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1
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84927881284
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All features of quantum mechanics needed in this paper can be found, for instance, in A. Messiah, Quantum Mechanics (Wiley, New York, 1958).
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2
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84927881283
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The calculation is carried out in more detail and in a broader context in Sec. III B. Notice that the Hamiltonian function H is quadratic in the xk's and pk's. To reduce it to a simple form, take for lcurl | φk> rcurl a proper basis of H hat. Then H= sum Ek| λk|2= sum Ek(pk 2+ xk 2) /2 hbar, where the Ek's are the energy levels. H appears thus as a sum of Hamiltonian functions of harmonic oscillators with pulsations ωk=Ek/ hbar.
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3
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84927881282
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Hamilton's equations express the existence of an underlying Poisson bracket (or symplectic) structure. See Sec. II A and references therein.
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5
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84927881281
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More precisely, the space of states is a smooth manifold, and regular means smooth. Notice that we exclude the case of explicitly time-dependent observables.
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6
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0346918708
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Such a structure is called symplectic by mathematicians. The even dimension of the space of states is a necessary condition for that structure to exist. For details on the Poisson bracket, see, for instance, Benjamin, New York, Chap. 1.
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(1973)
Nonrelativistic Mechanics
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Finkelstein, R.J.1
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7
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84927881280
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This is the Darboux theorem. See for a proof V. I. Arnold, Mathematical Methods in Classical Mechanics (Mir, Moskow, 1975) [Also published as Springer Graduate Texts in Mathematics, No. 60 (Springer, New York, 1978)], Chap. 8.
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8
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84927881279
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We exclude the case of time-dependent canonical transformations. For a proof of the equivalence with the usual definition of canonical transformations (also called symplectomorphisms), see Arnold (Ref. 7).
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9
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84927881278
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See, for instance, Finkelstein (Ref. 6), Chap. 3.
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10
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84927881277
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The reason for this restriction to infinitesimal transformations is that the set of canonical transformations is a Lie group: Up to topological details, such a group is characterized by its Lie algebra, i.e., by its infinitesimal elements.
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11
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84927881276
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Other important examples are the components of the linear (angular) momentum, defined as the generators of translations (rotations).
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13
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84927881275
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This is a consequence of the fact that the set of observables is a Lie algebra under the Poisson bracket, closely related to the Lie algebra of the group of automorphisms of the space of states. See Heslot (Ref. 12).
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14
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84927881274
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This happens, for instance, in the Hamiltonian description of the classical electromagnetic field, starting from a Lagrangian description in terms of potentials, with no choice of gauge. The conjugate variables of the potentials obey, then, constraint relations, which play the role of a complementary structure on the space of states. Our definition identifies observables with gauge-invariant quantities, and states which differ from a gauge transformation are thus equivalent. See for details, A. Heslot, The`se, de troisie`me cycle, Université Paris VI, 1979.
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15
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84927881273
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We do not consider antiunitary transformations which describe, for instance, time reversal as true automorphisms. The situation is very similar to that of ordinary classical mechanics, where time reversal changes the sign of the Poisson bracket, and hence is not a true canonical transformation.
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16
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84927881272
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See, for instance, Arnold (Ref. 7).
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17
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84927881271
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The tool used in Sec. III A, i.e., decomposition in real and imaginary parts, ``explains'' the occurrence of complex numbers in quantum mechanics: They ensure the even real dimension of the space of states which is necessary for it to carry a Poisson bracket structure.
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18
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84927881270
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The result of the Introduction (Sec. I) is recovered with g=H.
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19
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84927881269
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An observable is thus necessarily quadratic in the xk's and pk's: For instance, these canonical variables are not themselves observables. Compared to classical mechanics, there are very few observables in quantum mechanics. This explains why the description of most physical systems requires an infinite-dimensional Hilbert space.
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20
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84927881268
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The phase arbitrariness is thus very similar to the gauge invariance of classical electromagnetism. See Ref. 14.
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23
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84927881267
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The linearity of unitary transformations need not be assumed: It is a consequence of the fact that they preserve < | >.
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24
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84927881266
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In mathematical terms, omega is a nondegenerate closed two-form, and defines what is properly called a symplectic structure (mathematicians usually consider the Poisson bracket as a derived notion).
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25
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84972568888
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Some notes on the group of automorphisms of contact and symplectic structures
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It can be shown that a classical phase space is necessarily homogeneous and isotropic: See, : Our assumptions of homogeneity and isotropy are natural extensions to the general case. The assumption of positiveness of the curvature is a physical requirement on Planck's constant. The connexity and simple connexity are technical requirements.
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(1966)
Tohoku Mathematical Journal
, vol.18
, pp. 338
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Hatakeyama, Y.1
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27
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84927881265
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This result has been obtained independently by V. Cantoni (private communication).
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28
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84927881264
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See, for instance, S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Wiley, New York, 1963), Vol. 2, Chap. 9. The result holds at least when the space of states has finite dimension. Extension to the more realistic case of infinite dimension is now under investigation.
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29
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84927881263
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Heslot (Ref. 14).
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30
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84927881262
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Let lcurl , rcurl1 and lcurl , rcurl2 be the Poisson brackets on the two spaces of states. Then the Poisson bracket on their Cartesian product is lcurl , rcurl = lcurl , rcurl1+ lcurl , rcurl2.
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