Duffin–Kemmer–Petiau Particles are Bosons

Abstract

The parametrized Duffin–Kemmer–Petiau wave equation is formulated for many relativistic particles of spin-0 or spin-1. The first-quantized formulation lacks the fields of creation and annihilation operators which satisfy commutation relations subject to causality conditions, and which are essential to the Quantum Field Theoretic proof of the spin-statistics connection. It is instead proved that the wavefunctions for identical particles must be symmetric by extension of the nonrelativistic argument of Jabs (Found Phys 40:776–792, 2010). The causal commutators of Quantum Field Theory restrict entanglement to separations of the order of the Compton wavelength $\hbar /mc$ . The entanglement manifest in the symmetric Duffin–Kemmer–Petiau wavefunctions is unrestricted.

Appendices

Appendix A: Scattering Amplitudes

The Møller operator $\omega _+$ acting on the positive–mass, free incident wavefunction $\phi _i$ is defined by

$$\begin{aligned} \omega _+ \phi _i (x,\tau )=\lim _{\tau ' \rightarrow -\infty }\frac{+1}{{\mathrm {i}}}\int d^4x'\, \Gamma _+(x,\tau ,x',\tau ')\phi _i(x',\tau ')\,. \end{aligned}$$

(74)

Thus the wavefunction $\omega _+\phi _i(x,\tau )$ is the propagation to $\tau$ , through (52), of the asymptotic wavefunction $\phi _i(x',\tau ')$ . Similarly, for the positive–mass, free final wavefunction $\phi _f$ ,

$$\begin{aligned} \omega _- \phi _f(x,\tau )=\lim _{\tau '' \rightarrow +\infty }\frac{-1}{{\mathrm {i}}}\int d^4x''\, \Gamma _-(x,\tau ,x'',\tau '')\phi _f(x'',\tau '')\,, \end{aligned}$$

(75)

and thus the wavefunction $\omega _-\phi _f(x,\tau )$ is the precursor at $\tau$ of the asymptotic wavefunction $\phi _f(x'',\tau '')$ . It follows from the psuedo–unitary propagation that the Stueckelberg scattering matrix

$$\begin{aligned} S_{fi} = \int d^4x\, \overline{\omega _- \phi _f(x,\tau )}\omega _+ \phi _i (x,\tau ) \end{aligned}$$

(76)

is independent of the parameter $\tau$ . It may be shown that the Møller operators have the Markov property, and so

$$\begin{aligned} S_{fi}= \lim _{\tau \rightarrow +\infty } \int d^4x\, \overline{\phi _i(x,\tau )}\omega _+ \phi _i (x,\tau )\,. \end{aligned}$$

(77)

It is eventually inferred that the incoming and final masses $m_i$ and $m_f$ are the same, owing to the gauge field ${\mathcal A}^\mu$ being massless. The scattering amplitudes and cross-sections so obtained agree, after a conventional scaling for dimensionality, to those obtained by integrating with respect to coordinate time $x^0$ at fixed mass $m=m_i=m_f$ . The cross sections may be calculated perturbatively, with radiative corrections and anomalies recovered as in [24] .

As indicated in Sect. 3 the parameter $\tau$ may be eliminated in favour of $X^0$ , the center of a wave packet in coordinate time $x^0$ . Following the convention in scattering calculations [26] it is assumed that as $\tau ' \rightarrow -\infty$ , or equivalently as ${X^0}' \rightarrow -\infty$ , the spacetime support of the wave packet does not overlap that of the scattering field ${\mathcal A}^\mu (x)$ and so the packet is asymptotically free, with a similar assumption and a similar conclusion as $\tau ''\rightarrow +\infty$.

Appendix B: Spin-0, Spin-1/2

1.1 (i) Spin-0

The DKP five-spinor for a spin-0 particle is $\psi =(\partial _{0}\rho ,\partial _{1}\rho ,\partial _{2}\rho , \partial _{3}\rho ,-mp)$, where $\rho$ satisifies the Klein–Gordon equation [26, p5]

$$\begin{aligned} \partial _\mu \partial ^\mu \rho +m^2\rho =0. \end{aligned}$$

(78)

The four $5 \times 5$ DKP matrices are

$$\begin{aligned} \beta _0= {\mathrm {i}} \left( \begin{array}{lll} 0 &{} \quad \mathbf 0^t &{} \quad -1 \\ \mathbf 0&{} \quad Z &{} \quad \mathbf 0\\ 1 &{} \quad \mathbf 0^t &{} \quad 0 \end{array} \right) \end{aligned}$$

(79)

and (for $\mathrm {j}=1,2,3)$

$$\begin{aligned} \beta _\mathrm {j}= {\mathrm {i}} \left( \begin{array}{lll} 0 &{} \quad \mathbf 0^t &{} \quad 0 \\ \mathbf 0&{} \quad Z &{} \quad \mathbf e_\mathrm {j} \\ 0 &{} \quad \mathbf e_\mathrm {j}^{\;t} &{} \quad 0 \end{array} \right) . \end{aligned}$$

(80)

These obey the meson algebra (8), and also (11) where now

$$\begin{aligned} \eta _0=2\beta _0^{\;2}-1= \left( \begin{array}{lll} 1 &{} \quad \mathbf 0^t &{} \quad 0\\ \mathbf 0&{} \quad -I &{} \quad \mathbf 0\\ 0 &{} \quad \mathbf 0^t &{} \quad 1 \end{array} \right) . \end{aligned}$$

(81)

The single eigenstates of the projections $\Lambda _u$ and $\Lambda _v$ are

$$\begin{aligned} \mathsf {w}_\pm = \frac{1}{\sqrt{2}}\left( \begin{array}{l} 1\\ \mathbf 0\\ \pm {\mathrm {i}} \end{array} \right) , \end{aligned}$$

(82)

in the rest frame, while the three eigenstates of $\Lambda _z$ are

$$\begin{aligned} \mathsf {z}_0=\left( \begin{array}{l} \mathbf {0}^t\\ I\\ \mathbf {0}^t \end{array} \right) , \end{aligned}$$

(83)

also in the rest frame.

The rotation operator $O({\mathbf s}, \theta )$ is now

$$\begin{aligned} O({\mathbf s},\theta )= \left( \begin{array}{lll} 1 &{} \quad \mathbf 0^t &{} \quad 0\\ \mathbf 0&{} \quad \exp ({\mathrm {i}}\theta {\mathbf s} \cdot {\mathbf V}) &{} \quad \mathbf 0\\ 0 &{} \quad \mathbf 0^t &{} \quad 1 \end{array} \right) . \end{aligned}$$

(84)

As pointed out by Duffin [1], the spatial gradient of $\rho$ responds to a rotation. The five eigenvectors of O are

$$\begin{aligned} X= \left( \begin{array}{lllll} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad {\mathbf v}_0 &{} \quad 0 &{} \quad {\mathbf v}_{-1}&{}{\mathbf v}_{+1}\\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad 0\\ \end{array} \right) . \end{aligned}$$

(85)

1.2 (ii) Spin-1/2

The influence function $\Gamma _+$ for the parametrized Dirac wave equation may be found in [23, 25]. The influence function for two particles, also given by (60), preserves any symmetry or antiysmmetry of the two-particle wavefunction. The rotation operator is, in terms of the Dirac matrices,

(86)

Its two eigenvalues $\exp ({\mathrm {i}} l \theta )$ , where $l=\pm 1/2$ , are doubly–degenerate. The projection operators for $l=\pm 1/2$ are

(87)

That is, $P({\mathbf s},l)$ projects any 4-spinor $\psi$ onto a rotational eigenstate:

$$\begin{aligned} O({\mathbf s},\theta )P({\mathbf s},l)\psi =\exp ({\mathrm {i}} l \theta )P({\mathbf s},l)\psi \,. \end{aligned}$$

(88)

The wavefunction at $\tau =0$ for two identical particles is also given by (66). The factor before the second summand in (66) is now $\exp ( 2 \pi {\mathrm {i}} l)=-1$ and so Fermi–Dirac statistics are inferred.