The covariance of the Schr\"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr\"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
The covariance of the Schr"odinger equation under Galilei boosts and the compatibility of nonrelativistic quantum mechanics with Einstein's equivalence principle have been constrained for so long to the existence of a superselection rule which would prevent a quantum particle from being found in superposition states of different masses. In an effort to avoid this expedient, and thus allow nonrelativistic quantum mechanics to account for unstable particles, recent works have suggested that the usual Galilean transformations are inconsistent with the nonrelativistic limit implied by the Lorentz transformation. Here we approach the issue in a fundamentally different way. Using a formalism of unitary transformations and employing quantum reference frames rather than immaterial coordinate systems, we show that the Schr"odinger equation, although form variant, is fully compatible with the aforementioned principles of relativity.
We show that the superselection rule in the Galilean transformation, forbidding the superposition of states of different mass, in inconsistent with the nonrelativistic limit of the Lorentz transformation. We also point out that the extra Galilean phase is merely the residue of the "twin-paradox" effect, which does not vanish nonrelativistically. In general, there are phase effects due to proper time differences, and effects due to mass-energy equivalence, that do not vanish nonrelativistically but that are not handled adequately by the Galilean transformation.
The observational basis of quantum theory in accelerated systems is studied. The extension of Lorentz invariance to accelerated systems via the hypothesis of locality is discussed and the limitations of this hypothesis are pointed out. The nonlocal theory of accelerated observers is briefly described. Moreover, the main observational aspects of Dirac's equation in noninertial frames of reference are presented. The Galilean invariance of nonrelativistic quantum mechanics and the mass superselection rule are examined in the light of the invariance of physical laws under inhomogeneous Lorentz transformations.
In physics, every observation is made with respect to a frame of reference. Although reference frames are usually not considered as degrees of freedom, in all practical situations it is a physical system which constitutes a reference frame. Can a quantum system be considered as a reference frame and, if so, which description would it give of the world? Here, we introduce a general method to quantise reference frame transformations, which generalises the usual reference frame transformation to a "superposition of coordinate transformations". We describe states, measurement, and dynamical evolution in different quantum reference frames, without appealing to an external, absolute reference frame, and find that entanglement and superposition are frame-dependent features. The transformation also leads to a generalisation of the notion of covariance of dynamical physical laws, to an extension of the weak equivalence principle, and to the possibility of defining the rest frame of a quantum system.
The Schroedinger equation is up-to-a-phase invariant under the Galilei group. This phase leads to the Bargmann's superselection rule, which forbids the existence of the superposition of states with different masses and implies that unstable particles cannot be described consistently in non-relativistic quantum mechanics. In this paper we claim that Bargmann's rule neglects physical effects and that a proper description of non-relativistic quantum mechanics requires to take into account this phase through the Extended Galilei group and the definition of its action on spacetime coordinates.
Einstein's equivalence principle (EP) states the complete physical equivalence of a gravitational field and corresponding inertial field in an accelerated reference frame. However, to what extent the EP remains valid in non-relativistic quantum mechanics is a controversial issue. To avoid violation of the EP, Bargmann's superselection rule forbids a coherent superposition of states with different masses. Here we suggest a quantum simulation of non-relativistic Schrödinger particle dynamics in non-inertial reference frames, which is based on the propagation of polarization-entangled photon pairs in curved and birefringent optical waveguides and Hong-Ou-Mandel quantum interference measurement. The photonic simulator can emulate superposition of mass states, which would lead to violation of the EP.
In (1) a Galilean transformation x’ = x-vt and t’=t is applied to the Schrodinger equation id/dt (partial) W(x,t) = -1/2m d/dx d/dx W(x,t) for a free particle, where W(x,t) is the wavefunction, and it is argued that W’(x’,t’) does not equal W(x(x’t,t’),t(x’,t’)). Instead it is shown that W’(x’,t’) = W(x,t) exp( -i/bar [mvx’+.5mvvt’] with x=x’+vt’ and t=t’. The author’s of (1) suggest this is due to remnants of special relativity creeping into the nonrelativistic situation. They then argue that this issue is already found in nonrelativistic classical mechanics, in particular in the Hamiltonian-Jacobi equation H(x, dS/dx) + dS/dt = 0 where S is the free particle action= t *Lagrangian and H is the Hamiltonian i.e 1/2m dS/dx dS/dx. They show that S’(x’,t’) = S(x,t) - (mvx’ +.5mvvt’) and also argue that this is due to a remnant of special relativity creeping into nonrelativistic classical mechanics. In this note, we consider the free particle action S= -Et+px nonrelativistically. (This form also holds for the relativistic free particle). It may be seen that such a solution satisfies the Hamilton-Jacobi equation, but the catch seems to be that one must transform both (x,t) and (p,E) to find a solution in a moving frame. The Galilean transformation leads to two different transformations, one for (x,t) and another for (p,E). A Lorentz transformation on the other hand treats (x,t) and (p,E) in the same way and -Et+px behaves as a dot product with a (1,-1) type metric. We argue that given the physical example of a particle with rest mass mo at x=0 at t=to, a moving frame transforms this into a new energy (including mo and kinetic energy), a p and x’,t’ such that x’/t’=v where -v is the velocity of the frame. One may suggest a more general transformation than the Galilean i.e. replace t’=t with t’=g(v) t. This then leads to -Et+px written in terms of E’,t’, p’,x’ equalling -E’t’ + p’x’ if g(v) = 1/sqrt(1-vv/bb) where b is a constant to make v/b dimensionless. Applying the transformation, which we assume holds for both (x,t) and (p,E) to light yields b=c. One may then show that -Et+px, written in terms of E’,t’,p’,x’, yields -E’t’ + p’x’,