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\topmatter
\title
$p$-Adic commutation relations
\endtitle
\author{\rm Anatoly N. Kochubei}
\endauthor
\affil
{\sl Institute of Mathematics,Ukrainian National Academy of Sciences, \newline
vul.Tereshchenkivska
3,Kiev,252601 Ukraine}
\endaffil
\abstract\nofrills {\bf Abstract}
Representations of the canonical and deformed commutation relations
by bounded operators on $p$-adic Banach spaces are constructed.
Functions from the Mahler basis of the space of $p$-adic continuous
functions and their multiplicative analogues are shown to be the
$p$-adic counterparts of the Hermite and $q$-Hermite functions.
The analogue of the Stone-von Neumann uniqueness theorem fails
in the $p$-adic case.
\endabstract
\endtopmatter
\document
\bigpagebreak
It is well known that no bounded operators $A,B$ on a Hilbert space
may satisfy the commutation relation $[A,B]=I$ (see e.g. [1]).
The unbounded irreducible representation of this relation (understood
in a proper way) is unique up to the unitary equivalence within
reasonable classes of operators and constitutes one of the main
building blocks of quantum mechanics and quantum field theory.
In this letter we show that a $p$-adic version of this problem possesses
quite different features. Note that most of the recent concepts of the
$p$-adic quantum mechanics [2-5] deal with complex-valued wave
functions of $p$-adic arguments, thus with conventional Hilbert spaces.
The problem under consideration arises in quantum mechanics with
$p$-adic valued wave functions initiated in [6,7], and its deeper
understanding will hopefully contribute to further development of
$p$-adic methods in physics.
Let $p$ be a prime number, $Q_p$ the field of $p$-adic numbers,
$\bold Z_p$ the ring of $p$-adic integers (here and below we use
standard notions and notations of $p$-adic analysis; see e.g. [2,8]).
Denote by $C(\bold Z_p,Q_p)$ the Banach space of continuous functions
on $\bold Z_p$ with values in $Q_p$ equipped with sup-norm. The
sequence of functions
$$
P_n(x)=\frac{x(x-1)\ldots (x-n+1)}{n!},\quad n\ge 1;\quad P_0(x)\equiv 1,
$$
forms an orthonormal basis of $C(\bold Z_p,Q_p)$ [9,10]. This means
that every function $f\in C(\bold Z_p,Q_p)$ admits a unique uniformly
convergent expansion
$$
f(x)=\sum \limits _{n=0}^\infty c_nP_n(x),\quad c_n\in Q_p,
$$
with $|c_n|_p\to 0$, and $\Vert f\Vert =\sup \limits _n|c_n|_p.$
Here $|\cdot |_p$ denotes the absolute value of a $p$-adic number.
Let us consider on $C(\bold Z_p,Q_p)$ the operators
$$
(a^+f)(x)=xf(x-1),\qquad (a^-f)(x)=f(x+1)-f(x),\quad x\in \bold Z_p.\tag 1
$$
It follows from the non-Archimedean property of the absolute value
and the norm that $a^\pm $
are defined correctly and $\Vert a^\pm \Vert \le 1$. Calculating the
commutator we find easily that
$$
[a^-,a^+]=I.\tag 2
$$
Simple calculations also show that
$$
a^-P_n=P_{n-1},\quad n\ge 1;\qquad a^-P_0=0;
$$
$$
a^+P_n=(n+1)P_{n+1},\quad n\ge 0,
$$
so that $a^\pm $ are clear analogues of the creation and annihilation
operators. Next, putting $H=a^+a^-$ so that
$$
Hf(x)=x\{f(x)-f(x-1)\}
$$
we come to an operator with the properties:
$$
HP_n=nP_n,\quad n\ge 0;\qquad \left[ H,a^\pm \right] =\pm a^\pm .
$$
Note that though the operator $H$ has a complete system of eigenvectors,
and its point spectrum coincides with the set $\bold Z_+$ of non-negative
integers, the whole spectrum of $H$ equals $\bold Z_p$, that is the
closure of $\bold Z_+$ in $Q_p$.
The density of $\bold Z_+$ in $\bold Z_p$ implies also that the kernel
of $a^-$ consists of constant functions. Using this fact and the standard
argument from e.g. [11] we find that $a^-,a^+$ form an irreducible couple.
Every Banach space over $Q_p$ having an infinite countable orthonormal
basis is isomorphic to $C(\bold Z_p,Q_p)$. Thus the $p$-adic analogue
of the harmonic oscillator Hamiltonian constructed in [6] as a result
of a complicated integration theory is in fact equivalent to our much
simpler model (since both operators possess orthonormal eigenbases with
the same eigenvalues).
It is not difficult to compute also the operator $a^-a^+$. In particular,
$a^-a^+P_n=(n+1)P_n$, so that $\{P_n\}$ is an orthonormal eigenbasis
of $a^-a^+$.
A small modification of the above construction shows the non-uniqueness
of our representation.Namely, let us take instead of $a^-$ the operator
$$
(a'f)(x)=f(x+1)
$$
so that $a'=a^-+I$. Of course, $[a',a^+]=I$. Consider the operator
$H'=a^+a'$. We have $(H'f)(x)=xf(x)$ so that $H'$ has no eigenvectors
in $C(\bold Z_p,Q_p)$ and is not equivalent to $H$ in any reasonable
sense.
The problem of complete characterization of irreducible representations
remains open. Its solution will probably require further development
of $p$-adic spectral theory [12].
The basis $\{P_n\}$ called Mahler basis plays a significant role in
$p$-adic analysis. Other objects related to the representation (1)
are also well known. For example, the coherent states (eigenfunctions
of the annihilation operator $a^-$) are precisely the functions
$f_\lambda (x)=(1+\lambda )^x,\quad x\in \bold Z_p$, where $|\lambda |_p
<1$. The proof of this fact immediately follows from the results of
[10] where the definition of the function $f_\lambda $ can also be
found.
Another example is a version of the Bargmann-Fock representation
realized in the Banach space of power series
$$
\varphi (z)=\sum \limits _{n=0}^\infty \varphi _nz^n, \quad z\in \bold
Z_p,\tag 3
$$
with coefficients $\varphi _n\in Q_p, |\varphi _n|_p\to 0$, and the
norm $\Vert \varphi \Vert =\sup \limits _n|\varphi _n|_p\quad $ (see
[10]). As usual, we may set
$$
(b^-\varphi )(z)=\varphi '(z),\qquad (b^+\varphi )(z)=z\varphi (z),
$$
with $[b^-,b^+]=I$. Note that in contrast to the classical situation
the power series (3) are not necessarily entire functions; they have
to be defined only on $\bold Z_p$. Some other versions of a $p$-adic
Bargmann-Fock representation have been given in [13,14].
Now let us turn to the deformed commutation relation
$$
a_q^-a_q^+-qa^+_qa^-_q=I \tag 4
$$
where $q\in Q_p,\ |q|_p=1,\ q^N\ne 1$ for any $N\in \bold Z$. In purely
algebraic terms such a relation over an arbitrary field was studied in
[15]. We are interested in constructing a representation of (4) by
bounded operators on a $p$-adic Banach space.
Let $G_q$ be a closure of the multiplicative cyclic subgroup of the ring
$\bold Z_p$ generated by $q$. The sequence $\{q^{-n},\ n>0\}$ is dense in
$G_q$ [16]. The explicit description of $G_q$ for some special cases is
given in [10]. For example, if $p\ne 2,\ |q-1|_p=p^{-1}$, then $G_q=1+
p\bold Z_p$.
Denote by $C(G_q,Q_p)$ the Banach space of all continuous functions on
$G_q$ with values in $Q_p$. An orthonormal basis of this space may be
constructed as follows [10]:
$$
P_n^{(q)}(x)=\frac{R_n^{(q)}(x)}{R_n^{(q)}(q^{-n})},\ n\ge 1;\quad
P_0^{(q)}(x)\equiv 1,\tag 5
$$
where
$$
R_n^{(q)}(x)=(x-1)(x-q^{-1})\cdots (x-q^{-n+1}),\ n\ge 1.
$$
A representation of the relation (4) by bounded operators on \linebreak
$C(G_q,Q_p)$ is given by
$$
\aligned \left( a_q^+f\right) )(x)&=(x-1)f(qx),\\ \left(
a_q^-f\right) (x)&=q(1-q)^{-1}
x^{-1}\{f(q^{-1}x)-f(x)\}. \endaligned \tag 6
$$
Calculating the action of the operators (6) upon the basis (5) we find
that
$$
a_q^+P_n^{(q)}=(q^{-n-1}-1)P_{n+1}^{(q)},\qquad a_q^-P_n^{(q)}=q^n(1-q)^{-1}
P_{n-1}^{(q)},
$$
$$
\left( a_q^-a_q^+\right) P_n^{(q)}=(q^n+q^{n-1}+\cdots +1)P_n^{(q)},\ n\ge
1;\qquad a_q^-a_q^+P_0^{(q)}=P_0^{(q)},
$$
so that $\left\{ P_n^{(q)}\right\}$ is an orthonormal eigenbasis for the
operator $a_q^-a_q^+$.
If we introduce an operator $N_q$ by the relation $N_qP_n^{(q)}=nP_n^{(q)}
\quad $ ($\Vert N_q\Vert \le 1$ since $|n|_p\le 1$) we find that
$$
\left[ N_q,a_q^\pm \right] =\pm a_q^\pm .
$$
Thus we have obtained a $p$-adic version of the deformed oscillator
algebra (see e.g. [17]; for some related work in the $p$-adic setting
see [18,19]).
Just as in the conventional quantum mechanics, the existence of a "vacuum
vector" (like $P_0$ or $P_0^{(q)}$) and the assumption of a "Hermitian"
property of the operator $a^-a^+$ (resp. $a_q^-a_q^+$) imply the
essential features of $p$-adic representations.
Suppose that $a_q^\pm $ are bounded operators on a $p$-adic Banach space
$E$ satisfying the relation (4) with $q\in Q_p$ (here we do not exclude
the case $q=1$). Assume that $a_q^-\varphi _0=0$ for some $\varphi _0\in E$,
and that the eigenvectors corresponding to different eigenvalues of
$a_q^-a_q^+$ (if they exist) are orthogonal (in the $p$-adic sense).
By the induction argument we have
$$
a_q^-\left( a_q^+\right) ^m-q^m\left( a_q^+\right) ^ma_q^-=(q^{m-1}+
q^{m-2}+\cdots +1)\left( a_q^+\right) ^{m-1},\quad m\ge 1.\tag 7
$$
If the couple $a_q^\pm $ is irreducible then it follows from (7) that the
vectors $\left( a_q^+\right) ^{m-1}\varphi _0,\ \ m=1,2,\ldots $, form
an orthogonal eigenbasis of the operator $a_q^-a_q^+$ in $E$. The
corresponding eigenvalues are $q^{m-1}+q^{m-2}+\cdots +1$ so that
necessarily $|q|_p\le 1$. After a suitable renormalization we obtain the
expressions for the action of the operators $a_q^\pm $ upon the basis
vectors similar to the ones found above for our function space
representations.
\bigpagebreak
The work was supported in part by grants from the International Science
Foundation and the Ukrainian Fund for Fundamental Research.
\bigpagebreak
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