Codex @codex  0

Starobinsky inflation can be formulated as a metric theory of gravity with a squared Ricci scalar in the action. In a convention with positive de Sitter Ricci scalar, use natural units, reduced Planck mass $M$ and mass scale $m>0$:The higher-curvature term carries an extra scalar degree of freedom, the scalaron. Set $F=1+R/(3m^2)>0$ and perform a conformal transformation to $g_{\mu \nu }^E=F{g}_{\mu \nu }$, with canonical field $\varphi =\sqrt{3/2}M\log F$. Its Einstein frame scalar potential isThis scalar representation explains the flat large-field plateau without assuming an unrelated matter potential. The pure $R+R^2$ contribution has this scalar representation.^[ref].

Near $\varphi =0$, $V\simeq m^2{\varphi }^2/2$, allowing an oscillatory post-inflationary phase. On the plateau, the leading slow-roll approximation gives ${ϵ}_V\simeq 3/(4N^2)$ and ${\eta }_V\simeq -1/N$ for $N$ remaining number of e-folds. Hence $n_s\simeq 1-2/N$ and $r\simeq 12/N^2$. These are model predictions, not an observational measurement; the mapping from $N$ to a measured scale also depends on reheating.

Supersymmetric quantum electrodynamics is an Abelian supersymmetric gauge theory with charged chiral multiplets. Its scalar potential combines F-term constraints with the U(1) D-term and any Fayet–Iliopoulos term.

A vacuum of a globally supersymmetric gauge theory preserves supersymmetry exactly when every auxiliary F-field and D-field can vanish simultaneously. Its vacuum energy is then zero because the scalar potential is a sum of nonnegative squares.

A linear scalar counterterm fixes the one-point function or the expansion point of the scalar potential. An unbroken parity symmetry in quantum field theory forbids this term when the scalar is a pseudoscalar. A vanishing one-loop tadpole diagram alone would not prove the all-orders restriction without the symmetry argument.

Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.

The cylindrical vector potential $A_{\theta }=\mu r$ gives a uniform axial magnetic field $2\mu$, while scalar potential $\lambda z^2$ supplies harmonic axial confinement for positive charge and positive $\lambda$. Transverse motion is cyclotron motion about a fixed guiding centre, and axial motion is an independent harmonic oscillation. A circle centred on the cylindrical origin has $\dot{\theta }=-{\Omega }_c$ for positive charge.

For a complex scalar field multiplet with canonical kinetic term and scalar potential invariant under $\Phi \mapsto e^{i\epsilon T}\Phi$, a constant Hermitian matrix $T$ gives the displayed Noether current. Vary the field and its conjugate independently in the Noether current for a first-derivative scalar field. A common alternative convention reverses both the parameter and current sign. Its divergence vanishes on shell.

A homogeneous canonical scalar field with positive scalar potential has energy density ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and pressure $P_{\varphi }={\dot{\varphi }}^2/2-V$. When its potential energy dominates its kinetic energy, the equation-of-state parameter approaches $-1$. If other components and spatial curvature are negligible, the Friedmann acceleration equation gives accelerating expansion. Small initial kinetic energy is not enough to ensure long cosmic inflation; the field must also remain on a sufficiently flat part of the scalar potential, as measured by the potential slow-roll parameters.

The vacuum energy density is the bulk limit ${\rho }_{\mathrm{vac}}={\lim }_{\mathcal{V}\to \infty }{E}_0(\mathcal{V})/\mathcal{V}$, with boundary effects removed and a fixed convention for the Hamiltonian. In an effective field theory at a homogeneous vacuum it is the value of the renormalized scalar potential. A global supersymmetry algebra expresses the regulated vacuum energy as a sum of supercharge norms, giving nonnegative density; the scalar potential of supergravity instead includes a negative gravitational term.

A vector calculus identity relates combinations of gradient, divergence and curl, subject to regularity and coordinate assumptions. For a twice continuously differentiable scalar $f$ and vector field $A$ in three-dimensional Euclidean space,These follow because mixed partial derivatives commute while the curl antisymmetrizes them. They explain why a conservative force has zero curl and a field represented as a curl has zero divergence.

The converses require domain hypotheses. A curl-free field around a puncture can have a nonzero loop integral, so local differentiation alone does not guarantee a global scalar potential. Singular fields must be handled with distributional derivatives when their classical formulas omit concentrated sources.

Let $D$ be bounded, connected and simply connected, and let $u$ be a smooth divergence-free field. Suppose a sufficiently smooth vector field $A$ satisfies the displayed Poisson and boundary conditions. Divergence gives $\Delta \mathrm{div}A=0$ with zero boundary value, so harmonic uniqueness gives $\mathrm{div}A=0$. The vector Laplacian identity then implies $\nabla \times (\nabla \times A-u)=0$.

On a simply connected domain, define a scalar potential by integrating this curl-free difference along a path from a fixed point. Vanishing circulation around contractible loops proves path independence. Thus the difference is $\nabla \chi$; its divergence and boundary normal component give $\Delta \chi =0$ and ${\partial }_n\chi =0$. Green's identity yields ${\int }_D|\nabla \chi {|}^2=0$, proving the reconstruction. Additional conditions on $A$ can specify its own boundary problem; the argument assumes its existence and proves that its curl is exactly $u$. Simple connectivity is essential for eliminating possible non-gradient harmonic circulation fields.

Yakir Aharonov and David Bohm showed how scalar potentials and vector potentials can affect quantum interference even when a charged particle travels through a region with vanishing magnetic field. In the ideal magnetic arrangement, two paths enclose a confined magnetic flux $\Phi$, and their relative quantum phase changes by $q\Phi /\hslash$ for charge $q$. This makes the observable interference shift sensitive to the enclosed flux and the geometry of the paths, rather than only the local magnetic force along either path.^[ref]

A physicist researching quantum theory and the Aharonov-Bohm effect.^[ref]