Abstract
Space and time have a double role in quantum mechanics. On the one hand, they form the arena for physical events; physical systems occur in spacetime. On the other hand, the symmetries of the underlying spacetime specify the description of the system and dictate its basic properties. In the relativistic case spacetime symmetries are expressed with respect to the Poincaré group, while in the nonrelativistic case the relevant symmetry group is the Galilei group. This Chap. 17 presents an analysis of covariant time observables in the case of classical Hamiltonian mechanics and analogous considerations in quantum mechanics.
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Notes
- 1.
The one-dimensional case of this theorem extends easily to the three-dimensional case.
- 2.
In Galilei relativistic quantum mechanics the structure of the evolution generator, the energy operator, is determined to a large extent by the symmetry requirements [1–3]. Here we recall only that for a spinless free particle moving in \(\mathbb R^3\), \(H=\frac{1}{2m}\varvec{P}^2\) whereas it is of the form \(H=\frac{1}{2m}\sum _{i=1}^3(\varvec{P}-\varvec{A}(\varvec{Q}))^2 + V(\varvec{Q})\) if the particle is moving in an external field described by vector and scalar potentials \(\varvec{A}:\mathbb R^3\rightarrow \mathbb R^3\) and \(V:\mathbb R^3\rightarrow \mathbb R\) (measurable functions), respectively. Here \(\varvec{P}=(P_1,P_2,P_3), \; \varvec{Q}=(Q_1,Q_2,Q_3)\).
- 3.
This is the set of states \(\varrho \) whose eigenvectors belong to the domain of \(T^{(0)}\).
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Busch, P., Lahti, P., Pellonpää, JP., Ylinen, K. (2016). Time and Energy. In: Quantum Measurement. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-43389-9_17
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