View metadata, citation and similar papers at core.ac.uk brought to you by CORE Quantization, group contraction and zero point energy provided by CERN Document Server M. Blasone1;4, E. Celeghini2, P. Jizba3 and G. Vitiello4 1 Blackett Laboratory, Imperial College, London SW7 1BZ, U. K. 2 Dipartimento di Fisica, and Sezione INFN, Universit`a di Firenze, I-50125 Firenze, Italy 3 Institute of Theoretical Physics, University of Tsukuba, Ibaraki 305-8571, Japan 4 Dipartimento di Fisica \E.R. Caianiello", INFN and INFM, Universit`a di Salerno, I-84100 Salerno, Italy 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. I. INTRODUCTION the limit to the continuum which, according to ’t Hooft, leads to the quantum systems. + Recently, the “close relationship between quantum We then study the Dk representation of SU(1; 1) and harmonic oscillator (q.h.o.) and the classical particle find that it naturally provides the non-vanishing zero moving along a circle” has been discussed [1] in the frame point energy term. Due to the remarkable fact that h(1) + of ’t Hooft conjecture [2] according to which the dis- and the Dk representations share the same Hilbert space, sipation of information which would occur at a Planck we are able to find a one-to-one mapping of the deter- + scale in a regime of completely deterministic dynamics ministic system represented by the D1=2 algebra and the would play a role in the quantum mechanical nature of q.h.o. algebra h(1). Such a mapping is realized without our world. In particular, ’t Hooft has shown that in a recourse to group contraction, instead it is a non-linear certain class of classical, deterministic systems, the con- realization similar to the Holstein-Primakoff construction straints imposed in order to provide a bounded from be- for SU(2) [8]. low Hamiltonian, introduce information loss and lead to Our treatment sheds some light on the relationship “an apparent quantization of the orbits which resemble between the dissipative character of the system Hamil- the quantum structure seen in the real world”. tonian (formulated in the two-mode SU(1; 1) representa- Consistently with this scenario, it has been explicitly tion) and the zero point energy of the q.h.o., in accord shown [3] that the dissipation term in the Hamiltonian with the conclusions presented in Ref. [3]. 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 II. 'T HOOFT'S SCENARIO obtained after reduction. Such a dissipative term mani- fests itself as a geometric phase and thus the appearance As far as possible we will closely follow the presentation of the zero point energy in the spectrum of q.h.o can be and the notation of Ref. [1]. We start by considering the related with non-trivial topological features of an under- discrete translation group in time T1. ’t Hooft considers lying dissipative dynamics. the deterministic system consisting of a set of N states, The purpose of this paper is to further analyze the (ν) (0); (1); :::(N 1) , on a circle, which may be relationship discussed in [1] between the q.h.o. and the represented{ }≡{ as vectors: − } classical particle system, with special reference to the algebraic aspects of such a correspondence. 0 1 0 ’t Hooft’s analysis, based on the SU(2) structure, uses 0 0 1 0 0 1 0 . 1 (0) = ;(1)= ; ;( 1) = . (1) finite dimensional Hilbert space techniques for the de- . ::: N ; B . C B . C − B 1 C scription of the deterministic system under considera- B C B C B C B 1 C B 0 C B 0 C tion. Then, in the continuum limit, the Hilbert space @ A @ A @ A becomes infinite dimensional, as it should be to repre- and (0) (N). The time evolution takes place in discrete sent the q.h.o.. In our approach, we use the SU(1; 1) time steps≡ of equal size, ∆t = τ structure where the Hilbert space is infinite dimensional from the very beginning. t t + τ :(ν) (ν +1 modN)(2) → → We show that the relation foreseen by ’t Hooft between and thus is a finite dimensional representation ( ) classical and quantum systems, involves the group con- DN T1 of the above mentioned group. On the basis spanned by traction [7] of both SU(2) and SU(1; 1) to the common the states (ν), the evolution operator is introduced as [1] limit h(1). The group contraction completely clarifies (we use ~ =1): 1 (2) 1 N 2l +1;n m + l; m l; :::; l ; (5) (1) ≡ ≡ ≡− so that, by using the more familiar notation l; m for the | i 0.5 states n in Eq.(4) and introducing the operators L+ and (3) | i L and L3, we can write the set of equations − H 1 (0 l; m =(n + ) l; m : (6) -1 - 0.5 0.5 1 ! | i 2 | i L l; m = m l; m ; 3 | i | i L+ l; m = (2l n)(n +1) l; m +1 ; (4) - 0.5 | i − | i L l; m = p(2l n +1)n l; m 1 : (7) − | i p − | − i (6) with the su(2) algebra being satisfied (L L1 iL2): -1 ± (5) ≡ ± [Li;Lj ]=iijkLk ; i;j;k=1; 2; 3: (8) FIG. 1. ’t Hooft’s deterministic system for N =7. ’t Hooft then introduces the analogues of position and momentum operators: 01 τ 2 π xˆ αLx; pˆ βLy,α ,β − ; (9) 0 10 1 ≡ ≡ ≡ rπ ≡ 2l +1r τ iHτ i π 10 U(∆t = τ)=e− = e− N (3) B . C satisfying the “deformed” commutation relations B .. .. C B C B 10C τ @ A [ˆx; pˆ]=αβiLz = i 1 H : (10) − π This matrix satisfies the condition U N =1I and it can be diagonalized by a suitable transformation. The phase The Hamiltonian is then rewritten as factor in Eq.(3) is introduced by hand. It gives the 1 2 = 1 1 τ !2 term contribution to the energy spectrum of the eigen- H = !2xˆ2 + pˆ2 + + H2 : (11) 2 2 2 4 states of H denoted by n , n =0; 1; :::N 1: π | i − H 1 2π The continuum limit is obtained by letting l and n =(n + ) n ;! : (4) τ 0with! fixed for those states for which the→∞ energy ! | i 2 | i ≡ Nτ stays→ limited. In such a limit the Hamitonian goes to the The Hamiltonian H in Eq.(4) seems to have the same one of the harmonic oscillator, thex ˆ andp ˆ commutator spectrum of the Hamiltonian of the harmonic oscillator. goes to the canonical one and the Weyl-Heisenberg al- However it is not so, since its eigenvalues have an upper gebra h(1) is obtained. In that limit the original state bound implied by the finite N value (we have assumed space (finite N) changes becoming infinite dimensional. a finite number of states). Only in the continuum limit We remark that for non-zero τ Eq.(10) reminds the case (τ 0andl with ! fixed, see below) one will get of dissipative systems where the commutation relations a true→ correspondence→∞ with the harmonic oscillator. are time-dependent thus making meaningless the canon- The system of Eq.(1) is plotted in Fig. 1 for N =7. ical quantization procedure [4]. An underlying continuous dynamics is introduced, where We now show that the above limiting procedure is x(t)=cos(αt)cos(βt)andy(t)= cos(αt)sin(βt). At nothing but a group contraction. One may indeed de- − the times tj = jπ=α,withj integer, the trajectory fine a† L+=√2l, a L =√2l and, for simplicity, 2 2 2 ≡ ≡ − touches the external circle, i.e. R (tj)=x (tj )+y (tj)= restore the n notation (n = m + l) for the states: 1, and thus π/α is the frequency of the discrete (’t Hooft) | i H 1 system. At time tj, the angle of R(tj) with the posi- n =(n + ) n (12) tivexaxisisgivenby: θ = jπ βt = j(1 β/α)π. ! | i 2 | i j − j − When β/α is a rational number, of the form q = M=N, (2l n) a† n = − √n +1 n +1 ; the system returns to the origin (modulo 2π)afterN | i r 2l | i steps. To ensure that the N steps cover only one circle, 2l n +1 we have to impose α(tj )=j 2π/N, which gives the con- a n = − √n n 1 : (13) dition M = N 2. Thus, in order to reproduce ’t Hooft’s | i r 2l | − i system for N =7,asinFig.1,wechoose− q =5=7. For The continuum limit is then the contraction l (fixed N =8,wehaveq =3=4andsoon. !): →∞ The system of Eq.(1) can be described in terms of an SU(2) algebra if we set 2 (6)A (1)A (6)B (1)B H 1 1 1 n =(n + ) n : (14) A B ! | i 2 | i = √ +1 +1 .0 5 .0 5 a n n n ; A † (2) (2)B (5)A | i | i (5)B a n = √n n 1 ; (15) | i | − i (0)A (0)B and, by inspection, 1- -050.