PramaBa, Vol. 8, No. 6, 1977, pp. 545-~62. © Printed in India. Tensor product composition of algebras in Bose and Fermi canonical formalism DEBENDRANATH SAHOO Reactor Research Centre, Kalpakkam 603102, Tamil Nadu MS received 21 O~tober 1976; in revised form 25 April 1977 Abstract. We consider a graded algebra with two products O, a) over an c-factor commutation. One of the products O) is E-commutative, but, in general non- associative; and the other (a) is a graded Lie E-product and a graded E-derivative with respect to the first (o). Using the obvious mathematical condition, namely-- the tensor product of two graded algebras with the same E-factors is another with the same ~-factor, we determine the complete structure of a two-product (o, a) graded algebra. When the E-factors aretaken to be unity and the gradation structure is ignoied, we recover the algebras of the physical variables of classical and quantum systems, considered by Grgin and Petersen. With the retention of the gradation structure and the possible choice of two E-factors we recover the algebras of the canonical formalism of boson and fermion systems for the above classical and quantum theories. We also recover in this case the algebra of anticommutative classical systems considered by Martin along with its quantum analogue. Keywords. Canonical formalism, graded algebra; two-product algebra; algebraic structure; composition axiom. 1. Introduction The algebras of physical variables (or observables) of both classical and quantum mechanical systems are two-product algebras. In classical mechanics, the obser- vables are real-valued functions on the phase space. One of the two products is commutative and associative. The other product is a Lie product which is the Poisson bracket. In quantum mechanics, the observables are self-adjoint operators on a Hilbert space. The two corresponding products are: the Jordan product which is the anti-commutator of two operators (commutative, but non-associative) and the Lie product which is the commutator of two operators. In both the cases, •the two products are interrelated by the derivation law. If f, g, h are classical variables and .... and {, } denote the two classical products, then {f, g. h} = {f, g}- h + g. {f, h}. A structurally similar derivation law can be written in quantum mechanics also. These common features associated with the set of observables of classical and quantum 545 P--8 546 Debendranath Sahoo systems suggest the abstract concept of a two-product algebra (TPA) ~4 = {ej~, 6, ~} defined as a linear space over a (commutative) field cj, equipped with two products 6, ~: ¢J~ (~) cjj ~ cj~ and which satisfy the following identities: fo~g ~ -- g~f (1 a) (fctg) cth + (g~h) ctf + (h:tf) ctg = 0 (1 b) fct(g6h) = (fag) 6h + ga(f~h) (1 c) for all f, g, hEC-~. The assumption ofa TPA with identities (1, a-c) as the basic structure of mechanics can be traced back to a more basic axiom, to be referred to hereafter as tee duality axiom, which states that the physical variables of a theory play a dual role as both observables and generators of infinitesimal transformations in the invariance group of the theory. Using duality as the main axiom, Grgin and Petersen (1974) have obtained two results: (i) the algebraic structure, that has the observable-gene- rator duality as a fundamental property, is a TPA. (ii) furff,er, if the algebra of generators (i.e., the subst:ucture {c~, ~}) is central-simple, the observable-generator duality restricts the a-product of the algebra of observables (i.e., the substructure {c~, a}) to two cases: eitker 6 is a commutative, associative product as in classical mechanics, or 6 is a central-simple special Jordan product as in quantum mechanics. Proceeding in a manner identical to that of these authors it can be shown easily that if c~ is a graded linear space, the duality axiom implies a graded two-product algebra (GTPA) structure on c~, i.e., the result stated in (i) can be generalised to the graded space. This is a trivial exercise and will not be carried out here. Grgin and Petersen (1976) have also obtained the full algebraic structure of ~A relevant to classical and quantum mechanical systems in a different manner. Defin- ing a composition class as a set cj of TPA's equipped with a product o : cA × cA cA, they assume the following: A. ff ~41, ~42 e c5, and ~Atz : ~i o ~2, then ~12 : cJ~l (~ c3~2, (2 a) B. Associ~tivity: (~Al o ~A~)o ~A3 : ~A1 0(o42 o ~A3), (2b) C. Existence of unity: The field ~, consideied as a TPA {if, 6, ~}, where 6 is the product in c3: and ~t = 0, is a unit for the composition product "o ", Lea o ~ : o4 =o4 o c3:, for all ~4 E c5. We obtain the same result as that of Grgin and Petersen (1976) with what we consider to be a more natusal procedure, namely the consideration of the tensor product of two graded algebras. This gives the algebraic structure of the entire canonical formalism that includes both the Bose and the Fermi types of mechanics. Since, for masons to be mentioned further on (see sec. 4), one cannot generalise the result stated in (ii) for the GTPA using the original method of Grgin and Petersen (1974), we shall carry out this task by using the tensor product composi- tion technique instead. The plan of the paper is as follows: in section 2 we introduce some basic algebraic concepts and notations which are required for the rest of the paper. In section 3, which is the heart of the paper, we introduce and use the composition axiom to derive the complete algebraic structure of the Bose and the Fermi canoni- cal formalisms. Section 4 contains a discussion and the summary of our results. Algebraic structure of canonical formalism 547 2. Algebraic preliminaries The algebraic concepts introduced here in brief are not the most general possible, but they are adequate for our present purpose. 2.1. A Graded Linear Space cd3 is a linear space over the field of the reals R, tl where n belongs to the set of integers Z, and cjj._ is a subspace of c~ spanned by homogeneous elements of degree n. By setting cj~, = 0 for n < 0, and cjjo = R, we convert c6J into a positively graded linear space, i.e., ® nEZ + where Z + denotes the set of positive integers including zero. The degrce of an element f~ ¢'jJ~' will sometimes be written as deg f (i.e., m = deg f). 2.2. Factors of commutation Let • be a mapping ,:Z + X I, +--* {-- 1, + 1} satisfying the following properties: • (m, n) = • (n, m) (3 a) • (m+m', n)= •(m,n) • •(m', n) (3b) • (2m, n) = • (m, 2n) = 1, (3 c) where m, m', n ~ Z +. Then, there are two and only two types of factors of commutation, namely E(m,n) = + 1, for all m,n Z+ (4a) and • (m,n) = (-- 1) ran, for all m,n Z ~, (4b) which can be derived from (3, a-c). For a more general definition of the E-factors we refer tl~.e reader to Bourbaki (1964). The e-factors given in (4 a) and (4 b) respectively may be called the Bose type and the Fermi type factors. Since the solution (4 b) involves the product of the degrees, we shall adopt a simpler notation: ,( m, n) = , (mn). (5) Then the properties (3, a-c) can be expressed as: ,(mn) = ,(nm) (3 a') ,(mn + m'n) = , (ran)" e (m'n) (3 b') • (2 mn) = 1. (3 c') 548 Debendranath Sahoo Note that E(mn) :- ~ (-- mn). [This is seen by multiplying both sides of (3 c') by e(--mn) and using (3b').] 2.3. A graded two-product algebra (GTPA) -~ {c3J, a, ~, ~} is a positively graded linear space c~ over R equipped with two products a, c¢: c6J C~ c3~ -~ ~ such that iff~ c3~", gEC~ ", h~C~ p, then fag E c'~+. (6 a) and f~g ~ ci~.+, -* . (6b) s is called the shift degree of gradation. Analogous to a TPA, the following rela- tions hold in a GTPA: (i) The substructure {c3J, ~} is a graded Lie algebra in which focg -~ -- ~ (mn) g~f (7) E(pm)fo~ (go~h) + e (mn) gc~ (h~J') + c (np) hcz (fotg) = O. (8) The relation (7) will be referred to as the ~-anticommutative relation and the rela- tion (8), as the E-Jacobi identity. (ii) The product ~ is a graded derivation with iespect to the a-product: fo~ (gab) = (f~g) ah + c (ran) ga (f~h). (9) This relation will be referred to as the ~-derivation law. It will be assumed that there is a unit element in the substructure {c~, a}, i.e., there exists e ~ c~ such that eaf=f=fae, for any f~ C-d~. (10) It tken follows from (9) (by putting g = h-----e) that fc~e = O, for any f~ c3J. (11) It is also easy to see from (5) and (10) that e ~ C e, i.e., the unit element is of degree zero. It may be noted that the null element 0 is a member of every sub- space c3J". Further, note that if c~,= 0, then c-~-~. = O, fol all m ~ Z +. 2.4. Tensor product of graded linccr spaces Let c3Jl=Qc3J1 ~1 and ~2=@cJ~2~% tll I In2 (ml, m2 E Z +) be two graded linear spaces ovc.r R. Define m where ~12 '~ = ® 9~i "~ Q 9~:'. 1/11 -I- m..-~tli Algebraic structure of canonical formalism 549 We shall make it a convention to be followed throughout this paper, unless other- wise stated, that f, g, h are three homogeneous elements of degree m, n and p respectively.