Nonrelativistic particles and wave equations

Abstract

This paper is devoted to a detailed study of nonrelativistic particles and their properties, as described by Galilei invariant wave equations, in order to obtain a precise distinction between the specifically relativistic properties of elementary quantum mechanical systems and those which are also shared by nonrelativistic systems. After having emphasized that spin, for instance, is not such a specifically relativistic effect, we construct wave equations for nonrelativistic particles with any spin. Our derivation is based upon the theory of representations of the Galilei group, which define nonrelativistic particles. We particularly study the spin 1/2 case where we introduce a four-component wave equation, the nonrelativistic analogue of the Dirac equation. It leads to the conclusion that the spin magnetic moment, with its Lande factor g = 2, is not a relativistic property. More generally, nonrelativistic particles seem to possess intrinsic moments with the same values as their relativistic counterparts, but are found to possess no higher electromagnetic multipole moments. Studying "galilean electromagnetism" (i.e. the theory of spin 1 massless particles), we show that only the displacement current is responsible for the breakdown of galilean invariance in Maxwell equations, and we make some comments about such a "nonrelativistic electromagnetism". Comparing the connection between wave equations and the invariance group in both the relativistic and the nonrelativistic case, we are finally led to some vexing questions about the very concept of wave equations.

Arbitrary spin Galilean oscillator

“…Making use of the usual result for the energy levels of the SHO in spherical coordinates (namely, ω[2n + ℓ + 3 2 ] where n, ℓ = 0, 1, 2...), this yields the eigenvalue spectrum for the spin one-half Galilean oscillator as…”

Index Theory and Adiabatic Limit in QFT

“…27 Yet the full version (necessity and sufficiency) would be very desirable as we expect in this case that the integrality of the index (necessity and sufficiency condition) puts strong limitations on the allowable values of the deformation parameter, i.e. on the coupling constant g. This goes outside our proposal, but we expect that in general situation (not only for QED) an analog of Fedosov theorem holds: namely that the actual asymptotic representation does exist (and so the adiabatic limit exist) whenever the index induced by (A, D, H) 28 fulfills some integrality conditions.…”

“…Perhaps it is worth to emphasize that the inclusion of the algebra of spacetime coordinates as a structural ingredient of the 27 In fact Fedosov proved his theorem on sufficiency for the existence of asymptotic representation for compact manifolds only. But an analogue theorem is certainly true for the non-compact case as well (after some reasonable assumptions of course).…”

“…In fact if we want to describe the Galilean spacetime spectrally and explain in addition its connection to non-relativistic quantum fields, it is the central Bargmann extension 39 G of the inhomogeneous Galilean group which is more natural here as a symmetry group. Indeed the appropriate Dirac operator 40 we should use here is the non relativistic Dirac operator −i∂ t ⊗ A − i∂ i ⊗ B i + 1 ⊗ C found by Lévy-Leblond [27], where A, B i , C are elements of a Clifford algebra over the (five dimensional) extension 41 of tangent space with a positive definite and singular quadratic form in it. Indeed, in this 37 Although action of the extension on the Galilean spacetime degenerates to the ordinary action of the inhomogeneous Galilean group, using of the Bargmann extension is essential if one intents to describe spectrally the Galilean spacetime manifold, see below for some further comments.…”

“…Galilean case the Krein-type space H corresponding to the more degenerate "metric" structure of M is slightly different from the ordinary Krein space and may be reduced to a positive definite inner product space with non-trivial closed subspace of zero norm vectors, which cannot be quotiened out, as the kernel subspace reflects the degenerate character of the "metric" structure on M. This can be immediately seen from the results of [27], as all the algebra elements A, B i , C can be (algebraically) generated from the standard Clifford algebra over a positive definite metric of rank 4. Indeed as follows from [27], we may put: …”