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The absolutely quantal face of the coherent states reappears here Indeed, the additional term 1 , called symplectic correction or vacuum energy according to 2 the context, has no classical counterpart Finally, from the respective variances of position and momentum, ( Q)2 = z|Q 2 Q 2 |z = z

= ( P )2 ,

58 59

we infer the saturation of the uncertainty relation or, more properly, Heisenberg inequality: Q P = 1 2 = (329)

This is one of the most important features exhibited by coherent states: their quantal face is the closest possible to its classical counterpart

From the eigenvalue equation H|n = n + 1 |n for the Hamiltonian of the oscil2 lator, one deduces the time evolution of a coherent state: e iHt |z =e

zn e iHt |n n!

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62 64 72 72

e it z |n n! (330)

We thus ascertain the temporal stability of the family {|z , z C} or {|z , z C}

since a quantum state is de ned up to a phase factor There follows the illuminating form of this time evolution in position representation:

qp 1 1 2 x |z = e ix p e i 2 e 2 (x q) , 4 q p 1 1 2 x |e iHt |z = e ix p e i 2 e 2 (x q ) , 4

(331)

75 77 78

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Hence, at given z = (q, p), the time evolution of the Gaussian localization of the particle is a harmonic motion taking place between |z| and |z| We exactly recover the classical behavior of the harmonic oscillator such as it is described by its phase diagram in which, at constant energy, 2E = p 2 + q 2 = 2|z|2 = p + q , the phase trajectory is a circle (or, depending on the units utilized, an ellipse) The complex number z = q+i2p can unambiguously be identi ed as a phase space point (a classical state ) and the Liouville measure or 2-form = idz d z = dq d p on the phase space is naturally present in the de nition of the Fock Bargmann space Given an initial state z = z 0 , say, at t = 0, its time evolution is simply given by z = z(t) = z 0 e it The alternative phase space formalism angle action becomes apparent here The angle is precisely the argument = t + 0 of z, whereas its canonical conjugate is the action I0 = 1 2 = |z 0 |2 (332)

, , Z,

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(164)

with | Tr A| = | + | > 2 and det A = = 1 As solutions of 2 Tr A + 1 = 0, the eigenvalues =

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that | + | > 1 and = 1/ + This action on the torus, known as Anosov diffeomorphism, stretches in the eigendirection corresponding to + , whereas it contracts in the direction determined by It gives rise to deformations of gures such as the famous Arnold cat The set of such transformations forms the subgroup of SL(2, R) leaving invariant the lattice This subgroup is isomorphic to SL(2, Z) There exist periodic points in the torus for such hyperbolic actions They are given by x= q p , a b Q2 /Z2

We have shown three examples of invertible transformations of the torus that are measure-preserving The triplet = (T2 , , ) is a dynamical system It is said to be ergodic if for all integrable functions on the torus, f L1 (T2 , ), its temporal average is equal to its spatial average: ( f ) =

T 1

(165)

79 80 82 83

Now, there holds a general result for characterizing ergodicity of a dynamical system (X , , ), where X is a measure space with measure , here X = T2 , and is an invertible and measure-preserving map

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