The Inconsistency of the Usual Galilean Transformation in Quantum Mechanics and How To Fix It Daniel M. Greenberger City College of New York, New York, NY 10031, USA Reprint requests to Prof. D. G.; [email protected] Z. Naturforsch. 56a, 67-75 (2001); received February 8, 2001 Presented at the 3rd Workshop on Mysteries, Puzzles and Paradoxes in Quantum Mechanics, Gargnano, Italy, September 17-23, 2000 It is shown that the generally accepted statement that one cannot superpose states of different mass in non-relativistic quantum mechanics is inconsistent. It is pointed out that the extra phase induced in a moving system, which was previously thought to be unphysical, is merely the non-relativistic residue of the “twin-paradox” effect. In general, there are phase effects due to proper time differences between moving frames that do not vanish non-relativistically. There are also effects due to the equivalence of mass and energy in this limit. The remedy is to include both proper time and rest energy non-relativis- tically. This means generalizing the meaning of proper time beyond its classical meaning, and introduc­ ing the mass as its conjugate momentum. The result is an uncertainty principle between proper time and mass that is very general, and an integral role for both concepts as operators in non-relativistic physics. Key words: Galilean Transformation; Symmetry; Superselection Rules; Proper Time; Mass. Introduction We show that from another point of view it is neces­ sary to be able to superpose different mass states non- It is a generally accepted rule in non-relativistic quan­ relativistically. Then we show that there is indeed a tum theory that one cannot coherently superpose parti­ physical interpretation to the phase shift described cles of different masses. This rule comes from a demon­ above, namely that it is the “twin paradox” effect [3], stration by Bargman [1,2] that, if one makes a series of the residue of the different proper times elapsed transformations to moving coordinate systems, using between the original inertial system, and the set of ac­ the Galilean transformation, and eventually arrives back celerating systems used to describe the particle. So these at the original system, one will have produced a phase two sets of coordinate systems are not physically equiv­ shift between the components of the wave function rep­ alent. The amazing thing is that the difference in proper resenting different mass states. However, the argument times between them produces a residue in the non-rela­ goes, these transformations are unphysical. How can tivistic limit. This is a phase effect and would not be merely looking at a wave function from a different co­ seen in classical mechanics, but it shows that there are ordinate system possibly induce a physically meaning­ non-relativistic effects due to proper time in quantum ful phase shift that could be detected in an interference theory, and a correct treatment of non-relativistic quan­ experiment? In order to eliminate this possibility, one tum theory must include the concept of proper time, and imposes the superselection rule that one cannot super­ the equivalence of mass and energy. The superselection pose wave functions of different masses. rule prohibiting the superposition of different mass But there is a very puzzling feature to this result. Rel- states is inconsistent with the non-relativistic limit of ativistically one can coherently combine wave func­ the Lorentz transformation. tions of different mass states, and the relevant transfor­ mation here is the Lorentz transformation. In the non- relativistic limit this reduces to the Galilean transforma­ Necessity of Non-Relativistic Mass Superpositions tion, but the phase shift does not disappear in this limit. How can a meaningful relativistic effect that has a non­ Consider the case of a particle of mass M at rest in an zero non-relativistic limit be eliminated by a superselec­ inertial system S decaying into two particles of mass m, tion rule, and why should it? Shouldn’t instead this non- flying off in opposite directions (along the x axis) at relativistic limit have a physical interpretation? speed v. Non-relativistically, since mass is conserved, 0932-0784 7 01 / 0100-0067 $ 06.00 © Verlag der Zeitschrift für Naturforschung, Tübingen • www.znaturforsch.com D. M. Greenberger • The Inconsistency of the Galilean Transformation in Quantum Mechanics k,co M (a) M r\Lr\j y (a) InS L, In S I*. □ v □ (b) (b) In S’ ] u InS’ |u 1 1 Fig. 1. Decay of a particle into two equal daughter particles, Fig. 2. Decay of an excited particle into its ground state plus (a) A particle of mass M decays into two equal particles of a photon, (a) The particle of mass M decays into one of mass mass m, seen from a system S where M is at rest. Non-relativ- m plus a photon. The internal energy e supplies the recoil and istically, since mass is conserved, M = 2m. The internal ener­ photon energies. Here, e and the relativistic mass change are gy e supplies the velocity v of the daughters, and e is of order of order v/c. (b) The same decay, as seen from the system S', ( f 2/c2), as is the relativistic mass change, (b) The same decay, as before. as seen from a system S', moving along the -y axis at speed u. one would have (see Figure la). Numerically, both of these give the same information as (2a, b, c). However, ( a) says that what looked like en­ M = 2m, 6 ergy in the rest system S (in (2a)), looks like inertial s - mv2. (1) mass in the moving system S', and is essentially one of Here, e is the internal energy of the particle M. Non-rel- Einstein’s early derivations of, and is the true meaning ativistically, the mass and enery of the particle are con­ of, the equivalence of mass and energy. Again, because served separately. Relativistically, of (3), there are no surprises here. However, the situation is very different if one consid­ M = 2 my, ers M to be the mass of an atom in an excited state, and y= (1 - v2/c2)~112, (2a, b, c) it decays to its ground state with mass m, emitting a pho­ M = 2 m + etc2. ton of frequency co (see Figure 2a). Classically, photons do not exist, but this is a situation that one frequently There is no conflict here since relativistically, treats non-relativistically in quantum mechanics. In this M =2m + 0(v2/c2). (3) case, non-relativistic momentum conservation along the ^-direction in the S system gives Now consider the same decay from a system S', mov­ ing downward along the - y axis at velocity u (see Fig­ mv = hk. (7) ure lb). Here, looking at momentum conservation (non- relativistically) along the y axis, In the S' system momentum conservation in the y direc­ tion gives (see Figure 2b) Mu = 2 mu. (4) Mu = mu + hku/c, ( ) This is perfectly consistent with (1). Relativistically, us 8 ing = (x, y, z, ict), (5) because both particle and the photon carry momentum '1 0 0 0 ' ~vyv' vYv in the y direction. But this gives the relation 0 0 - iyuu/c 0 Yu uYuYv M = m + hco/c2, (9) 0 0 1 0 0 0 _0 iyuu/c 0 Yu . icYv. .icYuYv. which, together with (7), gives and momentum and energy conservation in the y direc­ M = m (1 + v/c), (10) tion gives a first order vie effect, in direct contrast to (3). So even Muyu = 2 muyuyv, non-relativistically, one cannot ignore the increase in Mcyu = 2mcyuyv. (6a, b) mass, as well as in energy, of the excited state. D. M. Greenberger • The Inconsistency of the Galilean Transformation in Quantum Mechanics 69 Relativistically, (5) gives for the photon, (11) h (16) '1 0 0 0 ' ’ k ' ~ k ' iyuulc kuyu/ c 0 Yu 0 - 0 Here we have replaced t’ by t. One can eliminate the un 0 0 1 0 0 0 wanted Vip term by the substitution 0 iyuu/c 0 Yu ico/c icoyu / c ip(r, t) = eiJ{r,t) (p(r', ).t (17) and relativistic jc-momentum and energy conservation in Then, the system S becomes mvyv = fik, V'ip = (V'ip + iV'f) eif, Me = mcyv + fico/c, (12) V'2ip = {V'2ip + 2iV'f- V'(p+(pV'2f + <p(V'f)2)e‘f, while in the S' system, y-mo men turn and energy become i>= (f)+ if<p) eif, (18) Muyu - muyuyv + Tikuyjc, and the Schrödinger equation becomes Mcyu = mcyuyv + Ticuyjc. (13) n 2 Here it is very clear that energy in the S system becomes (V,z(p + 2 iV'f-V'(p+ i(pV f - (V 'fY <p) 2 m inertial mass in the S' system, and that the increase of mass in the excited state is necessary for the consisten­ = ifi [(<p +if(p) -g.(V'(p + i(pv' f )]. (19) cy of the theory. Non-relativistically, (12) reduces to (9). Quantum mechanically there are many situations One can choose/such as to eliminate the terms in V'(p, where one has coherent combinations of the excited and which gives f = — %-r'+ g(t). Then one can choose ground state, which are generally thought of as superpo­ n g{t) such as to eliminate the purely time-dependent sitions of different energy states. But it is clear that for terms, and one finally arrives at the non-relativistic Galilean transformation to be con­ sistent with the Lorentz transformation, they must also , = * ( be considered as superpositions of different mass states. V'2(p + mf; ■ r'(p = ih(p, 2m The Quantum-Mechanical Galilean Transformation ipir, t) —= ea h J(pir',t).