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Quantization, Group Contraction and Zero Point Energy

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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.Jizbac, 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 which resemble the quantum structure seen in the real world”. Consistently with this scenario, it has been explic- Recently, the “close relationship between quantum itly shown [3] that the dissipation term in the Hamil- harmonic oscillator (q.h.o.) and the classical particle tonian for a couple of classical damped-amplified os- moving along a circle” has been discussed [1] in the cillators [4–6] is actually responsible for the zero point frame of ’t Hooft conjecture [2] according to which energy in the quantum spectrum of the 1D linear har- the dissipation of information which would occur at a monic oscillator obtained after reduction. Such a dis- Planck scale in a regime of completely deterministic sipative term manifests itself as a geometric phase and dynamics would play a role in the quantum mechan- thus the appearance of the zero point energy in the ical nature of our world. ’t Hooft has shown that, in spectrum of q.h.o. can be related with non-trivial topo- a certain class of classical deterministic systems, the logical features of an underlying dissipative dynamics. constraints imposed in order to provide a bounded- The purpose of this Letter is to further analyze the from-below Hamiltonian introduce information loss. relationship discussed in [1] between the q.h.o. and the This leads to “an apparent quantization of the orbits classical particle system, with special reference to the algebraic aspects of such a correspondence. ’t Hooft’s analysis, based on the SU(2) structure, * Corresponding author. uses finite-dimensional Hilbert space techniques for E-mail address: [email protected] (M. Blasone). 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 and (0) ≡ (N). The time evolution takes place in space becomes infinite-dimensional, as it should be discrete time steps of equal size, t = τ to represent the q.h.o. In our approach, we use the SU(1, 1) structure where the Hilbert space is infinite- t → t + τ : (ν) → (ν + 1) modN (2) dimensional from the very beginning. and thus is a finite-dimensional representation D (T ) We show that the relation foreseen by ’t Hooft N 1 of the above-mentioned group. On the basis spanned between classical and quantum systems, involves the by the states (ν), the evolution operator is introduced group contraction [7] of both SU(2) and SU(1, 1) as [1] (we use h¯ = 1): to the common limit h(1). The group contraction completely clarifies the limit to the continuum which, U( t = τ)= e−iHτ according to ’t Hooft, leads to the quantum systems. + 01 We then study the D representation of SU(1, 1) k 10 and find that it naturally provides the non-vanishing −i π = e N 10 . (3) 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 map- 10 ping of the deterministic system represented by the N = + This matrix satisfies the condition U 1 and it D1/2 algebra and the q.h.o. algebra h(1). Such a map- can be diagonalized by a suitable transformation. The ping is realized without recourse to group contrac- phase factor in Eq. (3) is introduced by hand. It gives tion, instead it is a non-linear realization similar to the the 1/2 term contribution to the energy spectrum of the Holstein–Primakoff construction for SU(2) [8]. eigenstates of H denoted by |n , n = 0, 1,...,N − 1: Our treatment sheds some light on the relationship between the dissipative character of the system Hamil- H 1 2π |n = n + |n ,ω≡ . (4) tonian (formulated in the two-mode SU(1, 1) repre- ω 2 Nτ sentation) and the zero point energy of the q.h.o., in The Hamiltonian H in Eq. (4) seems to have the accord with the conclusions presented in Ref. [3]. 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 2. ’t Hooft’s scenario (we have assumed a finite number of states). Only in the continuum limit (τ → 0andl →∞with ω fixed, see below) one will get a true correspondence with the As far as possible we will closely follow the harmonic oscillator. presentation and the notation of Ref. [1]. We start by The system of Eq. (1) can be described in terms of considering the discrete translation group in time T1. an SU(2) algebra if we set ’t Hooft considers the deterministic system consisting of a set of N states, {(ν)}≡{(0), (1),...,(N− 1)},on N ≡ 2l + 1,n≡ m + l, m ≡−l,...,l, (5) a circle, which may be represented as vectors: so that, by using the more familiar notation |l,m for the states |n in Eq. (4) and introducing the operators 0 1 L+ and L− and L3, we can write the set of equations 0 0 = ; = ; ; (0) . (1) . ... H 1 . |l,m = n + |l,m . (6) 1 0 ω 2 0 L |l,m =m|l,m , . 3 − = . (N 1) , (1) L+|l,m = (2l − n)(n + 1) |l,m + 1 , 1 0 L−|l,m = (2l − n + 1)n|l,m − 1 (7) M. Blasone et al. / Physics Letters A 310 (2003) 393–399 395 √ † with the su(2) algebra being satisfied (L± ≡ L ± a |n = n + 1 |n + 1 , 1 √ iL2): a|n = n |n − 1 , (15) [Li ,Lj ]=iij k Lk, i,j,k= 1, 2, 3. (8) and, by inspection, ’t Hooft then introduces the analogues of position and a,a† |n =|n , (16) momentum operators: 1 a†,a |n =2 n + |n . (17) xˆ ≡ αL , pˆ ≡ βL , 2 x y τ −2 π [ †]= = 1 { † } α ≡ ,β≡ , (9) We thus have a,a 1andH/ω 2 a ,a on π 2l + 1 τ the representation {|n }. With the usual definition of † satisfying the “deformed” commutation relations a and a , one obtains the canonical commutation relations [ˆx,pˆ]=i and the standard Hamiltonian of τ the harmonic oscillator. [ˆx,pˆ]=αβiLz = i 1 − H . (10) π We note that the underlying Hilbert space, origi- The Hamiltonian is then rewritten as nally finite-dimensional, becomes infinite-dimensional, under the contraction limit. Then we are led to con- 1 1 τ ω2 H = ω2xˆ2 + pˆ2 + + H 2 . (11) sider an alternative model where the Hilbert space is 2 2 2π 4 not modified in the continuum limit. 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 3. The SU(1, 1) systems goes to the one of the harmonic oscillator, the xˆ and pˆ commutator goes to the canonical one and the Weyl– The above model is not the only example one Heisenberg algebra h(1) is obtained. In that limit may find of a deterministic system which reduces the original state space (finite N) changes becoming to the quantum harmonic oscillator. For instance, infinite-dimensional. We remark that for non-zero we may consider deterministic systems based on τ Eq. (10) reminds the case of dissipative systems the non-compact group SU(1, 1). An example is the where the commutation relations are time-dependent system which consists of two subsystems, each of thus making meaningless the canonical quantization them made of a particle moving along a circle in procedure [4]. discrete equidistant jumps. Both particles and circle We now show that the above limiting procedure is radii might be different, the only common thing is nothing but a group√ contraction.√ One may indeed de- that both particles are synchronized in their jumps. fine a† ≡ L+/ 2l, a ≡ L−/ 2l and, for simplicity, We further assume that for both particles the ratio restore the |n notation (n = m + l) for the states: (circumference)/(length of the elementary jump) is an irrational number (generally different) so that particles H 1 |n = n + |n , (12) never come back into the original position after a finite ω 2 number of jumps. We shall label the corresponding states (positions) as (n)A and (n)B , respectively.
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