The Dirac Path Integral in Biquaternionic Form

Introduction

The path integral is the third formulation of quantum mechanics, alongside the operator formulation and the wave-equation formulation. For a relativistic spin-$\tfrac12$ particle it presents a specific difficulty that the scalar case does not: the Dirac operator is first order, and a path integral whose phase is a sum over a Lagrangian is naturally a device for a second-order operator. The history of the subject is the history of two complementary resolutions of that difficulty — Feynman's original "checkerboard", in which the first-order structure is built into a discrete lattice of directed paths, and the proper-time/worldline formulation, in which the first-order operator is applied to a scalar path integral or is generated by an additional Grassmann (spin) degree of freedom on the worldline.

This article develops the biquaternionic reading of both. The square of the biquaternionic Dirac operator is the Klein–Gordon operator (the companion article Klein–Gordon from the Dirac Square in Biquaternionic Form derives this), so the Dirac propagator is obtained from the scalar path integral by applying the first-order operator; and the reason the spin is carried by a path integral over an odd variable is that the odd variable is precisely what generates the Clifford algebra whose even part is the biquaternion algebra $\mathbb{B}$. The two statements are the algebraic content of the round trip: the algebra is the even part of the Clifford algebra generated by the spin, and the square of the Dirac operator is the scalar operator whose path integral is the scalar one.

Three boundaries are respected. The Klein–Gordon path integral and the scalar propagator belong to the spin-$0$ subcategory and are cited, not re-derived. The spin-half path integral and the fermionic field path integral are different objects: the former is the non-relativistic spinning path integral and the latter is the quantised-field (Grassmann-functional) integral belonging to Biquaternion Quantum Fields; neither is the single-particle Dirac path integral treated here. The general apparatus of the d'Alembertian and its Green's functions belongs to the generalities.

The conventions are the series conventions. The Clifford generators satisfy $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}I_4$ with $g = \mathrm{diag}(+1,-1,-1,-1)$, the Dirac equation is $(i\not\partial - mc/\hbar)\psi = 0$ with $\not\partial = \gamma^\mu\partial_\mu$, and the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ is the even subalgebra of $\mathrm{Cl}_{1,3}$. The worldline parameter is denoted $\tau$ and the proper time $T$; $x(\tau)$ is the material four-position $\tilde{Q}\in\mathbb{M}_-$.

The Dirac Propagator and the Square

The object the path integral must compute is the Feynman propagator for the spin-$\tfrac12$ field,

$$ S_F(x-x') = \int\frac{d^4p}{(2\pi)^4}\, \frac{i\left(\not p + mc/\hbar\right)}{p^2 - m^2c^2/\hbar^2 + i\epsilon}\, e^{-ip\cdot(x-x')}, $$

the matrix-valued Green's function satisfying

$$ \left(i\not\partial - \frac{mc}{\hbar}\right)S_F(x-x') = i\,\delta^{(4)}(x-x'). $$

This is standard relativistic quantum mechanics (see the Further Reading); the companion article The Feynman Propagator in Biquaternionic Form treats the same object in the biquaternion framework. Two structural facts are needed here.

The propagator is the scalar kernel acted on by the first-order operator. Write

$$ \Delta_F(x-x') = \int\frac{d^4p}{(2\pi)^4}\, \frac{1}{p^2 - m^2c^2/\hbar^2 + i\epsilon}\, e^{-ip\cdot(x-x')}, $$

for the scalar (Klein–Gordon) Feynman propagator, the spin-$0$ object of the companion subcategory. Then, formally,

$$ S_F = \left(i\not\partial + \frac{mc}{\hbar}\right)\Delta_F \quad\text{up to the overall factor } i, $$

and the verification is the same one-line Dirac square used in the companion article: in momentum space,

$$ \left(\not p - mc/\hbar\right)\left(\not p + mc/\hbar\right) = \not p^2 - m^2c^2/\hbar^2 = p^2 - m^2c^2/\hbar^2, $$

which is the denominator of $\Delta_F$. The Dirac propagator is therefore the Klein–Gordon propagator with the first-order operator applied; if the scalar kernel is available as a path integral, the Dirac propagator is available from it by a differential operation. This is one of the two standard resolutions of the first-order problem, and it is the one the biquaternion algebra states most directly: the inverse of the first-order operator is the conjugate first-order operator times the inverse of its square, and its square is central.

The propagator does not contain the dynamics of a path integral in a single phase. The worldline proper-time representation makes the same point in a Lorentz-invariant way. Since

$$ \frac{i}{p^2 - m^2c^2/\hbar^2 + i\epsilon} = \int_0^\infty dT\,e^{\,iT\left(p^2 - m^2c^2/\hbar^2 + i\epsilon\right)}, $$

the scalar propagator is the integral over a proper time $T$ of a kernel $\langle x'|e^{\,iT(\Box - m^2c^2/\hbar^2)}|x\rangle$, and that kernel is a sum over worldlines,

$$ G_F(x,x') = \int_0^\infty dT\int_{x(0)=x}^{x(T)=x'}\mathcal{D}x(\tau)\; e^{\,iS_0[x]}, \qquad S_0[x] = \int_0^T d\tau\left(\frac{1}{4}\dot{x}^\mu\dot{x}_\mu - m^2c^2/\hbar^2\right), $$

up to the standard normalisation of the measure and the einbein gauge choice. The tangent is contracted with the level-2 form $\eta = \mathrm{diag}(-1,+1,+1,+1)$ — the $ict$-coordinate metric of the conventions article, read here on the real-time components $(x^0,\mathbf{x})$ — so that for a straight worldline $\dot{x}^\mu\dot{x}_\mu = -(x-x')^2/T^2$ and the phase reduces to $\exp\left(-i\left[\frac{(x-x')^2}{4T}+\mu^2T\right]\right)$, with $(x-x')^2 = (x^0-x'^0)^2-|\mathbf{x}-\mathbf{x}'|^2$; this is the sign that the explicit free kernel of the next section displays. The Dirac propagator is then the operator $(i\not\partial + mc/\hbar)$ applied to $G_F$, with the appropriate $i$. The worldline is a curve in the material sector $\mathbb{M}_-$: its tangent $\dot{\tilde{Q}}$ is a material four-velocity, and the phase $S_0$ is central. This is the biquaternion content of the scalar worldline: a central phase over a material curve.

The deficiency of this representation is that the spin is nowhere to be seen until the differential operator is applied at the end. The second resolution supplies the spin directly.

The Free Kernel, Explicitly

It is worth evaluating the scalar worldline kernel in the free case, because the calculation is the whole content of the proper-time method and fixes the sense in which a first-order object is a square root. The proper-time kernel is the Fourier transform

$$ G_F(x,x') = \int_0^\infty dT\int\frac{d^4p}{(2\pi)^4}\, e^{\,iT\left(p^2 - \mu^2 + i\epsilon\right)}\,e^{-ip\cdot(x-x')}, \qquad \mu = \frac{mc}{\hbar}, $$

where the $i\epsilon$ prescription makes the $T$-integral converge at large $T$. The $p$-integral is a Gaussian: completing the square in each of the four components gives the standard free kernel

$$ G_F(x,x') = \int_0^\infty dT\;\mathcal{N}(T)\; \exp\left(-i\left[\frac{(x-x')^2}{4T} + \mu^2 T\right]\right), \qquad (x-x')^2 = (x^0-x'^0)^2 - |\mathbf{x}-\mathbf{x}'|^2, $$

with $\mathcal{N}(T)$ the standard Gaussian normalisation $\propto T^{-2}$ whose precise constant depends on the measure convention and is not needed below. Three features of this expression carry the physics.

The proper time $T$ is the worldline length parameter, and the exponent is the classical action of a relativistic free particle, $\frac{(x-x')^2}{4T}$ being the kinetic term and $\mu^2T$ the mass term; the path integral over $x(\tau)$ is the statement that the exponent is extremised by a straight line, with the fluctuations around it producing the prefactor.
The mass is a constant in the exponent. There is no spin and no matrix structure: $G_F$ is a scalar, and it obeys $(\Box - \mu^2)G_F = i\delta^{(4)}$ up to the measure convention.
The first-order structure enters only through the differential prefactor. The Dirac propagator is obtained by applying $(i\not\partial + \mu)$ to $G_F$, which is why the spin is invisible until that operation is performed.

The same proper-time integral is the bridge to the worldline of the next section: replacing the constant mass by a general worldline interaction, and adding the odd variables, converts this scalar kernel into the matrix kernel of the Dirac propagator.

The Spinning Worldline and the Spin Factor

The worldline that generates the Dirac propagator itself, rather than the Klein–Gordon kernel it acts on, carries an additional odd (Grassmann) degree of freedom $\psi^\mu(\tau)$ at each point. The standard supersymmetric worldline action is

$$ S[x,\psi] = \int_0^T d\tau\left[ \frac{1}{4}\dot{x}^\mu\dot{x}_\mu + \frac{i}{2}\psi^\mu\dot{\psi}_\mu - m^2c^2/\hbar^2 \right], $$

with the flat metric, the mass term normalised to the proper-time parameter $T$ introduced above, so that the phase of the free kernel is the one displayed in the previous section; the term odd in the Grassmann variables is the kinetic term $\frac{i}{2}\psi^\mu\dot\psi_\mu$, and its presence is what makes the worldline supersymmetric. Quantization of the odd variables gives the Clifford algebra

$$ \{\hat\psi^\mu,\hat\psi^\nu\} = \frac{1}{2}g^{\mu\nu}\,I , $$

in the normalisation of the worldline literature — the one in which the spin factor below takes the quoted form — so that the Hilbert space of the odd sector is the spinor representation, of dimension $2^{[4/2]} = 4$ in four dimensions. The path integral over the odd variables, with the boundary conditions appropriate to the propagator (anti-periodic for the fermionic trace, periodic for the matrix element), is a spin factor $\Phi[x]$ multiplying the scalar measure:

$$ K(x,x') = \int_0^\infty dT\int\mathcal{D}x\,\Phi[x]\,e^{\,iS_0[x]}, \qquad \Phi[x] = \int_{\text{b.c.}}\mathcal{D}\psi\,e^{\,i\int_0^T \frac{i}{2}\psi^\mu\dot\psi_\mu\,d\tau}. $$

The spin factor is the path-ordered exponential of the odd kinetic term; expanded, it is a sum of products of pairs $\psi^\mu\psi^\nu$, i.e. of Clifford bivectors, which is why the path integral over $\psi$ produces a Clifford-algebra-valued weight. The complete worldline path integral, with the correct measure and boundary conditions, reproduces the Dirac propagator $S_F$; this is the Berezin–Marinov / Brink–Di Vecchia–Howe construction, and it is standard. The same construction in a background electromagnetic field replaces the free kinetic term by the minimally coupled one and produces the spin factor $\Phi[x]=\mathrm{Tr}\,\mathcal{P}\exp\left(-\frac{i}{2}\int \sigma_{\mu\nu}F^{\mu\nu}\,d\tau\right)$ quoted in the literature (with the sign conventions of the cited works).

The Biquaternion Reading of the Spin Factor

The spin factor is where the algebra sits, and its structure is exactly the algebra's defining relation to the Clifford algebra.

The odd variable generates $\mathrm{Cl}_{1,3}$, and the biquaternion algebra is its even part. The quantized $\hat\psi^\mu$ generate the Clifford algebra with the metric $g$. Products of an even number of $\hat\psi$'s close under multiplication and form the even subalgebra $\mathrm{Cl}_{1,3}^{+}$, which is isomorphic to $\mathbb{B}$: this is the dictionary of The Dirac Algebra and Biquaternions — A Dictionary, $\Phi(\mathbb{B}) = \mathrm{Cl}_{1,3}^{+}$. A single $\hat\psi^\mu$ is Clifford-odd — it exchanges the two chiral halves and is the generator of the reflection — and it is exactly the kind of object the biquaternion algebra does not contain. The worldline spin variable is therefore the odd object whose even products are the biquaternions.

The spin factor is the Clifford-algebra path-ordered product. The weight $\Phi[x]$ is, order by order, a sum of Clifford elements: the identity, then bivectors, then quadrilinears. Its even part lies in $\Phi(\mathbb{B})$, its odd part is the Clifford-odd remainder. In the free case the odd part renormalises the boundary spinor and the even part is what multiplies the biquaternion-valued field; in a background field the bivector term is the spin–field coupling $\sigma_{\mu\nu}F^{\mu\nu}$ of the standard spin factor. The biquaternion framework thus reads the spin factor as the algebraic object that converts a material curve into a spinor transport: the curve is $\mathbb{M}_-$-valued, and the transport is $\Phi(\mathbb{B})$-valued with an odd dressing.

The square of the operator is the scalar path integral. The worldline action has a bosonic kinetic term $S_0$ and a fermionic term; integrating out the odd variables returns a factor that is the square root's worth of spin content. The statement that the Dirac propagator is the Dirac operator applied to the scalar kernel, and the statement that the worldline spin variable generates the Clifford algebra, are two faces of the same relation: the odd variable is the spin, and the even products are the algebra; squaring the first-order operator removes the odd variable and returns the scalar operator whose path integral is $G_F$.

The Spin Factor in a Background Field

The spin factor earns its name in a background electromagnetic field, and the calculation shows how the biquaternion algebra's even/odd split organises the result. Minimal coupling adds the gauge term $qA_\mu(x)\dot{x}^\mu$ to the worldline action,

$$ S[x] = \int_0^T d\tau\left[\frac{1}{4}\dot{x}^\mu\dot{x}_\mu - \mu^2 + qA_\mu(x)\dot{x}^\mu\right], $$

and the odd kinetic term is unchanged. The path integral over the odd variables now produces the spin factor

$$ \Phi[x] = \mathrm{Tr}\,\mathcal{P}\exp\left(-\frac{i}{2}\int_0^T d\tau\;\sigma_{\mu\nu}F^{\mu\nu}(x(\tau))\right), \qquad \sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu,\gamma_\nu], $$

the path-ordered trace of the bivector coupling to the field strength. This is the standard Berezin–Marinov spin factor; the trace is over the spinor indices and the path ordering is required because the bivectors at different times do not commute. The same structure is what produces the anomalous magnetic moment in the worldline computation of the electron's form factors, and it is the worldline form of the spin–field coupling that the companion articles The Electron in Biquaternionic Form and The Minimal Coupling of the Biquaternion Dirac Field to Electromagnetism treat from the operator side.

In the biquaternion framework the bivector $\sigma_{\mu\nu}$ decomposes into the two pieces of the algebra. The combination $\frac{i}{2}[\gamma_\mu,\gamma_\nu]$ is a Clifford bivector; the bivectors split into the timelike ones, which lie in $\Phi(\mathbb{M}_+)$ and are the boost generators, and the spacelike ones, which lie in the image of the real-quaternion sector and are the rotation generators (the reflection article's decomposition). The spin factor therefore exponentiates a sum of a frame-even $\mathbb{M}_+$ piece and a frame-odd piece; the trace at the end keeps the scalar part of the path-ordered exponential alone, since the bivectors and the pseudoscalar $\gamma_5$ are both traceless in four dimensions, so that the worldline integral is built from the Lorentz invariants of the field strength: the scalar $F_{\mu\nu}F^{\mu\nu}$, which the second-order term of the expansion produces, and the pseudoscalar combination that carries the $\mathbf{E}\cdot\mathbf{B}$ invariant. The path-ordered exponential is the worldline image of the algebra's rotor construction: at each instant the spin is transported by a bivector, and the product of all the transports is the spin factor.

Feynman's Checkerboard

Before the continuum worldline there was a discrete model, and in 1+1 dimensions it exhibits the biquaternion structure with unusual clarity. Feynman's checkerboard (1949) represents the Dirac particle as moving at the speed of light along the two null directions of two-dimensional spacetime, with the amplitude for a path built from two rules: a directed step carries a phase, and each corner (reversal of direction) carries a factor $i\epsilon m/\hbar$ (in units where the step length is $\epsilon$ and $m$ is the mass; the standard convention of the checkerboard literature). The propagator is the sum over all such paths, and in the continuum limit the sum satisfies the two-dimensional Dirac equation.

The reason is the one on which this whole article turns. In 1+1 dimensions the Dirac spinor has two components, and the two components are exactly the amplitudes for the two directions of motion. Writing the propagator as a two-by-two matrix $K = \begin{pmatrix} K_{RR} & K_{RL} \\ K_{LR} & K_{LL}\end{pmatrix}$, where the indices record the initial and final directions, the corner weight builds the mass mixing: the off-diagonal entries $K_{RL},K_{LR}$ require an odd number of corners, so they are the entries proportional to $m$; the diagonal entries require an even number of corners. The continuum limit is

$$ K \;\longrightarrow\; \text{the two-component Dirac propagator in } 1+1 \text{ dimensions}, $$

a matrix of Bessel functions whose small-$\epsilon$ expansion satisfies $(i\not\partial - mc/\hbar)K = i\delta$ in two dimensions; the standard derivations of the checkerboard (see the Further Reading) give the explicit form. The two directions are the two chiral components, and the mass is the corner operator that flips them.

The biquaternion reading is then immediate. In 1+1 dimensions the biquaternion algebra reduces to the even part of $\mathrm{Cl}_{1,1}$, whose two minimal ideals are the two chiral halves; the two directions of the checkerboard are those two halves, and each corner is an insertion of the off-diagonal mass of the chiral pair $\tilde{\nabla}\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\tilde{\Psi}_R$. The checkerboard is thus the discrete, real-time picture of the mass pair: the free propagation is diagonal (each chirality propagates on its own), and the mass is the reversal that connects them. The continuum limit is the statement that the off-diagonal mass pair, iterated on the lattice, reproduces the flat-space Dirac propagator.

The Recurrence and Its Limit

The discrete structure can be written as a pair of coupled recurrences and taken to the continuum, and doing so exhibits the mass pair explicitly. Let $R(x,t)$ and $L(x,t)$ be the amplitudes to arrive at the lattice point $(x,t)$ having last moved in the $+x$ and $-x$ directions respectively, with the lattice spacing and time step both equal to $\epsilon$ (units $c = 1$). A step carries the particle one cell in its direction of motion, and a corner contributes the weight $i\epsilon m/\hbar$; the recurrences are therefore

$$ R(x,t+\epsilon) = R(x-\epsilon,t) + \frac{i\epsilon m}{\hbar}\,L(x-\epsilon,t), $$

$$ L(x,t+\epsilon) = L(x+\epsilon,t) + \frac{i\epsilon m}{\hbar}\,R(x+\epsilon,t), $$

where the first term on each right-hand side is the straight continuation and the second is the corner. Two features are visible by inspection. The free propagation is diagonal: the term that carries $R$ to $R$ and $L$ to $L$ involves no mixing, and each direction propagates independently. The mass is off-diagonal: the corner terms couple $R$ to $L$ and $L$ to $R$ with the weight $i\epsilon m/\hbar$, i.e. with an explicit factor of the mass and an explicit factor of $i$.

Taylor-expanding and taking $\epsilon \to 0$ with the lattice constraint $|\dot{x}| = 1$ turns the pair into the two-component Dirac equation in 1+1 dimensions, whose two components are $R$ and $L$. The corner weight $i\epsilon m/\hbar$ becomes the mass term, and the two directions become the two chiral components. This is the discrete origin of the off-diagonal mass pair that the biquaternion algebra writes as $\tilde{\nabla}\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\tilde{\Psi}_R$: the checkerboard is that pair before the continuum limit has identified the directions with the chirality eigenspaces. The order of the two factors in a corner is the path ordering that survives as the ordering of the operators $\tilde{\nabla}$ and $\tilde{\nabla}^{\natural}$ in the continuum.

Composition, the Propagator Equation, and the Scalar Reduction

The path integral must satisfy the two defining properties of a propagator, and both are transparent in the biquaternion framework.

Composition. The kernel obeys $\int d^4y\,K(x,y)K(y,x')=K(x,x')$, the statement that a sum over paths is a sum over intermediate points. In the worldline picture this is the semigroup property of the proper-time evolution, and it is why the kernel is built from an operator exponential. In the biquaternion algebra the composition is the multiplication of biquaternion-valued amplitudes in the module, with the central phase composing additively; the algebra's non-commutativity means that the order of the intermediate points matters, and the path-ordered spin factor is the price of that non-commutativity.

The propagator equation. The kernel satisfies $(i\not\partial - mc/\hbar)K = i\delta^{(4)}$. In the biquaternion framework this is the statement that the propagator is the Green's function of the first-order pair. Applying the conjugate operator and using the Dirac square $(i\not\partial + \mu)(i\not\partial - \mu) = \Box - \mu^2$ gives

$$ \left(\Box - \frac{m^2c^2}{\hbar^2}\right)K = \left(i\not\partial + \frac{mc}{\hbar}\right)i\delta^{(4)} , $$

so the scalar part of the Dirac propagator obeys the Klein–Gordon equation with a delta-function source. This is the path-integral form of the Dirac square: the first-order propagator, acted on by its conjugate, obeys the second-order equation. The path integral for the first-order object is therefore not an independent construction; composing it with the conjugate operator returns the scalar path integral of the companion subcategory, and the spin factor is what distinguishes them.

It is worth stating what the square does not do in this setting. It does not remove the spin: the squared propagator is matrix-valued and carries the Clifford structure in its free indices, even though its differential operator is central. The spin-0 theory is recovered only after the spin indices are contracted or traced. In the algebra this is the statement that the even part of the spin factor is what survives squaring, while the odd part has been absorbed into the boundary spinors.

The Measure and the Einbein

The worldline integrals above are written with a proper-time parameter $T$ that has already been gauge-fixed, and the fixing is worth stating because it is the reason the four-dimensional propagator carries a proper-time integral at all. A relativistic worldline is reparametrisation invariant: the action

$$ S[x,e] = \int_0^1 d\tau\left[\frac{\dot{x}^\mu\dot{x}_\mu}{4e(\tau)} - e(\tau)\,\mu^2\right] $$

depends on the worldline only through its image, the auxiliary field $e(\tau)$ (the einbein) transforming as a density; the parameter $\tau$ has no physical meaning. In the path integral one does not fix $e$ but integrates over it, and the integral over the constant modes of $e$ is precisely the proper-time integral: writing $e(\tau) = T$ for the constant mode (the other modes can be gauged to zero and divide out the volume of the reparametrisation group), the action becomes

$$ S = \frac{1}{4T}\int_0^1 \dot{x}^\mu\dot{x}_\mu\,d\tau - \mu^2 T , $$

which is the proper-time exponent of the previous sections, with $T$ identified as the total proper time. The fluctuations of the material trajectory $x(\tau)$ about the straight line — equivalently, in the Fourier form used above, the four-dimensional Gaussian integral over $p$ — produce the $T^{-2}$ prefactor $\mathcal{N}(T)$ of the free kernel.

The algebraic reading is a statement about the two sectors. The einbein is a central scalar (it commutes with the odd variables and carries no spinor index), so its integral only reorganises the material part of the worldline; it is the same for every spin and is the reason the spin-0 and spin-$\tfrac12$ theories share the proper-time structure. The spin enters entirely through the odd variables and the spin factor, not through the measure. In the biquaternion framework the einbein is a central element of the worldline's phase, and the spin factor is the $\Phi(\mathbb{B})$-valued remainder; the split of the worldline integral into a central measure and an algebra-valued weight is the path-integral image of the algebra's centre-versus-module distinction.

The Chiral Pair on the Worldline

The two constructions of the Dirac path integral can be expressed in one biquaternionic statement, and it is worth writing it explicitly because it is the point at which this article meets the rest of the subcategory.

A worldline carries two structures at once: a material trajectory $x^\mu(\tau)$, whose tangent is a four-velocity in $\mathbb{M}_-$, and a spin transport, whose generator is the odd variable $\hat\psi^\mu$. The material trajectory is the object the scalar path integral sums over; the spin transport is the object the spin factor sums over. Squaring the first-order operator pairs them: the square of the Dirac operator is central and acts only on the material trajectory, while the spin indices are carried along by the Clifford structure. The chiral pair

$$ \tilde{\nabla}\tilde{\Psi}_R = m\tilde{\Psi}_L, \qquad \tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\tilde{\Psi}_R $$

is the operator statement of the same split: the two gradients act on the two chiral halves, the material trajectory is common to both, and the mass is the off-diagonal corner that turns one into the other. In the worldline language the two chiral halves are the two possible "spins" of the transport, and the mass term is the reversal; in the checkerboard the same reversal is the lattice corner. The three descriptions — the operator pair, the discretised checkerboard, and the continuous spinning worldline — are the same off-diagonal mass structure in three registers.

There is a further consistency check that ties this article to Klein–Gordon from the Dirac Square in Biquaternionic Form. The scalar kernel $G_F$ is the path integral of the squared theory; the spin factor is what the first-order theory adds. If the spin factor is expanded in the coupling, its leading term is the identity (no bivector insertion), and the worldline path integral reduces to the scalar $G_F$; this is why the square of the Dirac propagator's equation is the Klein–Gordon equation with a delta source, as computed above. The higher terms of the spin factor are the spin–field couplings, and they have no spin-0 counterpart. The spin-0 path integral of the companion subcategory is therefore not a separate construction with an analogous appearance; it is the trivial-spin-factor case of the worldline integral, and the full Dirac path integral is what the spin degrees of freedom add to it.

What Is Standard and What the Algebra Adds

The Feynman propagator, the proper-time representation, the worldline supersymmetric action, the spin factor and the checkerboard limit are all standard results, transcribed here and cited. The biquaternion reading contributes three placements:

The spin variable is the odd Clifford element $\hat\psi^\mu$, whose even products close to form $\mathbb{B}$. The biquaternion algebra is not an auxiliary description of spin; it is the even part of the algebra that the spin variable generates.
The Dirac propagator is the Dirac operator applied to the scalar propagator, and the reason is the Dirac square: the first-order operator's square is the central Klein–Gordon operator, whose path integral is the spin-0 object of the companion subcategory. The first-order path integral is thus a "square root" of the scalar one, in the same sense as the Dirac equation itself.
The checkerboard is the chiral pair on a lattice. Its two directions are the two minimal ideals of the reduced algebra, and its corners are the off-diagonal mass insertions; the continuum limit is the biquaternion mass pair.

The worldline and the checkerboard are complementary descriptions of the same object, and the algebra shows why both work: a first-order operator can be realised as a sum over directed paths (the checkerboard, which builds in the chirality), or as a second-order path integral dressed by the Clifford-valued spin factor (the worldline, which builds in the square). The two agree because the odd variable and the mass term are the two faces of the same off-diagonal structure.

Summary

The Dirac path integral is the sum over worldlines that computes the propagator $S_F$, the matrix-valued Green's function satisfying $(i\not\partial - mc/\hbar)S_F = i\delta^{(4)}$. Because the Dirac operator is first order, the path integral is organised in one of two standard ways. In the proper-time way, the scalar Klein–Gordon kernel is written as a worldline sum over material curves in $\mathbb{M}_-$ with central phase $S_0$, and the Dirac propagator is obtained by applying the conjugate first-order operator: $S_F \propto (i\not\partial + mc/\hbar)\Delta_F$, the identity following from $(\not p - mc/\hbar)(\not p + mc/\hbar) = p^2 - m^2c^2/\hbar^2$. In the spinning-worldline way, an odd Grassmann variable $\psi^\mu$ is added at each point with kinetic term $\frac{i}{2}\psi^\mu\dot\psi_\mu$; quantisation gives $\{\hat\psi^\mu,\hat\psi^\nu\}=\frac12 g^{\mu\nu}I$, whose products generate the Clifford algebra $\mathrm{Cl}_{1,3}$; the path integral over $\psi$ is the Clifford-valued spin factor $\Phi[x]$; and the full worldline integral reproduces the Dirac propagator (Berezin–Marinov, Brink–Di Vecchia–Howe).

In the biquaternion framework, the odd variable is the generator of the Clifford algebra and the biquaternion algebra is its even part, $\Phi(\mathbb{B}) = \mathrm{Cl}_{1,3}^{+}$; the spin factor is the path-ordered Clifford product, even part in $\Phi(\mathbb{B})$, odd part a dressing of the boundary spinors; and composing the first-order operator with its conjugate returns the scalar Klein–Gordon operator — in momentum space $(\not p - \mu)(\not p + \mu) = p^2 - \mu^2$, the identity of the notation table below; the scalar path integral of the companion subcategory is the kernel that the first-order factor acts on, a consequence of this identity rather than an independent construction. Feynman's checkerboard is the discrete realisation of the same structure in 1+1 dimensions: the two directions are the two chiral components, the corners carry the weight $i\epsilon m/\hbar$ and are the off-diagonal mass insertions, and the continuum limit is the two-dimensional Dirac propagator.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong \mathrm{Cl}_{1,3}^{+}$
$\mathbb{M}_-$	Anti-Hermitian (material) sector; the worldline position
$g = \mathrm{diag}(+1,-1,-1,-1)$	Clifford metric (level-3 tool)
$\not\partial = \gamma^\mu\partial_\mu$	Slash (first-order Dirac operator)
$S_F(x-x')$	Dirac (Feynman) propagator
$\Delta_F(x-x')$	Scalar (Klein–Gordon) Feynman propagator
$(i\not\partial - mc/\hbar)S_F = i\delta^{(4)}$	Propagator equation
$(\not p - mc/\hbar)(\not p + mc/\hbar) = p^2 - m^2c^2/\hbar^2$	Dirac square (momentum space)
$T$	Proper time (Schwinger parameter)
$x(\tau)$, $\dot x^\mu$	Worldline and its tangent
$S_0[x] = \int_0^T d\tau\left(\tfrac14\dot x^\mu\dot x_\mu - m^2c^2/\hbar^2\right)$	Scalar worldline action
$\psi^\mu(\tau)$	Grassmann (spin) worldline variable
$\{\hat\psi^\mu,\hat\psi^\nu\} = \tfrac12 g^{\mu\nu}I$	Clifford algebra from the odd variable
$\Phi[x]$	Spin factor (Clifford-valued path-ordered weight)
$K_{RR}, K_{RL}, K_{LR}, K_{LL}$	Checkerboard propagator matrix
$i\epsilon m/\hbar$ per corner	Checkerboard corner weight
$\tilde{\nabla}\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\tilde{\Psi}_R$	Biquaternion mass pair
$\Box = \partial_{ict}^2 + \Delta$	Series d'Alembertian