Quantization, group contraction and zero point energy

Physics Letters A 310 (2003) 393–399 www.elsevier.com/locate/pla Quantization, group contraction and zero point energy M. Blasone a,d,∗ , E. Celeghini b , P. Jizba c , G. Vitiello d a Blackett Laboratory, Imperial College, London SW7 1BZ, UK b Dipartimento di Fisica, and Sezione INFN, Università di Firenze, I-50125 Firenze, Italy c Institute of Theoretical Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan d Dipartimento di Fisica “E.R. Caianiello”, INFN and INFM, Università di Salerno, I-84100 Salerno, Italy Received 6 January 2003; received in revised form 6 January 2003; accepted 21 February 2003 Communicated by A.P. Fordy Abstract We study algebraic structures underlying ’t Hooft’s construction relating classical systems with the quantum harmonic oscillator. The role of group contraction is discussed. We propose the use of SU(1, 1) for two reasons: because of the isomorphism between its representation Hilbert space and that of the harmonic oscillator and because zero point energy is implied by the representation structure. Finally, we also comment on the relation between dissipation and quantization.  2003 Elsevier Science B.V. All rights reserved. 1. Introduction Recently, the “close relationship between quantum harmonic oscillator (q.h.o.) and the classical particle moving along a circle” has been discussed [1] in the frame of ’t Hooft conjecture [2] according to which the dissipation of information which would occur at a Planck scale in a regime of completely deterministic dynamics would play a role in the quantum mechanical nature of our world. ’t Hooft has shown that, in a certain class of classical deterministic systems, the constraints imposed in order to provide a boundedfrom-below Hamiltonian introduce information loss. This leads to “an apparent quantization of the orbits * Corresponding author. E-mail address: [email protected] (M. Blasone). which resemble the quantum structure seen in the real world”. Consistently with this scenario, it has been explicitly shown [3] that the dissipation term in the Hamiltonian for a couple of classical damped-amplified oscillators [4–6] is actually responsible for the zero point energy in the quantum spectrum of the 1D linear harmonic oscillator obtained after reduction. Such a dissipative term manifests itself as a geometric phase and thus the appearance of the zero point energy in the spectrum of q.h.o. can be related with non-trivial topological features of an underlying dissipative dynamics. The purpose of this Letter is to further analyze the relationship discussed in [1] between the q.h.o. and the classical particle system, with special reference to the algebraic aspects of such a correspondence. ’t Hooft’s analysis, based on the SU(2) structure, uses finite-dimensional Hilbert space techniques for the description of the deterministic system under con- 0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00374-8 394 M. Blasone et al. / Physics Letters A 310 (2003) 393–399 sideration. Then, in the continuum limit, the Hilbert space becomes infinite-dimensional, as it should be to represent the q.h.o. In our approach, we use the SU(1, 1) structure where the Hilbert space is infinitedimensional from the very beginning. We show that the relation foreseen by ’t Hooft between classical and quantum systems, involves the group contraction [7] of both SU(2) and SU(1, 1) to the common limit h(1). The group contraction completely clarifies the limit to the continuum which, according to ’t Hooft, leads to the quantum systems. We then study the Dk+ representation of SU(1, 1) and find that it naturally provides the non-vanishing zero point energy term. Due to the remarkable fact that h(1) and the Dk+ representations share the same Hilbert space, we are able to find a one-to-one mapping of the deterministic system represented by the + algebra and the q.h.o. algebra h(1). Such a mapD1/2 ping is realized without recourse to group contraction, instead it is a non-linear realization similar to the Holstein–Primakoff construction for SU(2) [8]. Our treatment sheds some light on the relationship between the dissipative character of the system Hamiltonian (formulated in the two-mode SU(1, 1) representation) and the zero point energy of the q.h.o., in accord with the conclusions presented in Ref. [3]. 2. ’t Hooft’s scenario As far as possible we will closely follow the presentation and the notation of Ref. [1]. We start by considering the discrete translation group in time T1 . ’t Hooft considers the deterministic system consisting of a set of N states, {(ν)} ≡ {(0), (1), . . . , (N − 1)}, on a circle, which may be represented as vectors:   0 0  (0) =   ...  ; 1   1 0  (1) =   ...  ;   0  ...   (N − 1) =  1, 0 ...; and (0) ≡ (N). The time evolution takes place in discrete time steps of equal size, t = τ   t → t + τ : (ν) → (ν + 1) mod N (2) and thus is a finite-dimensional representation DN (T1 ) of the above-mentioned group. On the basis spanned by the states (ν), the evolution operator is introduced as [1] (we use h̄ = 1): U ( t = τ ) = e−iH τ  =e π −i N 0 1     1 0 1 0 .. . .. . 1 0 (1)   .   (3) This matrix satisfies the condition U N = 1 and it can be diagonalized by a suitable transformation. The phase factor in Eq. (3) is introduced by hand. It gives the 1/2 term contribution to the energy spectrum of the eigenstates of H denoted by |n , n = 0, 1, . . . , N − 1: 1 H |n = n + |n , ω 2 ω≡ 2π . Nτ (4) The Hamiltonian H in Eq. (4) seems to have the same spectrum of the Hamiltonian of the harmonic oscillator. However it is not so, since its eigenvalues have an upper bound implied by the finite N value (we have assumed a finite number of states). Only in the continuum limit (τ → 0 and l → ∞ with ω fixed, see below) one will get a true correspondence with the harmonic oscillator. The system of Eq. (1) can be described in terms of an SU(2) algebra if we set N ≡ 2l + 1, n ≡ m + l, m ≡ −l, . . . , l, (5) so that, by using the more familiar notation |l, m for the states |n in Eq. (4) and introducing the operators L+ and L− and L3 , we can write the set of equations 1 H |l, m = n + |l, m . ω 2 0  (6) L3 |l, m = m|l, m , L+ |l, m = (2l − n)(n + 1) |l, m + 1 , L− |l, m = (2l − n + 1)n |l, m − 1 (7) M. Blasone et al. / Physics Letters A 310 (2003) 393–399 with the su(2) algebra being satisfied (L± ≡ L1 ± iL2 ): [Li , Lj ] = iǫij k Lk , i, j, k = 1, 2, 3. (8) ’t Hooft then introduces the analogues of position and momentum operators: x̂ ≡ αLx , p̂ ≡ βLy , α≡ β≡ τ , π −2 2l + 1 π , τ (9) satisfying the “deformed” commutation relations [x̂, p̂] = αβiLz = i 1 − τ H . π (10) The Hamiltonian is then rewritten as 1 1 τ H = ω2 x̂ 2 + p̂2 + 2 2 2π ω2 + H2 . 4 (11) The continuum limit is obtained by letting l → ∞ and τ → 0 with ω fixed for those states for which the energy stays limited. In such a limit the Hamiltonian goes to the one of the harmonic oscillator, the x̂ and p̂ commutator goes to the canonical one and the Weyl– Heisenberg algebra h(1) is obtained. In that limit the original state space (finite N ) changes becoming infinite-dimensional. We remark that for non-zero τ Eq. (10) reminds the case of dissipative systems where the commutation relations are time-dependent thus making meaningless the canonical quantization procedure [4]. We now show that the above limiting procedure is nothing but a group √ One may indeed de√ contraction. fine a † ≡ L+ / 2l, a ≡ L− / 2l and, for simplicity, restore the |n notation (n = m + l) for the states: H 1 |n = n + |n , ω 2 (12) (2l − n) √ n + 1 |n + 1 , 2l 2l − n + 1 √ n |n − 1 . a|n = (13) 2l The continuum limit is then the contraction l → ∞ (fixed ω): a † |n = 1 H |n = n + |n , ω 2 (14) √ a † |n = n + 1 |n + 1 , √ a|n = n |n − 1 , and, by inspection,  a, a † |n = |n ,  †  1 a , a |n = 2 n + |n . 2 395 (15) (16) (17) We thus have [a, a †] = 1 and H /ω = 21 {a † , a} on the representation {|n }. With the usual definition of a and a † , one obtains the canonical commutation relations [x̂, p̂] = i and the standard Hamiltonian of the harmonic oscillator. We note that the underlying Hilbert space, originally finite-dimensional, becomes infinite-dimensional, under the contraction limit. Then we are led to consider an alternative model where the Hilbert space is not modified in the continuum limit. 3. The SU(1, 1) systems The above model is not the only example one may find of a deterministic system which reduces to the quantum harmonic oscillator. For instance, we may consider deterministic systems based on the non-compact group SU(1, 1). An example is the system which consists of two subsystems, each of them made of a particle moving along a circle in discrete equidistant jumps. Both particles and circle radii might be different, the only common thing is that both particles are synchronized in their jumps. We further assume that for both particles the ratio (circumference)/(length of the elementary jump) is an irrational number (generally different) so that particles never come back into the original position after a finite number of jumps. We shall label the corresponding states (positions) as (n)A and (n)B , respectively. The synchronized time evolution is by discrete and identical time steps t = τ as follows: t → t + τ; (1)A → (2)A → (3)A → (4)A · · · , (1)B → (2)B → (3)B → (4)B · · · . This evolution is, of course, completely deterministic. A practical realization of one of such particle subsystem is in fact provided by a charged particle in the 396 M. Blasone et al. / Physics Letters A 310 (2003) 393–399 cylindrical magnetron, which is a device with a radial, cylindrically symmetric electric field that has in addition a perpendicular uniform magnetic field. Then the particle trajectory is basically a cycloid which is wrapped around the center of the magnetron. The actual parameters of the cycloid are specified by the Larmor frequency ωL = qB/2m. We confine ourself only to observation of the largest radius positions of the particle, disregarding any information concerning the actual underlying trajectory. If the Larmor frequency and orbital frequency are incommensurable then the particle proceeds via discrete time evolution with τ = 2π/ωL and returns into its initial position only after infinitely many revolutions. The actual states (positions) can be represented by vectors similar in structure to the ones in Eq. (1) with the important difference that in the present case the number N of their components is infinite. It might be worthwhile to observe that the set of vectors  {|n1 , n2 , . . . , ni , . . . } with infinite number N = i ni of their components is an uncountable set and it may be put in one-to-one correspondence with the set of real numbers. This is best seen by adopting the binary number system where the set of real numbers is {A = 0.n1 n2 · · · ni · · ·} with ni = 0, 1 for each i. In such case the set of real numbers {A} covers the interval (0, 1) of the real line and it is, indeed, an uncountable set. As usual, one assumes then to be able to select a countable subset {ξn } for the basis of the Hilbert state space H, namely one assumes that H is a separable space. Under such an assumption, any vector ξ in H can be approximated by a linear combination of ξn to any accuracy, i.e., for any ξ in H and  any ǫ > 0, it exists a sequence {cn } such that |ξ − n cn ξn | < ǫ. The one-time-step evolution operator acts on (n)A ⊗ (m)B and in the representation space of the states it reads U (τ ) ≡ e−iH τ  0 1 = 0 = e−iHA τ 0 ... 0 ... 1 ... .. .. . . 0 1 ⊗ 0  0 0 1 .. . ⊗ e−iHB τ  1 0  0 ... ... ... .. . A  1 0  0 . B (18) We stress that Eq. (18) symbolically represents infinitedimensional (square) matrices. As customary, however, one works with finite-dimensional matrices and at the end of the computations the infinite-dimensional limit is considered. Such a limiting procedure is the one by which any vector ξ of our space may be represented to any accuracy by the countable basis {ξn }, as said above. The advantage with respect to the previous SU(2) case is now that the non-compactness of SU(1, 1) guarantees that only the matrix elements of the rising and lowering operators are modified in the contraction procedure. Since the SU(1, 1) group is well known (see, e.g., [9]), we only recall that it is locally isomorphic to the (proper) Lorentz group in two spatial dimensions SO(2, 1) and it differs from SU(2) only in a sign in the commutation relation: [L+ , L− ] = −2L3 . SU(1, 1) representations are well known, in particular, the discrete series Dk+ is L3 |n = (n + k)|n , L+ |n = (n + 2k)(n + 1) |n + 1 , L− |n = (n + 2k − 1)n |n − 1 , (19) where, like in h(1), n is any integer greater or equal to zero and the highest weight k is a non-zero positive integer or half-integer number. In order to study the connection with the quantum harmonic oscillator, we set H = L3 − k + ω L+ a† = √ , 2k 1 , 2 L− a= √ . 2k (20) (21) The SU(1, 1) contraction k → ∞ again recovers the quantum oscillator Eqs. (15) and (17), i.e., the h(1) algebra. From (19), as announced, we see that the contraction k → ∞ does not modify L3 and its spectrum but only the matrix elements of L± . The relevant point is that, while in the SU(2) case the Hilbert space gets modified in the contraction limit, in the present SU(1, 1) case the Hilbert space is not modified in such a limit: a mathematically well founded perturbation theory can be now formulated (starting from Eq. (19), with perturbation parameter ∝ 1/k) in order to recover the wanted Eq. (15) in the contraction limit. M. Blasone et al. / Physics Letters A 310 (2003) 393–399 397 4. The zero point energy 5. The dissipation connection We now concentrate on the phase factor in Eq. (3), which fixes the zero point energy in the oscillator spectrum. It is well known that the zero point energy is the true signature of quantization and is a direct consequence of the non-zero commutator of x̂ and p̂. Thus this is a crucial point in the present analysis. The SU(2) model considered in Section 2 says nothing about the inclusion of the phase factor. On the other hand, it is remarkable that the SU(1, 1) setting, with H = ωL3 , always implies a non-vanishing phase, since k > 0. In particular, the fundamental representation has k = 1/2 and thus Eqs. (19) and (22) suggest to us one more scenario where we may recover the already known connection [2,3] between dissipation and quantization. Indeed, by introducing the Schwinger-like two mode SU(1, 1) realization in terms of h(1) ⊗ h(1), the square roots in the eigenvalues of L+ and L− in Eq. (22) may also be recovered. We set: L3 |n = n + 1 |n , 2 L+ |n = (n + 1)|n + 1 , L− |n = n|n − 1 . (22) We note that the rising and lowering operator matrix elements do not carry the square roots, as on the contrary happens for h(1) (cf., e.g., Eq. (15)). Then we introduce the following mapping in the universal enveloping algebra of su(1, 1): 1 a= √ L− ; L3 + 1/2 1 (23) a † = L+ √ L3 + 1/2 which gives us the wanted h(1) structure of Eq. (15), with H = ωL3 . Note that now no limit (contraction) is necessary, i.e., we find a one-to-one (non-linear) mapping between the deterministic SU(1, 1) system and the quantum harmonic oscillator. The reader may recognize the mapping Eq. (23) as the non-compact analog [10] of the well-known Holstein–Primakoff representation for SU(2) spin systems [8,11]. We remark that the 1/2 term in the L3 eigenvalues now is implied by the used representation. Moreover, after a period T = 2π/ω, the evolution of the state presents a phase π that it is not of dynamical origin: e−iH T = 1, it is a geometric-like phase (remarkably, related to the isomorphism between SO(2, 1) and SU(1, 1)/Z2 (ei2·2πL3 = 1)). Thus the zero point energy is strictly related to this geometric-like phase (which confirms the result of Ref. [3]). L+ ≡ A† B † , L− ≡ AB ≡ L†+ ,  1 L3 ≡ A† A + B † B + 1 , 2 (24) with [A, A† ] = [B, B † ] = 1 and all other commutators equal to zero. The Casimir operator is C 2 = 1/4+L23 − 1/2(L+ L− + L− L+ ) = 1/4(A†A − B † B)2 . We now denote by {|nA , nB } the set of simultaneous eigenvectors of the A† A and B † B operators with nA , nB non-negative integers. We may then express the states |n in terms of the basis |j, m , with j integer or half-integer and m
|j |, and 1 j = (nA − nB ), (25) 2 1 1 |j, m , m = (nA + nB ). L3 |j, m = m + 2 2 (26) Here m − |j | = n and |j | + 1/2 = k (cf. Eq. (19)). Clearly, for j = 0, i.e., n = nA = nB , we have the fundamental √ √ representation (22) and L− |n = AB|n = n n |n = n|n (and similarly for L+ ). This accounts for the absence of square roots in Eq. (22). In order to clarify the underlying physics, it is π convenient to change basis: |φj,m ≡ e 2 L1 |j, m . By exploiting the relation [4] C|j, m = j |j, m , π π ie 2 L1 L3 e− 2 L1 = L2 , (27) we have L2 |φj,m = i m + 1 |φj,m . 2 (28) Here it is necessary to remark that one should be careful in handling the relation (27) and the states |φj,m . In fact Eq. (27) is a non-unitary transformation in SU(1, 1) and the states |φj,m do not provide a unitary irreducible representation (UIR). They are 398 M. Blasone et al. / Physics Letters A 310 (2003) 393–399 indeed not normalizable states [12,13] (in any UIR of SU(1, 1), L2 should have a purely continuous and real spectrum [14], which we do not consider in the present case). It has been shown that these pathologies can be amended by introducing a suitable inner product in the state space [4,6,12] and by operating in the Quantum Field Theory framework. In the present case, we set the Hamiltonian to be H = H0 + HI , (29)   H0 ≡ Ω A† A − B † B = 2ΩC,   HI ≡ iΓ A† B † − AB = −2Γ L2 . (30) Here we have also added the constant term H0 and set 2Γ ≡ ω. In Ref. [4] it has been shown that the Hamiltonian (29) arises in the quantization procedure of the damped harmonic oscillator. On the other hand, in Ref. [3], it was shown that the above system belongs to the class of deterministic quantum systems à la ’t Hooft, i.e., those systems who remain deterministic even when described by means of Hilbert space techniques. The quantum harmonic oscillator emerges from the above (dissipative) system when one imposes a constraint on the Hilbert space, of the form L2 |ψ = 0. Further details on this may be found in Ref. [3]. See also Refs. [15,16] for related ideas. 6. Conclusions In this Letter, we have discussed algebraic structures underlying the quantization procedure recently proposed by G. ’t Hooft [1,2]. We have shown that the limiting procedure used there for obtaining truly quantum systems out of deterministic ones, has a very precise meaning as a group contraction from SU(2) to the harmonic oscillator algebra h(1). We have then explored the role of the non-compact group SU(1, 1) and shown how to realize the group contraction to h(1) in such case. One advantage of working with SU(1, 1) is that its representation Hilbert space is infinite-dimensional, thus it does not change dimension in the contraction limit, as it happens for the SU(2) case. However, the most important feature appears when we consider the Dk+ representations of SU(1, 1), and Fig. 1. Different quantization routes. The left route represent ’t Hooft procedure, with contraction of su(2) to h(1). + in particular D1/2 : we have shown that in this case the zero-point energy is provided in a natural way with the choice of the representation. Also, we realize a oneto-one mapping of the deterministic system onto the quantum harmonic oscillator. Such a mapping is an analog of the well-known Holstein–Primakoff mapping used for diagonalizing the ferromagnet Hamiltonian [8,11]. A shematic representation of quantization routes explored in this Letter is shown in Fig. 1. Finally, we have given a realization of the SU(1, 1) structure in terms of a system of damped-amplified oscillators [4] and made connection with recent results [3]. Acknowledgements We acknowledge the ESF Program COSLAB, EPSRC, INFN and INFM for partial financial support. References [1] G. ’t Hooft, hep-th/0104080; G. ’t Hooft, hep-th/0105105. [2] G. ’t Hooft, in: Basics and Highlights of Fundamental Physics, Erice, 1999, hep-th/0003005. [3] M. Blasone, P. Jizba, G. Vitiello, Phys. Lett. A 287 (2001) 205. [4] E. Celeghini, M. Rasetti, G. Vitiello, Ann. Phys. 215 (1992) 156. M. Blasone et al. / Physics Letters A 310 (2003) 393–399 [5] M. Blasone, E. Graziano, O.K. Pashaev, G. Vitiello, Ann. Phys. 252 (1996) 115. [6] M. Blasone, P. Jizba, quant-ph/0102128; M. Blasone, P. Jizba, Can. J. Phys. 80 (2002) 645. [7] E. Inönü, E.P. Wigner, Proc. Nat. Acad. Sci. Am. 39 (1953) 510. [8] T. Holstein, H. Primakoff, Phys. Rev. 58 (1940) 1098. [9] A.M. Perelomov, Generalized Coherent States and Their Applications, Springer, Berlin, 1986. [10] C.C. Gerry, J. Phys. A 16 (1983) L1. 399 [11] M.N. Shah, H. Umezawa, G. Vitiello, Phys. Rev. B 10 (1974) 4724. [12] H. Feshbach, Y. Tikochinsky, Trans. N.Y. Acad. Sci. Ser. II 38 (1977) 44. [13] Y. Alhassid, F. Gursey, F. Iachello, Ann. Phys. 148 (1983) 346. [14] G. Lindblad, B. Nagel, Ann. Inst. H. Poincaré A 13 (1970) 346. [15] A.J. Bracken, quant-ph/0210164. [16] H.T. Elze, O. Schipper, Phys. Rev. D 66 (2002) 044020.