para-Fermi Statistics Of Order Research Articles

Parastatistics and Uniquantization

A connection between the scheme of unitary quantization (uniquantization) and para-Fermi statistics of order 2 is considered. An appropriate generalization of the Green’s ansatz is suggested based on incorporation of the additional operator Ω which allows one to transform into the identity the bilinear and trilinear commutation relations of unitary quantization for the creation and annihilation operators of two different para-Fermi fields φа and φb. The way of incorporating para-Grassmann variables ξk into the general scheme of unitary quantization necessary for the definition of coherent states is suggested. For parastatistics of order 2, the new fact of existence of two alternative definitions of the coherent state for the para-Fermi oscillators is established.

Unitary Quantization and Para-Fermi Statistics of Order 2

A connection between a unitary quantization scheme and para-Fermi statistics of order 2 is considered. An appropriate extension of Green's ansatz is suggested. This extension allows one to transform bilinear and trilinear commutation relations for the annihilation and creation operators of two different para-Fermi fields $\phi_{a}$ and $\phi_{b}$ into identity. The way of incorporating para-Grassmann numbers $\xi_{k}$ into a general scheme of uniquantization is also offered. For parastatistics of order 2 a new fact is revealed, namely, the trilinear relations containing both the para-Grassmann variables $\xi_{k}$ and the field operators $a_{k}$, $b_{m}$ under a certain invertible mapping go over into the unitary equivalent relations, where commutators are replaced by anticommutators and vice versa. It is shown that the consequence of this circumstance is the existence of two alternative definitions of the coherent state for para-Fermi oscillators. The Klein transformation for Green's components of the operators $a_{k}$, $b_{m}$ is constructed in an explicit form that enables us to reduce the initial commutation rules for the components to the normal commutation relations of ordinary Fermi fields. A nontrivial connection between trilinear commutation relations of the unitary quantization scheme and so-called Lie-supertriple system is analysed. A brief discussion of the possibility of embedding the Duffin-Kemmer-Petiau theory into the unitary quantization scheme is provided.

Conformal invariance and parastatistics in two dimensions

We show that the conformally invariant O(N) nonlinear σ-models with a Wess-Zumino term with coefficient k, have the same current and conformal algebras with a theory of N Majorana fermions with a hidden (gauged) O(k) quantum number. These latter are interpreted as N particles obeying the free Dirac equation but quantized with (nonlocal) para-Fermi statistics of order k, an interpretation that we verify by deriving the exact finite-temperature propagator, thereby explicitly relating the anomalous scaling dimensions to nonstandard statistics.

Quantization of Fields in Accordance with Modular Statistics

A new generalization of quantum statistics is described, which is different from parastatistics, though it includes Fermi statistics and parafermi statistics of order two. It can be applied to the quantization of nonlinear field theories, without violating the correspondence principle. Like parastatistics, it allows the occupation of a given dynamical state by more than one particle of half-odd-integral spin. A special feature is that quark-like particles are naturally associated in modules, which have many of the characteristics of baryons and mesons. The group theoretical properties of the new statistics, and the implied classification of states, are briefly examined.