8 January 2001

Physics Letters A 278 (2001) 239–242www.elsevier.nl/locate/pla

On the ubiquity of non-unitary quantum canonicaltransformations

E.D. Davis∗, G.I. GhandourDepartment of Physics, Faculty of Science, University of Kuwait, P.O. Box 5969, Safat, Kuwait

Received 25 May 2000; accepted 17 November 2000Communicated by P.R. Holland

Abstract

We present examples in support of the conjecture that quantum canonical transformations which are used to establish theequivalence or duality of different quantum systems and which have non-linear classical counterparts will be non-unitary or,more generally, linear isomorphisms but not isometries. 2001 Elsevier Science B.V. All rights reserved.

PACS:03.65.CaKeywords:Quantum canonical transformations; Unitarity; Duality

The founding fathers of quantum mechanics identi-fied the quantum analogues of canonical transforma-tions asunitary transformations of the position andmomentum operators which leave the canonical com-mutation relations unchanged [1,2]. However, it hasbeen observed that the natural definition of quantumcanonical transformations as any changes of the non-commuting phase space variables which preserve thecanonical commutation relations is purely algebraic incharacter and hence such transformations are not in-trinsically unitary [3]. In fact, several of the familiartools for obtaining closed form solutions in quantummechanics and quantum field theory can be construedas quantum canonical transformations and few of theseare unitary [3–6].

In the last few years, there have been several in-vestigations using canonical transformations to provethe equivalence or duality of different field and string

* Corresponding author.E-mail address:davis@kuc01.kuniv.edu.kw (E.D. Davis).

theories [7–11]. It has been argued [3] that, for thepurpose of establishing equivalence of two mod-els, a quantum canonical transformation should bea linear norm-preserving isomorphism between therespective Hilbert spaces (an isometry). Althoughthis is certainly a sufficient requirement, we demon-strate in this Letter that it is somewhat too restric-tive by constructing counterexamples. To this end,we consider non-linear (time-independent) classicalcanonical transformationsq,p → Q,P for whichthe transformed Hamiltonian functionH(Q,P ) ≡H(q(Q,P ),p(Q,P )) is the same as the originalHamiltonian functionH, i.e., H(Q,P ) = H(Q,P ).For systems of one degree of freedom, these trans-formations map trajectories (level curves ofH) ontothemselves and, thus, amount to evolutions. Althoughthe correspondingquantum theories are trivially equiv-alent, we show that the quantum canonical transforma-tions which relate them can be non-isometric, or, moreprecisely, non-unitary (since these transformations actwithin a single Hilbert space).

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(00)00785-4

240 E.D. Davis, G.I. Ghandour / Physics Letters A 278 (2001) 239–242

Specialization to a state of definite energy (withwavefunctionΨα(q, t)= e−(i/h)Eαtψα(q)) of the stan-dard integral equation for the time evolution of wave-functions in terms of the propagatorK(q, t;q ′, t ′) =〈q, t|q ′, t ′〉 yields the integral equation

(1)

ψα(q)= e(i/h)Eα(t−t ′)∫K(q,q ′|t − t ′)ψα(q ′) dq ′,

where use has been made of the fact that, for con-servative systems (for which stationary states exist),the propagatorK(q, t;q ′, t ′) must be a function ofthe time differencet − t ′ — i.e., K(q, t;q ′, t ′) =K(q,q ′|t − t ′). The existence of a class of canoni-cal transformations (labelled by the continuous para-meter τ ) which leave the Hamiltonian function un-changed also implies an integral relation for stationary-state wavefunctionsψα , namely

(2)ψα(q)=Nα(τ )∫e(i/h)F (q,Q|τ )ψα(Q)dQ,

whereF(q,Q|τ ) is a (c-number) quantum generatingfunction which reduces in the limith→ 0 to the clas-sical generating functionF(q,Q|τ ) of these canonicaltransformations [12]. Although (2) is of the same formas (1) (withτ replacing the time differencet − t ′), it isthe purpose of this Letter to show that there can be dif-ference in the nature of the kernels: whereas the kernelK(q,q ′|t − t ′) in (1) must be unitary, we conjecturethat the kernelK(q,Q|τ ) = e(i/h)F (q,Q|τ ) in (2) doesnot have to be.

This distinction will be visible in the nature of thereciprocalsNα(τ ) of the eigenvalues ofK(q,Q|τ ).If the kernelK(q,Q|τ ) is unitary, then, as in (1),the dependence ofNα(τ ) on the choice of state|α〉must appear in a (complex) multiplicative factor ofmagnitude unity.

For simplicity, we limit ourselves to quantum sys-tems with Hamiltonian operator

Hq ≡ H(q,h

i

∂

∂q

)=− h

2

2m

∂2

∂q2 + V (q)

corresponding to a particle of massm moving in onedimension in the potentialV . If we substitute forthe energy eigenfunctionψα(q) (eigenenergyEα) inHqψα(q) = Eαψα(q) using (2), replace the productEαψα(Q) by HQψα(Q), and then, after two integra-tions by parts, appeal to the completeness of theψα ,

we find that theF(q,Q|τ ) should satisfy

H

(q,h

i

∂

∂q

)e(i/h)F (q,Q|τ )

(3)= H(Q,− h

i

∂

∂Q

)e(i/h)F (q,Q|τ )

provided the bilinear concomitant

(4)e(i/h)F (q,Q|τ )∂

∂Qψα −ψα ∂

∂Qe(i/h)F (q,Q|τ )

vanishes at the endpoints of integration.To facilitate identification of solutions of (3), we

introduce the decompositionF = F + ihf , wherethe classical generating functionF is, by assumption,a solution of

(5)1

2m

(∂F∂q

)2

+ V (q)= 1

2m

(−∂F∂Q

)2

+ V (Q).

Eq. (5) is the form the conditionH(Q,P ) ≡H(q(Q,P ),p(Q,P )) takes in terms ofF whenH = H. The equation determining the quantum cor-rectionf is

2

(∂F∂q

∂f

∂q− ∂F∂Q

∂f

∂Q

)+ ih

[(∂f

∂q

)2

−(∂f

∂Q

)2]

(6)− ih(∂2f

∂q2− ∂2f

∂Q2

)= ∂

2F∂q2− ∂

2F∂Q2

.

Difficulties associated with the construction off canbe side-stepped if the inhomogeneity on the right-hand side of (6) vanishes: thenf can be taken tobe independent ofq andQ and can, without loss ofgenerality, be set to zero (by, in effect, absorbing itintoNα(τ )). Accordingly, we look for solutions of (5)of the form (q± ≡ (q ±Q)/2)

(7)F(q,Q|τ )=F+(q+)+F−(q−),which, for any choice ofF+ and F−, guaranteesthat the inhomogeneous term in (6) disappears. Theexpression in (7) is the most general consistent withthe requirement that (6) be homogeneous.

Substitution of (7) into (5) yields the followingrelation betweenV and theF±:

(8)

1

2mF ′+(q+)F ′−(q−)= V (q+ − q−)− V (q+ + q−).

E.D. Davis, G.I. Ghandour / Physics Letters A 278 (2001) 239–242 241

From inspection of derivatives of (8) with respectto q− evaluated atq− = 0, we infer that for a po-tentialV with non-zero derivatives of all orders to becompatible with (8), it must be such that

(9)V ′′′(x)= ρV ′(x),whereρ is a real non-zero constant of either sign. Us-ing (9) and its derivatives to sume a Taylor series ex-pansion of the right-hand side of (8) in powers ofq−,we deduce that the right-hand side of (8) can be rewrit-ten as the separable form−(2/√ρ)V ′(q+)sinh

√ρq−.

After dropping constants of integration (which couldinstead be absorbed intoNα(τ )),

(10)F+(x)=− 4m

F (2)− (0)V (x)

and

(11)F−(x)= F(2)− (0)ρ

cosh√ρx.

The non-vanishing of V ′(x) assumed impliesF (2)− (0) 6= 0 but it is otherwise not fixed (because (8)involves the productF ′+F ′−).

As anticipated above, we have found for any po-tential V satisfying (9) not one but an infinite classof generating functionsF = F+ + F− distinguishedby the value of the arbitrary factorF (2)− (0). We iden-

tify F (2)− (0) with the continuous labelτ introduced inconnection with (2). We note that, with this choice,the identity transformation is recovered in the limitτ→∞.

Other potentials V and generating functionsF(q,Q|τ ) are consistent with (8) if we relax our as-sumption about the number of finite non-zero deriv-atives ofV [13]. However, the corresponding trans-formations are linear and so their quantum equiva-lents are unitary [14,15]. By contrast, the transforma-tions with the generating functions implied by (7), (10)and (11) are non-linear and so could conceivably benon-unitary at the quantum level [16,17].

In (10), we have the freedom to fixV so that, con-sistent with (9),V ′′(x) = ρV (x). The dependence ofour generating functions onq andQ is then such that

(12)

∂2

∂q2F(q,Q|τ )=ρ

4F(q,Q|τ )= ∂2

∂Q2F(q,Q|τ ).

Settingτ = τ (z)≡ τ0e√ρz/2, we also have that

(13)∂2

∂z2F(q,Q|τ (z))= ρ

4F(q,Q|τ (z)).

As the similarity of (12) and (13) suggests, it ispossible to chooseV andτ0 so thatF(q,Q|τ (z)) isunchanged whenz is interchanged with either ofqandQ, i.e.,

(14)F(q,Q|τ (z))=F(z,Q|τ (q))=F(q, z|τ (Q)).

Among such choices [13] is the sinusoidal potentialVs(q)= V0 cos2aq with τ0= 4a

√mV0.

Used in conjunction with (2), (14) implies that

ψα(q)

Nα(τ (z))=∫e(i/h)F(q,Q|τ (z))ψα(Q)dQ

=∫e(i/h)F(z,Q|τ (q))ψα(Q)dQ

= ψα(z)

Nα(τ (q)),

or that

(15)Nα(τ )= Cα

ψα(z),

where z = z(τ ) ≡ ln(τ/τ0)2/√ρ . The constantCα

can be fixed by the requirement that we recover theidentity transformation in the limitτ→∞, i.e.,

(16)limτ→∞Nα(τ )e

(i/h)F(q,Q|τ ) = δ(q −Q)for each state|α〉. The nature of the energy eigenfunc-tionsψα for the potentials for which (15) holds is suchthat the state dependence ofNα(τ ) will not be con-fined to a factor of modulus unity. The correspondingkernelsK(q,Q|τ ) are thus non-unitary.

By way of illustration, we consider the sinusoidalpotentialVs(q) alluded to above. For the denumerableset of energy eigenfunctionsψα(q) proportional to theMathieu functions [18]ceα(aq,U0) (α = 0,1,2, . . .)and se|α|(aq,U0) (α = −1,−2, . . .), where U0 =mV0/(ha)

2, application of (15) yields (with√ρ =

2ai)

(17)Nα(τ )= Cα

M(1)α (ln[τ/(4ha2

√U0)],U0)

,

whereM(1)α (x, η) denotes (in the notation of [18]) the

modified Mathieu functionsMc(1)α (x, η) (α > 0) and

242 E.D. Davis, G.I. Ghandour / Physics Letters A 278 (2001) 239–242

Ms(1)|α| (x, η) (α < 0). The limit in (16) is achieved

if Cα = i |α|a/(2π), but irregardless of the choiceof Cα , it should be apparent that the state dependenceof Nα(τ ) in (17) cannot be transcribed as a factorof modulus unity. The integral equations betweenMathieu functions implicit in this analysis turn out tobe equivalent to (20.7.34) and (20.7.35) in [18].

We believe that the non-unitarity of the quantumcanonical transformations we have discussed is a man-ifestation of their non-linearity at the classical level.We conjecture that any other quantum canonical trans-formation relating equivalent or dual systems whichhas a non-linear classical counterpart will also be non-unitary or, more generally, isomorphic but not isomet-ric. As classical canonical transformations are generi-cally non-linear, there should be many quantum trans-formations of this nature although their explicit con-struction may be difficult (because the quantum cor-rectionf will, in general, be non-zero). These obser-vations should apply to systems of any number of de-grees of freedom including field theories. In fact, manyof the specific results in this Letter generalize automat-ically [13] to theories of a scalar fieldϕ(σ, τ ) in 1+ 1dimensions with the first quantized Hamiltonian func-tional

H [ϕ,π] = 1

2

∫ [π2+ (∂ϕ/∂σ)2]dσ + ∫ V (ϕ) dσ,

where π is the field momentum conjugate toϕ(σ denotes the spatial dimension andτ the time innatural units such thath= 1= c).

References

[1] M. Born, W. Heisenberg, P. Jordan, Z. Phys. 35 (1926) 557.[2] P.A.M. Dirac, Proc. R. Soc. London Ser. A 110 (1926) 561.[3] A. Anderson, Ann. Phys. 232 (1994) 292.[4] G.I. Ghandour, Phys. Rev. D 35 (1987) 1289.[5] T. Curtright, Quantum Bäcklund transformations and confor-

mal algebras, in: L. Chau, W. Nahm (Eds.), Differential Geo-metric Methods in Theoretical Physics: Physics and Geometry,Plenum Press, New York, 1989, pp. 279–289.

[6] T. Curtright, G.I. Ghandour, Using functional methods tocompute quantum effects in the Liouville model, in: T.Curtright, L. Mezincescu, R. Nepomechie (Eds.), QuantumField Theory, Statistical Mechanics, Quantum Groups andTopology, World Scientific, Singapore, 1992, hep-th/9503080.

[7] T. Curtright, C. Zachos, Phys. Rev. D 52 (1995) 573.[8] J. Maharana, H. Singh, Phys. Lett. B 380 (1996) 280.[9] Y. Lozano, Phys. Lett. B 399 (1997) 233;

Y. Lozano, Mod. Phys. Lett. A 11 (1996) 2893.[10] A.J. Bordner, Phys. Rev. D 55 (1997) 7739.[11] M.M. Sheikh-Jabbari, Phys. Lett. B 474 (2000) 292.[12] P.A.M. Dirac, The Principles of Quantum Mechanics, 4th ed.,

Clarendon Press, Oxford, 1959, Section 32.[13] E.D. Davis, G.I. Ghandour, in preparation.[14] C. Itzykson, Comm. Math. Phys. 4 (1967) 92.[15] M. Moshinsky, C. Quesne, J. Math. Phys. 12 (1971) 772.[16] P.A. Mello, M. Moshinsky, J. Math. Phys. 16 (1975) 2017.[17] J. Deenen, M. Moshinsky, T.H. Seligman, Ann. Phys.

(NY) 127 (1980) 458.[18] G. Blanch, Mathieu functions, in: M. Abramowitz, I.A. Stegun

(Eds.), Handbook of Mathematical Functions, NBS, Washing-ton, 1964.