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Star Product for Para-Grassmann Algebra of Order Two

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Abstract

In para-Grassmann algebra a noncommutative, associative star product \(*\) (the Moyal product), which is a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Isomorphism between the algebra of para-Grassmann numbers of order two equipped with the star product and the algebra of creation and annihilation operators \(a_{k}^{\pm }\) obeying the para-Fermi statistics of the same order is established.

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Notes

  1. For the completeness of the picture it is necessary to mention another way for a generalization of Grassmann algebra based on bilinear relations of the type

    $$\begin{aligned} \xi _{i}\xi _{j} = q \xi _{j}\xi _{i}, \end{aligned}$$

    where q is a primitive root of unity such that \(q^{p + 1} = 1,\, p = 1, 2,\ldots \;\). This extension of a Grassmann algebra sometimes is called the generalized Grassmann algebra to distinguish it from the proper para-Grassmann algebra (see, for example, [2, 10, 12, 13, 18, 21]). In the present work we will not use the generalized Grassmann algebra because the para-Grassmann algebra of order \(p=2\) is still quite visible for concrete calculations, however, probably in the situation of higher order (\(p\ge 3\)) algebras this generalization is more suitable. The latter case implies the use of bilinear commutation relations for the so-called q-fermion creation and annihilation operators \(a^{\pm }_{k}\) [2].

  2. We follow the notations of the right and left derivatives adopted in [4].

  3. We recall that N designates the number of the generating elements in a para-Grassmann algebra and the number of the parafermion creation and annihilation operators of para-Fermi algebra. For \(N = 2\) we have two pairs of Grassmann variables in involution, \((\xi _1, \xi _2)\) and \((\bar{\xi }_1, \bar{\xi }_2)\), and two pairs of the parafermion creation and annihilation operators, \((a_{1}^{+}, a_{2}^{+})\) and \((a_{1}^{-}, a_{2}^{-})\).

  4. The only thing we have to use is a general expression for the measure of integration, namely, \((d\mu )_{p}(d\bar{\mu })_{p}\), where now

    $$\begin{aligned} (d\mu )_{p}\equiv d^{p\!}\mu _{N} d^{p\!}\mu _{N-1}\ldots d^{p\!}\mu _{1}, \qquad (d\bar{\mu })_{p}\equiv d^{p}\bar{\mu }_{1} d^{p}\bar{\mu }_{2}\ldots d^{p}\bar{\mu }_{N}. \end{aligned}$$

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Acknowledgements

The authors thank the reviewer for careful reading of the first version of the article and for valuable comments.

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  1. Matrosov Institute for System Dynamics and Control Theory SB RAS, Lermontov str. 134, P.O. Box 1233, Irkutsk, 664033, Russia

    Yuri Markov & Margarita Markova

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Appendix: Para-Grassmann differentiation

Appendix: Para-Grassmann differentiation

In this Appendix we derive a number of particular differentiation rules. In the special case of \(p = 2\) from the formulae (2.5)–(2.7) the expression for the right derivative follows

$$\begin{aligned} \frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, ( \bar{\mu }_{j}\xi _{i}) \equiv -\frac{1}{2}\,\frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, [\xi _{i}, \bar{\mu }_{j}] + \frac{1}{2}\,\frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, \{\xi _{i}, \bar{\mu }_{j}\} = -\delta _{ki}\bar{\mu }_{j} + \delta _{ki}\bar{\mu }_{j} = 0, \end{aligned}$$

and similarly for the left derivative we get

$$\begin{aligned} (\xi _{i} \bar{\mu }_{j}) \frac{\overleftarrow{\partial }}{\partial \xi _{k}} \equiv \frac{1}{2}\,[\xi _{i}, \bar{\mu }_{j}]\, \frac{\overleftarrow{\partial }}{\partial \xi _{k}} + \frac{1}{2}\,\{\xi _{i}, \bar{\mu }_{j}\}\, \frac{\overleftarrow{\partial }}{\partial \xi _{k}} = -\bar{\mu }_{j}\delta _{ki} + \bar{\mu }_{j}\delta _{ki} = 0. \end{aligned}$$

By this mean we have

$$\begin{aligned} \frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, ( \bar{\mu }_{j}\xi _{i}) = 0, \quad (\xi _{i} \bar{\mu }_{j}) \frac{\overleftarrow{\partial }}{\partial \xi _{k}} = 0. \end{aligned}$$
(A.1)

The fact that these derivatives vanish is a distinguishing feature of the case \(p = 2\). For \(p\ne 2\) these derivatives always are different from zero, for example,

$$\begin{aligned} \text{ for }\;\; p = 1:\quad \frac{\overrightarrow{\partial }}{\partial \xi _{1}}\, (\bar{\mu }_{j}\xi _{1}) = -\bar{\mu }_{j}; \qquad \text{ for }\;\; p = 3:\quad \frac{\overrightarrow{\partial }}{\partial \xi _{1}}\, (\bar{\mu }_{j}\xi _{1}) = \bar{\mu }_{j} \end{aligned}$$

etc. In addition to (A.1) for \(p = 2\) we also can write

$$\begin{aligned} \frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, (\xi _{i}\bar{\mu }_{j}) = 2\delta _{ik}\bar{\mu }_{j}, \quad (\bar{\mu }_{j}\xi _{i}) \frac{\overleftarrow{\partial }}{\partial \xi _{k}} = 2\delta _{ik}\bar{\mu }_{j} \end{aligned}$$
(A.2)

and as a consequence of (A.1) and (A.2) we have

$$\begin{aligned} \frac{\overrightarrow{\partial }}{\partial \xi _{k}}\, (\xi _{j}\xi _{i}) = 2\delta _{kj}\xi _{i}, \quad (\xi _{i}\xi _{j}) \frac{\overleftarrow{\partial }}{\partial \xi _{k}} = 2\delta _{kj}\xi _{i}. \end{aligned}$$
(A.3)

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Markov, Y., Markova, M. Star Product for Para-Grassmann Algebra of Order Two. Adv. Appl. Clifford Algebras 31, 27 (2021). https://doi.org/10.1007/s00006-021-01125-8

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