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A scalar potential whose negative spatial gradient is the material acceleration. For a smooth global potential, the spatial curl of the material acceleration vanishes. On a simply connected domain the converse holds under the usual smoothness assumptions. Its physical dimension is squared length divided by squared time.

For zero scalar potential, the static field equations are $D_i{D}_i\Phi =0$ and $D_i{F}_{ij}=[\Phi ,D_j\Phi ]$. If $B_i=-D_i\Phi$, the gauge-theory Bianchi identity gives the first equation. For the second,using $F_{ik}={ϵ}_{ik\ell }{B}_{\ell }$ and ${ϵ}_{ijk}{ϵ}_{ik\ell }=-2{\delta }_{j\ell }$. Thus the first-order equations imply the full second-order equations locally, independently of the radial ansatz.

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.

For one canonical chiral field at $W^{\prime }(v)=0$, the quadratic scalar potential is $|W^{\prime \prime }(v){|}^2|\delta \phi {|}^2$. Writing $\delta \phi =(a+ib)/\sqrt{2}$ gives two canonically normalized real scalars with squared mass $|W^{\prime \prime }(v){|}^2$. The chiral-superfield fermion mass matrix has the same physical mass. A complex bilinear coefficient or its algebraic square is not itself a signed physical squared mass.

Consider two real scalar fields $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2)$ with positive $\lambda ,v$, standard kinetic terms, and the displayed scalar potential. At $h=0$, the circle group rotates one field into the other: this is a global symmetry in field theory, with the same rotation at every point in space. The vacuum manifold is the circle ${\mathbf{\varphi }}^2=v^2$. Choose $(v,0)$ and differentiate the scalar potential twice:The scalar mass matrix assigns a restoring force to the radial displacement, whereas its tangent is one of the Goldstone directions in the scalar mass matrix. Moving around the circle costs no potential energy; moving between neighbouring points in space can still cost gradient energy. Flat angular potential energy therefore means a soft collective excitation, not that every spatial pattern is free.

A positive source $h$ explicitly selects the positive ${\varphi }_1$ direction. The minimum is $(w,0)$, where $w>v$ is the positive solution of $\lambda w(w^2-v^2)=h$. The two eigenvalues of the scalar mass matrix there areFor example, $\lambda =v=1$ and $h=0.15$ give $w\simeq 1.06792$, $m_{\mathrm{radial}}^2\simeq 2.42138$ and $m_{\mathrm{angular}}^2\simeq 0.140459$. The light angular excitation becomes a Goldstone boson as $h\downarrow 0$, while the radial mass stays finite. The source changes both the preferred direction and the radius; expanding about the old point $(v,0)$ at nonzero $h$ would miss that shift.

Without a symmetry excluding odd scalar powers, a real-scalar Yukawa interaction with a massive Dirac field requires a general scalar potential through degree four. For a constant background, the fermion mass becomes $M+y\varphi$, and its divergent local determinant contribution contains $(M+y\varphi {)}^4$. Linear and cubic terms are therefore generated when $M\ne 0$. A zero renormalized tadpole is a renormalization condition, not a prohibition of its counterterm. An exact discrete chiral symmetry in a massless fermion theory can instead enforce an even scalar potential.

Under a superpotential critical-point shift, $m\mapsto M=m+gs$ and the linear coefficient vanishes, while $M^2=\Delta$ is unchanged. At either supersymmetric root $v$, the complex fermion mass parameter is $W^{\prime \prime }(v)=\pm \sqrt{\Delta }$. The physical scalar and fermion masses are both $\sqrt{|\Delta |}$, independent of the additive superpotential constant. A repeated critical point has a massless multiplet and a quartic leading scalar potential.

For constant mass density ${\rho }_0$, a frame with constant angular velocity $\mathbf{\Omega }$ absorbs centrifugal acceleration into reduced pressure $\Pi =p/{\rho }_0-|\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Indeed $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})=-\nabla |\mathbf{\Omega }\times \mathbf{r}{|}^2/2$. Transforming the material acceleration in the Euler equations gives the displayed equation. The Coriolis acceleration remains velocity dependent and cannot be absorbed into a scalar potential. A conservative physical body force can be included by adding its potential to $\Pi$.

In a spatially flat universe dominated by a homogeneous canonical scalar field, use natural units and reduced Planck mass $M$. For $\lambda >0$, ${\varphi }_0>0$ and $t>0$, takeThis is an exact solution for a particular scalar potential, not for an arbitrary prescribed potential. The identities $\dot{H}=-{\dot{\varphi }}^2/(2M^2)$ and $V=3M^2{H}^2-{\dot{\varphi }}^2/2$ reconstruct it asTo check the reconstruction, $\dot{\varphi }=2\lambda {\varphi }_0/\sinh (2\lambda t)$ and $\sinh (\varphi /{\varphi }_0)=-1/\sinh (2\lambda t)$. The scalar equation then follows from the Friedmann equation and energy conservation where $\dot{\varphi }\ne 0$.

For $2/3<\beta <2$, the potential is positive and expansion begins as a decelerating power law $a\propto t^{\beta /2}$. At late times $H\to \lambda \beta$ and expansion becomes exponential. More precisely ${ϵ}_H=-\dot{H}/H^2=(2/\beta ){\mathrm{sech}}^2(2\lambda t)$ decreases through one. This illustrates entry into late accelerated expansion rather than a graceful exit from inflation.

A local minimum of the scalar potential that has higher energy than a global minimum. It can be stable against infinitesimal classical fluctuations but metastable to a transition to a lower-energy vacuum, including quantum tunnelling. The stability of a cubic-quartic scalar potential gives an elementary example distinguishing local curvature from global stability.

In vacuum analysis, an exact flat direction of a scalar potential is a nonconstant continuous family of degenerate stationary vacua. The scalar potential stays constant along this family and its gradient vanishes there. An ordinary level set does not suffice: $V(x,y)=x^2+y^2$ is constant on every circle, but its vacuum at the origin has no flat direction. A vanishing eigenvalue of the quadratic Hessian matrix also does not suffice: $V(x)=x^4$ has zero quadratic mass at zero but no family of nearby vacua. Exact flat directions can be lifted by corrections absent from the classical scalar potential.

A force field specifies a force vector as a function of position and, when necessary, time, for a specified test body. It is a vector field. A time-independent conservative vector field has $\mathbf{F}=-\nabla V$, where $V$ is the body's potential energy; its work along a path equals the decrease in $V$. Position-dependent forces can have nonzero curl, so a scalar potential need not exist.

For a smooth invariant scalar potential, differentiating ${\partial }_{I}V(i{t}^a\varphi {)}^I=0$ at a classical vacuum gives $M^2(i{t}^a{\varphi }_0)=0$. The symmetry tangent space has dimension $\dim G-\dim H_0$, where $H_0$ is the full stabilizer subgroup. With a positive nonsingular scalar kinetic term, these are massless directions of the quadratic theory. Further zero eigenvalues are possible unless the Hessian matrix is positive on the normal complement.

For ${\mu }_h^2>0$ and $\lambda >0$, the minimum has ${\varphi }^{†}\varphi =v^2/2$ with $v^2={\mu }_h^2/\lambda$. In unitary gauge, $\varphi =(0,v+H{)}^T/\sqrt{2}$ gives $V=\text{constant}+\lambda v^2{H}^2+\lambda v{H}^3+\lambda H^4/4$. The canonically normalized Higgs boson therefore has $m_H^2=2\lambda v^2$. The angular modes produce the Goldstone bosons absorbed in the Higgs mechanism.

Set $r=\sqrt{2{\varphi }^{†}\varphi }$. The same Higgs potential becomesThe origin has negative radial curvature, while $r=v$ minimizes the scalar potential. Changing the field's angular coordinates at fixed $r$ costs no potential energy; changing its radial magnitude does. This geometric distinction explains why the surviving physical scalar field has a mass even though the angular directions are flat in the scalar potential. Local angular variations still enter the gauge-covariant kinetic term.

For a chosen model, not fitted physical constants, take $v=240\mathrm{GeV}$ and $\lambda =1/8$ in units $\hslash =c=1$. Then ${\mu }_h^2=7200{\mathrm{GeV}}^2$ and $m_H=120\mathrm{GeV}$. Direct expansion gives a quadratic coefficient $7200{\mathrm{GeV}}^2$, cubic coefficient $30\mathrm{GeV}$ and quartic coefficient $1/32$. The quadratic coefficient is $m_H^2/2$, not $m_H^2$. Keeping the same named $\lambda$ in the Higgs radial mass for a half-normalized quartic potential would instead give $m_H^2=7200{\mathrm{GeV}}^2$; these are different potentials. The shifted quartic Higgs potential normalization agrees with this section when $\kappa =\lambda$.

For a spatially homogeneous minimally coupled scalar field with zero scalar potential, the stress-energy tensor has $\rho =P={\dot{\varphi }}^2/2$. Its field equation is $\ddot{\varphi }+3H\dot{\varphi }=0$, so $a^3\dot{\varphi }$ is constant and its energy density decreases as $a^{-6}$. Thus its homogeneous stress is precisely that of a stiff fluid. Spatial gradients would introduce directional stresses, so this equivalence requires homogeneity.

A differential one-form on a specified domain is inexact if it is not the differential of a single-valued scalar potential there. For $Pdx+Qdy$, unequal cross derivatives $P_y\ne Q_x$ prove inexactness locally. Equality is sufficient on a simply connected plane domain under the usual smoothness hypotheses, but holes can obstruct a global potential. The form $42ydx+xdy$ has cross derivatives $42$ and $1$ and counterclockwise unit-square integral $-41$, so it is inexact. A conservative vector field has an exact differential as its work form.

Adding a holomorphic function $f$ and its conjugate does not change the Kähler metric. In global supersymmetry, these terms integrate to zero in the D-term action. In Planck-unit supergravity, the simultaneous transformation $W\mapsto e^{-f}W$ leaves the scalar potential invariant; shifting $K$ while holding $W$ fixed generally does not.

A spinless charged particle in $B_z=\alpha x$ admits vector potential $A_y=\alpha x^2/2$. Separating a plane wave along y produces a one-dimensional quartic potential energy. For nonzero $q\alpha$, $k/(q\alpha )>0$ gives two minima at $\pm \sqrt{2\hslash k/(q\alpha )}$; the opposite sign gives one minimum at zero. The field gradient creates confinement even though there is no scalar potential. The quartic potential is even, so its nondegenerate eigenstates have definite spatial parity.

In a simply connected current-free region of constant magnetic permeability, Ampère's circuital law makes the magnetic field curl-free, so it admits a scalar potential. The absence of magnetic monopoles then gives Laplace's equation for that potential. Some conventions define the potential for $\mathbf{H}$ instead of $\mathbf{B}$, differing by the constant permeability. Boundary conditions, including imposed fields at infinity, determine the potential up to an additive constant.

Across an interface with no free surface current, the normal component of $B$ and tangential component of $H$ are continuous. For an axisymmetric spherical shell, harmonic scalar potentials have dipole form $(ar+b/r^2)\cos \theta$ in each radial region.