15 Coherent State Quantization of Motions on the Circle, in an Interval, and Others

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and, in particular, for the angle operator itself, [A J , A ] = i

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to be compared with the classical { J, } = 1 We have already encountered such dif culties in s 12 and 14, and we guess they are only apparent They are due to the discontinuity of the 2 -periodic saw function B( ) which is equal to on [0, 2 ) They are circumvented if we examine, like we did in those chapters, the behavior of the corresponding lower symbols at the limit 0 For the angle operator, J 0,

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where we recognize at the limit the Fourier series of B( 0 ) For the commutator, J 0,

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ie 2 n

2 +in

~ i + i

2 n) (1517)

So we (almost) recover the canonical commutation rule except for the singularity at the origin mod 2

153 From the Motion of the Circle to the Motion on 1 + 1 de Sitter Space-Time

The material in the previous section is now used to describe the quantum motion of a massive particle on a 1 + 1 de Sitter background, which means a one-sheeted hyperboloid embedded in a 2 + 1 Minkowski space Here, we just summarize the content of [198] The phase space X is also a one-sheeted hyperboloid, J 2 + J 2 J 2 = 2 > 0 , 1 2 0 (1518)

with (local) canonical coordinates ( J, ), as for the motion on the circle Phase space coordinates are now viewed as basic classical observables, J 0 = J, J 1 = J cos sin , and obey the Poisson bracket relations { J 0, J 1} = J 2 , { J 0, J 2} = J 1 , { J 1, J 2} = J 0 (1520) J 2 = J sin + cos , (1519)

154 Coherent State Quantization of the Motion in an In nite-Well Potential

They are, as expected, the commutation relations of so(1, 2) sl(2, R), which is the kinematical symmetry algebra of the system Applying the coherent state quantization (154) at = 0 produces the basic quantum observables: / A J0 =

n|e n e n | , 1 e 4 2 1 e 4 2i n+

(1521a) 1 + i |e n+1 e n | + cc , 2 1 + i |e n+1 e n | cc 2 (1521b) (1521c)

A J 1 = A J 2 =

The quantization is asymptotically exact for these basic observables since [A J 0 , A J 1 ] = iA J 2 , [A J 0 , A J 2 ] = iA J 1 , [A J 1 , A J 2 ] = ie 2 A J 0 Moreover, the quadratic operator C = (A J 1 )2 + (A J 2 )2 e 2 (A J 0 )2

(1522)

(1523)

commutes with the Lie algebra generated by the operators A J 0 , A J 1 , A J 1 , that is, it is the Casimir operator for this algebra In the representation given in (1521a) (1521c), its value is xed to e 2 ( 2 + 1 )I and so admits the limit 4 C ~ 2 + 1 4 Id

Hence, we have produced a coherent state quantization that leads asymptotically to the principal series of representations of SO 0 (1, 2)

154 Coherent State Quantization of the Motion in an In nite-Well Potential 1541 Introduction

Even though the quantum dynamics in an in nite square well potential represents a rather unphysical limit situation, it is a familiar textbook problem and a simple tractable model for the con nement of a quantum particle On the other hand, this model has a serious drawback when it is analyzed in more detail Namely, when one proceeds to a canonical standard quantization, the de nition of a momentum operator with the usual form i d/dx has a doubtful meaning This subject has been discussed in many places (see, eg, [134]), and the attempts to circumvent this anomaly range from self-adjoint extensions [134 137] to PT symmetry approaches [204] First of all, the canonical quantization assumes the existence of a momentum operator (essentially) self-adjoint in L2 (R) that respects some boundary conditions at