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. (2l − n) √ The synchronized time evolution is by discrete and a†|n = n + |n + , 1 1 identical time steps t = τ as follows: 2l 2l − n + 1√ → + ; a|n = n |n − 1 . (13) t t τ 2l (1)A → (2)A → (3)A → (4)A ···, The continuum limit is then the contraction l →∞ (1)B → (2)B → (3)B → (4)B ···. (fixed ω): This evolution is, of course, completely deterministic. H 1 |n = n + |n , (14) A practical realization of one of such particle subsys- ω 2 tem 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, We stress that Eq. (18) symbolically represents infinite- cylindrically symmetric electric field that has in ad- dimensional (square) matrices. As customary, how- dition a perpendicular uniform magnetic field. Then ever, one works with finite-dimensional matrices and the particle trajectory is basically a cycloid which is at the end of the computations the infinite-dimensional wrapped around the center of the magnetron. The ac- limit is considered. Such a limiting procedure is the tual parameters of the cycloid are specified by the Lar- one by which any vector ξ of our space may be repre- mor frequency ωL = qB/2m. We confine ourself only sented to any accuracy by the countable basis {ξn},as to observation of the largest radius positions of the said above. particle, disregarding any information concerning the The advantage with respect to the previous SU(2) actual underlying trajectory. If the Larmor frequency case is now that the non-compactness of SU(1, 1) and orbital frequency are incommensurable then the guarantees that only the matrix elements of the rising particle proceeds via discrete time evolution with τ = and lowering operators are modified in the contraction 2π/ωL and returns into its initial position only after procedure. Since the SU(1, 1) group is well known infinitely many revolutions. (see, e.g., [9]), we only recall that it is locally isomor- The actual states (positions) can be represented phic to the (proper) Lorentz group in two spatial di- by vectors similar in structure to the ones in Eq. (1) mensions SO(2, 1) and it differs from SU(2) onlyina with the important difference that in the present case sign in the commutation relation: [L+,L−]=−2L3. the number N of their components is infinite. It SU(1, 1) representations are well known, in particular, + might be worthwhile to observe that the set of vectors the discrete series Dk is {| } = n1,n2,...,ni ,... with infinite number N i ni of their components is an uncountable set and it may L |n =(n + k)|n , 3 be put in one-to-one correspondence with the set of L+|n = (n + 2k)(n + 1) |n + 1 , real numbers. This is best seen by adopting the binary number system where the set of real numbers is {A = L−|n = (n + 2k − 1)n|n − 1 , (19) 0.n n ···n ···} with n = 0, 1 for each i.Insuch 1 2 i i where, like in h(1), n is any integer greater or equal case the set of real numbers {A} covers the interval to zero and the highest weight k is a non-zero positive (0, 1) of the real line and it is, indeed, an uncountable integer or half-integer number. set. As usual, one assumes then to be able to select a In order to study the connection with the quantum countable subset {ξ } for the basis of the Hilbert state n harmonic oscillator, we set space H, namely one assumes that H is a separable space. Under such an assumption, any vector ξ in H H 1 = L − k + , (20) can be approximated by a linear combination of ξn to ω 3 2 H any accuracy, i.e., for any ξ in and any >0, it L+ L− { } | − | † = √ = √ exists a sequence cn such that ξ n cnξn <. a ,a . (21) The one-time-step evolution operator acts on 2k 2k (n)A ⊗ (m)B and in the representation space of the The SU(1, 1) contraction k →∞again recovers the states it reads quantum oscillator Eqs. (15) and (17), i.e., the h(1) algebra. From (19), as announced, we see that the − − − U(τ)≡ e iHτ = e iHAτ ⊗ e iHB τ →∞   contraction k does not modify L3 and its 00... 1 spectrum but only the matrix elements of L±.The  10... 0  relevant point is that, while in the SU(2) case the =    01... 0  Hilbert space gets modified in the contraction limit, .. .. in the present SU(1, 1) case the Hilbert space is . . A   not modified in such a limit: a mathematically well 00... 1 founded perturbation theory can be now formulated  10... 0  ⊗   (starting from Eq. (19), with perturbation parameter  01... 0  . (18) ∝ 1/k) in order to recover the wanted Eq. (15) in the ...... B 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), Eqs. (19) and (22) suggest to us one more scenario which fixes the zero point energy in the oscillator where we may recover the already known connection spectrum. It is well known that the zero point energy [2,3] between dissipation and quantization. Indeed, is the true signature of quantization and is a direct by introducing the Schwinger-like two mode SU(1, 1) ⊗ consequence of the non-zero commutator of xˆ and pˆ. realization in terms of h(1) h(1), the square roots in Thus this is a crucial point in the present analysis. the eigenvalues of L+ and L− in Eq. (22) may also be The SU(2) model considered in Section 2 says recovered. We set: nothing about the inclusion of the phase factor. † † † L+ ≡ A B ,L− ≡ AB ≡ L+, On the other hand, it is remarkable that the SU(1, 1) = 1 † † setting, with H ωL3, always implies a non-vanish- L3 ≡ A A + B B + 1 , (24) ing phase, since k>0. In particular, the fundamental 2 representation has k = 1/2 and thus with [A,A†]=[B,B†]=1 and all other commutators C2 = + 2 − equal to zero. The Casimir operator is 1/4 L3 1 1/2(L+L− + L−L+) = 1/4(A†A − B†B)2. L3|n = n + |n , {| } 2 We now denote by nA,nB the set of simultane- ous eigenvectors of the A†A and B†B operators with L+|n =(n + 1)|n + 1 , nA, nB non-negative integers. We may then express | | L−|n =n|n − 1 . (22) the states n in terms of the basis j,m , with j integer or half-integer and m |j|,and We note that the rising and lowering operator matrix 1 elements do not carry the square roots, as on the C|j,m =j|j,m ,j= (nA − nB ), (25) 2 contrary happens for h(1) (cf., e.g., Eq. (15)). 1 1 Then we introduce the following mapping in the L |j,m = m + |j,m ,m= (nA + nB ). 3 2 2 universal enveloping algebra of su(1, 1): (26) Here m −|j|=n and |j|+1/2 = k (cf. Eq. (19)). = √ 1 ; † = √ 1 a L− a L+ (23) Clearly, for j = 0, i.e., n = nA = nB ,wehave L3 + 1/2 L3 + 1/2 the fundamental√ √ representation (22) and L−|n = which gives us the wanted h(1) structure of Eq. (15), AB|n = n n |n =n|n (and similarly for L+). with H = ωL3. Note that now no limit (contraction) This accounts for the absence of square roots in is necessary, i.e., we find a one-to-one (non-linear) Eq. (22). In order to clarify the underlying physics, it is mapping between the deterministic SU(1, 1) system π | ≡ L1 | and the quantum harmonic oscillator. The reader may convenient to change basis: φj,m e 2 j,m .By recognize the mapping Eq. (23) as the non-compact exploiting the relation [4] analog [10] of the well-known Holstein–Primakoff π L − π L ie 2 1 L e 2 1 = L , (27) representation for SU(2) spin systems [8,11]. 3 2 We remark that the 1/2termintheL3 eigenvalues we have now is implied by the used representation. Moreover, 1 after a period T = 2π/ω, the evolution of the state L2|φj,m =i m + |φj,m . (28) presents a phase π that it is not of dynamical origin: 2 e−iHT = 1, it is a geometric-like phase (remarkably, Here it is necessary to remark that one should be related to the isomorphism between SO(2, 1) and careful in handling the relation (27) and the states i2·2πL SU(1, 1)/Z2 (e 3 = 1)). Thus the zero point |φj,m . In fact Eq. (27) is a non-unitary transformation energy is strictly related to this geometric-like phase in SU(1, 1) and the states |φj,m do not provide (which confirms the result of Ref. [3]). 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) H ≡ Ω A†A − B†B = 2ΩC, 0 † † HI ≡ iΓ A B − AB =−2ΓL2. (30)

Here we have also added the constant term H0 and set 2Γ ≡ ω. In Ref. [4] it has been shown that the Hamil- Fig. 1. Different quantization routes. The left route represent tonian (29) arises in the quantization procedure of ’t Hooft procedure, with contraction of su(2) to h(1). the damped harmonic oscillator. On the other hand, in Ref. [3], it was shown that the above system be- + in particular D1/2: we have shown that in this case the longs to the class of deterministic quantum systems à zero-point energy is provided in a natural way with the la ’t Hooft, i.e., those systems who remain determin- choice of the representation. Also, we realize a one- istic even when described by means of Hilbert space to-one mapping of the deterministic system onto the techniques. The quantum harmonic oscillator emerges quantum harmonic oscillator. Such a mapping is an from the above (dissipative) system when one im- analog of the well-known Holstein–Primakoff map- poses a constraint on the Hilbert space, of the form ping used for diagonalizing the ferromagnet Hamil- L2|ψ =0. Further details on this may be found in tonian [8,11]. A shematic representation of quantiza- Ref. [3]. See also Refs. [15,16] for related ideas. tion 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 6. Conclusions oscillators [4] and made connection with recent results [3]. 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 Acknowledgements limiting procedure used there for obtaining truly quantum systems out of deterministic ones, has a very pre- We acknowledge the ESF Program COSLAB, EP- cise meaning as a group contraction from SU(2) to the SRC, INFN and INFM for partial financial support. 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 References contraction to h(1) in such case. One advantage of working with SU(1, 1) is that its representation Hilbert [1] G. ’t Hooft, hep-th/0104080; space is infinite-dimensional, thus it does not change G. ’t Hooft, hep-th/0105105. dimension in the contraction limit, as it happens for [2] G. ’t Hooft, in: Basics and Highlights of Fundamental Physics, Erice, 1999, hep-th/0003005. the SU(2) case. [3] M. Blasone, P. Jizba, G. Vitiello, Phys. Lett. A 287 (2001) 205. However, the most important feature appears when [4] E. Celeghini, M. Rasetti, G. Vitiello, Ann. Phys. 215 (1992) + we consider the Dk representations of SU(1, 1),and 156. M. Blasone et al. / Physics Letters A 310 (2003) 393–399 399

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