It is shown that the assumptions of Poincar\'e invariance and manifest covariance in quantum mechanics requires an interaction-dependent scalar product. Properties of the kernel of the inner product based on these assumptions are derived. Additional assumptions further restrict the form of this kernel. The relationship of this result to local quantum field theory and covariant constraint dynamics is discussed. A formal representation of a kernel satisfying all of these properties is given in terms of two-body constraints.
It is shown that the assumptions of Poincar'e invariance and manifest covariance in quantum mechanics requires an interaction-dependent scalar product. Properties of the kernel of the inner product based on these assumptions are derived. Additional assumptions further restrict the form of this kernel. The relationship of this result to local quantum field theory and covariant constraint dynamics is discussed. A formal representation of a kernel satisfying all of these properties is given in terms of two-body constraints.
The concepts of instantaneous observables and dynamical variables are analyzed and generalized to arbitrary spacelike hyperplanes. A formalism is developed which gives the basic equations of relativistic quantum mechanics for dynamical variables on arbitrary hyperplanes a manifestly covariant form. A covariant linear transformation on the Poincaré generators introduces the hyperplane generators which yield commutation relations displaying a clear separation of the kinematical and dynamical properties of dynamical variables. An axiomatic study of the center-of-mass position operator yields the uniqueness of the operator and completes the physical interpretation of the hyperplane generators. The Poincaré invariance and hyperplane independence of the scattering operator is related to asymptotic conservation laws in the hyperplane formalism, and finally, a nonlocal, hyperplane-dependent, field theory of free spinless particles is considered.
The problem of describing a quantum mechanical system is considered. For a system which is invariant under the Poincaré (inhomogeneous Lorentz) transformations this description is provided by the generators of those transformations, which satisfy the usual commutation relations. General expressions for the centre of mass position and internal angular momentum operators, in terms of these generators, are obtained. The generators of the Poincaré transformations are written in terms of the fundamental dynamical variables for several systems. The systems considered are those consisting of a single free spinless particle, a single free particle with spin, a single free Dirac particle, n noninteracting particles, and ft interacting particles. In each case, the centre of mass position and internal angular momentum are given in terms of the fundamental dynamical variables of the system. For the first two systems listed above, these two operators are found to be equal to the Cartesian coordinates and spin of the particle, respectively. In the case of the Dirac particle, these operators are seen to be related to the Cartesian coordinates and spin of the particle via the Pryce-Foldy-Wouthuysen transformation. Following Bakamjian and Thomas, interaction is introduced to the n particle system via a single operator which depends only on internal variables. The condition of "asymptotic covariance" of the scattering operator is discussed for two particle scattering. The scattering operator for a two particle system with no bound states and with Poincaré generators of the Bakamjian-Thomas form is seen to be asymptotically covariant.
In a recent paper a relativistically covariant Schr\"odinger equation was derived for particles of arbitrary spin s, locally covariant wave functions without redundant components being used to describe states of a particle. Here we determine the invariant scalar product with respect to which the representation of Poincar\'e transformations on these wave functions is unitary. It is shown that the conventional position and spin operators, not being Hermitian with respect to this scalar product, cannot be observables. New operators which can represent these observables are constructed with the aid of a generalized Foldy-Wouthuysen transformation which is determined explicitly for arbitrary spin.
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 [Formula: see text], which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to [Formula: see text]. Position on [Formula: see text] is determined by a 4-vector [Formula: see text]. The corresponding quantum position operator, formed in terms of the operators [Formula: see text], [Formula: see text], that create/annihilate particles on [Formula: see text], has thus well-behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with [Formula: see text], 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.
We present a covariant quantum formalism for scalar particles based on an enlarged Hilbert space. The particular physical theory can be introduced through a timeless Wheeler DeWitt-like equation, whose projection onto four-dimensional coordinates leads to the Klein-Gordon equation. The standard quantum mechanical product in the enlarged space, which is invariant and positive definite, implies the usual Klein-Gordon product when applied to its eigenstates. Moreover, the standard three-dimensional invariant measure emerges naturally from the flat measure in four dimensions when mass eigenstates are considered, allowing a rigorous identification between definite mass history states and the standard Wigner representation. Connections with the free propagator of scalar field theory and localized states are subsequently derived. The formalism also allows the superposition of different theories and remains valid in the presence of a fixed external field, revealing special orthogonality relations. Other details such as extended identities for the current density, the quantization of parameterized theories and the nonrelativistic limit, with its connection to the Page and Wooters formalism, are discussed. A related consistent second quantization formulation is also introduced.
A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in H that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in H becomes the usual Schrödinger evolution with t as a parameter. The relativistic invariance of the construction is proved. The usual theory of relativity on Minkowski space-time is shown to be “isometrically and equivariantly embedded” into H. That is, classical space-time is isometrically embedded into H, Poincaré transformations have unique extensions to isomorphisms of H, and the embedding commutes with Poincaré transformations.