In quantum mechanics, each physical variable ("observable")
is associated with a Hermitian operator. In Heisenberg
representation, the operators evolve in time, while the state vector
does not depend on time.
In our case, the operatirs act in the vector space of dimension *N*
and are represented by matrices of size
*N*x*N*.

Let us start with a trivial remark that
the classic Arnold cat map may be reformulated avoiding the
modulus operation. To do this we introduce new variables
*K*=exp(2p*ip*)
and *A*=exp(2p*ix*),
instead of *p* and *x*. Then, the map
*p'=p+x*, *x'=p'+x=p*+2*x*
will be rewritten as *K'*=*KA*, *A'*=*AK'*.

Accounting the remark, it is worth formulating
the Heisenberg representation of the dynamics in terms
of the operators associated with the variables
*K* and *A*.
*These are the operators of finite shift for position and momentum,
known as the Weyl-Heisenberg operators.*

Indeed, let us consider the operator
**K*** _{a}*=exp((2p

**K**_{a}y(*x*)=
exp((2p*i/h*)*a***p**)y(*x*)
+*a*y'(*x*)
+(1/2)*a*^{2}y''(*x*)
+(1/6)*a*^{3}y'''(*x*)+...
= y(*x+a*).

In the momentum representation
action of the operator
**K*** _{a}* corresponds simply to
multiplication by

The operator of the momentum shift by
*b* in the position representation
is defined as multiplication by factor
**A*** _{b}*(

The result of action of operators
**K**_{a}
and **A**_{b}
onto function y(*x*)
depends on their order:

**K**_{a}**A**_{b}y
=exp((2p*i/h*)(*bx*+*ba*))y(*x*+*a*),
**A**_{b}**K*** _{a}*y
=exp((2p

This may be written in a form of the commutative relation
**K**_{a}**A*** _{b}*=exp((2p

As mentioned, the wave function of unit periods for
coordinate and momentum is a comb of delta-functions
with step of
1/*N* both in the position and momentum representations.
Let us define the shift operators
taking
1/*N* as the shift constant and setting *h*=1/*N*:

Then, the commutation relation becomes

**KA** =a**AK**, a=exp(2p*i/N*).

Now we have all necessary to construct the quantum analog
of the Arnold cat map in terms of operators **K** and **A**.
We simply change *K* and *A* with the respective
operators and obtain the operator map

**K**'=**KA**, **A**'=**AK**'

or

**K**'=**KA**, **A**'=**AKA**.

So, the operators **K**'
and **A**' corresponding to the next time step
are expressed via the operators defined at the previous step.
To interpret them again as the shift operators it is necessary
to have the same commutation relation. As may be checked easily,
this is indeed true:

**K**'**A**'=**(KA)(AKA)**=a**(AKA)(KA)**=a**A**'**K**'.

As the state vector is represented by
a set of *N* coefficients
y* _{m}*,
the operators

(**K**y)* _{m}*=y

It corresponds to the matrices

*K _{mn}*=d

where d* _{mn}*=1, if

As it will be clear in the next section,
**the quantization schema based on the above relations is appropriate for odd ***N* **only** .

(Generalization for even *N* see here.)

In the Heisenberg representation the state vector
does not depend on time, which is the iteration number,
while the operators associated with the dynamical variables,
vary at each iteration.
Setting initial conditions for the operator map
with the matrices
**K**^{(0)}=**K** and **A**^{(0)}=**A**,
we obtain recursively the matrices
**K**^{(k)} and
**A**^{(k)} for subsequent
time steps *k*. For example, for *N*=5,

and so on. Each matrix in this sequence is a product of two previous ones.

It is evident from
this example that each time nonzero elements are arranged
along the line parallel to the main diagonal, i.e. the
matrices are of the form

**K**^{(k)}* _{mn}*=d

s'=*s*+*q*, *q'=q+s'*, *r'=r+p*, *p'=r+p'* (mod *N*).

Two additional variables m
and n
obey equations, in which *s*, *r*, *p*, *q*
play a role of external periodic force:

n'=n+m-*rq*,
m'=n'+m-*ps'* (mod *N*).

**The dynamics takes place on a finite set of
integers, and, hence, is not chaotic, it is periodic.**

The period *T*(*N*) of the operator
sequence **K**, in the Heisenberg representation is
also a period of the
wave funcnction evolution in the Schrodinger representation.
For example, *T*(5)=10.
The figure shows a plot of the quantum period *T*(*N*)
for the Arnold cat map. Observe that the dependence is extremely
irregular, although in average the period grows with *N*.

**Remark I.** As the operators **K** and **A**
do not commute, our selection of the operator map
is not unique. As well, we could define a map with rearranged operators
**K** and **A** (say, **K**'=**AK**,
**A**'=**KAA** or others). Although the commutation relation allows
returning to the original disposition of operators, it leads to
appearance of factor a in some power.
The dynamics then is modified, but the quantum period remains
unchanged.

**Remark II** relates to
quantization of more general maps on torus.
If a matrix is of a form

there exists a natural logical way to select the version of the operator map
using the symmetrized combinations of **K** and **A**.
In the first case the operator map may be chosen as
**K'**=**K**^{a/2}**A**^{b}**K**^{a/2},
**A'**=**A**^{d/2}**K**^{c}**A**^{d/2},
and in the second as
**K'**=**A**^{a/2}**K**^{a}**A**^{b/2},
**A'**=**K**^{c/2}**A**^{d}**K**^{c/2}.
This class of maps was named as quantizable in the original
work of Hannay and Berry. The Arnold cat map does not relate to this class.
In fact, this does not mean that the quantization is impossible at all,
but the consequence is that the basic relations for the quantum Arnold cat map
are distinct for odd and even *N*.

**Remark III.**

Operator map of parabolic type describing free motion
of the particle may be defined as
**K**'=**K**,
**A**'=**AK**.

Quantum analog of the elliptic map, the canonical transformation
exchanging coordinate and momentum variables look as follows:
**K**'=**A**, **A**'=**K**^{+},
where cross means the Hermitian conjugate.

of theoretical nonlinear

dynamics

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