Noncommutative quantization in 2D conformal field theory

Physics Letters B 546 (2002) 157–161 www.elsevier.com/locate/npe

Noncommutative quantization in 2D conformal field theory Agapitos Hatzinikitas, Ioannis Smyrnakis University of Crete, Department of Applied Mathematics, L. Knosou-Ambelokipi, 71409 Iraklio Crete, Greece Received 5 July 2002; received in revised form 3 September 2002; accepted 3 September 2002 Editor: L. Alvarez-Gaumé

Abstract The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions compactified on the torus. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time by a factor that depends on the noncommutativity parameter and change the compactification radius appropriately. The four-point function and certain 2n-point functions are deformed, preserving, nevertheless, the sl(2, C) covariance.  2002 Elsevier Science B.V. All rights reserved.

In this Letter we explore the possibility of quantizing a string based on a noncommutative harmonic oscillator as opposed to the ordinary commutative one [1]. This noncommutativity is neither the one in the D brane worldvolume that arises in the quantization of open strings ending on D branes with background B field [2], nor the one introduced in M-theory compactified on T 2 [3]. A noncommutative harmonic  oscillator in d dimensions is a system described by the d-dimensional harmonic
osc = 1 di=1 (pˆ 2 + ω2 qˆ 2 ) in which different components of the position and momentum oscillator Hamiltonian H i i 2 operators do not necessarily commute. Here the nontrivial commutators among different components are c-numbers. Note that we need at least two dimensions for the noncommutative harmonic oscillators. To study the modifications that occur in bosonic string theory, the simplest possible noncommutative harmonic oscillator (with ω = n) in two dimensions is employed. We assume that the position and momentum operators satisfy the following commutation relations [4]:  1 2  1 2  i j qˆn , qˆn = iθ/n; pˆn , pˆ n = −inθ. qˆn , pˆn = i h¯ δij ; (1) An operator representation of the qˆni (pˆ ni ) that realizes the commutation relations (1) is:  α  1 † 1 a + an1 , qˆn = 2n n    1†  n  1 1 2† pˆ n = −i h¯ an − iθ an2 − ha ¯ n + iθ an , 2α E-mail addresses: [email protected] (A. Hatzinikitas), [email protected] (I. Smyrnakis). 0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 2 ) 0 2 6 2 9 - 1

(2) (3)

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   2†   1  2 α  1† An + A†n , ≡ + iθ a (4) h¯ an − iθ an1 + ha ¯ n n 2nα 2n  nα  2 † an − an2 , pˆ n2 = −i (5) 2  j† where α = h¯ 2 + θ 2 . The creation and annihilation operators satisfy the usual commutation relations: [ani , an ] = δij , from which one deduces that [An , A†n ] = 1. The Hamiltonian of this harmonic oscillator is qˆn2

=

H =α

 2 1 † n ani ani + 2

(6)

i=1

a

a

and the time evolution of the modes ani , ani is found to be ani (t) = ani e−i h¯ nt and ani (t) = ani ei h¯ nt . Consider now the free closed bosonic string in two dimensions. It is necessary to have at least a two-dimensional target space to employ a noncommutative harmonic oscillator quantization procedure. The coordinates that satisfy both the equations of motion and the boundary conditions admit a real expansion of the form †

1 (σ, t) = X01 + X

†

†

N1 P01 t+ 0σ+ qn1 (t) cos nσ + q¯n1 (t) sin nσ, 2 2

(7)

N2 P02 t+ 0σ+ qn2 (t) cos nσ + q¯n2 (t) sin nσ. 2 2

(8)

n>0

2 (σ, t) = X02 + X

n>0

We treat qn1 , qn2 and q¯n1 , q¯n2 as two separate systems of two-dimensional harmonic oscillators. Upon quantization the operators qˆn1 , qˆn2 and qˆ¯ 1n , qˆ¯ 2n are expressed in terms of the operators an , an† and bn , bn† , respectively, obeying √ √ 1 = −a 1 † , identical commutation relations (1). Rescaling the modes by ani → ani / n, bni → bni / n and defining a−n n †

†

†

2 = a 2 , b 1 = b 1 , b 2 = −b 2 we have a−n n −n n −n n

1 (σ, t) = X01 + X

 a a N1 P01 i 1

n e−in( h¯ t +σ ) , t+ 0σ+ Cn e−in( h¯ t −σ ) + C 2 2 2 n

(9)

 a a N2 P02 i 1

n e−in( h¯ t +σ ) , t+ 0σ+ Dn e−in( h¯ t −σ ) + D 2 2 2 n

(10)

n =0

2 (σ, t) = X02 + X

n =0

where



   α 1 α 1 1

b + ian , b − ian1 , Cn = Cn = − 2 n 2 n   α α

sign(n)(Bn + iAn ), sign(n)(Bn − iAn ), Dn = Dn = − 2 2 h¯ a 2 − iθ an1 h¯ b2 − iθ bn1 An = n , Bn = n . α α The C and D modes obey the following commutation relations: [Cn , C−n ] = nα,

[Cn , D−n ] = inθ sign(n),

[Dn , D−n ] = nα,

(11) (12) (13)

(14)

where the corresponding barred operators satisfy the same commutation relations and commute with the unbarred operators. Note that we cannot use the noncommutative harmonic oscillator to quantize the zero modes because the commutation relations (1) are not well defined for n = 0. The zero modes are taken to satisfy the commutation

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159

relations  i  i j j XL , PL = XR , PR = iαδij ,

(15)

i + X i )/2 and where X0i = (XL R

PLi =

Ni h¯ P0i + 0, α 2 2

PRi =

Ni h¯ P0i − 0. α 2 2

(16)

The reason for adopting the commutation relations (15) will be justified when we consider propagators in Euclidean space–time (26). The regularized Hamiltonian, which is the sum of the Hamiltonians of the individual noncommutative harmonic oscillators (6) together with the Hamiltonian for the zero modes, is:  2  1  i 2  i 2  α 2 

−n C

−n D

n + D−n Dn + D

n H= PL + PR + C−n Cn + C 2 2 h i=1 n>0 ¯ +i

  α αθ 



D + . C D C − C D + D − C −n n −n n −n n −n n 12 h¯ 2

(17)

The term α/12 stems from the normal ordering. This has been constructed so as to generate time translations on the coordinates Xi . It is similarly possible to construct the generator of spatial translations which is:  2  h¯  i 2  i 2  α 

n + D−n Dn − D

n

−n C

−n D PR − PL + P= C−n Cn − C 2α h¯ i=1 n>0 +i

  θ

−n C

−n D

n + C

n . D−n Cn − C−n Dn − D h¯

(18)

Note that the momentum and the Hamiltonian operator commute, as expected. Performing the Wick rotation and passing to the complex plane (t → −it, θ → −iθ , τ = −i αh¯ t, w = τ + iσ , z = e−w ), we obtain 1 X1 (z) = xR1 − ipR ln z + i

1 n =0

1

1 (¯z) = xL1 − ipL X ln z¯ + i

1 n =0

2 ln z + i X2 (z) = xR2 − ipR

n

1 n =0

2

2 (¯z) = xL2 − ipL X ln z¯ + i

n

n

1 n =0

n

Cn z−n ,

(19)

n z¯ −n , C

(20)

Dn z−n ,

(21)

n z¯ −n , D

(22)

i (¯z))/2. where Xi (σ, τ ) = (Xi (z) + X 1 (σ, t), X 2 (σ, t) on a circle of radius R. Since X i (σ + 2π, t) = X i (σ, t) + 2πnR Suppose now we compactify X j the possible eigenvalues of N0i are N0i = 2nR. Using now the commutation relations (15) we get [X0i , P0 ] =

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2

i αh¯ δij .1 This means that the plane waves e

i

h¯ Xi P i α2 0 0

have to be well defined. Hence the possible eigenvalues of the

α2 m h¯ R .

momentum are = The partition function on the torus for this prototype theory is given by: τ1   h¯ τ2 Z = Tr e2πi h¯ P e−2π α h¯ H , P0i

(23)

and D, D

operators subjected to where the trace is taken over the module generated by the negative modes of C, C the commutation relations (15). Also a sum over the eigenvalues of the zero modes is assumed in the trace. After evaluating this trace we get the usual expression as if noncommutativity was absent: Z=

Z02 (τ, τ¯ ) , |η(τ )|4

(24)

where Z0 (τ, τ¯ ) =

∞

1 √

q2

m nR α 2R −√ α

2

1 √

q¯ 2

m nR α 2R +√ α

2

,

(25)

n,m=−∞

q = e2iπτ , and R is the compactification radius. Note that the zero-mode contribution for each dimension is Z0 (τ, τ¯ ) while the oscillatory part contribution for both dimensions together is 1/|η(τ )|4 . To make further progress we need to determine the propagators. These turn out to be:  2  1   2    2  X (z)X2 (w) = xR2 − α ln(z − w), X (z)X1 (w) = xR1 − α ln(z − w),



   1   1 2   2 1  2 w w 2 1 X (z)X (w) = xR xR − θ ln 1 − (26) , X (z)X (w) = xR xR + θ ln 1 − . z z In determining (26) we have imposed the commutation relations (15) among the zero modes which guarantee the ln(z − w) dependence of X1 (z)X1 (w) and X2 (z)X2 (w). Note that there is a new singularity introduced at z = 0, but it is not possible to avoid it by changing the zero-mode commutation relations because then it becomes impossible to find generators of time and space translations. Identical relations hold in the antiholomorphic sector. Regarding the stress-energy tensor, there is no longer a unique generator of conformal transformations for both 1 :(∂z X1 (z))2 : that generates conformal transformations Xi components. Rather there is the usual tensor T1 (z) = − 2α on X1 (z) but not on X2 (z) and conversely for T2 (z). The algebra of the moments of each stress-energy tensor is the Virasoro algebra with central charge one. Nevertheless the moments of T1 , T2 do not commute. The primary fields are the usual ones for each string component. Our next task is to compute the correlation functions on the sphere. The only interesting two-point function is the mixed one:

 w θk1 k2 ik1 X 1 (z) ik2 X 2 (w) ik1 xR1 ik2 xR2 ::e :|0 = 0|:e ::e :|0 1 − = 0. 0|:e (27) z It vanishes because of the expectation value of the zero modes, unless both k1 , k2 are 0. The zero-mode expectation value also indicates that in higher correlation functions charge conservation must be maintained for each primary field separately. This implies that the first correlation function that will differ from the commutative case is the four-point function. It takes the form: θkλ   ikX1 (u) −ikX1 (v) iλX2(w) −iλX2 (z)  2 2 (u − w)(v − z) :e (28) ::e ::e ::e : = (u − v)−αk (w − z)−αλ . (u − z)(v − w) 1 We should stress that the zero modes X1 , P 1 commute with X2 , P 2 in contrast to what happens with qˆ 1 , qˆ 2 . This is because we have n n 0 0 0 0 treated separately the zero modes due to the breakdown of the commutation relations (1) for n = 0. In fact, we have only slightly modified the zero-mode commutation relations we had for the usual bosonic string.

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This four-point function is covariant under global sl(2, C) transformations, a fact that indicates the dependence of the correlation functions on the complex structure only. Thus the noncommutative quantization procedure we apply gives a nontrivial deformation of the original theory, since the four-point function is deformed, while the nice geometric properties of the four-point functions are preserved. It is possible to generalize the above formula to the case of the 2n-point functions of the form below:   n n     2  k k αδ iki X σ (i) (zi ) −iki X σ (i) (wi ) :e ::e : = (zi − wi )−αki Cr(i, j ) i j σ (i),σ (j)+θ 0σ (i),σ (j) , (29) i=1

i
i=1 (z −z )(w −w )

where σ : {1, . . . , n} → {1, 2} and Cr(i, j ) = (wjj −zi i )(zjj −wii ) . Note that when a particular pair of insertions zi , wi is far from the rest of the insertions then the 2n-point function factorizes into a product of a two-point function and a (2n − 2)-point function as expected. As a conclusion, we have shown that, in the case of two target space dimensions, following a quantization procedure based on a particular noncommutative harmonic oscillator leads to a modular-invariant partition function where the modular parameter depends on the noncommutativity parameter θ . This new theory is shown to preserve sl(2, C) covariance of the four-point function and certain 2n-point functions on the sphere. Quantization using a more general noncommutative harmonic oscillator is currently under investigation.

References [1] J. Scherk, Rev. Mod. Phys. 47 (1975) 123; M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Cambridge Univ. Press, Cambridge, 1987; P. Ginsparg, Applied conformal field theory, in: E. Brézin, J. Zinn-Justin (Eds.), Les Houches, 1988, Elsevier, Amsterdam, 1989. [2] C.-S. Chu, P.-M. Ho, Nucl. Phys. B 550 (1999) 151. [3] A. Connes, M.R. Douglas, A. Schwarz, JHEP 02 (1998) 003. [4] A. Hatzinikitas, I. Smyrnakis, J. Math. Phys. 43 (2002) 113, and references therein.