Relativistic probability densities for location

Abstract

Abstract Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes <?CDATA $\psi ({\boldsymbol{k}},t)=\psi ({\boldsymbol{k}})\exp (-{\rm{i}}{\omega }_{{\boldsymbol{k}}}t)$?> ψ ( k , t ) = ψ ( k ) exp ( − i ω k t ) yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ μ ( x , t) = (ℏ/2i)[ψ +( x , t)∂ μ ψ( x , t) − ∂ μ ψ +( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities <?CDATA ${ \mathcal H }({\boldsymbol{x}},t)=c{{ \mathcal P }}^{0}({\boldsymbol{x}},t)$?>  ( x , t ) = c  0 ( x , t ) and <?CDATA ${ \mathcal P }({\boldsymbol{x}},t)$?>  ( x , t ) differ from ℘ μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density <?CDATA ${ \mathcal H }({\boldsymbol{x}},t)$?>  ( x , t ) , the canonical charge density ϱ( x , t), the energy pseudo-density <?CDATA $\tilde{{ \mathcal H }}({\boldsymbol{x}},t)=c{\wp }^{0}({\boldsymbol{x}},t)$?>  ˜ ( x , t ) = c ℘ 0 ( x , t ) , and the Born density ∣ψ(