Translational Invariance of the Quantum Skeleton

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.

For
the first contribution to ,
the kick propagator
,

Next
I consider the propagator
which describes
the free harmonic oscillator dynamics.
At this stage of the
present
discussion, is not yet restricted to one of the
resonance values specified in equation (1.23), but
can take on any value.
Since
brings about
clockwise rotation in
phase space through the angle ,
corresponding to the free part
of the
classical web map (1.20c),
the operators
and
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

(4.5) |

and the new in terms of the old is

Combination of equations (4.10) and (4.11b) with the
splitting (2.28) then gives

(4.6) |

for a

where -- with equations (4.12) -- the real parameters

have been introduced. Expressing in terms of where this seems appropriate, and using equation (4.15), equation (4.17) can also be formulated as

with

(4.8) |

Concluding from equation (4.19), the
operators
and commute
if and only if
there are integers
such that

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
(1.22).
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
condition
implies

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
definitions (4.18),

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

(4.7) |

Note that
-- with any
, and in fact with any
--
the following
two
assertions
are equivalent:

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

(4.7) |

Finally, for general values of
one has the
result that

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
(4.25-4.30),
it is
now
shown that if satisfies the *classical*
resonance condition
(1.23) with
,
then integers
can in fact be
found
which satisfy equation (4.25a),
and therefore
equation (4.24a) as well,
provided the are chosen suitably.
This being granted, the result

(4.9) |

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 and 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 case, the translations

are obtained, reproducing the classical square grid of figure 1.10b and of the quantum stochastic webs for in figures 4.2, 4.3, 4.7 and C.1-C.17.

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.