Philips’ Lorentz-covariant localized states and the extended-type position operator

On a Time–Space Operator (and other Non-Self-Adjoint Operators) for Observables in QM and QFT

“…It is noteworthy [3,4] that, as we are going to see, operator (42a) is nothing but the usual Newton-Wigner operator, while (42b) can be interpreted [52][53][54][55][56]3,4,31] as yielding the sizes of the localization-region (an ellipsoid) via its average values over the considered wave-packet. Let us underline that the previous formalism justifies from the mathematical point of view the treatment presented in papers like [52][53][54][55][56][57][58].…”

“…Following, e.g., the ideas in Refs. [52][53][54][55][56], we are going to show that the mean values of the hermitian (selfadjoint) part of ẑ will yield a mean (point-like) position [57,58], while the mean values of the anti-hermitian (anti-selfadjoint) part of ẑ will yield the sizes of the localization region [3,4].…”

“…The position operator i ∇ p , is known to be actually non-hermitian, and may be in itself a good candidate for an extended-type position operator. To show this, we want to split [52][53][54][55][56] it into its hermitian and anti-hermitian (or skew-hermitian) parts.…”

“…It is noteworthy [3,4] that, as we are going to see, operator (42a) is nothing but the usual Newton-Wigner operator, while (42b) can be interpreted [52][53][54][55][56]3,4,31] as yielding the sizes of the localization-region (an ellipsoid) via its average values over the considered wave-packet.…”

“…Let us underline that the previous formalism justifies from the mathematical point of view the treatment presented in papers like [52][53][54][55][56][57][58]. We can split [3,4] the operator ẑ into two bilinear parts, as follows:…”