A note on broken dilatation symmetry in planar noncommutative theory
aa r X i v : . [ h e p - t h ] J a n Remarks on noncommutativity and scaleanomaly in planar quantum mechanics
Partha Nandi ∗ , Sankarshan Sahu † , Sayan Kumar Pal ‡ S. N. Bose National Centre for Basic Sciences,JD Block, Sector III, Salt Lake, Kolkata-700106, India. Indian Institute of Engineering Science and Technology, Shibpur,Howrah, West Bengal-711103, India.January 21, 2021
Abstract
A study of a riveting connection between noncommutativity and theanomalous scale or dilatation symmetry is presented for a generalizedquantum Hall system due to time dilatation transformations. On us-ing the "Peierls substitution" scheme, it is shown that noncommutativitybetween spatial coordinates emerges naturally at a large magnetic fieldlimit. Thereafter, we derive a path-integral action for the correspondingnoncommutative quantum system and discuss the equivalence betweenthe considered noncommutative system and the generalized Landau prob-lem thus rendering an effective commmutative description. By exploitingthe path-integral method due to Fujikawa, we derive an expression forthe unintegrated scale or dilatation anomaly for the generalized Landausystem, wherein the anomalies are identified with Jacobian factors aris-ing from measure change under scale transformation and is subsequentlyrenormalised. In fact, we derive exact expressions of anomalous Wardidentities from which one may point out the existence of scale anomalywhich is a purely quantum effect induced from the noncommutative struc-ture between spatial coordinates. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Scale or dilatation symmetry has quite a long history [1, 2] in particle physicsand has become an interesting concept in present day physics for its usefulnessin understanding a variety of physical phenomena ranging from phase transi-tions in statistical mechanics, stochastic processes, many-body condensed mat-ter systems to construction of conformal quantum field theories and even inplane gravitational waves [3, 4, 5, 6, 7]. A classical example in mechanics whichrespects this symmetry is the inverse square potential in and di-mensions [8]. Further in dimensions, the delta function potential δ ( ~r ) isalso a scale invariant theory [9]. On the other hand in classical field theories,the massless Klein Gorden as well as massless phi-four theory also respect thissymmetry [10]. However, the phenomenon of conservation of certain currents orcharges valid in classical domain may not continue to hold upon quantization- the appearance of anomalies. Anomaly is one of the three possible types ofbreaking of a symmetry exhibited by a physical system at the quantum levelwith the other two being classical/explicit and spontaneous symmetry breakingeffects. As a matter of fact, nature does not respect scale invariance as far asparticle physics is concerned [1, 2]. The most famous example of an anomalywhich created much debate and attention in high energy physics is the so-calledABJ anomaly and the PCAC (partial conservation of axial-vector current) hy-pothesis [11, 12, 13]. In this regard, in anomalous gauge theories, it has beenshown from algebraic consistency in [14] that the electric fields must be non-commutative. Also it is worthwhile to mention here that the explicit structurefor the chiral anomaly in non-commutative gauge theories has been already wellstudied in the literature [15, 16].In modern physics, the study of symmetries of physical systems and also on theirbreaking has helped to gain deep insights into their physical content and also ontheir underlying mathematical (geometrical) structure. Study of anomalies hasbecome indispensable to explore novel features in the study of quantum phasetransitions, see for example [17] and references therein. Moving ahead, it hasbeen shown by R. Jackiw in [18] that a charged particle interacting with a vor-tex magnetic field enjoys the scale or time dilatation symmetry, as well as othertime reparametrization symmetry namely, conformal transformation and trans-formation due to time translation. The invariance group of this system turnsout to be SO (2 , × U (1) and it admits a hidden SO (2 , dynamical symmetrygroup of conformal transformations, where generators of this dynamical symme-try are generators of the above three time reparametrization transformations.Furthermore, this leads to a deformed Heisenberg algebra i.e. noncommutativemomenta as a by-product. In this context, it is important to mention herethat recently one of us [19] has shown that due to deformed planar Heisenbergalgebra with phase space non-commutativity, the system Hamiltonian of a 2Dharmonic oscillator can be recast as an algebra element of SO (2 , × U (1) whichis responsible for the emergence of Berry phase in such systems under adiabatic2volution of the system Hamiltonian. A relation between ABJ anomalies andoccurrence of Berry phase has been unfolded in [20] for Fermi liquids. Again, ithas been investigated in [21] for the dilatation anomaly in the Landau problem.In this regard, it is interesting to notice that the problem of charged particleinteracting with a point magnetic vortex differs from the Landau problem un-der scaling transformations even though the form of interactions is the same.Now, as a matter of fact, it is well-known now that in the Landau problem,guiding-center coordinates satisfy a noncommutative commutation relation. Itis therefore quite natural to investigate the fate of scale symmetry breaking andoccurrence of an anomaly, if any, in deformed (noncommutative) planar systemstaking the harmonic oscillator as a first study which has been lacking in the lit-erature so far. We will carry out the analysis both in the classical as well asin quantum domain using path-integral techniques. In this context, recently, apath-integral derivation of the nonrelativistic scale anomaly in usual commuta-tive space has been carried out in [22]. It may be useful to mention here that in[23], a connection between axial anomaly of QED in a strong magnetic field andthe non-planar axial anomaly of the conventional non-commutative U (1) gaugetheory has been observed.Deformed (noncommutative) quantum physics has gained much attention in thelast two or three decades as a promising route towards constructing a consis-tent theory of gravity. Actually the concept of noncommutativity dates backmuch earlier in the works of [24, 25] in an attempt to unify quantum mechan-ics with gravity. But soon it faded out with the success of renormalizationprogramme. Howbeit starting from 90’s, different approaches in high energyphysics have pointed out at the existence of a minimal length in Planck scaleregime and noncommutative physics has revived from this confluence of ideas[26, 27, 28, 29, 30, 31]. However, the concept of noncommutativity has re-mained no more just a theoretical interest relevant in very high energy physicsbut has emerged naturally in condensed matter systems and has phenomenolog-ical consequences as well. For an example, in semi-classical dynamics of Blochelectrons exhibiting Berry curvature, the coordinates become noncommutative[32] and lead to anomalous Hall conductance in ferromagnetic semiconductors[33]. Closely related to this development, there has been other advances innoncommutative physics from the perspective of Galilean symmetry in planarnonrelativistic mechanics. In (2+1) dimensions, the Galilean algebra admitsa unique second central extension which can be identified with nonrelativisticspin [34, 35] and also this can be corroborated from a consistent nonrelativisticlimit of Poincare group in (2+1) dimensions. Now, it has been shown explicitlythat the deformation parameter in noncommutative mechanics can be identifiedwith the spin of particles in two dimensions (anyonic spin) [36, 37] and there-fore noncommutative quantum mechanics helps in the understanding of spin aswell as fractional quantum Hall effect [38, 39, 40] where the exotic statistics ofanyons play an important role. Moreover, spatial noncommutativity has beensuggested to emerge in the physics of cold Rydberg atoms [41, 42]. Very recently3t has been reported in [43] that there have been anomalous frequency shifts inultracold Rydberg atoms implying breaking of scale symmetry in such systems.This particular observation adds further to the motivation of the present study.Here in the current work, we investigate existence of anomaly under scalingtransformations in deformed (noncommutative) quantum system, in particularin exotic oscillators. The form of the effective action of exotic oscillators (72)will be reminiscent of the action of a charged particle interacting with magneticpoint vortex under appropriate variables [18]. It will be shown that the systemdoes not admit scaling symmetry classically on account of the non-zero springconstant. Even so strikingly, at the quantum level there appears an anomalouscontribution to the non-conservation of dilatation charge. This is an interestingresult that we report here and to the best of our knowledge, this paper is a firstdirection towards the study of scale anomalies in deformed quantum systems.This paper also pursues the current problem in the spirit of a (0+1) dimen-sional noncommutative field theory [44] employing methods generally used infield theories as quantum mechanics can be considered to be a (0+1) dimen-sional quantum field theory [45].The paper has been arranged as follows. We start by demonstrating in section 2how deformed harmonic oscillators can arise in the classical setting by consider-ing Peierls-substitution scheme [46, 47] and show that this represents a systemof electric dipole under a very strong magnetic field with a harmonic interac-tion between the charges along with an extra harmonic potential part of one ofthe charges. In section 3, we then move on to the quantum setting where weformulate the noncommutative quantum mechanics of a two-dimensional har-monic oscillator in the framework of Hilbert-Schmidt operators, an operatorformulation of noncommutative quantum mechanics. Then we will constructthe coherent-state path integral of the noncommutative quantum system andcompute the phase-space action for the system where a resemblance with Chern-Simons quantum mechanics shall be made in section 4. Symplectic brackets canbe computed with this action and the noncommutative structure is figured outbetween the position coordinates. In section 5, with the help of the generatingfunctional for Green’s functions, we explicitly compute Ward-Takahashi iden-tities as a (0+1) dimensional noncommutative field theory. These identitieswill get plagued with anomalies as quantum corrections owing to the existenceof noncommutativity. The anomaly term needs to regularized and is done insection 6 using Fujikawa’s regulator. The anomaly term will be found to be pro-portional to the product between the spring constant of the harmmonic oscillatorand the noncomutative parameter. Note that initially during the constructionof generating function in presence of source, we have worked with noncommu-tative variables. Then we have switched to canonical phase-space variables forfurther development in the computation of Ward identities.4 The classical picture of noncommutative space
We consider, inspired by the "Peierls-substitution" scheme, a pair of non-relativisticinteracting opposite charged particles having same mass m moving on the planein a constant magnetic field B along z axis (within the approximation wherewe can ignore Coulomb and radiation effects). The coordinates of negative andpositive charges are denoted in components form by x i and y i ( i = 1 , ) re-spectively. The dynamics of the system is confined in a plane and thereforethe z coordinate can be suppressed. The system is described by the followingstandard Lagrangian in C.G.S units : L = 12 m ( ˙ x i + ˙ y i ) + eB c ǫ ij ( x j ˙ x i − y j ˙ y i ) − k x i − y i ) − k x i (1)where c denotes the speed of light in vacuum. The first term of the above La-grangian (1) represent the kinetic term of the charges and the second term rep-resent their interaction with the external magnetic field B . Further, to prescribea vector potential ~A , satisfying ~ ∇ × ~A = B ˆ z , we chose rotationally symmetricgauge. The third term is the harmonic interaction between the two charges andfinally the fourth term describes additional interactions of the negative chargewith an impurity.Introducing magnetic length scale l B = q ~ ceB [48, 49] and the dimensionlesscoordinates ξ i = l − B x i and ρ i = l − B y i , the Lagrangian (1) reduces to - L = ~ (cid:18) mceB ( ˙ ξ + ˙ ρ ) + ǫ ij ( ξ j ˙ ξ i − ρ j ˙ ρ i ) − ck eB ( ξ i − ρ i ) − ck eB ρ i (cid:19) . (2)In what follows we will be interested in the limit of strong magnetic field B and small mass m as meB → , in which the kienetic term from (2) can beeffectively ignored [50]. Also, upon quantization of the system (1), in the absenceof a harmonic interaction potential between two charges and the additionalinteractions ("impurities"), the quantum spectrum consists of the well-knownLandau levels [51, 52]. Since the separation between the successive Landau levelsis O ( eBm ) , if the magnetic field is strong, only the lowest Landau level (LLL) isrelevant for the dynamics. The higher states are essentially decoupled to infinity.So the Lagrangian can be modified so as to describe only high magnetic fieldeffects as mentioned just above by setting meB to zero in (2) [53, 54, 55]. Thus theLagrangian of interest here, in terms of the dimensionful Cartesian co-ordinates x i and y i , is the following :- L = eB c ( ǫ ij x j ˙ x i − ǫ ij y j ˙ y i ) − V ( x i , y i ) , (3)with V ( x i , y i ) = k ( x i − y i ) + k x i . Thus the Euler-Lagrange equations asso-ciated with the above first-order Lagrangian (3) is given by - ˙ x i = ceB ǫ ij ∂V∂x j ; ˙ y i = − ceB ǫ ij ∂V∂y j . (4)5he Hamiltonian corresponding to (3) is constructed by the usual Legendretransformation : H = ∂L ∂ ˙ x i ˙ x i + ∂L ∂ ˙ y i ˙ y i − L = V ( x i , y i ) . (5)In order to show the equivalence between the Lagrangian and Hamiltonian for-malisms [56, 57] , we consider the Hamilton’s equations of motion: ˙ x i = { x i , H } = { x i , V } , (6) ˙ y i = { y i , H } = { y i , V } , (7)with the potential V ( x i , y i ) playing the role of the Hamiltonian. The symplecticstructure can readily be obtained now by comparing the Lagrangian equationsof motion (4) with the form of Hamilton’s equations of motion (6,7) to yield thefollowing brackets : { x i , x j } = ceB ǫ ij ; { y i , y j } = − ceB ǫ ij ; { x i , y j } = 0 . (8)Let us define a new pair of canonical variables as, p i = eBc ǫ ij ( x j − y j ) and x i , (9)which satisfies the following symplectic structure : { x i , x j } = ceB ǫ ij ; { x i , p j } = δ ij ; { p i , p j } = 0 , (10)where p i play the role of canonical conjugate momentum of cordinates x i . Thusat very high magnetic field and low mass limit, the canonical Hamiltonian (5)can be rewritten as - H = p i m B + 12 k x i , (11)where m B = e B k . Therefore at strong magnetic field limit, the dynamics ofsystem (1) is governed by a two dimensional harmonic oscillator with the de-formed symplectic structure (10). Notice that Poisson-noncommutativity of thecoordinates has been already established at the classical level as the symplecticbracket between coordinates x i is nonvanishing. In the next section we shalldiscuss the quantum version of the theory. Now to describe quantum theory of the above model at stong magnetic field limitin a systematic manner, we start with the hermitian Hamiltonian operator: ˆ H = ˆ p i m B + 12 k ˆ x i , (12)6here phase-space variables/operators ( ˆ x i , ˆ p i ) satisfy the following noncommu-tative Heisenberg algebra (NCHA) : [ˆ x i , ˆ x j ] = iθǫ ij , [ˆ x i , ˆ p j ] = i ~ δ ij , [ˆ p i , ˆ p j ] = 0; f or i, j = 1 , (13)with θ = ~ ceB > . So, the system of interest (12) is nothing but a two-dimensional harmonic oscillator system placed in the ambient noncommutativespace [58], which we refer here as exotic oscillators.Recently in [59, 60], it was pointed out that noncommutative quantum me-chanics should be formulated as a quantum system on the Hilbert space ofHilbert-Schmidt operators acting on the classical configuration space. Here, wepresent a very brief review of the formulation in order to dispense it with anappropriate physical setting and also to pave the road for path-integral scheme.In two dimension, the coordinates of configaration space follow the NC algebra: [ˆ x i , ˆ x j ] = iθǫ ij ; f or i = j = 1 , (14)and they act on a Hilbert space H c , which is referred as configuration space orclassical Hilbert space. One can realize it by introducing the annihilation andcreation operators, ˆ b = 1 √ θ (ˆ x + i ˆ x ) , ˆ b † = 1 √ θ (ˆ x − i ˆ x ) , (15)satisfying the commutation relation [ˆ b, ˆ b † ] = I c . The noncommutative configu-ration space is isomorphic to boson Fock space : H c = span {| n i = 1 √ n ! (ˆ b † ) n | i} , (16)This H c furnishes the representation of the coordinate algebra (14). Now quan-tum states of our noncommutative Hamiltonian system (12) are operators whichare elements of the algebra generated by the coordinate operators i.e. ψ (ˆ x , ˆ x ) .These states live in a Hilbert space, called the quantum Hilbert space H q [60, 61]in which the entire NCHA (13) should be represented. In fact, one can identify H q as H c ⊗ H ∗ c , with H ∗ c being the dual of H c (16). An important notationthat we adopt is the following: states in noncommutative configuration spaceare denoted by | i and states in the quantum Hilbert space are denoted by around " bra-ket" ψ (ˆ x , ˆ x ) ≡ | ψ ) . The natural choice for the basis of quantumHilbert space is the space of Hilbert-Schmidt operators - H q = span { ψ (ˆ x , ˆ x ) = | ψ ) : ψ (ˆ x , ˆ x ) ∈ B ( H c ) , tr c ( ψ † (ˆ x , ˆ x ) ψ (ˆ x , ˆ x )) < ∞} (17)where the subscript c refers to tracing over H c and B ( H c ) is the set of boundedoperators acting on H c . Now we search for a unitary representation of the non-commutative Heisenberg algebra (13) on quantum Hilbert space in terms ofoperators ˆ X i , ˆ P i in the following way - ˆ X i | ψ ) = | ˆ x i ψ ) , ˆ P i = ~ θ ǫ ij | [ˆ x j , ψ ]) = ~ θ ǫ ij [ ˆ X i − ˆ X Rj ] || ψ ) . (18)7ere the capital letters ˆ X i and ˆ P i have been used to denote the representationsof the operators ˆ x i and ˆ p i acting on H q . Note that we have taken the actionof ˆ X i to be left action by default and the momentum operator acts adjointly.Also, ˆ X Ri implies the right action on the quantum Hilbert space in the followingway : ˆ X Ri | ψ ) = | ψ ) ˆ X i = | ψ ˆ x i ); [ ˆ X Ri , ˆ X Rj ] | ψ ) = | ψ [ˆ x j , ˆ x i ]) = − θǫ ij | ψ ) ∀ ψ ∈ H q (19)It can be checked easily that the right action satisfies [ ˆ X i , ˆ X Rj ] = 0 (20)We can thus identify ˆ X Ri as quantum mechanical counterpart of y i in (9) andnote (18, 19) are analogous to (8, 9) in the quantum setting. Therefore, theclassical setting discussed in previous section provides a clear physical inter-pretation of the abstract formulation of noncommutative mechanics in terms ofHilbert-Schmidt operators. Having pointed this, we now introduce one furthernotational convention - for any operator ˆ O acting on the quantum Hilbert space(17), we may define left and right action (denoted by superscripted L and R) asfollows: ˆ O L | ψ ) = ˆ O | ψ ) , ˆ O R | ψ ) = | ψ ) ˆ O ; ∀| ψ ) = ψ (ˆ x , ˆ x ) ∈ H q (21)Thus the operators acting on the quantum Hilbert space obey the commutationrelations - [ ˆ X i , ˆ X j ] = iθǫ ij , [ ˆ X i , ˆ P j ] = i ~ δ ij , [ ˆ P i , ˆ P j ] = 0 f or i, j = 1 , (22)It is now useful to define the following operators on the quantum Hilbert spacewhich we shall require later: ˆ B = ˆ X + i ˆ X √ θ , ˆ B ‡ = ˆ X − i ˆ X √ θ , ˆ P = ˆ P + i ˆ P , ˆ P ‡ = ˆ P − i ˆ P ; [ ˆ B, ˆ B ‡ ] = I q (23)where we have used the symbol ‡ specifically for the operator adjoint on H q . Itcan be easily checked that all these phase space operators are self adjoint withrespect to the inner product - ( ψ | φ ) = tr c ( ψ † φ ) < ∞ ∀ ψ, φ ∈ H q , (24)and stems from the fact that the product of two HS operators has finite trace-class norm. Therefore, the representation of our system Hamiltonian (12) on H q is given by ˆ H = 12 m B ( ˆ P + ˆ P ) + 12 k ( ˆ X + ˆ X ) , (25)where phase-space operators ˆ X i and ˆ P i satisfy the same i.e. isomorphic algebra(22) while acting on the quantum Hilbert space H q . Here we do not embark8pon the canonical quantization of (25) as this has been studied extensivelyin the literature [61, 62] where the system gets diagonalized as two decoupledoscillators of frequency ω ± : ˆ H | n , n ) = E n n | n , n ) ; E n ,n = ( n ~ ω + + n ~ ω − ) + E , (26)where the characteristic frequencies ω ± are given by, ω ± = ω q m B ω θ ~ ± m B ω θ ~ and E , = ~ ( ω + + ω − ) is the finite zero point energy of the systemHamiltonian. This however, being a c -number and not an operator will yieldthe same value in all states and is displaced off by simply defining the zero ofenergy as, ˆ e H := ˆ H − E , I q , (27)Therefore ˆ˜ H | n , n ) = ˜ E n n | n , n ) , (28)with ˆ e H | ,
0) = 0 and ˜ E n n = n ~ ω + + n ~ ω − . From now on, we only work with ˆ e H. As is the case always, the energy eigenvectors of H q constitute a completeset- X n ,n | n , n )( n , n | = I q (29)We now, as discussed at the outset, proceed to carry out the path-integral ofthe system in order to study its scaling properties. We would like to introduce the notion of position states, however, in view of theabsence of common eigenstates of ˆ x and ˆ x , the best one can do is to introducethe minimal uncertainty states i.e. maximally localized states (coherent states)on H c [63] as, | z i = e − ¯ z ˆ b + z ˆ b † | i = e − | z | e z ˆ b † | i ∈ H c , (30)where z = x + ix √ θ is a dimensionless complex number. These states provide anover-complete basis on H c and it is possible to write the identity as, Z dzd ¯ zπ | z i h z | = I c , (31)Using these states, we can construct a state in quantum Hilbert space H q asfollows : | z, ¯ z ) = | x , x ) = 1 √ πθ | z i h z | ∈ H q ; ( w. ¯ w | z, ¯ z ) = 12 πθ e −| w − z | , (32)9atisfying ˆ B | z, ¯ z ) = z | z, ¯ z ) . Now, the coherent state ("position") representationof a state | ψ ) = ψ (ˆ x , ˆ x ) can be expressed as, ψ ( x , x ) = ( z, ¯ z | ψ ) = 1 √ πθ tr c ( | z i h z | ψ (ˆ x , ˆ x )) = 1 √ πθ h z | ψ (ˆ x , ˆ x ) | z i . (33)We now introduce the normalized momentum eigenstates [64] such that - | p, ¯ p ) = r θ π ~ e i q θ ~ (¯ p ˆ b + p ˆ b † ) ; ˆ P i | p, ¯ p ) = p i | p, ¯ p ) , p = p + ip (34)with these states satisfying usual resolution of identity and orthogonality con-ditions, one has the coherent state representation of the momentum states onthe noncommutative plane to be given by [64]- ( z, ¯ z | p, ¯ p ) = 1 √ π ~ e − θ ~ p ¯ p e i q θ ~ ( p ¯ z +¯ pz ) . (35)Now the completeness relations for the "position basis" (coherent state basis) | z, ¯ z ) on H q , given by Scholtz et al.[60] reads, Z θdzd ¯ z | z, ¯ z ) ⋆ V ( z, ¯ z | = Z dx dx | x , x ) ⋆ V ( x , x | = I q . (36)This involves the Voros star product [65], where the star-product between twofunctions f ( z, ¯ z ) and g ( z, ¯ z ) is defined as- f ( z, ¯ z ) ⋆ V g ( z, ¯ z ) = f ( z, ¯ z ) e ← ∂ ¯ z → ∂ z g ( z, ¯ z ) (37)In fact, in [66, 67], it has been pointed out that the low energy dynamics ofrelativistic quantum field theories in the presence of strong magnetic field canbe described by the Voros star product. The finite time propagation kernel canbe splitted up into n steps, ( z f , t f | z , t ) = lim n →∞ Z ∞−∞ (2 θ ) n n Y j =1 dz j d ¯ z j ( z f , t f | z n , t n ) ⋆ V n ( z n , t n | ........... | z , t ) ⋆ V ( z , t | z , t ) (38) Writing down the propagation over an infinitesimal step τ = t f − t n +1 : ( z j +1 , t j +1 | z j , t j ) = ( z j +1 | e − i ~ τ ˆ e H | z j )= Z + ∞−∞ d p j e − θ ~ ¯ p j p j e i q θ ~ [ p j (¯ z j +1 − ¯ z j )+¯ p j ( z j +1 − z j ) ] × e − i ~ ǫ [ ¯ pjpj mB + k θ (2¯ z j +1 z j + c )] (39) where c = 1 − θk E , . Now substituting the above expression in (38) and10omputing star products explicitly, we obtain (apart from a constant factor) - ( z f , t f | z , t )= e − ~∂ zf ~∂ ¯ z lim n →∞ Z n Y j =1 ( dz j d ¯ z j ) n Y j =0 d p j exp n X j =0 τ " i ~ r θ h p j n ¯ z j +1 − ¯ z j τ o + ¯ p j n z j +1 − z j τ o − √ θk ¯ z j +1 z j i exp n X j =0 τ (cid:20) − i m ~ ¯ p j p j + θ ~ ( p j +1 − p j ) τ ¯ p j (cid:21) , (40) where, σ = − i ( τ m B ~ − i θ ~ ) and we treat z n +1 = z f . Finally, in the ǫ → or n → ∞ limit, symbolically, we write the phase space form of the path integral: ( z f , t f | z , t ) = e − ~∂ zf ~∂ ¯ z Z z ( t f )= z f z ( t )= z D ¯ z ( t ) D z ( t ) Z D ¯ p ( t ) D p ( t ) e i ~ S [ z ( t ) , ¯ z ( t ) ,p ( t ) , ¯ p ( t )] (41) where S is the phase-space action given by - S = Z t f t dt "r θ p ˙¯ z + ¯ p ˙ z ) − iθ ~ ¯ p ˙ p − m B ¯ pp − θk ¯ zz . (42)Note that this action contains an exotic ¯ p ˙ p term which is a Chern-Simons liketerm in momentum space and is responsible for rendering the configuration spaceaction to be non-local [68]. On recognizing that the first order system (42) canbe described in the extended phase space, where z ( t ) and p ( t ) are treated asconfiguration space variables of the system, the action can be written in theso-called symplectic form [69] as - S = Z t f t i dt (cid:20) ξ α f αβ ˙ ξ β − V ( ξ ) (cid:21) , ξ α := { z, ¯ z, p, ¯ p } ; α = 1 , .., (43)with f = − q θ σ q θ σ − θ σ . (44)where σ , σ are the usual Pauli matrices, and V ( ξ ) = 12 m B ¯ pp + θk ¯ zz (45)One can easily read off the symplectic brackets :- { z, ¯ z } = − i ~ , { z, ¯ p } = { ¯ z, p } = r θ , { p, ¯ p } = 0 (46)The symplectic brackets are consistent with the classical version of the non-commutative Heisenberg algebra given in (10). Having done this brief review todescribe the path-integral action of the noncommutative harmonic oscillator, weare all set to dive into the main theme of the paper where now we will build-up11n this action to study its scaling properties.Looking at the expression of the path integral kernel, we can think of noncom-mutative quantum mechanics as a complex scalar field theory in dimen-sions, the coordinates are field which depend on time z ( t ) . Therefore we writethe matrix elements of the time-ordered product of source term for all t with t i < t < t f as a path integral - ( z f , t f | z , t ) J, ¯ J = ( z f , t f | T exp " i ~ Z t f t i r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | z , t )= e − ~∂ zf ~∂ ¯ z Z z ( t f )= z f z ( t )= z D ¯ z D z D ¯ p D p exp " i ~ ( S + Z t f t i r θ z ( t ) J ( t ) + z ( t ) ¯ J ( t )] = e − ~∂ zf ~∂ ¯ z Z z ( t f )= z f z ( t )= z D ¯ z D z D ¯ p D p exp (cid:18) i ~ S J, ¯ J (cid:19) , (47) with S J, ¯ J = Z t f t dt "r θ p ˙¯ z + ¯ p ˙ z ) − iθ ~ ¯ p ˙ p − m B ¯ pp − θk ¯ zz + r θ z ( t ) J ( t ) + z ( t ) ¯ J ( t ) , (48)where J ( t ) = J ( t ) + iJ ( t ) is a time dependent source vanishing at t → ±∞ .Now in quantum theories, the object of prime usefulness is the vacuum to vac-uum persistence amplitude in the presence of an external source. A simple wayto obtain this is to go back to the transition amplitude in the coordinate space(47) and introduce complete sets of energy as :- ( z f , t f | z , t ) J, ¯ J = X n ,n X m ,m ( z f , t f | n , n )( n , n | T exp " i ~ Z t f t r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | m , m ) × ( m , m | z , t )= X n ,n X m ,m ( n , n | T exp " i ~ Z t f t r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | m , m ) × ( m , m | e i ~ t ˆ˜ H | z )( z f | e − i ~ t f ˆ˜ H | n , n )= (0 , | T exp " i ~ Z t f t r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | , × (0 , | z )( z f | , X n ,n =0 X m ,m =0 ( n , n | T exp " i ~ Z t f t i r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | m , m ) × ( m , m | z )( z f | n , n ) e − i ~ t f ˜ E n ,n + i ~ t ˜ E m ,m , (49) where we have used the fact of (28). To accomplish the projection onto thevacuum, we now introduce a variable T with units of time. One can for instantreplace - t f = T (1 − iǫ ) , t = − T (1 − iǫ ) , (50)12nd take T → ∞ , with ǫ being a small positive constant. At the end we willtake the limit ǫ → , but only after the infinite limit of T is carried out. Inthe limit T → ∞ , the exponentials in (49) oscillate out to zero except for theground state. Thus in this asymptotic limit, we obtain lim ǫ → lim t f →∞ (1 − iǫ ) t →−∞ (1 − iǫ ) ( z f , t f | z , t ) J, ¯ J = (0 , | T exp " i ~ Z ∞−∞ r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | , × (0 , | z )( z f | , . (51) However, one can also observe that - lim ǫ → lim t f →∞ (1 − iǫ ) t →−∞ (1 − iǫ ) ( z f , t f | z , t ) J = ¯ J =0 = (0 , | z )( z f | , (52)Consequently, we can write - (0 , | T exp " i ~ Z ∞−∞ r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | ,
0) = lim ǫ → lim t f →∞ (1 − iǫ ) t →−∞ (1 − iǫ ) ( z f , t f | z , t ) J, ¯ J ( z f , t f | z , t ) J = ¯ J =0 (53) As the left hand side of (53) is independent of the boundary conditions z and z f imposed at t → ±∞ , so the right hand side is also independent of the boundaryconditions imposed at t → ±∞ provided that one chooses the same boundaryconditions in the numerator and the denominator. Furthermore, the right handside has the structure of a functional integral and we can write (53) also as, (0 , | T exp " i ~ Z ∞−∞ r θ J ( t ) ˆ B † + ¯ J ( t ) ˆ B ) dt | ,
0) = (0 , ∞| , −∞ ) J, ¯ J = N − Z D ¯ z D z D ¯ p D p e i ~ (cid:20) S + R tf = ∞ t −∞ q θ [¯ z ( t ) J ( t )+ z ( t ) ¯ J ( t ) dt ] (cid:21) , (54) where the normalization factor N in (54) is fixed so as to ensure (0 , | ,
0) = 1 .Therefore, the generating functional for connected Green’s functions or the vac-uum persistence amplitude in presence of external source J ( t ) and ¯ J ( t ) is definedas, Z ( J, ¯ J ) = Z ( J , J ) = N − Z D ¯ z D z D ¯ p D p e i ~ h S + R tf = ∞ t −∞ √ θ [¯ z ( t ) J ( t )+ z ( t ) ¯ J ( t )] dt i . (55)Now, notice that we can realize the symplectic brackets (46) in terms of canonicalvariables as, z ( t ) = α ( t ) + i ~ r θ p ( t )¯ z ( t ) = ¯ α ( t ) − i ~ r θ p ( t ) , (56)13atisfying { α, ¯ α } = 0 , { α, ¯ p } = { ¯ α, p } = r θ , { p, ¯ p } = 0 (57)In order to compute the generating functional in configuration space in a localform, it will be beneficial at this stage to use this change of variables (56) (thisleaves the integration measure invariant) in the noncommutative phase-spacegenerating functional (55) to re-write it as : Z ( J, ¯ J ) = Z ( J , J ) = N − Z D ¯ α D α D ¯ p D p exp (cid:18) i ~ S J, ¯ J [ α, ¯ α, p, ¯ p ] (cid:19) , (58)where S J, ¯ J [ α, ¯ α, p, ¯ p ] = Z ∞−∞ dt "r θ α − iθ ~ k ¯ α + i ~ r θ J ) p + r θ α + iθ ~ k α − i ~ r θ J )¯ p − m B (1 + k θ m B ~ )¯ pp − θk ¯ zz + r θ α ¯ J + ¯ αJ ) (59) Furthermore for simplicity, on rescaling the dynamical variables as- α ( t ) → Z ( t ) = 1 q k θ m B ~ α ( t ) p ( t ) → P ( t ) = r k θ m B ~ p ( t ) , (60)the functional integral (58) yields :- Z ( J, ¯ J ) = Z ( J , J ) = N − Z D ¯ Z D Z D ¯ P D P e i ~ S J, ¯ J [ Z, ¯ Z,P, ¯ P ] (61)with S J, ¯ J [ Z, ¯ Z, P, ¯ P ] = Z ∞−∞ dt "r θ Z − iθ ~ k ¯ Z + iλ ~ r θ J ) P + r θ Z + iθ ~ k Z − iλ ~ r θ J ) ¯ P − m B ¯ P P − θk (1 + k θ m B ~ ) ¯ ZZ + λ (1 + k θ m B ~ ) r θ Z ¯ J + ¯ ZJ ) (62) where λ = (1 + k θ m B ~ ) − . The generating function in configuration space isnow easily derived by integrating over the momenta. Indeed, the dependenceon momenta in the exponent of (61) is at most quadratic and one may performthe gaussian integrations over the momenta to obtain - Z ( J, ¯ J ) = Z ( J , J ) = ˜ N − Z D ¯ Z D Z e i ~ S eff [ Z, ¯ Z ] e i ~ S I [ Z, ¯ Z,J, ¯ J ] , (63)14here ˜ N is a normalization constant which arises from the integration of mo-menta, and S eff [ Z, ¯ Z ] = Z ∞−∞ dt (cid:20) m B θ ˙¯ Z ˙ Z − im B θ k ~ ( ¯ Z ˙ Z − ˙¯ ZZ ) − k θ ¯ ZZ (cid:21) , (64) S I [ Z, ¯ Z, J, ¯ J ] = Z ∞−∞ dtλ r θ " ¯ Z ( J + im B θ ~ ˙ J ) + Z ( ¯ J − im B θ ~ ˙¯ J ) + m B λθ ~ r θ JJ (65)By introducing a "renormalized" (modified) source (and its corresponding com-plex conjugate)- J R ( t ) = (1 + im B θ ~ ∂ t ) J ( t ) , (66)and after ignoring the boundary contribution, the above interacting part of theaction (65) can be casted into the standard form - S I [ Z, ¯ Z, J R , ¯ J R ] = Z ∞−∞ dtλ r θ " ¯ ZJ R + Z ¯ J R + m B λθ ~ r θ J R (1 + im B θ ~ ∂ t ) − J R (67)Here the interaction part of the action (67) contains standard source terms aswell as a new quadratic source term. Now writing, Z ( t ) = q ( t )+ iq ( t ) √ θ , J R ( t ) = J (1) R + iJ (2) R , the functional integral (63) becomes with the irrelevant source-independent normalization factor ˜ N − being ignored, Z (cid:2) J R , ¯ J R (cid:3) = e imB ~ ( λθ ~ ) R ∞−∞ dt h ¯ J R (1+ imBθ ~ ∂ t ) − J R i Z eff h J (1) R , J (2) R i (68)where Z eff h J (1) R , J (2) R i = Z D q ( t ) D q ( t ) e i ~ R ∞−∞ dt h L eff + λ ( q J (1) R + q J (2) R ) i (69)with L eff = 12 m B ˙ q i − m B θk ~ ǫ ij q j ˙ q i − k q i (70)Therefore at strong magnetic field, the classical dynamics of our system (12)can be described by the effective equivalent Lagrangian (70), which can beinterpreted as the Lagrangian of a charged particle moving in the commutative q - q plane under the influence of a constant effective magnetic field with anadditional quadratic potential where the second term in the right hand side of(70) describes the interaction of an electrically charged particle (of charge e)with a constant magnetic field ( B eff ) pointing along the normal to the plane.The components of the corresponding vector potential ~A eff in symmetric gaugecan be read off from (70), ( ~A eff ) i = − m B θk c ~ e ǫ ij q j , (71)15ith ~B eff = ~ ∇ × ~A eff . Thus, we have an exact mapping between the pla-nar noncommutative system and (commutative) generalized Landau problem[70, 71], providing an effective commutative description of the noncommutativesystem. Therefore, the classical dynamics of the noncommutative system iseffectively described by the following action : S eff = Z dt (cid:20) m B ˙ q i + ec ( A eff ) i ˙ q i − k q i (cid:21) . (72)This form of the action reminds us of that of a charged particle interacting withmagnetic point vertex written in some appropriate variables for facilitating analternate consistent quantization procedure [18]. Also, the corresponding actionwas shown there to be scale-invariant. We now move on to study the scaletransformation properties of the action (72). dimensional non-commutative fields In this section, we study dilatation symmetry on noncommutative space by us-ing the equivalent commutative description. Further we will be deriving theanomalous Ward-Takahashi (W-T) identities and an explicit evaluation of thedilatation anomaly will be carried out in the path integral approach. The meth-ods developed here capture the quantum effects.
In a series of papers, Jackiw [18, 72] has shown the existence of time dilatationsymmetry for a charged point particle interacting with magnetic point vortexand magnetic monopole at the classical level. On the other hand, in [73] the di-latation symmetry of the non-relativistic Landau problem has been investigated.However it has been found that there is no dilatation symmetry associated withthe non-relativistic Landau problem, thus giving rise to a non-conserved dilata-tion current at the classical level and further a dilatation anomaly at the quan-tum level. Note that all these problems has a similar form for the Lagrangianbut their scaling properties are different. Motivated by this observation, weinvestigate whether the generalized Landau problem respect the symmetry andif any dilatation anomaly pop up. Now, by inspection we see that the effectiveaction (72) is not invariant under a global time dilatation: t → t ′ = e − γ t, (73)16here γ is a real parameter. Following Jackiw [18, 74], the form variation of thefield q i ( t ) due to infinitesimal dilatation transformation is given by - δ q i = γ ( t ˙ q i − q i ); (74)where δ q i ( t ) = q ′ i ( t ) − q i ( t ) is the functional change. It is now possible to writethe non-conserved dilatation generator [75] from a Noether analysis of (70). Toderive the generator of the broken dilatation symmetry, it will be convenient toallow the constant γ in (74) to depend on time: γ = γ ( t ) i.e., infinitesimal localvariation viz. δ L q i ( t ) = γ ( t ) Q q i ( t ) , (75)where Q q i ( t ) = ( t ∂∂t − ) q i ( t ) . The change in the action resulting from thesetransformations (74) is δ L S eff = Z dt (cid:20) ∂L eff ∂ ˙ q i ˙ γ Q q i ( t ) + γ ( t ) (cid:18) ∂L eff ∂ ˙ q i Q ˙ q i ( t ) + ∂L eff ∂q i Q q i ( t ) (cid:19)(cid:21) (76) = Z dt (cid:20) p i ˙ γ Q q i + γ ( t ) (cid:18) ddt ( tL eff ) + k q i + m B θk ~ ǫ ij q j ˙ q i (cid:19)(cid:21) , (77)where p i = ∂L eff ∂ ˙ q i is the canonical momentum. Here, we choose γ ( t ) to decayasymptotically so that one can safely discard the surface term and the action S eff then changes as : δ L S eff = Z dtγ ( t ) (cid:18) k q i + m B k θ ~ ǫ ij q j ˙ q i − dD ( t ) dt (cid:19) , (78)where D ( t ) = tH eff − ( q i p i ) , and H eff = ˙ q i p i − L eff . The expression (78) holds for any field configuration q i ( t ) with the specificchange δ L q i ( t ) . However, when q i ( t ) obeys the classical equations of motionthen δS eff = 0 for any δq i including the symmetry transformation (75) with γ ( t ) a function of time. This means that at the on-shell level, we have thenon-conserved dilatation charge ( D ) corresponding to the (broken) dilatationsymmetry as - dD ( t ) dt = ∆ ( t ) (79)where ∆ ( t ) := ∆ ( q i , ˙ q i ; t ) = k ( q i + m B θ ~ ǫ ij q j ˙ q i ) . Thus, we identify the non conserved dilatation charge D which is nothing butthe generator of the infinitesimal global transformation (74). Here, we observethat scale invariance is broken explicitly by the presence of parameter k in theaction (72); the dilatation charge D acquires a non vanishing time derivative(79). If k vanishes, the scale transformation (73) has no effect on the dynamicsand therefore corresponds to a symmetry at the classical level.17 .2 Anomalous Ward-Takahashi Identities Now we will move ahead and discuss about the path integral formulation ofWard-Takahashi (W-T) identities associated with the action described in (72).At zeroth order, it represents the quantum mechanical version of Noether’s cur-rent conservation theorem. Here in this section, we will explicitly derive theW-T identities upto 2nd order and during the course of derivation of these W-Tidentities, it will be evident that the contribution from Jacobian to be non-trivial. It is precisely this contribution from the Jacobian which gives rise tothe anomalous terms in the Ward identities. The anomalous term arising hereis eventually regularized using Fujikawa’s prescription [76].The starting point is to consider the generating functional for connected Green’sfunctions in dimensional QFT defined in equivalent commutative descrip-tion (unconstrained variables) (68). Here we switch on the source adiabaticallyand thus the higher order derivatives of J (1) R and J (2) R ) can be left out (we alsoassume J ( i ) R to vanish at boundaries), leaving (68) to be : Z h J (1) R , J (2) R i = Z D X ( t ) e i ~ h S eff [ X ( t )]+ λ R ∞−∞ dtJ TR .X + m B ( λθ ~ ) R ∞−∞ dt ( J TR .J R + iθmB ~ J TR σ y ˙ J R ) i (80)Note that in this expression, we have rewritten the action S eff as, S eff [ X ( t )] = Z ∞−∞ dt X T ( t ) R X ( t ) (81)by defining the column vectors, X ( t ) = 1 √ (cid:0) q ( t ) q ( t ) (cid:1) T ; J R ( t ) = √ (cid:16) J (1) R ( t ) J (2) R ( t ) (cid:17) T , (82)and the differential operator R - R := − ( m B d dt + k ) m B k θ ~ ddt − m B k θ ~ ddt − ( m B d dt + k ) ! (83) = − ( m B d dt + k ) I + im B σ y k θ ~ ddt (84)Here R is itself hermitian and its eigenvalues are being considered as discretehere - R φ k ( t ) = λ k φ k ( t ) , (85)where the eigenfunctions φ k ( t ) satisfy the usual orthogonality and completenessconditions: Z dtφ † k ( t ) φ j ( t ) = δ kj ; ∞ X k =1 φ k ( t ) φ † k ( t ′ ) = δ ( t − t ′ ) I . (86)18n order to properly specify the functional measure, the ( ) dimensional field X ( t ) are expanded in terms of these eigenfunctions, X ( t ) = ∞ X k =1 a k φ k ( t ) (87)Now, the functional-integral measure is then defined as, D X = k = ∞ Y k =1 da k = D a (88)To obtain the Ward identities, we study the behaviour of the commutative equiv-alent action (72) and the measure defined in (93) under infinitesimal transfor-mation (75) of the field q i specified by a local parameter γ ( t ) . Therefore, underthe local transformation - X ( t ) → X ′ ( t ) = X ( t ) + δ L X ( t ) , (89)with δ L X ( t ) = γ ( t )( t ∂∂t − ) X ( t ) , the expansion coefficients of (87) change to- a j → a ′ j = a j + X k a k Z dtγ ( t ) φ † j ( t )( t ∂∂t −
12 ) φ k ( t )) . (90)Thus, the measure changes as [77, 78] - D X → D X ′ = j = ∞ Y j =1 da ′ j = (det c jk ) k = ∞ Y k =1 da k = e T r ( ln c jk ) D X (91)where det ( c jk ) can be read as the Jacobian of the transformation (90), whereas,the transformation matrix c jk can be obtained by, c jk = ∂a ′ j ∂a k = δ jk + Z dtγ ( t ) φ † j ( t )( t ∂∂t −
12 ) φ k ( t ) (92)Now, the above generating functional of connected Green’s functions (80) maybe rewritten as, Z h J (1) R , J (2) R i = Z D X ′ ( t ) e i ~ R ∞−∞ dt h X ′ T R X ′ + λJ TR .X ′ + m B ( λθ ~ ) ( J TR .J R + iθmB ~ J TR σ y ˙ J R ) i = Z e T r ( ln c jk ) D X e i ~ R ∞−∞ dt h X T R X + λJ TR .X + m B ( λθ ~ ) ( J TR .J R + iθmB ~ J TR σ y ˙ J R ) i × e i ~ [ δ L S eff + λ R ∞−∞ dtJ TR ( t ) δ L X ( t )] , (93)The first equality here is a triviality: we have simply re-labelled X ( t ) by X ′ ( t ) as a dummy variable in the functional integral (80). The second equality is19ontrivial and uses the assumed transformations (89) and (91). For infinitesimal γ ( t ) the generating functional transforms as, Z h J (1) R , J (2) R i = Z D X ( t ) e i ~ R ∞−∞ dt h X T R X + λJ TR .X + m B ( λθ ~ ) ( J TR .J R + iθmB ~ ǫ ij J TR σ y ˙ J R ) i × e i ~ [ δ L S eff + R ∞−∞ dt ( λJ TR δ L X − i ~ γ ( t ) A ( t ))] (94)where, A ( t ) = X k φ † k ( t )( t ∂∂t −
12 ) φ k ( t ) . (95)This A ( t ) is referred to as the quantum anomaly term. Now reverting backto our previous co-ordinates and performing the Taylor series expansion of theexponential factor in (94) upto terms of O ( γ ) , we obtain :- Z h J (1) R , J (2) R i = Z D q ( t ) D q ( t ) e i ~ h S eff + R ∞−∞ dt [ λJ ( i ) R q i + m B ( λθ ~ ) ( J ( i ) R + θmB ~ ǫ ij J ( i ) R ˙ J R ( j ) ) i × (cid:18) i ~ Z dtγ ( t )(∆ − dDdt + λJ ( i ) R ( t ) Q q i ( t ) − i ~ A ( t )) + O ( γ ) (cid:19) , (96)where in the last line we have made use of eq.(78). The W-T identities aresummarized by the variational derivative (the change of integration variablesdoes not change the integral itself). In particular the variation with respect to γ ( t ) must vanish, since it holds for any value of γ , i.e. δZ [ J (1) R , J (2) R ] δγ ( t ′ ) = 0 (97)Thus Taylor expanding the exponential in (96) containing source terms and thenmaking their variation w.r.t. γ ( t ′ ) go to zero, we have (where | t ′ | < ∞ ):- Z D q ( t ) D q ( t ) e i ~ S eff × (1 + iλ ~ Z dt J ( i ) R ( t ) q i ( t ) − λ ~ Z Z dt dt J ( i ) R ( t ) J ( k ) R ( t ) q i ( t ) q k ( t )+ λ ~ Z Z dt dt im B ~ δ ( t − t ) (cid:20) ( J ( i ) R ( t )) + θm B ~ ǫ ij J ( i ) R ( t ) ˙ J R ( j ) ( t ) (cid:21) + ........ ) × i ~ (cid:18) ∆ − dDdt ′ + λJ ( i ) R ( t ′ ) Q q i ( t ′ ) − i ~ A ( t ′ ) (cid:19) , (98)20earrangement of the previous expression yields: Z D q ( t ) D q ( t ) e i ~ S eff ( i ~ (∆ ( t ′ ) − dDdt ′ − i ~ A ( t ′ ))+ iλ ~ [ i ~ Z dt J ( i ) R ( t ) q i ( t )(∆ ( t ′ ) − dDdt ′ − i ~ A ( t ′ ))+ Z dt J ( i ) R ( t ) Q q i ( t ) δ ( t − t ′ )] − λ ~ [ Z Z dt dt J ( i ) R ( t ) J ( k ) R ( t ) q i ( t ) Q q k ( t ) δ ( t − t ′ )+ Z Z dt dt J ( i ) R ( t ) J ( k ) R ( t ) Q q i ( t ) q k ( t ) δ ( t − t ′ )+ i ~ Z Z dt dt J ( i ) R ( t ) J ( k ) R ( t ) q i ( t ) q k ( t )(∆ ( t ′ ) − dDdt ′ − i ~ A ( t ′ )) − Z Z dt dt im B ~ δ ( t − t )(( J ( i ) R ( t )) + θm B ~ ǫ ij J ( i ) R ( t ) ˙ J ( j ) R ( t )) × i ~ (∆ ( t ′ ) − dDdt ′ − i ~ A ( t ′ ))] + ............... ) (99)The different orders of Ward-Takahashi identities are obtained in terms of meanvalues of dynamical quantities by taking the variation of right hand side of(99) with respect to the source term i.e. δδJ ( i ) l ( t ′ ) δδJ ( m ) R ( t ′ ) ..... and then setting J ( l ) R ( t ) ’s to .Thus the zeroth-order W-T identity:- ddt ′ h D ( t ′ ) i = h ∆ ( t ′ ) i − i ~ h A ( t ′ ) i (100)First-order W-T identity:- ddt ′ h T ∗ [ D ( t ′ ) q i ( t ′ )] i = h T ∗ [∆ ( t ′ ) q i ( t ′ )] i − i ~ h T ∗ [ A ( t ′ ) q i ( t ′ )] i− i ~ δ ( t ′ − t ′ ) h Q q i ( t ) i (101)Second-order W-T identity:- ddt ′ h T ∗ [ D ( t ′ ) q k ( t ′ ) q i ( t ′ ))] i = h T ∗ [∆ ( t ′ ) q k ( t ′ ) q i ( t ′ )] i− i ~ δ ( t ′ − t ′ ) h T ∗ [ q k ( t ′ ) Q q i ( t ′ ) i− i ~ δ ( t ′ − t ′ ) h T ∗ [ Q q k ( t ′ ) q i ( t ′ )] i− i ~ h T ∗ [ A ( t ′ ) q k ( t ′ ) q i ( t ′ )] i (102) where the mean values are defined as: h T ∗ [ ............ ] i = R D q ( t ) D q ( t )[ ............ ] e i ~ S e ff apart from the aforementioned normalization factor. k in the Lagrangian(70), which is consistent with that from Noether’s prescription (79). But atthe quantum level, there is also an additional second term on the right whichis the contribution from the existence of anomaly at the quantum level; it isthe anomaly term A ( t ) which is responsible for modifying the rate of change ofdilatation charge D ( t ) . The higher-order identities also indicate this anomalousbehavior up to "contact terms".It is worthwhile to note that in this derivation of W-T identities, the ill-definedexpression of the anomaly term in (95) apparently seems divergent. This isbecause, at each time t we are summing over an infinite number of modes φ k ( t ) ,however it is possible to give a meaning to this expression by the method ofregularization and subsequently extract out a non divergent result as we willsee in the next section. Now we are in a position to compute the anomalous term in W-T identities.Following Fujikawa’s prescription [78], the anomaly can be regularized by cor-recting each contribution from φ k ( t ) by a factor e − λ k /M as below - A ( t ) → A Reg. ( t ) := lim M →∞ X k φ † k ( t )( t ∂∂t −
12 ) φ k ( t ) e − λ k /M (103)Here φ k ( t ) φ † k ( t ) is an ill-defined object and can be interpreted consistently byfirst separating time points and then taking the coincident limit at a suitablestep. A simple manipulation therefore yields: A Reg. ( t ) = lim M →∞ lim t → t ′ tr X k ( t ∂∂t −
12 ) e − R /M φ k ( t ) φ † k ( t ′ )= lim M →∞ lim t → t ′ tr (cid:20) ( t ∂∂t −
12 ) e − R /M δ ( t − t ′ ) I (cid:21) = lim M →∞ lim t → t ′ tr Z ∞−∞ dz π ( t ∂∂t −
12 ) e − R /M e iz ( t − t ′ ) I (104)Note that in above, tr refers to only the × matrix indices whereas the Trappearing earlier in (91) refers to both functional and matrix indices. Herewe have used the completeness relation of φ k ( t ) (86) and in the last step, theintegral representation of δ -function has been used. Now we define θ ′ = θ ~
22o be used from now onwards in order to remove any ~ dependency of θ ren-dering θ ′ a classical parameter. The anomaly expression (104) after evaluating lim t → t ′ e − R M e iz ( t − t ′ ) and then taking the limit t → t ′ becomes - A Reg. ( t ) = lim M →∞ tr Z ∞−∞ dz π ( izt −
12 ) e − a I + b σ y I , (105)where a = 1 M ( m B z + 4 m B k θ ′ z − m B k z + k ) b = 4 m B k θ ′ M ( m B z − k z ) Using e − a I = e − a I , e bσ y = I cosh b + σ y sinh b and completing the trace opera-tion, we have :- A Reg. ( t ) = 2 lim M →∞ Z ∞−∞ dz π ( izt −
12 ) e − a cosh b (106)Since e − a cosh b is an even function in z , the first term in the integral doesn’tcontribute. Therefore, the regularized version of the anomaly term (106) isrewritable in a much simpler form as, A Reg. ( t ) = − lim M →∞ π ( I + I ) , (107)where the integrals I and I are given by - I = Z ∞ dze − a + b I = Z ∞ dze − a − b For simplicity in notation, let z − k θ ′ z = s , m B M = p , pω ′ = q (where ω ′ = k θ ′ + k m B ) and r = p s + k θ ′ . The integral I becomes : I = Z ∞ ds p s + k θ ′ exp (cid:8) − p ( s − ω ) (cid:9) = Z ∞ k θ ′ dr √ r exp (cid:8) − p ( r − ω ′ ) (cid:9) = e − pω ′ Z ∞ k θ ′ dr √ r e − pr +2 qr (108)Following similar steps, one can show :- I = e − pω ′ Z ∞ k θ ′ dr √ r e − pr +2 qr (109)23hus the regularised anomaly factor A Reg. ( t ) simplifies to - A Reg. ( t ) = − lim M →∞ π e − pω ′ Z ∞ k θ ′ dr r − / e − pr +2 qr = − π lim M →∞ e − pω ′ "Z ∞ dr r − / e − pr +2 qr − Z k θ ′ dr r − / e − pr +2 qr (110)Upon taking the limit M → ∞ , the 2nd integral in (110) is convergent and givesa finite result of k θ ′ . The first integral can be evaluated as in [79]: lim M →∞ e − pω ′ Z ∞ dr √ r e − pr +2 qr = lim M →∞ (2 p ) − Γ (cid:18) (cid:19) D − ( y ) e y e − pω ′ (111)where y = pω ′ √ p . We can also write D − / ( y ) in terms of K / ( y ) , where D − / ( y ) and K / ( y ) represent the parabolic cylindrical functions and modifiedBessel function respectively. lim M →∞ (2 p ) − Γ (cid:18) (cid:19) D − ( y ) e y = lim M →∞ (2 p ) − e y ( 14 y ) K / ( 14 y ) (112)Using K / ( x ) ≈ Γ( )( x ) − as x → , we have :- lim M →∞ e − pω ′ Z ∞ dr √ r e − pr +2 qr = 12 lim M →∞ Γ (cid:18) (cid:19) p − = 12 lim M →∞ Γ (cid:18) (cid:19) ( M/m B ) / (113) A Reg. ( t ) = − π (cid:20)
12 lim M →∞ Γ (cid:18) (cid:19) ( M/m B ) / − k θ ′ (cid:21) (114)The first term in the expression is independent of the coupling k and is the sameas in the non-interacting case of our equivalent effective commutative theory,which is usually considered to be non-anomalous and only the second term isindependent of the cutoff M . Therefore, following conventional wisdom [80], wenow concentrate our attention to the nontrivial finite contribution coming from(114). To obtain a non-divergent anomaly contribution, we need to renormalizethe relation (114) in free particle theory limit k → by simply adding a term i ~ γ ( t ) A f in the Lagrangian L eff (94), thus cancelling the divergence in (114).Here, A f is equal to the Fujikawa factor for the free particle theory, A f = − π lim M →∞ Γ (cid:18) (cid:19) ( M/m B ) / (115)Therefore, after renormalization the correct dilatation anomaly is given by : A renormalized = A Reg. − A f = 1 π k θ ′ (116)24aking into account the above renormalization scheme, only the finite part of theanomaly term contributes to obtain renormalized version of the set of bare Wardidentities given in (100-102). One can also notice the fact that the anomalouscorrection to the W-T identities is a first order in θ , where θ = ~ ceB , whichsuggests that for commutative limit i.e. θ → , one must have ~ → , thusimplying that the commutative limit coincides with the classical limit. Thisindeed attests to the fact that in this setting, the correct quantum corrections arefully taken into account only when we switch to the noncommutative scenario. Firstly, we summarize our key results. By considering a pair of non-relativisticinteracting opposite charged particles in a region of high constant magneticfield and low mass limit, we show how the Lagrangian of the system can be di-rectly mapped to a harmonic oscillator in noncommutative space, referred hereas exotic oscillators. A quantum picture is also provided for the above usingHilbert-Schimdt operators, thereby unleashing a physical setting of this abstractformulation of noncommutative quantum mechanics. Therefore, the noncom-mutativity is given a physical model here unlike in many other works whereit is postulated as a fundamental parameter. We then construct an effectivepath integral action for the noncommutative system using coherent states in amethod similar to the one followed by Gangopadhyay and Scholtz in [64], how-ever we have included the source term here. The effective commutative actionderived through path-integral describes a charged particle moving in a constantmagnetic field and under the influence of a harmonic oscillator potential - thegeneralized Landau problem. Also the form of this action is reminiscent of thatof a charged particle interacting with point magnetic vortex studied in [18] wherescale symmetry exists in contrast to the present case, suggesting that thoroughtreatment is required for each individual case. We compute the anomalous W-Tidentities which necessarily includes contribution from the Jacobian. This hasbeen unexplored in the literature from the point of view of noncommutativequantum mechanics under scaling transformations. The anomalous term hasbeen calculated and regularised following Fujikawa’s method.We now make certain pertinent observations in connection with the problem.It is to be pointed out that our investigation suggests the emergence of scaleanomalies in interacting quantum Hall systems as a natural consequence of im-purity interactions [50] upon quantization in large magnetic field limit. This isbecause we have built our noncommutative model starting from the Lagrangian(1). We also mention that the computation of anomalous correction in W-T identities is independent of the status of noncommutativity parameter θ -whether it being an independent fundamental constant or a derived one as isthe case here. The anomalous correction is directly proportional to the non-commutative parameter θ , which suggests that quantum corrections arise dueto the noncommutativity of spatial coordinates. One can show, by considering25he noncommutativity parameter to be fundamental, that the dilatation cur-rent for a harmonic oscillator in commutative space is not inflicted with ananomaly upon quantization. Also in this regard, we mention that the anomalycontribution in the non-conservation of dilatation current (i.e. in zeroth-orderW-T identity) (100) being imaginary can be traced to the fact that domain ofdefinition of the Hamiltonian does not remain invariant under quantum correc-tions [81]. It is also worth mentioning that the effect of noncommutativity canbe traced back to a curved momentum space [82] and our analysis hints at anemergence of scale anomaly in such a curved momentum space. In the end, wewant to point out a conceptual resemblance of the present problem with themassive scalar φ theory where a scale anomaly appears upon quantization be-sides the explicit broken charge owing to the mass of field in the sense that herealso there is explicit breaking of scale symmetry due to the spring constant withan additional anomalous correction occurring upon quantization. This featuremotivates an ambitious task of probing the effects of Planck scale in "contact"interacting two-dimensional ultracold Bose gas by starting from a noncommu-tative (2+1) massive relativistic scalar φ theory and then taking some suitablenon-relativistic limit [83, 84].Finally, our present work can be extended in the following directions: (i) Veryrecently, in [85], it has been shown that there exists a connection between con-formal symmetry breaking and production of entropy and also in a slightly dif-ferent perspective in [86]. We intend to investigate along this line as deformedquantum system has been shown to produce entropy in [87]. (ii) Another im-portant avenue of further study will be to look into the modified Virial theoremin the Euclidean version theory of the scale anomaly studied here, so that onecan find the pressure-energy relation in this system. It is generally believedthat anomalies are UV effects whereas Berry phase is an infrared effect - hereit seems interesting to explore any mixing between anomalies and geometricalphases in noncommutative spaces in the adiabatic approximation as hinted fromthe following studies [88, 89, 19]. Acknowledgements
The authors would like to extend their gratitudeness to Prof. Biswajit Chakrabortyfor a careful reading of the manuscript and for his questions and critical com-ments on the paper. One of the authors (P.N.) thanks Professor Kazuo Fujikawafor a correspondence and for enlightening him about quite a few subtle points inconnection with this paper. Also he has benefited from conversations with Fred-erik G. Scholtz, Alexei Deriglazov. We sincerely thank Prof. Rabin Banerejeeand Debasish Chatterjee for introducing us to the subject. Special thanks arealso due to Prof. A. P. Balachandran for his constant encouragement. S. S. isfunded by Jagadis Bose National Science Talent Search (JBNSTS) scholarshipand S. K. P. thanks UGC-India for providing financial assistance in the form offellowship during the course of this work.26 eferences [1] S. Coleman and R. Jackiw, “Why dilatation generators do not generatedilatations,”
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