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For this three-site Abelian Higgs chain, the potential is minimized by $|{\varphi }_1|=|{\varphi }_2|=v$. Choose their phases real in a local unitary gauge. The scalar kinetic terms then givewhere Lorentz contractions are understood andThe nonzero eigenvalues of $K$ equal those ofnamely $1+q^2-q$ and $1+q^2+q$. The third eigenvalue is zero because $R$ has rank two. Both nonzero eigenvalues are positive for $q>0$.

Mass-normalization correction: the canonical vector mass term is $\frac{1}{2}{B}^{\mathsf{T}}{\mathsf{M}}^{2}B$, so the Lagrangian actually gives ${\mathsf{M}}^2=2g^2{v}^{2}K$. Its physical masses are thereforeThe printed dimensionless eigenvalue pattern is correct, but its common scale must be $m=\sqrt{2}gv$, not $gv$, for the written kinetic terms and vacuum value. For a concrete counterexample set $q=1$ and $B=(A,0,-A)/\sqrt{2}$. This is a canonically normalized eigenmode and its mass term is $g^2{v}^2{A}_{\mu }{A}^{\mu }$, giving physical squared mass $2g^2{v}^2$ rather than the printed $g^2{v}^2$. The printed scale $gv$ would require a vacuum modulus $v/\sqrt{2}$ or a corresponding change in scalar normalization, neither of which is the specified potential minimum.

The unbroken transformations preserve both vacuum phases:Thus the local breaking pattern isTwo scalar phases supply the longitudinal polarizations of the two massive vector bosons; one massless vector remains. This is the Higgs mechanism, with its mass matrix fixed by the scalar charge matrix and the vacuum normalization.^[ref]

A massless eigenvector must lie in the null space of $R$, so its coefficients satisfy $c_1=q{c}_2$ and $c_2=q{c}_3$. Hence they are proportional to $(q^2,q,1)$. The original gauge kinetic terms have the Euclidean unit matrix in species space, so normalize this vector by its ordinary squared length. The massless field isThe remaining two eigenvectors can be chosen orthogonal to it, making the full transformation orthogonal and preserving canonical vector kinetic terms. At $q=1$, the formula is exactly $(B_1^{\mu }+B_2^{\mu }+B_3^{\mu })/\sqrt{3}$ as required. This null direction and its normalization are unaffected by the common mass-scale correction in part b.

For $q>1$, the massless mode has geometrically unequal components; a matter field coupled to one end can therefore couple more weakly to this surviving vector than a field coupled to the other end. This follows directly from the eigenvector coefficients, without changing the equal microscopic gauge coupling $g$.

Goldstone's theorem states that a spontaneously broken continuous internal global symmetry in field theory of a local relativistic quantum field theory produces massless Goldstone bosons. For a Lorentz invariance vacuum breaking $G$ to its stabilizer $H$, there are $\dim G-\dim H$ independent symmetry-required massless directions. Additional accidentally massless modes are possible. A gauge redundancy does not obey this physical-state counting, and nonrelativistic systems need not have one independent mode per broken generator.

For the classical proof, consider real scalar fields ${\varphi }_i$ with a positive canonical kinetic term and an invariant smooth scalar potential $V$. An infinitesimal generator $T^a$ satisfiesLet $v$ be a stationary vacuum and differentiate this identity with respect to ${\varphi }_j$. At $v$, the term proportional to ${\partial }_{i}V$ vanishes, leavingThe Hessian matrix ${\mathsf{M}}^2$ is the scalar mass matrix. An unbroken generator has $T^{a}v=0$; broken generators span the tangent space of the vacuum manifold $G/H$. Each independent nonzero tangent vector is a zero eigenvector of ${\mathsf{M}}^2$. The quadratic Euler-Lagrange equations in these directions are $\square {\pi }^a=0$, giving the massless dispersion relation ${\omega }^2={\mathbf{p}}^2$. This proves the classical result and explains why the Goldstone bosons describe motion along the degenerate minima rather than against a restoring curvature.

Both the canonical kinetic term and $V({\varphi }^{\mathsf{T}}\varphi )$ are invariant under the orthogonal group $O(n)$. The stated nonzero vacuum expectation value has length $|v|$. Its stabilizer is the orthogonal group $O(n-1)$ acting on the perpendicular subspace. HenceThe two nonzero coordinates of this particular vacuum do not break more generators than an axis-aligned vacuum: the two choices are related by a common orthogonal transformation.

Explicitly, with $s={\varphi }^{\mathsf{T}}\varphi$,stationary point at nonzero $v$ requires $V^{\prime }(v^2)=0$. The $n-1$ transverse eigenvalues vanish, while the radial squared mass is $4v^2{V}^{\prime \prime }(v^2)$. If this curvature also vanishes, there is an additional accidental massless radial mode; the number of symmetry-protected Goldstone bosons remains $n-1$. If $v=0$, the symmetry is unbroken and none is required.

The action is dimensionless and $[d^{4}x]=-4$, so $[\mathcal{L}]=4$. A canonical scalar kinetic term then requires $2+2[\varphi ]=4$, and the electromagnetic kinetic term requires $2[F_{\mu \nu }]=4$. ThusThe brackets here denote mass dimension. The Lagrangian has no explicit coordinate dependence, and all indices are contracted as Lorentz scalars. Its continuous spacetime symmetry is the Poincaré group. Noether's theorem gives conserved four-momentum from translations and conserved angular momentum, including the boost generators, from Lorentz transformations. The nonzero mass $M$ and dimensionful couplings prevent a dilation symmetry.

Sign qualification: the printed electromagnetic kinetic term has the opposite healthy relative sign to the scalar kinetic terms. With the stated mostly-minus Minkowski metric, a physical Maxwell Lagrangian is $-F_{\mu \nu }{F}^{\mu \nu }/4$. The printed plus sign instead gives photon Hamiltonian density $-({\mathbf{E}}^2+{\mathbf{B}}^2)/2$; a nonzero purely electric configuration already has negative energy, unbounded below as its amplitude grows. The physical decay widths below use the healthy Maxwell sign with the given interaction normalization. The dimensional and spacetime-symmetry conclusions themselves do not depend on this sign correction.

Group the nonzero modes into independent pairs $p>0$. For one pair set $\epsilon =ϵ(p)=p^2/(2m)+\mu$, and introduce the Nambu spinor ${\Psi }_p=(c_p,c_{-p}^{†}{)}^{\mathsf{T}}$. Normal ordering using $c_{-p}{c}_{-p}^{†}=1-c_{-p}^{†}{c}_{-p}$ givesThis exactly contains both kinetic terms and the pairing terms once; there is no additional factor of one half because only $p>0$ is summed.

Insert ${\Psi }_p=U_p{\mathcal{A}}_p$, where ${\mathcal{A}}_p=({\alpha }_p,{\alpha }_{-p}^{†}{)}^{\mathsf{T}}$ and $U_p$ is the Bogoliubov transformation matrix. Direct matrix multiplication yieldsChoose the diagonalizing branchThen the off-diagonal terms vanish and the upper diagonal entry is $E_p>0$. Extend the angle to negative momenta as in part b; the sine relation just written applies to the positive representative of each pair. The BCS coherence factors can be chosen asHere $u_p$ denotes $\cos {\theta }_p$, and $v_p$ denotes $\sin {\theta }_p$; these expressions are nonsingular because $\mu >0$ and $m>0$. The minus sign of $v_p$ for positive $\Delta$ is fixed by the pairing phase in this Hamiltonian.

A second use of the canonical anticommutation relations givesThe zero-momentum term was never paired and is simply $\mu {\alpha }_0^{†}{\alpha }_0$. HencewithThe constant is the ground-state energy. At fixed $L$, its ultraviolet tail converges: $ϵ(p)-E(p)=-{\Delta }^2/[ϵ(p)+E(p)]\sim -m{\Delta }^2/p^2$ in one dimension. Replacing $E(0)$ by $\sqrt{{\mu }^2+{\Delta }^2}$ would incorrectly pair a spinless mode with itself.

Expanding the covariant kinetic term of scalar electrodynamics gives a one-photon two-scalar vertex and a two-photon two-scalar seagull vertex. Their tree amplitudes and Ward identities depend on a consistent choice of charge-flow and gauge covariant derivative convention.

Near a nonzero vacuum expectation value, the radial variable $h$ describes the Higgs mode and the angular variable $\pi$ describes the would-be Goldstone boson. For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, their local gauge transformation is $h\mapsto h$, $\pi \mapsto \pi +f\alpha$, and $A_{\mu }\mapsto A_{\mu }-{\partial }_{\mu }\alpha /e$. The scalar kinetic term is $(\partial h{)}^2/2+(1+h/f{)}^2(\partial \pi +efA{)}^2/2$. These coordinates simplify gauge orbits but obscure ordinary power counting through higher-dimensional derivative vertices and fail at $\varphi =0$.

For a complex scalar field with canonical kinetic term $|\partial \varphi {|}^2$ and potential $V=F(|\varphi {|}^2)$, take a broken-symmetry minimum $\varphi =v>0$, where $F^{\prime }(v^2)=0$. Writing $\varphi =v+(h+i\pi )/\sqrt{2}$ gives quadratic potential $F(v^2)+v^2{F}^{\prime \prime }(v^2)h^2$. Hence the canonical squared masses are $m_h^2=2v^2{F}^{\prime \prime }(v^2)$ and $m_{\pi }^2=0$. Coupling $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$ gives gauge-boson squared mass $m_A^2=2e^2{v}^2$; the factor of two depends on defining $v$ as the complex field's vacuum value rather than its canonically normalized real component.

For $V=m^2{\varphi }^2/2+\lambda ({\varphi }^2{)}^2/4$, $m^2<0$, the vacuum manifold is a sphere of radius $v=\sqrt{-m^2/\lambda }$. At a chosen vacuum the Hessian matrix is $2\lambda v^{2}n{n}^T$, where $n$ is its unit direction. The radial scalar has mass squared $2\lambda v^2$; the $n-1$ angular modes are physical Goldstone bosons when the symmetry is global.

For the two-component case, write $\mathbf{\varphi }=(v+\rho )(\cos \theta ,\sin \theta )$. The kinetic term becomes $\frac{1}{2}({\partial }_{\mu }\rho {)}^2+\frac{1}{2}(v+\rho {)}^2({\partial }_{\mu }\theta {)}^2$. At quadratic order the canonically normalized angular field is $\pi =v\theta$, with no mass term. Its Euler-Lagrange equation is the wave equation, so ${\omega }^2=|\mathbf{k}{|}^2$ in units where the propagation speed is one. The radial field instead obeys ${\omega }^2=|\mathbf{k}{|}^2+2\lambda v^2$. Thus a common rotation costs no energy, whereas a slowly varying rotation propagates as a collective wave. With the selecting source of continuous vacuum selection by an external source, use $\pi =w\theta$ and obtain ${\omega }^2=|\mathbf{k}{|}^2+h/w$. This elementary dispersion relation separates a Goldstone boson, a pseudo-Goldstone boson and a massive radial excitation within one example.

Expanding the covariant scalar kinetic term gives $ie{A}_{\mu }(\varphi {\partial }^{\mu }{\varphi }^{\ast }-{\varphi }^{\ast }{\partial }^{\mu }\varphi )$. With $D=\partial +ieA$, the vertex for an incoming charged particle of momentum $p$ and an outgoing one of momentum $p^{\prime }$ is $-ie(p+p^{\prime }{)}^{\mu }$. The antiparticle has the opposite charge-flow sign. The quadratic term $e^2{A}^2{\varphi }^{\ast }\varphi$ gives the separate seagull vertex $2i{e}^2{\eta }^{\mu \nu }$. On equal-mass external shells, contracting the three-point momentum with the photon momentum gives zero, the tree scalar quantum electrodynamics Ward identity.

The scalar mass matrix is the Hessian matrix of the scalar potential at a vacuum. Its eigenvalues are the tree-level squared masses; broken symmetry directions are zero eigenvectors before a Higgs mechanism or explicit symmetry breaking removes the corresponding massless modes.

For one charged chiral superfield, canonical kinetic terms, no nonconstant gauge-invariant superpotential, and $g>0$, the sign of $q\xi$ selects the vacuum. If $q\xi >0$, the minimum has $\varphi =0$, positive vacuum energy, unbroken internal gauge symmetry and a gaugino Goldstino; the scalar squared mass is $gq\xi$ while its chiral fermion is massless. If $q\xi <0$, a scalar condensate cancels the auxiliary field, preserving supersymmetry while Higgsing the gauge group. These are classical statements; a consistent quantum charged spectrum must cancel its gauge anomalies.

For repulsive contact interactions in a three-dimensional isotropic harmonic trap, neglecting the kinetic term in the Gross–Pitaevskii equation gives $n(r)=(\mu -m{\omega }^2{r}^2/2)/U_0$ inside $R=\sqrt{2\mu /(m{\omega }^2)}$ and zero outside. Normalization yields $N=8\pi \mu R^3/(15U_0)$, hence $\mu =(\hslash \omega /2)(15Na/a_0{)}^{2/5}$ and $R=a_0(15Na/a_0{)}^{1/5}$, with oscillator length $a_0=\sqrt{\hslash /(m\omega )}$. The density is parabolic but the wavefunction is its square root. Neglected kinetic energy rounds the edge and supplies a tail.

A spin coherent-state path integral inserts group-averaged resolutions of the identity between short time steps. Its kinetic term is the Spin coherent-state Berry phase, equal to $S$ times the oriented solid angle swept on the sphere.

The string-frame metric is the target metric appearing directly in the kinetic term of the string nonlinear sigma model. With conventional Euclidean dilaton $\phi$, its leading spacetime gravitational action contains $\sqrt{-g^{(s)}}{e}^{-2\phi }(R+4|\nabla \phi {|}^2)$. The dilaton-dependent Ricci coefficient distinguishes it from the Einstein-frame metric. Frame changes are local field redefinitions where the conformal factor is finite and nonzero; the Lorentzian dilaton coupling sign convention fixes which named field is $\phi$.

Two canonically normalized complex scalar fields of vacuum modulus $v$ and charge rows $R_1=(1,-q,0)$, $R_2=(0,1,-q)$ couple to three canonically normalized Abelian vector fields. The scalar kinetic terms give $g^2{v}^2|RB{|}^2$, so the physical squared-mass matrix includes the factor two shown. Its nonzero eigenvalues are $2g^2{v}^2(1+q^2\pm q)$ and its normalized null vector is $(q^2,q,1)/\sqrt{q^4+q^2+1}$. The two scalar phases are absorbed into the massive vector bosons, leaving one unbroken local Abelian generator. A compact global gauge-group interpretation additionally requires a compatible charge lattice.

For a Hamiltonian operator with kinetic term ${\hat{p}}^2/2$, insert position-basis resolutions between Trotter product formula factors. The measure is the limit of $(2\pi i\Delta {)}^{-M/2}{\prod }_{j=1}^{M-1}d{q}_j$, and the exponent contains ${\sum }_j[(q_j-q_{j-1}{)}^2/(2\Delta )-\Delta V(q_{j-1})]$. Its continuum notation is $\int \mathcal{D}q{e}^{iS[q]}$, with fixed endpoint values. A damping prescription defines the oscillatory limit.

For canonical positive kinetic terms in global supersymmetry, $V={\sum }_k|W_k{|}^2$ has mixed Hessian matrix $V_{i\bar{j}}={\sum }_k{W}_{ki}\overline{W_{kj}}$ and holomorphic block $V_{ij}={\sum }_k\overline{W_k}{W}_{kij}$. The $2n$ real scalar squared-mass sum is $2\mathrm{tr}(m^{†}m)$ for the chiral-superfield fermion mass matrix $m=W_{ij}$. Each Weyl spinor contributes two states, giving the same fermion weighted sum. Thus their boson-minus-fermion supertrace vanishes even with F-term breaking. The holomorphic Hessian splits real scalar masses without changing their sum. Noncanonical Kähler metrics, supergravity, and radiative corrections require different formulas.

A real scalar triplet of weak hypercharge zero with vacuum expectation value $(0,0,v)$ breaks $T^1,T^2$ but preserves both $T^3$ and the independent weak hypercharge generator. With canonical kinetic terms its gauge-boson mass matrix in $(W^1,W^2,W^3,B)$ is $g^2{v}^2\mathrm{diag}(1,1,0,0)$. The two massless neutral gauge bosons prevent this field alone from reproducing Standard Model electroweak symmetry breaking.