What can we learn from the conformal noninvariance of the Klein–Gordon equation?

Abstract

It is well known that the Klein–Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. We show that such a noninvariance provides nontrivial physical insights at different levels, first within the fully relativistic regime, then in the nonrelativistic regime leading to the Schrödinger equation, and then within the de Broglie–Bohm causal interpretation of quantum mechanics. The conformal noninvariance of the Klein–Gordon equation coupled to a vector potential is confronted with the conformal invariance of Maxwell’s equations in the presence of a charged current. The conformal invariance of the nonminimally coupled Klein–Gordon equation to gravity is then examined in light of the conformal invariance of Maxwell’s equations. Finally, the consequence of the noninvariance of the equation on the Aharonov–Bohm effect in curved space–time is discussed.

Quantum conformal symmetries for spacetimes in superposition

“…e.g. [30,31]]. In particular, we offer a new perspective by viewing the description in terms of fields on a modified Minkowski spacetime as a physically instantiated situation instead of a mere computational tool [32,33].…”

Revisiting the Schrödinger-Dirac equation

“…(2.1), one cannot help but think of the conformally invariant version of the non-minimally coupled Klein-Gordon equation in curved spacetime: (g µν ∇ µ ∇ ν + m 2 + 1 6 R) φ = 0. This equation is conformally invariant only because of the presence of the specific factor 1 6 in front of the Ricci scalar R. Any other factor in front of R in the latter equation would not render it conformally invariant [9]. One might then naturally expect that while the Schrödinger-Dirac equation (2.1) is not conformally invariant, a simply different numerical factor in front of R in that equation could render the latter conformally invariant.…”

“…On the other hand, it is also a known fact that when conformally deforming spacetime the Dirac equation remains conformally invariant whereas one easily shows that the Schrödinger-Dirac equation does not. When recalling that the Klein-Gordon equation in curved spacetime is not conformally invariant but its non-minimally coupled version is [9], it becomes of great interest to seek a generalization of the already non-minimally coupled Schrödinger-Dirac equation that would also be conformally invariant. We show in this paper that such a generalization does indeed exist and that it requires the spinor field to conformally transform by bringing in a conformal factor that is different from the one required by the Dirac equation.…”

Plane-wave and wave-packet neutrino flavor oscillations in vacuum in conformal coupling models

“…Thus, when a free neutrino mass eigenstate of mass m j and of energy E j is conformally coupled to the spacetime, it actually propagates in the curved spacetime of metric g µν = Ω −2 gµν as a 'free' neutrino mass eigenstate of mass m j and of energy Ẽ j . Under such a conformal deformation of the metric, the masses and energies of quantum particles are related, as usual [41], by m j = Ωm j and Ẽ j = ΩE j , respectively. Therefore, since the characteristic length L jk osc of neutrino flavor oscillations is 4π Ē/|m 2 j −m 2 k | (where Ē is the average energy of the neutrinos and j and k stand for the two mass eigenstates 1 ), we expect that the conformal coupling would simply lead to a new oscillation length L jk osc given by 4π Ē/| m2 j − m2 k |.…”

“…(A1) and (A2), the effective squared-masses of the flavor states |ν e and |ν µ readm 2 e = ν e | M 2 |ν e = m 2 1 cos 2 θ + m 2 2 sin 2 θ, m 2 µ = ν µ | M 2 |ν µ = m 2 1 sin 2 θ + m 2 2 cos 2 θ. (A3)From these two equations, one deduces that the single mixing matrix parameter θ is given in terms of the squared-masses by the any mass m of a conformally coupled particle should be related to the mass m of its non-conformally coupled counterpart by m = Ω(φ)m[41,63], we deduce that the dimensionless mixing parameter θ corresponding to the conformally coupled neutrinos is given by cos 2 θ = m2 e…”