Bosonization in Biquaternionic Form

Introduction

Bosonization is the statement that a theory of fermions in two spacetime dimensions is the same theory as a theory of bosons. The equivalence is exact and non-perturbative: the massless Dirac field is equivalent to a free scalar field, the massive Thirring model is equivalent to the sine-Gordon model, and the correspondence extends to the current algebra, the stress tensor, and the solitons of the bosonic side, which are the fermions of the fermionic side. The construction is standard (Coleman 1975; Mandelstam 1975; and, in the non-abelian case, Witten 1984), and its two-dimensionality is not an accident: it is tied to the fact that a two-component spinor and a scalar have the same number of degrees of freedom in two dimensions, and that the current algebra of the fermion closes on the scalar's derivatives.

This article asks what the biquaternion framework contributes to that statement. The findings are the following.

Bosonization is a correspondence between the framework's module and its center. This is the genuinely algebraic observation of the article. The fermionic side of the duality is carried by the spinor module $\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$: the Dirac field is module-valued, its bilinears are built from the module's inner product, and its current algebra is a module-valued current algebra. The bosonic side is carried by a central scalar: the boson field is an element of the center $\mathbb{C}_{\mathbb{B}}$ (or of the real scalars), its exponentials are central, and its stress tensor is a central scalar. Bosonization is thus, in the framework's terms, an isomorphism between the operator algebra of the module and the operator algebra generated by the exponential of a central field, and the equivalence's "one complex fermion equals one real boson" is a statement about the dimension of the module's complex structure against the central field's single real component. Stating the correspondence this way makes the framework's contribution precise and modest: it names which two objects are being identified.
The framework's linear, chirality-off-diagonal mass is what bosonization requires. The series convention for the Dirac mass is linear and chirality-off-diagonal, $\tilde\nabla\tilde\Psi_R=m\tilde\Psi_L$, $\tilde\nabla^{\natural}\tilde\Psi_L=m\tilde\Psi_R$, and it is a right multiplication, which is exactly the structure that couples the two minimal left ideals of $\mathbb{B}$. In two dimensions this is the standard Dirac mass, whose bosonized image is the cosine of the sine-Gordon model. The framework's mass term therefore maps to the bosonic interaction without a change of form, and the phase of the mass term — the framework's central phase $e^{i\alpha}$ multiplying the mass — maps to a shift of the boson field, which is the standard statement that the fermion mass's phase is the bosonic field's vacuum angle. The equivalence is thus exact at the level of the mass term, and the framework's mass convention is compatible with it rather than competing with it.
The anomaly is the reason the correspondence is not trivial. A naive operator correspondence between a fermion bilinear and a boson derivative fails by a c-number; the bosonization dictionary is exact only with that c-number included, and the c-number is the two-dimensional anomaly of The Trace Anomaly in Biquaternionic Form. The framework's contribution here is again a counting one: the central charges of the two sides must match, and the matching fixes the normalisation of the boson. The framework supplies the module's dimension on the fermionic side and the trace on the bosonic side.
The two-dimensional arena is a restriction, and the framework must be reduced to it. The algebra $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ is the even Clifford algebra $\mathbb{B}\cong\mathrm{Cl}^+_{1,3}$ of four-dimensional Minkowski spacetime; two-dimensional bosonization uses the two-dimensional Clifford algebra and its chirality operator, and the reduction from four to two dimensions is a restriction of the framework, not a property of it. The reduction is performed explicitly below, and it is the reason this article is a statement about a reduced framework. The lesson is the general one: bosonization is a feature of two dimensions, and the biquaternion approach carries it because it carries the spinor module, not because the algebra is four-dimensional.
The non-abelian and the Wess–Zumino term are inherited. Non-abelian bosonization identifies fermions in a representation of a group with a sigma model carrying a Wess–Zumino term, and the level of the term is fixed by the fermion's representation; the framework's non-commutative gauge-field material is the appropriate place for the group theory, and this article cites the result rather than deriving it.

The article proceeds as follows. The next section reduces the framework to the two-dimensional arena and fixes the two-dimensional Clifford data. A section states the abelian bosonization dictionary in the framework's notation. A section identifies the module–center correspondence and states the counting it implies. A section shows how the framework's mass term and its central phase bosonize, and a section treats the anomaly and the central-charge matching. A section states the non-abelian result and its status. A section separates what is established from what is interpretation, and the article closes with open questions.

Conventions. We use those of the companion articles unless a reduction is stated. The algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, and central scalar imaginary $i$, $i^2=-1$. The sectors are $\mathbb{M}_-$ (anti-Hermitian, material) and $\mathbb{M}_+$ (Hermitian, informational), with $\mathbb{M}_-=i\mathbb{M}_+$; the center is $\mathbb{C}_{\mathbb{B}}=\mathrm{span}_\mathbb{R}\{e_0,ie_0\}$. The trace pairing is $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$, $\mathrm{Tr}(e_0)=2$. The series d'Alembertian is $\Box=\tilde\nabla\tilde\nabla^{\natural}=\partial^2_{ict}+\Delta$. In the two-dimensional reduction the Clifford generators are $\gamma^0,\gamma^1$ with $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}I_2$ and the level-3 tool metric $g=\mathrm{diag}(+1,-1)$, so that $(\gamma^0)^2=+I_2$ and $(\gamma^1)^2=-I_2$; the chirality operator is $\gamma^5=\gamma^0\gamma^1$ and $(\gamma^5)^2=+I_2$; the antisymmetric symbol is $\epsilon^{01}=+1$. These are the conventions of Conventions in the Biquaternion Universe, The Feynman Propagator in Biquaternionic Form, The Trace Anomaly in Biquaternionic Form, and The Theta Vacuum in Biquaternionic Form.

The Two-Dimensional Arena

Bosonization lives in two dimensions, and the framework must be reduced before it can be stated.

The reduction. The framework's algebra is the even subalgebra of the four-dimensional Clifford algebra; in two dimensions the analogous object is the even subalgebra of $\mathrm{Cl}_{1,1}$. The two-dimensional even Clifford algebra is $\mathrm{span}\{I,\gamma^5\}\cong\mathbb{R}\oplus\mathbb{R}$, generated by the chirality operator, and it is two-real-dimensional; the reduction of the framework's "even" structure is therefore a real pair rather than a complex pair, and the complexification that produces a spinor module in four dimensions degenerates to it. The framework's four-dimensional structure is retained as the two-dimensional Dirac spinor's two complex components, and the spinor module remains two-complex-dimensional, which is what gives the counting below.

The two-dimensional Clifford data. With $g=\mathrm{diag}(+1,-1)$, $$ \gamma^0 = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}, \qquad \gamma^1 = \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}, \qquad \{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}I_2 , $$ the chirality operator is $$ \gamma^5 = \gamma^0\gamma^1 = \begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}, \qquad (\gamma^5)^2 = +I_2 , $$ and the chiral projectors $\tilde\Pi_{L,R}=\tfrac12(1\pm\gamma^5)$ project onto the two components. Two facts are worth recording because they differ from the four-dimensional case: $(\gamma^5)^2=+I_2$ here, whereas in four dimensions $(\gamma^5)^2=-I_4$, so the two-dimensional chirality operator is a real structure rather than a complex one; and $\mathrm{tr}\,\gamma^\mu=0$ while $\mathrm{tr}\,\tilde\Pi_{L,R}=1$. Both were verified explicitly and are quoted in the established list below. The distinction matters for the framework: a two-dimensional chirality operator that squares to $+1$ has real eigenspaces, so the two chiralities of a two-dimensional Dirac field are a real splitting, and the irreducible two-component spinor is the sum of two real (Majorana–Weyl) spinors. This is the fermionic side's true content, and the bosonic side must match it.

The boson. The boson is a single real scalar field $\phi(x)$, whose exponentiated field $e^{2i\sqrt{\pi}\phi}$ is, in the framework's terms, a central object: the exponent carries the central $i$, and a central scalar commutes with every element of $\mathbb{B}$. The boson's stress tensor is a central scalar, and its free action $$ S_{\mathrm{b}} = \frac{1}{2}\int d^2x\;\partial_\mu\phi\,\partial^\mu\phi $$ is the two-dimensional restriction of the framework's scalar action with the series $\Box$ and the $ict$ metric. The bosonic side is therefore the central sector's dynamics, and this is the identification the next sections use.

The Abelian Bosonization Dictionary

The dictionary is stated first in its standard form, then in the framework's notation.

The standard dictionary. For a massless Dirac field $\psi$ and a real scalar $\phi$, with a normalisation constant $\alpha$ setting the short-distance cutoff, $$ \bar\psi\gamma^\mu\psi \;\longleftrightarrow\; -\frac{1}{\sqrt{\pi}}\,\epsilon^{\mu\nu}\partial_\nu\phi , \qquad \bar\psi\psi \;\longleftrightarrow\; -\frac{1}{\pi\alpha}\cos\big(2\sqrt{\pi}\,\phi\big), $$ and the fermion number current's divergence vanishes while the axial current's does not: $$ \partial_\mu j^\mu = 0 , \qquad \partial_\mu j^\mu_5 = -\frac{1}{\pi}\,\Box\phi . $$ The first relation is the current correspondence; the second is the mass correspondence, which is the sine-Gordon interaction; and the third is the anomaly, which is the statement that the axial current is not conserved. These are standard (Coleman 1975; Mandelstam 1975; von Delft and Schoeller 1998), and the framework does not alter them.

The dictionary in the framework's notation. The two bilinears are module objects traced into the center: $$ \bar\psi\gamma^\mu\psi = \mathrm{Sc}\big(\tilde\Psi^{\natural}\gamma^\mu\tilde\Psi\big) , \qquad \bar\psi\psi = \mathrm{Sc}\big(\tilde\Psi^{\natural}\tilde\Psi\big), $$ where the trace pairing extracts the central scalar. The correspondence is therefore a map $$ \text{module bilinear}\;\longrightarrow\;\text{central field} , $$ and the bosonization dictionary is the explicit realisation of that map. Read in the framework's terms, the map's domain is the tensor square of the spinor module (which decomposes into the current and the mass bilinear) and its target is the center's derivative and exponential. The framework's contribution to the dictionary is thus not a new identity but the identification of the two sides' value spaces; the identities themselves are the standard ones and are cited.

The cutoff and the normal ordering. The constant $\alpha$ is a short-distance cutoff, and the correspondence holds after normal ordering of both sides. This is worth recording because it is the first appearance of the regularisation that the anomaly section will need: the dictionary is exact only for normal-ordered operators, and the normal-ordering constant is the same anomaly coefficient that appears in the central-charge matching. In the framework the normal ordering is not algebraic — it is a subtraction of the short-distance behaviour of a two-point function — and it is transcribed as such.

The Module–Center Correspondence

This section states the article's genuinely algebraic observation and its counting consequence.

The two objects. On the fermionic side stands the spinor module, $$ \mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu}) \cong \mathbb{C}^2 , $$ two complex dimensions, carrying the Dirac field, its Clifford action, and the current algebra built from its bilinears. On the bosonic side stands the center, $$ \mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\{e_0,ie_0\} \cong \mathbb{C} , $$ one complex dimension, and the real boson is its real part $\mathbb{R}e_0$ — the fixed space of coefficient conjugation — a single real dimension carrying the exponentiated field. Bosonization identifies an operator algebra built on the first with an operator algebra built on the second.

The counting. The equivalence's counting is the standard match between a complex fermion and a real boson. A complex (Dirac) fermion in two dimensions has four real components, which assemble into two Majorana fields of central charge $c=\tfrac12$ each; a real boson has $c=1$. The two sides therefore agree, and the agreement is the arithmetic $$ 2\times\tfrac12 = 1 , $$ the factor $2$ being the number of Majorana fields in the Dirac field. In the framework's terms the fermionic side is the spinor module $\mathbb{C}^2$, four real components, and the bosonic side is the center's single real scalar; the module's two complex dimensions carry the two chiral eigenspaces of $\gamma^5$, each of one complex dimension, and the boson carries the one real dimension of the central field. The counting is a component count of the kind the partition-function and harmonic-oscillator articles perform, and it is the statement that the two sides' operator algebras agree, not that the two spaces are isomorphic.

What the correspondence is not. It is not an identity of the algebras: $\mathbb{C}^2$ is not isomorphic to $\mathbb{C}$ as a module over $\mathbb{B}$, and the identification is between operator algebras built on them, not between the spaces. It is not a statement that the biquaternion algebra "contains" a boson; the boson is the center's field and the fermion is the module's field, and both are placed in the algebra rather than derived from it. And it is not a four-dimensional statement: the equivalence fails in four dimensions because the counting fails there, and the framework's four-dimensional algebra is where the counting fails.

Why the module and the center. The reason the correspondence takes this form is the one stated in the arena section: the four-dimensional framework's spinor module and its center are the two objects whose two-dimensional reductions have the same size in real components, and bosonization is the statement that the two reduced theories are the same. The framework's contribution is to make the two objects named and their dimensions explicit; it does not derive the equivalence.

The Mass Term and the Central Phase

The framework's mass convention is used here rather than merely cited, because it is where the framework's structure meets the dictionary.

The series mass term. As Conventions in the Biquaternion Universe fixes, the Dirac mass is linear and chirality-off-diagonal, $$ \tilde\nabla\tilde\Psi_R = m\tilde\Psi_L , \qquad \tilde\nabla^{\natural}\tilde\Psi_L = m\tilde\Psi_R , $$ and it is a right multiplication, which is why it can relate the two minimal left ideals while a central scalar cannot. In two dimensions, with the chiral projectors $\tilde\Pi_{L,R}$, the mass term is $$ \bar\psi\psi = \bar\psi_R\psi_L+\bar\psi_L\psi_R , $$ which is the bilinear the dictionary sends to the cosine.

The bosonization of the mass. Under the dictionary, $$ m\,\bar\psi\psi \;\longleftrightarrow\; -\frac{m}{\pi\alpha}\cos\big(2\sqrt{\pi}\,\phi\big) = -\frac{2\mu}{\beta^2}\cos(\beta\phi) , \qquad \mu = \frac{2m}{\alpha}, $$ the second form exhibiting the sine-Gordon parameters $\beta=2\sqrt\pi$ and $\mu\propto m/\alpha$, and the fermion's mass $m$ becoming the coupling of the bosonic cosine. The equivalence is therefore a statement about the mass: the fermion's mass is the boson's interaction strength, and the massless fermion is the free boson. This is the standard content of the Coleman–Mandelstam equivalence and is cited.

The phase of the mass. If the mass carries a phase, $m\to|m|e^{i\alpha}$, then the bosonized cosine shifts: $$ \bar\psi\psi \;\longleftrightarrow\; \cos\big(2\sqrt{\pi}\,\phi+\alpha\big), $$ i.e. the fermion mass's phase is a shift of the boson field. In the framework the mass's phase is a central phase $e^{i\alpha}e_0$, the same object as the $\theta$ vacuum's weighting of The Theta Vacuum in Biquaternionic Form, and the shift it induces is the two-dimensional instance of the statement that a $\theta$ angle is a bosonic vacuum angle. The framework's statement is therefore exact and structural: the central phase of the mass is the central phase of the vacuum, and bosonization realises it as a field shift.

The $\gamma^5$ structure. The two-dimensional $\gamma^5$ squares to $+I_2$, so the chiral projectors are real and the two chiralities are Majorana–Weyl spinors. This is why the bosonized mass is a cosine rather than a phase: the mass term pairs the two real chiralities, and its bosonized image is a real, doubly-periodic function of the field. In four dimensions the corresponding object squares to $-I_4$ and the chiralities are complex; the change of the sign of $(\gamma^5)^2$ between two and four dimensions is thus the algebraic reason the bosonized mass is a cosine in two dimensions. This is a genuine, checkable statement, and it was verified.

The Anomaly and the Central-Charge Matching

The equivalence is exact only with the anomaly included, and the framework's counting enters through the central charges.

The anomaly's role. The dictionary's current relation and mass relation cannot both hold with the free-field stress tensor: the fermionic stress tensor and the bosonic one differ by a c-number, and the difference is the anomaly. Equivalently, the two-point function of the fermion current has a contact term that the boson's two-derivative two-point function reproduces only with the correct normalisation, and that normalisation is the central charge. In two dimensions the anomalous trace is $$ T^\mu{}_\mu = \frac{c}{24\pi}R $$ from The Trace Anomaly in Biquaternionic Form, and the matching of $c$ on the two sides is the consistency condition the dictionary must satisfy.

The matching. The matching is the arithmetic of the previous section: $$ c_{\text{Dirac}} = 2\,c_{\text{Majorana}} = 2\times\tfrac12 = 1 = c_{\text{boson}} . $$ The framework supplies the factor $2$ as the two Majorana fields into which the module's four real components assemble, and the boson's $1$ as the real part of the center. The matching is therefore a component count, and it is the same kind of count that appears in the partition-function and harmonic-oscillator articles; once again, it is a count and not a claim of a multiplicity of independent fields.

What the anomaly forbids. The anomaly forbids a bosonization that preserves both the vector and the axial current, and it forbids a strictly local and free dictionary for a massive fermion. The bosonized mass is a cosine rather than a free term for this reason, and the sine-Gordon model's solitons — the fermions — are its consequence. The framework adds nothing to these statements; it supplies the value spaces and the counting, as before.

The Schwinger model as the standard illustration. A fermion coupled to a two-dimensional abelian gauge field with a massless fermion has a massive bosonic spectrum, and the anomaly is what supplies the mass; the framework's reading is that the gauge field's longitudinal mode is eaten by the would-be Goldstone mode of the boson, with the anomaly providing the mass. The statement is standard (Schwinger 1962; Coleman, Jackiw, and Susskind 1975) and is recorded only as the illustration of the anomaly's role.

Vertex Operators and the Current Algebra

The fermionic side of the dictionary is built from vertex operators, and their dimensions are the sharpest arithmetic of the equivalence.

The current algebra. In the bosonized language the fermion current is $\partial_\mu\phi/\sqrt{\pi}$ (the dictionary above), and its equal-time commutator in $1+1$ dimensions is $$ \big[j^0(x),j^1(y)\big] = \frac{i}{\pi}\,\delta'(x-y), $$ the Schwinger term. The term is the two-dimensional anomaly's coordinate-space form, and its coefficient is the same $1/\pi$ that appears in $\partial_\mu j^\mu_5=-\tfrac1\pi\Box\phi$. In the framework the normalisation of the current algebra is a statement about the $U(1)$ current built from the module's bilinear; the Schwinger term lives in the coefficient and not in the algebra, exactly as before.

Vertex operators. The bosonized fermion is an exponential of the boson field, the vertex operator $$ \psi_{L,R}(x)\;\sim\;:\!e^{\,i\beta\phi_{L,R}(x)}\!:, \qquad \Delta(\beta) = \frac{\beta^2}{4\pi}, $$ with $\phi_{L,R}$ the chiral halves of $\phi$ and $\Delta$ the scaling dimension. Two values of the coupling are fixed, and the arithmetic was verified: $\beta=\sqrt{2\pi}$ gives $\Delta=\tfrac12$, the dimension of a free fermion field, and $\beta=2\sqrt\pi$ gives $\Delta=1$, the dimension of the mass operator. The fermion vertex carries the first, $\beta_{\mathrm f}=\sqrt{2\pi}$, and the sine-Gordon cosine of the mass term carries twice it, $2\beta_{\mathrm f}=2\sqrt\pi$, because the mass is a bilinear of two vertices. So the same vertex at two couplings is the fermion of the theory and the perturbation that gives it a mass, and the two couplings differ by the factor $\sqrt2$. This is the clearest form of the equivalence: the fermion is the vertex with the fermion's dimension, and the mass is the vertex with the mass's dimension.

The framework's reading of a vertex. The operator $:\!e^{i\beta\phi}\!:$ is built from the exponential of a central field: $\phi$ is central, so the vertex operator is the exponential of a central element, and it is the only kind of local operator the bosonic side has. The fermion on the other side is a module-valued, anticommuting object. The equivalence therefore identifies a central exponential with an anticommuting, module-valued field, and the identification is not an isomorphism of the algebras as graded objects: the anticommutation is the spin–statistics statement, carried on the bosonic side by the cocycle of the vertex algebra, and the cocycle's coefficient is the same anomaly as before. Read this way, the bosonization dictionary is a map between algebras with different gradings, made an isomorphism of operator algebras by the vertex algebra's central extension. This is the algebraic content of the equivalence and it is worth stating plainly, because it shows the dictionary's non-triviality: an associative central exponential and an anticommuting module field are not the same object, and only the extension makes them equivalent.

Sugawara. The bosonic stress tensor is the Sugawara form of the current, $$ T = \tfrac12\,:\!\partial\phi\,\partial\phi\!:, $$ whose central charge is $c=1$ in the standard normalisation, matching the fermionic $c=1$ of the central-charge section. The framework adds nothing beyond naming the current's $U(1)$ as the abelian part of the module's symmetry; the non-abelian Sugawara construction belongs to the group theory of the next section.

The Non-Abelian Case and the Wess–Zumino Term

The non-abelian extension is stated and its status fixed.

The result. A theory of fermions transforming in a representation of a group is equivalent to a sigma model on the group manifold with a Wess–Zumino term whose coefficient — the level $k$ — is fixed by the representation, $k=1$ for the fundamental representation of $SU(N)$ at the appropriate normalisation. The correspondence maps the fermion's symmetry currents to the sigma model's currents and the fermion's stress tensor to the Sugawara form of the current algebra. This is the standard non-abelian bosonization (Witten 1984), and it is cited.

The framework's status regarding it. The group theory of the non-abelian case belongs to the framework's gauge-field material, not to this subcategory, and the level's quantisation belongs to the topology that the $\theta$-vacuum article discusses. What the framework contributes here is the same as in the abelian case: the fermionic side is module-valued and the bosonic side's group-valued field is exponentiated from elements whose logarithms lie in the appropriate sector. The framework does not derive the Wess–Zumino term; it places it.

The level and the anomaly. The level's integrality and the anomaly's coefficient are the same kind of statement — a quantised coefficient read from a topological density — and the $\theta$-vacuum and instanton articles supply the topological objects. This enters the article only as a pointer, because the non-abelian bosonization is a gauge-field statement and is out of this subcategory's scope.

What Is Established and What Is Interpretation

Established (framework and algebra).

Bosonization is a correspondence between the operator algebra of the spinor module $\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$ and the operator algebra generated by the exponential of a central scalar field; the module's four real components assemble into two Majorana fields and the center's real part supplies the single real boson, the two sides matching in the counting $2\times\tfrac12=1$.
In the two-dimensional reduction with $g=\mathrm{diag}(+1,-1)$, $(\gamma^5)^2=+I_2$ (verified: $\gamma^5=\mathrm{diag}(-1,+1)$, square $I_2$), so the chiral projectors are real and the two-dimensional chirality is a real structure; this is the algebraic reason the bosonized mass is a cosine. In four dimensions $(\gamma^5)^2=-I_4$, and the counting that makes bosonization possible fails.
The chiral projectors $\tilde\Pi_{L,R}=\tfrac12(1\pm\gamma^5)$ satisfy $\tilde\Pi_{L,R}^2=\tilde\Pi_{L,R}$, $\tilde\Pi_R\tilde\Pi_L=0$, $\mathrm{tr}\,\tilde\Pi_{L,R}=1$, and $\sum_\pm \tilde\Pi_{L,R}=I_2$; verified.
The framework's mass term is the right-multiplication, chirality-off-diagonal term, and it bosonizes to the cosine; its central phase bosonizes to a shift of the boson field, $\cos(2\sqrt\pi\phi)\to\cos(2\sqrt\pi\phi+\alpha)$.
The vertex-operator arithmetic is exact: with $\Delta(\beta)=\beta^2/(4\pi)$, $\Delta(\sqrt{2\pi})=\tfrac12$ is the free-fermion dimension and $\Delta(2\sqrt\pi)=1$ is the mass dimension; the two couplings differ by $\sqrt2$. The bosonized fermion is therefore a central exponential while the fermion itself is module-valued and anticommuting, so the dictionary maps algebras with different gradings.
The central charges match, $c_{\text{Dirac}}=2\times\tfrac12=1=c_{\text{boson}}$, as a component count.

Standard, and transcribed.

The abelian dictionary: $\bar\psi\gamma^\mu\psi\leftrightarrow-\tfrac{1}{\sqrt\pi}\epsilon^{\mu\nu}\partial_\nu\phi$, $\bar\psi\psi\leftrightarrow-\tfrac{1}{\pi\alpha}\cos(2\sqrt\pi\phi)$; the axial anomaly $\partial_\mu j^\mu_5=-\tfrac1\pi\Box\phi$; the sine-Gordon parameters $\beta=2\sqrt\pi$.
The Schwinger term $[j^0(x),j^1(y)]=\tfrac{i}{\pi}\delta'(x-y)$; the vertex operators $\psi\sim\,:\!e^{i\beta\phi}\!:$ with $\Delta=\beta^2/4\pi$; the Sugawara stress tensor and its $c=1$.
The Coleman–Mandelstam equivalence of the massive Thirring model and sine-Gordon; the soliton–fermion correspondence.
The central charge $c=1$ for a complex fermion and for a real boson, and $c=\tfrac12$ for a Majorana–Weyl fermion.
The Schwinger model's bosonic spectrum; the non-abelian bosonization and the Wess–Zumino level.

Interpretation.

Reading bosonization as the module–center correspondence is the framework's presentation of the standard equivalence; the equivalence is not derived here, and the dictionary's identities are the standard ones.
Reading the mass phase as a central phase of the same kind as the $\theta$ vacuum's weighting is the framework's identification of two appearances of the center; the physical statement (a mass phase shifts the boson) is standard.

Open.

Whether the framework has a natural bosonization in the presence of its full four-dimensional structure — for instance whether a two-dimensional reduction of a biquaternion gauge theory bosonizes with a framework-specific level — is not addressed.
The precise operator-algebraic isomorphism between the module's current algebra and the center's exponentiated algebra is stated but not proved; the article gives the correspondence and the counting, and cites the standard construction for the identity itself.
Whether the two-dimensional chirality's reality ($(\gamma^5)^2=+I_2$) has a counterpart in the framework's sector structure, as opposed to being a dimensional accident, is not resolved here.

Summary

Bosonization in biquaternionic form is the standard two-dimensional equivalence, with the framework's naming of the two sides. In the two-dimensional reduction the Clifford data are $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}I_2$ with $g=\mathrm{diag}(+1,-1)$, $\gamma^5=\gamma^0\gamma^1=\mathrm{diag}(-1,+1)$, and $(\gamma^5)^2=+I_2$, so the chiralities are real; the dictionary is $$ \bar\psi\gamma^\mu\psi\longleftrightarrow-\frac{1}{\sqrt\pi}\epsilon^{\mu\nu}\partial_\nu\phi , \qquad \bar\psi\psi\longleftrightarrow-\frac{1}{\pi\alpha}\cos\big(2\sqrt\pi\,\phi\big), \qquad \partial_\mu j^\mu_5=-\frac{1}{\pi}\Box\phi , $$ and the framework's contribution is to name the two sides: the fermion is the spinor module $\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$ and the boson is a central scalar, so that the equivalence is a correspondence between a module-valued operator algebra and the center's exponentiated algebra. The counting $c_{\text{Dirac}}=2\times\tfrac12=1=c_{\text{boson}}$ matches the central charges, the framework's linear chirality-off-diagonal mass bosonizes to the cosine, and the mass's central phase bosonizes to a shift of the boson field, the same central phase that weights the $\theta$ vacuum. The dictionary, the sine-Gordon equivalence, the Schwinger model, and the non-abelian case with its Wess–Zumino term are standard and transcribed.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra
$\mathbb{C}_{\mathbb{B}}=\mathrm{span}_\mathbb{R}\{e_0,ie_0\}$	Center; home of the boson field
$\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$	Spinor module; home of the Dirac field
$\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$	Trace pairing; extracts central scalars from bilinears
$g=\mathrm{diag}(+1,-1)$	Two-dimensional level-3 Clifford metric
$\gamma^5=\gamma^0\gamma^1$	Chirality operator; $(\gamma^5)^2=+I_2$ in $d=2$
$\tilde\Pi_{L,R}=\tfrac12(1\pm\gamma^5)$	Chiral projectors; real in $d=2$
$\phi$	Real boson field
$\alpha$	Short-distance cutoff in the dictionary
$\epsilon^{\mu\nu}$	Antisymmetric symbol, $\epsilon^{01}=+1$
$\beta=2\sqrt\pi$	Sine-Gordon coupling
$:\!e^{i\beta\phi}\!:$	Vertex operator; scaling dimension $\Delta=\beta^2/4\pi$
$c$	Central charge; $c=1$ for a Dirac fermion or a real boson