FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS Thomas Curtright, David Fairlie, and Cosmas Zachos

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1 1 DTP/97/61 MIAMI-TH-97-3 ANL-HEP-PR FEATURES OF TIME-INDEPENDENT WIGNER FUNCTIONS Thomas Curtright, David Fairlie, and Cosmas Zachos Department of Physics, University of Miami, Box 48046, Coral Gables, Florida 3314, USA Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK High Energy Physics Division, Argonne National Laboratory, Argonne, IL , USA Abstract The Wigner phase-space distribution function provides the basis for Moyal s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. General features of time-independent Wigner functions are explored here, including the functional star eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux supersymmetric isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Pöschl-Teller potential, and the Liouville potential. 1 Introduction Wigner functions have been receiving increasing attention in quantum optics, dynamical systems, and the algebraic structures of M-theory [1]. They were invented by Wigner and Szilard [], and serve as a phase-space distribution alternative to the density matrix, to whose matrix elements they are related by Fourier transformation. The diagonal, hence real, time-independent pure-state Wigner function fx, p corresponding to the eigenfunction ψ of Hψ = Eψ, is fx, p = 1 dy ψ x π y e iyp ψx + y. 1 These functions are not quite probability distribution functions, as they are not necessarily positive illustrated below. However, upon integration over p or x, they yield bona-fide positive probability distributions, in x or p respectively. Wigner functions underlie Moyal s formulation of Quantum Mechanics [3], through the unique [4, 5] oneparameter associative deformation of the Poisson-Bracket structure of classical mechanics. Expectation values can be computed on the basis of phase-space c-number functions: given an operator Ax, p, the corresponding phase-space function Ax, p obtained by p p, x x yields that operator s expectation value through A = dxdp fx, pax, p, assuming the usual normalization, dxdpfx, p = 1, and further assuming Weyl ordering, as addressed by Moyal, who took matrix elements of all such operators: Ax, p = 1 π dτdσdxdp Ax, pexpiτp p+iσx x. 3

2 Curtright, Fairlie, & Zachos Wigner functions hep-th/ Wigner functions are c-numbers, but they compose with each other nonlocally. Properties of these compositions were explored in, e.g., [6, 7], and were codified in an elegant system in [5]: to parallel operator multiplication, the Wigner functions compose with each other through the associative star product e i x p p x. 4 Recalling the action of a translation operator expa x hx =hx+a, it is evident that the -product induces simple Bopp -shifts: fx, p gx, p =fx, p i x gx, p + i x =fx+ i p,p i x gx, p, 5 etc., where and here act on the arguments of f and g, respectively. This intricate convolution samples the Wigner function over the entire phase space, and thus provides an alternative to operator multiplication in Hilbert space. Antisymmetrizing and symmetrizing the star product, yields the Moyal Sine Bracket [3], and Baker s [6] Cosine Bracket respectively. Note [7, 8] that {{f,g}} f g g f, 6 i f,g f g+g f, 7 dpdx f g = dpdx fg. 8 Further note the Wigner distribution has a -factorizable integrand, fx, p = 1 dy ψ x e iyp ψx e iyp. 9 π In general, systematic specification of time-dependent Wigner functions is predicated on the eigenvalue spectrum of the time-independent problem. For pure-state static distributions, Wigner and, more explicitly, Moyal, showed that {{Hx, p,fx, p}} =0, 10 i.e. H and f -commute. However, there is a more powerful functional equation, the star-genvalue equation, which holds for the time-independent pure state Wigner functions Lemma 1, and amounts to a complete characterization of them Lemma. We will explore the features of this -genvalue equation, and illustrate its utility on a number of solvable potentials, including both the harmonic oscillator and the linear one. The -multiplications of Wigner functions will be seen to parallel Hilbert space operations in marked detail. The Pöschl-Teller potential will reveal how the hierarchy of factorizable Hamiltonians familiar from supersymmetric quantum mechanics finds its full analog in -space. We determine the Wigner function s transformation properties under phase-space volume-preserving canonical transformations, which we finally elaborate in the context of the Liouville potential. The -genvalue Equation Lemma 1: Static, pure-state Wigner functions obey the -genvalue equation, Hx, p fx, p =Efx, p. 11

3 Curtright, Fairlie, & Zachos Wigner functions hep-th/ Without essential loss of generality, consider Hx, p =p /m+vx. Hx, p fx, p = 1 π = 1 π = 1 π = 1 π p i dy x /m+vx dy e iyp+i x ψ x y ψx + y 1 p i dy e iyp i y +i x /m + V x + y e iyp ψ x y ψx + y x /m + V x + y ψ x y ψx + y dy e iyp ψ x y Eψx+ y=efx, p, since the action of the effective differential operators on ψ turns out to be null; and, likewise, f H= 1 dy e iyp y π Thus, both of the above relations 10 and Lemma 1 obtain. x /m + V x y ψ x y ψx + y=efx, p. 13 This time-independent equation was introduced in ref [7], such that the expectation of the energy Hx, p in a pure state time-independent Wigner function fx, p isgivenby Hx, pfx, p dxdp = E fx, p dxdp. 14 On account of the integration property of the star product, 8, the left hand side of this amounts to dxdp Hx, p fx, p. Implicitly, this equation could have been inferred from the Bloch equation of the temperature- and time-dependent Wigner function, in the early work of [9]. -genvalue equations are discussedinsomedepthinthesecondofrefs[5],andin[10]. By virtue of this equation, Fairlie also derived the general -orthogonality and spectral projection properties of static Wigner functions [7]. His results were later formalized in the spectral theory of the second of refs [5] e.g. eqn 4.4. Consider g corresponding to the normalized eigenfunction ψ g corresponding to energy E g. By Lemma 1, and the associativity of the -product, Then, if E g E f, this is only satisfied by f H g=e f f g=e g f g. 15 f g=0. 16 N.B. the integrated version is familiar from Wigner s paper, dxdp f g = dxdp fg =0, 17 and demonstrates that all overlapping Wigner functions cannot be everywhere positive. The unintegrated relation introduced by Fairlie appears local, but is, of course, highly nonlocal, by virtue of the convolving action of the -product. Precluding degeneracy, for f = g, f H f=e f f f=h f f, 18 which leads, by virtue of associativity, to the normalization relation [6], f f f. 19

4 Curtright, Fairlie, & Zachos Wigner functions hep-th/ Both relations 16 and 19 can be checked directly: fx, p gx, p =fx, p i x gx, p + i x 0 = 1 π dy ψf x y ψ f x + y e iyp i x dy e iy p+ i x ψg x Y ψ gx + Y = 1 π dydy e iy+y p ψf x y + Y ψ f x + y + Y ψ g x Y y ψ gx + Y y= 1 dy +ye iy+yp ψ gx π Y +y ψ fx + 1 y + Y y dy ψ f h Y yψg Y y. The second integral factor is 0 or 1/h, depending on f g or f = g, respectively, specifying the normalization f f = f/h in 19. In conclusion, Corollary 1: f a f b = 1 h δ a,b f a. These spectral properties are summoned up by their own necessity; much of their meaning, nevertheless, resides in their margins: For non-normalizable wavefunctions, the above second integral factor may diverge, as illustrated below for the linear potential, but the orthogonality properties still hold. Thus, e.g., for an arbitrary functional F z, F [f ]f = F1/hf, 1 and, for -genfunctions of Lemma 1, F [H ]f = FEf. Baker s converse construction extends to a full converse of Lemma 1, namely Lemma : Real solutions of Hx, p fx, p =Efx, p = fx, p Hx, p must be of the Wigner form, f = dy e iyp ψ x yψx + y/π, such that Hψ = Eψ. As seen above, the pair of -eigenvalue equations dictate, for fx, p = dy e iyp fx, y, dy e iyp 1 m ± y x + V x ± y E fx, y =0. 3 This constrains fx, y to consist of bilinears ψ x y ψx + y of unnormalized eigenfunctions ψx corresponding to the same eigenvalue E in the Schrödinger equation with potential V. These two Lemmata then amount to the statement that, for real functions fx, p, the Wigner form is equivalent to compliance with the -genvalue equation real and imaginary part. 3 Example: The Simple Harmonic Oscillator The eigenvalue equation of Lemma 1 may be solved directly to produce the Wigner functions for specific potentials, without first solving the corresponding Schrödinger problem as in, e.g. [11]. Following [7], for the harmonic oscillator, H =p +x /with=1,m= 1, the resulting equation is x + i p +p i x E fx, p =0. 4 By virtue of its imaginary part, x p p x f =0,fis seen to depend on only one variable, z =4H= x + p, so the equation reduces to a simple ODE, z 4 z z z E fz =0. 5

5 Curtright, Fairlie, & Zachos Wigner functions hep-th/ Moreover, setting fz = exp z/lz, this yields z z +1 z z +E 1 Lz=0, 6 which is the equation satisfied by Laguerre polynomials, L n = e z n e z z n, for n = E 1/ =0,1,,..., so that the un-normalized eigen-wigner-functions are f n = e H L n 4H; L 0 =1, L 1 =1 4H, L =16H 16H +,... 7 Note the eigenfunctions are not positive definite, and are the only ones satisfying the boundary conditions, f0 finite, and fz 0, as z. In fact, Dirac s Hamiltonian factorization method for algebraic solution carries through cf. [5] intact in -space. Indeed, H = 1 x ip x + ip+1, 8 motivating definition of Thus, noting andalsothat,byabove, a 1 x + ip, a 1 x ip. 9 a a a a=1, 30 a f 0 = 1 x+ip e x +p =0, 31 provides a -Fock vacuum, it is evident that associativity of the -product permits the entire ladder spectrum generation to go through as usual. The -genstates of the Hamiltonian, s.t. H f= f H, are thus f n a n f 0 a n. 3 These states are real, like the Gaussian ground state, and are thus left-right symmetric -genstates. They are also transparently -orthogonal for different eigenvalues; and they project to themselves, as they should, since the Gaussian ground state does, f 0 f 0 f 0. It will be seen below that even the generalization of this factorization method for isospectral potential pairs goes through without difficulty. 4 Further Example: The Linear Potential For simplicity, take m =1/, = 1. Recall [1] that the problem readily reduces to a free particle: Hx, p = p + x H free = P is accomplished by canonically transforming through the generating function F x, X = 1 3 X3 xx. The energy eigenfunctions are Airy functions, + ψ E x = 1 π dx e if x,x e iex = Aix E. 33 The -genvalue equation in this case is x + i p+p i x E fx, p =0, 34 whose imaginary part, 1 p p x fx, p = 0, gives fx, p =fx+p =fh. The real part of the equation is then an ordinary second order equation, just as in the above harmonic oscillator case. Moreover, here the real part of the -genvalue equation is essentially the same as the usual energy eigenvalue equation: z 1 4 z E fz =0, 35

6 Curtright, Fairlie, & Zachos Wigner functions hep-th/ where z = x + p. Hence, the Wigner function is again an Airy function, like the above wavefunctions, except that the argument has a different scale and shift 1. fx, p = /3 π Ai/3 z E = /3 π Ai/3 x + p E = 1 π dy e iye x p y /1. 36 The Airy functions are not square integrable, so that the conventional normalization f f= 1 πf does not strictly apply. On the other hand, the energy eigenfunctions are non-degenerate, and the general Corollary 1 projection relations f a f b δ a,b f a still hold for the continuous spectrum: f E1 f E = f E1 x + i p +p f E x i p +p 1 = π 4 dy dy e iye1 x p Y/ y /1 e iy E x p+y/ Y /1 1 = = π 4 1 π by virtue of the direct definition 36. E1+E iy+y x p dy + Y e y+y /1 d y Y e i y Y E1 E δe1 E f E1+E x + p, 37 5 Darboux Construction of Wigner Function Recursions Analogous ladder operators for eigenstates corresponding to essentially isospectral pairs of partner potentials [14] familiar from supersymmetric quantum mechanics can also be defined mutatis-mutandis for Wigner functions and -products. They faithfully parallel the differential equation structures. Consider a positive semidefinite Hamiltonian Thiscanbewrittenasa -product of two operators, p H = Q Q= +iw x m H = p /m + V x. 38 p iw x m, 39 provided W m x W = V x. 40 This Riccati equation, familiar from ssqm, can be Darboux-transformed by changing variable for the superpotential W x: W = xψ 0, 41 mψ0 which reduces the condition to the Schrödinger equation for zero eigenvalue: m x ψ 0 + V xψ 0 =0. 4 Also note Q f 0 = 0 for the corresponding Wigner function. It is easy to generalize this by adding a constant to H to shift the ground state eigenvalue from zero. 1 This case is similar to the Gaussian wave function, i.e. the harmonic oscillator ground state encountered above, whose Wigner function is also a Gaussian, but of different width. S Habib kindly informed us that this solution is also given in ref [13], eqn 9.

7 Curtright, Fairlie, & Zachos Wigner functions hep-th/ By virtue of associativity, it is evident that the partner Hamiltonian H = Q Q =H+ m x W, 43 i.e. the one with a partner potential V = W + m x W, 44 has Wigner function -genstates of the same energy as those of H. Specifically, H f=q Q f =f Q Q=Ef 45 implies that the real functions Q f Q are -genfunctions of H with the same eigenvalue E: H Q f Q =Q Q Q f Q =EQ f Q, 46 unless f is the Wigner function corresponding to ψ 0,sinceQ f 0 =0. In consequence, E n = E n+1 for n 0. Conversely, for g -genfunctions of H, Q g Q are - genfunctions of H with the same eigenvalues. Moreover, ψ 0 1/ψ 0 will be an invalid zero mode eigenfunction of H, as seen from the sign flip in 41 and 44. Consequently, an unnormalized, runaway zero energy solution of the Schrödinger equation with V x will invert to the legitimate ground state of H and will permit construction of V given V. For example, starting from the trivial potential with a continuous unnormalizable spectrum, V =1, 47 and the solution mx mx ψ 0 =cosh, = W = tanh, 48 results via 40 in the symmetric, reflectionless Pöschl-Teller potential [15], V = 1 /cosh mx. Conversely, starting from this potential, V x =1 there is a single bound state normalizable to ψ 0 =, cosh mx, 49 so that mx mx ψ 0 =sech, = W = tanh, 50 V =1. 51 Thus, the Wigner function ground state for m =1/ is f 0 x, p = 1 dy π e iyp coshx/ y/ coshx/ + y/ = 1 π 0 dy cosyp coshx/+coshy 5 sinxp/ = sinhx/ sinhπp. N.B. Not positive definite, nor a function of just Hx, p. It may be verified directly that Q f 0 = p i x itanh x + i p f 0 x, p =0. 53

8 Curtright, Fairlie, & Zachos Wigner functions hep-th/ This appendage of bound states to a potential generalizes [16] to the hierarchy associated with the KdV equation. Specifically, mx W n =ntanh 54 connects the reflectionless Pöschl-Teller potential mx mx V x =n nn 1/cosh to its contiguous V x =n nn+1/cosh, 55 which has one more bound state shape-invariance. Recursively then, one may go in N steps, with the suitable shifts of the potential by n 1 in each step, from the constant potential to mx V N; x =N NN+1/cosh. 56 Shifting this potential down by N assigns the energy E = N to the corresponding ground state ψ 0 N =sech N x unnormalized, which is the null-state of m x + W N. The corresponding unnormalized Wigner function is the -null state of QN, f 0 N; x, p = 1 π 0 dy cosyp coshx/+coshy N = 1 N 1! where the integral only need be evaluated from the above f 0 1; x, p. Alternatively, N 1 sinhx/ x f 0 1; x, p, 57 f 0 N; x, p =sechx/ N 1 f 0 1; x, p sechx/ N The unnormalized state above the ground state at E = N 1 is m x W N ψ 0 N 1, and its corresponding Wigner function setting m =1/ is found recursively from the ground state of HN 1, through Q N f 0 N 1 QN, p f 0 N 1 + in tanh x f 0N 1 QN= p f 0 N 1 + N N 1 p f 0N 1 QN 59 by virtue of = N 1 N 1 p f 0 N 1 p, QN 1 f 0 N 1 = 0 = f 0 N 1 Q N The state above that, at E = N, is found recursively through Q N Q N 1 f 0 N QN 1 QN, 61 andsoforth. Thus,theentireWigner -genfunction spectrum of HN is obtained with hardly any reliance on Schrödinger eigenfunctions. 6 Canonical Transformation of the Wigner Function For notational simplicity, take = 1 in this section. The area element in phase space is preserved by Canonical Transformations x, p Xx, p,px, p 6

9 Curtright, Fairlie, & Zachos Wigner functions hep-th/ which yield trivial Jacobians dxdp = dxdp {X, P} by preserving Poisson Brackets {u, v} xp u v x p u p They thus preserve the canonical invariants of their functions: Equivalently, v x. 63 {X, P} xp = 1 hence {x, p} XP =1. 64 {x, p} = {X, P}, 65 in any basis. Motion being a canonical transformation, Hamilton s classical equations of motion are preserved, for HX, P Hx, p, as well [17]. What happens upon quantization? Since, in deformation quantization, the Hamiltonian is a c-number function, and so transforms classically, HX, P Hx, p, the effects of a canonical transformation on the quantum -genvalue equation of Lemma 1 will be carried by a suitably transformed Wigner function. Predictably, the answer can be deduced from Dirac s quantum transformation theory. Consider the canonical transformations generated by F x, X: Fx, X Fx, X p =, P = x X. 66 Following Dirac s celebrated exponentiation [18] of such a generator, in the implementation of [1, 19], the energy eigenfunctions transform canonically through a generalization of the representation-changing Fourier transform. Namely, ψ E x =N E dx e if x,x Ψ E X. 67 Thus, fx, p = N E dy π dx 1 e if x y/,x 1 Ψ EX 1 e iyp dx e if x+y/,x Ψ E X. 68 The pair of Wigner functions in the respective canonical variables, fx, p and FX, P = 1 dy Ψ X π Y e iy P ΨX + Y, 69 are connected by a transformation functional Tx, p; X, P, fx, p = dx dp Tx, p; X, P FX, P = dx dp Tx, p; X, P FX, P, 70 where is with respect to the variables X and P. To find this functional, let X = 1 X 1 + X andy =X X 1,sothat dx 1 dx = dx dy. Noting that Ψ X Y ΨX+ Y= dp e iy P FX, P, 71 it follows that 68 reduces to fx, p = N π = N π = N π dy dx 1 e if x y/,x 1 Ψ X 1 e iyp dx e if x+y/,x ΨX 7 dxdy dy e iyp e if x y/,x Y/ Ψ if x+y/,x+y/ X Y/ ΨX + Y/ e dxdp dy dy e iyp+ip Y if x y/,x Y/+iF x+y/,x+y/ FX, P,

10 Curtright, Fairlie, & Zachos Wigner functions hep-th/ which leads to Lemma 3: Tx, p; X, P = N π dy dy exp iyp + ip Y if x y,x Y +if x + y,x + Y. Corollary : This phase-space transformation functional obeys the two-star equation, Hx, p Tx, p; X, P =Tx, p; X, P HX, P, 73 as follows from Hx, i x expif x, X = HX, i X expif x, X. If F satisfies a genvalue equation, then f satisfies a -genvalue equation with the same eigenvalue, and vice versa. Note that, by virtue of the spectral projection feature 16,19, this equation is also solved by any representation-changing equal-energy bilinear in real Wigner -genfunctions of H and H, Tx, p; X, P = E gef E x, p F E X, P, 74 for arbitrary real ge. Such a bilinear transformation functional is nonsingular invertible if and only if ge has no zeros on the spectrum of either Hamiltonian. As an example, consider the linear potential again, which transforms to a free particle H = P through F = 1 3 X3 xx = p = X, x = P X. 75 By direct computation, Tx, p; X, P = /3 Ai /3 x + X P δp + X =π de f E x, pf E X, P δp + X. 76 Note N E =1/ πfor the free particle energy eigenfunction normalization choice Ψ E X =π 1/ expiex. Thus, indeed, the free particle Wigner function, F E X, P =δe P/π, transforms to fx, p = 1 dp dx T δe P = /3 π π Ai /3 x + p E, 77 as it should; and 73 is seen to be satisfied directly, by virtue of the linearity of the respective Hamiltonians in the variables P, x, conjugate to those of the arguments of δp + X. The structure of the result in 76 underscores that the linear potential is as close to classical as one can get, in simple quantum mechanics. It has been noted before [1] that the transformation functional for linear potential wave functions is exactly the exponential of the classical generating function for the canonical transformation to a free particle, and that this is not the case for any other potential. The present result for the transformation functional for Wigner functions is further evidence for this close to classical behavior. The delta function δp+x in 76 is half of the classical story. Were the Airy function In general, if the transformation functional effects a map to a free particle, the P integration is trivial in 70, and the result for the Wigner function of the x, p theory is just an average over X of the transformation functional. That is, if FX, P = δp ke, where ke is the momentum-energy relation for the free particle theory in question, fx, p = dx dp Tx, p; X, P FX, P = dx Tx, p; X, ke. One might then be tempted to wonder if just Tx, p; X, P =ψ Px X/ e ixp ψ P x+x//π Gx, p; X, P. However, what determines the allowed range for P? It is always possible to embed any real energy spectrum into the real line, but knowing this does not help at all to determine what points are to be embedded. From the point of view of this paper, even when the spectrum is obvious, such a choice for the transformation functional in general does not satisfy the two- equation 73. Rather, the equation fails by total derivatives that vary contingent on particularities of the case. E.g., for free-particle plane waves, ψ Ex =expiex, so that p G G P= XG. This choice for T, then, does not yield useful information on the Wigner functions.

11 Curtright, Fairlie, & Zachos Wigner functions hep-th/ also a delta function of its argument, we would have an exact implementation of the X, P x, p classical correspondence. As it is, there is some typically quantum mechanical spread around the classical constraint x + X P = 0, in the form of oscillations of the Airy function, and, in consequence, the Wigner functions of the free particle do not retain their delta-function form under the canonical transformation to the linear potential Wigner functions. Reinstating into 36 3, and taking the limit 0 converts the Airy function to a delta function, δx + X P, thereupon producing the complete classical correspondence between the two sets of phase space variables, in that limit. As already seen, there is substantial non-uniqueness in the choice of transformation functional. For example, for the linear potential again, 73, x + p Sx, p; X, P = Sx, p; X, P P 78 is also satisfied by a different and somewhat simpler choice: Sx, p; X, P =exp i 3 X3 +x+p PX. 79 This transformation functional also converts the free particle Wigner function, F E X, P =δe P/π, into an Airy function as above after integrating over the free particle phase space, dxdp. Actually, it is not necessary to integrate over the phase space. In general, -multiplying a delta function spreads it out, and yields a Fourier transform with respect to the conjugate variable. Thus, for the example considered, e i 3 X3 x+p P X δp E = e ixp E 1 dz e izp E e i 3 Z3 x+p P Z π = e ixp E 1 dz e i 3 Z 3 x+p EZ = e ixp E /3 Ai /3 x + p E. 80 π Hence, dx dp e i 3 X3 x+p P X δp E= /3 πai /3 x + p E. 81 Compare this to the action of the above Tx, p; X, P, Ai /3 x + X P δp + X δp E = 8 e ixp E 1 dz e izp E Ai /3 x + Z P δp + Z = e ip+xp E 1 π π Ai/3 x + p P. Aside from innocuous normalizations, the difference in the two transformation functionals acting on the free particle Wigner function is just the phase factor e ipp E, and the argument of the Airy function, where E has been replaced by P. Indeed, the phase factor precisely compensates for the different energyeigenvalue occurring in the argument of Ai, when acted upon by x + p. Such simple phase factors may be used to shift a -genvalue whenever the Hamiltonian is linear in any variable. 7 Illustrations using Liouville Quantum Mechanics Summary illustration of all the above, in particular the canonical transformation effects on Wigner functions, is provided by the Liouville model [0]. Our conventions for the model which are essentially those of [1], with their m 1/4π andtheirg 1 are given by 3 The exponent of the integrand turns to iye x p y /1. H Liouville = p + e x. 83

12 Curtright, Fairlie, & Zachos Wigner functions hep-th/ The energy eigenfunctions are then solutions of d dx + ex ψ E x =Eψ E x. 84 The solutions are Kelvin modified Bessel K functions, for 0 <E<, ψ E x= 1 sinh π E K π i E e x, 85 which are normalized such that + dx ψ E 1 x ψ E x =δe 1 E. There is no solution [0] for E =0. For completeness, consider the Fourier transform including a convergence factor, necessary for x to control plane wave behavior, but not for x : + Φ E p + iɛ = dx e ixp+iɛ ψ E x 86 = 1 sinhπ i p + iɛ+i E ip+iɛ i E E ip+iɛ Γ Γ. 4π This follows, e.g., from a result in [], Vol II, p 51, eqn 7: + 1+µ+ν 1+µ ν dz z µ K ν z = µ 1 Γ Γ, 87 0 valid for R 1 + µ ± ν > 0 i.e. the previous transform is valid for ɛ>0. The right hand side of this last relation clearly displays the symmetry ν ν, which just amounts to the physical statement that the energy eigenfunctions are non-degenerate for the transmissionless exponential potential of the Liouville model. Further note the effect on Φ E p + iɛ ofshiftingp p+i,usingγ1+z=zγz, ip+iɛ+i E ip+iɛ i E Φ E p +i+iɛ = 4 Φ E p + iɛ = E p + iɛ Φ E p + iɛ. 88 So, as ɛ 0, Φ E p+i= E p Φ E p. But this simple difference equation is just the Liouville energy eigenvalue equation in the momentum basis, p E Φ E p+e i p Φ E p =0. 89 Such first order difference equations invariably lead to gamma functions [3]. Below, it turns out that the Wigner functions also satisfy momentum difference equations, but of second order. Many, if not all, properties of the Liouville wave functions may be understood from the following integral representation [4], Ch VI, 6., eqn 10. Explicitly emphasizing the abovementioned non-degeneracy, K ik e x =K ik e x = 1 + eπk/ dx e iex sinh X e ikx. 90 Also see [5], This integral representation may be effectively regarded as the canonical transformation of a free particle energy eigenfunction, e ikx, through use of the generating function F x, X = e x sinh X. Classically, p = F/ x = e x sinh X, andp= F/ X = e x cosh X, sop p =e x.that is, H Liouville = H free P under the classical effects of the canonical transformation. The quantum effects are detailed below, by -acting with the Liouville and free Hamiltonians on the suitable transformation functional.

13 Curtright, Fairlie, & Zachos Wigner functions hep-th/ The Liouville Wigner function may be obtained from the definition 1 in terms of known higher transcendental functions, fx, p = 1 + dy 1 π π sinhπ E K i E e x y/ e iyp K i E e x+y/ 91 = 1 e 4π 3 sinhπ E ip e 1 ipx G 40 4x 1+i E 04, 1 i E, 1+i E+4ip, 1 i E +4ip The following K-transform was utilized to express this result in closed form, a dw wz 1/ w σ 1 K µ a/w K ν wz = σ 5/ a σ G 40 z µ σ 04, µ σ, ν, 1 4 ν. 9 0 The right hand side involves a special case of Meijer s G-function, G mn a pq z i,i=1,...,p, 93 b j,j=1,...,q cf. [], 5.3, which is fully symmetric in the parameter subsets {a 1,...,a n },{a n+1,...,a p }, {b 1,...,b m }, and {b m+1,...,b q }. It is possible to re-express the result as a linear combination of generalized hypergeometric functions of type 0 F 3, but there is little reason to do so here. This transform is valid for Ra >0, and is taken from [6], p 711, eqn The transform is complementary to [7], 10.3, eqn 49, in an obvious way, a K-transform which appears in perturbative computations of certain Liouville correlation functions [1]. The result 91 may be written in slightly different alternate forms, fx, p = sinhπ Ee x e 4π 3 G 40 4x 1+i E ip 04, 1 i E ip, 1+i E+ip, 1 i E +ip = sinhπ E e 8π 3 G 40 4x i E ip 04, i E ip, i E + ip, i E + ip, by making use of the parameter translation identity for the G-function [] 5.3.1, eqn 8: z λ G mn a pq z r = G mn a pq z r + λ. 95 b s b s + λ Yet another way to express the result utilizes the Fourier transform of the wave function, 86, in terms of which the Wigner function reads, in general, 1 + fx, p = dk Φ π Ep 1 k eixk Φ E p + 1 k. 96 The specific result 86 then gives, as ɛ 0, 1 fx, p = 8π sinhπ + i p k/ iɛ i E E dk e ixk 4 ik/+iɛ Γ 97 i p k/ iɛ+i E ip+k/+iɛ+i E ip+k/+iɛ i E Γ Γ Γ. However, this is a contour integral representation of the particular G-function given above. Because of the ɛ prescription, the contour in the variable z = k/ +iɛ runs parallel to the real axis, but slightly 4 There is an error in this result as it appears in [7], Vol II, 10.3, eqn 58, where the formula has a z /4 instead of a z /16 as the argument of the G-function. The latter argument is correct, and appears in Meijer s original paper cited here.

14 Curtright, Fairlie, & Zachos Wigner functions hep-th/ above the poles of the Γ-functions located on the real axis at z = p E, z = p + E, z = p + E, and z = p E.Changing variables to s = 1 iz yields fx, p = 1 8π 3 sinhπ 1 e 4x s ip i E E ds Γ s 98 πi C 16 ip + i E ip + i E ip i E Γ s Γ s Γ s, where the contour C in the s-plane runs from i to +i, just to the left of the four poles on the imaginary s axis at ip + E/, ip E/, i p + E/, and i p E/. This is recognized as the Mellin-Barnes type integral definition of the G function cf. [], 5.3, eqn 1 in agreement with the second result above, 94. The translation identity 95 is seen to hold by virtue of 98, through simply shifting the variable of integration, s. Moreover, deforming the contour in 98 to enclose the four sequences of poles s n = n + i±p ± E/ reveals the equivalence of this particular G-function to a linear combination of four 0 F 3 functions, one for each of the sequences of poles. Evaluating the integral by the method of residues for all these poles produces the standard 0 F 3 hypergeometric series. It should now be straightforward to directly check that the explicit result for fx, p is indeed a solution to the Liouville -genvalue equation, H Liouville fx, p = p i x +e x+ i p fx, p =Efx, p. 99 For real E and real fx, p, the imaginary part of this -genvalue equation is p x + e x sin p fx, p =0, 100 while the real part is p E 14 x + e x cos p fx, p = The first of these is a first-order differential/difference equation relating the x and p dependence, e x x fx, p = 1 fx, p + i fx, p i. 10 ip Similarly, the real part of the -genvalue equation is a second-order differential/difference equation, e p x E 1 4 x fx, p+ 1 fx, p + i+fx, p i = The previous first-order equation may now be substituted twice into this last second-order equation, to convert it from a differential/difference equation into a second-order difference only equation in the momentum variable, with non-constant coefficients. That is, 0 = p E fx, p+ e x fx, p +i fx, p+fx, p i 4p +i ex 4p fx, p + i fx, p i + ex fx, p + i+fx, p i. 104 We leave it as an exercise for the reader to exploit the recursive properties of the Meijer G-function and show that this difference equation is indeed obeyed by the result 91. Rather than pursue this in detail, we turn our attention to the transformation functional which connects the above result for f to a free particle Wigner function.

15 Curtright, Fairlie, & Zachos Wigner functions hep-th/ Given 90, it follows that ψ E x = 1 π sinhπ E K i E e x = 1 π sinhπ E e π E/ + dx e iex sinh X e i EX, 105 hence N E =4π Ee π E sinhπ E 1/ /π, ifwechooseaδe 1 E normalization for the free particle plane waves as well as for the Liouville eigenfunctions. Therefore, Lemma 3 yields Tx, p; X, P = N dy dy exp iyp + ip Y if x y/,x Y/ + if x + y/,x +Y/ 106 π = 1 π 3 4π Ee π E sinhπ E dy dy exp iyp + ip Y ie x y/ sinh X Y + ie x+y/ sinh X + Y = 1 4π 3 d We thus conclude, 4π Ee π E sinhπ E Y y exp y + Y d exp i P + p Y y + ie x X sinh i P p y + Y Y y. y + Y + ie x+x sinh Tx, p; X, P = 4 π Ee π E sinhπ E e πp K ip p e x+x K ip+p e x X. 107 We now check that this result obeys 73 and, in so doing, carry out the nontrivial steps needed to show the Liouville Wigner functions satisfy the Liouville -genvalue equation 99. That is to say, we shall show p i x + e x+ i p Tx, p; X, P =Tx, p; X, P P + i X, 108 or, equivalently, p i x + e x+ i p P + i X K ip p e x+x K ip +p e x X = Specifically, 1 4 x X K ip p e x+x K ip +p e x X = e x K ip p ex+x K ip+p ex X, 110 ip x ip X K ip p e x+x K ip +p e x X 111 = i p + P e x+x K ip p ex+x K ip +p e x X i p P e x X K ip p e x+x K ip +p ex X, and e x+ i p K ip p e x+x K ip +p e x X =e x K 1+iP p e x+x K 1+iP +p e x X. 11 Now, recall the recurrence relations [5], K 1+iP p e x+x = K ip p ex+x +ip pe x X K ip p e x+x, 113 K 1+iP +p e x X = K ip+p ex X ip +pe x+x K ip+p e x X. 114 So the previous relation 11 becomes e x+ i p K ip p e x+x K ip +p e x X = e x K ip p ex+x K ip+p ex X 115 +i P + p e x+x K ip p ex+x K ip +p e x X i P p e x X K ip p e x+x K ip +p ex X + P p K ip p e x+x K ip +p e x X.

16 Curtright, Fairlie, & Zachos Wigner functions hep-th/ The sum of 110, 111, and 115, shows that 109 is, indeed, satisfied. Integrating over X and P the product of Tx, p; X, P and the free particle Wigner function, as given here by 4π E 1 δp E, yields another expression for the Liouville Wigner function which checks against the previous result, 91. Using 9 and the parameter translation identity for the G-function, this other expression is just 94. Supersymmetric Liouville quantum mechanics is obtained by carrying through the Darboux construction detailed above with =1=m, for the choice Conventions essentially follow [8]. The first Hamiltonian of the essentially isospectral pair is then W x =e x. 116 H = p + e x e x, 117 and the allowed spectrum is 0 E<, including zero-energy, for which there is a bounded wave function normalized as part of the continuum, ψ 0 x = 1 π e ex. 118 The other, E>0, eigenfunctions are ψ E x = 1 4π E ex cosh again normalized so that + dx ψ E 1 x ψ E x =δe 1 E. The second Hamiltonian of the pair is π 1/ E K 1 i E ex +K1 +i E ex, 119 H = p + e x + e x, 10 and the allowed spectrum is 0 <E<, excluding zero energy 5.TheE>0eigenfunctions are then ψ Ex 1 = 4π E ex cosh π 1/ E ik 1 i E ex ik 1 +i E ex, 11 and may be obtained from the previous E>0 eigenfunctions, as ψ E x = 1 E x +Wψ E x. For both Hamiltonians, the Wigner functions are straightforward to construct directly, once again leading to the K-transform 9 and particular Meijer G-functions. We find it sufficient here to consider only the ground state for H, f 0 x, p = 1 π + dy e ex coshy/ iyp = π K ipe x, 1 a single modified Bessel function. It smoothly satisfies p iw x f 0 = 0, and hence the -genvalue equation H f 0 =0. Acknowledgements We thank Y Hosotani for helpful discussions. This work was supported in part by NSF grant PHY , and by the US Department of Energy, Division of High Energy Physics, Contract W ENG The candidate ψ 0x =1/ψ 0x = πexpe x solves the Schrödinger equation, but is obviously unbounded, as expected.

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