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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Noncommutative Geometry and Dynamical Models on U(u(2)) Background

Gurevich D1* and Saponov P2

1LAMAV, Universit de Valenciennes, France

2National Research University Higher School of Economics, Russia

*Corresponding Author:
Gurevich D
LAMAV, Universit de Valenciennes, 59313 Valenciennes, France
Tel: +33 (0)3 27 51 12 34
E-mail: [email protected]

Received date: January 17, 2015; Accepted date: June 16, 2015; Published date: June 18, 2015

Citation: Gurevich D, Saponov P (2015) Noncommutative Geometry and Dynamical Models on U(u(2)) Background. J Generalized Lie Theory Appl 9: 215. doi:10.4172/1736-4337.1000215

Copyright: © 2015 Gurevich D, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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In our previous publications we have introduced a differential calculus on the algebra U(gl(m))based on a new form of the Leibniz rule which differs from that usually employed in Noncommutative Geometry. This differential calculus includes partial derivatives in generators of the algebra U(gl(m))and their differentials. The corresponding differential algebra omega(U(gl(m)))is a deformation of the commutative algebra omega(Sym(gl(m))). A similar claim is valid for the Weyl algebra W(U(gl(m)))generated by the algebra U(gl(m))and the mentioned partial derivatives. In the particular case m=2 we treat the compact form U(u(2))of this algebra as a quantization of the Minkowski space algebra. Below,we consider non-commutative versions of the Klein-Gordon equation and the Schrodinger equation for the hydrogen atom. To this end we define an extension of the algebra U(u(2))by adding to it meromorphic functions in the so-called quantum radius and quantum time. For the quantum Klein-Gordon model we get (under an assumption on momenta)an analog of the plane wave, for the quantum hydrogen atom model we find the first order corrections to the ground state energy and the wave function.


Weyl algebra; Leibniz rule; Quantum radius; Plane wave; Klein-Gordon model; Schrodinger model; Hydrogen atom model

AMS Mathematics Subject Classification, 2010: 46L65, 46L87, 81T75


One of the basic problems of the Noncommutative (NC) Geometry is to construct a differential calculus on a given NC algebra A. This calculus should involve the differential algebra Ω(A), generated by A and differentials da, a∈A, and vector fields (in appropriate cases, analogs of partial derivatives). The core of the problem is a proper definition of the Leibniz rule for these vector fields and the corresponding de Rham operator. Especially, this problem is interesting for algebras arising from quantization of commutative ones since in this case the question of deformation properties of the algebra Ω (A) can be addressed.

The most common approach to this problem is based on the socalled Kahler forms which generate the universal differential algebra. In this algebra the usual Leibniz rule d(ab)=da b + a db is preserved but the commutativity relation a db=db a is canceled. As a result, the differential algebra becomes bigger than the usual one, even if A is a commutative algebra, say the coordinate algebra of a regular algebraic variety.

Sometimes, the Hochschild homology Hi(A) is considered as a convenient analog of the space Ωi(A) (for the relations between these two objects for NC algebras A we refer the reader to Loday [1]. Whereas the Hochschild cohomology Hi(A) plays the role of the space of polyvector fields on A (this is motivated by the famous Hochschild- Kostant-Rosenberg theorem). In particular, the first group H1(A) is identified with the space of derivations (quotiented over the inner ones), i.e. operators subject to the usual Leibniz rule. An advanced calculus based on this treatment of the Hochschild (co)homology can be found in the papers [2-4] and others [1,5]. Nevertheless, if A is a coordinate algebra of a regular variety, the space Hi(A) differs from that Ωi(A) considered in its classical sense.

The same is valid if we replace Hi(A) here by the cyclic homology HCi(A) due to Connes. He also introduced another complex in which the role of the de Rham operator is played by the commutator with a Dirac operator. However, it differs from the usual de Rham complex as well.

Also, observe that in all mentioned approaches no analogs of partial derivatives are introduced, even if a given NC algebra is a deformation of the coordinate algebra of a linear space.

A couple of years ago we suggested a new type of differential calculus [6,7] on the algebras U(gl(m)), the super-algebras U(gl(m|n)), and their braided analogs the so-called modified Reflection Equation algebra L(R) related to a Hecke symmetry R. The calculus is based on so-called permutation relations between elements of the NC algebra in question and partial derivatives on it.

If such a Hecke symmetry R is a deformation of the usual flip P, the corresponding Reflection Equation algebra L(R) admits a construction of the differential algebra which is a deformation of its classical counterpart. Moreover, the corresponding Weyl algebra, generated by L(R) and the mentioned partial derivatives, is also a deformation of its classical counterpart. Besides, if R is involutive (R2=I) the mentioned permutation relations can be expressed via a new form of the Leibniz rule which in its turn is related to a coproduct in the algebra generated by the partial derivatives.

Below we assume R to be the usual flip P. Moreover, we deal with a particular case m=2, i.e. our basic NC algebra is U(gl(2)h) or rather its compact counterpart U(u(2)h). The latter algebra can be treated as a NC (quantum) analog of the Minkowski space algebra. Emphasize that the algebra U(u(2)h) is a deformation (quantization) of the commutative algebra Sym(u(2)) which is considered as the Minkowski space algebra provided it is endowed with a proper metric.

In this connection we want to mention other deformations of the Minkowski space algebra such as q-Minkowski space algebra (in fact it is a particular case of the Reflection Equation algebra). There is a large number of publications devoted to these NC versions of the Minkowski space algebra.

Note, that our version of the quantum Minkowski space algebra, namely U(u(2)h) has a lot of advantages with respect to the other ones.

First of all, the algebra U(u(2)h) is an SO(3)-covariant enveloping algebra of the reductive algebra u(2)h whose simple component is the Lie algebra su(2)h. In virtue of the Kostant's theorem (see section 2) the algebra U(su(2)h) admits a decomposition in the direct sum of isotypic components.

image(1.1) compo

where Z(U(su(2)h)) is the center of the algebra U(su(2)h) and Vk ⊂ U(su(2)h) is the irreducible SU(2)-module (dim V k=2k + 1). This decomposition provides us with a basis in the algebra U(su(2)h) convenient for our computations.

Besides, in the algebra U(u(2)h) there is a version of the Cayley- Hamilton (CH) identity for a matrix N composed of the U(u(2)h) generators. The roots μ1 and μ2 of this CH identity ("eigenvalues" of the matrix N) are related to a quantity which can be treated as a quantum analog of the radius image. This "quantum radius" image together with the quantity image (treated as a quantum analog of the time) is very useful for a parametrization of the center Z(U(su(2)h) ). It also enables us to construct a central extension of the algebra U(su(2)h) (and consequently, of U(u(2)h)) needed for constructing NC versions of certain physical models.

In the present paper we are dealing with the model of the free massive scalar field described by the Klein-Gordon equation and with the model of a particle in a central potential field (the hydrogen atom) described by the Schrodinger equation. In our previous publication we have considered the Klein-Gordon equation in nontrivial rotationally invariant metric. Below we study the simplest form of the NC Klein- Gordon equation and find its plane wave type solution. For this solution we exhibit (under an assumption on momenta) a quantum deformation of the so-called relativistic dispersion relation E2=m2c4 +p2c2 connecting the energy E of a particle of the mass m and the square of its momentum p2. As for the NC Schrodinger equation, we find the first order corrections to the ground state energy and to the wave function of the corresponding hydrogen atom model.

Emphasize that there are known numerous attempts to generalize some models of mathematical physics to quantum algebras. Besides the aforementioned versions of the Minkowski space algebra, we want to cite the papers [8,9] where some rotationally invariant models were quantized. The essential dissimilarity between the cited papers and ours consists in different treatment of the differential operators (such as the Laplacian Δ= ∂2x + ∂2y + ∂ 2y) appearing in dynamical equations of the models. In the cited papers the partial derivatives are treated in the classical sense, only the usual commutative product of functions is replaced by its deformed version (called image -product). By contrary, in our approach the partial derivatives are directly defined on the NC algebra in question and their properties differ drastically from those of the usual derivatives.

First, the partial derivatives in the spacial generators x, y, z (and consequently, the Laplacian) act nontrivially on the time variable and vice versa.

Second, we treat the quantized models in terms of the quantum algebra without using any image -product. The only problem arising in our approach is pulling an operator which describes the model forward to the quantum algebra. We call this procedure the quantization of operators. For the operators we are dealing with, this quantization is easy: the derivatives should be treated in the new sense (i.e. according to our definition of the derivatives on the algebra U(u(2)h)) and the radial variable r in a potential V (r) should be replaced by its quantum counterpart γr , as well as the time variable t is replaced by its quantum counterpart τ . In more detail the problem of quantization of differential operators is considered in Gurevich et al. [7].

Third, the models in question can be restricted to any component coming in the sum (1.1) and then to some lattices in 2-dimensional space "time-quantum radius". This property is due to the fact that the partial derivatives in time and quantum radius are difference operators. In this sense we speak about a discretization of the space-time.

The ground field image is assumed to be image or image , its concrete form is always clear from the context.

The paper is organized as follows. In the next section we introduce a quantum version of the Weyl algebra. Also, we define its extended version by adding meromorphic functions in the quantum radius and the time to it. In the section 3 we give an operator meaning to elements of the quantum Weyl algebra they become operators acting on the algebra U(u(2)h). In section 4 we introduce the partial derivative in quantum radius and clarify its role in discretization of models we are dealing with. Section 5 is devoted to our quantization of the aforementioned rotationally invariant dynamical models. In section 6 we present some concluding remarks and discuss the perspective of our study. In Appendix we get together some technical results used in the text.

The Weyl Algebra W(U(u(2)h)) and its Extension

First, recall that according to Gurevich D et al., [6,7] the Weyl algebra W(U(gl(m)h)) is a unital associative algebra generated by 2m2 elements image and image. Having defined the m × m generating matrices image we present the multiplication rules of the W(U(gl(m)h)) generators in the following matrix form:

PN1PN1 - N1PN1P = h(PN1 N1P ),

PD1PD1 = D1PD1P,

D1PN1P - PN1P D1 = P + h D1P (2.1) sett

Where h is a numeric parameter, image, image matrix P is the usual flip.

As is clear from the first line of the system (2.1), nji are generators of the sub algebra U(gl(m)h) ⊂ W(U(gl(m)h)). The second line expresses the commutativity of generators dji. The corresponding Abelian subalgebra will be denoted D. The relations coming in the third line are called the permutation relations. The permutation relations allow us to treat the generators {dji} as (quantum) partial derivatives. In particular, in the classical limit (h=0) the element djican be identified with the partial derivative image. From now on we shall use the term “partial derivatives" for the elements dji, while the precise operator meaning will be given to them in the next section.

Note that the algebra W(U(gl(m)h)) can be treated as the enveloping algebra of a semidirect product of the Lie algebra gl(m)h with the basis njiand the commutative algebra D=gl(m)0 with the basis dji. In fact we have a slight modification of the semidirect product because of the constant terms, entering the flip P in the third line of (2.1), but this does not impact the claim that in virtue of the PBW theorem the family of elements


is a basis of the algebra W(U(gl(m)h)).

Remark 1: Meljanac and Skoda have suggested the realization of relations (2.1) analogous to their construction from [MS] (also, see [S]). Let image be the generating matrix of the algebra Sym(gl(m)) (i.e. it satisfies the first line relations (2.1) with h=0) and image be the matrix composed of the usual partial derivatives in image Then the matrices imageand D satisfy the system (2.1).

Now, we write down the relations (2.1) in the particular case m=2 and pass to the compact counterpart U(u(2)h) of the algebra U(gl(2)h). To simplify formulae we use the following notation for the U(gl(2)h) generators:


The generators of the algebra U(u(2)h)

image (2.2) abcd are subject to the system of relations


The center Z(U(u(2)h)) is generated by the element t and quadratic Casimir element

Cas=x2 + y2 + z2.

The partial derivatives in t, x, y and z will be denoted as ∂t ,∂x,∂ y and ∂z respectively.

Besides, it is convenient to introduce a shifted derivative image

In terms of generators t, x, y, z and the corresponding partial derivatives the third line of the system (2.1) reads

image(2.3) leib-r

Our next aim is to introduce quantum analogs of "radial variables". To this end we need a version of the Cayley-Hamilton (CH) identity for the generating matrix N

image(2.4) matrN

In this case the corresponding characteristic polynomial χ (λ ) is quadratic and the CH identity explicitly reads


Note, that the coefficients of the above identity are central in the algebra U(u(2)h). Recall, that this is also true in the general case U(gl(m)h).

The roots μ1 and μ2 of the characteristic polynomial are called the eigenvalues of the matrix N. This means that they solve the system of two algebraic equations

image(2.6) mmu

and, consequently, the eigenvalues should be treated as elements of an algebraic extension of the center Z(U(u(2)h)).

Equations (2.6) allow us to express the center generators (and, therefore, any element of Z(U(u(2)h))) in terms of μ1 and μ2:

image(2.7) cas-mm

Hereafter, we use the notation μ=μ12.

Remark 2: Note that the quantity μ is defined up to a sign since the ordering of the eigenvalues is arbitrary.

Now, we compute the permutation relations of the partial derivatives ∂x , ∂ y , ∂ z and the Casimir element. In contrast with the commutative case, the other elements of the algebra W(U(u(2)h)) turn out to be involved in these permutation relations. We need the following operators

image(2.8) Qx

Theorem 3:

image(2.9) form

The permutation relation of ∂ y , (respectively, ∂z) with the Casimir element are obtained from (2.9) by replacing x and X for y and Y (respectively, for z and Z).

We denote the 4 × 4 matrix in formula (2.9) by Ψ(Cas) . Being expressed in terms of μ (see (2.7)) it reads


The matrix Ψ (Cas) is semisimple with the following spectrum:


Its spectral decomposition reads


where I is the unit 4 × 4 matrix. The explicit matrix form of the complementary projectors Pi(μ) is as follows:


The permutation relations of the column entering (2.9) with the element Casp, p ≥2, can be obtained in the same way but the matrix Ψ (Cas) must be replaced by its p-th degree. Using the spectral decomposition, we calculate Ψp (Cas) as


So, we can find the permutation relations of the set of elements image, imageimage and X with any polynomial or even a series f(u) in u=Cas. The only thing we need for this purpose is


However, we want to get the permutation relations of the above operators with μ and its powers image . Since μ=f(Cas) where f(u)=image , We get


The sign is motivated by the classical limit. Finally, we arrive to the following permutation relations

image (2.11) formm

In the same way the permutation relations of the partial derivatives ∂y and ∂z with image can be found. As for the permutation relations of the operators image and Q with μp, they can be extracted from formula (2.11): it suffices to cancel x in the second and third lines of this matrix.

Thus, we arrive to the result (also, see [GS2] where these formulae were first obtained)

image (2.12) formm

Note that in the classical case (h=0) the variable μ is a multiple of the radius image namely, μ=2ir. In the general case (h ≠ 0) we introduce an analog of the radius by a similar formula.

Definition 4: The quantity image is called the quantum radius.

Below, we use the element image as well as μ. However, dealing with image we prefer to use the renormalized deformation parameter image in order to avoid complex shifts of the type image imageimage+ia(as for the factor 2 it is introduced for the future convenience). For the same reason we will use the renormalized time τ = −it . All our formulae represented in terms of the initial quantities (μ, t, h) can be easily rewritten in terms of ( image,τ , image) and vice versa. For example, the Casimir element in terms of new quantities has the form


We also define the partial derivative image and the corresponding "shifted derivative” image. Note, that image

Introduce now the commutative algebra image of functions in τ and image any image we assume the function f(z1, z2) to be meromorphic functions in two complex variables.

Next, consider the algebra

image(2.13) quot

The generators τ and image are assumed to commute with any element of the algebra image Thus, the subalgebra imageforms evidently the center Z(image) of this algebra

Our next aim is to find the permutation relations of the partial derivatives image and elements of the algebra image

First, consider in image polynomial functions in two variables. Since τ and image , any polynomial image can be presented as a finite sum image where gi and qi are some polynomials in one variable. Then, as was shown in [GS2], the permutation relations of the partial derivatives with a polynomial g(τ ) are

image (2.14) Fo

As for the permutation relations of the partial derivatives with image (and, consequently, with any polynomialimage they can be easily deduced from (2.11).

Therefore, the above considerations together with (2.14) and (2.11) allow us to get for any polynomial image in two variables the following result

image (2.15) dxf

and similarly for the derivatives ∂y and ∂z .

For the partial derivative image we find

image (2.16) dtf

We extend these formulae from polynomials to arbitrary functions image by definition.

Now, define the algebra W( image ) to be the algebra generated by and the partial derivatives image endowed with the permutation relations described by formulae (2.3), (2.15) with its y- and z-analogs and (2.16). We call W( image ) the extended Weyl algebra.

Below, we define the derivative ∂rh in W( image) but it will be expressed in terms of already existing elements of W( image) so we shall not need its new extension.

By resuming this section, we want to emphasize that the algebra W(image) is introduced as a quotient-algebra. Intuitively, it is clear that it is a deformation of its classical counterpart, de fined by formula (2.13) with h=0. However, the fact that the algebra W(U(u(2)h)) is a deformation of its classical counterpart W(Sym(u(2))) can be shown rigorously. Indeed, it follows from the discussed above treatment of the algebra W(U(gl(m)h)) as the enveloping algebra of a semidirect product.

Differential Operators on the Algebra image

Now, we are going to define an action of the algebra W( image ) onto image . To this end we first define this action for the “partial derivatives” dji (recall that at the limit h=0 we have image )


Hereafter, the action of an operator D on an element n € U(u(m)h) will be denoted D (n), while we keep the notation image for the permutation relation.

Below we need the counit map image introduced by

image (3.1) coun

The symbol 1A stands for the unit element of the (sub) algebra A.

Then we proceed as follows. Given a product of elements d n (here after, we omit the sign image ), we first apply the permutation relations (the third line of (2.1)) to the elements d and n in order to move elements of D to the most right position


where in the right hand side we have a finite sum of elements. After that, we apply the counit map ε to all elements of D in the above sum and get an element from U(gl(m)h):


where we identify U(gl(m)h) imageimage with U(gl(m)h).

Thus, all partial derivatives image become operators acting on U(u(2)h). Emphasize that the action of the partial derivatives is not subject to the classical Leibniz rule. By contrary, these derivatives meet a new version of the Leibniz rule which was found by Meljanac and Skoda [10] (see also [6] where we have given another but equivalent form of this Leibniz rule). In order to describe this form of the Leibniz rule, consider the following coproduct in the algebra D

image(3.2) cco

This coproduct and the counit (3.1) endow D with a structure of a coassociative coalgebra. Moreover, since this structure is coordinated with the algebraic structure of D, this algebra becomes a bi-algebra.

Now, the standard action image the derivatives image on the generators image of the algebra U(gl(m)h) can be extended to the whole algebra U(gl(m)h) via the above coproduct. In order to prove that the extended action respects the algebraic structure of U(gl(m)h), it suffices to check that the ideal generated in this algebra by the elements

PN1PN1 - N1PN1P- h(PN1 - N1P)

is invariant with respect to the action of the derivatives. The detail is left to the reader.

Remark 5: Note that by using the method of [11] it is possible to recover the permutation relations (the third line of (2.1)) on the base of the coproduct (3.2). However, in a more general case related to the Reflection Equation algebra [7] we have not succeeded in finding the coproduct corresponding to the permutation relations studied in this paper. Thus, the permutation relations are in a sense a more fundamental structure enabling us to de ne an action of the algebra D on that U(gl(m)h) without any coproduct.

Now, pass to the case m=2 and extend the derivatives image onto the algebra image (here we deal with the compact form of the algebra gl(2)h and also we use the generator τ instead of t). First, we exhibit the coproduct Δ in the generators image. Thus, we have



By applying the cyclic permutations to the latter formula, we get Δ(∂ ) and Δ(∂z) .

Now, using formulae (2.16) and (2.15) we can compute the action of the derivatives image and image

image (3.3) patt

image (3.4) pax

Similar formulae are valid for image

Extend the action of the derivatives on the algebra image via the above coproduct. In order to show that the extended derivatives are well defined on this algebra it sufficed to check that they map the ideal generated in this algebra by the elements

image (3.5) elem

to itself. The following proposition entails this property.

Proposition 6: Each element

image(3.6) that

is a linear combinations of the elements of the form (3.5).

Proof: In fact, it suffices to check (3.6) with v=x and u=y. Other relations are trivial or can be obtained by the cyclic permutation of the generators x, y, z. Checking (3.6) with v=x, u=y can be done with the help of formula (3.4) or its equivalent form (4.1) below.

Note that any element of the algebra W(U(u(2)h)) can be presented as a finite sum of elements image, image coefficients from U(u(2)h) (we put them on the left hand side). In a similar way we can realize elements of the algebra W( image ). This realization of any element of the algebra W(U(u(2)h)) or W( image ) is called canonical. A canonical form of a given element image allows us to assign to A an operator DA:image (for this assignment we use the operator meaning of elements from D discussed above).

Let us denote Diff the vector space of all such differential operators. Thus, we have an isomorphism

Iso : W( image ) → Diff

which is a representation of the algebra W( image )

Now we construct a special basis in the algebra U(u(2)h) useful in the sequel. First, note that U(u(2)h)=U(su(2)h) image. Second, we use the fact that the algebra U(su(2)h) is free over its center Z(U(su(2)h) ) (Kostant's theorem, see [D]). Let Vk ⊂ U(su(2)h) , k ≥ 0 (we set image ), be an irreducible adjoint U(su(2)h) submodule such that bk = (−ix − y)k is a highest weight element in its complexification VCk. Thus, Vk is the submodule of the spin k, dim Vk=2k + 1.

Then we have


As is known, Z(U(su(2)h)) consists of all polynomials in Cas. By fixing a basis {vik ≤ 1 ≤ i ≤ 2k + 1} in each submodule Vk and the basis {Casp}, p > 0 in Z(U(su(2)h)), we get a basis {Casp image vik} of the whole algebra U(su(2)h) .

A similar decomposition hold true for the algebras U(u(2)h) and image :


Consider now a natural action of the rotation group SO(3) on the algebra W( image ) (and consequently on Diff). This group acts on the "spacial generators" x, y, z of the algebra W( image ) and on the corresponding derivatives ∂x, ∂ y and ∂z in the classical way, but it affects neither the factor image nor the derivative ∂T.

Definition 7: We call a differential operator DA∈ Diff to be invariant if it is the image of an SO(3)-invariant element A∈W (image ) of the algebra W( image ) under the isomorphism W( image ) → Diff.

Important examples of the first order invariant operators are

image(3.7) first

Some the second order differential operators to be used below are


All operators from the lists (3.7) and (3.8) will be called basic. The operator Δ is called Laplacian. All polynomials in the basic operators with coefficients fromimage are also invariant operators.

Proposition 8: Let D : imageimage be a basic invariant differential operator. Then it maps the component

image(3.9) isoco

to itself. Moreover, its action on any element imagev, v ∈ Vk alters only the central function image. . Thus,we have


for some image

Proof: First, observe that the evaluation map


is SO(3)-covariant. This entails that if D is an invariant differential operator it commutes with the action of the group SO(3). Therefore, its complexification commutes with the action of the group SL(2, image ).

Now we use the following fact, proved in [GS2].

Lemma 9: The action of the partial derivatives image onto the elements image is subject to the classical Leibniz rule.

Due to lemma 9 we can conclude that all basic differential operators act on the elements bk by the classical formula. Namely, we have



Now, the claim of the proposition follows immediately from the invariancy of the basic operators.

As follows from the above consideration, the restriction of any basic operator to the components Vk can be well defined. Note that being restricted to the component Vk such an operator acts as a difference operator on the factor image in an elementimage The reader is referred to [7] for explicit examples and detail.

Thus, the spectral problem for a basic differential operator D


reduces to the spectral problem of a family of its restrictions to the components Vk.

Definition 10: The restriction of a basic operator D on the component image is called the radial part of the operator D and is denoted Drad.

As follows from [GS2], the radial part of the Laplacian Δ has the form:

image (3.10)L3

Another important example is the radial part of the operator Q

image (3.11) rad Q

This result immediately follows from (3.4) and its y- and z-analogs.

Quantum Radial Derivative and its Applications

Let us introduce the operator imageby the following rule


This operator is well defined on the component image and we call it the quantum radial derivative. It is a difference operator which turns into the usual derivative in r as image :


With this notation we rewrite formula (3.4) in the form:


This formula is similar to the classical one but with another meaning of the derivative. Formula (3.11) can also be rewritten in terms of the new derivative:


Since the function image is an arbitrary element of image we have an operator equality on the algebra image

image (4.2) parh

We use this relation as the definition of an extension of the quantum radial derivative image on the whole algebra image. Note that this definition differs from that from [7] by the factor at the operator Q.

As the next step we express the operator Δrad in terms of the quantum radial derivative. A straightforward calculation on the base of relation (3.11) leads to the result:

image (4.3) rr Lap

Remark 11: Formula (4.3) is analogous to the classical one

image (4.4) r Lap

and turns into this as image . Actually, formula (4.3) has motivated our definition of the quantum radial derivative.

In contrast with the classical case, the Laplace operator Δ affects the variableτ , though it does not explicitly contain the derivative ∂T . This effect disappears as image .

Our next aim is to solve the equation

image (4.5) solv

for a τ -independent function image. In the classical case (image) the vector space of solutions of the equation image in the region r>0 consists of all linear combinations of two functions f1(r)=1 and f2(r) = r−1 .

In the quantum case as follows from expression (4.3) the functions image and image are solutions of the equation (4.5). Moreover, all linear combinations image are also solutions provided that the coefficients image i=1, 2, are 2 image periodical functions in image

Remark 12: In the classical case the function image satisfies the equation


on the whole space image Here, x=(x, y, z) and δ(x - a) stands for the δ-function located at the point a=(a1, a2, a3) ϵ image . In section 6 we shall discuss a quantum version of this equation.

Note that the operator (4.3) is well defined on a lattice

image (4.6) lat

where a and b are some fixed real numbers. Thus, we can consider the restriction of this operator to the lattice (4.6). A similar restriction is possible for any operator depending only on the quantities τ and image and the derivatives ∂T and ∂rh . In this sense we speak about the discretization of a dynamical odel escribed by such an operator.

This discretization reduces the freedom for coefficients in the space of solutions of the equation Δ(f(τ , image ))=0. Thus, the aforementioned coefficients image entering the general solution become constant on each of the two sublattices imageand


Let us assume the algebra U(gl(2)) to be represented in a Verma module image of the highest weight image (for the definition we refer the reader to the book [12]. The module image has a finite dimensional U(gl(2)) submodule if image is a non-negative integer. On passing to the algebra U(gl(2)) we get (after rescaling of the generators x → x/h etc.) the corresponding representation image , which can also be factorized up to a finite dimensional representation provided image. (Note that image is the spin of this finite dimensional representation).

It is known that the image of any central element image under the representation image is a scalar operator (it will be identified with its eigenvalue). In particular, we have


This enables us to compute the corresponding value of the quantum radius image . Namely, we have


The sign here is motivated by our wish to have a positive quantum radius(at image >0) on finite dimensional modules. Thus, on the finite dimensional module of the spin image the quantum radius takes the values


So, the operator Δrad admits a discretization well defined on a family of the Verma modules, some of them can be factorized to finite dimensional ones. This is also true for the isotypic components Vk (3.9) with k ≠ 0.

We complete this section with the following observation. Let

a : sym(u(2))→U(u(2)h) (4.7)

be an SU(2) -covariant quantizing map [13]. Then any operator

D: sym(u(2))→Sym(u(2))

can be pulled forward to

image (4.8) D-al

However, the image of a usual partial derivative in the algebra Sym(u(2)) with respect to the map (4.8) does not coincide with the quantum partial derivatives considered in the present paper. Nevertheless, the images of the infinitesimal rotations in Sym(u(2)) remain to be infinitesimal rotations in U(u(2)h).

As for the boosts, their analogs are not naturally defined on the algebra U(u(2)h). Let us introduce the infinitesimal "radial boost" in the classical case ( image =0)

image (4.9) boo

The corresponding one parameter group exp image preserves the Lorentz interval image since image

We define the quantum analog of the "infinitesimal radial boost" as


and the "quantum Lorentz interval" squared as image. However, now the action


is proportional to h and disappears only as image

Klein-Gordon and Schrodinger Equations on U(u(2)h)

In this section we consider the NC (quantum) versions of two dynamical models playing an important role in physics. Namely, we are dealing with the Klein-Gordon equation (as an example of a relativistic model) and the Schrodinger equation with the hydrogen atom potential (as an example of a non-relativistic model). For the former equation we exhibit an analog of the plane wave solution under an assumption on momenta parameterizing the solution. With this assumption we get a deformed version of the relativistic dispersion relation. For the latter equation we are looking for a value of the ground state energy.

The Klein-Gordon equation

The Klein-Gordon equation describes evolution of a free massive scalar field. It is the second order partial differential equation of the form (we assume the speed of light and the Planck constant to be equal to the unity)

image (5.1) eq : Kl-G

Here m is a real constant (the mass of the eld). This equation possesses the plane wave solutions

image(5.2) plw

where the energy E and the momenta (px, py, pz) are subject to the relativistic dispersion relation


Let us pass to a NC version of the Klein-Gordon equation. Since the generators x, y, z do not explicitly enter equation (5.1), this passage is straightforward we have to assign a new meaning to the derivatives expressed in the relations (2.3).

Note that in the NC case we meet an additional difficulty. The symbol φ(τ ,x,y,z) is not well defined now, since we have to explicitly fix some ordering of the NC variables x, y and z. To avoid this technical difficulty, we consider a solution depending on the central element τ and the only spacial variable, say x. Otherwise stated, we assume the momentum to be of the form p=(px,0, 0).

Remark 13: This simplification enables us to compute the action of the Klein-Gordon operator on a plane wave (5.2). Note that since the element t, x, y, z belong to the algebra U(u(2)h), this plane wave can be treated as an element of the group U(2) (at least for the small enough coefficients E and p) presented in the coordinates of the first kind. The computation becomes easier if we present elements of this group in the coordinates of the second kind. This means that the function image has the form exp(iEt) exp(-ipxx) exp(-ipy y) exp(-ipz z) . An example of computation with such a function is given in [6].

First of all, we note, that similarly to the classical case


This immediately follows from formula (7.1) of Appendix. So, our equation becomes

image (5.3) eq:KG-1

We look for solutions of this equation in the form


where E and p=px are numeric parameters of the solution (analogs of the energy and momentum of the commutative case). Our goal is to find the explicit form of the functionsimage and image and the dispersion relation, connecting E and p.

Let us demand the function image to be an eigenfunction of the derivative τ ∂ with the eigenvalue E. The action of the derivative ∂τ on a function in τ can be extracted from (2.14). So, we have


The solution of this difference equation reads:

image (5.4) sol

Note that the function image can be presented in the form image where image In the classical limit image the function α(E) turns into E.

In a similar way, the function gp(x) is taken to be an eigenfunction of the operator ∂x with the eigenvalue p:


We take the solution of this equation in the form


In the classical limit we get image

Let us now turn to solving equation (5.3) with the above substitution image. First of all, permuting image and image we have:


Also (see Appendix for detail)


In the above formulae we omit the arguments E and p in the notations x (E) and η( p) respectively. All these formulae follow from the permutation relations of the partial derivatives in τ, presented in Appendix, and the particular form of the functions imageand gp(x) . This allows us to calculate the term image in (5.3).

In order to find the value of the second term image, we use the relation


Finally, we come to the equation:


which, in turn, leads to the dispersion relation, connecting the parameters E and p


On taking into account the explicit form of ξ(E) and η( p) , we simplify the dispersion relation to the form


On solving the equation in E we find


This is the final form of the "quantum dispersion relation" under the above assumption p=px, py=pz=0.

The Schrodinger type equation

Now, let us consider a differential equation of the Schrodinger type

image(5.5) eq-schr

where imageis a quantum radius and a, b, q are some numeric constants. It is convenient to look for image in the form


where v∈Vk and image and image is defined by formula (5.4).

Below we are dealing with the particular case k=0 of this problem. Using the properties of image we get the equation on image:


where ξ(E) is defined in (5.4).

Now, we are interested in question: is there a ground state of the image (similarly to the commutative case) Substituting this exponent into the equation, we get


Collecting separately the terms with and those without it, we get a system of equations to determine σ and E:



image (5.6) gr-energy

This system reduces to the equation on image :

image (5.7) eq-tha



In the commutative limit image the above equation gives in the leading order in image

image (5.8) lim-a

while the leading order of the ground state energy can be extracted from the first line of the system (5.6)

image (5.9) lim-E

These limit values coincide with the corresponding solution of the differential equation


which is the commutative limit of (5.5) with zero value of the orbital momenta. Below, we compute the first order corrections in image to the values E0 and image . We do not know any "physical meaning" of other solutions of the equation (5.7).

The first order corrections in image to the classical results (5.8) and (5.9) can be found from the system (5.6). We present the result of calculations by passing to the physical meaning of the parameters a, b, q entering the equation (5.5). More Precisely, we set image , where pl h is the Planck constant, c is the speed of light, m is the electron mass, image is the electron charge squared. Besides, we must renormalize the energy parameter image . It is convenient to introduce the dimensionles fine structure constant α and the electron De Broglie wave length λB by the relations


With this constant our answer has the following simple form:

image (5.10) E-first

image (5.11) sig-first

The above relations show that noncommutativity of the space-time leads to increasing of the ground state energy as well as to increasing of an average size of the hydrogen atom which is roughly proportional to σ-1 (we assume image to be positive).

As for the eigen function image, the first order correction to it can be easily obtained with the use of (5.10) and (5.11) and is left for the reader.

Concluding Remarks, Discussion

1. First, let us go back to the remark 12. The relation exhibited there means that for any sufficiently smooth compactly supported function f(x, y, z) defined on image we have

image (6.1) int

where the above integral is treated in the sense of the principal value.

A proof of (6.1) is usually based on the Green identities. We slightly modify this proof in order to adapt it to the quantum case.

Since we consider the function image as a distribution, the integral in the left hand side of (6.1) is by definition

image (6.2)

where image and B(0, a) is the ball with the center at 0 and the radius image.

Note that it suffices to assume the function f to be dependent on the radius r only. Otherwise, it can be replaced by


Where dΩ is the surface 2-form of the unit sphere. Note that g(0)=f(0, 0, 0).

Then, in virtue of (4.4), the integral (6.2) can be written as


On passing to the limits R→∞ , ε →0 and assuming the function f to be continuously differentiable at (0, 0, 0) we get the claim.

Heuristically, we can reproduce the computation of (6.2) in the quantum case (using of course (4.3) instead of (4.4)) and get the same result under the following assumptions (actually, definitions)

image and image

Thus, the notion of the integral (including that "in spherical coordinates") can be adapted to the derivative image in the classical manner. We plan to consider a multidimensional analog of this formula related to the algebra U(gl(m)h) in the subsequent publications.

2. Note that we have also considered another basic physical model, namely, 3-dimensional harmonic oscillator. Compared with the hydrogen atom model, the harmonic oscillator is more difficult to deal with. The main difficulty here is in the asymptotic of the wave function of the commutative case image. The direct substitution of the non-commutative analog imageinto the corresponding equation leads to an equation which we cannot solve. Apparently, one should find here some another function in image (with the commutative limit image ) which would be more adequate to the problem in the NC case. Another possible way out is a modification of the harmonic oscillator potential (with recovering its classical form at the commutative limit image→0 ) in such a way that the exponent er~2 would be an eigen function of the modified Hamiltonian.

3. The renormalization of the time variable leads to a change of the step of the corresponding lattice. We do not know what normalization of the time variable and the corresponding step could be "physically motivated".

4. Now, let us compare once more our approach to quantization of dynamical models with that arising from deformation quantization. As we noticed in Introduction, the models introduced in the frameworks of the latter approach make use of the classical derivatives. Only, the usual commutative product of functions on a given Poisson variety is replaced by the corresponding image -product. Note that sometimes such a image -product is introduced via a map (4.7). For instance, this map can be constructed with the help of isotypic components as done in [13]. In this case it can be naturally extended to the completion similar to that

Then the operator describing a given dynamical model can be pulled forward to the quantum algebra by means of this map (see formula (4.8)). However, this way of quantizing dynamical models is equivalent to the previous one and does not give rise to partial derivatives which could be difference operators. Consider the operator image defined in this way on the center of the algebra image (see [GS1]). Since this operator is the image of the derivative image under the map image, it is also a differential operator. By contrast, the operator image constructed in the current paper is a difference operator.


This study (research grant No 14-01-0173) was supported by The National Research University Higher School of Economics Academic Fund Program in 2014/2015.


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