Grassmann algebras of the usual kind are generalized to what are to be called para‐Grassmann algebras. Para‐Grassmann numbers are defined as those satisfying the trilinear commutation relations that resemble the para‐Fermi commutation relations. Basic mathematical properties of these algebras are studied in detail on the basis of generalized Grassmann algebras discussed in the preceding paper. Applications are made to description of para‐Fermi systems, and it is found that such systems can completely be described in what we call the para‐Grassmann representation.
Topics
Fermion systems
REFERENCES
1.
2.
See, for example, F. A. Berezin, The Method of Second Quantization (Academic, New York, 1966);
Theor. Math. Phys.
6
, 194
(1971
);3.
Preliminary results were reported in
M.
Omote
and S.
Kamefuchi
, Lett. Nuovo Cimento
24
, 345
(1979
).To our knowledge the earliest work that considers this kind of generalization is
J. L.
Martin
, Proc. R. Soc. (London) Ser. A
251
, 543
(1959
),and the first author who introduced the term “para‐Grassmann” is
A. J.
Kálnay
, Rep. Math. Phys.
9
, 9
(1976
).4.
Y.
Ohnuki
and S.
Kamefuchi
, J. Math. Phys.
21
, 601
(1980
), to be referred to hereafter as (I). The present paper employs mostly the same notation and conventions as in (I). Equation (3.2) of (I), for example, will be quoted as (I.3.2).5.
See, for example, Ref. 2 of (I).
6.
Theorem 1 in
Y.
Ohnuki
and S.
Kamefuchi
, Phys. Rev.
170
, 1279
(1968
).7.
8.
Y.
Ohnuki
and S.
Kamefuchi
, Quantum Field Theory and Parastatistics, Soryushiron Kenkyu (Kyoto)
55
, (1977
), special issue.9.
10.
K.
Drühl
, R.
Haag
, and J. E.
Roberts
, Commun. Math. Phys.
18
, 204
(1970
);
This content is only available via PDF.
© 1980 American Institute of Physics.
1980
American Institute of Physics
You do not currently have access to this content.
