Stress–Energy, Conservation Laws and the Field Action in Biquaternionic Form

Introduction

The preceding articles of this subcategory treat the particle's kinematics, the causal cone, the Lorentz group and its topology. Each of these is a statement about a single object or a single transformation. The present article treats the quantity that ties the objects together under evolution: the stress–energy tensor, and the action from which both the field equations and the conservation laws follow.

The organizing fact is that in a Lagrangian field theory the stress–energy tensor is not an independent input. It is the Noether current of translation invariance of the action,

$$ S[\psi] = \int \mathcal{L}(\psi,\partial_\mu\psi)\,d^4x , $$

so that the same action that yields the field equations through the Euler–Lagrange equations also yields the ten components of $T^{\mu\nu}$ through Noether's theorem, and the conservation law

$$ \partial_\mu T^{\mu\nu} = 0 $$

is the on-shell statement of that invariance. The biquaternion framework adds nothing to this logic, but it gives the objects a definite home and makes two constructions explicit: the stress–energy of a biquaternion-valued field is built from the algebra's bilinear form, and the electromagnetic stress–energy is a biquaternion bilinear in the field-strength biquaternion, with the basis elements of $\mathbb{M}_-$ supplying the two vector directions of the two indices.

Three threads are developed.

The field action. The scalar action whose Euler–Lagrange equation is the biquaternionic Klein–Gordon equation, and the electromagnetic action whose Euler–Lagrange equation is the biquaternionic Maxwell equation. Both are transcribed from standard field theory; the algebra supplies the packaging.
The Noether construction. The general stress–energy of a field, its conservation on shell, and the explicit scalar stress–energy with its positive energy density, verified on a superposition of on-shell modes.
The electromagnetic tensor as a biquaternion bilinear. The construction $T^\mu{}_\nu = \tfrac12\mathrm{Sc}(\tilde{F}\mathcal{E}_\mu\tilde{F}^{*}\mathcal{E}_\nu)$, its component table, its symmetry, its tracelessness, and its conservation, including the source term that couples it to matter.

Boundaries. This article is classical and non-quantum: it concerns the classical field action, its Noether currents, and the classical conservation laws. The quantisation of these fields, the associated Hilbert-space structures and the generating functionals belong to the sibling categories and to the companion article Canonical Quantization of the Biquaternion Maxwell Field, and they are not developed here. The informational reading of any of these quantities is likewise outside the scope; the stress–energy tensor constructed here is an element of the field's tensor algebra, and its bilinear form takes values in the Hermitian sector $\mathbb{M}_+$ only in the sense that the contraction $\tfrac12\tilde{F}\tilde{F}^{*}$ is Hermitian, which is recorded in the companion article Maxwell's Equations in the Biquaternionic Formulation. The multipole and spin-direction effects are treated elsewhere in the category.

Conventions. We use those of the read list unchanged. The algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with quaternion basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$ and $e_je_k = -\delta_{jk}e_0+\varepsilon_{jkl}e_l$, and central scalar imaginary $i$. The subspaces are $\mathbb{M}_-$ (anti-Hermitian: imaginary scalar, real vector — the material sector, with basis $\mathcal{E}_\mu\in\{ie_0,e_1,e_2,e_3\}$), $\mathbb{M}_+$ (Hermitian: real scalar, imaginary vector — the informational sector), $\mathbb{H}_{\mathbb{B}}$ (real quaternions) and $\mathbb{C}_{\mathbb{B}}$ (the center). The conjugations are ${}^{\natural}$ (quaternion), $\bar{\cdot}$ (complex), ${}^{*} = ({}^{\natural})^{\,*}$ (Hermitian) and ${}^\flat=-{}^{*}$ (anti-Hermitian). The gradient is $\tilde{\nabla}=e_0\partial_{ict}+e_1\partial_x+e_2\partial_y+e_3\partial_z$, with $\tilde{\nabla}^{\natural}=e_0\partial_{ict}-e_1\partial_x-e_2\partial_y-e_3\partial_z$ and

$$ \Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \partial_{ict}^2+\Delta = -\frac{1}{c^2}\partial_t^2+\Delta . $$

The biquaternion norm is $N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=\sum_\mu Q_\mu^2$ — level 1, the identity $\mathrm{diag}(+1,+1,+1,+1)$ on $\mathbb{C}$ — and its restriction to the real material slice is the level-2 form $\eta=\mathrm{diag}(-1,+1,+1,+1)$. The scalar mass parameter is $\mu = mc/\hbar$, so that the scalar equation is $(\Box-\mu^2)\tilde{\Phi}=0$, matching the companion article The Klein–Gordon Equation in Biquaternionic Form. The field strength is $\tilde{F}=i\sqrt{\epsilon}\,\mathbf{E}-\sqrt{\mu}\,\mathbf{H}$, the source is $\tilde{R}=\frac{i\rho}{\sqrt{\epsilon}}e_0+\sqrt{\mu}\,\mathbf{J}$, and the Maxwell equation is $\tilde{\nabla}\tilde{F}=-\tilde{R}$, all as fixed in the companion article Maxwell's Equations in the Biquaternionic Formulation. The energy density is $W=\tfrac12(\epsilon\mathbf{E}^2+\mu\mathbf{H}^2)$ and the Poynting vector is $\mathbf{S}=\mathbf{E}\times\mathbf{H}$. The trace formula is $\mathrm{Tr}(\tilde{P}\tilde{H})=2\,\mathrm{Sc}(\tilde{P}\tilde{H})$. Throughout, $c=1/\sqrt{\epsilon\mu}$ is the speed of light in the medium and $c_0$ its vacuum value.

A sign warning. As in the companion exercise Exercise: The Electromagnetic Energy–Momentum Tensor, the mixed component $T^0{}_0$ carries a sign in the $ict$ convention: for the electromagnetic field $T^0{}_0=-W$, while the physical energy density is the contravariant $T^{00}=W$. This article keeps the same convention for the scalar field and states it at each tensor, so that the two fields can be compared without a sign discrepancy.

The Field Action in Biquaternion Form

The action of a field theory is a functional of the field and its first derivatives,

$$ S = \int \mathcal{L}(\psi,\partial_\mu\psi)\,d^4x , \qquad d^4x = c\,dt\,dx\,dy\,dz , $$

and the field equation is its stationarity condition. For each field component $\psi_a$ the Euler–Lagrange equation is

$$ \frac{\partial\mathcal{L}}{\partial\psi_a} - \partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi_a)} = 0 , $$

and the statement that a continuous transformation of the field leaves $S$ unchanged yields a conserved current by Noether's theorem. The framework adds no new principle here; it supplies the algebra in which the fields are written, and, as the next two subsections show, the equations come out in the biquaternionic forms already fixed by the companion articles.

The Scalar Action

Let $\tilde{\Phi}$ be a biquaternion-valued field and take the Lagrangian density

$$ \mathcal{L}_\phi = -\tfrac12\,\partial_\mu\tilde{\Phi}\,\partial^\mu\tilde{\Phi}^{\natural} - \tfrac12\mu^2\,\tilde{\Phi}\tilde{\Phi}^{\natural} , \qquad \mu = \frac{mc}{\hbar} . $$

Because $\Box$ is central and scalar, the field decomposes into its four real components and the Euler–Lagrange equation is componentwise the scalar equation. For a real scalar component $\phi$ the variation gives

$$ \partial_\mu\partial^\mu\phi - \mu^2\phi = \Box\phi-\mu^2\phi = 0 , $$

that is,

$$ \left(\Box-\mu^2\right)\tilde{\Phi} = 0 , $$

the biquaternionic Klein–Gordon equation of the companion article. The overall sign of $\mathcal{L}_\phi$ is fixed by the requirement that the energy be positive, as the stress–energy section shows; in the corpus's level-2 signature, the kinetic term of $\mathcal{L}_\phi$ written in coordinates is

$$ \mathcal{L}_\phi = -\tfrac12\left(-\frac{1}{c^2}\dot{\phi}^2+(\nabla\phi)^2\right)-\tfrac12\mu^2\phi^2 = \frac{1}{2c^2}\dot{\phi}^2-\frac12(\nabla\phi)^2-\frac12\mu^2\phi^2 , $$

and it is this sign that makes the energy density of the next section positive. Two independent real fields, or equivalently one complex field, are needed for the general solution, exactly the doubling the Klein–Gordon article records.

The Electromagnetic Action

For the field strength $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ the free action is

$$ S_{\mathrm{em}} = -\frac14\int F_{\mu\nu}F^{\mu\nu}\,d^4x , \qquad F_{\mu\nu}F^{\mu\nu} = 2\mu\,\mathrm{Re}\,N(\tilde{F}) , $$

the second equality being the statement, established in the companion article The Field-Strength Biquaternion and Its Invariants, that the two Lorentz invariants of the free field are the real and imaginary parts of the biquaternion norm $N(\tilde{F})=\sum_kF_k^2$. Varying with respect to $A_\nu$ gives the source-free Maxwell equation

$$ \partial_\mu F^{\mu\nu} = 0 , $$

and adding the minimal-coupling term $S_{\mathrm{int}}=-\int A_\nu J^\nu\,d^4x$ with the four-current $J^\nu=(\rho c,\mathbf{J})$ gives the inhomogeneous Maxwell equations, whose biquaternion form is the single equation

$$ \tilde{\nabla}\tilde{F} = -\tilde{R}, \qquad \tilde{R} = \frac{i\rho}{\sqrt{\epsilon}}e_0+\sqrt{\mu}\,\mathbf{J}, \qquad \mathrm{Sc}\!\left(\tilde{\nabla}^{\natural}\tilde{R}\right)=0 , $$

the form fixed in the companion article Maxwell's Equations in the Biquaternionic Formulation. Its content is exactly the four component equations together with the integrability condition $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{R})=0$, equivalent to charge conservation.

The two actions exhibit the same pattern. The kinetic term is the biquaternion norm of the derivative, $-\tfrac12\partial_\mu\tilde{\Phi}\partial^\mu\tilde{\Phi}^{\natural}$ for the scalar and $-\tfrac14 F_{\mu\nu}F^{\mu\nu}\propto N(\tilde{F})$ for the electromagnetic field; the equation of motion is the vanishing of the appropriate first-order or second-order operator; and the conservation of the source current is the integrability condition of the equation, not a separate law.

Noether's Theorem and the Stress–Energy of Translation

The stress–energy tensor is the Noether current of invariance under spacetime translations. Under an infinitesimal translation $x^\mu\to x^\mu+\epsilon^\mu$ a Lagrangian that depends on $x$ only through the fields changes by

$$ \delta\mathcal{L} = \epsilon^\nu\partial_\nu\mathcal{L} = \epsilon^\nu\partial_\mu\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi_a)}\partial_\nu\psi_a\right) - \epsilon^\nu\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi_a)}\partial_\nu\psi_a , $$

where the second equality uses the Euler–Lagrange equation to replace the derivative with respect to $\psi_a$. Combining the two terms gives the divergence of the canonical stress–energy tensor

$$ T^\mu{}_\nu = \sum_a \frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi_a)}\,\partial_\nu\psi_a - \delta^\mu_\nu\,\mathcal{L}, \qquad \partial_\mu T^\mu{}_\nu = 0 \ \text{on shell} . $$

The ten conserved quantities are the four components of energy–momentum (from $\nu=0$) and the six components of angular momentum (from the spatial part, after improvement to a symmetric tensor). The biquaternion framework's contribution is not to this derivation but to the interpretation of the result: for a field whose values lie in $\mathbb{B}$, the sum over components makes $T^\mu{}_\nu$ the bilinear of the field with its conjugate, and for the electromagnetic field the bilinear has a closed algebraic form.

Symmetry and improvement. The canonical tensor need not be symmetric or gauge invariant; the physical stress–energy is obtained by adding a divergence of an antisymmetric tensor, which does not change the conserved charges. For the electromagnetic field the improved tensor is symmetric and gauge invariant, and it is the one constructed in the next sections. For a single real scalar the canonical tensor is already symmetric. The improvement is standard and is recorded here because the biquaternion bilinear automatically gives the improved form.

Overall sign. The canonical tensor is defined up to an overall sign by the sign of the Lagrangian, and this article follows the convention of the companion exercise: the tensor is taken with the sign that makes the physical energy density positive, $T^{00}>0$, which for the electromagnetic field means $T^0{}_0=-T^{00}=-W$. Since an overall constant does not affect $\partial_\mu T^{\mu\nu}=0$, the conservation law is unaltered.

The Scalar Field Stress–Energy

For a real scalar field with the Lagrangian $\mathcal{L}_\phi$ of the previous section, the conserved, symmetric stress–energy tensor with positive energy is

$$ T_{\mu\nu} = \partial_\mu\phi\,\partial_\nu\phi + \eta_{\mu\nu}\mathcal{L}_\phi , $$

the sign being the one that makes $T_{00}>0$; the canonical Noether tensor is the negative of this expression, and the two differ only by the overall sign just discussed. Its components are, with $\eta=\mathrm{diag}(-1,+1,+1,+1)$ and $x^0=ct$,

$$ T^{00} = T_{00} = \frac{1}{2c^2}\dot{\phi}^2 + \frac12(\nabla\phi)^2 + \frac12\mu^2\phi^2 > 0 , $$

$$ T^{0k} = T^{k0} = -\frac{1}{c}\,\dot{\phi}\,\partial_k\phi , \qquad T^{jk} = T_{jk} = \partial_j\phi\,\partial_k\phi + \delta_{jk}\left(\frac{1}{2c^2}\dot{\phi}^2-\frac12(\nabla\phi)^2-\frac12\mu^2\phi^2\right) . $$

For a right-moving massless wave, $\phi=f(x-ct)$, these reduce to $T^{00}=T^{xx}=f'^2$ and $T^{0x}=T^{00}$, the null-field pattern of a single traveling wave.

The energy density is manifestly positive: it is the sum of three squares, one for the kinetic term, one for the gradient, and one for the mass. The conservation law

$$ \partial^\mu T_{\mu\nu} = 0 \qquad\text{on shell} $$

is the statement that the four-momentum

$$ P_\nu = \frac{1}{c}\int T_{0\nu}\,d^3x $$

is time-independent, which follows by integrating the conservation law over a spatial volume and using the vanishing of the flux at infinity.

Verification on a superposition. The conservation law was checked numerically on a superposition of two on-shell real modes rather than on a single plane wave, for which the identity would hold mode by mode and would not test the cross terms. The representation used was

$$ \phi(\mathbf{x},t) = \cos\!\left(k_1x-\omega_1t\right) + 0.6\cos\!\left(k_2y-\omega_2t\right), \qquad \mathbf{k}_1 = (1.1,0,0), \quad \mathbf{k}_2 = (0,0.9,0), \quad \mu = 1.3, $$

with $\omega_i^2 = \mathbf{k}_i^2+\mu^2$ for each mode, so that each mode is on shell and the two are on shell simultaneously. Evaluating $\partial^\mu T_{\mu\nu}$ by second-order finite differences with step $10^{-3}$ at the event $t=0.7$, $\mathbf{x}=(0.3,-0.5,0.2)$ gave residuals of order $10^{-6}$ for all four $\nu$, which is the discretisation error $O(h^2)$; the energy density at the same event was $T^{00}=3.37>0$. The cross terms between the two modes, which are the content of the check, cancel as they must for a conserved bilinear of on-shell fields.

The Electromagnetic Stress–Energy as a Biquaternion Bilinear

For the electromagnetic field the improved stress–energy tensor has a closed algebraic form in the biquaternion algebra. With the material basis $\mathcal{E}_\mu\in\{ie_0,e_1,e_2,e_3\}$,

$$ \boxed{\; T^\mu{}_\nu = \frac12\,\mathrm{Sc}\!\left(\tilde{F}\,\mathcal{E}_\mu\,\tilde{F}^{*}\,\mathcal{E}_\nu\right) = \frac14\,\mathrm{Tr}\!\left(\tilde{F}\,\mathcal{E}_\mu\,\tilde{F}^{*}\,\mathcal{E}_\nu\right) \;} $$

The two factors of $\mathcal{E}$ are what turn one biquaternion into a rank-two object: each supplies one vector direction, and the scalar part reads off the component. The construction is that of the companion exercise Exercise: The Electromagnetic Energy–Momentum Tensor, and its components are

$$ T^0{}_0 = -W, \qquad T^0{}_j = \frac{1}{c}S_j, \qquad T^j{}_0 = -\frac{1}{c}S_j, \qquad T^j{}_k = -\sigma_{jk}, $$

with the Maxwell stress

$$ \sigma_{jk} = \epsilon\,E_jE_k + \mu\,H_jH_k - W\,\delta_{jk}, \qquad W = \tfrac12\left(\epsilon\mathbf{E}^2+\mu\mathbf{H}^2\right), \qquad \mathbf{S} = \mathbf{E}\times\mathbf{H}. $$

Raising the second index with $\eta=\mathrm{diag}(-1,+1,+1,+1)$ gives the symmetric contravariant tensor

$$ T^{00} = W, \qquad T^{0j} = T^{j0} = \frac{1}{c}S_j, \qquad T^{jk} = -\sigma_{jk} = W\delta_{jk}-\epsilon E_jE_k-\mu H_jH_k , $$

which is the standard electromagnetic energy–momentum tensor in a medium: $W$ is the energy density, $\tfrac1c\mathbf{S}$ is the energy flux and (times $c$) the momentum density, and $-\sigma_{jk}$ is the momentum flux. The mixed tensor is deliberately not symmetric; the asymmetry is exactly the metric factor between the energy flux and the momentum density, and it disappears on raising the index.

Tracelessness. The trace in the Minkowski pairing is

$$ \eta_{\mu\nu}T^{\mu\nu} = -T^{00}+T^{kk} = -W - \sigma_{kk} = -W + W = 0 , $$

because $\sigma_{kk} = \epsilon\mathbf{E}^2+\mu\mathbf{H}^2-3W = 2W-3W=-W$. The tensor is traceless in the medium as well as in vacuum, and that is the classical statement that the source-free theory is invariant under the conformal rescaling of the metric. Verification. The trace was evaluated on three hundred random field configurations and vanished to machine precision, at the level of $10^{-15}$, in each case; the component table was checked against the bilinear definition on the same sample.

Conservation. The divergence of the tensor is the Lorentz four-force density,

$$ \partial_\mu T^{\mu\nu} = -f^\nu , \qquad f^\nu = \left(\frac{1}{c}\mathbf{E}\cdot\mathbf{J},\ \rho\mathbf{E}+\mathbf{J}\times\mathbf{B}\right), $$

so that for a source-free field $\partial_\mu T^{\mu\nu}=0$, and in the presence of charges the field momentum is exchanged with the matter at the rate given by the Lorentz force. The companion article The Lorentz Force in Biquaternion Form treats the matter side of this exchange. Verification on a superposition. The source-free conservation law was checked on a superposition of two plane waves in two different directions with two different polarisations,

$$ \mathbf{E} = \mathbf{e}_1\cos(k_1x-\omega_1t) + 0.5\,\mathbf{e}_2\cos(k_2y-\omega_2t), \qquad \mathbf{H} = \hat{\mathbf{k}}_i\times\mathbf{E}\ \text{per mode}, $$

with $\omega_i = c|\mathbf{k}_i|$, $c=1$, $\epsilon=\mu=1$. Evaluating $\partial_\mu T^{\mu\nu}$ by second-order finite differences with step $10^{-3}$ at the event $t=0.6$, $\mathbf{x}=(0.3,-0.7,0.2)$ gave $|\partial_\mu T^{\mu\nu}|$ of order $10^{-7}$, and falling as $h^2$, for all four $\nu$ — the discretisation error — confirming conservation for the interacting cross terms and not merely mode by mode.

Where the algebra enters. The bilinear form realizes the two-index object by the two inserted basis elements; the scalar projection extracts the component. This is why the natural object is the mixed tensor $T^\mu{}_\nu$ and not a doubly covariant one, and why no single biquaternion can carry the full tensor — the four-component object $\tilde{W}=\tfrac12\tilde{F}\tilde{F}^{*}$ carries only the time row, as the exercise shows. The rank-two object with two independent vector directions is the smallest algebraic structure that holds all the components, and the algebra supplies it through the bilinear rather than through a new field.

The Conservation Law for Field and Matter

The conservation law of the preceding section is not closed: the field's four-momentum changes in the presence of charges. The total energy–momentum of field and matter,

$$ T^{\mu\nu}_{\mathrm{tot}} = T^{\mu\nu}_{\mathrm{em}} + T^{\mu\nu}_{\mathrm{matter}} , $$

is conserved. The matter tensor of a pressureless dust of proper density $n$ and four-velocity $\tilde{U}$ is the tensor form of the particle flux,

$$ T^{\mu\nu}_{\mathrm{matter}} = m\,n\,U^\mu U^\nu , \qquad m = \text{particle mass}, $$

whose divergence is the negative of the field's:

$$ \partial_\mu T^{\mu\nu}_{\mathrm{matter}} = +f^\nu \qquad\Longleftrightarrow\qquad \partial_\mu T^{\mu\nu}_{\mathrm{tot}} = 0 . $$

The component $\nu=0$ of this statement is the Poynting theorem,

$$ \partial_t W + \nabla\cdot\mathbf{S} + \mathbf{E}\cdot\mathbf{J} = 0 , $$

the $\nu=k$ components are the momentum-balance equations that underlie radiation pressure, and the biquaternionic form of the whole statement is the single equation

$$ \mathrm{Sc}\!\left(\tilde{\nabla}\tilde{W}\right) = \frac{i}{c}\,\mathbf{J}\cdot\mathbf{E} $$

for the Hermitian form $\tilde{W}=\tfrac12\tilde{F}\tilde{F}^{*}$, which the companion article Maxwell's Equations in the Biquaternionic Formulation derives and the exercise re-derives by contrast with the failed real form. The pattern is general: the divergence of the field's stress–energy equals minus the four-force density, the divergence of the matter's tensor equals plus it, and the sum is conserved. The field action is what fixes both the field tensor and, through the coupling term, the force that appears on the matter side.

What the Algebra Supplies and What Is Transcribed

Supplied by the algebra. The biquaternion bilinear form of the electromagnetic stress–energy, $T^\mu{}_\nu=\tfrac12\mathrm{Sc}(\tilde{F}\mathcal{E}_\mu\tilde{F}^{*}\mathcal{E}_\nu)$, with the basis elements of $\mathbb{M}_-$ supplying the two index directions; the identification of the free-field Lagrangian with the biquaternion norm of the field strength, $\tfrac14F_{\mu\nu}F^{\mu\nu}\propto\mathrm{Re}\,N(\tilde{F})$, so that the action is built from the same biquaternion norm that defines the cone; the packaging of the scalar action, whose equation is the biquaternionic Klein–Gordon equation because $\Box=\tilde{\nabla}\tilde{\nabla}^{\natural}$ is central and scalar; and the interchange law of the divergence, field action and conserved current that lets the divergence of the field tensor be read as the source term of the matter.

Standard field theory transcribed. The Euler–Lagrange equations, Noether's theorem, the canonical stress–energy tensor, its improvement to a symmetric gauge-invariant form, the positivity of the scalar energy, the Poynting theorem and the Lorentz four-force are standard. The algebra reproduces them in its own notation; it does not add a conservation law or modify one.

Interpretation. The reading of the bilinear as an element whose Hermitian part is a positive energy density, and of the algebra's basis elements as the index directions, is the framework's structural reading; the tensor components themselves are the standard ones. The one genuinely algebraic statement is that the free Lagrangians of the scalar and electromagnetic fields are both the biquaternion norm of their respective objects — the derivative for the scalar and the field strength for the gauge field — so that the action and the causal cone are constructed from the same quadratic form.

Open Questions

The gravitational coupling. The stress–energy tensor is the source of gravity in any theory that couples gravity to matter. The framework so far is flat; whether the biquaternion algebra, whose biquaternion norm supplies the flat metric, can carry a dynamical metric in which $T^{\mu\nu}$ acts as source is the same open question the foundational articles record for curved spacetime.
The scalar field's trace and conformal coupling. The free electromagnetic tensor is traceless, and the free massless scalar tensor is not. The framework does not by itself distinguish the conformal coupling of the scalar; whether the algebra's structure prefers the conformally coupled scalar, whose tensor is traceless in four dimensions, is not settled here.
The biquaternion packaging of the general tensor. A rank-two tensor has ten independent components; a biquaternion has eight real dimensions. The bilinear construction solves the counting by using two basis-vector insertions, but the general classification of which biquaternion-valued bilinears produce which tensors is not developed in this article.
The medium and the action. The speed $c=1/\sqrt{\epsilon\mu}$ is local. The action written here treats $\epsilon$ and $\mu$ as constants; a dispersive medium would require a non-local action and would change both the conservation law and the tensor. The framework's treatment of a varying medium is an open question shared with the foundational article on the local complex structure.
Conserved currents of the other symmetries. Translation invariance gives the stress–energy; Lorentz invariance gives the angular-momentum tensor, and the internal symmetries of a complex scalar field give the vector current. A systematic biquaternionic account of all the Noether currents of these theories, and of their algebra, is left to the companion article Noether's Theorem in Biquaternionic Form and beyond it.

The conventions of the construction are those of the following companion articles:

Companion article Conventions in the Biquaternion Universe, for the algebra and basis, the conjugations, the real subspaces, the metric levels and the d'Alembertian.
Companion article The Klein–Gordon Equation in Biquaternionic Form, for the scalar equation and its mass parameter.
Companion article Maxwell's Equations in the Biquaternionic Formulation, for the field strength, the source biquaternion and the biquaternion Maxwell equation.
Companion article The Field-Strength Biquaternion and Its Invariants, for the biquaternion norm of the field strength and the Lorentz invariants.
Companion article Exercise: The Electromagnetic Energy–Momentum Tensor, for the biquaternion bilinear form of the tensor and its component table.
Companion article The Lorentz Force in Biquaternion Form, for the matter side of the field–matter exchange.
Companion article Noether's Theorem in Biquaternionic Form, for the general construction of the conserved currents.

Summary

The action of a field theory yields both its equations of motion and, by Noether's theorem, its conserved stress–energy. In biquaternion form the scalar action

$$ \mathcal{L}_\phi = -\tfrac12\partial_\mu\tilde{\Phi}\,\partial^\mu\tilde{\Phi}^{\natural}-\tfrac12\mu^2\tilde{\Phi}\tilde{\Phi}^{\natural}, \qquad \mu=\frac{mc}{\hbar}, $$

has the Euler–Lagrange equation $(\Box-\mu^2)\tilde{\Phi}=0$, and the electromagnetic action

$$ S_{\mathrm{em}} = -\frac14\int F_{\mu\nu}F^{\mu\nu}\,d^4x , \qquad F_{\mu\nu}F^{\mu\nu} = 2\mu\,\mathrm{Re}\,N(\tilde{F}), $$

has the equation $\tilde{\nabla}\tilde{F}=-\tilde{R}$, with $\tilde{R}=\frac{i\rho}{\sqrt{\epsilon}}e_0+\sqrt{\mu}\mathbf{J}$. The translation Noether current is the canonical tensor $T^\mu{}_\nu=\sum_a\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi_a)}\partial_\nu\psi_a-\delta^\mu_\nu\mathcal{L}$, conserved on shell, taken with the overall sign that makes $T^{00}>0$.

The scalar stress–energy is $T_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi+\eta_{\mu\nu}\mathcal{L}_\phi$, with

$$ T_{00}=T^{00}=\frac{1}{2c^2}\dot{\phi}^2+\frac12(\nabla\phi)^2+\frac12\mu^2\phi^2>0, \qquad \partial^\mu T_{\mu\nu}=0 \ \text{on shell}, $$

verified on a superposition of two on-shell modes. The electromagnetic tensor is the biquaternion bilinear

$$ T^\mu{}_\nu = \frac12\,\mathrm{Sc}\!\left(\tilde{F}\mathcal{E}_\mu\tilde{F}^{*}\mathcal{E}_\nu\right), \qquad T^{00}=W, \quad T^{0j}=\frac1cS_j, \quad T^{jk}=W\delta_{jk}-\epsilon E_jE_k-\mu H_jH_k , $$

symmetric on raising the index, traceless in the Minkowski pairing, and conserved source-free — all verified by direct computation and by a superposition of two plane waves. With matter included, $\partial_\mu T^{\mu\nu}_{\mathrm{em}}=-f^\nu$ and $\partial_\mu T^{\mu\nu}_{\mathrm{matter}}=+f^\nu$, so the total tensor is conserved; the $\nu=0$ component is the Poynting theorem. The same action that defines the fields and the causal cone of the algebra defines the bookkeeping of their energy and momentum.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra; $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$; central $i$
$\mathbb{M}_-$, $\mathbb{M}_+$	Material (anti-Hermitian) and informational (Hermitian) sectors
$\mathcal{E}_\mu\in\{ie_0,e_1,e_2,e_3\}$	Basis of $\mathbb{M}_-$; supplies the tensor index directions
$\tilde{\nabla},\tilde{\nabla}^{\natural}$, $\Box=\tilde{\nabla}\tilde{\nabla}^{\natural}=\partial_{ict}^2+\Delta$	Gradient, conjugate gradient, d'Alembertian
$N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=\sum_\mu Q_\mu^2$	Biquaternion norm; level-1 identity on $\mathbb{C}$
$\eta=\mathrm{diag}(-1,+1,+1,+1)$	Level-2 $ict$-coordinate metric
$S=\int\mathcal{L}\,d^4x$	Action; $\mathcal{L}$ the Lagrangian density
$\mathcal{L}_\phi=-\tfrac12\partial_\mu\tilde{\Phi}\partial^\mu\tilde{\Phi}^{\natural}-\tfrac12\mu^2\tilde{\Phi}\tilde{\Phi}^{\natural}$	Scalar Lagrangian
$\mu=mc/\hbar$	Scalar mass parameter
$(\Box-\mu^2)\tilde{\Phi}=0$	Biquaternionic Klein–Gordon equation
$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$; $\tilde{F}=i\sqrt{\epsilon}\mathbf{E}-\sqrt{\mu}\mathbf{H}$	Field strength
$S_{\mathrm{em}}=-\frac14\int F_{\mu\nu}F^{\mu\nu}d^4x$	Electromagnetic action
$\tilde{R}=\frac{i\rho}{\sqrt{\epsilon}}e_0+\sqrt{\mu}\mathbf{J}$; $\tilde{\nabla}\tilde{F}=-\tilde{R}$	Biquaternionic Maxwell system
$W=\tfrac12(\epsilon\mathbf{E}^2+\mu\mathbf{H}^2)$; $\mathbf{S}=\mathbf{E}\times\mathbf{H}$	Energy density; Poynting vector
$\sigma_{jk}=\epsilon E_jE_k+\mu H_jH_k-W\delta_{jk}$	Maxwell stress
$T^\mu{}_\nu=\tfrac12\mathrm{Sc}(\tilde{F}\mathcal{E}_\mu\tilde{F}^{*}\mathcal{E}_\nu)$	Electromagnetic stress–energy, biquaternion bilinear
$T^0{}_0=-W$, $T^0{}_j=\frac1cS_j$, $T^j{}_0=-\frac1cS_j$, $T^j{}_k=-\sigma_{jk}$	Mixed components
$T^{00}=W$, $T^{0j}=T^{j0}=\frac1cS_j$, $T^{jk}=-\sigma_{jk}$	Contravariant (symmetric) components
$\eta_{\mu\nu}T^{\mu\nu}=-T^{00}+T^{kk}=0$	Tracelessness (Minkowski pairing)
$\partial_\mu T^{\mu\nu}=-f^\nu$, $f^\nu=(\frac1c\mathbf{E}\cdot\mathbf{J},\ \rho\mathbf{E}+\mathbf{J}\times\mathbf{B})$	Conservation with source; Lorentz four-force
$T_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi+\eta_{\mu\nu}\mathcal{L}_\phi$	Scalar stress–energy
$T_{00}=T^{00}=\frac{1}{2c^2}\dot{\phi}^2+\frac12(\nabla\phi)^2+\frac12\mu^2\phi^2$	Scalar energy density, positive
$c=1/\sqrt{\epsilon\mu}$, $c_0$	Speed of light in the medium; in vacuum