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166 Quantization of Motions on the Torus
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(iv) In the limit N , we have for the trace
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N N
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tr Op W ( f ) = ,
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dq dp f (q, p) ab
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(v) There exists the following estimate for the difference between Weyl and coherent state quantizations:
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S Op W ( f ) Op C, ( f ) , L(H ( ))
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= O N 1
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(1643)
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In consequence, it can be asserted that the Weyl quantization and the coherent state quantization are identical up to : Op W ( f ) Op C S ( f ) = O ( ) (1644)
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166 Quantization of Motions on the Torus
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We are now equipped to proceed with the quantization of the symplectic transformations of the torus introduced in the rst section of this chapter
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1661 Quantization of Irrational and Skew Translations
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These transformations are proved to be ergodic and uniquely ergodic in the sense that there exists one and only one -invariant probability measure Any irrational translation , = ( 1 , 2 ), 1 / 2 Q, is unitarily represented by / the operator M ( ) on H ( ) de ned as M ( ) = e i with
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2 def M (0, 2 ) e = e i ( 2 + b j ) 1 e , j j 1 2 2
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M ( 1 ,0) M (0, 2 ) ,
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(1645)
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and M ( 1 ,0)
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The decomposition formula and the eigenequations are consistent with U ( 1 , 2 ) = 1 2 e i 2 U ( 1 , 0) U (0, 2 ) and the fact that U ( 1 , 0) (resp U (0, 2 )) is diagonal in the momentum (or spatial) representation Concerning the skew translations, we have the representation
M
= M (0
M (K ) ,
(1646)
where M (K ), for K = 1 0 k 1
realizes a quantization of the group SL(2, Z) that is explained in the next paragraph
16 Quantizations of the Motion on the Torus
1662 Quantization of the Hyperbolic Automorphisms of the Torus
The task now is to build a unitary representative of the hyperbolic transformation of the torus: A= b a
unitary M(A)
It is proven that for all > 0 and for all T 2 , there exists T 2 such that M(A) H ( ) H ( ), where 1 2 =A 1 2 + N
b a
2 b 2 a
and where the map A= a1 a3 a2 a4 M(A)
is, in general, de ned on any function S(R) by (M(A) )(x) =
i i e S(x,y ) ( y ) d y , 2 1 1 a1 2 def 1 a 4 2 x xy + y S(x, y ) = 2 a2 a2 2 a2
(1647)
The operator M(A) is the quantum propagator associated with the classical discrete dynamics de ned by the matrix A It is extended by duality to the space of tempered distributions S (R) It ful lls the important intertwining property M(A) U (q, p) (M(A)) = U A q p (1648)
Let us now restrict the transformations A to be the element of the group SL(2, Z) and such that | Tr A| > 2 The following was proved in [208]: Proposition 161 For all > 0, there exists T 2 such that the unitary representatives M(A) stabilize the Hilbert space H ( ) of quantum states on the torus, M(A) H ( ) H ( ) , if and only if A SL(2, Z) has the form A= even odd ~ b a
a odd ~ b even
or A =
odd even ~
even ~ odd
Note that the 2-component momentum can be chosen independently of
166 Quantization of Motions on the Torus
1663 Main Results
The applications of the previous results concern particularly the control of the perturbations of the hyperbolic automorphisms of the torus T2 , the demonstration of a semiclassical Egorov theorem on the relation between quantization and temporal evolution, the propagation of coherent states on the torus, and the semiclassical behavior of the spectra of evolution operators of the type M ( ) U H ( ) Let us say more about this semiclassical Egorov theorem It means that quantization and temporal evolution commute up to More precisely, Theorem 162 Let H be a smooth observable on the torus, that is, H C (T2 ) For all f C (T2 ), for all t R, and for all T 2 , we have the following estimate: e i
, (H)
, (
f ) e i
, (H)
Op
f H t
L(H ( ))
= O(N 1 ) , (1649)
where H is the classical ow associated with the observable H, and the notation t Op , stands for both types of quantization (coherent state or Weyl) This theorem is exact in the case of hyperbolic automorphisms of the torus and the Weyl quantization: Op W ( f A) = (M(A)) Op W ( f )M(A) , , (1650)
Concerning the spectral properties of the evolution operator, one gets precise N information on the semiclassical behavior of the eigenvalues e i j , 1 u j u N 1 , N and the corresponding eigenfunctions i j , 1 u j u N 1 , of the unitary operators M ( ) representative of automorphisms : T2 T2 Proposition 162 [[211]] Let a hyperbolic automorphism of the torus Then the eigenvalues of the unitary representative M ( ) are Lebesgue uniformly distributed on the unit circle at the semiclassical limit: lim # { j, N [ 0 , 0 + ]} j N = [ 0 , 0 + ] , (1651)
N
where is the Lebesgue measure This result also holds for the Hamiltonian perturbations of Finally, let us end this chapter by formulating the essential equipartition result in regard to the initial purpose to examine ergodicity on the torus on a semiclassical