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The DKP equation is, allegedly, a relativistic spin-0 or spin-1 equation. In the spin-0 case the defining algebra is a 5 dimensional representation. For spin-1 the four matrices are 10 dimensional, in either case the E.O.M. looks superficially just like the Dirac equation.

Resources online about the equation are scarce. Is it in reducible representation of the Lorentz group? If not, how is it possibly Lorentz invariant? I.e. how does a real scalar field have more than a single component?

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  • $\begingroup$ A five-dimensional object can probably be decomposed as a scalar and a 4-vector. $\endgroup$
    – Buzz
    Aug 13, 2023 at 22:14
  • $\begingroup$ While I don't have a complete answer to your question, I think it's worth noting that $\beta_a\beta_b=\delta_{ab}$ for the matrices given in that Wikipedia article, and there are only four matrices, so in fact that equation just seems to give four independent components. $\endgroup$ Aug 13, 2023 at 22:15
  • $\begingroup$ @Buzz there are only four matrices so the equation seems to consist of four components. $\endgroup$ Aug 13, 2023 at 22:15
  • $\begingroup$ @SuzuHirose yes they couple to the gradient, so there should be 4. But they act on a wavefunction which must be components $\endgroup$
    – Craig
    Aug 13, 2023 at 22:54
  • $\begingroup$ While this "looks like" the Dirac equation the fact is that the matrices don't interact with each other so it seems to just revert to the Klein-Gordon equation, it appears to just be a first-derivative version of that. I might have missed something but I can't see what. $\endgroup$ Aug 14, 2023 at 0:18

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After some research I was able to answer this question. These references (1), (2) were extremely useful.

The elements $\beta$ belong to what is known nowadays as the meson algebra, and these are studied more contemporaneously alongside the usual Clifford algebras by mathematicians. Very loosely one may think of them as being 'reducible' generalizations of the usual Clifford algebra.

Speaking of, from the fact that $[\beta^\mu,\beta^\nu] = M^{\mu\nu}$ are generators of the Lorentz Lie algebra, we may by direct computation show the following facts about the two cases:

Spin-$0$:

In the $5$ dimension spin-$0$ DKE-equation, the wavefunction belongs in the reducible representation of the Lorentz group labelled $$(0,0)\oplus(\frac{1}{2},\frac{1}{2}).$$ That is to say, a scalar stacked atop of a four-vector, as guessed by user Buzz. Let us abuse notation and write the wavefunction via

$$ \psi = \begin{pmatrix} \phi \\ A_\mu \end{pmatrix}, $$with $\phi$ the Lorentz scalar field and $A_{\mu}$ the components of the vector field. In more familiar notation the DKP-equation can be written as:

$$ \begin{pmatrix} \partial_\mu A^\mu \\ \partial_\mu \phi \end{pmatrix} = \frac{mc}{i\hbar}\begin{pmatrix} \phi \\ A_\mu \end{pmatrix}. $$

Spin-$1$: In the $10$ dimensional spin-$1$ DKE-equation, the wavefunction belongs in the reducible representation of the Lorentz group labelled $$(1,0)\oplus(0,1)\oplus(\frac{1}{2},\frac{1}{2}).$$ I.e. the sum of the adjoint (field strength tensor) and fundamental (vector) rep's. Let us abuse notation and write the wavefunction via

$$ \psi = \begin{pmatrix} \vec{E} \\ \vec{B} \\ A^\mu \end{pmatrix}, $$with $\vec{E},\vec{B}$ the timelike and spacelike $3$-vectors associated to an anti-symmetric tensor of rank-$2$ (E&M fields from a typical Faraday tensor $F^{\alpha \beta}$), and $A^{\mu}$ the components of a vector field. In the familiar notation the DKP-equation can be re-written as:

$$ \begin{pmatrix} -\partial_t\vec{A}-\vec{\nabla}A^0 \\ \vec{\nabla}\times \vec{A} \\ \vec{\nabla}\cdot \vec{E} \\ \vec{\nabla}\times \vec{B} - \partial_t \vec{E} \end{pmatrix} = -\frac{mc}{i\hbar}\begin{pmatrix} \vec{E} \\ \vec{B} \\ A^\mu \end{pmatrix}. $$ (Despite the rather gross mismatch of notation it should be clear how the components line up if one counts the arguments dimensions.)

Thus we see the meson algebra is a clever way to embed the structure of coupled 'Proca' like equations into a single equation superficially identical to the Dirac equation.

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  • $\begingroup$ To future readers: I am relatively confident about the signs and placement of indices, but if you plan to use this for anything important please verify these yourself. As well, if it matters I am implicitly using the mostly minus metric convention. $\endgroup$
    – Craig
    Aug 15, 2023 at 6:13
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    $\begingroup$ Also Fushchich, W., Nikitin, A. - Symmetries of the equations of quantum mechanics (AP, 1994) is worth a read. $\endgroup$
    – DanielC
    Oct 23, 2023 at 16:02

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