Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8, 913-930 2025 Publisher: Learning Gate DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 11 June 2025; Revised: 14 June 2025; Accepted: 18 June 2025; Published: 15 August 2025 * Correspondence: abdelmoumene.medadib@umc.edu.dz Parafermionic open string theory in noncommutative phase-space Mohamed Adib Abdelmoumene1*, Nadir Belaloui2 1,2Physics Department, Laboratoire de Physique Mathématique et Physique Subatomique, LPMPS, University Mentouri Constantine 1, Constantine, Algeria; abdelmoumene.medadib@umc.edu.dz (M.A.A.) belaloui.nadir@umc.edu.dz (N.B.) Abstract: We study how a free, open parafermionic string theory behaves when its target space is noncommutative. We consider noncommutativity in both space and momentum. We find new trilinear commutation relations for the string’s oscillating modes and modified Virasoro superalgebras with additional anomaly terms. This noncommutativity breaks Lorentz invariance and makes the mass operator non-diagonal. To address these issues, we propose a new Fock space that diagonalizes the noncommutativity parameter matrices, leading to a diagonalized mass operator. We also impose constraints on the noncommutativity parameters to eliminate the anomalies and recover the usual mass spectrum. This allows for the GSO projection, which restores spacetime supersymmetry. Finally, we impose additional constraints on the zero modes of the noncommutativity parameters to recover Lorentz invariance. In general, our work provides a clearer picture of how noncommutative structures can be meaningfully included in string theory and suggests that even subtle deformations can carry significant consequences for the theory’s symmetry and dynamics. Keywords: GSO projection, Lorentz algebra, Noncommutativity, Parafermionic strings, Virasoro super-algebra. 1. Introduction The idea of paraquantization began in 1950 by Wigner [1] who showed that the bilinear canonical commutation relations are a particular solution in order to satisfy the wave-particle duality. In 1953, Green [2] generalized the creation-annihilation operator algebra for bosons and fermions based on trilinear commutation relations. This paraquantization is parametrized by the order Q such that Q = 1 represents the usual canonical quantization. One can check as an application of the paraquantization in string theory done by Ardalan and Mansouri [3] in which they considered the center-of-mass coordinates obey the ordinary commutation relations, however, the oscillations are written in the paraquantum mode. Another application was investigated [4-6] where the string variables verify the trilinear commutation relations. As results for the two approaches, new possibility of a critical dimensions are obtained: 24 2D Q = + for the parabosonic string, 2 8 D Q = + for paraspining string and 24 3D Q = + for parabosonic membrane. A bosonic string in noncommutative space-time [7] can be generalized into the paraquantum case [8] indeed, it was found that the Virasoro algebra contains a new term of anomaly and the reconstruction of the Fock space is needed, in order to restore the photon state. For a parabosonic string between two parallel Dp and Dq brane, and by taking some restriction on the noncommutative parameters, the model will be free of tachyon. Finally, the closed parabosonic string has been investigated, where they demonstrate a reduction in the spectrum, specifically restoring the critical massless graviton state. 914 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate For the Fermionic case Hamam and Belaloui [9] and Hammam and Belaloui [10] between two parallel Dp-, Dq-branes in Ramond and Neveu–Schwarz sectors, the study examines the possible existence of a free tachyon model within this system under specific conditions related to paraquantization and brane dimensions. To validate the model, the partition function is calculated and compared to the results of degeneracies, showing a perfect match. This consistency also confirms the integrity of the Virasoro superalgebra. The purpose of this paper, is to study the paraquantum extension of a free fermionic string propagating in a noncommutative target phase-space [11] we will define the paraquantization generalization. Then, we calculate the Virasoro para-super-algebra, evaluate the mass spectrum and examine the Lorentz invariance. Finally, we summarize our work and conclude. 2. Open Fermionic Strings We begin by considering the dynamics of free fermionic strings propagating within a noncommutative target space [7, 8, 11-15]. The action governing these strings is described by: ( ) ( ) ( ) ( )  1 , , , , 2 S d d X X i                     = −   −  (1) where the noncommutation relations are given by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) , , , , , , , , , , , , X P i X X i P P i                                                  = −     = −     = −   = − and where ( ) ( ) 1 , , 2 P X       =   ,  represent the noncommutativity parameters of the space part and  the ones of the momentum part of the phase-space. One can write the Fourier expansions for the variables ( ), ( )      −  −  [7] and ( , ), ( , )X       [16-21]: ( ) ( )in n n e       + − =− − =  (3) ( ) ( )in n n e       + − =− − =  (4) ( ) 0 1 , 2 2 cos( ) in n n X x p i n e n            −   = + +  (5) (6) (2) ( ) ( ) 1 2 1 sec : ( , ) 2 1 sec : ( , ) 2 ir r r Z in n n Z NS tor b e R tor d e             − −  + − −  − = − =   915 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate Using (3), (4), (5) and (6), we can verify that the equations (2) are equivalent to the following modified commutation relations of the oscillator algebra [7]: (7) 2 2 ,0 (2 ) , 2 2 m n n n n m n m i i             +     = + +          ,0 ,0 , , m n m n r s r s d d b b           + +  =  = These modifications will impact the Virasoro super-algebra for both the Ramond and Neveu- Schwarz sectors, introducing new anomalies terms 3. Modified Virasoro Super-Algebra The Virasoro generators in a quantized system are given by: For Ramond sector: (10) (11) (12) (13) which represent the bosonic sector. Given the modifications to the oscillator algebra in (8), one can derive the modified Virasoro super- algebras for both sectors [11]. which represent the fermionic sector. For Neuveu-Schwarz sector: 1 : : 2 1 1 : : 2 2 m n m n n Zd m m m d m n m n n Z L L L L L n m d d    − +  − +   =  = + =    = +      m n m n n Z F d− +  = 1 2 1 : : 2 1 1 : : 2 2 m n m n n Z b m m m b m r m r r Z L L L L L r m b b    − +  − +  +  =  = + =    = +        r n r n n Z G b− +  = 2 0 2 0 2 2 2 0 0 , , 2 , 4 p p i x p i i x x i i                         =    = −    = −  (8) (9) 916 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( )( ) ( ) ( ) 2 ,0[ , ] ( ) 1 12 m n n m m n mn d L L m n L m m R   + += − + − + (14) where Rmn represent the anomaly part due to the noncommutativity, defined by: ( ) ( ) ( )( )2 221 2 2 2 mn p n m p p n m p p m n p p i R i p n m p              + − − − − + − =−   = − + + − + −    (15) The super-algebra then, is: For Neuveu-Schwarz sector: (16) 8   1 , 2 m r m r mrL G m r G V+   = − +    (17)   2 1 , 2 2 4 r s r s r s rs D G G L r B+ +   = + − +    (18) where Brs, Vmr are given by: ( ) ( ) ( )( )2 221 2 2 2 rs q s r q q s r q q r s q q i B i q s r q b b            + − − − − + − =−   = − + + − + −    (19) ( ) ( ) ( )( )2 221 2 2 2 mr q r m q q r m q q r m q q i V i q r m q b             + − − − − + − =−   = − + + − + −    (20) For Ramond sector: (21)   1 , 2 m n m n mnL F m n F W+   = − +      2, 2 2 r s r s r s rs D F F L r D+ += + + with again Drs, Wmn are given by: (23) ( ) ( ) ( )( )2 221 2 2 2 rs q s r q q s r q q r s q q i D i q s r q d d            + − − − − + − =−   = − + + − + −    (24) ( ) ( ) ( )( )2 221 2 2 2 mn q n m q q n m q q n m q q i W i q n m q d             + − − − − + − =−   = − + + − + −    (25) 4. Modified Lorentz Algebra The angular momentum Mµν is given by: ( )2 ,0[ , ] ( ) 1 8 m n n m m n mn D L L m n L m m R+ += − + − + 3 ,0[ , ] ( ) 8 m n n m m n mn D L L m n L m R+ += − + + 917 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( ) ( ) ( ) ( ) 1 1 1 N-S sector 4 1 R-sector 4 n n n n n r r r r r n n n n n m m m m m x p x p i n i b b b b M x p x p i n i d d d d                                   − − = + − − =−  − − = + − − =−  − − − −   − →  =   − − − −    − →      (26) Using (7) (8) (9), a direct calculation leads to the following modified Lorentz algebra [11]: (27) [ , ]p M i p i p K       = − + (28) 2 0,p p i     =  (29) where Tµνρλ, Kνµρ represent the anomalies due to the noncommutativity and which are given by: ( ) ( ) 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 2 2 0 0 2 2 2 0 0 0 2 4 4 4 x x x x T i x x x x x p x p x p x p i x p x p x p x p i i p p i i p p i i                                                                           + + = +  +   − +   − +  +  − +   −  − + − + −( ) ( ) ( ) ( ) ( ) 2 2 2 0 2 2 2 0 0 2 2 2 2 2 2 2 4 (2 ) 2 2 (2 ) 2 2 (2 ) 2 2 (2 ) 2 n n n n n n n n n n n n n n n n n n n p p i i p p n i i n i i n i i i                                                                  − − − − − −  + − +   + + +       + + +       + + +      +  ( ) 2 2 n n n n n n i           − −   +    [ , ]                  M M i M i M i M i M T                = − + + − + (30) 918 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate 5. Paraquantization Generalization Now, we are going to generalize the theory into the para-quantization, where we have a trilinear commutation relation, instead of the usual commutations. The generalization will be done in the light cone coordinates. By defining the Green’s ansatz [6, 8-10] one can present the Green representation of the bosonic and fermionic string coordinates: ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , , , , Q ii Q jj A A X X               = = = =   (32) Where i = 2..D−2 are the light cone coordinates, α = 1,2...Q which represent the Green indices, and Q represent the paraquantization order (taking Q = 1 we find results of the ordinary case). So, the generalization of canonical variables for the parabosonic and parafermionic coordinates are given by: The equations (33) are equivalent to the following tri-linear commutation relations: 2 2 0 0 2 2 0 0 2 2          K i p i p i p i p                     = − + − (31) (33) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ( , ), ( , )] ( ) , , , 0;     , , , , , , 0;     , , , , , , 0;     , i j ij i j i j ij i j i j ij A B AB i j A B X X i X X P P i P P x p i x                                                             + + + − − + −  = −   =      = −     =      = −     =      =   ( ) , 0;    p    + +   =    919 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) , , , , , 2 , , , , , , , 2 , , , , , , , 2 , , , , , , i j k ij k ik j i j k ij k ik j i j k ij k ik j i j X X X i X X X P X i X P X P P i P P P X P                                                        + + +         = − + −           = − + −           = − + −    ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 2 , , , , , , , 2 , , , , , , , 2 , , , , , , 2 k ij k ik j i j k ij k ik j i j k ik j A A i j k ik A i P X P P P i P P X X i P P i                                                      + + + +        = − − + −           = − + −         = −        = −   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , 2 , , , , , , 2 , , , , , , , 2 , , , , , , j A i j k ik j A A i j k ij k ik j A B C AB A AC B i j k ik j A C AC i j k A C X P i X X P                                                                + − +        = −           = − − −         = −    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , , , , , 2 , , , , 2 , , , , 2 , , 2 , , 4 ik j AC i j ij i j ij i j ij P X X A i A X P A i A P P A i A x p B iB x p p ip                              + + + + − + + − + + + +      = −       = −       = −       = −      =      =   (34) Now, in terms of modes, the trilinear equations take this form: 920 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( ) ( ) ( ) 2 22 2 22 2 2 , , 2 2 2 2 2 2 , , 2 2 2 , , i j k ij ij ij k ik ik ik j m n l n n m n l l l m l n i j ij ij ij m n n n m n n n m i i m i i n C m i i C x p                              + ++ ++ − +          = + + + + +                         = + +           ( )   ( ) ( ) ( ) 2 0 0 2 2 0 0 2 0 2 2 , , 2 2 2 21 1 , , 2 2 2 2 2 21 , , 2 2 2 i i m m i j k ij k ik j m n m n i j k ij ij k ik ik j i j k ij ij i p p p i p p x p p i p p p x x i                    + + ++ + +    =       = +             = − + −                      = − +      ( ) ( )( ) ( )( )  ( ) ( ) 2 0 2 22 2 0 0 0 0 21 2 2 , , 2 2 2 , , 2 , , 2 k ik ik j i j k ij ij k ik ik j i j k ij k ik j n m l n m l n l m i j k ij k ik j n m l n m l n l m x x x x x i x x d d d d d b b b b b                     + + +− + +−     + +                = − + −      = −      = −   (35) 6. Virasoro Para-Super-Algebra The Virasoro generators in the parafermionic strings are defined by: (36) as a result, the equations (16) to (25) will take this form: For Ramond sector, we have: [ , ]m n m n n Z F d− + +  = which represent the parafermionic sector. For Neuveu-Schwarz sector, we have: (37) (38) 1 2 1 , 4 D i m m p pi i p L   −  ⊥ − + = =−  =   2 2 2 2 1 , 4 1 1 [ , ] 2 2 D i m m p pi i pd m m m D d i m n im n i n Z L L L L L n m d d     −  − + = =− − − + − =    =    = + =    = +       921 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate (39) (40) which represent the parabosonic sector. By using the trilinear commutations relations (35) and by analogy with calculation done in (21), (22) and (23), one can find that: for Ramond sector:   ( )     3 ,0 2 ,0 1 , 8 1 , 2 2 1 , 2 n m n m n m nm n m n m n m rs n m n m mn D L L n m L Q n R D F F L Q n D L F n m F W   + + + ++ + − = − + + − = + +   = − +    (41) where Rnm, Dmn and Wmn are given by: ( ) ( ) ( )( )2 221 2 [ , ] 2 2 ij ij ij ij i j mn p n m p p n m p p m n p p i R i p n m p         + − − − − + − + =−   = − + + − + −    (42) ( ) ( ) ( )( )2 221 2 , 2 2 ij ij ij i j rs q s r q q s r q q r s q q i D i q s r q d d       + − − − − + − − =−     = + + − + −      (43) ( ) ( ) ( )( )2 221 2 , 2 2 ij ij ij i j mn q n m q q n m q q n m q q i W i q n m q d        + − − − − + − + =−     = + + − + −      (44) for Neuveu-Schwartz sector:   ( )     2 ,0 2 ,0 1 , ( 1) 8 1 1 , 2 ( ) 2 4 1 , 2 n m n m n m nm n m n m n m rs n m n m mr D L L n m L Q n n R D G G L Q n B L G n m F V   + + + ++ + − = − + − + − = + − +   = − +    (45) where Rnm is given by (42), and Brs and Vmr are given by: (46) [ , ]r n r n n Z G b− + +  = 922 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( ) ( ) ( )( )2 221 2 , 2 2 ij ij ij ij i j mr q r m q q r m q q r m q q i V i q r m q b        + − − − − + − + =−     = + + − + −      (47) 7. Mass Spectrum and GSO Projection The mass operator will be written in the paraquantized form as [9, 10]: Ramond sector: 1 1 2 2 1 2 1 1 , , 2 D D i i i i n n r r i n i R r M r dd   −  −  − −+ − = = = =      = +         (48) (49) Neuveu-Schwarz sector: We note that [3-5]. We need to diagonalize the antisymmetric matrices θm and γm by introducing the unitary matrix Um such that: ( ) ( )1 ij mij ij m m m m iU i U D  − = = (50) and, ( ) ( )1 ij mij ij m m m m iU i U T  − = = (51) With [θm , γm] = 0. This last can be obtained through a redefinition of the Fock space [8, 22] in order to get a diagonal mass in this new basis. The redefinition takes this form: ( )  ( ) ( ) ( )  ( ) ( ) , , , , , , 1 1 1 32 1 2 1,2.. , ... 2 2 1 1 1 1 32 1 2 1,2.. , ... 2 2 1 , ! 1 , ! m i r j n j m i r j n j D D i j j T m r n i m j n r D D i j j T m m r n i m j n r b or d p p h U b or d p p h         −  − + − − − = = = =+ = − −  − − + − − − = = = =+ = −       →                              Where ⟨...⟩± represent the symmetrized (anti-symmetrized) form of the bosonic (Fermionic) oscillators product, , , m i m i h = represent its different possible permutations of the oscillators and ,n k takes either zero or one. In order to get an equivalent of a GSO projection, one can use the usual way to get the following steps in the table below (Table 1) and (Table 2). 1 1 2 12 1 2 2 1 2 , , 2 16 D D i i i i NS n n r r i n i r D M r b b Q   −  −  − −+ − = = = =   −     = + −              8 2D Q − = (52) 923 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate The results of GSO projection for the two sectors are grouped in (Table 3). (See (appendix A) as an example of calculations of mass spectrum). Table 1. This table represents the mass spectrum in terms of redefined modes. Level N-S Sector state Mass 0 |0⟩ 1 1 2 | 0ib −  0 2 1 1 2 2 1 , | 0 2! i jb b − − −       1 2α′ U 3 1 1 1 2 2 2 1 , , | 0 3! i j kb b b − − − −    3 2 | 0ib −  1 1 1 1 2 1 , | 0 2! i jU b− − − +       4 1 1 1 1 2 2 2 2 3 1 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 1 , , , | 0 4! 1 , | 0 2! | 0 1 , | 0 2! 1 , , | 0 3! i j k l i j i j k j k l b b b b b b U U U U b b     − − − − − − − − − − − − − − + − − + − −                  924 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate 5 1 1 1 1 1 2 2 2 2 2 5 2 1 1 1 2 2 2 1 2 2 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 2 2 1 1 1 3 2 1 , , , , | 0 5! | 0 1 , , 3! 1 , | 0 2! 1 , , | 0 3! 1 , , , | 0 4! 1 , | 0 2! i j k l m i i j k j k j k l j k l m j k b b b b b b b b b U b U U b U b b b U b      − − − − − − − − − − − − − − + − − − − + − − − + − − − − − − +                         Table 2. This table represents the mass spectrum in terms of redefined modes. Level R-Sector State Mass 0 |0⟩ 0 1 1 1 1 1 | 0 | 0 j j d U  − − −   1 α′ 2 2 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 | 0 1 , | 0 2! | 0 1 , | 0 2! 1 , | 0 2! j j k j j k j k d d d U U U U d     − − − − − − − − − − + − − − +               925 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate Table 3. This table represents the GSO projection for the two sectors. Level N-S Sector R-Sector State Masse State Masse 1 1 2 | 0ib −  0 |0⟩ 0 3 1 1 1 2 2 2 1 , , | 0 3! i j kb b b − − − −    3 2 | 0ib −  1 1 1 1 2 1 , | 0 2! i jU b− − − +       1 1 1 1 | 0 | 0 j j d U  − − −   5 1 1 1 1 1 2 2 2 2 2 5 2 1 1 1 2 2 2 1 2 2 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 2 2 1 1 1 3 2 1 , , , , | 0 5! | 0 1 , , 3! 1 , | 0 2! 1 , , | 0 3! 1 , , , | 0 4! 1 , | 0 2! i j k l m i i j k j k j k l j k l m j k b b b b b b b b b U b U U b U b b b U b      − − − − − − − − − − − − − − + − − − − + − − − + − − − − − − +                         2 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 | 0 1 , | 0 2! | 0 1 , | 0 2! 1 , | 0 2! j j k j j k j k d d d U U U U d     − − − − − − − − − − + − − − +               926 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate One can then impose: ( ) ( ) ( )1 1 2 1 2 i i   − =  (53) to restore the value of the mass for the first excited state (for example), and in general: ( ) ( ) ( ) 2 2 2 m m i i m    − =  (54) equivalent to: to restore those of the other levels, where m > 0 represents the number of state level. Finally, we obtain (Table 4). Table 4. This table represents the first levels of the mass spectrum after GSO projection and the application of the equation (54). Level N-S Sector R-Sector State Mass State Mass 1 1 2 | 0ib −  0 |0⟩ 0 3 d U 5 d 5 2 1 1 1 2 2 2 1 2 2 1 2 | 0 1 , , 3! 1 , | 0 2! i i j k j k b b b b U b − − − − − − − − +          U By applying (UmUm −1) on the both sides of (50) and (51), one can show that the equation (55) can be expressed with respect to θm and γm. ( ) 2 ( ) ( )2 2 ij ij m m m    − =  (56) From this result, we can fix our starting model (2) by imposing to θµν and γµν the following relation: ( ) 2 ( ) ( )2 2 ij ij m m m T D  − =  (55) 927 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate (57) where m ̸= 0 and µ,ν = 0,1...,D−1. With this condition (57), one can easily verify that all the anomaly terms (15), (19), (20), (24) and (25) of the modified Virasoro algebra due to the noncommutativity are eliminated. This result is a direct consequence of the fact that we considered noncommutativity between coordinates and moments instead of only between coordinates. In the other hand, the Lorentz algebra’s anomaly term (30) is simplified to: For the zero mode noncommutativity parameters, if we impose that 0 0 0  = = , the Lorentz algebra is restored where (27), (28) and (29) become as the ordinary ones, despite of the fact that the noncommutativity is still present in the relations (2) and (8). 8. Summary and Results To conclude, we have investigated the free open fermionic string theory in a noncommutative target phase-space. We postulated the noncommutation relations (2) and derived the ones in the paraquantum case (34). We found that the modification in the commutation relations in terms of oscillating modes introduce a new anomaly terms in the Neuveu-Schwarz and Ramond Virasoro super-algebras. The Lorentz covariance is affected by the noncommutativity and the mass operator becomes non-diagonal in the usual Fock space. A redefinition of this latter is possible and a diagonalized mass operator is obtained. We then imposed specific constraints on the noncommutativity parameters θµν and γµν to cancel the anomaly terms and recover the standard mass spectrum. Under these conditions, the GSO projection becomes applicable, allowing for the restoration of spacetime supersymmetry. Finally, to recover Lorentz invariance, we required the vanishing of the zero modes of the noncommutativity parameters, namely γ0 µν = 0 and θ0 µν = 0. As a result, equations (27), (28), and (29) reduce to their standard forms, while noncommutativity still remains encoded in equations (2) and (8). Overall, our analysis shows complete consistency between the results obtained from standard quantization and those from the paraquantum approach. ( ) 2 ( ) ( )2 2 m m m    − =  (58) ( ) ( ) 2 0 0 0 0 0 0 0 02 0 0 0 0 2 2 2 0 0 2 2 2 0 0 0 2 4 4 4 x x x x T i x x x x x p x p x p x p i x p x p x p x p i i p p i i p p i i                                                                           + =   + +   −   + − + +  + −    + −  − + − + −( ) ( ) 2 2 2 0 2 2 2 0 04 p p i i p p                  + − 928 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate Funding: This work is supported by the Algerian Ministry of High Education and Research under the PRFU project (Grant Number: B00L02UN250120220011). Transparency: The authors confirm that the manuscript is an honest, accurate, and transparent account of the study; that no vital features of the study have been omitted; and that any discrepancies from the study as planned have been explained. This study followed all ethical practices during writing. Copyright: © 2025 by the authors. This open-access article is distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). References [1] E. P. Wigner, "Do the equations of motion determine the quantum mechanical commutation relations?," Physical Review, vol. 77, no. 5, pp. 711-712, 1950. https://doi.org/10.1103/PhysRev.77.711 [2] H. S. Green, "A generalized method of field quantization," Physical Review, vol. 90, no. 2, pp. 270-273, 1953. https://doi.org/10.1103/PhysRev.90.270 [3] F. Ardalan and F. 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Wang, "Strings in noncommutative spacetime," arXiv preprint hep-th/0503111, 2005. https://arxiv.org/abs/hep-th/0503111 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.1103/PhysRev.77.711 https://doi.org/10.1103/PhysRev.90.270 https://doi.org/10.1016/0370-2693(86)90931-7 https://doi.org/10.1023/A:1025970432658 https://doi.org/10.1023/B:CJOP.0000029691.20924.71 https://doi.org/10.1590/S0103-97332009000600007 https://doi.org/10.1590/S0103-97332009000600007 https://doi.org/10.1007/s10773-009-9983-3 https://doi.org/10.1142/s0217751x15501754 https://doi.org/10.1142/s0217751x18500483 https://arxiv.org/abs/2307.07060 https://doi.org/10.1140/epjc/s2002-01044-y https://doi.org/10.1016/S0370-2693(02)02847-2 https://doi.org/10.1142/s0217732304013015 https://arxiv.org/abs/hep-th/0503111 929 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate Appendix A. As previously discussed, we now compute part of the mass spectrum in the paraquantized framework. Specifically, for the Ramond sector, we present the calculation of the second excited state as a representative example. The corresponding mass operator is given in equation (48), and by using (35): ( ) ( ) 2 2 2 2 2 2 2 , , 2 2 2 2 ij ij n k m n l ij n i j k m n l ik ik l j m l n ik l n m i i n m i                    + + +    + +      +            =        + +              ( ), , 2i j k ij k ik j n m l n m l n l md d d d d   + +−    = −   we get: 1 1 2 1 1 1 1 1 1 1 1 2 1 2 1 1 1 , | 0 , , , | 0 2! 2 D D j k i i i i j k R n n r r i n i r M U d r d d U d     −  −  − − − − − − − −+ + − + = = = =           = +              (62) We start with the first part of the right-hand side of the equation (62): ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 , , | 0 , | 0 D D i i j k i i j k k j n n n n i n i n U d U d d U       −  −  − − − − − − − − − − −+ + + = = = =       = +        (59) 1 1 2 2 1 2 1 1 , , 2 D D i i i i R n n r r i n i r M r d d   −  −  − −+ − = = = =      = +         (60) (61) (63) 930 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 8: 913-930, 2025 DOI: 10.55214/2576-8484.v9i8.9474 © 2025 by the authors; licensee Learning Gate ( ) ( ) 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 12 1 1 , 1 2 2 2 , 2 | 0 1 2 2 2 i i n n ji ji n i n n ji n j i i k k n n ji ji n i n n ji n U U U i U i U d d U U i U i                      − − + − − − − − − − − − −+ − − − + − −        − + +      +            −      − + +              = ( ) ( ) 1 11 2 1 |0 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 2 2 2 , 2 1 2 2 2 D j k i n d ji ji n i n n ji n k j i i k n n ji ji n ji n U U i U i d U d U U i U i                    −  −− = =  − − − − − − − − − −+ − − − −    − + +    +        + −    − + +          1 1 1 1 1 2 1 1 2 1 , |0 | 0 D ji i k n n i n D i n i n n d U    −  − − − − + = = −  = = + −                                                                               and for the second term of the right-hand side of the equation (62): 1 1 1 1 1 1 1 1 1 2 1 , , | 0 4 , D i i j k ji j k r r i r r d d U d U d   −  − − − − − − −− + + = =       =      (65) Combining (64) and (65), one can get the final result: ( ) 2 2 1 1 1 1 1 1 1 1 1 1 21 1 1 , | 0 4 1 1 , | 0 2! 4 2 2 j k ji ji j k R i iM U d U d           − − − − − − − −+ +       = − − +           (66) So, ( ) ( ) ( )( )21 12 1 1 1 1 1 1 1 1 1 1 1 , | 0 2 2 , | 0 2! 2 j k j k R j jM U d U d       − − − − − − − −+ +       = − −        (67) (64)