2D NC harmonic oscillator arXiv:1207.3303v1 [hep-th] 13 Jul 2012 The two dimensional harmonic oscillator on a noncommutative space with minimal uncertainties Sanjib Dey and Andreas Fring Centre for Mathematical Science, City University London, Northampton Square, London EC1V 0HB, UK E-mail: sanjib.dey.1@city.ac.uk, a.fring@city.ac.uk Abstract: The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken. In [1] we demonstrated how canonical relations implying minimal uncertainties can be derived from a q-deformed oscillator algebra for the creation and annihilation operators A†i , Ai Ai A†j − q 2δ ij A†j Ai = δ ij , [A†i , A†j ] = 0, [Ai , Aj ] = 0, for i, j = 1, 2, 3; q ∈ R, (1) as investigated for instance in [2, 3, 4, 5, 6]. Starting from the general Ansatz X = κ̂1 (A†1 + A1 ) + κ̂2 (A†2 + A2 ) + κ̂3 (A†3 + A3 ), Y = Z = Px = Py = Pz = iκ̂4 (A†1 − A1 ) + iκ̂5 (A†2 − A2 ) + iκ̂6 (A†3 − A3 ), κ̂7 (A†1 + A1 ) + κ̂8 (A†2 + A2 ) + κ̂9 (A†3 + A3 ), iκ̌10 (A†1 − A1 ) + iκ̌11 (A†2 − A2 ) + iκ̌12 (A†3 − A3 ), κ̌13 (A†1 + A1 ) + κ̌14 (A†2 + A2 ) + κ̌15 (A†3 + A3 ), iκ̌16 (A†1 − A1 ) + iκ̌17 (A†2 − A2 ) + iκ̌18 (A†3 − A3 ), (2) (3) (4) (5) (6) (7) p √ with κ̂i = κi ~/(mω) for i = 1, . . . , 9 and κ̌i = κi mω~ for i = 10, . . . , 18 we constructed some particular solutions and investigated the harmonic oscillator on these spaces. Here we provide an additional two dimensional solution previously reported in [6]. Setting κ3 = κ6 = κ7 = κ12 = κ15 = κ16 = κ17 = κ18 = 0 in equations (2)-(7), employing the constraints 2D NC harmonic oscillator reported in [6] together with the subsequent nontrivial limit q → 1, the deformed oscillator algebra
[X, Y ] = iθ 1 + τ̂ Y 2 , 2 [Px , Py ] = iτ̂ ~θ Y 2 ,
[X, Px ] = i~ 1 + τ̂ Y 2 ,
[Y, Py ] = i~ 1 + τ̂ Y 2 , [X, Py ] = 0, [Y, Px ] = 0, (8) was obtained, with τ̂ = τ mω/~ having the dimension of an inverse squared length. By the same reasoning as provided in [7, 8, 5, 9, 6, 1], we find the minimal uncertainties q ∆Ymin = 0, ∆ (Px )min = 0, ∆ (Py )min = ~ τ̂ + τ̂ 2 hY i2ρ , ∆Xmin = |θ| τ̂ + τ̂ hY (9) where h.iρ denotes the inner product on a Hilbert space with metric ρ in which the operators X, Y, Px and Py are Hermitian. So far no representation for the two dimensional algebra (8) was provided. Here we find that it can be represented by q 2 i2ρ , X = x0 + τ̂ y02 x0 , Y = y0 , Px = px0 , ~ and Py = py0 − τ̂ y02 x0 , θ (10) where the x0 , y0 , px0 , py0 satisfy the common commutation relations for the flat noncommutative space [x0 , y0 ] = iθ, [px0 , py0 ] = 0, [x0 , px0 ] = i~, [y0 , py0 ] = i~, [x0 , py0 ] = 0, [y0 , px0 ] = 0, for θ ∈ R. (11) Clearly there exist many more solutions one may construct in this systematic manner from the Ansatz (2)-(7), which will not be our concern here. Instead we will study a concrete model, i.e. the two-dimensional harmonic oscillator on the noncommutative space described by the algebra (8). Using the representation (10), the corresponding Hamiltonian reads mω 2 2 1 (Px2 + Py2 ) + (X + Y 2 ) (12) 2m 2     τ̂ τ̂ 2 ~ ~2 2D 2 2 2 2 2 2 = Hf ncho + {y x0 , py0 } + mω {y0 x0 , x0 } − mω + 2 y 0 x0 y 0 x0 2 mθ 0 2 mθ 2D Hncho = where we used the standard notation for the anti-commutator {A, B} := AB + BA. Evidently this Hamiltonian is non-Hermitian with regard to the standard inner product, but respects an antilinear symmetry PT ± : x0 → ±x0 , y0 → ∓y0 , px0 → ∓px0 , py0 → ±py0 and i → −i. This suggests that its eigenvalue spectrum might be real, or at least real in parts [10, 11, 12]. Let us now investigate the spectrum perturbatively around the solution of the standard harmonic oscillator. In order to perform such a computation we need to convert flat noncommutative space into the standard canonical variable xs , ys , pxs and pys . This is achieved by means of a so-called Bopp-shift x0 → xs − ~θ pys , y0 → ys , px0 → pxs and –2– 2D NC harmonic oscillator py0 → pys . The Hamiltonian in (12) then acquires the form   mθ2 ω 2 2 mθω 2 τ̂ ~ 2D 2D 2 2 2 Hncho = Hho + p − {xs , pys } + {y xs , pys } (13) mω {ys xs , xs } − 2~2 ys 2~ 2 mθ s   
τ̂ mθ2 ω 2 1 mθω 2 2 2 2 + + {ys pys , pys } − {ys pys , xs } + {ys xs , pys } 2 m ~2 ~    
τ̂ 2 1 ~ mθ2 ω 2 2 τ̂ 2 mθω 2 2 2 2 2 + y s p y s y s xs + y s xs y s p y s + + ys pys ys2 pys − 2 ~ mθ 2 m ~2   τ̂ 2 ~2 2 2 2 + mω + 2 y s xs y s xs 2 mθ 2D 2D = Hho (xs , ys , pxs , pys ) + Hnc (xs , ys , pxs , pys ). In this formulation we may now proceed to expand perturbatively around the standard two dimensional Fock space harmonic oscillator solution with normalized eigenstates (a†1 )n1 (a†2 )n2 √ |00i , n1 !n2 ! ai |00i = 0, |n1 n2 i = a†i |n1 n2 i = √ ni + 1 |(n1 + δ i1 )(n2 + δ i2 )i , √ ai |n1 n2 i = ni |(n1 − δ i1 )(n2 − δ i2 )i , (14) (15) (0) 2D |nli = E for i = 1, 2, such that Hho nl |nli. The energy eigenvalues for the Hamiltonian 2D Hncho then result to (p) (0) (1) (2) Enl = Enl + Enl + Enl + O(τ 2 ) X (0) 2D |nli + = Enl + hnl| Hnc (16) 2D |pqi hpq| Hnc (0) (0) Enl − Epq 2D hnl| Hnc p,q6=n+l=p+q |nli + O(τ 2 ) 1 = ω~ (n + l + 1) + ~ωΩ [2n − (2l + 1)Ω + 10l + 6] 16 
 1 + ~τ ω Ω 8nl + 4n + 6l2 + 10l + 5 + 10nl + 5n + 5l2 + 10l + 5 + O(τ 2 ), 8 where Ω = m2 θ2 ω2 /~2 . We notice the minus sign in one of the terms, which might be an indication for the existence of an exceptional point [13, 14] in the spectrum. Naturally it would be very interesting to obtain a more precise expression for the eigenenergies, but nonetheless the first order approximations will be very useful for the computation of coherent states [15]. Acknowledgments: SD is supported by a City University Research Fellowship. References [1] S. Dey, A. Fring, and L. Gouba, PT-symmetric noncommutative spaces with minimal volume uncertainty relations, arXiv:1205.2291 (2012). [2] L. C. Biedenham, The quantum group group SU (2)q and a q-analogue of the boson operators, J. Phys. A22, L873–L878 (1989). [3] A. J. Macfarlane, On q-analogues of the quantum harmonic oscillator and the quantum group SU (2)q , J. Phys. A22, 4581–4588 (1989). –3– 2D NC harmonic oscillator [4] C.-P. Su and H.-C. Fu, The q-deformed boson realisation of the quantum group SU (n)q and its representations, J. Phys. A22, L983–L986 (1989). [5] B. Bagchi and A. Fring, Minimal length in Quantum Mechanics and non-Hermitian Hamiltonian systems, Phys. Lett. A373, 4307–4310 (2009). [6] A. Fring, L. Gouba, and B. Bagchi, Minimal areas from q-deformed oscillator algebras, J. Phys. A43, 425202 (2010). [7] A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35, 4483–4496 (1994). [8] A. Kempf, G. Mangano, and R. B. Mann, Hilbert space representation of the minimal length uncertainty relation, Phys. Rev. D52, 1108–1118 (1995). [9] A. Fring, L. Gouba, and F. G. Scholtz, Strings from dynamical noncommutative space-time, J. Phys. A43, 345401(10) (2010). [10] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry, Phys. Rev. Lett. 80, 5243–5246 (1998). [11] A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Maths. Phys. 43, 202–212 (2002). [12] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rept. Prog. Phys. 70, 947–1018 (2007). [13] C. M. Bender and T. T. Wu, Anharmonic Oscillator, Phys. Rev. 184, 1231–1260 (1969). [14] T. Kato, Perturbation Theory for Linear Operators, (Springer, Berlin) (1966). [15] S. Dey and A. Fring, Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations, in preparation –4–