Post on 22-Sep-2020
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.
2
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
11
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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