http://www.ictp.trieste.it/~pub_offIC/97/143

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUANTIZED HAMILTONIAN SYSTEMS

A. Shafei Deh AbadInstitute for Studies in Theoretical Physics and Mathematics (IPM),

P.O. Box 19395-5746, Tehran, Iran1

Department of Mathematics, Faculty of Sciences, University of Tehran,Tehran, Iran2

andInternational Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTESeptember 1997

Mailing address. E-mail: shafei@netware2.ipm.ac.ir2Permanent address.

ABSTRACT

In this paper we first characterize the Lie algebra of derivations of the 3-dimensional

Manin quantum space as the semi-direct product of the Lie algebra of its inner derivations

and the 3-fold generalized Virasoro algebra with central charge zero. Then we consider

Hamiltonian systems on the quantum plane and we prove that the set of Hamiltonian

derivations is a Virasoro algebra with central charge zero. Moreover, we show that the

only possible motions on the quantum plane come from quadratic Hamiltonians and we

find the solutions of the correspondig Hamilton equations explicitly.

0 Introduction

Classical and quantum mechanics on q-deformed spaces have been studied by some

authors. Most of these works are concerned with Hamiltonian systems [4,5,6], but there

are also some works about Lagrangian formalism on the quantum plane [5,6]. To costruct

the classical mechanics on the quantum plane one can use the q-deformed symplectic

structure obtained by the q-deformation of the natural symplectic structure of the plane

and obtain the equations of motion in the form dx/dt = {H,x}q, dp/dt = {H,p}q [4,5].

Unfortunately the q-deformed Poisson bracket has nothing in common with the usual

Poisson bracket, unless it is bilinear, and the only use of it is in writing the equations of

motion as above. But most of the very interesting facts of classical mechanics are absent

here. It is unfortunate that, in general it is not true that {H, H}q = 0, and {H, f}q = 0

does not imply {H2, f}q = 0.

It is well known that there are two approaches to classical mechanics based on the

symplectic structure of the phase space [1,2]. The first is the state approach and the

second is the observable approach. In these approaches the coordinate and the momentum

functions appear like other observables and they all satisfy the same equation. A new

interpretation of the quantum spaces is given in [3]. According to this interpretation the

two approaches to classical mechanics are also suitable for the case of quantum spaces. To

be more precise, let Mq - for the notations and conventions see next pages - denote the A-

algebra of Q-meromorphic functions and π be the canonical q-deformed Poisson structure

on Mq. By a Hamiltonian system we mean a triple (Mq, π, z), where z is a π-Hamiltonian

element of Mq. Here also, just like the ordinary case, in the observable approach by the

motion of the above system we mean a strongly differentiable one-parameter local group

of automorphisms of the system φt, satisfying the following condition

VfeMq - ^ 1 = {z,φt(f)} , φ0(f) = f,

and in the state approach by the motion of the system with initial values (x,p) we mean

a path t —> (x(t) p(t)) in R2 such that for each f e Mq

df(x(t) p(t)) = {z,f}(x(t) p(t)) , x(0) = x , p(0) = p.

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In this paper following [4,5] we follow the observable approach, and accept that the

mass m of a particle moving on a stright line, like its coordinate and momentum is an

operator and we have the following commutation relations

mx = qxm , mp = qpm.

See also [4,5]. We mention that according to the state approach on quantum plane, the

mass m is a real number. So, the two above mentioned approaches to classical mechanics

on quantum spaces are not equivalent. As we will see, we cannot consider an arbitrary

element of Mq as a Hamiltonian. Indeed, to each Hamiltonian element on Mq there

corresponds a unique Hamiltonian derivation and the set of all these derivations is a

Virasoro algebra with central charge zero V. But as we will see for a general Hamiltonian

in our sense the corresponding Hamilton equations do not define any motion in general.

But when we restrict ourselves to the Hamiltonian systems (Mq,π,z), with z in the

subalgebra sl (2 , A) of V and only in this case the Hamilton equations give the motion

of the system and then one can easily see that the only possible motions on the quantum

spaces are those coming from quadratic Hamiltonians. Clearly these motions are of very

restricted types. So, we should look for other quantum manifolds to have motions of other

types.

This paper consists of 3 sections. The structure of the Lie algebra of derivations of

the 3-dimensional Manin quantum space Mq is considered in section 1. In this section we

show that the Lie algebra of derivations of Mq is the semi-direct product of the Lie algebra

of inner derivations of this algebra and the 3-fold generalized Virasoro algebra with central

charge zero, i.e. a Lie algebra generated by the family {Lmi | m = 1,2,3 and i E Z} and

the multiplication rule

[Lmi, Ln j] = jLn i+j — i Lm i+j.

Section 2 is devoted to Hamiltonian systems on Mq. In this section we will define the

notion of Hamiltonian derivation. Then we will prove that the set of all such derivations

is a Virasoro algebra with central charge zero. Finally, the Hamiltonian equations on

(A { , }q) will be solved in Section 3.

Before going further we recall that in this paper by A we mean the C-algebra of all

absolutely convergent power series J2i>_oo ciqi on ]o , 1] with values in C. Consider the

following commutation relations between x,p,m,

xaxb = xa+b,papb = pa+b, mamb = ma+b,paxb = qabxbpa, maxb = qabxbma, mapb = qabpbma,

where a and b are in Z. By Mq we mean the A-algebra of Q-meromorphic functions of

the form

where the sign " 3> " under the " Σ " means that the indices i,j, k are bounded below. By

using the above commutation relations one can easily see that Mq is closed under Cauchy

product. Therefore, it is an A-algebra. The value of an element z G Mq written in the

above form at a point (r s t) in R3 is

ij,fc>-oo

which is absolutly convergent by definition. The concept of " Q-meromorphic function" is

a generalisation of the concept of "Q-analytic function" given in [3].

The subalgebra of Mq consisting of all

z = 12 aijk(q)xipjmk,ij>0,fc>-oo

and the subalgebra consisting of all

z= 12 ci(q)mk

fc>-oo

will be denoted respectively by A and M. We also recall that all the above mentioned

function spaces are endowed with the natural locally convex structures. Finally, through-

out the paper the sign " — " on a " Σ " means that the " Σ " is with finite support.

1 Derivations of Mq

Let D : M q> Mq be a derivation. Assume that

and D(m) = Y,i,j,k->-oocijk{q)xlp' mk+1. Then, direct calculation shows that,

(qk ~ qi)aijk = (qk+j - 1)bijk, (qi+j - 1)bijk = (qi - qk)cijk, (qj+k - 1)cijk = (1 - qi+j)a

Therefore, for q ^ 0,

k =£ 0 and aijk =£Q4^k + j=£Q and bijk =£Q4^i+j=£Q and cijk =0 0.

The derivation D is called of type 1 if, for i = —j = k, aijk = bijk = cijk = 0. The set

of all derivations of type 1 will be denoted by D1.

Let D b e a derivation of type 1 and let D(x) , D(p) and D(m) be as above.Let

z= > . . . , „ \q I J - 1 ankxr/m

Clearly z is a well-defined element of Mq and we have

= D(m).

Therefore, for each y e Mq , D(y) = [z, y]. Since elements ofMq of the form z = Y,i<^iX%p~

are in the center of Mq, each derivation of type 1 is an inner derivation and vice versa.

Let D : Mq —> Mq be a derivation. Assume that

D(x) = J2 alxl+lp-lml D(p) = TI btx

lpl-lml D(m) = Ti ctxlp-lml+l.

Then D is called a derivation of type 2. Direct calculation shows that in this case we have

D(x)x = xD(x) qD(x)p = pD(x) qD(x)m = mD(x) qxD(p) = D(p)x D(p)p = pD(p)

qD(p)m = mD(p) qxD(m) = D(m)x qpD(m) = D(m)p D(m)m = mD(m).

Moreover, as one can easily see, a derivation D : Mq —> Mq is of type 2 if and only if it

satisfies one of the above 9 relations.

The set of all derivations of type 2 will be denoted by D2

Let

A = ^ . ai(q)xl+lp~lml , B = J^. bi(q)xlpl~lml and C = J^. Ci(q)xlp~lml+l.

Then the linear operators

given by

D1(xapbmc) = ]T x

iAxjpbmc , D2(p) = ]T xapiBpjmc , D3(m) = J2 xapbmiCmj,

and

D1(p) = D1(m) = 0 , D2(x) = D2(m) = 0 , D3(x) = D3(p) = 0,

are derivations of type 2.

We call D1 an x-derivation, D2 a p-derivation and D3 an m-derivation. It is clear

that each derivation of type 2 , D : M q —M q can be written uniquely as the sum of an

x-derivation D1, a p-derivation D2, and an m-derivation D3 , and

D1(x) = D(x) , D2(p) = D(p) , D3(m) = D(m).

We mention that if D : Mq —>• q is a derivation of type 2, then for each monomial

z = xapbmc we have zD(z) = D(z)z.

Let

O d ddxJ dp'1 :M q q

be linear operators given by

— (xipjmk) = ixt~lp3mk , —(xtpjmk) = q~tjxtpj~lmk.

and

d m " l J~H

Assume that D : Mq -^ Mq is a derivation. Then D can be written uniquely as

dD = D(x)— + Dip)— + Dim)- .

dx dp dm

Because, for a, b and c in N we have

D(xapbmc) = xaD(pb)mc + D(xa)pbmc + xapbD(mc)

= bxapb-lD{j))mc + axa-1D(x)pbmc + cxapbmc-lD{m)

= q-a[bD(p)xapb-1mc] + axb-1D(x)pbmc + q-{i+j)cD(m)xapbmc-1

= D(p)lp + D(x)^ + % ) A ] ( ^ ) .

If D : M q -̂ - q is a derivation of type 2, then

. D2 = B ( P ) | . D M .

-j^ , D(p)^are derivations. To prove this, it is sufficient to prove that D(x)-j^ , D(p)^- and

are derivations. Let a, b, c, r, s and t be in N. Then

D{x)^{xapbmcxrpsmt) = qr^+sc

= qr(b+c)+sc(a + r)D(x)xa+r~ 1pb+smc+t

= xapbmcD{x)^{xrpsmt) + D(x)-^(xapbmc)xrpsmt.

In the same way we see that D(p)^- and D(m)-£^ are derivations.

The set of all derivations of Mq which is clearly an A-Lie algebra will be denoted by

D.

Now it is clear that D2 with

is an A-Lie algebra. Moreover, each derivation of Mq can be written uniquely as D = D1 + D2,

where D 1 2?i and D2 e D2. Furthermore, D2 is an ideal ofD. Therefore, D is the semi-

direct product ofD1 andD2.

Let Xi = (q2ix)i , Pj = (q2jp)j , Mk = (qk2m)k and

L1i = XiP~iMi , L2_j = XjP~jMj and L3k = XkP~kMk.

Then clearly we have

\jl jn ] ATn nil

Therefore D2 is a 3-fold Virasoro algebra with central charge zero.

2 Hamiltonian systems on the quantum plane

In this section we endow Mq with the canonical q-deformed Poisson structure

_i d d 1 d dox op op ox

The associated q-deformed Poisson bracket will be denoted by { , } q . More precisely, for

each two elements f,g£ Mq we have

U>g^-q dxdp q dpdx

An element z G Mq is called Hamiltonian if the mapping

Xz:Mq^> Mq

defined by

Xz(f) = U , f}

is a derivation. In this case Xz is called a Hamiltonian derivation.

Assume that for z £ Mq, Xz is a derivation. Since Xz(m) = 0 it is necessarily a

derivation of type 2. Let Xz be a derivation. Suppose that z = J2i,j,k->-ooaijk(Q)x'lp'mk.

ThenXz(x) = - E*J)fc jaijk(q)q-i+1/2xip>-1mk , Xz(p) = Y,l,],kial]k{q)q-1l2x%-1p'mk, and

Xz(m) = 0. But since Xz is a derivation of type 2 as we have seen i = k + 1 and j = 1 — k.

Conversely assume that

Then

A = Xz( x ) = , ak(q)q 2 (1 - k)x + p mk ,

B = Xz(p) = V ak(q)q^L(l + k)xkp1~kmk,

and C = Xz(m) = 0. Hence Xz is a derivation of type 2. Therefore z £ Mq , Xz is

Hamiltonian if and only if it is of the above form.

The set of all Hamiltonian elements of Mq will be denoted by H(Mq). It is clear that

H(Mq) is an A-module. Let z1 = xk+lpl~kmk and z2 = xl+lpl~lml. Direct computation

shows that

{z1,z2}q = 2(k - i)q-W+mxi+{k+i)pi-{k+Dmk+i_

Therefore for each two elements z1 , z2 in H(Mq), {z1, z2}q E H(Mq). Moreover {z1, z2}q =

-{z2 , z1}q. Now let z1 and z2 be as above and z3 = xn+1p1-nmn. Then

{{z1, z2}q , z% + {{z2 , z% , z% + {{z3 , z% , z2}q = 0.

Therefore (H(Mq) { , }q) is an A-Lie algebra, with centre M. Let z1 and z2 be as above.

Then

- k - Z)g-(1+fe+z+fcZ)x1+(fc+zV"(fc+0^fc+z = [Xzi Xz2](x),

,z2}q(p) = 2(fc l ) ( 1 + k + l)q

and

From the above considerations we see that the mapping

X : H(Mq) -+ V

given by X(z) = Xz is a homomorphism of A-Lie algebras with kernel M.

Let zn E H(Mq) be defined as follows

zn = l/2q1^1xl+npl-nmn, n E Z.

Then {zm , zn} = (m - n)zm+n. Therefore

The Lie algebra of Hamiltonian derivations of Mq is the Virasoro algebra with central

charge zero.

This algebra will be denoted by V.

Let z be in H(Mq). Then for each f in Mq we have

df , x dfq qdx qdp

Because,

By a Hamiltonian system on the quantum plane we mean a triple (Mq,π,z), where

vr is the canonical q-deformed Poisson structure on Mq and z G H(Mq). Let φt be

a strongly differentiable one-parameter local group of automorphisms of the q-deformed

Poisson structure (Mq, π). We say that φt defines the motion of the system (Mq, π, z), if

for each f G Mq

dtt = {zt, ft}q , f0 = f,

where for each f G Mq, φt(f) is denoted by ft.

Proposition A necessary and sufficient condition for φt to define the motion of the

Hamiltonian system (Mq,π,z) is that for each t, zt G H(Mq) and xt , pt and mt satisfy

the following equations

dxt ( -, dpt ( -,

Proof. The condition is clearly necessary. To prove that the condition is sufficient,

assume that xt, pt and mt satisfy the above equations and zt G H(Mq). Then

dxt r -, dpt dmt

xt{zt,xt}q Pt-jj: =Pt{zt,Pt\q mtdmdtt = mt{zt,mt}q.

As we have seen earlier for each t, Xzt is a derivation. Therefore

dxxt{zt,xt}q = xtXzt(xt) = Xzt(xt)xt = dxdttxt

pt{zt ,pt}q = ptXzt(pt) = Xzt(pt)pt = dpdttpt

mt{zt,mt}q = mtXzt(mt) = Xzt(mt)mt = dmt

Now let f = xipjmk. Then ft = xt

iptjmt

k and

dftdt dt dt dt

fO

^t fa + X z t { m t ) f a t = Xzt(ft) = {zt, ft}q.Therefore for each f e Mq we have ^ = {zt, ft}q.

Now since for each t, {zt, zt}q = 0 therefore ^ = 0. This means that z is an invariant

of motion. It is easy to see that any analytic function of z is also an invariant of motion.

Notice that the Hamilton equations on Mq in general do not define a motion of the

corresponding Hamiltonian system.

3 Hamiltonian Systems on Aq

Clearly the subalgebra of V generated by

zo = 1/2q1/2xp , z1 = 1/2x2m , z_x = l/2p2m"1,

is sl(2 , A). Now consider the Hamiltonian system (Aq, π, z), where z = az-\ + βz1 + γzo,

and α , β and γ G A. The corresponding Hamilton equations are

or in the matrix form

dxt dpt(xt pt)

By solving this linear differential equation with constant coefficients we obtain

xt = coshθtx - 0-lsipt = cosh θtp + e-hinh et(q-1/2f3xm + q~1/2rp)

where θ = (g~1/272 — αβ)1/2 , x = x0 , p = p0.

Now let q be a constant complex number and let z1/2 denote the non-principal branch

of the second root of z. Then

1. Let α = 1 and β = γ = 0. In this case we have

xt = x- ql/2prn~l , pt=p.

2. Let α = 1 β = ω2 and γ = 0. In this case we have

xt = xcosωt — q~1l2p{iom)~ sinωt

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pt = pcos ωt + q l' 2ωxmsin ωt.

Notice that in these two special cases the slight difference between our results and

those in [4] comes from the difference between the definitions of the q-deformed Poisson

structures given in [4] and also the difference between Hamiltonians come from different

rules of differentiation.

Acknowledgements.

This work was supported by the University of Tehran and the Institute for Studies

in Theoretical Physics and Mathematics(IPM), Tehran, Iran. Part of this work has been

done during the author's stay at the ICTP. He would like to thank Prof. Virasoro, Prof.

Narasimhan, Prof. Kuku and others for their hospitality.

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References

[1] R. Abraham and E. Marsden, Foundations of Mechanics, (1978).

[2] V. Arnold, Mathematical Methods of Classical Mechanics, Mir Publ.(1975).

[3] A. Shafei Deh Abad and V. Milani, J M P 35 (1994) 5074.

[4] I. Ya. Arefeva and I. V. Volovich, Phys. Lett. B, 268 (1991) 179.

[5] J. R. P. Malik, Phys.letter B, 316 (1993) 257.

[6] M Lukin, A Stern and I Yakushin, J. Phys. A: Math. Gen. 26 (1993) 5115.

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