real. At $\nu =0$ the roots coincide and a second solution involves $z^{d/2}\log z$. Resonant expansions can also contain logarithms. In the appropriate mass window, Alternative quantization of an AdS scalar field changes which coefficient acts as source. ^[ref]

For a scalar field of mass $m$ on ${\mathrm{AdS}}_{d+1}$ of radius $L$, the scaling dimension of the dual scalar primary obeys $m^2{L}^2=\Delta (\Delta -d)$. The two roots correspond to the two independent near-boundary falloffs.

Advection transports a quantity with a velocity field. For a scalar field $f$, its local advective rate is $\mathbf{u}\cdot \nabla f$, the spatial contribution to the material derivative. The same term transports the components of a vector field.

Alexei Starobinsky investigated how energy stored in an oscillating inflaton can become particles after cosmic inflation. With Lev Kofman and Andrei Linde, he developed a treatment of preheating through parametric resonance in an expanding cosmological background.

Repeated oscillations can amplify a coupled scalar field much more rapidly than independent perturbative decays would suggest. The calculation follows both the growth of the produced particles and their backreaction on the oscillating field. That feedback, together with scattering, changes and eventually terminates the resonance.^[ref]

An anomalous dimension is the interaction-generated difference between a full scaling dimension and its engineering dimension. For a scalar field, momentum-dependent self-energy corrections require wave-function renormalization and produce ${\Delta }_{\varphi }=(d-2+\eta )/2$.

For an affine connection, define geometric torsion tensor by $\mathcal{T}(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$. The covariant Hessian of a scalar field is ${\nabla }^{2}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Antisymmetrizing gives $[X,Y]f-({\nabla }_{X}Y-{\nabla }_{Y}X)f=-\mathcal{T}(X,Y)f$. Thus all scalar Hessians are symmetric exactly when the connection is torsion-free. In derivative-first coordinates, ${\nabla }_{\mu }{\nabla }_{\nu }f={\partial }_{\mu }{\partial }_{\nu }f-{\Gamma }_{\mu \nu }{}^{\lambda }{\partial }_{\lambda }f$, which makes the sign explicit. Differentials of local coordinate functions span the cotangent space, so vanishing of this antisymmetric part for every $f$ forces the torsion itself to vanish.

For an ordinary scalar field, its derivative is a polar covector, while the axial current is a pseudovector. Their contraction is invariant under the proper orthochronous Lorentz group but changes sign under parity. If the field is a pseudoscalar, the extra transformation sign makes this coupling parity even. In four spacetime dimensions its mass dimension is five, so Lorentz invariance alone does not imply power-counting renormalizability.

An axisymmetric flow is invariant under rotations about one fixed axis, so scalar fields and cylindrical-coordinate velocity components do not depend on the azimuthal angle.

The first Bianchi identity follows by applying cyclic curvature commutators to the gradient of an arbitrary scalar field: its symmetric Hessian makes the third-derivative terms cancel, leaving $R_{abc}{}^d+R_{bca}{}^d+R_{cab}{}^d=0$. For the second Bianchi identity, write the vector-bundle derivative as $D_a={\partial }_a+{\Gamma }_a$, so $[D_b,D_c]={\mathcal{R}}_{bc}$. The operator Jacobi identity gives the cyclic sum of ${\partial }_a{\mathcal{R}}_{bc}+[{\Gamma }_a,{\mathcal{R}}_{bc}]$ equal to zero. Restoring the connection terms on the two curvature-form indices gives precisely ${\nabla }_a{R}_{bcd}{}^e+{\nabla }_b{R}_{cad}{}^e+{\nabla }_c{R}_{abd}{}^e=0$: the additional terms cancel because the connection is torsion-free and curvature is antisymmetric in its first pair.

For a free massless real scalar field with wave speed set to one and metric $(+,-)$, define $\mathcal{E}=({\varphi }_t^2+{\varphi }_x^2)/2$ and $p=-{\varphi }_t{\varphi }_x$. Its symmetric stress-energy tensor is $T^{\mu \nu }=\left(\begin{array}{cc}\mathcal{E}& p\\ p& \mathcal{E}\end{array}\right)$. The Lorentz current has density $tp-x\mathcal{E}$ and flux $t\mathcal{E}-xp$. Its conserved charge is the displayed expression when boundary flux vanishes. In one spatial dimension it generates boosts, not spatial rotations; there is no independent purely spatial angular-momentum component. Its conservation says that the center of energy moves uniformly. This is a symmetry of the wave-field model and need not survive boundaries or additional material effects.

For coincident D3-branes in the critical 26-dimensional bosonic string theory, the massless open-string oscillator has a four-dimensional gauge vector and 22 transverse scalar fields. Chan-Paton factors put all these fields in the adjoint of $U(k)$ for $k$ coincident branes. Ignoring the tachyon, their leading interactions are the dimensional reduction of 26-dimensional Yang-Mills theory. This contains a Yang-Mills sector but is not pure four-dimensional Yang-Mills theory and is not the supersymmetric six-scalar D3-brane theory of the ten-dimensional superstring.

The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.

The Breitenlohner-Freedman bound permits a scalar field in Anti-de Sitter spacetime to have a moderately negative squared mass without the instability that the same parameter would imply in flat spacetime. In ${\mathrm{AdS}}_{d+1}$ of curvature radius $L$, the bound is $m^2{L}^2\ge -d^2/4$ for the standard scalar dynamical stability problem with appropriate boundary conditions.

The near-boundary exponents satisfy $\Delta (\Delta -d)=m^2{L}^2$, hence ${\Delta }_{\pm }=d/2\pm \sqrt{d^2/4+m^2{L}^2}$. The bound keeps these exponents real. The spacetime geometry and boundary conditions are part of the dynamical stability statement; a negative squared mass cannot be classified without them.^[ref]

Neglecting metric perturbations, a free massive scalar field fluctuation obeys $\delta {\chi }^{\prime \prime }+2\mathcal{H}\delta {\chi }^{\prime }-{\nabla }^2\delta \chi +a^2{m}^2\delta \chi =0$. Rescaling it by the scale factor removes the conformal-time friction and gives $v^{\prime \prime }-{\nabla }^{2}v+(a^2{m}^2-a^{\prime \prime }/a)v=0$. Growth of $v$ need not mean growth of the physical fluctuation $v/a$.

A charged scalar field is a complex scalar field transforming nontrivially under a gauge symmetry. For an electromagnetic potential $A_a$, its gauge-covariant derivative is ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$.

Classical field theory describes physical quantities assigned throughout space and time, with field values that are ordinary numerical or geometric objects. A scalar field assigns a scalar to each point; the electromagnetic field assigns electric and magnetic components. Initial and boundary conditions together with field equations determine their evolution.

A local action $S=\int \mathcal{L}(\varphi ,\partial \varphi ,x)d^{d}x$ provides a systematic formulation. Varying the fields gives the Euler-Lagrange equations, while continuous symmetries lead through Noether theorem to conserved currents. Maxwell equations and the classical Klein-Gordon equation are examples. The field description makes finite-speed propagation and local energy transfer explicit. In quantum field theory, the corresponding fields become operator-valued objects and their correlations determine quantum predictions.

For a torsion-free connection, the commutator of coordinate covariant derivatives vanishes on a scalar field and acts on a vector field through the displayed Riemann curvature tensor. On a covector field it instead gives $[{\nabla }_{\alpha },{\nabla }_{\beta }]A_{\mu }=-R^{\nu }{}_{\mu \alpha \beta }{A}_{\nu }$. The curvature commutator is a tensor derivation, so each index of a general tensor contributes independently with the corresponding sign. Including the lower derivative index in the second covariant derivative is essential for this identity.

The scalar potential is a positive sum of squared commutators of the six adjoint scalar fields. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical flat directions of a scalar potential, with residual identification by the Weyl group. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.