UMN-TH-1720-98-TPI-MINN-98/15 UFIFT-HEP-97-18, DFPD98/TH/41 September 1998, hep-th/9809127 The Equivalence Postulate of Quantum Mechanics Alon E. Faraggi1 and Marco Matone2 1 Department of Physics University of Minnesota, Minneapolis MN 55455, USA e-mail: [email protected] 2 Department of Physics “G. Galilei” – Istituto Nazionale di Fisica Nucleare University of Padova, Via Marzolo, 8 – 35131 Padova, Italy e-mail: [email protected] Abstract The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parameterization is at the heart of the recently formulated Equivalence Principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). This is a third–order non–linear differential equation which provides a trajectory representation of Quantum Mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second–order linear differential equations. This reflects in an intrinsic ψD–ψ duality between linearly independent solutions of the Schr¨odinger equation. Unlike Bohm’s theory, there is a non–trivial action even for bound states and no pilot–wave guide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave–function is made. For example, tunnelling is a direct consequence of the arXiv:hep-th/9809127v2 6 Aug 1999 quantum potential which differs from the Bohmian one and plays the role of particle’s self–energy. Furthermore, the QSHJE is defined only if the ratio ψD/ψ is a local homeomorphism of the extended real line into itself. This is an important feature as the L2(R) condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave–function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ψD/ψ at q = as implied by the ±∞ EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non–stationary higher dimensional quantum HJ equation and the relativistic extension are derived. Contents 1 Introduction 1 2 Canonical and coordinate transformations 13 2.1 TheclassicalHJequation .............................. .... 13 2.2 Coordinate transformations and the distinguished frame . ............ 15 2.3 Time parameterization and space in Classical Mechanics . ......... 16 2.4 v–transformations.................................... ... 18 2.5 Ageometricalpicture................................ ..... 19 3 p–q duality and the Legendre transformation 20 3.1 Energy conservation and classical p–q duality........................ 20 3.2 Thedualreducedaction .............................. ..... 22 3.3 ThedualHJequation ................................. ... 23 4 M¨obius symmetry and the canonical equation 25 4.1 –transformation induced by v ............................... 25 T0 4.2 Thecanonicalequation............................... ..... 26 4.3 Legendre transformation as M¨obius transformation . ............. 27 4.4 M¨obius invariance of .................................... 27 U 4.5 Canonical equation, dynamics and initial conditions . ......... 28 4.6 Covariance........................................ ... 29 5 Legendre duality 31 5.1 u–transformations.................................... ... 31 5.2 Thedualcanonicalequation . ...... 32 5.3 Legendrebrackets................................. ...... 32 5.4 p and q asLegendrepair ................................... 33 6 Self–dual states 34 6.1 Where do the – picturesoverlap?............................ 34 S0 T0 6.2 Dilatations......................................... .. 36 u v 6.3 Changing sign: pq p qu = pq = pvq ......................... 36 v−→ u − 6.4 Self–dual states: q = qu, p = pv .............................. 37 7 The Equivalence Principle 39 7.1 Equivalence Principle and v–transformations . .. .. 39 7.2 Classical Mechanics and the Equivalence Principle . ........ 40 7.3 Modifying the classical HJ equation . ...... 40 7.4 Covariance as consistency condition . ........ 41 i 8 Implementing the Equivalence Principle 41 8.1 statesasinhomogeneousterms . .... 42 W 8.2 Thecocyclecondition................................. .... 43 8.3 Cocycle condition and M¨obius symmetry . ....... 44 8.4 Selectingaself–dualstate ............................ ...... 46 9 The quantum stationary HJ equation 46 9.1 Complex entries and real values of the Schwarzian derivative . ............ 47 2 9.2 (qa; qb)= β qa, qb ..................................... 48 − 4m { } 9.3 Uniquenessofthesolution ............................. ..... 52 10 The Schr¨odinger equation 53 10.1 QSHJEandSchr¨odingerequation . ........ 53 10.2 Wave–functionandCopenhagenSchool . .......... 54 10.3 Q andthedualSchr¨odingerequation. 56 W ←→ 10.4 sd = 0 andexistence oftheLegendre transformation . ... 57 W W 10.5 The trivializing map . .. 58 11 Quantum HJ equation and the reality condition 59 11.1 Quantum HJ equation and reality of R and ˆ ....................... 59 S0 11.2 The classical limit . ... 61 11.3 The WKB approximation and the quantum HJ equation . ........ 63 12 The wave equation of Classical Mechanics 63 12.1 The wave–function of Classical Mechanics . ......... 64 12.2 Probabilistic interpretation of the classical wave–function . .............. 64 12.3 statesinClassicalandQuantumMechanics . .... 65 W 13 The trivializing map and quantum transformations 66 13.1 Onthequantum canonicaltransformations . ........... 68 13.2 Analogy with uniformization theory . ....... 68 13.3 Quantumtransformations . ....... 72 13.4 Classical and quantum potentials and equivalence of curves . ............. 73 13.5 Areafunctionandcommutator . ....... 75 14 Canonical variables and M¨obius transformations 76 14.1 The canonical variables of the state 0 ........................... 76 W 14.2 Time and elliptic curve as moduli of the state 0 ..................... 77 W 14.3 The canonical variables of arbitrary states . ........... 78 14.4 Thesymmetriesofthewave–function . ......... 83 ii 15 Trajectories and the Equivalence Principle 87 15.1 Differential properties of the reduced action . ........... 87 15.2 The Equivalence Principle and the trivializing map . ...... 88 15.3 Equivalence Principle and local homeomorphicity of v–maps ............... 89 15.4 The M¨obius symmetry of the trivializing map . ....... 90 15.5Thefreeparticle ................................... .... 93 15.6 Timeparameterization ............................... ..... 94 15.7 Quantummassfield: dynamicalequation . ........ 95 16 Equivalence Principle and fundamental constants 96 16.1h ¯ 0, E 0andPlancklength ............................. 97 −→ −→ 16.2 Time,energyandtrajectories . ........102 17 Equivalence Principle, tunnelling and quantized spectra 104 17.1Tunnelling......................................... 104 17.2 Quantized spectra from the Equivalence Principle . ..........105 17.3 The index of the trivializing map . 111 17.4 Equivalence Principle and admissible potentials . ....... 111 18 The potential well and the harmonic oscillators 112 18.1Thepotentialwell.................................. 112 18.2 The infinitely deep potential well and the free particle . ...........116 18.3 The simple and double harmonic oscillators . ...... 118 18.4Thegeneralcase .................................. 122 19 Generalizations 124 19.1 Time–dependent Schr¨odinger equation . ..........124 19.2Higherdimension .................................... 127 19.3 Schr¨odinger equation and space compactification . .............131 19.4 Time–dependentcase............................... ...... 132 19.5 Relativistic extension . 132 20 Conclusions 133 iii 1 Introduction Twentieth century physics has revolutionized our view and understanding of the fundamental princi- ples underlying all physical phenomena. At its core, General Relativity (GR) and Quantum Mechanics (QM) reign unchallenged in their respective domains. Yet, the mutual coexistence of these two fun- damental theories remains elusive after many decades since their inception. While GR is based on underlying geometrical concepts, general covariance and the Equivalence Principle, the traditional for- mulation of QM does not follow from such principles. Rather, the conventional formulation of QM is based on concepts concerning the measurement process and the related axiomatic interpretation of the wave–function. This indicates that difficulties in understanding the synthesis of these two fundamental theories trace back to the apparently different nature of such principles. A deeper comprehension of such principles constitutes the basic theoretical challenge of the new millennium. Recently we showed in [1][2] that QM can in fact be derived from an Equivalence Principle (EP) which is reminiscent of Einstein’s EP [3]. In this paper we will expand and develop the approach initiated in Refs.[1][2]. We start in sect.2 with a critical investigation which will lead to formulate the EP. This includes a basic analysis of the role of time in Classical Mechanics (CM) and of the distinguished nature of the rest frame. In this process we will