International Journal of Theoretical Physics, Vol. 40, No. 1, 2001
Principal Loop Bundles: Toward Nonassociative Gauge Theories Alexander I. Nesterov1 Received November 15, 1999; revised December 12, 1999 We introduce a nonassociative gauge field theory with nonassociative symmetries. The approach is based on the nonassociative generalization of principal bundles theory.
1. PRELIMINARIES Difficulties in unifying all interactions prompt us to look for a mathematical structure beyond groups. The quantum group approach is a construction of this type where the Lie group symmetry is replaced by a quantum group symmetry and the latter reduces to the standard one in some limit (–Durd-evich, 1996, 1997, 2000). Another possibility is a nonassociative generalization of a Lie group, such as quasigroups and smooth loops. We developed gauge theories based on a nonassociative generalization of principal bundles theory (Nesterov, 1989, 1999, 2000; Nesterov and Stepanenko, 1986). The algebraic theory of quasigroups and loops may be found in Belousov (1967), Bruck (1971), Pflugfelder (1990), Chein et al. (1990), and Sabinin (1999). Let 具Q, ⭈典 be a groupoid with a binary operation (a, b) 哫 a ⭈ b. A groupoid 具Q, ⭈典 is called a quasigroup if equations a ⭈ x ⫽ b and y ⭈ a ⫽ b have unique solutions: x ⫽ a \ b, y ⫽ b/a. A loop is a quasigroup with a twosided identity a ⭈ e ⫽ e ⭈ a ⫽ a, ∀a 苸 Q. A loop 具Q, ⭈, e典 that is also a differential manifold and an operation (a, b) :⫽ a ⭈ b is a smooth map is called a smooth loop. We define 1
Departmento de Fı´sica, CUCE I, Universidad de Guadalajara, Blvd. M. Garcı´a Barraga´n y Calz. Olı´mpica, Guadalajara, Jalisco, C.P. 44460, Me´xico and L.V. Kirensky Institute of Physics, Siberian Branch Russian Academy of Sciences, Akademgorodok, 6600 36, Krasnoyarsk, Russia; e-mail:
[email protected] 339 0020-7748/01/0100-0339$19.50/0 䉷 2001 Plenum Publishing Corporation
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l(a,b) ⫽ L⫺1 a⭈b ⴰ La ⴰ Lb ,
Lab ⫽ Rba ⫽ a ⭈ b,
ˆl(a,b) ⫽ La⭈b ⴰ l(a,b) ⴰ L⫺1 a⭈b (1)
where La is a left translation, Rb is a right translation, l(a,b) is a left associator, and ˆl(a,b) is an adjoint associator, ˆl(a,b) ⫽ La ⴰ Lb ⴰ L⫺1 a⭈b The operation a ⭈ b need not be associative, a ⭈ (b ⭈ c) ⫽ (a ⭈ b) ⭈ c, but there exists quasiassociativity: (a) The left identity of quasiassociativity a ⭈ (b ⭈ c) ⫽ (a ⭈ b) ⭈ l(a,b)c where l(a,b) is the (left) associator defined above. (b) The right identity of quasiassociativity r(b,c) a ⭈ (b ⭈ c) ⫽ (a ⭈ b) ⭈ c where r(b,c) ⫽ R⫺1 (b,c) ⴰ Rc ⴰ Rb is the right associator. Let Te(Q) be the tangent space of Q at the neutral element e. Then for each Xe 苸 Te(Q), we construct a smooth vector field on Q Xb ⫽ Lb*Xe ,
b 苸 Q, Xe 苸 Te(Q), Xb 苸 Tb(Q)
where Lb*: Te(Q) 哫 Tb(Q) denotes the differential of the left translation. Note that Xb satisfies La*Xb ⫽ ˆl(a,b)*Xa⭈b,
∀a, b 苸 Q
Definition 1.1. A vector field X on Q which satisfies the relation La*Xb ⫽ ˆl(a,b)*Xa⭈b for any a, b 苸 Q is called a left fundamental or left quasiinvariant vector field. In the view of the noncommutativity of the right and left translations, ⫺1 ˜ La ⴰ Rb ⫽ Rb ⴰ La , the definitions Adgg⬘ ⫽ Lg ⴰ R⫺1 g g⬘ and Adgg⬘ ⫽ Rg ⴰ Lgg⬘ are not equivalent. We define a generalized adjoint map of Q on itself in the following way. Definition 1.2. A map ⫺1 Adb(a) ⫽ L⫺1 Q哫Q a ⴰ Rb ⴰ La⭈b:
is called an Ad-map. Remark 1.1. Adb(e) ⫽ R⫺1 b ⴰ Lb. The Ad-map (2) generates the map
(2)
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⫺1 Adb(a) :⫽ (Adb(a))* ⫽ L⫺1 Te(Q) 哫 Te(Q) a* Rb* La⭈b*:
Definition 1.3. A vector-valued 1-form is said to be canonical Adform if is defined ∀a 苸 Q through the relation (Va) ⫽ Ve, Ve ⫽ L⫺1 Ve 苸 Te(Q), Va 苸 Ta(Q) a* Va, and V 苸 T(Q) is a left fundamental vector field. Theorem 1.1 (Nesterov, 1989, 1999). The canonical form is a left fundamental form and it is transformed under left (right) translations as (L*b )Va ⫽ l(b,a)*(Va),
(R*b )Va ⫽ Ad⫺1 b (a)(Va)
2. PRINCIPAL LOOP BUNDLES Let M be a manifold and 具Q, ⭈, e典 a smooth two-sided loop. A principal loop Q-bundle (a principal bundle with the structure loop Q) is a triple (P, , M ), where P is a manifold and the following conditions hold: Q acts freely on P by the right map: ( p, a) 苸 P ⫻ Q 哫 R˜a p ⬅ pa 苸 P. 2. M is the quotient space of P by the equivalence relation induced by Q, M ⫽ P/Q, and the canonical projection : P → M is a smooth map onto. 3. P is locally trivial, that is, for each x 苸 M, there exists an open neighborhood U and a diffeomorphism ⌽: ⫺1(U ) 哫 U ⫻ Q such that for any point u 苸 ⫺1(U ), it has the form ⌽(u) ⫽ ((u), (u)), where is the map from ⫺1(U ) to Q satisfying (R˜a p) ⫽ Ra( p).
1.
We say P is a total space or bundle space, Q is a structure loop, M is a base or base space, and is a projection. For any x 苸 M, the inverse image ⫺1(x) is the Fx fiber over x. Any fiber is diffeomorphic to Q and the loop Q acts transitively on each fiber, satisfying the right identity quasiassociativity R˜a⭈b p ⫽ R˜a ⴰ R˜br˜(a, b)p,
(r˜(a,b) p) ⫽ r(a,b)( p)
Let us consider a covering {U␣} of M, which can be chosen in such a way that the restriction of the fibration to each open set U␣ is trivializable. This implies that there exists a diffeomorphism ⌽a: ⫺1(U␣) 哫 U␣ ⫻ Q. The set {U␣, ⌽␣} is called a local trivialization. Proposition 2.1 (Nesterov, 1999). The right map on the fiber does not ⫺1 ˜ ˜ depend on the trivialization: ⌽⫺1 ␣ (( p), ␣(Rq p)) ⫽ ⌽ (( p), (Rq p)).
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⫺1 Definition 2.1. The family of maps {q␣( p) ⫽ R⫺1 q␣ q: (U␣ 艚 U) 哫 ⫺1 Q}, where q␣ :⫽ ␣( p), q :⫽ ( p), p 苸 (U␣ 艚 U), is called the family of transition functions of the bundle P(M, Q) corresponding to the covering {U␣} on M.
Proposition 2.2. Transition functions q␣( p) change under right translations as Raq␣( p) ⫽ r(a, q␣)q␣( p) where r(a, q␣) is the right associator. Definition 2.2. Let {U␣, ⌽␣} be the local trivialization and e 苸 Q a neutral element; then ␣ ⫽ ⌽⫺1 ␣ (x, e), x 苸 U␣, is called the local section associated with this trivialization. Let u 苸 P; then one can describe the local section as follows: ␣ ⫽ R˜⫺1 q␣(u)u, where by definition, q␣(u) ⫽ ␣(u). Proposition 2.3. The section ␣ does not depend on the choice of the point in the fiber. Proof. Let u, p 苸 P and u ⫽ R˜q␣(u)␣, p ⫽ R˜q␣( p)⬘␣. There exists an element a 苸 Q such that p ⫽ R˜au. Then we have ⌽␣(R˜q␣( p)⬘␣) ⫽ ⌽␣( p) ⫽ ⌽␣(R˜au) ⫽ ⌽␣(R˜a ⴰ R˜q␣(u)␣) ⫽ (( p), Ra ⴰ Rq␣(u)␣(␣)) ⫽ (( p), Ra ⴰ Rq␣(u)e) ⫽ ⌽␣(R˜q␣(u)⭈a␣) Taking into account q␣( p) ⫽ q␣(u) ⭈ a, we obtain Rq␣( p)⬘␣ ⫽ Rq␣( p)␣. This gives ␣ ⫽ ⬘␣ and, hence, the section ␣ does not depend on the point of the fiber, while the dependence on the base, ␣ ⫽ ␣(x), holds. 䡲 Proposition 2.4. Let the point x lie in the intersection of the neighborhoods U␣ and U, x 苸 U␣ 艚 U; then the following formula holds: ␣(x) ⫽ R˜q␣(x)
(3)
Proof. Let u ⫽ R˜q␣␣; rewriting this formula in the chart U, we obtain ⌽(R˜q␣␣) ⫽ ⌽(R˜q) ⫽ ((u), Rq()) ⫽ ((u), Rq␣⭈q␣()) ⫽ ((u), Rq␣ ⴰ Rq␣e) ⫽ ((u), Rq␣ ⴰ Rq␣()) ⫽ ⌽(R˜q␣ ⴰ R˜q␣) where the relation () ⫽ e has been used. Finally, we obtain ␣(x) ⫽ R˜q␣(x) in the intersection U␣ 艚 U. 䡲
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Example 2.1. Principal QU(1) bundle over S 2. We define a smooth loop QU(1) as a loop of multiplication by unimodular complex numbers, with elements ei␣, 0 ⱕ ␣ ⬍ 2, and operation ei␣ ∗ ei ⫽ ei(␣⫹˙ )
(4)
˙  ⫽ ␣ ⫹  ⫹ F(␣, ), and F(␣, ) is a smooth function such where ␣ ⫹ that F(␣, ) ⫽ F(, ␣), F(␣, 0) ⫽ F(0, ) ⫽ 0. Keeping in mind only the nonassociative case of the operation (4), we assume further that F(␣, ) ⫹ ˙ , ␥) ⫽ F(, ␥) ⫹ F(␣,  ⫹ ˙ ␥). F(␣ ⫹ We construct the bundle by taking Base M ⫽ S 2
with coordinates 0 ⱕ ⬍ , 0 ⱕ ⬍ 2
Fiber QU(1) ⫽ S1
with coordinates ei␣
We break S 2 into two hemispheres H⫾ with H+ 艚 H⫺ being a thin strip parametrized by the equatorial angle . Locally the bundle looks like H⫺ ⫻ QU(1)
with coordinates (, , ei␣⫺)
H+ ⫻ QU(1)
with coordinates (, , ei␣⫹)
In ⫺1 (H⫺ 艚 H+) the elements ei␣⫹ and ei␣⫺ must be related by the transition function ei␥: ei␣⫹ ⫽ ei␥ ∗ ei␣⫺ ⫽ ei(␥⫹˙ ␣⫺) This implies ␣+ ⫽ ␣⫺ ⫹ ␥ ⫹ F(␥, ␣⫺) Taking into account that the resulting structure must be a manifold, we find that ␥ ⫹ F(␥, ␣⫺) ⫽ n
(5)
where n must be integer. Comment 2.1. The structure of the obtained manifold P is unknown. 3. CONNECTION, CURVATURE, AND BIANCHI IDENTITIES Let P(M, Q) be the principal loop Q-bundle over the manifold M. For any u 苸 P a tangent space at u, we denote as Tu(P) (or simply Tu) and the tangent to the fiber passing through u as ᐂu. We call ᐂu a vertical subspace. It is generated by the right translations on the fiber: u 哫 R˜au, a 苸 Q, u 苸 P: Xu ⫽
d ˜ Ra(t)u앚t⫽0, dt
Xu 苸 ᐂu , a 苸 Q, u 苸 P
(6)
The basic idea of connection is to compare the points in the “neighboring”
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fibers in a way that is not dependent on a local trivialization. Let ␥(t) 苸 M be a smooth curve. A horizontal lift of ␥ is a curve ␥˜ (t) 苸 P such that (␥˜ (t)) ⫽ ␥(t). Evidently, for determining ␥˜ (t), it is suffucient to define at any point of it a tangent vector X˜: *X˜ ⫽ X, where X is the tangent vector to ␥(t). A set {X˜} is called a horizontal subspace Ᏼu. Definition 3.1. A connection form on a principal Q-bundle is a vectorvalued 1-form taking values at Te(Q), which satisfies: (i) (Xp) ⫽ Xe , where Xp 苸 ᐂp , Xe 苸 Te(Q) are determined according to (7), (8). (ii) (R˜*a ) Xp ⫽ Ad⫺1 a (q)(Xp), where q ⫽ ( p). (iii) The horizontal subspace Ᏼp is defined as a kernel of : Ᏼp ⫽ {Xp 苸 Tp(P): (Xp) ⫽ 0} This definition implies that a connection in Q-bundle is determined by the left canonical Ad-form. The connection allows us to decompose any vector Z 苸 Tp(P) in the form Z ⫽ X ⫹ Y, where X ⫽ horZ 苸 Ᏼp is the horizontal component of the vector Z and Y ⫽ verZ 苸 ᐂp is the vertical one. The map induces the map of the vertical subspace ᐂp onto the tangent * space to Q, ᐂp → Tq(Q), q ⫽ ( p): Vp ⫽
* d ˜ d Ra(t) p앚t⫽0 → Vˆq ⫽ Ra(t)q앚t⫽0 dt dt
(7)
Note that the vector field Vˆq is left quasiinvariant. Indeed, it can be written as Vˆq ⫽ (Lq)*Ve , where Ve ⫽
冟
da(t) 苸 Te(Q) dt t⫽0
(8)
Definition 3.2. Let Ve 苸 Te(Q). The vector field Vp 苸 ᐂp connecting with Ve by means of (7), (8) is called a fundamental vector field. With the given connection form, a local 1-form taking values in Te(Q) can be associated as follows. Let : U 傺 M 哫 (U ) 傺 P, ⴰ ⫽ id, be a local section of a Q-bundle Q 哫 P 哫 M which is equipped with a connection 1-form . Define the local -representative of to be the vector-valued 1form [taking values a Te(Q)] U on the open set U 傺 M given by U :⫽ *. Theorem 3.1. (On reconstruction of the connection form). For a given canonical 1-form ˜ defined on U 傺 M with values in Te(Q) and the section : U 哫 ⫺1(U ), there exists one and only one connection 1-form on ⫺1(U ) such that * ⫽ ˜ .
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Proof. Let p0 ⫽ (x) and Z 苸 Tp0(P). We have Z ⫽ X1 ⫹ X2, where X1 :⫽ (* ⴰ *)Z and X2 苸 ᐂp0, *X2 ⫽ 0. Define at p0 the 1-form to be the vector-valued 1-form given by p0 ⫽ ˜ x(*X ) ⫹ Xˆ2. Continuation of the 1-form onto all points of the fiber is realized by means of right translations, namely, ∀p 苸 P, ∃a 苸 Q: p ⫽ R˜ap0. This implies p((R˜a)*X ) ⫽ Ad⫺1 a (q0)p0(X ),
q0 ⫽ ( p0)
It is easy to see that the obtained 1-form satisfies all conditions of Definition 3.1. 䡲 Let {U␣, ⌽␣} be a local trivialization and let Id: U␣ 哫 U␣ ⫻ Q by x 哫 (x, e). A trivialization ⌽␣ defines a canonical section ␣ by the equation ␣ ⫽ ⌽⫺1 ␣ ⴰ Id and vice versa. We denote by C a canonical Ad-form. Definition 3.3. Let ␣ ⫽ *␣ , where is the connection form. The form ␣ on U␣ is called the connection form in the local trivialization {U␣, ⌽␣}. Theorem 3.2. At U␣ 艚 U, local connection forms ␣ and , corresponding to the same connection on P, are related by  ⫽ Ad⫺1 q␣(q␣)␣ ⫹ l(q␣,q␣)*␣
(9)
where q␣ are transition functions, and ␣ ⫽ q*␣C denotes the pullback on U␣ 艚 U of the canonical 1-form C on Q. Vice versa, for any set of the local forms {␣} satisfying (9), there exists the unique connection form on P generating this family of the local forms, namely, ␣ ⫽ *␣ , ∀␣. Proof. 1. The direct theorem. Let x 苸 U␣ 艚 U. Applying (3), we obtain (x) ⫽ R˜q␣␣(x), ∀x 苸 U␣ 艚 U. The map ((x))* transforms any vector X 苸 Tx(U␣ 艚 U) into ()*X 苸 T(x)(P). Using the Leibniz formula (Kobayashi and Nomizu, 1963, Ch. I), we get ()* ⫽ (R˜q␣)*(␣)* ⫹ (Lq␣)*(q␣)* where q␣ :⫽ (␣). Applying to both sides of this relation, we find (X ) :⫽ (()*X ) ⫽ ((R˜q␣)*(␣)*X ) ⫹ ((Lq␣)*(q␣)*X ) ⫽ Ad⫺1 q␣(q␣)␣(X ) ⫹ l(q␣,q␣(*(q* ␣C)(X ) ⫽ Ad⫺1 q␣(q␣)␣(X ) ⫹ l(q␣,q␣)*␣C(X ) 2. The inverse theorem. Let us define the 1-form ˜ as follows:
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˜ ⫽ Ad⫺1 q␣ (e)(*␣) ⫹ q* ␣ C ,
q␣ :⫽ ␣( p)
(10)
Let X 苸 Tu(P) be an arbitrary vector and u ⫽ ␣((x)). Decompose X into horizontal Y and vertical Z components: X ⫽ Y ⫹ Z,
Y ⫽ (␣)*(*X ),
*Z ⫽ 0
This implies ˜ (X ) ⫽ Ad⫺1 q␣ (e)␣(*X ) ⫹ (q* ␣ C)(X ) ⫽ Ad⫺1 q␣ (e)((␣)**X ) ⫹ C((q␣)*X ) ⫽(R˜*q␣*␣ )(*Y ) ⫹ C((q␣)*Z ) ⫽ ((␣)**Y ) ⫹ C((q␣)*Z ) ⫽ (Y ) ⫹ Zˆ ⫽ (Y ) ⫹ (Z ) ⫽ (X ) and one sees that ˜ ⫽ at any point of the section ␣. So these forms are transformed in the same way under the right translations and therefore coincide on ⫺1(U ). 䡲 Corollary 3.1. For arbitrary sections 1 and 2 such that 2 ⫽ Rq1 and 1 ⫽ *1 , 2 ⫽ *2 , the following relation holds: 2 ⫽ Ad⫺1 q (q1)1 ⫹ l(q1,q))*(q*C) where q1 :⫽ (1). Remark 3.1. The local form * is called a gauge potential 1-form in the physics literature. 3.1. Covariant Derivative. Curvature Form Let {x, yi} be a local coordinate system in the neighborhood ⫺1(U␣); x are coordinates in U␣ 苸 M and yi are coordinates in the fiber. Locally ⫺1(U␣) can be presented as a direct production U␣ ⫻ Q. The connection form can written in the form ⫽ iLi , with {Li} being the basis of left fundamental fields and {i} the basis of 1-forms. Taking into account (10), we find that in the coordinates {x, yi}, i j i j i ⫽ (Ad⫺1 y (e))jA(x) dx ⫹ j( y) dy
(11)
i j where Ai(x) dx ⫽ *(i) and ij( y) dy j ⫽ (L⫺1 * )j dy .
Definition 3.4. A covariant derivative D in the principal Q-bundle is defined as follows:
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D ⫽ ⭸ ⫺ Ai(x)Li where the Li ⫽ (R*) ji⭸/⭸y j are generators of the left translations (right quasiinvariant vector fields). Comment 3.1. It is easy to show that (D) ⫽ 0. Indeed, (11) implies i j i j p (D) ⫽ [(Ad⫺1 y (e))jA(x) ⫺ pA(R*)j ]Li i i p ⫺1 i Noting that ip ⫽ (L⫺1 * )p, we obtain p(R*)j ⫽ (Ady (e))j, and hence (D) ⫽ 0. Let us compute the commutator [D, D]:
[D, D] ⫽ (⭸Ai ⫺ ⭸Ai)Li ⫹ AiAj[Li , Lj] Introducing [Li , Lj] ⫽ Cpij( y)Lp, we get [D, D] ⫽ ⫺F iLi ,
F i :⫽ ⭸Ai ⫺ ⭸Ai ⫺ AjApCijp
Definition 3.5. Let ⌿ be a vector-valued r-form in the principal Qbundle. An (r ⫹ 1)-form D⌿ defined by D⌿(X1, X2, . . . , Xr⫹1) ⫽ d⌿(horX1, . . . , horXr⫹1) is called a covariant differential of the form ⌿. Definition 3.6. A vector-valued 2-form ⍀(X, Y ) defined as ⍀(X, Y ) ⫽ D(X, Y ) ⫽ d(horX, horY ) where is a connection form, is called a curvature form. Lemma. 3.1. Let X, Y be horizontal fields; then the following relation holds: ([X, Y ]) ⫽ ⫺2⍀(X, Y ) Proof. Applying the exterior differentiation to 1-form , we obtain 2d(X, Y ) ⫽ X(Y ) ⫺ Y(X ) ⫺ ([X, Y ]) As X, Y 苸 Ᏼp , then (X ) ⫽ (Y ) ⫽ 0. This implies ([X, Y ]) ⫽ ⫺2⍀(X, Y ). 䡲 Corollary 3.2. The curvature form can be defined as ⍀(X, Y ) ⫽ ⫺–21 ([horY, horY]) where X, Y are any continuations of the vectors X, Y 苸 Tp(P), respectively. Corollary 3.3. The two-form ⍀ is an Ad-form and is transformed under the right translations as
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(R˜*a ⍀)(X, Y ) ⫽ Ad⫺1 a (q)⍀앚p(X, Y ),
where q ⫽ ( p)
Theorem 3.3. The curvature form ⍀ satisfies the structure equation ⍀ ⫽ d ⫹ ∧
(12)
Proof. To prove (12), one needs to consider all possible pairs X, Y. The same idea is used in the usual fiber bundle theory for proving that the curvature satisfies the structure equation (Kobayashi and Nomizu, 1963). 䡲 Comment 3.2. Choosing the family of the local sections ␣ associated with the trivialization U␣, ⌽␣, and taking into account that ␣(␣) ⫽ e, where ␣ is the restriction of ⌽␣ on ⫺1(U␣), we obtain *⍀앚␣ ⫽
1 F dx ∧ dx 2
where F ⫽ F iLˆj is introduced. Since ⍀ is an Ad-form, the following transformation law holds: ⍀앚 ⫽ Ad⫺1 q␣(q␣)⍀앚␣ Theorem 3.4. Bianchi identity: D⍀ ⫽ 0. Proof. It is sufficient to show that d⍀(X, Y, Z ) ⫽ 0 if X, Y, Z are horizontal vector fields. Applying the exterior derivative to (12), we obtain d⍀(X, Y, Z ) ⫽ 0 if X, Y, Z are horizontal vector fields. 䡲 4. NONASSOCIATIVE GAUGE THEORY ON A PRINCIPAL LOOP Q-BUNDLE Field theory is formulated as living not on spacetime M, but on the principal loop Q-bundle P. Choosing a natural coordinate system (x, q) on P, where x 苸 M and q 苸 Q, the Lagrangian ᏸg of the free gauge field is as follows: ᏸg ⫽ ⫺–41 具F, F 典 where 具 , 典 denotes the Ad-invariant scalar product on Q. ˜ (x, q) Fields are functions on P. With ⌿(x) 苸 M, we relate a function ⌿ [lift of the function ⌿(x)] on the principal Q-bundle in the following way. ˜ (x, q) transforms under a finite gauge transformation U(q) We assume that ⌿ of nonassociative representations of the loop Q (Nesterov and Stepanenko, 1986; Nesterov, 1989) according to the inverse rule ˜ (x, q⬘ ⭈ q) ⫽ U(q⫺1)⌿ ˜ (x, q⬘), ⌿ q苸Q ˜ (x, q) of the function ⌿(x, q) is defined as follows: Then the lift ⌿
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˜ (x, q) ⫽ U(q⫺1)⌿ ˜ (x, e) ⌿ ˜ (x, e) :⫽ ⌿(x), and e 苸 Q is a neutral element. where ⌿ The map of the element of quasialgebra X 苸 ᒎ is defined by ˜ (x, q) ⫽ d ⌿ ˜ (x, q ⭈ e⫺tX)앚t⫽0 X⌿ dt where e⫺tX is an exponential mapping on Q. This implies that the covariant ˜ is given by derivative D ˜ ⫽ (⭸ ⫹ A˜(x, q))⌿ ˜ dx ⫽ D ˜ dx ˜⌿ ˜ ⌿ D where we set A˜ ⫽ U(q)*앚q⫽eAd⫺1 q (e)A(x). The nonassociative gauge-invariant Lagrangian of matter has the structure ˜,D ˜ , x) ˜ ⌿ ᏸm ⫽ ᏸ(⌿ and describes a matter field ⌿ minimally coupling with the gauge field A. ACKNOWLEDGMENTS I am grateful to Lev Vasilievich Sabinin and Zbigniew Oziewicz for helpful discussions and comments. REFERENCES Belousov, V. D. (1967). Foundations of the Theory of Quasigroups and Loops, Nauka, Moscow. Chein, O., Pflugfelder, H., and J. D. H. Smith, Editors (1990). Quasigroups and Loops: Theory and Applications, Heldermann Verlag, Berlin. –Durd-evich, Micho (1996). Geometry of quantum principal bundles, Communications in Mathematical Physics, 175, 451–521. –Durd-evich, Micho (1997). Quantum principal bundles and corresponding gauge theories, Journal of Physics A, 30, 2027–2054. –Durd-evich, Micho (2001). Quantum spinor structures for quantum spaces, International Journal of Theoretical Physics, 40, 115–138. Kobayashi, S. K., and Nomizu, K. (1963). Foundations of Differential Geometry, Vol. 1. Interscience, New York. Kobayashi, S. K., and Nomizu, K. (1969). Foundations of Differential Geometry, Vol. 2, Interscience, New York. Nesterov, Alexander I. (1989). Methods of nonassociative algebra in physics, Dr. Sci. Dissertation, Institute of Physics, Estonian Academy of Science, Tartu. Nesterov, Alexander I. (2000). Principal Q-bundles, in: R. Costa, H. Cuzzo, A. Grishkov, Jr., and L. A. Peresi, Editors, Non Associative Algebra and Its Applications, Marcel Dekker, New York. Nesterov, Alexander I. (1999). Smooth loops and fibre bundles: Theory of principal loop Qbundles. Submitted to Letters in Mathematical Physics.
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Nesterov, Alexander I., and Lev V. Sabinin (1997). Smooth loops, generalized coherent states, and geometric phases, International Journal of Theoretical Physics, 36, 1981–1990. Nesterov, Alexander I., and Lev V. Sabinin (1997). Smooth loops and Thomas precession, Hadronic Journal, 20, 219–237. Nesterov, Alexander I., and V. A. Stepanenko (1986). On methods of nonassociative algebra in geometry and physics, L. V. Kirensky Institute of Physics, preprint 400F. Pflugfelder, H. (1990). Quasigroups and Loops: An Introduction, Heldermann Verlag, Berlin. Sabinin, Lev V. (1999). Smooth Quasigroups and Loops, Kluwer, Dordrecht.
