Medical, Pharma, Engineering, Science, Technology and Business

^{a}Department of Theoretical Physics, School of Engineering Sciences, Royal Institute of
Technology (KTH) { AlbaNova University Center, SE-106 91 Stockholm, Sweden, **E-mail:** [email protected]

^{b}Centre for Mathematical Sciences, Lund University, P.O. Box 118, SE-221 00 Lund,
Sweden, **E-mail:**[email protected]

**Received Date:** January 15, 2009; **Revised Date:** August 10, 2009

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We describe realizations of a Lie colour algebra with three generators and ve relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

The main object studied in this paper is the unital associative algebra with three generators
*A*_{1}, *A*_{2}, and *A*_{3} satisfying dening commutation relations

The main goal is to show how *A*_{1}, *A*_{2}, and *A*_{3} can be expressed, using elements *A* and *B*,
obeying Heisenberg's canonical commutation relation

The canonical representation of the commutation relation (1.6) is given by choosing *A* as the
usual dierentiation operator and *B* as multiplication by x acting on dierentiable functions
of one real variable *x*, on polynomials in one variable, or on some other suitable linear space
of functions invariant under these operators. In quantum mechanics, these operators, when
considered on the Hilbert space of square integrable functions, are essentially the same as
the canonical Heisenberg-Schrodinger observables of momentum and coordinate, diering
just by a complex scaling factor. The Heisenberg canonical commutation relation (1.6) is
also satised by the annihilation and creation operators in a quantum harmonic oscillator.

Since the 1970s, generalized (colour) Lie algebras have been an object of constant interest in both mathematics and physics [1,3-5,8-17,19-21,23,25-27]. Description of representations of these algebras is an important and interesting general problem. It is well known that representations of three-dimensional Lie algebras play an important role in the representation theory of general Lie algebras and groups, both as test examples and building blocks. Similarly, one would expect the same to be true for three-dimensional Lie colour algebras and superalgebras with respect to general Lie colour algebras and superalgebras. The representations of nonisomorphic algebras have dierent structure. In [26,27], three-dimensional Lie colour algebras are classied in terms of their structure constants, that is, in terms of commutation relations between generators. In [11,17,23], quadratic central elements and involutions on these algebras are calculated. In [16,25], Hilbert space *-representations are described for the graded analogues of the Lie algebra and of the Lie algebra of the group of plane motions, two of the nontrivial algebras from the classication. The classication of *-representations in [16,25] is achieved, using the method of dynamical systems based on generalized Mackey imprimitivity systems.

The colour Heisenberg Lie algebra is another important nontrivial algebra in the classi- cation of three-dimensional Lie colour algebras obtained in [26,27]. In the paper [24] we approached representations of this algebra in a totally dierent way than it was done in [16,25]. Namely, we studied those representations which can be obtained as power series in operator representations of Heisenberg's canonical commutation relations by rst obtaining in general realizations of the colour Heisenberg Lie algebra generators in terms of power series in elements of an associative algebra obeying the Heisenberg's canonical commutation relations and then combining these realizations with canonical representations of Heisenberg's canonical commutation relations.

In this paper we extend these investigations of realizations via Heisenberg's canonical
commutation relations to another colour Lie algebra with three generators and ve relations.
This algebra can be considered as another colour analogue of the Heisenberg Lie algebra.
However, we show that a structure of this algebra is quite dierent from that for the algebra
considered in [24] as far as realization via Heisenberg's canonical commutation relations is
concerned. In Section 2 we show that, with a natural choice for *A*_{1} as the rst generator of
the Heisenberg algebra corresponding to dierentiation, there are no nonzero power series in
Heisenberg generators which can be taken as *A*_{2} and *A*_{3} so that the three relations (1.1)-(1.3)
are satised. In Lemma 2.2, we describe all such formal power series solutions *A*_{2} and *A*_{3} for
the rst two relations (1.1)-(1.2) as in [24]. In Theorem 2.4, we present all such formal power
series solutions for *A*_{2} and *A*_{3} satisfying the three relations (1.1)-(1.3) showing that *A*_{3} = 0
is the only possibility. Using this result we get in Corollary 2.7 that *A*_{2} = *A*_{3} = 0 must hold
for such realizations of the ve relations (1.1)-(1.5). However, by considering 2 × 2 matrices
with entries chosen as formal power series in the noncommuting indeterminates *A* and *B*
satisfying Heisenberg's canonical commutation relations, we demonstrate how it is possible
to construct nontrivial realizations of (1.1)-(1.5). We also construct concrete operator
representations by applying this construction to the canonical representation of Heisenberg
canonical commutation relations and to the simple quantum mechanical harmonic oscillator.

Throughout this article denotes the eld of complex numbers and the set of nonnegative integers. By and we mean the ring of polynomials and formal power series over , respectively.

Consider a set {*A*_{1};*A*_{2};*A*_{3}} in some associative algebra over with unit element *I* satisfying
commutation relations (1.1)-(1.5). From relation (1.1) we obtain and hence by (1.2) it follows that

Using only (1.2) and (1.3) we may conclude that commutes with both *A*_{1} and *A*_{2}, that
is, . If merely (1.1) and (1.2) are satised and if for some
nonzero , then by (2.1) we have

Applying the famous Wintner-Wielandt theorem [18,28,29], we have that no elements in any unital normed algebra can satisfy the Heisenberg canonical commutation relation

So, we obtain from (2.2) the following result.

**Proposition 2.1.** *The commutation relations*

*together with cannot be satised by bounded operators on a Hilbert space
or even generally by elements in any unital normed algebra.*

When computing with power series in noncommuting elements *A* and *B* of an associative
unital algebra , we use the usual addition and multiplication rules of the Magnus algebra of
noncommutative formal power series in two indeterminates (see [2]). However, we assume that
*A* and *B* are not free, but satisfy at least the Heisenberg commutation relation as elements
in . We denote the algebra that we are working with by. In addition to the
subalgebra of generated by *A* and *B*, consisting of noncommutative polynomials in *A* and
*B*, the algebra may contain other elements which are innite noncommutative
power series in *A* and *B* not belonging to . The problem of equality of two elements in is a very complex matter in itself, deeply connected both to the properties of
noncommutative power series and Heisenberg's relation and to the structure of the algebra , and properties of A and B in . We say that an element of is in the (*B*, *A*)-
normal form (resp., (*A*, *B*)-normal form) if it is a noncommutative power series built of only
ordered monomials . In order to be able to
enjoy the equality properties in a similar way with formal power series as in the polynomial
case, we assume throughout this article that two formal power series in *A* and *B*, written
in the (*B*, *A*)-normal form (resp., in the (*A*, *B*)-normal form), are equal if and only if their
coecients are the same and in particular such a series is zero if and only if all coecients
are zero. This important equality assumption is actually an assumption on , on
the algebra as well as on *A* and *B* as elements in . In the particular case of polynomials
in *A* and *B*, that is, for the subalgebra of generated by *A* and *B*, the assumption yields
the same property as in , namely that are linearly independent as subsets of .

With the assumption above we may claim the equality of two elements of if they are equal to the same element in (*B*, *A*)-normal form (resp., in (*A*, *B*)-normal form).
However, it is important to observe that may well contain elements which cannot be represented on (*B*, *A*)-normal form, or (*A*, *B*)-normal form or even on either of
them. In most of the statements in this article we will adhere to the (*B*, *A*)-normal forms and
the corresponding equality assumptions. But, we will also comment and at some instances will
formulate the corresponding results when instead the (*A*, *B*)-normal forms and corresponding
equality of series is used. Which of these assumptions is used will be clear from the context.
We refer to [6,7] for further discussion on power series extensions of the Heisenberg algebra,
Diamond lemma, and normal forms.

In the paper [24] we have proved the following useful result.

**Lemma 2.2.** *Let* *A*_{1}=*A* *and assume that* *A*_{2} and *A*_{3} *are elements of the algebra* *written in the* (*B*, *A*)-*normal form, that is,*

*Then* *A*_{1}, *A*_{2}, *and* *A*_{3} *satisfy the commutation relations*

*if and only if*

**Remark 2.3.** It follows by this lemma that the commutation relations

can be satised by polynomial *A*_{2} and *A*_{3} (nite sums) only if *A*_{2} = *A*_{3} = 0:

**Theorem 2.4.** *Suppose that* *A*_{1}, *A*_{2}, *and* *A*_{3} *are elements of the algebra* *such
that* *A*_{1} = *A* *and* *A*_{2}, *A*_{3} *are formal power series in the* (*B*, *A*)-*normal form given as*

*Then* *A*_{1}, *A*_{2}, *and* *A*_{3} *will satisfy the commutation relations*

*if and only if*

*where*

**Proof.** By Lemma 2.2 we have, considering only the two anticommutation relations, a general
solution of the form

Applying the rules of Lemma 4 in [24] yields

and hence

which is a functional-dierential equation for *V* and *W*. Thus *A*_{2}*A*_{3} – *A*_{3}*A*_{2} = 0 implies
2*BW*(–*A*)*W*(*A*) = 0, which for a formal power series *W*(*A*) yields *W*(–*A*)*W*(*A*) = 0. The
set of complex formal power series is an integral domain and hence *W*(*A*) = 0. This
means that *A*_{2} = *T*(*B*, *A*)*V* (*A*) and *A*_{3} = 0.

**Remark 2.5.** Note that in Theorem 2.4, *A*_{1}*A*_{2} + *A*_{2}*A*_{1} = *A*_{3} = 0 is the only nontrivial
relation left. The other two relations are trivially satised when *A*_{3} = 0, independently of
*A*_{1} and *A*_{2}. So, under conditions of Theorem 2.4, one can say that

is equivalent to

Remark 2.6. In Lemma 2.2, the formal series *A*_{2} and *A*_{3} are expressed in the (*B*, *A*)-normal
form. This is a natural ordering when we think of *A* as the usual dierentiation operator
*∂* and *B* as a multiplication operator *M* acting on dierentiable functions on the real line,
given by . In other situations, it may be more appropriate to
consider the reversed order.

Taking *A*_{2} and *A*_{3} in Lemma 2.2 to be in the (*A*, *B*)-normal form, but keeping *A*_{1} = *A*,
the general solution will be changed to the following form:

where

If we consider a solution in the (A;B)-normal form satisfying the three relations in Theorem
2.4, a completely similar proof shows that also in this case *A*_{2} = *V* (*A*)*U*(*A*, *B*) and
*A*_{3} = 0.

**Corollary 2.7.** *Suppose that* *A*_{1}, *A*_{2} *and* *A*_{3} *are elements of* *such that* *A*_{1} = *A*,
*and* *A*_{2} *and* *A*_{3} *are formal power series in the* (*B*, *A*)-*normal form given as*

*Then* *A*_{1}, *A*_{2} *and* *A*_{3} *satisfy*

*if and only if* *A*_{2} = *A*_{3} = 0.

**Proof. **By Theorem 2.4, we know that *A*_{2} = *T*(*B*, *A*)*V* (*A*) and *A*_{3} = 0. Since

the relation holds only if *V* (–*A*)*V* (*A*) = 0, which, for a power series *V* (*A*), yields
*V* (*A*) = 0 and *A*_{2} = 0.

Now let *A*_{1}, *A*_{2} *and* *A*_{3} be 2 × 2 matrices on the form

where *A*_{11}, *A*_{12}, *A*_{21}, *A*_{22}, *L*, and *M* are elements of some associative algebra or ring .
Then

Therefore, *A*_{1}*A*_{2} + *A*_{2}*A*_{1} = *A*_{3} is equivalent to the conditions

and similarly we have *A*_{1}*A*_{3} + *A*_{3}*A*_{1} = 0 if and only if

**Remark 2.8.** Observe that for any elements *L* and *M* in the relations

are always satised, since moreover we have *A*_{2}*A*_{3} = *A*_{3}*A*_{2} = 0. Thus, under the conditions
(2.3) and (2.4), elements *A*_{1}, *A*_{2}, and *A*_{3} satisfy commutation relations (1.1)-(1.5). This gives
a method of construction for realizations of (1.1)-(1.5).

Let us assume that either *L* or *M* are left or right invertible. Then *A*_{21} = 0, and hence
*A*_{1} is upper triangular, that is,

So, we have the following useful statement.

**Lemma 2.9.** *Let L and M be elements of an associative algebra or ring , such that at
least one of them is left or right invertible in . Then the elements*

*of the algebra (ring)* of 2 × 2 *matrices over* *satisfy the commutation relations*

*if and only if*

*together with*

*Given* *A*_{1} *and* *A*_{2} *of the form in* (2.7) *and* (2.5), *respectively, there exists an M so that* *A*_{3}
*of the form in* (2.5) *satisfies* (2.6) *if and only if*

**Proof.** The rst part was proved before the statement. The last relation follows by eliminating
*M* using (2.8). On the other hand, if (2.9) holds, then *M* = *A*_{11}*L* + *LA*_{22} satises
(2.8).

**Lemma 2.10.** *Let the matrices*

*be elements of* , *the algebra (ring) of* 2 × 2 *matrices over an associative algebra (ring)* *, and assume * *Then*

*are satised if and only if* *B*_{1}, *B*_{2}, *and* *B*_{3} *satisfy*

Moreover, and *A*_{2}*A*_{3} =*A*_{3}*A*_{2} =*A*_{2}*A*_{3} –* νA*_{3}*A*_{2} =0, *independently on*

**Proof.** The proof follows from the following easily checked equalities:

By Lemma 2.2 we know that if *AB* – *BA* = *I* for *A* and *B* in some unital associative
algebra , then

satisfy commutation relations

and also if *B*_{1} = *A* and *B*_{2} and *B*_{3} are (*B*, *A*)-normally ordered power series in *A* and *B*,
then *B*_{1}, *B*_{2}, and *B*_{3} satisfy (2.11) if and only if *B*_{2} and *B*_{3} are of the form (2.10). A similar
result is obtained if we choose to write *B*_{2} and *B*_{3} in the (*A*, *B*)-normal form.

Note also that we can apply Lemma 2.2 and its version in the (*A*, *B*)-normal form on
a new set of generators . Observing that *T*(–*A*, *B*) = U(*A*, *B*) and U(*B*,–*A*) = T(*B*, *A*), the resulting general solutions are

Combining this with Lemma 2.10, we can construct a realization of (1.1)-(1.5) using *A* and
*B* satisfying the Heisenberg commutation relation *AB* – *BA* = *I*.

**Theorem 2.11.** *Assume A and B are two elements of an associative unital algebra or a
ring (with identity) , satisfying the Heisenberg commutation relation AB – BA = I. Let * *and define the following* 2 × 2 *matrices*:

*where*. *Then* *A*_{1}, *A*_{2}, *and* *A*_{3} *satisfy the
commutation relations*

*Moreover*, *A*_{2}*A*_{3} = *A*_{3}*A*_{2} = 0.

When the (A, B)-normal forms and the corresponding equality assumption in is used, the following analogue of Theorem 2.11 holds.

**Theorem 2.12.** *Assume A and B are two elements of an associative unital algebra or a
ring (with identity)* *, satisfying the Heisenberg commutation relation AB – BA = I. Let* , *and dene the following* 2 × 2 *matrices*:

where* *. *Then* *A*_{1}, *A*_{2}, *and* *A*_{3} *satisfy the
commutation relations*

*Moreover*, *A*_{2}*A*_{3} = *A*_{3}*A*_{2} = 0.

**Example 2.13.** Introduce the operators

acting on . These operators satisfy *AB – BA* = *I*. Note also that *U*(*A*, *B*) = *U*(*M*, –*∂*) =
*T*(*M*, *∂*). Let be any operator. Then *A*_{1}, *A*_{2}, *and* *A*_{3} defined in Theorem
2.12 are realized on by the operators

acting on . By the same theorem they satisfy the ve relations for
any power series *V* (*M*) and *W*(*M*) in *M*. Note that here is the
parity operator (see [24]). For example, if *V* (*t*) = 1 and *W*(*t*) = *t*^{2}, then

**Example 2.14.** Referring to [22] and [24, Example 5], we consider a sequence of functions defined by

where a^{*} is the linear operator

defined for any positive real constant *x*_{0}. It follows that

for *n* = 0, 1, 2, ..., where *H _{n}* are the Hermite polynomials. Defining the linear operator

it can be shown that Moreover,

for any The sequence of functions describes the energy eigenstates of the
simple quantum mechanical harmonic oscillator. Consider now the two dierential operators a* ("creation" operator) and *a* ("annihilation" operator) dened on the linear space Ω = consisting of all complex linear combinations of functions from the set of
eigenfunctions . Since now *aa** – *a***a* = 1, we can dene *A* and *B* in Theorem 2.11 as *A* = *a* and *B* = *a** acting on Ω. Suppose is any operator. Then *A*_{1}, *A*_{2}, and *A*_{3} introduced in Theorem 2.11 are realized as the operators

acting on .

Recall that . Hence

Moreover, for ,

Now let and, where. Since

it readily follows that

keeping in mind that .

Similarly, using the fact that , one obtains

This means that

In the first expression, we have to assume that. Introduce a basis given by

For , it follows that

And, for ,

Moreover, for all ,

This work was supported by the Swedish Research Council, The Crafoord Foundation, The Royal Swedish Academy of Sciences, The Royal Physiographic Society in Lund, and The Swedish Foundation for International Cooperation in Research and Higher Education (STINT). The rst author is grateful to the Centre for Mathematical Sciences, Lund University for hospitality and support during his visits in Lund.

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