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(x q)2 i 1 = e 2 q p e i px e 2 , 4 i i i
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z (x)
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= x |z = D ( z) x |0 ,
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(354)
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to be compared with (233)
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344 Symplectic Phase and the Weyl Heisenberg Group
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In fact, the coherent state family should be viewed as the orbit of any particular coherent state since D (z )|z = D (z )D (z)|0 = e iz z D (z + z)|0 = e iz z |z + z (356)
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3 The Standard Coherent States: the (Elementary) Mathematics
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One notices that the presence of the phase factor in (356) prevents us from viewing the displacement operator D (z) as the representation of a simple translation Indeed, it appears as the subtle mark of noncommutativity of two successive displacements: D (z 1 )D (z 2 ) = e 2iz 2 z 1 D (z 2 )D (z 1 ) (357)
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This equation is the integrated version of the canonical commutation rules [Q, P ] = iI d In this regard, one speaks of projective representation z D (z) of the Abelian group C, since the composition of two operators D (z 1 ), D (z 2 ) produces a phase factor, with phase Iz 1 z 2 = z 1 z 2 In other words and to insist on this important feature of the formalism, the unavoidable appearance of this phase compels us to work with a wider set than the complex numbers This set is precisely the (Lie) Weyl Heisenberg group W R ~ C, the Lie algebra of which is wm W
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g = (s, z) = (s, q, p) (358) X = isI d + za z a
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The group law is given by (s 1 , z 1 )(s 2 , z 2 ) = (s 1 + s 2 z 1 z 2 , z 1 + z 2 ) (s 1 , q 1 , p 1 )(s 2 , q 2 , p 2 ) = s 1 + s 2 1 (q 1 p 2 q 2 p 1 ), q 1 + q 2 , p 1 + p 2 2 The neutral element is (0, 0, 0) and the inverse is (s, z) 1 = ( s, z)
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345 Coherent States as Tools in Signal Analysis
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An intriguing question arises from the group-theoretical interpretation of the coherent states: what about transporting a state different from the vacuum Concretely, let us make D (z) act on an arbitrarily chosen state | H: |z
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def
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= D (z)|
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For instance, in position representation, these states read as x |z
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def
= z (x) = e 2 q p e i px (x q)
(361)
A question naturally arises: which genuine coherent state properties are still valid (i) The states |z
are normalizable: = |D (z) D (z)| =
(362)
(ii) They solve the identity Indeed, consider the operator A=
d2z =
D (z)| |D ( z)
d2z
(363)
34 Properties in the Group-Theoretical Context
This operator commutes with all operators e is D (z ) of the unitary irreducible representation of the Weyl Heisenberg group (exercise!) Hence, by applying the Schur lemma, 2)A = constant I d , with consequently the same reproducing properties The computation of the constant = c is straightforward: c
= |A| =
| |D (z)| |2
d2z
(iii) These coherent states enjoy the same covariance properties with respect to the action of W: D (z )|z
= e iz z |z + z
(364)
But then what have we lost (iv) We have lost an important property: a|z
= z|z /
(365)
since the equality only holds true for | |0 Nevertheless, we get a serious improvement with regard to the freedom in the choice of ! This is essentially the main interest in these states from the point of view of signal analysis The so-called Gabor (or windowed Fourier ) transform of signal analysis [22 24] precisely rests upon the resolution of the identity Id = |z
d2z c
(366)
This identity allows one to implement a Hilbertian analysis of any state | from the point of view of the continuous frame of coherent states |z : | =
z| |z
d2z c
(367)
The projection z| of the state | onto the state z (x) = e 2 q p e i px (x q) ,
(368)
which is the window or Gaboret or even wavelet , | , translated and modulated, is called the Gabor transform or the windowed Fourier transform or the time frequency representation of the signal This transform reads as
+