aa r X i v : . [ h e p - t h ] J u l Noncommutative Sugawara Construction
M. Ghasemkhani aa Department of Physics, Shahid Beheshti University,G.C., Evin, Tehran 19839, Iran
The noncommutative extension of the Sugawara construction for free massless fermionicfields in two dimensions is studied. We prove that the equivalence of the noncommutativeSugawara energy-momentum tensor and symmetric energy-momentum tensor persists inthe noncommutative extension. Some relevant physical results of this equivalence are alsodiscussed.
PACS numbers: 11.10.Nx, 11.10.Lm, 11.15.Bt
I. INTRODUCTION
One of the outstanding features of two dimensional field theories is bosonization where a free masslessfermionic field can be written as a bosonic filed. This property is rooted in the work of Jordanand Wigner [1] where it was shown that the fermionic creation and annihilation operators may berepresented as the bosonic counterparts. On the other hand, the idea of describing strong interactionprocess in terms of currents was proposed in [2–4]. In this approach, the dynamical variables aretaken to be the currents and the canonical formalism is abandoned. In other words, each particle doesnot correspond to a field which satisfies the canonical commutation relation but Hilbert space is builtupon current operators. Accordingly, it was shown that the energy momentum-tensor of these theoriescan be expressed as a quadratic function of the currents known as Sugawara construction [4]. Lateron, it was proved that the symmetric energy-momentum tensor of the two-dimensional free masslessfermionic theory is exactly equivalent to the Sugawara energy-momentum tensor which is bilinear infermionic currents [5].Indeed, this equivalence confirmed the result of [1], equivalence of free massless fermions and bosons, inan elegant way. Then generalization of this famous equivalence to the curved space-time was performedin [6] where boson-fermion correspondence was shown for a general metric in two dimensions.Our purpose is to study whether this equivalence is satisfied for noncommutative space, where thenature of the space-time changes at very short distances [7–9], which is not a trivial extension. Theauthors in [10] considered the noncommutative generalization of the Sugawara energy-momentumtensor and then used the Seiberg-Witten map. While in the present work, the correspondence betweenthe noncommutative Sugawara construction and the symmetric energy-momentum tensor for twodimensional free fermionic theory is addressed, without employing the Seiberg-Witten map. Applyingthe techniques described in [5], we demonstrate that the noncommutative Sugawara energy-momentumtensor exactly leads to the symmetric energy-momentum tensor. An interesting physical consequence e-mail: m − [email protected] f this equivalence is noncommutative bosonization that exhibits the relationship between the fermionicand bosonic fields in noncommutative space, as will be discussed in the last section. II. EQUIVALENCE OF THE SYMMETRIC ENERGY-MOMENTUM TENSOR ANDSUGAWARA ENERGY-MOMENTUM TENSOR IN NONCOMMUTATIVE SPACE
The first part of this section includes derivation of the symmetric energy-momentum tensor usingvariation of the action with respect to a generic metric. In the second part, we extend the Sugawaraconstruction to noncommutative space and demonstrate that it will be equivalent to the symmetricenergy-momentum tensor.
A. Symmetric energy-momentum tensor
Let us start from the noncommutative version of the free massless fermionic Lagrangian density, whichis obtained by replacing the ordinary product with the star-product L = i (cid:0) ¯ ψγ α ⋆ ∂ α ψ − ∂ α ¯ ψγ α ⋆ ψ (cid:1) , (II.1)where the star-product is defined as follows f ( x ) ⋆ g ( x ) ≡ exp (cid:18) iθ αβ ∂∂a α ∂∂b β (cid:19) f ( x + a ) g ( x + b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,b =0 , (II.2)here θ µν is an antisymmetric constant matrix. As is usual in two-dimensional field theory, we chooseto work in Euclidean signature. In the noncommutative version, this has the added virtue thatthe Euclidean theory does not have issues with unitarity. The noncommutative symmetric energy-momentum tensor T ⋆ µν is achieved by variation of the action S with respect to a generic metric g µν and setting g µν = δ µν in the end [11]: T ⋆ µν = 2 √ g δSδg µν ( x ) (cid:12)(cid:12)(cid:12)(cid:12) g µν = δ µν , (II.3)where g indicates the determinant of the metric with signature (+ , +). The variation of the actioncorresponding to the Lagrangian density (II.1) can be written as δS = i Z d y √ g ⋆ (cid:16) ¯ ψ ⋆ γ α δg αβ ⋆ ∂ β ψ + ¯ ψ ⋆ γ α δg βα ⋆ ∂ β ψ − ∂ α ¯ ψ ⋆ γ β δg αβ ⋆ ψ − ∂ α ¯ ψ ⋆ γ β δg βα ⋆ ψ (cid:17) + i Z d y ( δ √ g ) ⋆ (cid:0) ¯ ψγ α ⋆ ∂ α ψ − ∂ α ¯ ψγ α ⋆ ψ (cid:1) . (II.4)Using the relation (II.3) and the cyclic property of the star product under the integral, we have T ⋆ µν = − i (cid:16) ∂ ν ψ β ⋆ ¯ ψ α ( γ µ ) αβ + ∂ µ ψ β ⋆ ¯ ψ α ( γ ν ) αβ + ∂ µ ¯ ψ α ⋆ ψ β ( γ ν ) αβ + ∂ ν ¯ ψ α ⋆ ψ β ( γ µ ) αβ (cid:17) − i δ µν (cid:16) ¯ ψ α ⋆ ∂ λ ψ β ( γ λ ) αβ − ∂ λ ¯ ψ α ⋆ ψ β ( γ λ ) αβ (cid:17) . (II.5)Applying the equation of motion for free massless fermions, we find the energy-momentum tensor as T ⋆ µν = − i (cid:16) ∂ ν ψ β ⋆ ¯ ψ α ( γ µ ) αβ + ∂ µ ψ β ⋆ ¯ ψ α ( γ ν ) αβ + ∂ µ ¯ ψ α ⋆ ψ β ( γ ν ) αβ + ∂ ν ¯ ψ α ⋆ ψ β ( γ µ ) αβ (cid:17) , (II.6)which is completely symmetric under µ ↔ ν . 2 . Sugawara energy-momentum tensor The equivalence of the Sugawara construction and the symmetric energy-momentum tensor in commu-tative space has been shown in [5] and is reviewed in appendix A. In the present section, we constructthe noncommutative version of the Sugawara energy-momentum tensor to demonstrate that it is pre-cisely equivalent to (II.6).The Lagrangian (II.1) is invariant under global U(1) transformation which yields two different Noethercurrents [12] J µ ( x ) =: ¯ ψ α ( x ) ⋆ ψ β ( x ) : ( γ µ ) αβ , J µ ( x ) =: ψ β ( x ) ⋆ ¯ ψ α ( x ) : ( γ µ ) αβ , (II.7)where : : denotes normal ordering. Now, we extend the commutative Sugawara construction to thenoncommutative one as a bilinear function of J µ ( x ) with inserting star product instead of ordinaryproduct T s ⋆ µν = 12 c (cid:18) J µ ( x ) ⋆ J ν ( x ) + J ν ( x ) ⋆ J µ ( x ) − δ µν J λ ( x ) ⋆ J λ ( x ) (cid:19) , (II.8)where c is the Schwinger constant which appears in the equal-time commutator of currents. Sincethe mass dimension of energy-momentum tensor and of the currents in two dimensions is equal totwo and one respectively, the coefficient c should be dimensionless while in four dimensions, it is adimensionful quantity with dimension of a mass square. The detailed analysis of the current algebrain two dimensions shows that the value of c in noncommutative case is the same as the commutativeone, c = π [13]. To prove that (II.8) is exactly equivalent to (II.6), we need to regularize the operatorproducts in (II.8). To this end, we use the point-splitting technique [14] and replace J µ ( x ) ⋆ J ν ( x ) withlim ǫ → (cid:18) J µ ( x + ǫ ) ⋆ J ν ( x ) − h J µ ( x + ǫ ) ⋆ J ν ( x ) i (cid:19) , (II.9)which leads to T s ⋆ µν = π ǫ → (cid:18) J µ ( x + ǫ ) ⋆ J ν ( x ) + J ν ( x + ǫ ) ⋆ J µ ( x ) − δ µν J λ ( x + ǫ ) ⋆ J λ ( x ) − h J µ ( x + ǫ ) ⋆ J ν ( x ) i − h J ν ( x + ǫ ) ⋆ J µ ( x ) i + δ µν h J λ ( x + ǫ ) ⋆ J λ ( x ) i (cid:19) . (II.10)To perform some algebraic manipulations on (II.10), we employ the star product definition (II.2) J µ ( x + ǫ ) = F ab : ψ β ( x + ǫ + a ) ¯ ψ α ( x + ǫ + b ) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,b =0 ( γ µ ) αβ , (II.11)where F ab is an abbreviated notation for the exponential operator appearing in (II.2). Accordingly,the first term of the equation (II.10) can be written as J µ ( x + ǫ ) ⋆ J ν ( x ) = F fg F ab F cd : ψ β ( x + ǫ + f + a ) ¯ ψ α ( x + ǫ + f + b ) :: ψ σ ( x + g + c ) ¯ ψ ρ ( x + g + d ) : × ( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 . (II.12)3sing Wick’s theorem, (II.12) changes into J µ ( x + ǫ ) ⋆ J ν ( x ) = F fg F ab F cd (cid:18) : ψ β ( x + ǫ + f + a ) ¯ ψ α ( x + ǫ + f + b ) ψ σ ( x + g + c ) ¯ ψ ρ ( x + g + d ) : − : ψ σ ( x + g + c ) h ψ β ( x + ǫ + f + a ) ¯ ψ ρ ( x + g + d ) i ¯ ψ α ( x + ǫ + f + b ) : − : ψ β ( x + ǫ + f + a ) h ψ σ ( x + g + c ) ¯ ψ α ( x + ǫ + f + b ) i ¯ ψ ρ ( x + g + d ) : − h ψ β ( x + ǫ + f + a ) ¯ ψ ρ ( x + g + d ) ih ψ σ ( x + g + c ) ¯ ψ α ( x + ǫ + f + b ) i (cid:19) × ( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 . (II.13)Rewriting the other terms of (II.10) similar to (II.13) and substituting them again into (II.10), weobtain T s ⋆ µν = π ǫ → (cid:20) Q µν ( x, ǫ ) − R µν ( x, ǫ ) − R νµ ( x, ǫ ) − S νµ ( x, − ǫ ) − S µν ( x, − ǫ ) − δ µν [ R λλ ( x, ǫ ) + S λλ ( x, − ǫ )] (cid:21) , (II.14)with Q µν ( x, ǫ ) = F fg F ab F cd : (cid:18) ψ β ( x + ǫ + f + a ) ¯ ψ α ( x + ǫ + f + b ) ψ σ ( x + g + c ) ¯ ψ ρ ( x + g + d )( γ µ ) αβ ( γ ν ) ρσ + ψ β ( x + ǫ + f + a ) ¯ ψ α ( x + ǫ + f + b ) ψ σ ( x + g + c ) ¯ ψ ρ ( x + g + d )( γ ν ) αβ ( γ µ ) ρσ − ψ β ( x + ǫ + f + a ) ¯ ψ α ( x + ǫ + f + b ) ψ σ ( x + g + c ) ¯ ψ ρ ( x + g + d ) δ µν × ( γ λ ) αβ ( γ λ ) ρσ (cid:19) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 , R µν ( x, ǫ ) = F fg F ab F cd : ψ σ ( x + g + c ) h ψ β ( x + ǫ + f + a ) ¯ ψ ρ ( x + g + d ) i ¯ ψ α ( x + ǫ + f + b ) : × ( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 , S νµ ( x, − ǫ ) = F fg F ab F cd : ψ β ( x + ǫ + f + a ) h ψ σ ( x + g + c ) ¯ ψ α ( x + ǫ + f + b ) i ¯ ψ ρ ( x + g + d ) : × ( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 . (II.15)The field ordering appearing in the vacuum expectation value of the relation (II.15) is not time orderingand is defined as S (+) ( x − y ) = ψ ( x ) ¯ ψ ( y ) − : ψ ( x ) ¯ ψ ( y ) := h ψ ( x ) ¯ ψ ( y ) i . (II.16) We follow the convention used in [5].
4n view of the above definition, the quantities R µν ( x, ǫ ) and S νµ ( x, − ǫ ) can be represented as R µν ( x, ǫ ) = F fg F ab F cd : ψ σ ( x + g + c ) ¯ ψ α ( x + ǫ + f + b ) : × S (+) βρ ( ǫ + f + a − g − d )( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 , S νµ ( x, − ǫ ) = F fg F ab F cd : ψ β ( x + ǫ + f + a ) ¯ ψ ρ ( x + g + d ) : × S (+) σα ( g + c − ǫ − f − b )( γ µ ) αβ ( γ ν ) ρσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f,g,a,b,c,d =0 . (II.17)Converting equation (II.17) to the star product form, we obtain R µν ( x, ǫ ) = − (cid:16) γ µ S (+) ( ǫ ) γ ν (cid:17) αβ : ¯ ψ α ( x + ǫ ) ⋆ ψ β ( x ) : , S νµ ( x, − ǫ ) = (cid:16) γ ν S (+) ( − ǫ ) γ µ (cid:17) αβ : ψ β ( x + ǫ ) ⋆ ¯ ψ α ( x ) : , (II.18)where S (+) ( ǫ ) = − i π ǫ ξ γ ξ ǫ . (II.19)Note that the minus sign in R µν ( x, ǫ ) comes from the odd permutation of the fermionic fields.Expanding ψ and ¯ ψ up to the first order in ǫ yields R µν ( x, ǫ ) = iǫ ξ πǫ ( γ µ γ ξ γ ν ) αβ : [ ¯ ψ α ( x ) + ǫ η ∂ η ¯ ψ α ( x ) + O ( ǫ )] ⋆ ψ β ( x ) : , S νµ ( x, − ǫ ) = iǫ ξ πǫ ( γ ν γ ξ γ µ ) αβ : [ ψ β ( x ) + ǫ η ∂ η ψ β ( x ) + O ( ǫ )] ⋆ ¯ ψ α ( x ) : , (II.20)and using the following symmetric limitslim ǫ → ( ǫ α ǫ ) = 0 , lim ǫ → ( ǫ α ǫ β ǫ ) = 12 δ αβ , (II.21)we conclude that lim ǫ → R µν ( x, ǫ ) = i π ( γ µ γ ξ γ ν ) αβ : ∂ ξ ¯ ψ α ( x ) ⋆ ψ β ( x ) : , lim ǫ → S νµ ( x, − ǫ ) = i π ( γ ν γ ξ γ µ ) αβ : ∂ ξ ψ β ( x ) ⋆ ¯ ψ α ( x ) : . (II.22)Inserting the result (II.22) in (II.14), we arrive at T s ⋆ µν = π Q µν ( x ) − i (cid:20) ( γ µ γ ξ γ ν + γ ν γ ξ γ µ ) αβ (cid:16) ∂ ξ ¯ ψ α ( x ) ⋆ ψ β ( x ) + ∂ ξ ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:17) + δ µν ( γ λ γ ξ γ λ ) αβ (cid:16) ∂ ξ ¯ ψ α ( x ) ⋆ ψ β ( x ) + ∂ ξ ψ β ⋆ ¯ ψ α (cid:17) (cid:21) : . (II.23)The product of gamma matrices in (II.23) can be simplified using the Clifford algebra γ µ γ ξ γ ν + γ ν γ ξ γ µ = 2 ( δ µξ γ ν + δ νξ γ µ − δ µν γ ξ ) , γ λ γ ξ γ λ = (2 − d ) γ ξ . (II.24)5ubstituting (II.24) into (II.23) and applying the equation of motion γ ξ ∂ ξ ψ = 0 and ∂ ξ ¯ ψγ ξ = 0, weobtain the Sugawara energy-momentum tensor T s ⋆ µν = π Q µν ( x ) − i (cid:20)(cid:18) ∂ ν ¯ ψ α ( x ) ⋆ ψ β ( x ) + ∂ ν ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ( γ µ ) αβ + (cid:18) ∂ µ ¯ ψ α ( x ) ⋆ ψ β ( x ) + ∂ µ ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ( γ ν ) αβ (cid:21) : . (II.25)We notice that the last term of (II.23) vanishes in two dimensions as a result of the identity γ λ γ ξ γ λ =(2 − d ) γ ξ . In order to show that T ⋆ µν = T s ⋆ µν , it is enough to demonstrate Q µν = 0. For simplicity, wecarry out computations in the light-cone coordinate system, x ± = x ± ix . The representation of theEuclidean gamma matrices γ = − ii ! , γ = − − ! , (II.26)in the light-cone coordinates is given by γ + = γ + iγ = − i ! , γ − = γ − iγ = i ! , (II.27)with g µν = g ++ g + − g − + g −− ! = ! . (II.28)The equation of motion for two-dimensional massless fermions, which is described by iγ µ ∂ µ ψ = 0 with ψ = ψ ψ ! , in the light-cone coordinate system reduces to ∂ + ψ = ∂ − ψ = 0. Thus ψ = ψ ( x − ) , ψ = ψ ( x + ) , ¯ ψ = ¯ ψ ( x + ) , ¯ ψ = ¯ ψ ( x − ) . (II.29)Using [ x + , x − ] = 2 θ , the expression ψ β ( x ) ⋆ ¯ ψ α ( x )( γ µ ) αβ with on-shell Dirac fermions is rewritten as ψ β ⋆ ¯ ψ α ( γ + ) αβ = − i ψ ( x + ) ⋆ ¯ ψ ( x + ) = − i ψ ( x + ) ¯ ψ ( x + ) ,ψ β ⋆ ¯ ψ α ( γ − ) αβ = +2 i ψ ( x − ) ⋆ ¯ ψ ( x − ) = +2 i ψ ( x − ) ¯ ψ ( x − ) . (II.30)With equation (II.30), it would be possible to find all the components of Q µν . Since there is nosingularity in Q µν ( x, ǫ ), we have lim ǫ → Q µν ( x, ǫ ) = Q µν ( x ) , (II.31)and hence Q µν ( x ) = : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ µ ) αβ ( γ ν ) ρσ + : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ν ) αβ ( γ µ ) ρσ − : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : δ µν ( γ λ ) αβ ( γ λ ) ρσ . (II.32)6ne may then readily show that Q ++ = − (cid:18) ψ ( x + ) ¯ ψ ( x + ) (cid:19) ⋆ (cid:18) ψ ( x + ) ¯ ψ ( x + ) (cid:19) := − ψ ( x + ) ¯ ψ ( x + ) ψ ( x + ) ¯ ψ ( x + ) : . (II.33)Performing some straightforward permutations, we get Q ++ = 0. Similarly Q −− = − (cid:18) ψ ( x − ) ¯ ψ ( x − ) (cid:19) ⋆ (cid:18) ψ ( x − ) ¯ ψ ( x − ) (cid:19) := − ψ ( x − ) ¯ ψ ( x − ) ψ ( x − ) ¯ ψ ( x − ) := 0 . (II.34)Also for off-diagonal components Q ±∓ , we have Q ±∓ ( x ) = : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ± ) αβ ( γ ∓ ) ρσ + : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ∓ ) αβ ( γ ± ) ρσ − : 2 (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ λ ) αβ ( γ λ ) ρσ . (II.35)Inserting γ ± = γ ∓ results in Q ±∓ ( x ) = : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ± ) αβ ( γ ∓ ) ρσ + : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ∓ ) αβ ( γ ± ) ρσ − : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ± ) αβ ( γ ∓ ) ρσ − : (cid:18) ψ β ( x ) ⋆ ¯ ψ α ( x ) (cid:19) ⋆ (cid:18) ψ σ ( x ) ⋆ ¯ ψ ρ ( x ) (cid:19) : ( γ ∓ ) αβ ( γ ± ) ρσ = 0 . (II.36)Consequently Q µν = 0. This means that the equivalence of the Sugawara energy-momentum tensorand energy-momentum tensor T s ⋆ µν = T ⋆ µν in two-dimensional noncommutative space for free masslessfermions is still satisfied. Also this equivalence occurs for the Sugawara form in terms of the current J µ ( x ), as defined in (II.7). We have b T s ⋆ µν = π (cid:18) J µ ( x ) ⋆ J ν ( x ) + J ν ( x ) ⋆ J µ ( x ) − δ µν J λ ( x ) ⋆ J λ ( x ) (cid:19) . (II.37)To show this, let us write first all the components of the currents J µ ( x ) and J µ ( x ) using the represen-tation of the gamma matrices (II.26) as follows J ( x ) = i : (cid:18) ψ ( x − ) ⋆ ¯ ψ ( x − ) − ψ ( x + ) ⋆ ¯ ψ ( x + ) (cid:19) := i : (cid:18) ψ ( x − ) ¯ ψ ( x − ) − ψ ( x + ) ¯ ψ ( x + ) (cid:19) : ,J ( x ) = − : (cid:18) ψ ( x + ) ⋆ ¯ ψ ( x + ) + ψ ( x − ) ⋆ ¯ ψ ( x − ) (cid:19) := − : (cid:18) ψ ( x + ) ¯ ψ ( x + ) + ψ ( x − ) ¯ ψ ( x − ) (cid:19) : , (II.38)7s well as J ( x ) = i : (cid:18) ¯ ψ ( x − ) ⋆ ψ ( x − ) − ¯ ψ ( x + ) ⋆ ψ ( x + ) (cid:19) := i : (cid:18) ¯ ψ ( x − ) ψ ( x − ) − ¯ ψ ( x + ) ψ ( x + ) (cid:19) : , J ( x ) = − : (cid:18) ¯ ψ ( x + ) ⋆ ψ ( x + ) + ¯ ψ ( x − ) ⋆ ψ ( x − ) (cid:19) := − : (cid:18) ¯ ψ ( x + ) ψ ( x + ) + ¯ ψ ( x − ) ψ ( x − ) (cid:19) : . (II.39)We notice that the star product appearing in the noncommutative currents is removed. Applying thepermutation on the fermionic fields of the relation (II.38), we obtain J µ ( x ) = − J µ ( x ) . (II.40)This is an interesting result in two dimensions. Unlike the four-dimensional case, where J µ ( x ) and J µ ( x ) correspond to each other by the charge conjugation transformation [15], which is not conserved,in two dimensions the charge conjugation, as well as the Lorentz invariance, remain the symmetry ofthe theory as in their commutative case. Inserting (II.40) in (II.37) then leads to b T s ⋆ µν = T s ⋆ µν = T ⋆ µν . (II.41)One of the physical consequences of this equivalence is noncommutative bosonization, which is obtainedby writing the transformation of the field ψ under the spatial translation ∂ x ψ ( x ) = i (cid:20) P , ψ ( x ) (cid:21) , P = Z dx ′ T s ⋆ , (II.42)where T s ⋆ is the conserved current arising from translational invariance . We have ∂ x ψ ( x ) = i (cid:20) Z dx ′ T s ⋆ , ψ ( x ) (cid:21) x = x ′ , (II.43)and substituting the value of T s ⋆ from (II.8) in (II.43) yields ∂ x ψ ( x ) = iπ (cid:20) Z dx ′ (cid:18) J ( x ′ ) ⋆ J ( x ′ ) + J ( x ′ ) ⋆ J ( x ′ ) (cid:19) , ψ ( x ) (cid:21) x = x ′ . (II.44)To simplify (II.44), we insert R dx ′ δ ( x − x ′ ) = 1 to use the trace property of the star product, whichis given by Z dx ′ dx ′ J ( x ′ ) ⋆ J ( x ′ ) = Z dx ′ dx ′ J ( x ′ ) ⋆ J ( x ′ ) . (II.45)Inserting (II.45) in (II.44) and applying the operator definition of the star product (II.11), we have ∂ x ψ ( x ) = iπ Z dx ′ dx ′ δ ( x − x ′ ) F ab (cid:20) J ( x ′ + a ) J ( x ′ + b ) , ψ ( x ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,b =0 = iπ Z dx ′ dx ′ δ ( x − x ′ ) F ab (cid:18) J ( x ′ + a ) (cid:20) J ( x ′ + b ) , ψ ( x ) (cid:21) + (cid:20) J ( x ′ + a ) , ψ ( x ) (cid:21) J ( x ′ + b ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a,b =0 . (II.46) In two-dimensional Euclidean space x = ix . { ψ α ( x ) , ψ † β ( x ′ ) } = δ αβ δ ( x − x ′ ) as follows (cid:20) J ( x ′ + a ) , ψ ( x ) (cid:21) x = x ′ = ψ ( x ′ + a ) δ ( x − x ′ + a ) , (cid:20) J ( x ′ + a ) , ψ ( x ) (cid:21) x = x ′ = γ ψ ( x ′ + a ) δ ( x − x ′ + a ) , (II.47)with γ = iγ γ . Substituting (II.47) into (II.46) and then converting the result into the star productform, we have ∂ x ψ ( x ) = iπ (cid:18) J ( x ) + J ( x ) γ (cid:19) ⋆ ψ ( x ) . (II.48)The solution of this equation is represented by ψ ( x ) = P (cid:18) e iπ R x −∞ dx ′ [ J ( x ′ )+ J ( x ′ ) γ ] ⋆ (cid:19) ψ , (II.49)where P denotes the path-ordering operator and ψ is a constant spinor in two dimensions.Now, as a result of (II.38), we can use the bosonized form of the commutative current, which isintroduced in appendix A. Hence, we conclude ψ ( x ) = P (cid:18) e − i √ π [ γ φ ( x ) − R x −∞ dx ′ ˙ φ ( x ′ )] ⋆ (cid:19) ψ , (II.50)here ˙ φ = ∂ x ′ φ . III. DISCUSSION
In this paper, we established the noncommutative extension of the Sugawara construction in bilinearform of the currents for free massless fermions in two dimensions. It was shown that this constructionis precisely equivalent to the symmetric energy-momentum tensor.To prove the correctness of this equivalence, we determined the energy-momentum tensor in twoseparate methods. The first was the direct calculation using the symmetric definition of the energy-momentum tensor for on-shell Dirac fermions and the second contained a detailed analysis of non-commutative Sugawara construction by applying the point-splitting regularization. Furthermore, forsimplification in our calculation, we considered the light-cone system. In this coordinate, we realizedthat the currents J µ and J µ , apart from a minus sign, are actually the same in two dimensions whichleads to the charge conjugation symmetry restoration.Eventually, we presented a physical consequence of this equivalence, named as noncommutativebosonization (e.g. see [16–18]), that relates a fermionic field to a bosonic field through an exponen-tial function and demonstrated that a free massless fermion theory with a global U (1) symmetry innoncommutative space corresponds to a free massless boson theory. Also, the bosonized version ofa theory with local U (1) symmetry such as two-dimensional noncommutative QED (NC-QED ) wasaddressed in [19], where it was proven that the bosonized action contains a noncommutative Wess-Zumino-Witten (WZW) part, a gauge kinetic part and an interaction part between the WZW andgauge field. 9he physical significance of the bosonization procedure is that it specifies a duality between the strongand weak couplings for particular interacting quantum field theories. The most famous example of thisduality is the equivalence of the massive Thirring model and sine-Gordon model [20, 21], where theweak coupling β of the bosonic theory, that is the sine-Gordon model, is related to the strong coupling g of the fermionic theory, the massive Thirring model, through the bosonization rule described by πβ = 1 + gπ . Moreover, the duality between the noncommutative version of these models was studiedin [16–18] where it was shown that the strong-weak duality is also preserved. However, it is notablethat the strong-weak duality does not appear in the case of NC-QED and its bosonized version,because of the appearance of the same couplings in two theories. IV. ACKNOWLEDGMENTS
I am grateful to M.M. Sheikh-Jabbari for numerous fruitful discussions and careful reading of themanuscript. I would also like to thank M. Khorrami for his valuable comments and I appreciate theinsightful and constructive remarks of M. Chaichian and the referee, which led to improvement of themanuscript. Moreover, I acknowledge the School of Physics of Institute for research in fundamentalsciences (IPM) for the research facilities and environment.
Appendix A: Commutative Sugawara Construction
In this appendix, we present a detailed analysis on the proof of the relation T sµν = T µν in two dimen-sional commutative space for free massless fermions, as argued in [5], and describe some interestingconsequences of this equivalence [22–24]. The Lagrangian for the massless fermions is given by L = i (cid:0) ¯ ψγ µ ∂ µ ψ − ∂ µ ¯ ψγ µ ψ (cid:1) , (A.1)which is invariant under the global phase transformation ψ → e iα ψ and ¯ ψ → e − iα ¯ ψ that gives theconserved current j µ ( x ) =: ¯ ψ ( x ) γ µ ψ ( x ) : . (A.2)For this theory, the symmetric energy-momentum tensor is written as follows T µν = i (cid:0) ¯ ψγ µ ∂ ν ψ + ¯ ψγ ν ∂ µ ψ − ∂ µ ¯ ψγ ν ψ − ∂ ν ¯ ψγ µ ψ (cid:1) : . (A.3)The energy-momentum tensor in Sugawara form is described by a bilinear function of the currents as T sµν = π (cid:16) j µ ( x ) j ν ( x ) + j ν ( x ) j µ ( x ) − g µν j λ ( x ) j λ ( x ) (cid:17) . (A.4)To show T sµν = T µν , we start with (A.4) and replace j µ ( x ) j ν ( x ) withlim ǫ → (cid:18) j µ ( x + ǫ ) j ν ( x ) − h j µ ( x + ǫ ) j ν ( x ) i (cid:19) . (A.5)10pplying Wick’s theorem on (A.5) j µ ( x + ǫ ) j ν ( x ) = : ¯ ψ ( x + ǫ ) γ µ ψ ( x + ǫ ) :: ¯ ψ ( x ) γ ν ψ ( x ) := : ¯ ψ ( x + ǫ ) γ µ ψ ( x + ǫ ) ¯ ψ ( x ) γ ν ψ ( x ) :+ : ¯ ψ ( x + ǫ ) γ µ h ψ ( x + ǫ ) ¯ ψ ( x ) i γ ν ψ ( x ) :+ : ¯ ψ ( x ) γ ν h ψ ( x ) ¯ ψ ( x + ǫ ) i γ µ ψ ( x + ǫ ) : − tr (cid:0) γ µ h ψ ( x ) ¯ ψ ( x + ǫ ) i γ ν h ψ ( x + ǫ ) ¯ ψ ( x ) i (cid:1) , (A.6)and implementing a similar analysis for the other terms of (A.4), we arrive at T sµν = π ǫ → (cid:20) M µν ( x, ǫ ) + N µν ( x, ǫ ) + N νµ ( x, ǫ ) + N µν ( x, − ǫ ) + N νµ ( x, − ǫ ) − g µν [ N λλ ( x, ǫ ) + N λλ ( x, − ǫ )] (cid:21) , (A.7)where M µν and N µν are defined as M µν ( x, ǫ ) = : ¯ ψ ( x + ǫ ) γ µ ψ ( x + ǫ ) ¯ ψ ( x ) γ ν ψ ( x ) :+ : ¯ ψ ( x + ǫ ) γ ν ψ ( x + ǫ ) ¯ ψ ( x ) γ µ ψ ( x ) : − : ¯ ψ ( x + ǫ ) γ λ ψ ( x + ǫ ) ¯ ψ ( x ) γ λ ψ ( x ) : g µν , N µν ( x, ǫ ) = : ¯ ψ ( x + ǫ ) γ µ S (+) ( ǫ ) γ ν ψ ( x ) : , (A.8)and we have S (+) ( ǫ ) = h ψ ( x + ǫ ) ¯ ψ ( x ) i = − ( i π ) ǫ α γ α ǫ . (A.9)First, we concentrate on determining the value of M µν ( x ). M = M =: ( j ) + ( j ) : , M = M =: j j + j j : . (A.10)Choosing γ = σ z and γ = iσ y , we find j = ¯ ψ ψ − ¯ ψ ψ , j = ¯ ψ ψ − ¯ ψ ψ . (A.11)Therefore, all the components of M µν in terms of the fermionic fields are given by M = M =: ( ¯ ψ ψ − ¯ ψ ψ ) + ( ¯ ψ ψ − ¯ ψ ψ ) : , M = M =: ( ¯ ψ ψ − ¯ ψ ψ )( ¯ ψ ψ − ¯ ψ ψ ) + ( ¯ ψ ψ − ¯ ψ ψ )( ¯ ψ ψ − ¯ ψ ψ ) : . (A.12)Performing some permutations on fermionic fields yields M µν = 0. In the next step, our purpose isto obtain the value of N µν . To this end, let us start with expansion of the fermionic fields up to thefirst order in ǫ N µν ( x, ǫ ) = − iǫ ξ πǫ : [ ¯ ψ ( x ) + ǫ α ∂ α ¯ ψ ( x ) + O ( ǫ )] γ µ γ ξ γ ν ψ ( x ) : (A.13)11aking the symmetric limits (II.21),lim ǫ → N µν ( x, ǫ ) = i π : ∂ ξ ¯ ψ ( x ) γ µ γ ξ γ ν ψ ( x ) : , (A.14)putting (A.14) in (A.7) and using the identity (II.24) for on-shell fermions in two dimensions, we find T sµν = i ψγ µ ∂ ν ψ + ¯ ψγ ν ∂ µ ψ − ∂ µ ¯ ψγ ν ψ − ∂ ν ¯ ψγ µ ψ ) : , (A.15)which is exactly equal to T µν mentioned in (A.3). This equivalence suggests the existence of a canonicalmassless pseudo scalar field, satisfying [ φ ( x, t ) , ˙ φ ( y, t )] = iδ ( x − y ), which is related to the conservedcurrent j µ ( x ) through the following equation [23] j µ ( x ) = 1 √ π ǫ µν ∂ ν φ ( x ) . (A.16)If we substitute (A.16) into the Sugawara form (A.4) and use the identity ǫ µα ǫ νβ = g µβ g αν − g µν g αβ ,it is found that T sµν = 12 (cid:16) ∂ µ φ∂ ν φ + ∂ ν φ∂ µ φ − g µν ∂ λ φ∂ λ φ (cid:17) , (A.17)which describes the energy-momentum tensor for a free massless boson. Another interesting result ofthe Sugawara construction is that it is possible to find explicitly the fermionic filed in terms of thebosonic field, as mentioned in the introduction part. To show this, we consider the equation describingthe transformation of the field ψ under the spatial translation [24] ∂ x ψ ( x ) = i (cid:20) Z dx ′ T s , ψ ( x ) (cid:21) , P = Z dx ′ T s , (A.18)where T s is the Noether current of translational symmetry. Inserting T s from (A.4) into (A.18), wehave ∂ x ψ ( x ) = iπ (cid:20) Z dx ′ (cid:18) j ( x ′ ) j ( x ′ ) + j ( x ′ ) j ( x ′ ) (cid:19) , ψ ( x ) (cid:21) x = x ′ . (A.19)Applying the equal-time commutation relations (cid:20) j ( x ′ ) , ψ ( x ) (cid:21) x = x ′ = − ψ ( x ) δ ( x − x ′ ) , (cid:20) j ( x ′ ) , ψ ( x ) (cid:21) x = x ′ = − γ ψ ( x ) δ ( x − x ′ ) , (A.20)with γ = γ γ , we arrive at ∂ x ψ ( x ) = − iπ [ j ( x ) + j γ ( x )] ψ ( x ) . (A.21)Solving this equation yields ψ ( x ) = e − iπ R x −∞ dx ′ ( j ( x ′ )+ j ( x ′ ) γ ) ψ , (A.22)where ψ is a constant spinor in space-time. In the final step, we put the bosonized form of thecurrents from (A.16) in (A.22) ψ ( x ) = e i √ π [ γ φ ( x )+ R x −∞ dx ′ ˙ φ ( x ′ )] ψ . (A.23)As we see, the spinor field ψ is mapped to the bosonic field φ . [1] P. Jordan and E. P. Wigner, About the Pauli exclusion principle , Z. Phys. , 631 (1928).
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