Generalized (s-Parameterized) Weyl Transformation

a r X i v :q u a n t -p h /0208055v 3 11 N o v 2003Generalized (s-Parameterized)WeylTransformationAlex Granik ∗AbstractA general canonical transformation of mechanical operators of po-sition and momentum is considered.It is shown that it automat-ically generates a parameter s which leads to a generalized (or s -parameterized)Wigner function.This allows one to derive a general-ized (s -parameterized)Moyal brackets for any dimensions.In the clas-sical limit the s -parameterized Wigner averages of the momentum and its square yield the respective classical values.Interestingly enough,in the latter case the classical Hamilton-Jacobi equation emerges as a consequence of such a transition only if there is a non-zero parameter s1IntroductionThe Moyal transformation in the context of the Weyl transformation [1](the former being a particular case of the latter)was addressed by B.Leaf [2]who departing directly from quantum mechanics derived the Weyl-transform A w (Q ,K )of the quantum operator A (Q ,K )and investigated the properties of such a transform.Here Q and K denote the eigenvalues of the coordinate q and momentum k operators respectively.The well-known Moyal formula for the phase-space distribution function [3]readily followed from the de-rived expressions [2].However the work [2]did not yield a more general s -parameterized transformation unifying different quantization rules.Usually this transformation is introduced ”by hand”(cf.[4],[5],[6].A closer inspection of Leaf’s approach shows that it allows one to naturally arrive at the generalized (in a sense of s -parameterization)Weyl-Wigner-Groenewold-Moyal transformation without a need for apriori introduction of the param-eter s .This is associated with the fact that a shift of the operators K and Q (K ,Q →K ′,Q ′)under the only condition that the resulting transformation to be canonical automatically generates an arbitrary parameter s entering the resulting transformation.In Ref.[2]this shift was chosen in such a way as to satisfy the canonicity by a special choice of the numerical coefficients entering the transformation K ,Q →K ′,Q ′and ensuring the value of the parameter s =0.2Generalized Weyl TransformationLet us consider the Hilbert space of a quantum-mechanical system having n degrees of freedom.This space is spanned by the eigenkets |Q >and |K >of the Cartesian coordinate operator q (q 1,q 2,...,q n )and the conjugate momentum p =(¯h /i )(∂/∂q )=2π¯h k (k 1,k 2,...,k n )[with the commutation relations q i k j −k i q j =(i/2π)δij ;i,j =1,2,...,n ]:k |K =(2π¯h )−1¯h ∂q|K =K |K(1)which means|K =e 2πi q •K ;|Q =δ(q −Q )(2)The respective completeness relations are d Q |Q Q |=1, d K |K K |=1(3)Employing (1)and (2)we find that the scalar product <Q |K >isQ |K =e 2πi q •K δ(q −Q )d q =e 2πiQ •K(4)We represent an arbitrary quantum-mechanical operator A in the Hilbert space using the completeness relation (3):A ≡ ...d Q ′d Q ′′d K ′d K ′′|Q ′′ K ′′|K ′′ K ′′|A |K ′ K ′|Q ′ Q ′|(5)2Now we perform a linear transformation from the variables Q ′,Q ′′,K ′,K ′′to new variables K ,Q ,u ,v according to the followingQ ′′i =Q i +αi v i ,Q ′i =Q i +βi v i(6)K ′′i =K i +γi u i ,K ′i =K i +δi u i ;i =1,2..,.n(7)where αi ,βi ,γi and δi are some constants to be determined from an additional condition.If we require this transformation to be canonical then its Jacobian must be 1yielding the following relations:i =ni =1(αi −βi )(γi −δi )=1(8)Note that in [2]from the very beginning the coefficients are chosen in such a way as to identically satisfy the canonicity condition:αi =γi =1/2;βi =δi =−1/2.We rewrite identity (5)taking into account the transformation of variables (6),(7):A ≡...d Q d K d u d v |Q +αv Q +αv |K +γu× K +γu |A |K +δu K +δu |Q +βv Q +βv |(9)where according to (4)Q +αv |K +δu |Q +βv =exp {2πi [n i =1(γi −δi )u i Q i +(αi −βi )v i K i +u i v i (αi γi −βi δi )]}Inserting this expression into Eq.(9)we get A ≡ ...d Q d K d u d v K +γu |A |K +δu |Q +αv Q +βv |exp {2πi [ni =1(γi −δi )u i Q i +(αi −βi )v i K i +u i v i (αi γi −βi δi )]}(10)By a simple change of variables we can incorporate u i v i (αi γi −βi δi )into variables Q i .To this end we represent the power of the exponent in (10)as3follows(γi−δi)u i Q i+(αi−βi)v i K i+u i v i(αiγi−βiδi)=(γi−δi)u i[Q i+v iαiγi−βiδiγi−δi,dropping the superscript o,and using the fact thatexp{−2πk•a}|Q>=|Q+a>we obtain the following representation of the operator A:A= ... d Q d K A w(γ,δ,Q,K)F(α,β,Q,K)(12) whereA w(γ,δ,Q,K)= d u e2πi n k=1(γk−δk)u k Q k K+γu|A|K+δu ,(13) F(α,β,r,Q,K)= d v e−2πi(α−β)v(k−K)|Q+rv Q+rv|(14)andγir=r i=(r1,r2,...,r n)=Because both q and k are hermitian the last expression demonstrates that F(α,β,r,Q,K)is also Hermitian.Since∂()m d v e−2πi (αj−βj)v j(k j−K j)=∂K jd v(v j)m e−2πi (αj−βj)v j(k j−K j)(2πi)m(αj−βj)mEq.(16)yieldsF(α,β,r,Q,K)=e1αj−βj∂∂K j d v e−2πi (αj−βj)v j(k j−K j) d w e2πi w j(q j−Q j)(17)By introducing new variables k o j=k j(αj−βj),K o j=K j(αj−βj)the param-eters(αj−βj)are”absorbed”by these variables,which means that without any loss of generality we can set(αj−βj)=1.Therefore the operator F(α,β,r,Q,K)becomesF(r,Q,K)=e1∂Q j∂2πi r j∂2The last equation allows us tofind an explicit expression for the Weyl-transform of the quantum operator A w(γ,Q,K)which we represent as fol-lows:A w(r,Q′,K′)= d Q d K A w(r,Q,K)δ(Q−Q′)δ(K−K′)(20) Using(19)we calculate<Q′|F(r,Q,K)|K′>:Q′|F(r,Q,K)|K′ = Q′|K′ e1∂Q j∂K jδ(Q′−Q)δ(K′−K)(21) where we use the following identity|Q Q|Q′ ≡δ(Q′−Q)|Q′ ≡δ(q−Q)|Q′With the help of another identity Q′|K′ K′|Q′ ≡1we get from(21)δ(Q′−Q)δ(K′−K)=e−i∂Q′j∂K′j Q′|F(r,Q,K)|K′ K′|Q′ (22) Substitution of(22)into(20)yieldsA w(r,Q′,K′)=d Q d K A w(r,Q,K)e−i∂Q′j∂K′j Q′F(r,Q,K)|K′ K′|Q′ (23)On the other hand,from(12)follows thatQ′|A|K′ K′|Q′ =d Q d K A w(r,Q,K) Q′|F(r,Q,K)|K′ K′|Q′ . Thereforee−i∂Q′j∂K′j Q′|A|K′ K′|Q′ =d Q d K A w(r,Q,K)e−i∂Q′j∂K′j Q′|F(r,Q,K)|K′ K′|Q′ (24)6Combining(23)and(24)we get the following expression for A w(r,Q,K):A w(r,Q,K)=e−i∂Q′j∂K′j Q′|A|K′ K′|Q′ (25)If we replace K j→P j/2π¯h and take into account that for a system with N degrees of freedom|K =(2π¯h)−N/2|P then(25)takes the following form:A w(r,Q′,P′)=(2π¯h)N e−i¯hr j∂2∂Q′j∂P′j Q′|P′ P′|A|Q′ (26) To express Eq.(26)in the Schroedinger representation we consider an or-thonormal set of eigenkets|ψm and expand the operator A in terms of these eigenkets(eigenbras)A= mn w m|ψm w∗n ψn|(27)where w m are the respective coefficients of the expansion.Upon substitution of(27)into(26)we obtain the following expressionA w(r,Q,P)=e i¯h r j∂2d v d w e−πi(1+2r j )w j v j e−2πi(w j Q j −v j K j )e2πi(w j q j −v j k −j )This expression yieldsF (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )=d v d w d v ′d w ′e2πiw j q j −v j k j e−2πi[Q j (w ′j +w/2)+K j (v ′j−v/2)]×e2πi[Q ′j (w ′−w/2)+K ′(v ′+v/2)](29)Introducing new variablesv ′′=v ′+v/2;v ′′′=−(v −v/2);w ′′=−(w ′−w/2);w ′′′=w ′+w/2we obtain after some (rather lengthy)algebraF (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )=e−(i/4π)[∂∂K ′j−∂∂Q ′j]×e−(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]F (r ,Q ,K )F (r ,Q ′,K ′)Using this expression we readily obtain thatF (r ,Q ,K )F (r ,Q ′,K ′)=e (i/4π)[∂∂K ′j−∂∂Q ′j]×e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )Now we can write the product of two AB operators as followsAB =d Q d K d Q ′d K ′A w (r ,Q ,K )B w (r ,Q ′,K ′)F (r ,Q ,K )F (r ,Q ′,K ′)=d Q d K d Q ′d K ′A w (r ,Q ,K )B w (r ,Q ′,K ′)e (i /4π)[∂∂K ′j −∂∂Q ′j ]×e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )=d Q d K d Q ′d K ′F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )e(i/4π)[∂∂K ′j−∂∂Q ′j]×e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]A w (r ,Q ,K )B w (r ,Q ′,K ′)(30)8Quite similarly we obtain that BA isBA =d Q d K d Q ′d K ′A w (r ,Q ,K )B w (r ,Q ′,K ′)F (r ,Q ,K )F (r ,Q ′,K ′)=d Q d K d Q ′d K ′A w (r ,Q ,K )B w (r ,Q ′,K ′)e (i /4π)[∂∂K ′j −∂∂Q ′j ]×e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )=d Q d K d Q ′d K ′F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )e−(i/4π)[∂∂K ′j−∂∂Q ′j]×e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]A w (r ,Q ,K )B w (r ,Q ′,K ′)(31)Therefore the commutator [A ,B ]is[A ,B ]=2id Q d K d Q ′d K ′F (r ,Q ,K )δ(Q ′−Q )δ(K ′−K )e(i/4π)(1+2r j )[∂∂K ′j+∂∂Q ′j]sin {1∂Q ∂K ′−∂2∂t=[ρ,H ](33)where the operator ρis given by Eq.(12)ρ=d Q d K ρw (t ,r ,Q ,K )F (r ,Q ,K )(34)Inserting (32)and (34)into (33)we obtain∂ρw (r ,Q ,P )¯hd Q ′d K ′δ(Q ′−Q )δ(K ′−K )ei∂Q ′j∂K j+∂24π[∂2∂Q j ∂K ′j]}H w (Q ′,K ′)ρw (t ,Q ,K )(35)Performing integration and replacing parameter r by the followingr j =−(1+s j )we obtain the generalization of the Moyal bracket[3]:∂ρ(r,Q,P)2 s j(∂2∂Q j∂P′j)×22[∂2∂Q j∂P′j]}H w(Q′,P′)ρw(t,Q,P)The same result was presented in[6].However there the authors introduced parameter s by hand,without relating it to any transformation of the quan-tum states and simply treating it as a means to achieve a unified approach to different quantization rules.On the other hand,our approach(based on[2]) explicitly shows that such a parameter is a result of a linear transformation from one quantum state to another.Since the pure states are represented by rays,it is very natural to expect that the above transformation would result in the appearance of the phase,which is clearly seen in the exponential op-erator.3On a Physical Meaningof the s-ParameterIt is therefore interesting to investigate what role is played by this parameter in a transition to a classical case.To this end we restrict our attention to a1−D case and consider(following Moyal)space-conditional moments <p n>w(the Wigner averages of the powers of p n of the momentum):<p n>w= A w(p,q,s)p n dpAs a next step,we find Fourier-transform of A w (p,q,σ),Eq.(28).We denote thistransformby M (τ,θ,σ):M (τ,θ,σ)=e (τp +θq )e −1−σ∂p∂q [Ψ∗(q )Ψ(p )e ipq ]dpdq (38)Integration of (38)by parts yields:M (τ,θ,σ)=Ψ∗(q )e iθ[q +τ(1−σ)2(1−σ)Using q 1in (39)we obtain:M (τ,θ,σ)={e iθq 1Ψ∗[q 1−τ2(1+σ)]Ψ(p )dp }dq 1(40)Since1¯he ip [q 1+τ2(1+σ)]relation (40)takes the following form:M (τ,θ,σ)=Ψ∗[q 1−τ2(1+σ)]dq 1(41)Inverse Fourier-transform of M (τ,θ,σ)gives us the desired integral form of the phase-space distribution (a s -parameterized Wigner function)A w (p,q,σ)1:A w (p,q,σ)=12(1−σ)]e −iτp Ψ[q 1+τ1a simplified derivation of A w (p,q,σis given in the Appendix11In general,since the parameterσ(or s)is complex-valued,theσ-parameterized Wigner function A w(p,q,σ)is also complex−valued,in contradistinction to its conventional counterpart(withσ=0).However for the purely imaginary values of the parameterσ,theσ-parameterized Wigner function becomes real-valued again:i)A∗w(p,q,σ)=A w(q,p,−σ)ii)A∗w(p,q,σ)=A w(q,p,σ),Re(σ)=0For the following we rewrite A w(p,q,σ)in terms of the momentum wave functionΦ(p).After some algebra we obtainA w(p,q,σ)= dp′dp′′e−iq(p′′−p′)Φ∗(p′′)Φ(p′)δ[p−p′′(1+σ)+p′(1−σ)2]=dp′dp′′e−iq(p′′−p′)Φ∗(p′′)Φ(p′)[p′′(1+σ)+p′(1−σ) i∂i∂2i [(1−σ)∂∂q1]n e−iq1p′Φ∗(p′′)dp′′ e iq2p′}|q2→q1={1∂q2−(1+σ)∂Returning to the units with¯h and using(43),we calculate twofirst momenta <p>w and<p2>w:<p>w=12i[(1−σ)∂∂q1]Ψ∗(q1)Ψ(q)}|q2→q1=¯h∂q{Ln(Ψ4[(1−σ)2Ψ′′∂q∂LnΨ∗Ψ∗](48)Let us consider a semi-classical limitΨ(q,t)=√2∇(Lnρ)}=∇S=p classical(50) If we use the Schroedinger equation:i¯h ∂Ln(Ψ)2m∇2Ψ∂t−V+18m(∇Lnρ)2++σ2[−∂S2m(∇S)2−¯h22∂Ψ∗)}=m{−∂S2m(∇S)2+¯h2∂t−V−18m(∇Lnρ)2}(51)This limit must yield the classical value of the square of the classical momen-tumlim¯h→0<p2>w=p2classical=(∇S)213which is independent of the parameterσ.This is possible if the factor atσin(51)becomes0,that is−∂S2m(∇S)2+VBut amazingly enough this condition is nothing more than the classical Hamilton-Jacobi equation.Thus emergence of the parameterσin Wigner function is tied to an emergence of the classical Hamilton-Jacobi equation in transition to a classical regime.4ConclusionWe have demonstrated that s-parameterized Wigner function emerges as a result of a linear transformation from one quantum state to another.The respective change of the phase space coordinates accompanying such a trans-form must be necessarily canonical.This allows one to arrive in a natural way,without introducing”by hand”the s-parameter into the transformation either by suitably chosen displacement operator or by using it as a”missing link”between normal and anti-normal ordering operators.A transition to a classical regime demonstrates that parameter s plays an im-portant role,ensuring the emergence of the classical Hamilton-Jacobi equa-tion as a condition for the disappearance of this parameter in classical me-chanics.5AppendixSince the momentum representationΦ(p)and the coordinate representation Ψ(q)of the wave function(for simplicity sake we consider a1-D case)are related as follows:Φ(p)=12π¯he−ipq′/¯hΨ(q′)dq′(52)the momentum probability density is|Ψ(p)|2=1We introduce new variables q andτq′=q+ατ;q′′=q+βτwhereαandβare constants such that the Jacobian of transformation from q′′,q′to q,τis¯h which meansα−β=¯hAs a result,Eq.(53)yields|Φ(p)|2=12Upon substitution of this expression into(54)we obtain|Φ(p)|2=12]Ψ∗[q−¯h(1−s)2π Ψ[q+¯h(1+s)2]dτ(56)is the s-parameterized Wigner function found earlier,Eq.(42)6AcknowledgementThe author expresses his gratitude to C.McCallum for his help in preparation of this paper.15References[1]H.Weyl,The Theory of Groups and Quantum Mechanics,Dover Publ.,1950[2]B.Leaf,J.Math.Phys,9,No.1,65(1968)[3]J.E.Moyal,Proc.Cambridge Phil.Soc.45,99(1949)[4]K.E.Cahill,R.J.Glauber,Phys.Rev.,177,1857(1969);Phys.Rev.,177,1882(1969)[5]N.L.Balazs and B.K.Jennings,Physics Reports,104,347(1984)[6]T.Dereli,A.Vercin,J.Math.Phys.,38,5515(1997)[7]E.Wigner,Phys.Rev.,40,749(1932)16。