Manifestly covariant canonical quantization of the scalar field and particle localization

Abstract

Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface Σ, which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to Σ. Position on Σ is determined by a 4-vectorx µ . The corresponding quantum position operator, formed in terms of the operators a † (x), a(x), that create/annihilate particles on Σ, has thus well behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with a † (x), the superposition coefficients forming multiparticle wave packet profiles-wave functions, including a single particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center of mass operator is introduced and its expectation values in a generic multiparticle state calculated.

A novel view on successive quantizations, leading to increasingly more “miraculous” states

“…How the relativistic probability density and the probability current can be treated in a Lorentz covariant way is shown in Ref. [5]. At this point we can envisage that ϕ(t, x), satisfying the Klein-Gordon equation, is a field that corresponds to the expectation value of the corresponding operator.…”

“…The probability density of observing the particle at position x in a given reference frame is determined by |ψ(t, x)| 2 . The seeming intricacies concerning Lorentz covariance and other related concerns were clarified in [5,6], and references therein.…”

“…How the relativistic probability density and the probability current can be treated in a Lorentz covariant way is shown in Ref. [5].…”

“…We will present here a fresh and novel view on second quantization, stressing the fact that the state Φ[ψ] can be expanded in terms of the Fock space basis states, the superposition coefficients being multiparticle wave packet profiles f (t, x 1 , x 2 , ..., x r ). Each of them satisfies the corresponding Schrödinger equation, which for a given initial data gives a definite complex valued function, whose absolute square gives the probability density of observing at time t a multiparticle configuration (x 1 , ..., x r ) [5,6]. At this point we can envisage that the functions f (t, x 1 , x 2 , ..., x r ) are in fact the expectation values of some quantities which are operators of a more basic quantum theory.…”

Unified formalism for electromagnetic and gravitational probes: densities

“…It is controversial whether localization of the center-of-mass motion in coordinate space is relativistically possible (see Refs. [34][35][36] for attempts, however), and localization in momentum space produces infinite radii for all densities [21] owing to the Heisenberg uncertainty principle.…”