We show that the q-deformation of the Weyl-Heisenberg (q-WH) algebra naturally arises in discretized systems, coherent states, squeezed states and systems with periodic potential on the lattice. We incorporate the q-WH algebra into the theory of (entire) analytical functions, with applications to theta and Bloch functions

Celeghini, E.

Demartino, S.

Desiena, S.

Rasetti, M.

Vitiello, G.

English

NASA Technical Reports Server

N95- 13901
Q-DERIVATIVES, COHERENT STATES
AND SQUEEZING
E.Celeghini z, S.De Martino 2, S.De Siena 2, M.Rasetti s and G.Vitiello 2
i Dipartimento di Fisiea - Universitd di Firenze and INFN-Firenze, I501P.5 Firenze, Italy
2Dipartirnento di Fisica - Universitd di Salerno and INFN-Napoli, 184100 Salerno, Italy
s Dipartimento di Fisica and Unitd INFM- Politeenico di Torino, H0129 Torino, Italy
Abstract
We show that the q-deformation of the Weyl-Heisenberg (q-WH) algebra naturally arises
in discretized systems, coherent states, squeezed states and systems with periodic potential
on the lattice. We incorporate the q-WH algebra into the theory of (entire) analytical
functions, with applications to theta and Bloch functions.
1 Introduction
The general properties of q-algebras [1] [2] have been widely studied, in particular in connection
with specific physical models. In this paper we will show [3] that the q-deformation of the Weyl-
Heisenberg (q-WH) algebra naturally arises in discretized quantum systems, coherent states,
squeezed coherent states and systems with periodic potential on the lattice.
q-algebras are deformations of enveloping algebras of Lie algebras and, like the latter, they
have Hopf algebras properties. The q-deformation of the Weyl-Heisenberg algebra _q-WH),
as well as the WH algebra, is not even a Hopf algebra; it has only the properties of a Hopf
superalgebra [4].
In our study of q-deformations we want to preserve the analytic structure of the correspond-
ing Lie algebras and therefore we need to operate in a frame where analyticity is ensured: this
is guaranteed by working in the Fock-Bargmann representation (FBR). In this representation
it is immediate to show that finite difference operators possess the algebraic structure of q-WH
algebra: As a result we recognize that a q-deformation of the algebra occurs whenever a finite
length is involved in a physical system, the q-parameter being related with the finite spacing.
The q-deformation is also expected in the presence of periodic conditions, since periodicity is a
special form of invariance under finite difference operators.
We use the well known mapping of the q-algebra into the universal enveloping algebra of
a corresponding Lie structure; to be specific, the mapping of finite difference operators into
functions of differential operators, which can be indeed achieved only by operating on C °o
functions, namely by working in the FBR.
We would like to stress that we succeed into incorporating q-deformation of the WH algebra
into the theory of (entire) analytical functions, with specific applications to theta functions and
to Bloch functions, a result which may deserve by itself much attention.
39
https://ntrs.nasa.gov/search.jsp?R=19950007488 2020-06-16T10:28:00+00:00Z
In this paper we will use dimensionless units for all physical quantities.
2 Finite difference operators
The FBR operators, solution of the WH commutation relations [a,a t] = 1, [N,a] --- -a, [N,a t] -
a t , are [5]: d d
N --_ Z_zz , at--* z , a-+ d-'z" (2.1)
The Hilbert space _" is identified with the space of the entire analytical functions. Wave-
oo oo = _" (n • Z+) Thefunctions are expressed as ¢(z) = _,_=o c_,u,_(z), _.,_=o [c-I 2 1, u,_(z) - :_,_,
set {u,(z)} provides an orthonormal basis in _'. The finite difference operator Pq
f(qz)- f(z) f • _" (2.2)
Dq f(z) : (q-l) z '
with q -- c¢, _ • C, may be written on _" as Dq --- ((q- 1)z)-l(q _ - 1). D_ is the well known [6]
[7] [8] [9] q-derivative operator and, for q _ 1 (i.e. _ --_ 0), it reduces to the standard derivative.
We have the algebra
d g] : -D_ [z_ zl = z (2.3)ID_ , z] = qZ_ , [z_, , , ,
and observe that it is nothing but the q-deformation of the WH algebra. In fact, this can be
seen by introducing the following operators in the space 7
d
N--+ zwdz ' _q ---* z , aq ---* Pq , (2.4)
where clearly _q = &q=l = at and litre_.1 aq = a. The quantum version of the Weyl-
Heisenberg algebra is thus realized in terms of these operators {aq, _q, N; q • C) with relations
[1] [21"
[g, aq] = -a_ , [N,_q] = _q , [aq,_q]- aqaq -_qaq = qN. (2.5)
Equivalently, by introducing _q-aqq-N/2,theq-WHalgebraeq._!isrewritteninthemorefamiliar form as [N, aq] = -aq , [g,a,l = aq , a,a, - q _a_a, =
The finite difference operator algebra (2.3) in the FBR thus provides a realization of the
q-WH algebra (2.5).
The notion of hermiticity associated with (2.5) has been studied in ref. [10] in connection
with the discussion of the squeezing of the generalized coherent states (GCS)q, defined in the
usual Fock space 5.
We note that the commutator [aq, hq] acts in _" as follows
[aq,&q]f(z) = q_ f(z) = f(qz) . (2.6)
In conclusion, the strict relation of the q-WH algebra with the finite difference operator
Dq (q ¢ 1) suggests that whenever one deals with some lattice or discrete structure, then a
deformation of the operator algebra acting on 7 should arise.
40
3 Coherent states, theta functions and squeezing
We summarize now the relation of q-WH algebra with the customary coherent states (CS) Iz >
[5], with theta functions and with squeezing. Eq.(2.6) is the key relation to establish our results.
For sake of shortness we only report the relevant relations [3]:
< n[q_r[ z> = exp(-lz-_)un(qz) , (3.1)
< nl[aq,_q]lz> = exp(-(1-r/)(l+q)tZ-_l_) <nlP((q-1)z)[z> ' (3.2)
exp((1-1qJ2)lz-_)[aq,&q]lz > = .qz > , (3.3)
[aq,&q] f(z) = exp(-(1-g/)(l+q)_ -_) /9((1-?/)_)f(z), (3.4)
where P(z) denotes the usual CS generator.
We observe that [%,aq] acts as mapping operator from Iz > to lqz > up to a phase
factor. On the other hand, it acts the z-dilatation operator (z _ qz) in the space of entire
analytic functions. When q = e¢ , with f pure imaginary, f = iO, then [aq,_q] : z _ eiez ,
generates the U(1) group of phase transformations in the z-plane. We also observe that eqs.
(3.2) and (3.3) provide a non linear realization of the quantum algebra (2.5) in terms of a and
a t. Vice-versa, the nonlinear operator P(z) is represented by the linear form [aq, 5q].
In the framework of the formalism of CS on the von Neumann lattice L the defining functional
equation for the theta function is [5]
O_(z + zm) = exp(irF_(-m)) exp([_ _) exp(2mz) O_(z) , (3.5)
with zm = mxwx + m2w2 an arbitrary lattice vector and Fc(m) = rnlrn2 + mxq + m2e2 . A
solution of (3.5) can be expressed as
,.(,)= (,.o)
where f(z) is an arbitrary entire function such that the series (3.6) is converging.
To establish the relation between q-WH algebra and theta functions, we write q = qm = e;'"
with f,_ a vector on the lattice L and, by setting zm = (q,_ - 1)z, we have [3]
8,(q,.,.,z) = [aq,,,,hq,.] O,(z), (3.7)
[%=,5q,,,] O,(z) = exp(iTrF_(-m))exp(-(1-_/m)(l+qm )jz-_)O,(z) ,
O_(z) = _-_exp(-iTrFc(m))exp((1 ?:/m)(1 + qm) 1_ 2)- [aq,,,,aq,,,lf(z ) .
(3.s)
(3.9)
41
Eqs. (3.7-9) show that theta functions span indeed a space of representations for the q-
algebra (2.5).
Finally, we study the relation of q-WH algebra with squeezing. Let _, = -i_ and [_., _,] = i,
over a Hilbert space of states _(z) identified with the space of entire analytic functions _.
I _ _ [a, a t] I. It is immediate toIntroduce the operators a -- _Z_(z 4- i_,) , a t _(_ - il5_) , =
observe that
--exp(  
= =
with S(f) denoting the squeezing generator [11], f = logq the squeezing parameter and _,(z)
the squeezed state. We therefore conclude that the operator [aq,hq] is the squeezing generator
for CS in the FBR, thus confirming the conjecture previously [10] formulated whereby q-groups
are the natural candidates to study the squeezed CS.
4 Quantum mechanics on the lattice
Our purpose is now to show that q-WH algebra is underlying the physics of lattice quantum
systems. Lattice Quantum Mechanics (LQM) is characterized by the E(2) commutator algebra,
which in the momentum space is written as [3] [121
• d , _,sin(k_)] = icos(k_),
[k,,cos(k_)] = [id cos(kE)] = -iEsin(k_) . (4.1)
dk'
[ib,,cos(k,)] = [,-' sin(k,),cos(k,)] = O .
where _, and ib,, denote the (one-dimensional) lattice position operator and the lattice momen-
tum operator, respectively (extension to higher dimensions is straightforward). The correspond-
ing uncertainty relations are
1 (<cosCkE)>_) ' (4.2)
A_C_,)A2CcosCk,)) > _(_Z<sin(kE))_) , (4.3)
where A_(/i) = (/I 2> - (/I) 2 with = (_4) = f dkrP'(k)_4_P(k). We observe that these relations
go, in the continuum limit _ --. O, to the usual ones. In this connection we observe that the
continuum limit is, in fact, an isometric and conform_l mapping of the torus on the plane.
Following the usual procedure [13], the states minimizing the uncertainties (4.2) and (4.3)
are found to be, in momentum space
ffs(k) = G-_ exp ['_cos(_k)- fA_k] (4.4)
42
The normalization constant G is given by G = _I0(2_),/0 denoting the modified Bessel function
of the first kind of order 0. We adopted the notation: A = _-1, ,_ = _E-2 _ _. (_,) "b i_(p, / ,
and "y is connected with the mean square roots of position and momentum.
In the continuum limit, i.e. for small e, one recovers in the space of configurations,
_(z) = (.7_r)-_ exp {- [(2_/)-1(z -- (_;/) _ + i(_l(x -- (_;))] } , which is the minimum uncertainty
wave-function given by SchrSdinger [14]. The @(k)'s are the lattice coherent states.
In order to see the relation with the q-algebra we consider the conformal image _ of the
Hilbert space obtained upon introducing the variable z = e/÷ (¢ = kE , -Tr < _b < It), such that
• d
-s_ = -ie_ = ez_ . The functions in _ are assumed to be entire square-integrable analytic
functions. We have
L,f(¢) = -i_--_f(¢)--zd'f(z)- Nf(z) , f E _ , (4.5)
f(¢ + e) = eiELsf(¢) = qN'f(Z) = f(qz) , (4.6)
with q = e i'. The realization (2.4) has been adopted in the FBR, with z restricted t5 the unit
circle. The E(2) algebra (4.1) is realized by
[L1, Ls]'f (z) - -i L,'f (z) ,
with f E _, and the identifications
[L_,La]'f(z) = iLl'f(z), [L,,L2]'f(z) = O, (4.7)
z+_ z-_ d
L1- 2 , L2- 2i , L3=Z_z z, L+=z, L_ =_. (4.8)
One can see that [aq,aq] is nothing but the group element e_'L_ of E(2). The algebraic
structure of LQM is thus intimately related with the q-WH algebra, the deformation parameter
q being determined by the discrete lattice length e = -ilog q.
We finally note that z'* = e_n_, n integer, is the eigenfunction of Ls associated with the
eigenvalue n of the number operator in the FBR: Lsz'* = Nz n = nz n .
The functions z = e/_ play also a r61e in the Bloch functions theory. Suppose we have a
periodic potential V(xn) = V(xn + e) on the lattice. Bloch theorem ensures the existence of
solutions of the related SchrSdinger equation of the form ¢(x_) = e+'_z"v_(x,_), with v_(z,) =
vk(x,, + e). ¢(x,) is the Bloch function. Let us limit ourself to consider for simplicity the plus
sign in the exponentials. ¢(x_) has the property
(4.9)
The choice of the variable z = e ik_ turns out to be natural in the case of periodic potentials:
(4.10)
Since z" = (z,) k and qN(z,_)k = (qzn)_ = e,kd,+,) = z,+,,
(4.11)
which shows that the Bloch functions provide indeed a representation for the q-WH algebra.
43
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44


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