On Galilei Invariance in Quantum Mechanics and the Bargmann Superselection Rule

“This is done not for a free particle [1-6] but in the more 1 In field theory such an additional input is, for example, given by the principle of locality.”

“This is done not for a free particle [1-6] but in the more 1 In field theory such an additional input is, for example, given by the principle of locality.”

“It seems to be a generally accepted text-book wisdom that non-relativistic quantum mechanics has superselection rules for the total mass M [1-7].”

“Similar views were also expressed in [12].”

“tructive to compare the situation to ordinary quantum mechanics in a translation invariant context. In the latter case, 9 For some background material on continuous superselection rules, see e.g. [10][11]. 12 translation invariance is not interpreted to generally prevent us from forming superpositions of plane waves w…”

“Equivalently [9], we can have a proper representation of a slightly larger group, the so-called extended Galilei group Ḡ, which is a central extension of G by a group Z isomorphic to R(4) This means that all of Z lies in the centre of Ḡ and that the quotient group Ḡ/Z is isomorphic to G.”

“ervables to acquire a non-trivial centre. 2 Sometimes purely kinematical symmetry groups are invoked in the definition of quantum mechanical state spaces, before any dynamical laws are given. See e.g. [8]. This should be distinguished from the dynamical notion of symmetry used here. 2 general context of nspinless parti…”

“nsformation of a scalar function on spacetime under the diffeomorphism ˝ g. The g-dependent phase function : R4 !R is determined up to spacetime independent additions. A convenient choice is (compare [3]) g(t;~x) = m ~ h ~v(~x ~a) 1 2 ~v 2(t b) i : (9) It is immediate that the self-interaction term in (1) and (4) does…”

“From the point of view of non-relativistic quantum mechanics in the lower dimensional Newton-Cartan spacetime we have an example of the much discussed fact [42] that the mass in non-relativistic quantum mechanics may be used to label super-selection sectors.”

“There are certain analogs of this on the classical lev e ; see [24].”

“The new dual coordinate ζ is related to either the scalar mass M for the Schrödinger algebra (this fact was first noted by Giulini [18] for the Galilei algebra) or else to the vector of the rapidities γ for the conformal Galilei algebra.”

“34 Thedualisationidea[138]hasanotherapplication:byworkingoutthe n -pointfunctionindualspace,before back-transforming, the n -point functions can be shown to obey causality and therefore must be interpreted as response functions and not as correlators [104, 140].”

“In this paper we have shown that logarithmic terms may also appear in the NRCFT’s.”

“In order to apply our framework, we consider the central extension of the Galilei group, ~ G (see, for example, 66,67 ).”

“See [5] for elaborations on this simple procedure.”

“And in terms of the Fourier transform φ̃m, the previous equation coincides with the equation defined by Giulini [11].”

“(3) is the central extension of the Galilei group [39, 40].”

“How could it be that the equivalence principle, a cardinal tenet of modern physics, turns out to depend on a seemingly contingent rule? A rule which can only be considered as approximate since it lacks a relativistic counterpart (see [20])? Moreover, the imposition of the SSR itself is questionable because, as Giulini…”

“n caveat underling derivation of Bargmann‘s superselection rule is that it imposes restrictions on a kinematical level for states that, on the dynamical level, we would not know exactly how to handle [7,18]: if mass enters in the Schr¨odinger equation as a kinematical (rather than dynamical) variable, how can one cons…”

“Giulini [5] has stressed that Bargmann rule has a not well-defined status (comparing with other superselection rules).”

“14) is in agreement with its accepted form [6, 10].”

“Note that the (Bargmann) mass-superselection rule does not apply, because the masses of the particles are no dynamical variables [18].”

“…issues, as it implies the possibility of superposition of mass eigenstates, which (thus far) we have no experimental evidence for, and which complicates the dynamics already at the nonrelativistic level [30], due to the presence of the mass in the kinetic energy term of the Schr ¨ odinger equation.”