Dirac quantization of noncommutative Abelian Proca field
aa r X i v : . [ h e p - t h ] J a n Dirac quantization of noncommutative Abelian Proca field
F. Darabi ∗ and F. Naderi Department of Physics, Azarbaijan University of Tarbiat Moallem,Tabriz, Iran, P.O. Box: 53714-161, andResearch Institute for Astronomy and Astrophysics ofMaragha (RIAAM)- Maragha, Iran, P.O. Box: 55134-441. (Dated: November 6, 2018)
Abstract:
Dirac formalism of Hamiltonian constraint systems is studied for thenoncommutative Abelian Proca field. It is shown that the system of constraintsare of second class in agreement with the fact that the Proca field is not guageinvariant. Then, the system of second class constraints is quantized by introducingDirac brackets in the reduced phase space.
Keywords: Noncommutative Proca field, Second class constraint.
PACS numbers: 11.10.Ef, 11.10.Nx ∗ Electronic address: [email protected]
I. INTRODUCTION
The noncommutative geometry, pioneered by alain Connes, aims at a generalization ofgeometrical ideas to spaces whose coordinates fail to commute [1]. In this theory it ispostulated that space-time is noncommutative at very high energies and one tries to guessthe small scale structure of space-time from our present knowledge at the electroweak scale.String theory on the other hand aims at deriving the standard model directly from the Planckscale physics. Thus, noncommutative geometry may describe the low energy dynamics instring theory as a picture for the standard model, where symmetries act directly by a group ofcoordinate transformations on an underlying space-time manifold producing the electroweakand strong forces as pseudo-forces. Moreover, it may place gravity and the other forces onthe same footing by obtaining all forces as pseudo-forces from some general coordinatetransformations acting on some general spacetime.Quantum field theories on noncommutative spaces are usually formulated in terms of starproducts of ordinary functions [3]. The different field models on noncommutative spaces hasbeen recently of particular interest due to the recent development of the superstring theory.It was shown that noncommutative coordinates emerge naturally in the perturbative versionof the D-brane theory, namely low energy excitations of a D-brane, with the presence of theexternal background magnetic field [4]. Therefore, noncommutative Yang-Mills theoriesappear in the string theory in an effective way and that is why they are being so widelyinvestigated. It is known that there exists a Seiberg-Witten map between noncommutativefield theories and effective commutative theories in that both have the same degrees offreedom. Moreover, noncommutative gauge theories can be represented as ordinary gaugetheories with the same degrees of freedom, and with the additional deformation parameter θ [5]. The Seiberg-Witten map between field theory on noncommutative spaces and thecorresponding commutative field theory allows one to formulate an action principle in termsof ordinary field. The effective Lagrangian of this action is expanded as series of ordinaryfield and the noncommutative parameter θ which plays the role of coupling constant.Attempts to quantize the Maxwell theory on noncommutative spaces have already beendone. First, the corresponding commutative action in terms of ordinary fields and linear inthe deformation parameter has been derived in [6, 7]. Afterwards, the Dirac’s quantizationof Hamiltonian constraint systems [9] has been applied on this commutative action in [10]. ,In the present paper, motivated by the quantization of Maxwell theory on noncommutativespace, we attempt to quantize the noncommutative massive Abelian Maxwell theory, namelynoncommutative Abelian Proca field, using the Dirac quantization procedure in a similarway as in Ref.[10]. In section II, we first obtain the noncommutative action for the Proca fieldusing the Moyal product and then derive the corresponding commutative action in termsof ordinary fields and linear in the deformation parameter θ . In section III, we study thisaction in the context of the Dirac’s Hamiltonian constraint systems to find the correspondingconstraints which turn out to be the second class type. In section IV, the system of secondclass constraints are quantized in the reduced phase space. The paper ends with a conclusion. II. NONCOMMUTATIVE ABELIAN PROCA FIELD
The action of Abelian noncommutative Proca field is written S = Z ( −
14 ˆ F µν ∗ ˆ F µν + 12 m ˆ A µ ∗ ˆ A ν ) d x, (1)where ∗ denotes the Star product , ˆ A µ and ˆ F µν are the vector potential and field strengthtensor respectively, and m is the mass of the ˆ A µ field. The fields ˆ F µν and ˆ A µ may beexpressed in terms of the corresponding commutative quantities as follows [7]ˆ A µ = A µ − θ αβ A α ( ∂ β A µ + F βµ ) (2)ˆ F µν = ∂ µ ˆ A ν − ∂ ν ˆ A µ − i ˆ A µ ∗ ˆ A ν + i ˆ A ν ∗ ˆ A µ , (3)where θ αβ stands for the noncommutativity parameter that characterizes non-commutativitythrough the coordinate commutation relation [ x α , x β ] = iθ αβ [1], [2]. It is known that theintegral over the star product of quantities is equal to the corresponding integral over theordinary product [11], then we may rewrite the action (1) as S = Z ( −
14 ˆ F µν ˆ F µν + 12 m ˆ A µ ˆ A ν ) d x. (4)Using (2) and (3) the Lagrangian theory of the above noncommutative action may be ex-panded as the commutative theory with the same degrees of freedom, and with the additionalterms containing the noncommutative parameter θ αβ of the first orderˆ L = − F µν F µν + 18 θ αβ F αβ F µν − θ αβ F µα F νβ F µν (5)+ 12 m ( A µ − θ αβ A α ( ∂ β A µ + F βµ ) A µ ) , where the commutative field strength tensor is F µν = ∂ µ A ν − ∂ ν A µ . (6)Now we define the followings A µ = ( ~A, iA ) E i = iF i B i = ǫ ijk F jk θ i = ǫ ijk θ jk . (7)Then, the Lagrangian density (5) casts in the following formˆ L = 12 ( E − B )(1 + ~θ · ~B ) − ( ~θ · ~E )( ~E · ~B ) + m − A + A i ) (8)+ m ~θ × ~A ) · ~ ∇ ( A ) − m ~θ × ~A ) · ~E ] A + 3 m ~θ · ~B ) A j − ( ~θ · ~A )( ~A · ~B )] . The Euler-Lagrange equations for the noncummutative Proca field are obtained ∂ ρ F ρσ − ∂ ρ ( θ ρσ F µν ) − ∂ ρ ( θ αβ F αβ F ρσ ) + ∂ ρ ( θ σβ F νβ F ρν ) (9) − ∂ ρ ( θ ρβ F νβ F σν ) + 12 ∂ ρ ( θ αβ F ρα F σβ ) − ∂ ρ ( θ αβ F σα F ρβ ) − m θ σβ ( ∂ β A µ + F βµ ) A µ − m θ αβ ( ∂ β A σ + F βσ ) A α + m ∂ ρ ( θ αρ A α A σ ) − m ∂ ρ ( θ ασ A α ) A ρ + m A σ = 0 , where in the last line the Lorentz condition ∂ µ A µ = 0 has been used using the fact thatthis condition holds in the massive as well as massless Maxwell field [8]. Using (7), the fieldequations are divided into the following two set of equations ∂ ~D∂t − ~ ∇ × ~H = − ~J , (10) ~ ∇ · ~D = ρ, (11)where ( ~ ∇ × ~H ) i = ǫ ijk ∂ j H k , ~ ∇ · ~D = ∂ i D i , ∂/∂t = i∂ and ρ = m [ A + 12 ~ ∇ · ( ~θ × ~A ) A + 12 ( ~θ × ~A ) · ~E ] , (12) ~D = ~E + ( ~θ · ~B ) ~E − ( ~θ · ~E ) ~B − ( ~E · ~B ) ~θ − m ~θ × ~A ) A , (13) ~J = m [ ~A −
12 ( ~E × ~θ ) A + 32 ( ~θ · ~B ) ~A −
34 ( ~θ · ~A ) ~B −
34 ( ~A · ~B ) ~θ ] , (14) ~H = ~B + ( ~θ · ~B ) ~B + ( ~θ · ~E ) ~E −
12 ( E − B ) ~θ − m ( 14 A + A j ) ~θ + m ~A ( ~θ · ~A ) . (15)On the other hand, using the strength tensor F µν = ∂ µ A ν − ∂ ν A µ the source-less equationsare ∂ µ ˜ F µν = 0 , (16)where ˜ F µν = ǫ µναβ F αβ and ǫ µναβ is the Levi-Civita constant tensor ( ǫ = − i ). Therefore,equations (16) may be written as ∂ ~B∂t + ~ ∇ × ~E = 0 , (17) ~ ∇ · ~B = 0 . (18) III. HAMILTONIAN CONSTRAINT SYSTEM APPROACH
In this section, following Dirac, we will study the dynamics of the noncommutativeAbelian Proca field in the context of Hamiltonian constraint systems [9]. In so doing,we first obtain the conjugate momenta of A i and A , respectively as π i = ∂ ˆ L ∂ ( ∂ A i ) = − E i (1 + ~θ · ~B ) + ( ~θ · ~E ) B i + ( ~E · ~B ) θ i + m ~θ × ~A ) i A , (19) π = ∂ ˆ L ∂ ( ∂ A ) = 0 . (20)Equation (20) results in a primary constraint φ ≈ , (21)where φ ≡ π , and comparison of (13) with (19) leads to π i − = D i . (22)Therefore, we obtain the following commutation relations { A i ( x, t ) , D j ( y, t ) } = − δ ij δ ( x − y ) , (23) { B i ( x, t ) , D j ( y, t ) } = ǫ ijk ∂ k δ ( x − y ) . (24)Using the Legendre transformation as H = π µ ∂ A µ − L , we may obtain the Hamiltoniandensity ˆ H = 12 ( E + B )(1 + ~θ · ~B ) − ( ~θ · ~E )( ~E · ~B ) + m A − m A i (1 + 32 ~θ · ~B ) (25) − m ~θ × ~A ) i ( ∂ i A ) A + 3 m ~θ · ~A )( ~A · ~B ) − π i ∂ i A . Using (19), we may obtain E i in terms of π i to first order in θ as E i = − π i (1 − ~θ · ~B ) − ( ~π · ~B ) B i − ( ~π · ~B ) θ i + m ~θ × ~A ) i A . (26)Substituting for E i in Eq.(25) in terms of π i we obtainˆ H = 12 ( π + B ) + 12 ( B − π )( ~θ · ~B ) + ( ~π · ~θ )( ~B · ~π ) + m A − m ~θ × ~A ) · ~πA (27) − m A i (1 + 32 ~θ · ~B ) − m ~θ × ~A ) i ( ∂ i A ) A + 3 m ~θ · ~A )( ~A · ~B ) − π i ∂ i A + O ( θ ) . Now, we study the consistency condition for the primary constraint (21)˙ φ = { φ ( x ) , H } = 0 , (28)where H = Z ˆ H ( x ) d x. (29)The consistency condition (28) results in a secondary constraint φ = ∂ i π i + m A + m ~ ∇ · ( ~θ × ~A ) A − m ~θ × ~A ) · ~π, (30)or φ = ∂ i π i + m A + m ~ ∇ · ( ~θ × ~A ) A + m ~θ × ~A ) · ~E + O ( θ ) . (31)If we put π i = − D i , this equation casts in the form of generalized Gauss law which hasalready been obtained in (11). Now, the extended Hamiltonian is constructed by adding theprimary constraint φ up to an arbitrary coefficient u ( x ) H T = H + Z u ( x ) φ ( x ) d x. (32)The consistency condition for the secondary constraint φ is now considered as˙ φ = { φ ( x ) , H T } = 0 , (33)which introduces no new constraint and just fixes the unknown coefficient as u ( x ) = − ~ ∇ · ( ~θ × ~A ) A − ∂ i (cid:18) A i (cid:18) ~θ · ~B − ~ ∇ · ( ~θ × ~A ) (cid:19)(cid:19) (34)+ 34 ∂ i ( θ i ( ~A · ~B ) + B i ( ~θ · ~A )) + ( ~θ × ~A ) · ( ~ ∇ × ~B ) + O ( θ ) , where we have used the expansion ǫ ≃ (1 − ǫ ) due to the smallness of ~θ . It is easy to showthat the constraints φ and φ are second class, namely { φ ( x ) , φ ( x ′ ) } = − m (1 + 12 ~ ∇ · ( ~θ × ~A )) δ ( x − x ′ ) (35)or { φ i ( x ) , φ j ( x ′ ) } = − m (1 + ~ ∇ · ( ~θ × ~A )) m (1 + ~ ∇ · ( ~θ × ~A )) 0 δ ( x − x ′ ) . (36)It is of great importance to distinction between first and second class constraints. The firstclass constraints are defined as the constraints which commute (i.e. have vanishing Poissonbrackets) with all the other constraints. This situation brings to light the presence of somegauge degrees of freedom in the Dirac formalism. On the other hand, the second classconstraints have at least one non vanishing bracket with some other constraints, like (35).By manipulation and integration by parts in some appropriate terms in (27) the constraint φ appears in the Hamiltonian as H = Z [ 12 ( π + B ) + 12 ( B − π )( ~θ · ~B ) + ( ~π · ~θ )( ~B · ~π ) − m A (37) − m A i (1 + 32 ~θ · ~B ) + m ~ ∇ · ( ~θ × ~A ) A + 3 m ~θ · ~A )( ~A · ~B ) + A φ ] d x. Since the original Hamiltonian H includes the constraint φ with the known coefficient A ,it is not necessary to add once again this constraint with an unknown coefficient to theHamiltonian H T , and so we obtain the extended Hamiltonian H E = ¯ H + Z [ u ( x ) φ ( x ) + A ( x ) φ ( x )] d x, (38)where ¯ H = Z [ 12 ( π + B ) + 12 ( B − π )( ~θ · ~B ) + ( ~π · ~θ )( ~B · ~π ) − m A (39) − m A i (1 + 32 ~θ · ~B ) + m ~ ∇ · ( ~θ × ~A ) A + 3 m ~θ · ~A )( ~A · ~B )] d x. IV. QUANTIZATION OF SECOND CLASS CONSTRAINT SYSTEM
The problem of quantization of Hamiltonian constraint systems is twofold: quantization offirst class constraints and of second class constraints [9]. However, the problem of quantizingtheories with second class constraints is less ambiguous than quantizing theories with firstclass constraints. In the following analysis we will not deal with first class constraints dueto the fact that the Proca field is not gauge invariant. In the case of second class constraintswe can switch to new canonical brackets in order to set all of the second class constraintsstrongly equal to zero. This means that in any given quantity, such as the Hamiltonian, wecan set them to zero by hand. In such a case, we can safely change to the new canonicalbrackets, the so called Dirac brackets, defined as follows { A ( x, t ) , B ( y, t ) } DB = { A ( x, t ) , B ( y, t ) } (40) − Z { A ( x, t ) , φ i ( z, t ) } C − ij ( z, ω ) { φ j ( ω, t ) , B ( y, t ) } d z d ω, where C ij ( x, z ) = { φ i ( x ) , φ j ( z ) } is given by (36). We may obtain the inverse matrix C − ij using the relation Z C ij ( x, z ) C − jk ( z, y ) d z = δ ik δ ( x − y ) , (41)which leads to C − ( z, y ) = m − (1 + ~ ∇ · ( ~θ × ~A )) − − m − (1 + ~ ∇ · ( ~θ × ~A )) − δ ( z − y ) . (42)Therefore, the following Dirac brackets are obtained { π ( x, t ) , A ( y, t ) } DB = 0 , (43) { π ( x, t ) , A i ( y, t ) } DB = 0 , (44) { π µ ( x, t ) , π ν ( y, t ) } DB = 0 , (45) { A ( x, t ) , A ( y, t ) } DB = 0 , (46) { π i ( x, t ) , A j ( y, t ) } DB = − δ ij δ ( x − y ) , (47) { π i ( x, t ) , B j ( y, t ) } DB = − ǫ ijk ∂ k δ ( x − y ) , (48) { A ( x, t ) , A j ( y, t ) } DB = m − (1 − ~ ∇ · ( ~θ × ~A )) ∂ j ( x ) δ ( x − y ) (49) −
12 ( ~θ × ~A )) j ( x ) δ ( x − y ) + O ( θ ) , { A ( x, t ) , B k ( y, t ) } DB = ǫ klj ∂ l ( y )[ m − (1 − ~ ∇ · ( ~θ × ~A )) ∂ j ( x ) δ ( x − y ) (50) −
12 ( ~θ × ~A )) j ( x ) δ ( x − y )] , { π i ( x, t ) , A ( y, t ) } DB = 12 [ A ( y ) ǫ jli θ l ( y ) ∂ j ( y ) δ ( x − y ) (51) − ( ~π × ~θ ) i ( y ) δ ( x − y )] + O ( θ ) . It is known that the Dirac brackets of second class constraints with each arbitrary function f is strongly zero, namely { φ i ( x, t ) , f ( y, t ) } DB = 0. Therefore, the second class constraints aresupposed to be strongly zero [9] and so we obtain the physical Hamiltonian in the reducedphase space as H P h = ¯ H = Z [ 12 ( π + B ) + 12 ( B − π )( ~θ · ~B ) + ( ~π · ~θ )( ~B · ~π ) − m A (52) − m A i (1 + 32 ~θ · ~B ) + m ~ ∇ · ( ~θ × ~A ) A + 3 m ~θ · ~A )( ~A · ~B )] d x. The equations of motion are obtained˙ A i = ∂ A i = { A i , H P h } DB (53)= π i (1 − ~θ · ~B ) + θ i ( ~B · ~π ) + ( ~π · ~θ ) B i − ∂ i ( A − ~ ∇ · ( ~θ × ~A )) − m A ( ~θ × ~A ) i , ˙ A = ∂ A = { A , H P h } DB (54)= A ~ ∇ · ( ~π × ~θ ) −
12 ( ~θ × ~A ) · ( ~ ∇ × ~B ) − ∂ j ( A j (1 + 32 ~θ · ~B ))+ 12 ~ ∇ · ( ~θ × ~A ) ∂ j A j + 34 ∂ j ( θ j ( ~A · ~B ) + ( ~θ · ~A ) B j ) + O ( θ ) . ˙ π i = ∂ π i = { π i , H P h } DB (55)= − ~ ∇ × [ ~B (1 + ~θ · ~B ) + 12 ~θ ( B − π ) + ~π ( ~π · ~θ ) − m A j ~θ − m A ~θ − m ( ~θ · ~A ) A ] i + m A i (1 + 32 ( ~θ · ~B )) + 12 m A ( ~π × ~θ ) i − m ( θ i ( ~A · ~B ) + ( ~θ · ~A ) B i ) . π i in (55) in terms of E i through (19), ignoring the terms of the order O ( θ ),using (14), (15) and considering π i = − D i we obtain ∂ ~D∂t − ~ ∇ × ~H = − ~J , (56)which is the Ampere’s Law. This means the equation of motion for π i in noncommutativetheory obtained by Dirac formalism is in agreement with the Maxwell’s equation. On theother hand, equation (54) may be rewritten as ∂ j ( A ~π × ~θ ) j − A j − A j ( ~θ · ~B ) + 34 ( θ j ( ~A · ~B ) + ( ~θ · ~A ) B j )) (57)= ˙ A + 12 ( ~θ × ~A ) · ( ~ ∇ × ~B ) − ~ ∇ · ( ~θ × ~A ) ∂ j A j . The left hand side of above equation, using (14), is equal to − m − ~ ∇ · ~J . The time derivativeof ρ in (12) leads us to˙ ρ = m [ ˙ A + 12 ∂ j ( ~θ × ~ ˙ A ) j A + 12 ∂ j ( ~θ × ~A ) j ~ ˙ A + 12 ( ~θ × ~A ) · ~ ˙ E + 12 ( ~θ × ~ ˙ A ) · ~E ] , (58)which becomes the following form using (26) and ignoring terms of the order O ( θ )˙ ρ = m [ ˙ A + 12 ∂ j ( ~θ × ~ ˙ A ) j A + 12 ∂ j ( ~θ × ~A ) j ~ ˙ A −
12 ( ~θ × ~A ) · ~ ˙ π −
12 ( ~θ × ~ ˙ A ) · ~π ] . (59)Finally, using the equations of motion (53) and (55) we obtain˙ ρ = m (cid:18) ˙ A + 12 ( ~θ × ~A ) · ( ~ ∇ × ~B ) − ~ ∇ · ( ~θ × ~A ) ∂ j A j (cid:19) . (60)Therefore equation (57) casts in the following form ∂ρ∂t + ~ ∇ · ~J = 0 . (61)This shows that the equation of motion for A in noncommutative theory obtained by Diracformalism is in complete agreement with the charge conservation. The quantization ofsystem is then achieved by the standard replacement in the classical equations (53), (54)and (55), namely { , } DB → i ~ [ , ] QM . (62)1 V. CONCLUSION
The systems with constraints are usually described by the Dirac’s elegant formulationof Hamiltonian constraint systems. Two kinds of first and second class constraints playthe main role in this formulation. The first class constraints are responsible for gaugeinvariance whereas the second class constraints provides us with a reduced phase space.Linear combinations of second class constraints may result in the first class constraints. Fromclassical point of view no preference is made between these two kinds of constraints. However,the problem of quantizing theories with second class constraints is less ambiguous thanquantizing theories with first class constraints. We have studied the problem of quantizationof the noncommutative Abelian Proca field. The noncomutative Abelian Proca field hasbeen rewritten as a commutative theory containing the noncommutative tensor θ αβ as agiven external tensor (this is a breaking of manifest covariance) at the first order. Since thetheory is massive, gauge invariance is broken and instead of two first class constraints like inMaxwell theory (so that A and one component of A i are gauge variables) there is a pair ofsecond class constraints eliminating A and π (with a breaking of manifest covariance: thereduced phase space contains only A i and π i ). We then quantized the system by introducingDirac brackets in the reduced phase space. It is appealing to generalize this study for thenoncommutative non-Abelian Proca field [12]. Acknowledgment