Noncommutative scalar fields in compact spaces: quantisation and implications
Mir Mehedi Faruk, Mishkat Al Alvi, Wasif Ahmed, Md Muktadir Rahman, Arup Barua Apu
aa r X i v : . [ h e p - t h ] A ug Noncommutative scalar fields in compact spaces: quantisation andimplications
Mir Mehedi Faruk , ∗ , Mishkat Al Alvi , , Wasif Ahmed , , Md Muktadir Rahman , , and ArupBarua Apu Bose Centre for Advanced Study and Research in Natural Science, University of Dhaka,Bangladesh. Physics Department, McGill University Montreal, QC H3A 2T8, Canada Department of Theoretical Physics, University of Dhaka, Bangladesh Department of Physics and Astronomy, University of Minnesota, Duluth, Minnesota 55812. USA .Department of Physics, University of Dhaka, Bangladesh International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151 Italy Department of Physics, University of South Dakota, 414 E. Clark St., Vermillion, SD 57069. USA .Department of Physics, Western Illinois University, 1 University circle, Macomb, IL, 61455, USA August 31, 2017
Abstract
In this paper we consider a two component scalar field theory, with noncommutativity in its conjugate momentumspace. We quantize such a theory in a compact space with the help of dressing transformations and we reveal asignificant effect of introducing such noncommutativity as the splitting of the energy levels of each individual modethat constitutes the whole system. We further compute the thermal partition function exactly with predicteddeformed dispersion relations from noncommutative theories and compare the results with usual results. It is foundthat thermodynamic quantities in noncommutative models, irrespective of whether the model is more deformed ininfrared/UV region, show deviation from standard results in high temperature region.
It is a common concept that the usual picture of spacetime as a smooth pseudo-Riemannian manifold would breakdown due to quantum gravity effects at very short distances of the order of the Planck length. The deviation from theflat-space concept at the order of the Planck length is actually motivated from new concepts such as quantum groups[1],quantum loop gravity[2], deformation theories[3], noncommutative geometry[4], string theory[5] etc. Besides, the ideaof noncommutative spacetime was also discovered in string theory and in the matrix model of M theory, where in thecertain limit due to the presence of a background field B, noncommutative gauge theory appears[5]. Recently an in-creasing interest towards noncommutative theories has been triggered by the results in string theory[6]. A vast numberof papers dealing with the problem of formulating noncommutative field theory [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]has appeared in literature. Some implications of such noncommutativity in field theory including connections betweenLorentz invariance violation and noncommutativity of fields[14], deformed energy eigenvalue of the Hamiltonian[14, 16],replusive Casimir force[17], deformed Kac-Moody Algebra[16], Bose Einstein condensation[19], noncommutative fieldgas driven inflation[20], UV/IR mixing[21], GZK cutoff[22], path integral [23], matter-antimatter asymmetry[14] etchave appeared in literature. Due to the huge potential of noncommutative field theories to produce interesting results,such theories need extensive attention.Several groups have reported that quantum gravity relics could be seen from the Lorentz violating dispersion relations[28].Lorentz invariance is then considered as a good low-energy symmetry which may be violated at very high energies. Asour low-energy theories are quantum field theories (QFT), it is interesting to explore possible generalizations of the QFT ∗ Corresponding author: [email protected], [email protected], [email protected], [email protected] φ i ( i = 1 ,
2) and the canonical conjugate momentum π i are assumed to be operatorssatisfying the canonical commutation relations,[ φ i ( x , t ) , φ j ( y , t )] = 0 , (1a)[ π i ( x , t ) , π j ( y , t )] = 0 , (1b)[ φ i ( x , t ) , π j ( y , t )] = iδ ij δ ( x − y ) . (1c)Here, ( x , t ) are elements of base space. In their work, Balachandran et. al.[16] and Khelili[17] considered noncom-mutative massless scalar fields with commutative base space and noncommutative target space. Therefore, the abovecommutation relations take the form , [ ˆ φ i ( x , t ) , ˆ φ j ( y , t )] = iǫ ij θδ ( x − y ) , (2a)[ˆ π i ( x , t ) , ˆ π j ( y , t )] = 0 , (2b)[ ˆ φ i ( x , t ) , ˆ π j ( y , t )] = iδ ij δ ( x − y ) , (2c)where ǫ ij is an antisymmetric constant matrix and θ is a parameter with the dimension of length. After constructingthe Hamiltonian formulation of this theory and quantizing it in a compact space, Balachandran et. al. have obtained asplitting of the energy levels of each individual mode that constitutes the whole system. The resemblance of this effectto the well known Zeeman effect in a quantum system in the presence of a magnetic field is noticed[16]. Balachandranet. al. have considered a S × S × S × R type geometry, where all compactified spatial coordinates, i.e., S , areof same radius. In this manuscript we investigate a different type of noncommutativity in scalars. Here, we explorethe case where the fields are commutative but the conjugate momentum space is noncommutative. Therefore thecommutation relations are of the form, [ ˜ φ i ( x , t ) , ˜ φ j ( y , t )] = 0 , (3a)[˜ π i ( x , t ) , ˜ π j ( y , t )] = iǫ ij θδ ( x − y ) , (3b)[ ˜ φ i ( x , t ) , ˜ π j ( y , t )] = iδ ij δ ( x − y ) . (3c)At first, we canonically quantize a free massless boson theory with the commutation relation in Eq. 3 in (1 + 1)dimension following the regularization procedure shown in the seminal paper of Balachandran et. al.[16]. We thenconstruct a Fock space, since the Hamiltonian can be diagonalized using the Schwinger representation of SU(2).Afterwards, we generalize the results in arbitrary dimensions. Finally, we compute the thermal partition function forthe deformed Hamiltonian due to noncommutativity in Eq. (2) and (3) and compare them. Let us consider a theory in a ( d + 1)-dimensional base space and the target space is set in a commutative plane R .Here, we present a free massless boson theory with commutative base space and noncommutative target space. Thetarget space is the space where the field take its values. The spatial part of the base space is a d dimensional torus.Now if the field components are denoted by φ i where i = 1 , φ : S × S × ... × S d × R −→ R , (4a)( x , t ) φ ( x , t ) . (4b) We have used hat in field space noncommutativity and tilde for momentum space noncommutativity.
2e invoke that each spatial direction is compactified in S j with radius R j , which causes the field components to beperiodic in the spatial coordinates, φ ( x + R, t ) = φ ( x , t ) (5)Let us write the components of the field φ i ( x , t ) as a Fourier series, φ j ( x , t ) = X n ,j e − πiRj n . x ϕ j n ( t ) , (6)where, n = ( n , n , ...n d ) and thus the Fourier components of the field are, ϕ j n ( t ) = 1 M X j Z d d x e πiRj n . x φ i ( x , t ) . (7)Here, V = d Y j =1 R j , (8)One can notice a real condition φ ∗ n ( t ) = φ − n ( t ) from eq. (7). Now the Lagrangian is, L = 12 X j Z d d x (cid:2) ( ∂ t φ j ) − ( ∇ φ j ) (cid:3) . (9)The above Lagrangian in terms of Fourier mode for d dimensional target space is L = V X i, n ( ˙ ϕ i n ˙ ϕ i − n − ω n ϕ i n ϕ i − n ) . (10)Here, ω n is defined as, ω n = (2 π ) d X j =1 ( n j R j ) . (11)From the Lagrangian, we can now evaluate the expression for the momentum, π i n = ∂L∂ ˙ ϕ i n (12a)= V ˙ ϕ i − n . (12b)Armed with the Lagrangian and the momentum, we can finally write the Hamiltonian of the system, H = X i, n π i n ˙ ϕ i n − L (13a)= X i, n h V π i n π i − n + V ω n ϕ i n ϕ i − n i . (13b)So, ω n corresponds to the frequencies of the set of harmonic oscillators describing the system as defined in eq (11). It is well known that the phase space of a single particle in R has a natural group structure which is the semidirectproduct of R with R . The generators of its Lie algebra can be taken to be coordinates x a , with a = 1 ,
2, andmomenta being p a . [ x a , x b ] = 0 , (14a)[ x a , p b ] = iδ ab , (14b)[ p a , p b ] = 0 . (14c)One can now twist/deform the generators of the above algebra into x a , p b and thus obtain new algebras[16].3 .1 Model 1: Noncommutativity in momentum space, Quantisation in (1+1) Dimen-sion In this model, the algebra of derivatives is deformed, but the function algebra is not. Here we twist (or deform) thegenerators of the above algebra into ˜ x a , ˜ p b and thus obtain a new algebra.[˜ x a , ˜ x b ] = 0 (15a)[˜ p a , ˜ p b ] = iǫ ab θ = i ˜ θ ab (15b)[˜ x a , ˜ p b ] = iδ ab (15c)Here, θ is a parameter and ǫ ab is an antisymmetric constant matrix. We can relate the non-commutative coordinateswith their commutative counterparts in terms of the deformation parameter with the help of dressing transformation[16,30, 31, 32], ˜ p a = p b + 12 ˜ θ ab x b , (16a)˜ x b = x b . (16b)The above dressing transformation map (16) can be easily generalized to scalar field theory. We start with a free realmassless bosonic field. Its base space is a cylinder with circumference R and its target space is R . φ : S × R −→ R (17)( x, t ) −→ φ ( x, t ) (18)The compactification of the space coordinate makes each field component periodic, i.e. φ i ( x + R, t ) = φ i ( x, t ) . (19)As a result φ i ( x, t ) can be written as Fourier expansion as in Eq. (6), where the Fourier components can be rewrittenas eq (7). Now following the spirit of Balachandran et. al.[16], we rewrite the eq. (3),[ ˜ φ i ( x , t ) , ˜ φ j ( y , t )] = 0 , (20a)[˜ π i ( x , t ) , ˜ π j ( y , t )] = iǫ ij θ ( σ ; x − y ) , (20b)[ ˜ φ i ( x , t ) , ˜ π j ( y , t )] = iδ ij δ ( x − y ) , (20c)where, θ ( σ ; x − y ) = θσ √ π e − ( x − y )22 σ . (21)The redefinition of the term θδ ( x − y ) is to be seen as a regularization procedure (see reference [16]). It should be notedthat these new commutation relations reduce to those in eq. (3) in the limit σ →
0. And both in the limit θ → σ indicates a new distance scale in the equal time commutationrelations for the fields [16]. Here, σ has dimension of length, and θ has dimension of (length) . The novelty of theBalachandran et al’s work was to regularise the delta function in the commutation relation, which prevented the energydensity to diverge with respect to frequency. We will use their method in our work as well. Following eq (16), thedressing transformation for field theory in this model is,˜ π i ( x , t ) = π i ( x , t ) + 12 ǫ ij Z dyθ ( σ ; x − y ) φ j ( y , t ) , (22)˜ φ i ( y , t ) = φ i ( y , t ) . (23)Here, i = 1 , π ( x , t ) is the canonical conjugate momentum of the field ˜ φ ( x , t ). The map defined above reads inFourier modes, ˜ π in = π in + 12 R ǫ ij ϕ j − n θ ( n ) , (24)˜ ϕ in = ϕ in . (25) in d + 1 dimension θ has dimension of (length) d − , but σ always have dimension of length. ϕ im , ϕ jn ] = [ π im , φ jn ] = 0 , (26a)[ ϕ im , π jn ] = iδ mn δ ij , (26b)and the modified commutation relations in Fourier space are,[ ˜ ϕ im , ˜ ϕ jn ] = 0 , (27a)[˜ π in , ˜ π jm ] = iR ǫ ij θ ( n ) δ n + m, , (27b)[ ˜ ϕ in , ˜ π jm ] = iδ nm δ ij . (27c)Considering free massless noncommutative scalar fields, the Lagrangian can be written as, L = 12 X i Z dx [( ∂ t ˜ φ i ) − ( ∂ x ˜ φ i ) ] . (28)Now, the Hamiltonian in Fourier space,˜ H = X i (˜ π i ) R + 12 R X i,n =0 (cid:8) ˜ π in ˜ π i − n + (2 π | n | g ) ˜ ϕ in ˜ ϕ i − n (cid:9) = 12 R X i ( π i π i + 1 R θ (0) ǫ ij ϕ j π i + 14 R θ (0) ϕ i ϕ i )+12 R X i,n =0 (cid:8) π in π i − n + [(2 π | n | ) + θ ( n )4 R ] ϕ in ϕ i − n + 1 R θ ( n ) ǫ ij ϕ jn π in (cid:9) (29)Now, the standard harmonic oscillator Hamiltonian can be written as, H n = X i (cid:0) M π in π i − n + 12 M ¯ ω n ϕ in ϕ i − n (cid:1) (30)Comparing with the above equation we can write,¯ ω n = 1 M r (2 π | n | ) + θ ( n )4 R . (31)Now, we can define the creation and annihilation operators as, a in = r ∆ n ϕ in + i π i − n ∆ n ) , (32) a i † n = r ∆ n ϕ i − n − i π in ∆ n ) . (33)Here, ∆ n = R ¯ ω n = r (2 π | n | g ) + θ ( n )4 R . (34)Now, [ a im , a jn ] = [ a i † m , a j † n ] = 0 , (35)[ a im , a j † n ] = δ mn δ ij . (36)The original Hamiltonian can be written in terms of the creation and annihilation operators defined above, H n = X i (cid:0) R π in π i − n + R ω n ϕ in ϕ i − n (cid:1) (37)= X i
12 ¯ ω n (cid:0) a in a i † n + a i †− n a i − n (cid:1) (38)5ow, after normal ordering the Hamiltonian looks like H n = X i ¯ ω n a i † n a in . (39)Now, the ϕ jn π in term of the Hamiltonian can be written as ǫ ij ϕ jn π in = − i ǫ ij [ a j †− n a i − n − a jn a i † n ]= iǫ ij a i † n a jn . Normal ordering has been used to derive this expression. The complete Hamiltonian can be written as,˜ H = H + X i,n =0 ¯ ω n a i † n a in + i θ ( n ) M X i,j ; n =0 ǫ ij a i † n a jn . (40)Now lets define new creation and annihilation operators, A n = 1 √ a n − ia n ) , (41) A n = 1 √ a n + ia n ) . (42)So, if we write the Hamiltonian in terms of these new creation and annihilation operators, it looks like ,˜ H = H + X n ¯ ω n [ A † n A n + A † n A n ] − g M X n θ ( n )[ A † n A n − A † n A n ] (43)= H + X n (¯ ω n − R θ ( n )) A † n A n + X n (¯ ω n + 12 R θ ( n )) A † n A n . (44)This is how energy splitting occurs due to noncommutativity in momentum space. A consequence of such noncommu-tativity in momentum space is the appearance of a term proportional to a component of angular momentum in theHamiltonian of the theory. It affects the splitting of the energy levels. Splitting is also noticed if noncommutativity isintroduced in field space [16]. But the functional form of the two types of splitting are quite different. Now, we will have a look at the deformed conformal generators. The deformed Hamiltonian written with hattedoperators is, ˜ H = X i (˜ π i ) R + 12 R X i,n =0 (cid:8) ˜ π in ˜ π i − n + (2 π | n | ) ˜ ϕ in ˜ ϕ i − n (cid:9) . (45)The deformed creation and annihilation operators can be written as,˜ a in = 1 p π | n | (2 π | n | ˜ ϕ in + i ˜ π i − n ) , (46)˜ a i † n = 1 p π | n | (2 π | n | ˜ ϕ i − n − i ˜ π in ) . (47)So, [˜ a im , ˜ a jn ] = 14 π | n | − iR ǫ ij θ ( n ) δ n + m, , (48)[˜ a im , ˜ a j † n ] = δ ij δ mn + 14 π | n | iR ǫ ij θ ( n ) δ mn , (49)[˜ a i † m , ˜ a j † n ] = 14 π | n | − iR ǫ ij θ ( n ) δ n + m, . (50) following ref. [16] we ignore the zero mode. It is not relevant, since it is associated with the overall translation of the system.
6t should be noted that, if we make θ →
0, the creation-annihilation operators of noncommutative theories coincidewith the usual theory. The generators of the modified U (1) Kac-Moody algebra would be,For n > J in = − i √ n ˜ a in (51)¯ J in = − i √ n ˜ a i − n (52)For n < J in = i √− n ˜ a i †− n (53)¯ J in = i √− n ˜ a i † n (54)The commutators between the generators can be written as,[ J im , J jn ] = mδ ij δ n + m, + i πR ǫ ij θ ( n ) δ n + m, (55)[ ¯ J in , ¯ J jm ] = mδ ij δ n + m, + 14 π iR ǫ ij θ ( n ) δ n + m, (56)[ J im , ¯ J jn ] = 14 π iR ǫ ij θ ( n ) δ n,m (57)It can be easily observed that a term dependent upon noncommutative parameter θ has appeared in the commutationrelations of the U (1) Kac-Moody algebra. This deformed U (1) Kac-Moody due to momentum space noncommutativityis quite different compared to field space noncommutativity deformed U (1) Kac-Moody[16]. Now, the non-zero modeterms of the Hamiltonian can be written as, 2 πR X i,n> ( J i − n J in + ¯ J i − n ¯ J in ) (58)So, [ ˜ H, J k − m ] = 2 πR X i,m> (cid:8) mJ k − m + 12 πg iR θ ( n ) ǫ ik ( J i − m + ¯ J im ) (cid:9) (59)Now we can write the conformal generators,ˆ L = 12 X i J i + X i,n> J i − n J in (60)ˆ L n = 12 X i,m,n =0 J in − m J im (61)ˆ¯ L = 12 X i ˆ¯ J i + X i,n> ¯ J i − n ¯ J in (62)ˆ¯ L n = 12 X i,m ¯ J in − m ¯ J im (63)Here, J i = ¯ J i = 1 √ π r π i π i + 1 R θ (0) ǫ ij ϕ j π i + 14 R θ (0) ϕ i ϕ i (64)So, the Hamiltonian can be written as, ˜ H = 2 πR ( ˆ L + ˆ¯ L ) (65)7 .2 Momentum noncommutativity in (d+1) Dimension We will now generalize the results of the noncommutativity in (d+1) dimension. The spatial part of the base space isa d dimensional torus, just like eq. 4 but with non commutative scalar field. Therefore, the field can be written as,˜ ϕ i ( ~x, t ) = X ~n exp [2 πi ( n x R + n x R + · · · + n d x d R d )] ˜ ϕ i~n ( t ) (66)= X ~n exp [2 πi ( X j n j x j R j )] ˜ ϕ i~n ( t ) (67)The Fourier components can be written as,˜ ϕ i~n ( t ) = 1 R R · · · R d Z d d x exp [ − πi ( X j n j x j R j )] ˜ ϕ i ( ~x, t ) (68)So, the Lagrangian looks like L = 12 X i Z d d x [( ∂ t ˜ ϕ i ) − ( ∇ ˜ ϕ i ) ]If we write Lagrangian in terms of Fourier modes, then L = 12 R R · · · R d X i,~n (cid:8) ˙˜ ϕ i~n ˙˜ ϕ i~ − n − π [ X j n j R j ] ˜ ϕ i~n ˜ ϕ i~ − n (cid:9) (69)The canonical momentum is defined by: ˜ π i~n = ∂L∂ ˙˜ ϕ i~n = R R · · · R d ˙˜ ϕ i~ − n (70)Now, The deformation map can be written as: ˜ ϕ in = ϕ in (71)˜ π in = π in + 12 R R · · · R d ǫ ab ϕ j − n θ ( ~n ) (72)The commutation relationships between the deformed field modes are:[ ˜ ϕ im , ˜ ϕ jn ] = 0 (73)[˜ π im , ˜ π jn ] = iR R · · · R d ǫ ij θ ( ~n ) δ n + m, (74)[ ˜ ϕ an , ˜ π bm ] = iδ mn δ ab (75)Here the term θ ( ~n ) is defined as, θ ( ~n ) = θexp [ − π σ X j n j R j ] (76)If we write the Hamiltonian in terms of the hatted operators then, H = 12 R R · · · R d X i ˜ π i ˜ π i + 12 R R · · · R d X i,~n =0 (cid:8) ˜ π in ˜ π i~ − n + (2 πR R · · · R d ) [ X j n j R j ] ˜ ϕ i~n ˜ ϕ i~ − n (cid:9) (77)So, using the dressing transformation, H = H + 12 R R · · · R d X i,~n =0 π i~n π i − ~n + 12 R R · · · R d X i,~n =0 (cid:8) (2 πR R · · · R d ) [ X j n j R j ]+ θ ( ~n ) 14 R R · · · R d (cid:9) ϕ i~n ϕ i~ − n + θ ( ~n )2 R R · · · R d X i,~n =0 R R · · · R d ǫ ij π in ϕ jn (78)8ow, let us define, H ~n = X i R R · · · R d π i~n π i~ − n + X i R R · · · R d [(2 πR R · · · R d ) (cid:8) X j n j R j (cid:9) + θ ( ~n ) 14 R R · · · R d ] ϕ i~n ϕ i~ − n (79)(80)The standard harmonic oscillator Hamiltonian can be written as, H ~n = X i ( 12 V π i~n π i~ − n + V ω n ϕ i~n ϕ i~ − n ) (81)Comparing these two equations we can write V = R R · · · R d (82)¯ ω n = 1 M vuut (2 πR R · · · R d ) (cid:8) X j n j R j (cid:9) + θ ( ~n )4 R R · · · R d (83)Let us now define creation and annihilation operators, a i~n = r ∆ ~n ϕ i~n + i π i~ − n ∆ ~n ) (84) a i~n † = r ∆ ~n ϕ i~n − i π i~n ∆ ~n ) (85)Here, ∆ ~n = V ¯ ω ~n = vuut (2 πR R · · · R d ) (cid:8) X j n j R j (cid:9) + θ ( ~n )4 R R · · · R d (86)Using these operators, the Hamiltonian H ~n can be written as: H ~n = X i ω ~n a i~n a i~n † + ( a i − ~n a i − ~n † ) (87)The time-ordered form of this Hamiltonian is, X ~n =0 H ~n = X i,~n =0 ¯ ω ~n ( a i~n † a i~n ) (88)The last term of the Hamiltonian can be written as, θ ( ~n )2 R R · · · R d X i,~n =0 iR R · · · R d ǫ ij a i~n † a j~n (89)So, the total Hamiltonian can be written as, H = H + X i,~n =0 ¯ ω ~n a i~n † a i~n + i θ ( ~n ) V X i,j,~n =0 ǫ ij a i~n † a j~n We can repeat the Schwinger process done previously and define new creation and annihilation operators, A ~n and A ~n .Using these operators, the Hamiltonian, H = H + X ~n =0 (cid:8) (¯ ω ~n − V θ ( ~n )) A ~n † A ~n + (¯ ω ~n + 12 V θ ( ~n )) A ~n † A ~n (cid:9) (90)So, the energy levels can be read as, Λ n = ¯ ω ~n − V θ ( ~n ) (91)Λ n = ¯ ω ~n + 12 V θ ( ~n ) (92)9 .3 Model 2: Noncommutativity in field space, quantisation in (d+1) Dimension One can also consider noncommutativity in field space instead of the momentum space. Canonical quantisation ofsuch theories in compact space has already taken under consideration by Bal et. al . The twisted algebra reads in thismodel, [ˆ x a , ˆ x b ] = iǫ ab ¯ θ = i ¯ θ ab (93a)[ˆ p a , ˆ p b ] = 0 (93b)[ˆ x a , ˆ p b ] = iδ ab (93c)Also, the corresponding equal time commutation relations in field theory are,[ ˆ φ i ( x , t ) , ˆ φ j ( y , t )] = iǫ ij θ ( σ ; x − y ) (94a)[ˆ π i ( x , t ) , ˆ π j ( y , t )] = 0 (94b)[ ˆ φ i ( x , t ) , ˆ π j ( y , t )] = iδ ij δ ( x − y ) (94c)To see the the canonical quantisation procedure in compact space of this type of model see ref.[16]. The spatial part ofthe base space is a d dimensional torus. But in their paper, they considered compactified spatial coordinates, i.e., S ,all of them with the same radius R . But here we present the results invoking that each spatial direction is compactifiedin S with radius R j . Mimicking the calculation of Bal et al. [16], we find out the quantized Hamiltonian (normalordered), H = H + X n =0 ω ~n { Γ n A n † A n + Γ n A n † A n } (95)where, Γ n = Ω n − ω n θ ( n )2 (96)Γ n = Ω n + ω n θ ( n )2 (97) θ ( n ) = θexp [ − π σ ( n R + n R + .... + n d R d )] (98) ω n = 4 π ( n R + n R + .... + n d R d ) (99)Ω n = 1 + π θ ( n ) ( n R + n R + .... + n d R d ) (100)In the limit, R = R = ... = R d the above equations coincides with the result of Balachandran et. al.[16]. Thereforethe splitted deformed dispersion relation take the form below,Λ n = ω n (Ω n − ω n θ ( n )2 ) (101)Λ n = ω n (Ω n + ω n θ ( n )2 ) (102) In this section, we analyze the blackbody radiation due to deformed energy momentum relation coming from fieldspace (eq. 101 and 102) and momentum space noncommutativity (eq. 91 and 92). Although Balachandran et. al. hasbriefly discussed it, they did not calculate the thermodynamic quantities. However, we have numerically evaluated thethermodynamic quantities and we will present them in this section. We start from the partition function of quantumgases in grand canonical ensemble [33], Z = T r ( e − βH ) . (103)As there is a split in energy eigenvalues due to both types of noncommutativity, we find out from above eq. [16, 33],ln Z = − X k =0 (ln(1 − e − β Λ k ) + ln(1 − e − β Λ k )) . (104)10ere, Λ and Λ refer to two distinct classes of modes due to noncommutativity conditions, k is the momentum vectorand β = T . Therefore, the internal energy, U = − ∂∂β lnZ. (105)Now, the entropy S can be obtained from the partition function, S = − ∂F∂T , (106)where F = β ln Z is the free energy. U SR U Model2 U model1 × × × × × × × × × × × × × T U (a) Internal energy U , versus temperature S SR S Model2 S Model1 × × × × × × × × × × × × T S (b) Entropy S , versus temperature Figure 1: Plot of internal energy U (figure a) and entropy S (figure b) of blackbody radiation against temperature T for the special relativity theory and the noncommutative models. Following the ref. [14], [16], [20] we have chosen θ = 1 . × − and σ = 10 − .In the thermodynamic limit we consider all R i → ∞ which allows us to convert the sum in eq. (104) to an in-tegral. We performed the integrals numerically using mathematica[34], choosing specific values for θ and σ followingBalachandran et. al[16]. Finally, using eq (105) and (106) we evaluate the internal energy and entropy for both typeof noncommutative models and usual special relativity (SR). We have compared the results in fig. 1. We have foundthat all the three models agree in the lower temperature but the noncommutativity effects surely modify the Stefan-Boltzmann law ( U ∝ T ), which is clearly visible in the high temperature regime. But at high temperature a significantdifference is noticed between them (see figure 1a). In the high temperature regime it is seen that at any temperature T = T the internal energy coming from these models maintain a relation U SR ( T ) < U model ( T ) < U model ( T ). Thistrend is also noticed for other thermodynamic quantities such as entropy, specific heat etc. The dispersion relationpredicted from both types of noncommutative filed theories are clearly Lorentz violating. This trend of faster rate ofgrowth (with respect to temperature) of thermodynamic quantities at a high temperature regime compared to SR isalso noticed in other Lorentz violating studies on the thermodynamics of blackbody radiation[28]. As one can see inthe model 1 the modifications of the dispersion relations (91) and (92) occur for small wave number n and become theusual ones in the large n limit (more deformed in infrared region) whereas in model 2 the dispersion relations (101) and(102) have large modifications for large wave number n and becomes the usual one in small n limit (more deformed inUV region). But interestingly, thermodynamic quantities in both of the models show deviation from standard specialrelativity results in high temperature regime.In case of noncommutative models, due to Lorentz violating dispersion relations, number of available states grow.As a result when we do the integration over all the modes in eq. (104) we find the modified internal energy. ThePlanck distribution function picks up a smaller value in low temperature region compared to high temperature region.The noncommutative parameter makes some modification in Planck distribution but it is not extremely drastic. As aresult when we do integration over it no such significant change is noticed in low temperature due to noncommutativeparameter. Now as the temperature rises abruptly the Planck distribution attends higher values and a small changedue to noncommutative parameter makes the change big enough that when do integration over all the modes a differ-ence is noticed. The effect of noncommutative parameter in internal energy is less clear in low temperature region, as we have chosen Boltzmann constant k B = 1, c = 1
11n lower temperature region the (modified) Planck distribution picks up a very small value. As a result thermodynamicquantities in both of these models, irrespective of whether the model is more deformed in infrared/UV region, showdeviation from standard results in high temperature regime.
In this paper we have canonically quantized a noncommutative scalar field theory in a compact space with noncommuta-tivity in momentum space following the seminal work of Balachandran et. al.[16]. As a result of this noncommutativity,we have noticed the splitting of the energy levels of each individual mode that constitutes the whole system. Thistype of splitting in energy eigenvalue was also noticed in noncommutative scalars, where noncommutativity is in fieldspace[16]. But the functional forms of deformed dispersion relations due to two types of noncommutativity are quitedifferent and as a result their prediction are also quite different in blackbody radiation. We have paid special at-tention to the special case of 1+1 dimensional theory and found out the deformed conformal generators. We are ina process to evaluate the deformed Virasoro algebra for noncommutative theories and find out the status of centralcharge in such field theories. The central charge is a very significant concept in conformal theories as the theoriesare characterized by this number. A different central charge would imply a new interpretation of central extension.Such noncommutativity would be even more interesting for gauge fields as the cancellation of degrees of freedom withGupta-Bleuler quantization or Faddev-Popov method by appearance of ghost fields which can lead to new physics.The central charge of the ghost fields play a significant role in the critical dimension of string theory. Furthermore, itshould be noticed that we have considered the spatial part of the base space is a d dimensional torus where we invokedthat each spatial direction is compactified in S j with radius R j . The reason behind keeping the result more general iswe are in a process to compute Casimir force for noncommutative theories in compact space. The general result willhelp us to make any particular direction, say R to keep finite and other direction to put in the bulk limit. In thefuture, we would also like to investigate the finite temperature status of these type of theories. As a result we noticethat, thermdynamic quantities in both of these models show deviation from standard result One of the authors (MMF) would thanks the Bose Centre for Advanced Study and Research in Natural Sciences,University of Dhaka, Bangladesh for financial support. The authors would also like to thank Professor Dr. ArshadMomen for the fruitful discussions. The authors express their gratitude to Baptiste Ravina and Liana Islam for theirhelp to present the manuscript. We sincerely thank the referees for their fruitful comments.