LINEARIZING TRANSFORMATIONS AND RESULTING STRUCTURE. Arnold Sommerfeld Institute for Mathematical Physics. D Clausthal-Zellerfeld, Germany

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1 A FAMILY OF NONLINEAR SCHR OINGER EQUATIONS: LINEARIZING TRANSFORMATIONS AN RESULTING STRUCTURE H.-. oebner ;, G. A. Goldin 3, and P. Nattermann PostScript processed by the SLAC/ESY Libraries on 7 Feb 995. QUANT-PH Arnold Sommerfeld Institute for Mathematical Physics Institute for Theoretical Physics, Technical University of Clausthal Clausthal-Zellerfeld, Germany 3 epartments of Mathematics and Physics, Rutgers University New Brunswick, New Jersey 08903, USA Abstract We examine a recently-proposed family of nonlinear Schrodinger equations 4 with respect to a group of transformations that linearize a subfamily of them. We investigate the structure of the whole family with respect to the linearizing transformations, and propose a new, invariant parameterization.. INTROUCTION Previous work {5 on the representation theory of an innite-dimensional kinematical algebra on IR 3, and the corresponding innite-dimensional group, led to a Fokker- Planck type of equation for the quantum-mechanical probability density and current, t = ~ r ~ j + ; (.) and in turn to a family F of nonlinear Schrodinger equations. F is parameterized by the classication parameter of the unitarily inequivalent group representations (the diusion coecient in Eq. (.)), and ve real model parameters 0 c ;:::; 0 c 5 : 0 h ih t = m +V(~x) +i h +h 0 5X c j R j [ ] A : (.) Here 0 also has the dimensions of a diusion coecient (so that the c j are dimensionless), and the nonlinear functionals R j are complex homogeneous of degree zero, dened by: r R [ ]:= ~ J ~ ; R [ ]:= ; R J 3[ ]:= ~ ; J R 4 [ ]:= ~ r ~ (.3) ; R 5 [ ]:= (~ r) ; where := and ~ J := Im( ~ r ) =(m=h) ~ j. j=

2 A subfamily of these equations, characterized by 0 c = = 0 c 4, together with c +c 5 = 0 and c 3 = 0, satises Ehrenfest's theorem in quantum mechanics 4, and is linearizable via a nonlinear transformation 6,7. In this short note we sketch some ideas connected with the linearizing transformations; for a detailed derivation and description with emphasis on the physical interpretation, we refer to forthcoming articles 8.. LINEARIZATION can be transformed into linear Schro- = N( ), if the remaining unspecied The members of the Ehrenfest subfamily F Ehr dinger equations by a transformation 7 0 model parameter 0 c satises 4m h 0 c < 4m h : (.) Here N depends on two real parameters ; IR; 6= 0, and is given by 0 := N (;) ( ) = (++i) ( +i) = j j e i( ln j j+ arg ) : (.) This maps solutions of the Ehrenfest subfamily of (.) into solutions of the linear Schrodinger equation, i h h t 0 = m +V(~x) 0 ; (.3) for the choices = m h 4m 4m h 0 c ; = h 4m 4m h 0 c : (.4) h However, it should be noted that for non-integer values of, the map N is not actually well-dened by Eq. (.) for all. The statements in this paper therefore depend, in some cases, on an appropriate selection of wave functions. The transformations N are local, in that they depend only on the values of and. They also respect the projective structure of the Hilbert space H = L (IR 3 ;d 3 x); i. e. for any complex number c, N (;) (c )=jcje i( ln jcj+ arg c) N (;) ( ) ; (.5) whence (c ) 0 belongs to the same ray as 0. Furthermore the transformations leave the symplectic structure = ^ on the Hilbert space H invariant up to a factor: N = N (;) (;) + N (;) ^ = N (;) N (;) = N (;) N (;) N (;) + N (;) ^ (.6) This gives a Hamiltonian formulation for the Ehrenfest family F Ehr, as has been noted elsewhere 4,5, and establishes a connection to the framework of Weinberg 9. The set N := fn (;) g of these nonlinear transformations obeys the group law of the ane group A () in one dimension, N ( ; ) N ( ; ) = N ( ; + ) : (.7)

3 A further, essential property of N Nis, of course, that it leaves the probability density invariant: 0 (~x; t) =(~x;t); (.8) where 0 is the density transformed under N. In a certain sense N is a nonlinear generalization of a linear, U()-gauge transformation. We shall call N the set of local projective nonlinear gauge transformations. 3. GAUGE INVARIANCE AN REPARAMETERIZATION The transformation of the current ~ J 0 under N N is given by ~J 0 := Im 0 ~ r 0 = 0 ~ rarg 0 = ~ J+ ~ r: (3.) In order to show the invariance of the family F of equations (.), we rewrite F wholly in terms of densities and currents. Using the expansion of the Laplacian = fir [ ]+(=)R [ ] R 3 [ ] (=4)R 5 [ ]g, we obtain the general form, i t = i X j= j R j [ ] + 5X j= j R j [ ] + 0 V : (3.) From (.), (.8) and (3.), we deduce that 0 = N (;) ( ) again fullls (3.), but with primed parameters: 0 = ; 0 = + ; 0 = + ; 0 = + ; 0 = = 3+ 4 ; 0 5= ; 0 0= 0 : (3.3) Thus we have that the 8-parameter family F of Eq. (3.) is invariant under the action of N ; i.e., under the action of the ane group A (). An appropriate description of F is by means of the orbits of A (); since there are group parameters, we look next for 6 functionally independent parameters that are invariant under A (). These are the gauge invariants. After some calculations, we obtain gauge-invariant parameters: = ; = ; 3 =+ 3 = ; 4 = 4 3 = ; 5 = ( + 5 ) ( + 4 )+ 3= ; 0 = 0 : (3.4) Now ifwe likewe can choose and as our group parameters ( 6= 0); this condition is fullled for all modications of the linear Schrodinger equation, as derives from the Laplacian in the Schrodinger equation. Inverting (3.4), we have = ( ); = ( + ) ; 3 =( 3 ) ; 4 = = ( )+ ( )( 3 ) ; 0 = 0 : (3.5) With this reparameterization the family of nonlinear Schrodinger equations is foliated in leaves characterized by 0 ;:::; 5, of subfamilies depending on the two group parameters

4 ;. The group of nonlinear gauge transformations N acts eectively on each leaf of the foliation. Because of (.8), the time-evolving probability density for all points ( ; )inagiven leaf is the same. Let us identify the gauge invariants for some special leaves: a. The linear Schrodinger equation corresponds to the values = h=m, = h=4m, 3 =h=m, 5 =h=8m, 0 ==h, and = = 4 =0. We then have only two nonvanishing gauge invariants: = h 8m ; 0 = m ( for V 6 0) : (3.6) Note that 0 and 0 are indeterminate if V 0. So h and m (or their quotient, in the free case) are gauge-invariant quantities for the family of equations. b. The Ehrenfest subfamily F Ehr corresponds to the values = h=m, = =, =, = h=4m + c 0, 3 =h=m, 4 =, 5 =h=8m c 0 =, and 0 ==h. Again there are just two nonzero gauge invariants: 0 as before, and = h 8m 0 h c m : (3.7) Here we rediscover the linearization of the Ehrenfest family, as when the righthand side of (3.7) is positive, we can introduce a new constant h 0 such that a linear Schrodinger equation with h 0 replacing h is contained in the orbit. c. For the more general, Galilei (Schrodinger) invariant subfamily 4,0 F Gal of F, characterized by the conditions c + c 4 = c 3 = 0, wehave the values: = h=m, = =, = c 0, = h=4m + c 0, 3 =h=m, 4 = c 0, 5 =h=8m+c 5 0, and 0 ==h.nowwe get four nonvanishing gauge invariants, taking independent values: 0,,, and FINAL REMARK We have noted that the transformations linearizing the Ehrenfest subfamily F Ehr of a family F of nonlinear Schrodinger equations can be viewed as generalizing the usual U()-gauge transformations, and act as a gauge group A () on the parameter space of the family. Wehave calculated a parameterization by means of the gauge invariants, together with the group parameters of A (). In connection with a more physical interpretation of our foliation of F,we quote a remark by Feynman and Hibbs : Indeed all measurements of quantum-mechanical systems could be made to reduce eventually to position and time measurements. Because of this possibility a theory formulated in terms of position measurements is complete enough in principle to describe all phenomena. (p. 96) If one adopts this point of view, quantum theories for which the wave functions give the same probability density in space and time are \in principle" equivalent. Hence the leaves of our foliation consist of sets of \in principle" equivalent quantum-mechanical evolution equations.

5 References. H.-. oebner and G. A. Goldin: \On a general nonlinear Schrodinger equation admitting diusion currents", Phys. Lett. A 6:397 (99).. H.-. oebner and G. A. Goldin: \Group theoretical foundations of nonlinear quantum mechanics", in \Annales de Fisica, Monograas, Vol. II", p. 44, CIEMAT, Madrid (993). 3. H.-. oebner and G. A. Goldin: \Manifolds, general symmetries, quantization, and nonlinear quantum mechanics", in: \Proceedings of the First German-Polish Symposium on Paricles and Fields, Rydzyna Castle, 99", p. 5, World Scientic, Singapore (993). 4. H.-. oebner and G. A. Goldin: \Properties of nonlinear Schrodinger equations associated with dieomorphism group representations", J. Phys. A: Math. Gen. 7:77 (994). 5. P. Nattermann: \Struktur und Eigenschaften einer Familie nichtlinearer Schrodingergleichungen", iplom thesis; Technical University of Clausthal (993). 6. P. Nattermann: \Solutions of the general oebner-goldin equation via nonlinear transformations", in: \Proceedings of the XXVI Symposium on Mathematical Physics, Torun, ecember 7{0, 993", p. 47, Nicolas Copernicus University Press, Torun (994). 7. G. Auberson and P. C. Sabatier: \On a class of homogemeous nonlinear Schrodinger equations", J. Math. Phys. 35:408 (994). 8. H.-. oebner, G. A. Goldin and P. Nattermann: work in progress, to be submitted for publication. 9. S. Weinberg: \Testing quantum mechanics", Ann. Phys. (NY) 94:336 (989). 0. P. Nattermann: \Maximal Lie symmetry of the free general oebner-goldin equation in + dimensions", Clausthal{preprint ASI-TPA/9/94.. R. P. Feynman and A. R. Hibbs: \Quantum Mechanics and Path Integrals", McGraw-Hill Book Company, New York (965).