Abstract
Bargmann’s superselection rule, which forbids the existence of superpositions of states with different mass and, therefore, implies the impossibility of describing unstable particles in non-relativistic quantum mechanics, arises as a consequence of demanding Galilean covariance of Schrödinger’s equation. However, the usual Galilean transformations inadequately describe the symmetries of non-relativistic quantum mechanics since they can produce phases in the wavefunction which are relativistic time contraction remnants and therefore, cannot be physically interpreted within the theory. In this paper we review the incompatibility between Bargmann’s rule and Lorentz transformations in the low-velocities limit, we analyze its classical origin and we show that the Extended Galilei group characterizes better the symmetries of the theory. Furthermore, we claim that a proper description of non-relativistic quantum mechanics requires a modification of the notion of spacetime in the corresponding limit, which is noticeable only for quantum particles.
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Notes
The author thanks an anonymous referee for referring to these papers.
As it is well known, Galilean transformations leave the line elements dt 2 and d x 2 invariant, separately. The “metric” η μν introduced here corresponds to the dominant temporal part only, and the fact that Galilean transformations do not mix spatial and temporal coordinates is contained in its degeneracy.
It transforms properly not only the coordinates but the momentum and energy as well, cf. (24).
An action of a connected Lie group on a symplectic manifold is called a Poisson action if the Hamiltonian functions H a (where a is an element of the corresponding Lie algebra) for one-parameter groups are single-valued, and chosen so that the Hamiltonian function depends linearly on elements of the Lie algebra and so that the Hamiltonian function of a commutator is equal to the Poisson bracket of the Hamiltonian functions:
$$H_{[a,b]}=\{H_a,H_b\}, $$see Ref. [16], p. 372.
The author thanks an anonymous referee for bringing to his attention this remark.
The author thanks an anonymous referee for bringing to his attention this fact.
Notice that in the non-relativistic limit, quantum states ψ are such that |P ψ|≪mc|ψ| and therefore \(|H\psi|=mc^{2}|\psi|+\mathcal{O}(c^{0})\), that is, while H can generate non-relativistic time translations even for time intervals of \(\mathcal{O}(1/c^{2})\), P produces non-relativistic spatial translations only for spatial intervals of order c 0. And consequently a quantum particle can only keep track of corrections of order 1/c 2 in the time coordinate.
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Acknowledgements
It is a pleasure to thank C. Chryssomalakos, E. Okon, D. Sudarsky and Y. Bonder for helpful discussions. This work has been partially supported by the Czech Ministry of Education, Youth and Sports within the project LC06002.
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Hernandez-Coronado, H. From Bargmann’s Superselection Rule to Quantum Newtonian Spacetime. Found Phys 42, 1350–1364 (2012). https://doi.org/10.1007/s10701-012-9673-5
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DOI: https://doi.org/10.1007/s10701-012-9673-5