Scalar field

A scalar field assigns one scalar to each point of a specified space, such as Euclidean space or spacetime. For example, $F(x,y)=x^2+2y^2$ is a real-valued function on a plane, and its change along a moving observation is given by the derivative along a curve. Under a change of coordinates, a scalar field's value at the same physical point is unchanged.

A real scalar field is minimally coupled to a metric tensor when its kinetic term uses the metric contraction of first derivatives and its scalar potential depends only on the field, without an explicit scalar curvature coupling. For metric signature $(-,+,+,+)$ the displayed Lagrangian and volume density $\sqrt{-g}$ give $\square \varphi -V^{\prime }(\varphi )=0$. Its stress-energy tensor is $T_{ab}={\nabla }_a\varphi {\nabla }_b\varphi -g_{ab}[\frac{1}{2}(\nabla \varphi {)}^2+V]$. The scalar stress-energy divergence identity explains its conservation on the field equation.

For a minimally coupled scalar field with a Levi-Civita connection, metric compatibility and the symmetric Hessian of a scalar field give the displayed off-shell identity. It is a special case of the diffeomorphism Noether identity for a scalar field, and gives stress-energy conservation when the scalar equation is satisfied. No gravitational field equation is needed.

The mass of a canonically normalized scalar field is determined by its quadratic fluctuation potential at a stationary vacuum. With several real fields, physical squared masses are the eigenvalues of the scalar mass matrix. A negative squared mass signals an unstable direction rather than a stable massive particle.

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

A sextic scalar field theory includes a term proportional to ${\varphi }^6$ in its scalar potential. For a real field with reflection symmetry, a useful model isThe positive sextic coefficient stabilizes large field values even when the quartic coefficient $u$ is negative. This permits competition between a minimum at zero and symmetry-related nonzero minima, making the model useful for first-order phase transitions and a tricritical point.

At $r=u=0$ in a mean-field treatment, the sextic term is the leading restoring nonlinearity. Gradient terms determine interfaces between different minima. The classical stability statement is distinct from quantum power counting: in three spacetime dimensions a canonically normalized ${\varphi }^6$ interaction is marginal by engineering dimension, whereas in four it has a negative-dimension coupling.

A scalar field with no spacetime coordinates reduces its path integral to an ordinary integral. It preserves Wick contractions and Feynman-diagram symmetry factors, making it a useful model of perturbative quantum field theory.

For $m^2>0$, the zero-dimensional scalar field theory with Euclidean action $m^2{x}^2/2+\lambda x^4/4!$ has a Gaussian integral normalization $I(0)=\sqrt{2\pi /m^2}$. Its Feynman rules assign $1/m^2$ to each propagator, $-\lambda$ to each quartic vertex, and the reciprocal Feynman-diagram symmetry factor to each vacuum diagram. The integral itself includes disconnected vacuum diagrams; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with Feynman-diagram symmetry factors $128,16,48$ respectively. Their weights add to $35{\lambda }^2/(384m^8)$.

The Wick contractions give the general coefficient $I(0)(-1{)}^v(4v-1)!!/[v!(4!{)}^v{m}^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an asymptotic expansion as $\lambda \downarrow 0$, rather than a convergent power series. For every fixed truncation the Taylor theorem remainder estimate for $e^{-u}$, $u\ge 0$, can be integrated against the Gaussian distribution to justify the corresponding remainder bound.

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 ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$.

The densitycontains the term linear in the potential from minimal coupling, but omits $e^2{A}_{\mu }{A}^{\mu }|\varphi {|}^2$. It has exact global phase symmetry, not exact arbitrary local phase symmetry. For a prescribed background its field equation is $(\square +m^2)\varphi +ie(\partial \cdot A)\varphi +2ieA\cdot \partial \varphi =0$, together with its complex conjugate. The omitted quadratic interaction must not be added silently when solving this specified theory.

For $\delta \varphi =-iϵ\varphi$, the conserved current is $J^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2e{A}^{\mu }|\varphi {|}^2$. The conjugate field equations implywhich directly cancels the divergence of the explicit background term. Conservation requires no Lorenz gauge condition; constancy of total charge additionally requires vanishing boundary flux.

For the linear-in-potential charged-scalar Lagrangian, $\pi ={\dot{\varphi }}^{\ast }-ie{A}_0{\varphi }^{\ast }$ and ${\pi }^{\ast }=\dot{\varphi }+ie{A}_0\varphi$. With $A_{\mu }=(A_0,-\mathbf{A})$, the Hamiltonian density isThe first product includes $+e^2{A}_0^2|\varphi {|}^2$ after the Legendre transform; that term can arise in canonical variables even though the original Lagrangian has no interaction quadratic in $A$.

For electric charge $q$ and electromagnetic four-potential $A_a$, the derivative ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$ transforms covariantly under an electromagnetic gauge transformation.

The massless gauge-covariant wave equation is ${\mathcal{D}}_a{\mathcal{D}}^a\Phi =0$. In a stationary charged background, the electrostatic potential shifts the local frequency from $\omega$ to $\omega +q{A}_t$.

A scalar potential is the derivative-free energy density $V(\varphi )$ of a scalar field. Its derivative supplies the force term in the field equation, and its local minima describe candidate vacuum field values.

The special unitary group $SU(2)$ acts transitively on every sphere in its defining ${\mathbb{C}}^2$: for $|u{|}^2+|v{|}^2=1$, the matrix with columns $(u,v{)}^T$ and $(-\bar{v},\bar{u}{)}^T$ is in $SU(2)$. Thus any invariant real polynomial in a single commuting doublet and its complex conjugate depends only on ${\varphi }^{†}\varphi$. Through total degree four it is $V_0+m^2{\varphi }^{†}\varphi +\lambda ({\varphi }^{†}\varphi {)}^2$. A holomorphic polynomial invariant of positive degree does not exist for one doublet: ${\mathrm{Sym}}^{d}2$ is a nontrivial irreducible representation for every $d>0$.

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 scalar-field shift symmetry acts by $\varphi (x)\mapsto \varphi (x)+c$ for constant $c$. Its Noether charge is the spatial integral of the canonical momentum, up to the normalization of $c$, and interactions invariant under the symmetry contain derivatives of the scalar field.