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Fisica di Dipartimento efsmlrt;fatl;suee oeetsae;no states; coherent squeezed fractals; self-similarity; : I EFSMLRT N OEETSTATES COHERENT AND SELF-SIMILARITY II. nvri´ iSlro -48 icao(aen) Italy (Salerno), Fisciano I-84084 Salerno, di Universit´a q dfre ler fchrn tts efsmlrstruct Self-similar states. coherent of algebra -deformed e se per rca dimension fractal .INTRODUCTION I. o/ d ispeVitiello Giuseppe = fgettertcladpatclitrs u oteubiquitous the to due interest practical and theoretical great of n4 ln n3 ln ≈ 1 . 2619 d tstepeetatmtt rm nsc dynamical a such in frame to attempt present the ates eKc uv a edn yaotn h standard the adopting by done be can curve Koch he tn nwihfatl r osdrdudrtepoint the under considered are fractals which in xtent locle the called also , admfatl,adohrfatlpoete r not are properties fractal other and fractals, random sapa stersl fnnhmgnoscoherent non-homogeneous of result the as appear us nepann ayeprmna bevtos e.g., observations, experimental many explaining in l nesto rcse ttemcocpclvl[2–4]. level microscopic the at processes ondensation rdcdcytldfcsi h atc (dislocations) lattice the in defects crystal produced s oanwls n te ocle dfcs in “defects” so-called other and walls, domain ns, httednmclfraino fractals of formation dynamical the that d ttsaddsiaiessesi ierneof range wide a in systems dissipative and states t inVi eoe ocnlsos h esr of measure The conclusions. to devoted is V tion oeetsae,dsiainadnoncommutative and dissipation states, coherent d ntepeetpprweesl-iia structures self-similar where paper present the in ∗ aiet h arsoi aueof nature macroscopic The manifest. e . eomto processes. deformation h eeaoso h qezdcoherent squeezed the of generators the efaeoko h hoyo entire of theory the of framework he sthe is n ocmuaiegoer.Teresults The geometry. noncommutative and e fthe of e tde ntefaeoko h hoyof theory the of framework the in studied een te nw xmls uha h Sierpinski the as such examples, known other o h xmlso h ohcreand curve Koch the of examples The gneo rca ilcto tutrs 6 in [6] structures” dislocation fractal of rgence lt frltn oepoete ffatl to fractals of properties some relating of ility t yrcn bevtos[]soigta when that showing [6] observations recent by arsoi unu systems quantum macroscopic priso eemnsi rcas(hc are (which fractals deterministic of operties cmuaiegoer;dissipa- geometry; ncommutative v othe to rve otipratproperty important most lso h ohcre(i.1 n the and 1) (Fig. curve Koch the of ples nl iFsc Nucleare Fisica di onale q dfre ler fsuee coherent squeezed of algebra -deformed efsmlrt dimension self-similarity omttv geometry commutative q dfre oeetsae are states coherent -deformed rsaerelated are ures ffatl p 150 (p. fractals of r“extended or [5], (1) 2

FIG. 1: The first five stages of Koch curve.

d Let the n-th step or stage of the Koch curve construction (see Fig. 1) be denoted by un,q(α), with α = 4 and q =1/3 . Setting the starting stage u0 = 1, one has

n un,q(α) = (q α) =1, for any n, (2) from which Eq. (1) is obtained. It has to be stressed that self-similarity is properly defined only in the n limit (self-similarity does not hold when considering only a finite number n of iterations). → ∞ dθ By considering in full generality the complex α-plane, and putting q = e− , Eq. (1) is written as dθ = ln α and the functions un,q(α) are, apart the normalization factor 1/√n!, nothing but the restriction to real α of the functions (q α)n un,q(α)= , n + , α C , (3) √n! ∈ N ∈ which form a basis in the space of the entire analytic functions, orthonormal under the gaussian measure dµ(α)= 2 α F (1/π)e−| | dαdα¯. The study of the fractal properties may thus be carried on in the space of the entire analytic functions, by restricting, at the end, the conclusions to real α, α Re(α) [1, 8]. The connectionF between fractal self-similarity properties and coherent states is then readily established→ since one realizes that the space is the vector space which provides the so-called Fock-Bargmann representation (FBR) of the Weyl–Heisenberg algebraF [9] and is an useful frame to describe the (Glauber) coherent states. The “q-deformed” algebraic structure of which the space provides a representation is obtained by introducing the finite difference operator q, also called the q-derivative Foperator [10]: D f(qα) f(α) f(α)= − , (4) Dq (q 1)α − ζ N with f(α) , q = e , ζ C . q reduces to the standard derivative for q 1 (ζ 0). By applying q to the coherent state∈Fα (a α = α ∈α , withD a the annihilator operator), one finds that→ → | i | i | i qα 2 ∞ (qα)n qN α = qα = exp | | n (5) | i | i − 2  √ | i nX=0 n! where N α d/dα. The n-th iteration stage of the fractal now can be “seen” by applying (a)n to qα and restricting to real qα≡ | i qα (a)n qα = (qα)n = u (α), qα Re(qα). (6) h | | i n,q → The operator (a)n thus acts as a “magnifying” lens [1, 7, 8]. Thus the fractal n-th stage of iteration, with n = 0, 1, 2, .., , is represented, in a one-to-one correspondence, by the n-th term in the coherent state series Eq. (5). The operator ∞qN is called the fractal operator [1, 8]. Note that qα is actually a squeezed coherent state[11, 12]. ζ = ln q is called the squeezing parameter and qN acts in as the| squeezingi operator. The conclusion is that the self-similarity propertiesF of the Koch curve (and other fractals) can be described in terms of the coherent state squeezing transformation. Their functional realization in terms of the two mode SU(1, 1) representation is discussed in Section III. These results can be extended also to the logarithmic spiral. Its defining equation in polar coordinates (r, θ) is [5, 13]:

dθ r = r0 e , (7) 3

FIG. 2: The anti-clockwise and the clockwise logarithmic spiral.

with r0 and d arbitrary real constants and r0 > 0, whose representation is the straight line of slope d in a log-log plot with abscissa θ = ln eθ: r dθ = ln . (8) r0

1 The constancy of the angular coefficient tan− d signals the self-similarity property of the logarithmic spiral: rescaling n θ nθ affects r/r0 by the power (r/r0) . Thus, we may proceed again like in the Koch curve case and show the relation→ to squeezed coherent states. The result also holds for the specific form of the logarithmic spiral called the golden spiral (see Appendix A). The golden spiral and its relation to the Fibonacci progression is of great interest since the Fibonacci progression appears in many phenomena, ranging from botany to physiological and functional properties in living systems, as the “natural” expression in which they manifest themselves. Even in linguistics, some phenomena have shown Fibonacci progressions [14]. As customary, let in Eq. (7) the anti-clockwise angles θ’s be taken to be positive. The anti-clockwise versus (left- handed) spiral has q edθ > 1; the clockwise versus (right-handed) spiral has q< 1 (see Fig. 2); sometimes they are called the direct and ≡indirect spiral, respectively. In the next Section we consider the parametric equations of the spiral:

dθ x = r(θ) cos θ = r0 e cos θ , (9a) dθ y = r(θ) sin θ = r0 e sin θ . (9b)

III. SELF-SIMILARITY, DISSIPATION AND SQUEEZED COHERENT STATES

According to Eqs. (9), the point on the logarithmic spiral in the complex z-plane is given by

dθ iθ z = x + iy = r0 e e , (10)

The point z is fully specified only when the sign of dθ is assigned. The factor q = edθ may denote indeed one of dθ + dθ the two components of the (hyperbolic) basis e− , e . Due to the completeness of the basis, both the factors dθ { } e± must be considered. It is indeed interesting that in nature in many instances the direct (q> 1) and the indirect (q < 1) spirals are both realized in the same system (perhaps the most well known systems where this happens are found in phyllotaxis studies). The points z1 and z2 are then considered:

dθ iθ + dθ + iθ z1 = r0 e− e− , z2 = r0 e e , (11) where for convenience (see below) opposite signs for the imaginary exponent iθ have been chosen. By using the parametrization θ = θ(t), z1 and z2 are easily shown to solve the equations

m z¨1 + γ z˙1 + κz1 = 0 , (12a) m z¨ γ z˙ + κz = 0 , (12b) 2 − 2 2 respectively, provided the relation γ Γ θ(t)= t = t (13) 2 m d d 4 holds (up to an arbitrary additive constant c which we set equal to zero). We see that θ(T )=2 π at T =2 π d/Γ. In γ Eqs. (12) “dot” denotes derivative with respect to t; m, γ and κ are positive real constants. The notation Γ 2m iθ(t) ≡ also will be used. One can also show that ρ (t) r0 e± satisfy the equation ± ≡ m ρ¨ + Kρ =0 , (14) ± ± with 1 γ2 Γ2 γ2 K = mΩ2 , Ω2 = (κ )= , with κ> . (15) m − 4m d2 4m i Ω t Γt In conclusion, the parametric expressions z1(t) and z2(t) for the logarithmic spiral are z1(t) = r0 e− e− and + i Ω t +Γ t 2 πd m 2 πd m z2(t) = r0 e e , solutions of Eqs. (12a) and (12b). At t = mT , z1 = r0 (e− ) , z2 = r0 (e ) , with the integer m =1, 2, 3... These considerations and Eqs. (12) suggest to us that one can interpret the parameter t as the time parameter. As a result, we have that the time-evolution of the system of direct and indirect spirals is described by the system of equations (12a) and (12b) for the damped and amplified harmonic oscillator. The spiral “angular velocity” is given by dθ/dt = Γ/d . Note that the oscillator z1 is an open (non-hamiltonian) system and it is well known [15] that in order| to set| up| the| canonical formalism one needs to double the degrees of freedom of the system by introducing its time-reversed image z2: we thus see that the physical meaning of the mentioned mathematical necessity to consider dθ + dθ both the elements of the basis e− , e is that one needs to consider the closed system (z1, z2) [15] since then we are able to set up the canonical{ formalism.} The closed system Lagrangian is then given by 1 L = mz˙ z˙ + γ(z z˙ z˙ z ) κz z , (16) 1 2 2 1 2 − 1 2 − 1 2 from which, Eqs. (12a) and (12b)) are both derived by varying L with respect to z2 and z1, respectively. The canonical momenta are given by ∂L 1 ∂L 1 pz1 = = mz˙2 γz2 , pz2 = = mz˙1 + γz1 . (17) ∂z˙1 − 2 ∂z˙2 2

It is worth stressing that without closing the system z1 with the system z2 there would be no possibility to define the momenta pzi , i =1, 2, since the same Lagrangian L would not exist. The time evolution of the “two copies” (z1, z2) of the z-coordinate can be viewed as the forward in time path and the backward in time path in the phase space z,pz , respectively. We thus arrive at a crucial point in our discussion: it is known indeed that as far as z1(t) = z2(t) {the system} exhibits quantum behavior and quantum interference takes place [3, 16–18]. By following Schwinger6 [19], this can be explicitly proven, for example, by considering the double slit experiment in quantum mechanics [3, 16–18]. In the quantum mechanical formalism of the Wigner function and density matrix it is required to consider doubling the system coordinates, from one coordinate z(t) describing motion in time to two coordinates, z1(t) going forward in time and z2(t) going backward in time [16, 17]. In order to have quantum interference, the forward in time action A(z z ,t) must be different from the backward in time action A(z z ,t). The classical behavior of the system is − 1 − 2 obtained if the two paths can be identified, i.e. z1(t) z2(t) zclassical(t). When the forward in time and backward in time motions are (at the same time) unequal z (t≈) = z (≈t), then the system behaves in a quantum mechanical 1 6 2 fashion. Of course, when z is actually measured there is only one classical zclassical. This picture also agrees with ’t Hooft conjecture, which states that, provided some specific energy conditions are met and some constraints are imposed, classical, deterministic systems presenting loss of information (dissipation) might behave according to a quantum evolution [20, 21]. In the present case, this means that the logarithmic spiral and its time-reversed double (the anti-clockwise spiral and the clockwise spiral) manifest themselves as macroscopic quantum systems. ~ The canonical commutators [ z1, pz1 ] = i = [ z2, pz2 ], [ z1, z2 ] = 0 = [ pz1 ,pz2 ], and the corresponding sets of annihilation and creation operators are introduced in a standard fashion:

1 1 2 2 1 pz1 1 pz1 a i√mΩz ; a† + i√mΩz ; (18) ≡ 2~Ω √m − 1 ≡ 2~Ω √m 1 1 1 2 2 1 pz2 1 pz2 b i√mΩz ; b† + i√mΩz , (19) ≡ 2~Ω √m − 2 ≡  2~Ω √m 2 with [ a,a ] = 1 = [ b,b ], [ a,b ] = 0 = [ a,b ]. Performing the linear canonical transformation A 1 (a + b), † † † √2 1 ≡ B (a b), with commutation relations [A, A†] = 1 = [B,B†], [A, B] = 0 = [A, B†], from Eq. (16) the ≡ √2 − Hamiltonian = + is obtained [15], with H H0 HI = ~Ω(A†A B†B)=2~Ω , = i~Γ(A†B† AB)= 2~ΓJ , (20) H0 − C HI − − 2 5 which is called the fractal Hamiltonian and where and J2 are, respectively, the Casimir operator and one of the three generators of the SU(1, 1) group associated withC the system of quantum oscillators A and B. We also have [ , ]=0 . (21) H0 HI One can see that the eigenvalue of 0 is the constant (conserved) quantity ~Ω(nA nB). The ground state (the vacuum) is 0 n =0,n =0 , suchH that A 0 =0= B 0 , and 0 = 0; its time− evolution is given by | i ≡ | A B i | i | i H0| i 1 0(t) = exp itH 0 = exp tanh(Γt)J 0 , (22) | i − ~ | i cosh(Γt) + | i
with J+ A†B†, namely by a two-mode SU(1, 1) generalized coherent state [9, 15] produced by condensation of couples of≡ (entangled) A and B modes: (AB)n, n =0, 1, 2 .... . At every time t, 0(t) has unit norm, 0(t) 0(t) = 1, however as t , the asymptotic state turns out to be orthogonal∞ to the initial| statei 0 : h | i → ∞ | i lim 0(t) 0 = lim exp ln cosh(Γt) 0 , (23) t h | i t − → →∞ →∞
which expresses the decay (dissipation) of the vacuum under the time evolution operator (t) exp( it /~) (for U ≡ − HI large t, Eq. (23) gives the ratio r0/r(t) (cf. Eqs. (8) and (13)) and the limit consistently expresses the (unbounded) growth of r(t)). The state 0(t) is known to be a thermal state [15] and its time evolution may be written as [3, 15]: | i 1 1 0(t) = exp (t) exp A†B† 0 = exp (t) exp A†B† 0 , (24) | i −2SA  | i −2SB  | i

where (t) and (t) have the same formal expressions (with B and B† replacing A and A†): SA SB 2 2 (t) A†A ln sinh Γt AA† ln cosh Γt , (25) SA ≡− − n

o Since A’s and B’s commute, one simply writes for either A or B. in Eq. (25) is recognized to be the entropy for the dissipative system [15] (see also [2, 3]). It isS remarkableS thatS the sameS operator that controls the irreversibility of time evolution (breaking of time-reversal symmetry) associated with the choiceS of a privileged direction in time evolution (time arrow), also defines formally the entropy operator. The breakdown of time-reversal symmetry charac- teristic of dissipation is clearly manifest in the formation process of fractals; in the case of the logarithmic spiral the breakdown of time-reversal symmetry is associated with the chirality of the spiral: the indirect (right-handed) spiral is the time-reversed, but distinct, image of the direct (left-handed) spiral (Fig. 2). The Hamiltonian then turns out to be the fractal free energy for the coherent boson condensation process out of which the fractal isH formed. By identifying /~ =2Ω with the “internal energy” U and 2 J /~ with the entropy H0 C 2 S, from Eq. (20) and the defining equation for the temperature T (putting kB = 1), we have ∂S 1 = (26) ∂U T and obtain T = ~ Γ. Thus, /~ represents the free energy = U TS, and the heat contribution in is given H F − F by 2Γ J2. Consistently, we also have (∂ /∂T ) Ω = 2 J2/~. It is also interesting that the temperature T = ~ Γ is actually proportional to the background zeroF point| energy:− ~ Γ ~ Ω/2 [3, 18, 21]. One can also show that the Planck distribution for the A and B modes is obtained by extremizing∝ the free energy functional [2, 3, 15, 18]. So far for simplicity the problem has been tackled within the framework of quantum mechanics. However, it is necessary to stress that the correct mathematical framework to study quantum dissipation is the quantum field theory (QFT) framework [3, 15], where one considers an infinite number of degrees of freedom. This is also physically more realistic, because the realizations of the logarithmic spiral (and in general of fractals) in the many cases it is observed in nature involve an infinite number of elementary degrees of freedom, as it always happens in many-body physics. The operator (t) written in terms of the a and b operators is [3, 11] U Γt 2 2 2 2 (t) = exp a a† b b† , (27) U − 2 − − −  

 and we realize that it is the two mode squeezing generator with squeezing parameter ζ = Γ t. The SU(1, 1) generalized coherent state (22) is thus a squeezed state. In the case of the Koch curve, by doubling− the degrees of freedom in the operator qN one obtains the “doubled” exponential operator

2 2 2 2 (c c† ) (˜c c˜† ) = 2 C†D† CD . (28)  − − −  −  −  6

1 1 wherec ˜ andc ˜† denote the doubled degrees of freedom and C (c +˜c), D (c c˜). The second member of ≡ √2 ≡ √2 − this equation is of the same form as the one of the operator appearing in in Eq. (20). The fractal operator is thus HI proportional to the dissipative time evolution operator exp( it I /~) discussed above and the description in terms of generalized SU(1, 1) coherent state is recovered also in the− caseH of the Koch curve. Of course, one can obtain the relation (28) by starting since the begin from the analysis of the structure of Eq. (1); dθ use of q = e− , with d the fractal dimension, allows us to write the self-similarity equation q α = 1 in polar coordinates dθ as u = u0 αe , which is similar to Eq. (7). Then we may proceed in a similar fashion as we did for the logarithmic spiral, writing the parametric equations for the fractal in the z-plane and considering the closed system (cf. Eq. (11)), and so on. This leads us again to the fractal Hamiltonian (similar to the one in Eq.(20)) and fractal free energy and to SU(1, 1) generalized coherent state. The quantum dynamical scheme here presented seems therefore underlying the morphogenesis processes which manifest themselves in the macroscopic appearances (forms) of the fractals.

IV. NONCOMMUTATIVE GEOMETRY

I consider in this section the noncommutative geometry in the plane in the sense analyzed in Ref. [22] (see also [23]), namely considering that a quantum interference phase (of the Aharanov-Bohm type) can be shown [3, 18, 22] to be associated with the noncommutative plane, a phenomenon which appears to be related to dissipation and squeezing. There is an even deeper level of analysis which relates such noncommutative features to the noncommutativity of the Hopf algebra characterizing q-deformed algebraic structures of the type considered in this paper in connection with q-deformed coherent states. For brevity, such a level of analysis will not be presented here. The interested reader is referred to Refs. [3, 11, 24]. Let (x1, x2) represent the coordinates of a point in a plane and suppose that they do not commute:

2 [x1, x2]= iL . (29)

In order to understand the physical meaning of the geometric length scale L, one introduces

x1 + ix2 x1 ix2 z = , z∗ = − , (30) L√2 L√2 with non-vanishing commutator [z,z∗] = 1. Then, the Pythagoras’ distance δ in the noncommutative plane is

2 2 2 2 δ = x1 + x2 = L (2z∗z + 1). (31)

From the known properties of the oscillator destruction and creation operators z and z∗, it follows that δ is quantized in units of the length scale L according to

2 2 δn = L (2n +1) where n =0, 1, 2, 3,.... (32)

2 Thus, in the plane (x1, x2) we have quantized “disks” of squared radius δn and L gives the non-zero radius of the “smallest” of such disks (see also [25]). In the path integral formulation of quantum mechanics one may consider many possible paths with the same initial point and final point. The quantum interference phase ϑ (of the Aharanov-Bohm type) between two of such paths can be shown [3, 18, 22] to be determined by L and the enclosed area , ϑ = LA2 . In the case of the Koch curve, one can show that [1, 8] A 1 1 c x + ipx z,ˆ c† x ipx zˆ† , [ˆz, zˆ†]=1 , (33) → √2 ≡ → √2 − ≡

where px = id/dx.z ˆ† a† andz ˆ a are the usual creation and annihilation operators in the configuration representation.− Under the↔ action of the↔ squeezing transformation, this leads us to obtain [1, 8]

1 2 1 2 zˆ zˆq = x + iq px , zˆ† zˆ† = x iq px , [ˆzq, zˆ†]=1 . (34) → q√2 → q q√2 − q

In the “deformed” phase space, the coordinates x x and x q2p do not commute: 1 ≡ 2 ≡ x 2 [x1, x2]= iq (35) 7

(adimensional units and ~ = 1 are used). Thus the q-deformation introduces noncommutative geometry in the (x1, x2)- plane. We recognize that the q-deformation parameter (squeezing) plays the rˆole of the noncommutative geometric length L and the remarks made above again apply. The noncommutative Pythagora’s theorem holds:

2 2 2 2 1 δ = x + x =2q zˆ†zˆ + , (36) 1 2 q q 2
and in , in the y 0 limit, (i.e. in the limit α Re(α), α x + iy) the properties of creation and annihilation operatorsF give → → ≡ 1 δ2 =2q2 n + , n =0, 1, 2, 3... . (37) n 2
The unit scale in the noncommutative plane is now set by the q-deformation parameter. The quantum interference phase between two alternative paths in the plane is given by ϑ = /q2. The zero point uncertainty relation is also 2 A controlled by q, ∆x1 ∆x2 (q /2). ≥ 1 2 Eq. (37) actually is the expression of the energy spectrum of the harmonic oscillator and we can write 2 δn = q2 n +1/2 E , n =0, 1, 2, 3..., where E might be thought as the “energy” associated with the fractal n-stage. ≡ n n In the case
of the two mode description (of the logarithmic spiral as well as the Koch curve), the interference phase appears as a “dissipative interference phase” [16, 22], which provides a relation between dissipation and noncommu- tative geometry in the plane. In the (z ,z ) plane, by using for simplicity the index notation + 1 and 2 and 1 2 ≡ − ≡ Eq. (17) for the momenta pz± , the components of forward in time and backward in time velocity v =z ˙ are given by ± ± 1 1 v = (pz γz ) (38) ± m ∓ ∓ 2 ±

2 and they do not commute [v+, v ] = iγ/m . A canonical set of conjugate position coordinates (ξ+, ξ ) may be defined by putting ξ = (m/γ)v− , so− that − ± ∓ ± 1 [ξ+, ξ ]= i . (39) − γ

Equation (39) characterizes the noncommutative geometry in the plane (z+, z ). Thus, since in the present case L2 =1/γ, the quantum dissipative phase interference ϑ = /L2 = γ is associated− with the two paths and in A A P1 P2 the noncommutative plane, provided z+ = z . 6 − Without adding further details for brevity (see Refs. [3, 11, 18, 24])), a final remark is that the map 1 2 which duplicates the algebra is the Hopf coproduct map 1 + 1 . It is known that theA → Bogoliubov A ⊗ A A → A⊗ ⊗ A 1/2 transformations of “angle” Γ t are obtained by convenient combinations of the deformed coproduct ∆aq† = aq† q + 1/2 ⊗ q− aq†, where aq† are the creation operators in the q-deformed Hopf algebra [24]. These deformed coproduct maps are⊗ noncommutative and the q-deformation parameter is related to the coherent condensate content of the state 0(t) . This sheds some light on the physical meaning of the relation between dissipation (which is at the origin of q| -deformation),i noncommutative geometry and the non-trivial topology of paths in the phase space [24, 26]. It has to be remarked also that, finite difference operators (Eq. (4)) and q-algebras which are at the basis of squeezed states [11] are essential tools in the physics of discretized systems. Indeed, it has been recognized [11] that q-deformed algebras appear, independently on other coexisting mathematical properties, such as analyticity and differentiability, whenever some discreteness in space and/or time plays a rˆole in the formalism, which is consistent with the case of the fractal self-similarity properties here considered which are related to a limiting process involving discrete structures.

V. CONCLUSIONS

The relation established in this paper between the self-similarity properties of fractals and coherent states intro- duces dynamical considerations in the study of fractals and their origin. Like in the case of extended objects in condensed matter physics [2–4], also the self-similarity properties of fractals appear to be generated by the dynamical process controlling boson condensation at a microscopic level. Quantum dissipation, through quantum deformation, or squeezing, of coherent states appears at the root of the observed self-similarity properties at a macroscopic level. To the extent in which fractals are considered under the point of view of self-similarity, as done in this paper, fractals are thus examples of macroscopic quantum systems, in the specific sense that their macroscopic self-similarity properties cannot be derived without recurring to the underlying quantum dynamics controlled by the Hamiltonian (Eq. (27)), 8 or, more specifically, by the fractal free energy. The fact that the entropy controls the time-evolution is consistent with the breakdown of time-reversal symmetry (arrow of time) which is clearly manifest in the formation process of fractals; it is for example related to the chirality in the logarithmic spiral where the indirect (right-handed) spiral is the time-reversed, but distinct, image of the direct (left-handed) spiral (or vice-versa). These considerations suggest that the quantum dynamics here analyzed is actually at the basis of the morphogenesis processes responsible of the fractal macroscopic growth. The links, which also have been discussed, between fractal self-similarity and noncommutative geometry may open interesting perspectives in many applications where quantum dissipation cannot be actually neglected, e.g. in condensed matter physics, in quantum optics and in quantum computing. I finally recall that the q-deformation parameter acts as a label for the Weyl-Heisenberg representations of the canonical commutation relations. In the limit of infinite degrees of freedom (i.e. in QFT) different values of q label “different”, i.e. unitarily inequivalent representations [11, 26]. By tuning the value of the q-parameter one thus moves from a given representation to another one, unitarily inequivalent to the former one. Besides such a parameter one might also consider phase parameters and translation parameters characterizing (generalized) coherent states. By varying these parameters in a deterministic iterated function process, also referred to as multiple reproduction copy machine process, the Koch curve, for example, may be transformed into another fractal (e.g. into Barnsley’s fern [5]). In the frame here presented, these fractals are described by unitarily inequivalent representations in the limit of infinitely many degrees of freedom. Work in such a direction is still needed and is planned for the future. In the present paper, I have not discussed the measure of lengths in fractals, the Hausdorff measure, the fractal “mass”, random fractals, and other fractal properties. The discussion has been limited only to self-similarity properties of deterministic fractals. From a practical point of view, it may be used as a predictive tool: anytime experimental results lead to a straight line of given non-vanishing slope in a log-log plot, one may infer that a specific coherent state dynamics underlies the phenomenon under study. Such a kind of “theorem” has been positively confirmed in some applications in neuroscience [1, 8, 27]. Other applications are planned to be pursued in the future. In view of the rˆole played by coherent states in many applications, ranging from condensed matter physics to quantum optics and molecular biology, elementary particle physics and cosmology, it is also interesting that it is possible to “reverse” the arguments presented in this paper. Indeed, it appears that the conclusion that fractal self- similarity properties may be described in terms of coherent states may be reversed in the statement that coherent states have (fractal) self-similarity properties, namely there is a “geometry” characterizing coherent states which exhibits self-similarity properties.

Acknowledgements

This work has been supported financially by INFN and Miur. Useful discussions with Antonio Capolupo are acknowledged.

Appendix A: The golden spiral and the Fibonacci progression

dθ d (π/2) Consider the factor e in Eq. (7). Suppose that at θ = π/2 we have r/r0 = e = φ, where φ denotes the golden ratio, φ = (1+ √5)/2. Then the logarithmic spiral is called the golden spiral [5] and we may put dg (ln φ)/(π/2), dg≡θ where the subscript g stays for golden. Thus, the polar equation for the golden spiral is rg(θ)= r0 e . The radius of the golden spiral grows in geometrical progression of ratio φ as θ grows of π/2: rg(θ + nπ/2) = dg (θ+n π/2) dg θ n n r0 e = r0 e φ and rg,n rg(θ = nπ/2) = r0 φ . Here and in the following n =0, 1, 2, 3, .... A good “approximate” construction≡ of the golden spiral is obtained by using the so called Fibonacci tiling, obtained by drawing in a proper way [5] squares whose sides are in the Fibonacci progression, 1, 1, 2, 3, 5, 8, 13, .... (the Fibonacci generic number in the progression is Fn = Fn 1 + Fn 2, with F0 = 0; F1 = 1). The Fibonacci spiral is then made from quarter-circles tangent to the interior of each− square− and it does not perfectly overlap with the golden spiral. The reason is that the Fn/Fn 1 φ in the n limit, but is not equal to φ for given finite n and n 1. − → → ∞ − The golden ratio φ and its “conjugate” ψ = 1 φ = 1/φ = (1 √5)/2 are both solutions of the “quadratic formula”: − − −

x2 x 1=0 (A1) − − n n 1 n 2 and of the recurrence equation x x − x − = 0, which, for n = 2, is the relation (A1). We thus see that the geometric progression of ratio φ of− the radii− r = r φn of the golden spiral also satisfy for n 2 the recurrence g,n 0 ≥ 9 equation (A1). See ref. [28] for a recent analysis on q-groups and the Fibonacci progression. A final remark is that Eq. (A1) is the characteristic equation of the differential equation

r¨ +r ˙ r =0 , (A2) − i ω t +dθ(t) which admits as solution r(t) = r0 e e with ω = i √5/2 and θ = t/(2 d) + c, with c, r0 and d constants. √5 t/2 t/2 ±φ t − ψ t By setting c = 0, r(t)= r0 e∓ e− , i.e. rφ(t)= r0 e− and rψ(t)= r0 e− .

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