One of the defining properties of classical periodic stochastic webs is their translational invariance. Therefore it is natural to look for a similar property in the quantum realm. To this end, the commutation properties of the FLOQUET operator , as given by equations (2.28, 2.32a, 2.35a), with respect to the displacement operator (equations (4.1)) need to be considered.
the first contribution to ,
the kick propagator
I consider the propagator
the free harmonic oscillator dynamics.
At this stage of the
discussion, is not yet restricted to one of the
resonance values specified in equation (1.23), but
can take on any value.
clockwise rotation in
phase space through the angle ,
corresponding to the free part
classical web map (1.20c),
can be interchanged by introducing
a new set
of parameters which is obtained from
by anti-clockwise rotation through
with the real parameters
-- cf. equation (1.20c). Using the definition
Combination of equations (4.10) and (4.11b) with the
splitting (2.28) then gives
Concluding from equation (4.19), the
if and only if
there are integers
These conditions need to be analyzed further in order to understand the consequences for the quantum phase portrait, but some consequences can be read off from equations (4.21) directly.
Equation (4.21a) imposes a restriction on
the rotational part
of the FLOQUET operator
and thus leads to rotational symmetries.
For general values of , equation (4.21a)
is identical with the general classical resonance condition
For it restricts the values of to
In the context of equation (4.19), a lucid interpretation of the resonance condition (1.23/4.22) can be given: it refers to symmetries that come about after the successive applications of in equation (4.19) have accounted for exactly one full rotation in phase space. For general values of , belongs to the more general category of resonances as defined by equation (1.22), corresponding to rotations in phase space after applications of . Below I show that it suffices to consider the simplest case in order to discuss and explain the symmetries of the quantum stochastic webs that have been observed in the previous two sections.
Combination of equation (4.21a) with the
definition (4.18) for shows that the resonance
which is another way of expressing that a full rotation in phase space is obtained after not more than iterations of .
Equation (4.21b) determines the allowed
translations and therefore concerns the translational symmetries of
the phase portrait.
Substituting equation (4.21b) into the
is obtained; here, is used, which is consistent with equation (4.23a). Equation (4.24a) can be rewritten in the form
which holds for and with
-- with any
, and in fact with any
This is a useful observation, since it allows to discuss the rationality of for all by considering just the case of . The equivalence of (4.28a) and (4.28b) can be shown in the following way. Clearly, (4.28b) is implied by (4.28a) with . Conversely, assuming that is rational, the same is true for . Rationality of for all then follows by induction using
Finally, for general values of
one has the
For simplicity, from here on I consider the case of only. Higher order symmetries that are associated with are thus excluded from the following considerations. On the other hand, the choice is sufficient to discuss and explain the symmetries of the quantum stochastic webs that have been observed in the previous two subsections.
Combining the above arguments
shown that if satisfies the classical
can in fact be
which satisfy equation (4.25a),
equation (4.24a) as well,
provided the are chosen suitably.
This being granted, the result
In this way the classical resonance condition with is proven to be a necessary condition for the existence of periodic quantum stochastic webs; in this sense it plays the same role both in classical and quantum mechanics. This observation is supported by the fact that the results described above are obtained irrespective of the values of and . As in the classical case, the overall structure of the stochastic web is entirely and solely determined by the parameter .
In order to check that the determined in accordance with equation (4.25a) also satisfy (4.24a) it remains to check that equation (4.25b) is obeyed, too. In particular, it must be confirmed that the determined by equation (4.24a) are indeed integers, as required by equation (4.21b). This cannot be discussed in general terms, but needs to be checked for each individual .
Evaluating equation (4.24a) it turns out -- not surprisingly -- that the cases of , belonging to but not to , are in a characteristic way different from the cases of belonging to . In the following equations, both and are integers that can be chosen as desired.
The symmetric quantum phase space structures obtained for
are identical. The restrictions discussed above imply that the
parameters describing the translational symmetries
have to be chosen according to
The phase portrait is periodic in -direction with period ; translations by in -direction are admissible with any . This is exactly the symmetry pattern visible in the contour plot 1.9 of the classical time averaged Hamiltonians , . And while it is not confirmed -- due to the limited number of iterations -- by the quantum phase portraits C.38-C.40 for (see section C.3 of the appendix), these figures at least do not stand in contradiction to the symmetries described here.
For these values of , translational invariance is obtained with
respect to the translations given by
in agreement with both the skeleton of the classical stochastic webs displayed in figure 1.10a and the quantum phase portraits 4.4-4.6 and C.18-C.37 (in the appendix) for and .
In this way, the symmetries of the quantum stochastic webs that have been obtained in subsection 4.1.1 using numerical means are explained analytically by exploiting the translation invariance of the FLOQUET operator. This explanation of the infinitely extended eigenstates of the system works for all and for all values of and .
Note that this analytical explanation of the quantum skeletons nicely parallels the analytical explanation of the skeletons of classical stochastic webs (see subsection 1.2.2) in the following sense: the rotational and translational symmetries of the quantum webs arise from the combination of and . None of these operators alone gives rise to the symmetries of the webs. The same is true for and with respect to the classical webs -- cf. the footnote on page . In summary, both the classical and quantum stochastic webs are the result of the combination of both indispensable contributions and to the Hamiltonian. In particular, the symmetries of the quantum webs do not reflect any symmetries of the elements of the basis used for expanding the quantum states in the web.