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With the Minkowski metric $(+---)$, the momentum conjugate to the real field vector is $\mathbf{\pi }=\dot{\mathbf{\varphi }}$. The Hamiltonian density isAssume the potential has a global minimum $V_{\min }$ and the boundary conditions permit a constant minimizing field. ThenThe bound is attained byThe homogeneous vacuum minimum criterion proves existence of a homogeneous lowest-energy configuration under those assumptions. On infinite space one subtracts the constant vacuum energy before comparing finite energy differences. This identifies the classical vacuum configuration; it does not assert that the quantum field has no fluctuations.

The claim does not follow from “no derivative dependence in $V$” alone. For example, the allowed functional form $V(\varphi )=-{\varphi }^4$ is unbounded below, so there is no state of lowest energy to identify. Boundary conditions forcing different vacua at opposite ends can instead require an inhomogeneous minimum, such as a domain wall. The positive quartic potential used in the following parts does have homogeneous minima, so these caveats do not obstruct its vacuum calculation.

The Hamiltonian density is $\mathcal{H}=[{\dot{\varphi }}^2+({\partial }_x\varphi {)}^2+m^2{\varphi }^2]/2$. In the vacuum only $⟨a_n{a}_m^{†}⟩={\delta }_{nm}$ contributes. HenceSubstitution, using ${\omega }_n^2=k_n^2+m^2$, yieldsThis Dirichlet scalar vacuum energy density illustrates why a massive field's local unregularized energy density need not be uniform. The cosine integrates to zero over the interval, soThese divergent expressions are understood with a common regulator before exchanging sums, integrals or derivatives. Normal ordering the Hamiltonian operator separately in each geometry would erase the vacuum terms one is trying to compare; a common regulated subtraction is used instead in c and d.

For a static finite-energy field, the Hamiltonian density is $\frac{1}{2}({\varphi }^{\prime }{)}^2+V(\varphi )$. Multiply the static equation of motion by ${\varphi }^{\prime }$ and integrate once:At either vacuum endpoint both terms vanish, so $\frac{1}{2}({\varphi }^{\prime }{)}^2=V$. The total energy density is consequently $({\varphi }^{\prime }{)}^2=4k^2{\mathrm{sech}}^2[k(x-x_0)]/b^2$. Its integral givesFor the usual positive $a,b$ convention this isThe coefficient depends on the dimensionless parameter $b$; it is not a universal numerical constant independent of the potential. In two spacetime dimensions, $[\varphi ]=[b]=0$ and $[a]=2$, so the dimensions are consistent. If both $a$ and $b$ are negative while $ab>0$, the real answer is instead $8\sqrt{|a|}/|b{|}^{3/2}$; the literal factor $\sqrt{a}$ would not be real.

An equivalent positive-parameter calculation substitutes $z=b\varphi$ in $E_0={\int }_0^{2\pi /b}\sqrt{2V}d\varphi$ and uses the provided integral of $\sqrt{1-\cos z}$. The following original plot shows that the localized energy density sits where the field passes between the vacua.

The scalar potential is $V=-m^2|\phi {|}^2+\lambda |\phi {|}^4/4$. Completing the square givesThe Hamiltonian density also contains the nonnegative terms $|\dot{\phi }{|}^2+|\nabla \phi {|}^2$. Its lowest-energy configurations therefore have no time or spatial variation and minimize the displayed square. ThusThe minima form a circle of phases. Selecting the real positive representative $\phi =v$ chooses one vacuum and spontaneously breaks the continuous global symmetry in field theory. The stationarity at $\phi =0$ is a local maximum of the radial potential, not the ground state.

The Euler-Lagrange equation for the real scalar field givesThus the conjugate momentum and Hamiltonian density areThe spatial Fourier transform reduces the Klein-Gordon equation to ${\ddot{\varphi }}_{\mathbf{p}}+{\omega }_{\mathbf{p}}^2{\varphi }_{\mathbf{p}}=0$, with positive frequency ${\omega }_{\mathbf{p}}=\sqrt{{\mathbf{p}}^2+m^2}$. Hence a general real solution is the mode expansion of a free fieldThe two coefficients are complex conjugates because the field is real. Each pair of initial data $\varphi (0,\mathbf{x}),\pi (0,\mathbf{x})$ fixes these coefficients. The normalization anticipates canonical quantization and the relativistic normalization of one-particle states.

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.

Use the upper-index coordinates $B^{\mu },B^{\ast \mu }$ specified in the paper. Differentiation of the Lagrangian density gives the canonical momentaIn particular, ${\pi }_i=G^{\ast 0i}$ and ${\pi }_i^{\ast }=G^{0i}$. No time derivative of $B^0$ appears, so the zero momenta are primary constraints, not invertible equations for ${\dot{B}}^0$.

For spatial indices, $G^{0i}={\dot{B}}^i+{\partial }_i{B}_0$, so ${\dot{B}}^i={\pi }_i^{\ast }+{\partial }^i{B}_0$, with the conjugate relation for $B^{\ast i}$. AlsoThe Legendre transform for the complex Proca Hamiltonian, ${\mathcal{H}}_c={\pi }_i{\dot{B}}^i+{\pi }_i^{\ast }{\dot{B}}^{\ast i}-\mathcal{L}$ therefore givesRepeated spatial indices in the positive product ${\pi }_i^{\ast }{\pi }_i$ denote an ordinary three-component sum. A total Hamiltonian additionally includes multipliers for ${\pi }_0={\pi }_0^{\ast }=0$.

The Hamiltonian printed in the source cannot follow from the given complex Lagrangian density: its missing conjugations, mass coefficients and spatial mass sign are inconsistent. A concrete check is spatially uniform initial data $B^1=b\in \mathbb{R}$, $B^0=B^2=B^3=0$, with all canonical momenta zero. The correct Hamiltonian density is $M^2{b}^2$, while the printed expression gives $-M^2{b}^2/2$. These data satisfy the initial constraints and evolve as a spatially uniform massive oscillator. Thus the discrepancy is not a harmless surface term or a different complex-field normalization.

For $\delta x^0={ϵ}^0$ the Noether charge is $Q=-{ϵ}^{0}H$, with the Hamiltonian density $\mathcal{H}={\pi }_a{v}_a-\mathcal{L}(\varphi ,v,\nabla \varphi )$ and $v_a={\dot{\varphi }}_a$. Invertibility of the momentum-velocity relation allows $v_a=v_a(\varphi ,\pi ,\nabla \varphi )$. When taking a functional derivative at fixed $\varphi$, the implicit dependence of $v$ cancels in the Legendre transform:It follows thatThis proves the same generator relation for time translations. In a constrained theory the assumed inversion fails, so the appropriate reduced variables or Dirac brackets are needed instead. That is precisely the issue encountered for the Proca field in Question 4.

For the spatial momenta, integration by parts givesTheir Poisson bracket is thereforeFor the mixed case, $P_0=H$ and the spatial generator acts on both canonical fields by differentiation: $\{{\varphi }_a,P_i\}={\partial }_i{\varphi }_a$, $\{{\pi }_a,P_i\}={\partial }_i{\pi }_a$. Hence, for a translation-invariant local Hamiltonian density,Finally $\{H,H{\}}_{\mathrm{PB}}=0$ by antisymmetry. Together these proveThe proof requires commuting derivatives and no surviving boundary terms. It expresses the Abelian translation subgroup of Poincaré symmetry at the level of its canonical generators.