Tensor Product Composition of Algebras in Bose and Fermi Canonical Formalism

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 observables 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-generator 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 composition 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 canonical 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,

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 relations 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 relation (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 subspace 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. The elements of c~,,1 ~) c3j2,,, will be denoted as fl2, g12, "", etc., and elements of c~,,~ (i = 1, 2) will be denoted as f# go ..., etc. Then f12 =fl ~)f2, gl2 = gl @ g2,'.., etc. With the obvious convention of the degrees also made applicable to c3~2, c~ and c'dJ2, we have deg Az = m =mx + mz = deg f~ + deg f2" (12) Similarly,

n = nl + n z, p = Px + P2 and g12 e ~12 n, hi2 e c'~12v.

2.5. Tensor product of graded algebras To introduce some more notations, letus consider a graded algebra q] = {c-~, At} with only one product /z: q~ ~)q~-~ q~. /t may be identified with the product or ~ of a GTPA. Now let q~l ----- {c~1, Pt} and q~2 = {c6J2, P2} be two graded algebras. The tensor product of q~l and q~2 is an algebra q~12 = {c6J1~, P12}, where C~12 ~- C'~i (~ C~2 (13 and fli2 = ]'/1 @ /'/2" (14 a) The symbolic relation (14 a) has the following meaning: iffx 2 ~-fl (~f2, gl2 = gx ~ g2 E c~12 with fz e c~2m~, gl e c~1"~, then

f121t12 g12 = "(nxm2). (flPaga) @ (f2P2g2). (14 b) Relations like the one above will be abbreviated as (f/~g)12 ~- a (nxm2) . (flag)l (~ (fPg)2. (14 c) Note that in (14 b, c),f~ and g2 are not assumed to be homogeneous elements. This form of the tensor product will be called as the e-commutative tensor product. [Greub (1967) calls it an anticommutative tensor product.] In what follows, we shall write: [(fag) oth . ~ (ran)]1 ~ [(fl °qgl) oqhl] . E(mln~), etc.

2.6. Associators in a GTPA

Let us define two trilinear associator brackets for the tr and the ~ products: If, g, h]~ = (fug) oh -- f a (gah), (15) and Dr, g, h]a = (f~g) ah -- f~ (g~h), (16) 550 Debendranath Sahoo We now prove a lemma which will be used later. Lemma 1: If s, the shift degree of gradation is an even integer, then

If, g, h]a = ,(np). (fa h)ag. (17) The converse is also true. Proof: Using the E-anticommatativity property of a [eq. (7)] and the fact that the element fag e c-3J'~.... and fcch e c'JAm "-~ [eq. (6 b)], the ~-Jacobi identity of eq. (8) can be written as follows.

, (mp) . fo~ (geh) -- , (pm) . , (np -- sn) . (feh) eg

-- , (pn). e [p (m + n -- s)]. (f~g) eh = O. Since ~(pn) . ,(pm + pn--ps)= e(2pn) . e(pm) . e(ps)= e(pm). ,(ps), onecan cancel •(pm) throughout and write fe(gah)-- •(ps) (f~g)eh-- e(mn), ge(f~h)=0. Thus if s is an even integer, •(ps)= 1 and cq. (17~ follows. To prove the converse, we write (17) as fa(geh) -- (f~g) eh = ,(np) . (f~h) ~g. Using eqs (6b) and (7), one rewrites it as fa (gah) + • (mp +np -- sp) . h ~ (fo~g) + , (pm) . ~ (mn) . 8 (-- ns) • g~ (hoof) = O.

Then multiplying throughout by • (rap) one gets (rap). f~ (g~h) + • (np] . • (sp) . ha (fag) + , (ran). • (sn). g ~ (hotf) = O. Comparing this equation with the Jacobi idcntity (8), we obtain • (sp) --~ 1, and , (sn) = 1. Since this has to be true for arbitrary p (and n), s must be even, (Q.E.D.). Corollary: If s is even, the product a is a e-derivative with respect to itself, i.e.,

(f~g) ~h ~ f~(g~h) +,(np) . (f~h) ag. (18) This follows by using the definition (15) in (17). In what follows we shall use the abbreviated notation ([f, g, h]~. ,(mp))~ =--[fx, g~, hl]~. ~(mp). Lemma 2: If the shift degree of gradation is even, then [h, g,f]~ = -- • (ran + np + pro). If, g, h]a Proof : Using lemma 1, [h, g, f]~ = ~ (ran). (h~f) ~g. Algebraic structure of canonical formalism 551 and using (7) in the right side, we get

[h, g, f]~ = -- e (ran) . (fah) g . • (pm)

= -- e (mn + pm) . , (np) .[(np). (fceh) ~g]

: -- e(mn + np+ pm) . If, g, h]~, (Q.E.D.).

Lemma 3: If a is a e-commutative product, then

[h, g,f]~ = -- e (mn + np + pro) . [f, g, h]~.

Proof : Using the e-commutative property and (6 a) in the definition for [h, g,f] ~r we get

[h, g, f], = (hag) af -- ha (gaf)

= , (mn + np + pm) . [fa (gah) -- (fag) ah]

-~-- ,(mn + np + pm) . [f,g,h]a, (Q.E.D.).

3. Tensor product of two GTPA's

The tensor product of two algebras of the same type is not always an algebra of the same type. For example, if we take two algebras q~, ~ {,~¢'o/4} (i = 1, 2) with /q and P2 both commutative, the product Pl ~-)/~z in the tensor product algebra q~12 = {o~cl @ ,~cz, Pl ~/z~} is also commutative. However, if/q and /t2 are both anticommutative, the tensor product #x @ #3 is not anticommutative. It is a characteristic feature of the algebras of observables of classical (or quantum) systems that th~ tensor product composition of two classical (or two quantum) algebras of observables is again a classical (or a quantum) algebra of observables. From a physical point of view this is natural, since the coupling of two physical systems results in another physical system. This peculiar feature of classical and quantum mechanics was first recognised by Grgin and Petersen (1976) who studied the algebraic implications of composability of physical systems. As noted in the introduction, {see eqs (2 a-c)), the structure of the composition class {c5, o} which they have introduced is a monoid. In this section, we shall derive the graded version of their result which encompasses the entire canonical formalism and this includes both the Bose and the Fermi types of systems. However, we shallstart with a weaker axiom, which effectively amounts to relaxing the associativity property of the structure {cj, o}. In other words, we assume the above structure to be a groupoid instead. We formulate the axiom thus : the a-commutative tensor product of two GTPA's ~4, = {,~, at, ~, e} (i -- 1, 2) of the same e-type is a GTPA ~lz = {,7/'12, a12, ~12, ~}" of the same e-type. We shall make a natural assumption that the restriction of the algebra ~t~ to the algebra ~At (or ~2) yields ~41(~ez(or ei~ 049). This implies that for all elements of the type fl (~ e~, e I ~f~, e 1 ~) g~, gl @ el e ~¢'1 ~ g¢'~, the following conditions are satisfied:

(ft @ e~) 21z(gl (~ ez) = (f2g) 1 ~) e 4 (19) 552 Debendranath Sahoo and ~el ~)f~) Ala (ea (~ g2) = ex (~) (f(~ g)2. (20) (Here the symbol 2 stands for either ct or a). We now proceed to study the implications of our axiom on the structure of a GTPA. First, we prove a few lemmas. Lemma 4: The product al ~ az : ,%rlz@ ,7(1~ ,7912 is c-commutative. Proof: fj a (oq ~ ~a) giz = • (nlm.,,) . (fotg)l (~ (fctg)a = , (nlma) . [-- , (ran). (g~f)]l ~ [-- "(mn). (gaf)]a ~- • (mlna) . , (mini). e (manz). , (nima) . , (mtnz) . (g~f)l (~ (gotJ)a = • ((ml + ma) (ni + na)). [glz (sl @ ~dfld = •(mn). [gl~(~i ® ~)Ad, (Q.E.D.). Note that we have made use of (7) in going from the first equality to the second, and (12) in obtaining the final equality.

Lemma5: The product al~)a~: ~7¢'l~(~),YC'la--*~rla is E-commutative if and only if both a~ and az are either E-commutative or •-anticornmutative. Proof: Let a~: ,Y~.~<~),7/'~-* ,Y(~(i = 1, 2) be both •-commutative or both ~-anticommutative. Then

= • (nim2). (fag)l ~ (fag)~ -- • (n~m~) . [ + • (mn) . (g~D]~ ® [~ ~ (mn). (gaf)h = • (m~n~). ~ (mln~). ~ (n~m~) . • (m~n~). [• (m~.~). (g~f)~ ~ (gaf)d = • ((ml + na) (ml + n~)) . [gla (aa (~ o~)A~] : •(mn) . [gl~(a 1 Q a2)f12], thus proving the 'if' part. Note that the plus and tl=e minus ~igns in the third line of tee proof correspond to tee product a~ being •-commutative and ~-anticommutative respcctively. To prove the "only if' part, we start with tke equation f12 (al @ 0"2) gl '~ = "(mn) . gtz (al ® az)fla and reversing the procedure which led to tf~e previous proof, we obtain the following equation : • (ntmz) • (fag)l (~ (fag)a ----- e (nlm2) . [,(mn) . (gaf )]l ~ [• (mn) . (gaf)]a. Clearly, this equation holds only if (fag), = +_ [e (ran). (gaf)]~ (i = 1, 2), (Q.E.D.). Algebraic structure of canonical formalism 553

Lemma 6: If both al and as are neither e-commutative nor c-anticornmutative, then so is the product al ® as : -~C1~ ® "YCi~ -o ,~c,a. Proof: Replacing the equality sign by the inequality sign ':/: ' in the first part of the proof of lemma 5 (except in the first and the last equalities), the proof follows. Lemma 7: The products a 1 ® a~, al ® tea : °~fla ® "Yfi~ ~ ,7{1~ are e-anticommutative if and only if al and a s are e-commutative. Proof: We consider the product al ® as. The proof for the product a i @ az is similar. Let a~ : ~ ® ,7t'~ ~ ~ (i = 1, 2) be e-commutative. Then, fl~(~l ® a~)gl,~ = c(nlm.,) . (f~g)l ~(fag)~ = c (nlm.a) . [-- e (mn). (g~f)]l ® [c (mn) . (guf)]z

= -- a (nlml) . c (mzn~). c (nimz) . c (mlnz) . [c (mine). (g~f)l @ (gaf)z]

= - c(mn) . gi~(~ ® ~)A~, thus proving the ' if' part. To prove the ' only if' part, we start with the equation flz (~i ® u~) gla = -- c(mn), gl~ (~l O u2)A~ and reversing the steps of the above proof, we write it in the form , (nlmz). (f0~g)x ® (fag) z = a (nlma). [ -- e (mn). (g~f)it ® [c(mn) (gaf)]a. Using the c-anticommutative property of el and cancelling the non-zero factor c (nlma), this equation can be written as (f~g)l ® [(fag) -- e (mn) . (gaf)]~ = O. Since (fag)l is an arbitrary non-zero element of ,7f1, the square parenthesis must vanish, thus proving the 'only if' part (Q.E.D.). The composition laws for the products a~, a~ (i ----- 1, 2) are derived in the following theorem : Theorem 1: The products cqa , ai~: "Yfl~ ®,Y{~a-o g{le are given by:

(21) and

all = at ®aa + b . ~l ®~a, (22) where b ~ R is a constant. Proof: Let ~A,- {,Yr, a, ~, c} (i = 1, 2) be two e-type GTPA's. The most general combination of the products of a, ~ (i = 1, 2) which is bilinear in the elements of ,Yfi ¢9,7{2 is 554 Debendranath Sahoo

21~ = al (~l ~,~ ~) + a~ (a 1 ~.~ a~) + as (~l ® cr'z) + a4 (al ® ~). Thus ~18 and al~ must be special cases of 21a. According to our axiom, the tensor product algebra ~lz = {~ ® '7¢'z, al~, ~la, E} must also be a GTPA of the same ~-type. Thus, if 2is has to be identified with ~la, each of the terms with the coefficients a~ (i = 1..... 4) must be separately E-anticommutative. The first term in 21z is E-commutative because of lemma 4 and the second term is never E-anticommutative because of lemmas 5 and 6. This means ~ z must be of the form

Now, because of the restriction requirement expressed in (19) and (20) in conjunction with (10) and (11), as =a4 = 1, thus proving (21). Note that ax and a z are forced to obey the E-commutative property: (fag), = [, (mn). (gaf)], ~i---- 1, 2) (23) by virtue of lemma 7. In order to prove (22), note that since al and a~ are E-commutative, the first two terms of ;tl 2 are ~-commutative because of lemmas 2 and 3 and the last two terms are c-anticommutative because of lemma 7. Thus alz must be of the form alz = al • (~1 ® ~z) + a2 • (al ® a~). The restriction requirement of (19) and (20)in conjunction with (10)and (11) yields a~ ---- 1. The constant a I is undetermined and we put b = al, thus proving (22) (Q.E.D.). Note that (21) and (22) are same as the eqs (8a) and (8 c) of Grgin and Petersen (1976) respectively and the parameter b is their 'class parameter'. The only difference is that they are applicable for a GTPA instead of a TPA. The algebra ~Aiz is still not a GTPA, since the Jacobi identity (8) and the derivation law (9) remain to be investigated in the tensor product algebra ~4~. The following two theorems give the condition under which ~1~ becomes a GTPA. Theorem 2: The E-Jacobi identityis satisfied in ~Alz if and only if the shift degree of gradation is an even integer, and the following "canonical relation" holds in ~4, (i = 1, 2) (If, g, h]~), = a. (If, g, h]a), = a. [(f~h) ceg. c (np)],, (24) where a e R is a constant called the "canonical constant ". Proof: Let J=J1 + J2 + Js, where •/i = [¢(pm) - (f~(g~h))]lz and Jz and d~ are cyclic permutations in f, g, h and their respective degrees m, n, p. The vanishing of d is the e-Jacobi identity in ~l~. Direct computation of d 1 using (21), (6 a) and (6 b) yields 4 = "o. {" (sma) [(f~ (go~h). ~ (mp)) ~ ~ (ftr (g~rh). E (mp))a + (.fa(g~h). E(mp)~ ® (f~(gah). E(mp))~] Algebraic structure of canonical formalism 555 + [(fa(gah) . E(mp))l ® (fa(gah) . E(mp))z

+ (fa (gah). , (mp))l ® (fa (goth). ~ (mp))~]},

where % = c(mlp2 + Plnz + nlm~). Using the a-derivation property (9) to expand (fa(gah). E(mp))2 in the second term and (f~(gah) . E(mp))l in the third term of the above expression for Jx and writing similar expressions for Jz and J3, or can be reduced to the following form---- J= %.(J' +J").

Here J' ~- J" (fgh) + J' (ghf ) + J' (hfg)

and J" = J" (fgh) + J" (ghf) + J" (hfg), with S' (fgh) = e (sm2) . (fa (gah) . • (mp))x ® (fa (gah) . , (mp))2 + (fa (gah). , (mp))l ® (fa (gah) . , (rnp))2 and J" (fgh) = (E (sm 2 + sp,) -- 1). (fa (gah) . ~ (mp))a ® (ha (fag). , (np))2 + (e(smx) -- e(sm2)). (fa(gah). e(mp))l ® (ga(haf). E(mn))2. (25) The second and the third terms in J' and J" are obtained respectively from J'Oegh) and J" (fgh) by cyclic permutations in f, g, h and their respective degrees m, n, p. Since J' involves terms of the form (. #(. #.))1 ® (. v (. v .))3 (# = a, v = or # = a, v ---- a) and J" involves terms of the form (. a (. a .))1 ® (. a (. a. )),, there can be no cancellation of any term in J' with any term in J" (since a and a are different, the possibility that a ~-- a does not arise at all). This means that if J = 0, then J' = 0 and J" = 0. The latter condition implies that each of the six terms in J" must separately vanish. This can happen only if each of the e-factors containing s [like e (sm2), etc.] is equal to unity. This in turn implies that s must be an even integer, thereby proving that the first part of the thoorem is a necessary condition. Now, using the condition that s is an even integer, J' can be reduced to an expression involving only the associators of ~ and a~ (i ----- 1, 2). This is done in the following way. We expand both the terms in J' (ghf) using (18) for aa and as. This results in eight terms in J'. Then using the symmetry properties of the products ~ and a, (i = 1, 2) of (7) and (23) to rearrange the ordering of f~, g~, h~ in J', we group all the eight terms pair-wise and obtain the following: J' -: ((gah) af . ,(ran)) 1 ® ([g, h,fJa . ,(ran))2 + ([g, h,f],7. , (ran))1 ® ((gah) af. , (mn)) 2 --- (ha (fag). E(np))l ® ([h,f, g]o- . c(np))2 -- ([h,f, g]~, . ~ (np))l ® (ha (fag,). ~ (np))2. 556 Debendranath Sahoo Now, the factors involving two ~ products can be replaced by their respective ~cassociators (i = Is 2) by making use of (7) and (17) and the following e~xpression is obtained : J'= (e(np) . [h,f,g]a)~ ~)(,(mn) . [g, h,f]~)2 + (,(mn). [g,h,f],)x ~ (,(np). [h,f, g],,)2 -- (e(np). [h,f,g]~)l ® (,(ran). [g, h, fla)~ -- (,(mn) . [g, h,f]~,)x @ (,(np) . [h,f, g]cr)z. (26) Thus the Jacobi identity is satisfied in ~4~ if the above expression is identically zero. Interchanging hi and gl in this identity, we obtain a similar one and multiplying throughout by e (rnxnl + niP1 + plml), we obtain,

Y' = ([g,f, h]a . ,(ran + pro))1 ~ ([g, h,f]a . e(mn)), + ([h, g,f]a, e (mn + np))t ® ([h,J; g]~. • (np))~ -- ([g,f, h]¢. ,(mn + pro))1 ® ([g, h,f]a . ,(mn))z -- ([h,g,f]a . e(mp + np))l ® ([h,f, g]~. e(np))z = O. (27)

Now consider the identity J' + J' = 0, where J' is given by (26) and ~ is given by (27). Four of the terms in this new identity cancel because of lemma 2 and we obtain the following: (e (mn) . [g, h,f]~ + e (mn + np) . [h, g,f]~)l ® (" (np) . [h,f, g]o)2 = (e(mn) . [g,h,f]~ + e(mn + np). [h,g,f]a)l t~)(,(np). [h,f,g],)~. (28) This identity clearly implies that the ~ and the a asscciators are proportional, i.e.~

([h,f, g]~), = (a. [h,f, g]a), (29) where al, a2 E R. Substituting (27) in (28) we see that as -----as ---- a (say). Thus we have proved that the canonical relations (24) are necessary for the Jacobi identity to hold in ~412. It is easy to see now that the two conditions stated in the theorem are sufficient to guarantee that the Jacobi identity is satisfied in ~,. This is seen by using the first condition (i.e., s is an even integer) in (25) which implies J"= 0 and the second condition [i.e., (24)] in (26), which implies J'~ 0 (Q.E.D.). So far we have shown that the following conditions must be satisfied if ~4x, is to be a GTPA: (a) the product tr has to be E-commutative [i.e., (23)], (b) the shift degree of gradation, s, has to be an even integer, and (c) the canonical relation (24) has to hold in ~41 and ~ with the same canonical constant a. lit will be shown in lernma 8 that (24) holds also in ~41~ with the same canonical constant a.] In d~riving the following theorem, we shall make use of all the three conditions (a)-(e).) Algebraic structure of canonical formalism 557 Theorem 3: 0qg. is an c-derivation with respect to aa 2 if and only if b = -- a. Proof: The c-derivation law in ~AI~ is the vanishing of the expression D = ((fag) ah) 12 + (e (ran). ga (fah))l ~ -- (fa(gah))22. (30) A straightforward calculation, making use of the composition laws (2D and (22), yields for the first term in D the following: {(fag) ah)2~ = ,,. [((fag) ah)l (~ ((fag) ah)~ + b. ((fag) ah)l ~) ((fag) o~h)~ + ((fag) ah)2 ® ((fag) ah), + b. ((fag) ah)2 ~) ((fah) oth)~ (31) where •2 is given by

• 1 = • (nlm2 + p2m, + pin,). (32) Expanding the second and the fourth terms of the right side of (31) by making use of the •-derivaticn laws in ~,42 and ~A2 [eq. (9)], we obtain ((fag) o-h)~ s - e I . [((fc~g) ah)~ Q ((fag) ah)~ + ((fag) ah)l ~ ((fc, g) ah).,.] + b. e, • [((fag) ah)~ 0 (]o ($at,))~ + ((fag) ah)~ ® ((.fc~h) ag. • (rip))2 + ()ca (gah)) 1 ® ((fag) ~t~)2 + ((fah) ag. , (np)}l @ ((fczg) ah) 2]. (33) Proceeding in similar steps, we obtain for the second term in D, the following: [•(mn). (ga(fah))]l~--- el. tmlnl + m2n~) • [(g ,7 (fah))~ ~ (g~r (fah))~ (ga (fch))a Q (gcr (fah)) 2 + b (ga (fah))l ¢~ ((gaf) ah)2 + b (ga (fah)) 2 @ (re (goth))2 + b ((gaf) ah)~ @ (ga (fah))z + b (fa (gah'~)~ ~; (ga (f~th))3]. (34) The third term in D, after using (21) and (22) yields (f~ (gvhh)ls = -- •2. [(fa tgah)h ® (f~ (g~h)h + (f~ tgoh))~ ® (fa(g~h)),] + b. ,2. [(f~ (f~h)h ® (f~ (gah~)~ + (@ tg~h)h ® (fa (g~h))d. Using the ~-derivation law (9) for the first two terms and (18) for the last two t~rms, we obtain,

(fa (gah))~s : "2. [((fag)ah), ® (fa (gah))~ + (g~(f~h). • (,nn)), ® (f,~ ~g~h)), + (fo (gah))~ ~ ((fag) ah) + (fa(gah))~ ~ (ga(fixh). • (ran)) 2 558 Debendranath Sahoo

+ b. ,j . [((fxg) :oh), @ (fa (g~h))2 + (ga (fah). , finn))1 ® (fa (gah)) 2 + ()ca (gab))1 ~ ((fag) ah)2 + (fa ~gah))l ® (ga (fah). • (mn))d. (35) Now substituting the expressions (33), (34) and (35)in (30), and grouping the terms so as to form the a-associators, we obtain

D = ,, . [((fag) ~h)~ @ (If, g, h]a)2 + ([f, g, h]a), @ ((f~g) ah), -- ([g, h,f]~ . , (ran + pm))~ @ (ga (fah) . • (rnn))2 -- (ge (fah). E (mn))~ Q ([g, h,f],r. • (ran + pm))~]

+ b • ,,. [((f~g) ah)l @ (,f~h)ag. , (np))2 + ((fah)ag. , (np))l @ ((fag) ah)2 + (ga(fah). ~ (mn)) 1 G ((gaf) ah. • (mn))2 + ((gaf) ah. ~ (ran))1 ® (ga (fah). E (mn))s]. Now, making use of the canonical relation (24), and reordering the elements in some of the terms, [by using the symmetr~ properties (7) and (23)], we obtain D = "1 • (a + 6) . [((fag)ah)~ ® ((f~h) ag. • (np)),. + ((fah) ag., (~p))l ® ((fag) ~ hh + (ga(fah) . ~ (mn)) 1 G ((fag) ah)2 + ((fag) ah)l ® (g~ Cfah). ~ Cmn))d. Clearly D ~-- 0 if and only if b ------a (Q.E.D.). It now remains to show that the canonical :elation doe~ not induce any further identity in a GTPA. This is shown in the following lemma. Lemma 8: The canonical relation (24) holds in ~ls. Proof: Using the composition laws for a and a [eqs (21) and (22)], and putting b------a (theorem 3), we obtain for the a-assoeiator of three elements fl~, gl~, his ~ ~ls, the following: (If, g, h],)ts = (~fag) ah)~2 -- ~fo- (gah))l 2 • (nxms). [(fag)x ® (fag)2 -- a. (fag)l (~ (fag)~] alz (his) -- E (pin,) . (flz) ais [(gah)l ~)(gah)s -- a . (gah)l ® (gah)z] -- ,1. [((fag) ~h), ® ((fag) ~h)~ - a . ((fag) ahh ® ((f~g) ~h), -- (fa (gah))~ ® (fa (gah))2 + a. (fi tgah))a ® (fa (gah))s -- a . ((fag) ah) l ® ((fag) ah))~ + a s . (¢fag) ah)l ® ((fag) ah), + a. (fa (gab)) 1 (~ (fa (gah)),-- a s . (fa (gah)) x ® (fa (gah)) d, where e~ is given by (32). Algebraic structure of canonical formalism 559

Using the derivation law (9) in the second and the fourth terms and the canonical relation (24) (for J1 and ~A2) in the sixth and the eighth terms, we obtain (if, g, h]~)12

+ e 1 . [((fag) al;), @ (gaOeah) . , (mn))2 + (go(f~h~. E (mn)h ® ((J~g) ,~h)~ -- ((fah) ag . E (np))1 @ (fa(g~rh))z -- (fir (gtrh))l ® ((fah) ~rg . ,(np)) 2]

+ a. c 1 [((fag) ah)l @ (ga(foth) . c (ran)) 2 + (go (f~,h)., (mn))~ ® ((f~,g) oh)~ -- (fcr(go~h))l ® ((foth) trg . e (np))z -- ((foth) ag. , (np))~ ® (fa (g~h))2]. (36) We now calculate the right side of the canonical relation in a similar manner and obtain

a [(f~h) c~g . c (np)]l., --. a . (tqp 1 + n:p. + ;in, + Plm.. + nxm~) • [(fcd(h ® (fah)._, + (fah)~ @ (f0h)2] ~12 gx2 :- ¢~ . a. [(Ifuh) ~g. , (,p), @ ((fah) crg. E (np))~ + Off, h) ag., ~,p))~ ® ((f~h) ~g., ~,~p))o

+ ((fah) ag. , (np))l ® ((fah) trg. , (np))., + ((f~rh~ crg . , (rip))1 ® (If~h~: o~g. ~ (np])~2. Now, using the canonical relation for ~A1 and 043 [Ae., (24)] in the first and the fourth terms and the c-derivation law (9) in the second and the third terms of the last expression, we obtain an expression identical to that of the right side of (36). This proves that ([J;g, h]~)~.2 = a. (Ifoh) ctg . , (,7p))~2 (Q.I~.D..

4. Summary and discussion To summarise, a composable GTPA ~A ----- {H:, a, ~, ~} defined on the real field R has the following structure: (i) c¢ : ,7/'® He" ~ H: is a E-anticommutative product: fag -~ -- , (ran). (gcef). (ii) ~ fulfils the E-Jacobi identity , (mp] . (f~ (g~h)) + , (mn). (go~ (h~f)) + , (pn). (h c¢ (fag)) = O. (i) and (ii) together mean that {g:, c¢, ,} is a g aded Lie algebra. (iii) a : ¢7:® H:~ g¢" is a c-commutative product fag = , (mn) . gaf 560 Debendranath Sahoo (iv) a is a •-derivative with respect to 0.: fe (go.h) -~ (fc~g) 0.h + 4 (mn) . (go. (fc~h)). (v) The canonical relation holds: [f, g, h]~ = a. [f, g, hi, = a. [4 (np) . (feh) eg]. The composition properties of the e and the 0. products under the e-commutative tensor product are: and

al 2 = 0.1 (~ a2 -- a . cq @ cq. Definition: A GTPA with the above properties will be called as a graded Hamilton algebra (GHA). For a = 0, it will be called as a classical GHA and for aS 0, a quantum GHA. Whether a GHA (classical or q'~ntum) is of the Bose or of the Fermi type is dictated by the e-factor used in its definition. For a classical GHA, tl:e substructure {~r, 0., 4} is graded 4-commutative as well as associative by virtue of (23) arid (24). In the case of a quantum GHA, the substructure {,7/', 0., 4} is a graded Jordan algebra. This is seen by putting h =f0.f in eq. (24):

[f, g, faf]a := a . • (2ran) . (fe (faf)) otg = a. [ff~./) af+fa(faf)] ag ~0. (37) The second line follows from the e-derivation law (9) and the last line, from the 4-anticommutativity of a (7). In a quantum GHA of the Bose type, the substructure {c-~, a, 4 = 1} is a noncommutative Jordan algebra, i.e., in addition to the Jordan identity (37), the following 'flexibility law' also holds. [.f, g, f]~ -= O. As cart be easily seen from (24), the flexibility law does not hold in the case of a quantum algebra of the Fermi type. The Hamilton algebra of Grgin and Petersen (1974) is the algebra relevant to •the canonical formalism of the bosons. By irttroducing the gradation structure, and two types of factors of commutation, it has been possible to make the graded Hamilton algebla relevant to bosons (with a Bose type factor of commutation) as well as fermions (with a Fermi type factor of commutation). Comparing our axiom with that of Grgin and Peterscn (1976) (i.e., 2 a-c), it can be seen that the former is equivalent to the closure property of the o-product in the composition class c5. Our requiring (19) and (20) is equivalent to their assumption of the unit element in cj [i.e., (2 c)]. However, we have not made use of the associativity property of the o-product [i.e., (2 b)]. In fact, one can trivially derive the assoeiati- vity of the composition laws, Le.,

~{19.}3 ~ ~i{23} and o'(12) 3 ~ o"i(23 ) Algebraic structure of canonical formalism 561

(when one considers three GHA's) simply by making direct use of (21) and (22). A GHA of the Fermi type with a ---- 0 is the algebra relevant to anticommutative classical mechanics. This algebra was first introduced by Martin (1959) in connec- tion with the Feynman quantisation of the Fermi fields; he named it a "dynamics algebra " A GHA of the Fermi type with a ~: 0 is new, and to the best of our know- ledge, has not been proposed in the literature hitherto. As already stated, the canonical formalisms of bosons and fermions have the same algebraic structttre, except for the type of the E-factor. In spite of close structural similarity, there is one important difference between the two cases. A GHA of the Fermi type (classical or quantum) with finite number of generators z~' (p = 1, 2, ..., N) and which satisfies the "' c-symplectic condition" z~ ~ z ~ ___ g~'~ __-- g~t~ (g is a constant matrix) is necessarily of finite dimension (= 2s). On the other hand, a GHA of the Bose type (classical or quantum) with a finite number of generators z~(p----- 1, 2 .... N) which satisfies the usual symplectic condition z~ ~ z ~ = c~ = -- ~ (E is a constant matrix) is always of infinite dimension. In this respect, a Fermi type GHA is simpler than a Bose type GHA. There is another difference between the two types of GHA's and this concerns the applica- bility of the duality axiom of Grgin and Petersen (1974). In view of the fundamental significance of the duality axiom in the theory of measurement, a point which has been highly emphasised by Grgin and Petersen (1974), one would think that the same axiom should be applicable to a Fermi type of GHA. In parti- cular, starting with a GTPA one should be able to recover the full algebraic structure of a Fermi type of GHA by following the same line of arguments as those of Grgin and Petersen (1974). However, this does not seem to be pc)ssible because the identity

If, h~g, g],r + [h, fo~g, g]~ = 0

[eq. (11) of their paper] does not hold in a Fermi type GHA. A counter example invalidating the above identity is provided by the case when J; g and h are all homogeneous elements of odd degree. Because of this, the lemma following eq. (11) in the above-mentioned reference is not true for a Fermi type of GHA. It seems, therefore, that their result cannot be generalised to fermions using the duality axiom alone. We have, on the other hand, done so with the help of the composition axiom. The link between the present algebraic approach and the conventional approach to mechanics is already indicated in the work of Grgin and Petersen (1976). T~ey have observed that the quantum GHA ~ ----- {,Tf, a, ~, ~ -~ 1} (of the Bose type can be embedded in its associative envelope ~,~ = {~, t, ~} by defining an associative (but noncommutative) product fl ----- a + ~/-- a and this necessitates extension of the bas~ field to the complex number field C. The linear space ,~c is the complex extension of the space ,7/'. As noted by these authors, the constant a is proportional to the square of the quantum of action. This construction is also valid in the case of a Fermi type of GHA. In the case of a classical GHA the product a is already associative and hence the analogous construction is superfluous, though

P--9 562 Debendranath Sahoo

not meaningless. In the case of a quantum GHA, the product fl is an associative (but noncommutative) graded product, and the products tr and a are deri- vable from fl by respectively taking the e-anticommutator and the r-commutator: fag = ½ (fflg + ~ (mn) . gflf), and fctg ----- [1/(2x/-- a)]. (fflg -- 8 (mn) . gflf). The gradation structure of our algebra is reminiscent of the Fock space construction. In fact, we have been motivated to generalise the Hamilton algebra of Grgin and Petersen (1974) by enquiring whether a Fock type constiuction can be achieved in the space of the observables rather than the space of the states. This has lesulted in the concept of a GHA. However, what we have done is not the Fock space construction in the Heisenberg picture. The application aspects of the present formalism are too many and will be dealt with in future papers. However, one important application of the present approach is worth pointing out here. It has been shown by Grgin and Petersen (1976) that the compasition laws (21) and (22) for the ~ and the a products provide the basis for the structural relation between classical and quantum mechanics. Such a relation is manifest in the phase space formulation of quantum mechanics (Groenewold 1946, Moyal 1949) where one knows that the Lie product (the Moyal bracket) and the Jordan product (the cosine bracket) are the sine and the cosine "functiens " of the classical Lie product (the Poisson bracket). This structural relation shows how a classical GHA is obtained from a quantum GHA by taking th~ limit a ~ 0. Thus the present approach gives a deeper insight into the Bohr correspondence principle.

Acknowledgements

I am grateful to A Petersen for his comments on an earlier version of the manu- scIipt, and for sending me a preprint of his paper. I am thankful to G Anantha- krishna, V Balakrishnan and to one of the referees for suggesting improvements in the presentation.

References Bourbaki N 1964 Algebre (Paris, Hermann), ch. III, p. 44 Greub W 1967 Maltilinear Algebra (New York, Springer Verlag) Grgin E and Petersen A 1974 J. Math. Phys. 15 764 Orgin E and Petersen A 1976 Algebraic Implications of Composability of Physical Systems, Yeshiva University preprint Oroenewold H J 1946 Physica 12 405 Martin J L 1959 Prec. R. Soc. (London) 2,51 536 Moyal J E 1949 Prec. Camb. Phil. Soc. 45 99