The well-known difficulties with the relativistic localization in quantum theory of one particle are revisited. Among them the noncausal propagation is shown to be connected with the positivity of the time translations generator. A proposal for solving this problem is presented, namely searching for an operator probability density current being the 4-vector quantity. The operator of multiplication by the argument of the Dirac wavefunction is an example of a causal position operator.
The well-known difficulties with the relativistic localization in quantum theory of one particle are revisited. Among them the noncausal propagation is shown to be connected with the positivity of the time translations generator. A proposal for solving this problem is presented, namely searching for an operator probability density current being the 4-vector quantity. The operator of multiplication by the argument of the Dirac wavefunction is an example of a causal position operator.
A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac's position operator x . It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the four-vector density ( ( x , t ), j ( x , t )/ c ) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalized at the same time.
A causally well-behaved solution of the localization problem for the free electron is given, with natural space-time transformation properties, in terms of Dirac’s position operator x. It is shown that, although x is not an observable in the usual sense, and has no positive-energy (generalized) eigenstates, the 4-vector density (ρ(x,t),j(x,t)/c) is observable, and can be localized arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalized at the same time. 1
The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n
Causal localizations describe the position of quantum systems moving not faster than light. They are constructed for the systems with finite spinor dimension. At the center of interest are the massive relativistic systems. For every positive mass, there is the sequence of Dirac tensor-localizations, which provides a complete set of inequivalent irreducible causal localizations. They obey the principle of special relativity and are fully Poincaré covariant. The boosters are determined by the causal position operator and the other Poincaré generators. The localization with minimal spinor dimension is the Dirac localization. Thus, the Dirac equation is derived here as a mere consequence of the principle of causality. Moreover, the higher tensor-localizations, not known so far, follow from Dirac’s localization by a simple construction. The probability of localization for positive energy states results to be described by causal positive operator valued (PO-) localizations, which are the traces of the causal localizations on the subspaces of positive energy. These causal Poincaré covariant PO-localizations for every irreducible massive relativistic system were, all the more, not known before. They are shown to be separated. Hence, the positive energy systems can be localized within every open region by a suitable preparation as accurately as desired. Finally, the attempt is made to provide an interpretation of the PO-localization operators within the frame of conventional quantum mechanics attributing an important role to the negative energy states.
The principles of quantum mechanics impose certain restrictions on acceptable position operators. The probability density as derived from the position operator should agree with the fourth component of the current vector. The various relativistic wave equations are reexamined from the point of view of this consistency test. These and the group theoretical considerations lead to the conclusion that only a Dirac particle can be considered to be elementary. The position operator of the infinite-dimensional theories is investigated and the advantages of theories using the unitary representations of the Lorentz group are discussed.
We put forward an interpretation of scalar quantum field theory as relativistic quantum mechanics by curing well known problems related to locality. A probabilistic interpretation of quantum field theory similar to quantum mechanics is difficult if particle localization is defined using the Newton-Wigner position operator as it is non-local and non-covariant. An alternative bilinear covariant position operator is discussed which incorporates a time operator that can be exponentiated to a unitary operator. Moreover, it satisfies an algebra that unifies special relativity and quantum mechanics and has the same form for particles with spin. Higher power position operators are derived which yield Heisenberg's uncertainty relations. Our ideas are illustrated with a relativistic wave function whose probability density can be perfectly localized.