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Take the divergence of the Ampère-Maxwell equation and use that the divergence of a curl is zero:The last step uses Gauss's law. Thus the charge continuity equation is exactlyFor a direct justification that this four-current is a four-vector, take one locally comoving population of electric charges, with rest charge density ${\rho }_0$ and four-velocity $U^{\mu }=\gamma (c,\mathbf{v})$. Invariance of electric charge and length contraction give $\rho =\gamma {\rho }_0$ and $\mathbf{J}=\rho \mathbf{v}$, hence $J^{\mu }={\rho }_0{U}^{\mu }$. Summing populations preserves the four-vector rule. In particular, $c{\rho }^{\prime }=\gamma (c\rho -\beta J_x)$ and $J_x^{\prime }=\gamma (J_x-\beta c\rho )$. The Lorentz invariance of its four-divergence then explains why charge conservation holds in every inertial frame. A single zero-divergence condition alone would not determine a field's transformation law; the physical charge density and current density provide the additional content.

Use $x^{\mu }=(t,x)$ and Minkowski metric ${\eta }^{\mu \nu }=\mathrm{diag}(1,-1)$, with the wave speed equal to one. Regarding $\varphi$ as a scalar field, the symmetric canonical stress-energy tensor isIt obeys ${\partial }_{\mu }{T}^{\mu \nu }=0$ on the wave equation. For a scalar field the intrinsic spin current vanishes. The total orbital Lorentz current and its integrated angular-momentum tensor areBecause there is only one spatial dimension, its only independent possibly nonzero component iswhere $P=\int pdx$. The corresponding flux is ${\mathcal{J}}^{101}=t\mathcal{E}-xp$. Symmetry and conservation of $T$ giveThus $J^{01}$ is conserved provided the boundary flux vanishes and the integrals exist. For example, localized wave packets on an infinite line have this property. One can also check directly that $d\int x\mathcal{E}dx/dt=P$, so the two terms in $J^{01}$ have cancelling derivatives.

The significance is the boost charge of a one-dimensional scalar field. There is no independent spatial rotation or ordinary spatial angular-momentum component in one dimension. Instead, the action has the mathematical Lorentz invariance of a massless wave field with invariant speed $c_s=1$, and $J^{01}$ generates its boost. Its conservation says that the center of energy moves uniformly. This is a symmetry of the ideal wave-field model, rather than an assertion of fundamental vacuum Lorentz symmetry for a material rod. Fixed boundaries can break it, and a general solution with nonvanishing boundary flux need not conserve the integrated charge.

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.

All daughter particles are massless, so the parent mass $M$ supplies the only kinematic scale at tree level. Each vertex has two derivatives, making its scattering amplitude scale as ${\lambda }_i{M}^2$. For the four-$b$ vertex, symmetrizing over the derivative-bearing legs yields a sum of pairwise four-momentum products, which is proportional to $M^2$ by four-momentum conservation.

For $n$ massless daughters, the Lorentz invariance Lorentz-invariant phase-space measure scales as $M^{2n-4}$: each $d^{3}q/(2E)$ contributes two powers and the four-dimensional Dirac delta removes four powers. Including the parent factor $1/(2M)$ gives $\Gamma \sim |{\lambda }_i{|}^2{M}^{2n-1}$. ThusThe unspecified factors are dimensionless angular integrals and identical-particle factors. Each result has mass dimension one, as a decay width must in natural units.

Work in the parent rest frame, $p=(M,0)$. For massless daughters, the spatial Dirac delta fixes ${\mathbf{q}}_2=-{\mathbf{q}}_1$ and $E_1=E_2=q$. The Lorentz invariance two-body Lorentz-invariant phase-space measure becomesThe factor $1/2$ in the last step is the Jacobian of the energy-conservation Dirac delta. Consequently the labelled two-body decay width isFor two identical final particles, the full angular integral counts each unordered final state twice, so the physical decay width includes an additional factor $1/2!$. This factor is needed for both channels in part e; it is separate from any vertex combinatorics already included in the Feynman rule.

The real scalar field action has spacetime translations and Lorentz invariance, together giving Poincaré symmetry. With a real scalar transformation law it also has parity and time reversal. A generic polynomial potential has no continuous internal symmetry. The additional field reversal $\varphi \mapsto -\varphi$ is a symmetry precisely when every nonzero odd-power coupling vanishes. The free massless special case also admits a constant shift of $\varphi$; generic interactions remove it.

In mass units, $[{\partial }_{\mu }]=1$ and the dimensionless action requires $[\mathcal{L}]=D$. Normalizing the kinetic term fixes the classical mass dimension of the scalar field:For each interaction, $[{\lambda }_n]+n[\varphi ]=D$, soThese are classical or engineering scaling dimensions; quantum corrections can change the scaling of operators and couplings. The scale transformation prescribed later has field weight one, which equals this canonical weight only at $D=4$.

The joint energy-momentum spectrum lies in the closed forward light cone. In axiomatic quantum field theory, this condition permits analytic continuation of Wightman functions in suitable complex spacetime domains. Together with Lorentz invariance, microcausality and the positivity of the physical state space, it is one of the hypotheses behind the CPT theorem.

A scalar-valued function is a function whose codomain is a field, typically the real numbers or complex numbers. Its output is a single scalar, while its input can belong to an arbitrary set. For example, the square of the norm of a vector is real-valued. In a relativistic application, being scalar-valued alone does not establish Lorentz invariance: the transformation of the input and of the complete quantity must also be specified.

For electromagnetic deep inelastic scattering with an unpolarized target, Lorentz invariance, parity and current conservation leave two independent deep-inelastic structure functions. With $T^{\mu \nu }=g^{\mu \nu }-q^{\mu }{q}^{\nu }/q^2$ and $\tilde{P}=P-(P\cdot q/q^2)q$, both tensor structures are transverse to $q$. Terms with a free $q$ index disappear on contraction with a conserved leptonic tensor, although such terms can still be necessary to make the full hadronic tensor transverse.

For a massive spin-zero one-particle state with covariant normalization and a Lorentz scalar field, Lorentz invariance makes $F=⟨\Omega |\varphi (0)|p⟩$ independent of the on-shell four-momentum within a fixed species. A Lorentz boost to the rest frame, invariance of the vacuum state and scalar transformation at the origin prove this. Translation covariance of a quantum field then supplies the plane-wave factor. With noncovariant normalization, the boost law and the displayed momentum independence acquire energy factors.