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.There are nonetheless borderline cases where particles are prepared with awell defined momentum (according to a classical description of the preparationprocedure) and then these particles diffract like waves, so that de Broglie sformula »p = h is experimentally verifiable.For example, electrons launched11  2The conversion factor hc 7 × 10  55 kg/Hz is just ridiculous.12A.Zeilinger, Am.J.Phys.58 (1990) 103.The Euclidean group 229with an energy of 100 eV have a wavelength » = 1.23 Å, comparable to crystallattice spacings.Neutrons with energy around 0.03 eV are copiously producedin nuclear reactors (in thermal units, 0.03 eV/kB = 348 K).The correspondingwavelength is 1.65 Å.These neutrons are routinely used for solid state studies.Their wavelike diffraction by the crystalline lattice may be elastic, as for X-rays.However, the same waves may also be scattered inelastically, and then theneutrons behave just as ordinary particles, exchanging energy and momentumwith the elastic vibration modes of the lattice.Warning: The so-called principle of correspondence, which relates classical andquantum dynamics, is tricky and elusive.Quantum mechanics is formulated in aseparable Hilbert space and it has a fundamentally discrete character.Classicalmechanics is intrinsically continuous.Therefore, any correspondence betweenthem is necessarily fuzzy.I shall return to this problem in Chapter 10.8-5.The Euclidean groupThe Euclidean group consists of all possible rigid motions (translations androtations) in the ordinary Euclidean three dimensional space R3.If we ignoredistortions of the spacetime geometry due to gravitational effects, the physicalspace in which we live has a Euclidean structure.Therefore, the Euclideangroup corresponds to a physical symmetry, and rigid motions are representedby unitary matrices in the quantum mechanical Hilbert space.Dynamical variables vs external parametersWe have just created an exquisite fiction: a perfectly empty space which isrigorously symmetric.There is nothing in it to indicate where to put the originof a Cartesian coordinate system, and how to orient its axes.The laws ofphysics, Maxwell s equations say, are written in the same way in all these mentalcoordinate systems.However, when we clutter our pristine space with materialobjects (buildings, magnets, particle detectors and the like) we destroy thatsymmetry.It then becomes possible to say that the origin of the xyz axes islocated at this particular corner in our laboratory, and that the axes are parallelto specified walls.Yet the symmetry is not completely lost it only is more complicated.If wecarefully move the entire building, with the magnets and the particle detectors,and the coordinate system which was fastened to the walls of the experimentalhall (this Herculean job is a passive transformation) and if we likewise moveall the particles for which we are writing a Schrödinger equation (this is theactive transformation) then the new form of the Schrödinger equation is thesame as the old form.It is impossible to infer, just by observing the behaviorof the quantum particles, that the building and all its equipment have been230 Spacetime Symmetriestransported elsewhere, and the quantum particles too.We shall therefore distinguish two classes of physical objects.Those whosebehavior we are investigating are described by dynamical variables; they obeyNewton s equations, or the Schrödinger equation, or any other appropriate equa-tions of motion.For these dynamical variables, a symmetry is represented bya canonical transformation in classical physics, or a unitary transformation inquantum theory.And, on the other hand, there are auxiliary objects (magnets,detectors, etc.) whose properties are supposedly known, and whose behaviorcan be arbitrarily prescribed.These objects are not described by dynamicalvariables and they do not obey equations of motion.Their motion, if any, isspecified by us.Depending on the level of accuracy that we demand, the same object may beconsidered either as part of the dynamical system for which we write equationsof motion, or as something external to it, specified by nondynamical variables.For example, in the most elementary treatment of the hydrogen atom, thereis a point-like proton, located at a given position, R, and represented by afixed Coulomb potential, V =  e2 /| R  r |.Only the components of r (theposition of the electron) are considered as dynamical variables.Those of R areexternal parameters.In a more accurate treatment, the components of R tooare dynamical variables, and the proton is a full partner in the hydrogen atomdynamics.In that case, it is obvious that (R  r ) is invariant under a rigidtranslation of the atom: R ’! R + a and r ’! r + a.However, even in the hybrid description, where only r is dynamical and Ris an externally controlled parameter, there still is a well defined meaning totranslation invariance [ Pobierz caÅ‚ość w formacie PDF ]