Given an isolated system of either free or interacting particles and the associated realization of the ten conserved Poincaré generators its total conserved time-like 4-momentum defines its inertial rest-frame as the 3+1 splitting whose space-like 3-spaces (named Wigner 3-spaces) are orthogonal to it and whose inertial observer is the Fokker–Pryce 4-center of inertia. There is a discussion of the problem of the relativistic center of mass based on the fact that the 4-center functions “only” of the Poincaré generators of the isolated system are the following three non-local quantities: the non-canonical covariant Fokker–Pryce 4-center of inertia, the canonical non-covariant Newton–Wigner 4-center of mass and the non-canonical non-covariant Mőller 4-center of energy. At the Hamiltonian level one is able to express the canonical world-lines of the particles and their momenta in terms of the Jacobi variables of the external Newton–Wigner center of mass (a non-local non-covariant non-measurable quantity) and of Wigner-covariant relative 3-coordinates and 3-momenta inside the Wigner 3-spaces. This solves the problem of the elimination of relative times in relativistic bound states and to formulate a consistent Wigner-covariant relativistic quantum mechanics of point particles. The non-relativistic limit gives the Hamilton–Jacobi description of the system after the separation of Newtonian center of mass. Finally there is the definition of the non-inertial rest-frames whose 3-spaces are orthogonal to the total 4-momentum of the isolated system at spatial infinity.
Given an isolated system of either free or interacting particles and the associated realization of the ten conserved Poincaré generators its total conserved time-like 4-momentum defines its inertial rest-frame as the 3+1 splitting whose space-like 3-spaces (named Wigner 3-spaces) are orthogonal to it and whose inertial observer is the Fokker–Pryce 4-center of inertia. There is a discussion of the problem of the relativistic center of mass based on the fact that the 4-center functions “only” of the Poincaré generators of the isolated system are the following three non-local quantities: the non-canonical covariant Fokker–Pryce 4-center of inertia, the canonical non-covariant Newton–Wigner 4-center of mass and the non-canonical non-covariant Mőller 4-center of energy. At the Hamiltonian level one is able to express the canonical world-lines of the particles and their momenta in terms of the Jacobi variables of the external Newton–Wigner center of mass (a non-local non-covariant non-measurable quantity) and of Wigner-covariant relative 3-coordinates and 3-momenta inside the Wigner 3-spaces. This solves the problem of the elimination of relative times in relativistic bound states and to formulate a consistent Wigner-covariant relativistic quantum mechanics of point particles. The non-relativistic limit gives the Hamilton–Jacobi description of the system after the separation of Newtonian center of mass. Finally there is the definition of the non-inertial rest-frames whose 3-spaces are orthogonal to the total 4-momentum of the isolated system at spatial infinity.
A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.
After a review of the problems induced by the Lorentz signature of Minkowski space-time, like the need of a clock synchronization convention for the definition of 3-space and the complexity of the notion of relativistic center of mass, there is the introduction of a new formulation of relativistic quantum mechanics compatible with the theory of relativistic bound states. In it the zeroth postulate of non-relativistic quantum mechanics is not valid and the physics is described in the rest frame by a Hilbert space containing only relative variables. The non-locality of the Poincare' generators imply a kinematical non-locality and non-separability influencing the theory of relativistic entanglement and not connected with the standard quantum non-locality.
After a review of the few attempts to define quantum mechanics in noninertial frames, we introduce a family of relativistic nonrigid noninertial frames (equal-time parallel hyper-planes with differentially rotating 3-coordinates) as a gauge fixing of the description of N positive energy particles in the framework of parametrized Minkowski theories. Then we define a multitemporal quantization scheme in which the particles are quantized, but not the gauge variables describing the noninertial frames: they are considered as c-number generalized times. We study the coupled Schrödinger-like equations produced by the first class constraints and we show that there is a physical scalar product independent both from time and generalized times and a unitary evolution. Since a path in the space of the generalized times defines a nonrigid noninertial frame, we can find the associated self-adjoint effective Hamiltonian [Formula: see text] for the noninertial evolution: it differs from the inertial energy operator for the presence of inertial potentials and turns out to be frame-dependent like the energy density in general relativity. After a separation of the relativistic center of mass from the relative variables by means of a recently developed relativistic kinematics, inside [Formula: see text] we can identify the self-adjoint relative energy operator (the invariant mass) [Formula: see text] corresponding to the inertial energy and producing the same levels for the spectra of atoms as in inertial frames. Instead the (in general time-dependent) effective Hamiltonian is responsible for the interferometric effects signaling the noninertiality of the frame. It cannot be interpreted as an energy (there is no relativity principle and no kinematic group in noninertial frames) and generically, like in the case of time-dependent c-number external electromagnetic fields, it has no associated eigenvalue equation defining a noninertial spectrum. This formulation should help to find relativistic Bel inequalities and to define a quantization scheme for canonical gravity after having found a ultraviolet regularization of the Tomonaga–Schwinger formalism in special relativity as required by the Torre–Varadarajan no-go theorem.
Abstract Avoiding the assumption that relativistic scattering be describable by interacting fields we find in the Schrödinger picture relativistic scattering closely analogue to the non-relativistic case. On the space of scattering states the invariant mass operator M′ M ′ of the interacting time evolution has to be unitarily equivalent to the invariant mass M=P2 M = P 2 where P, acting on many-particle states, is the sum of the one-particle four-momenta. For an observer at rest P0 P 0 generates the free time evolution. Poincaré symmetry requires the interacting Hamiltonian H′ H ′ to Lorentz transform as 0-component of a four-vector and to commute with the four-velocity U=P/M U = P / M but not with P, else there is no scattering. Even though H′ H ′ does not commute with P, the scattering matrix does. The four-velocity U generates translations of states as they are seen by shifted observers. Superpositions of nearly mass degenerate particles such as a Klong K long are seen by an inversely shifted observer as a shifted Klong K long with an unchanged relative phase. In contrast, the four-momentum P generates oscillated superpositions e.g. a shifted Kshort K short with a changed relative phase. The probability of scattering of massive particles is shown to be approximately proportional to the spacetime overlap of their position wave functions. This is basic to macroscopic locality and justifies to represent the machinery of actual scattering experiments by the vacuum. In suitable variables the relativistic Hamiltonian of many-particle states is not the sum of a Hamiltonian for the motion of the center and a commuting Hamiltonian for the relative motion but factorizes as their product. They act on different variables of the wave functions.
Relativistic quantum theory poses a number of conceptual problems over and above the non-relativistic mechanics. From a purely mathematical point of view it is also much more sophisticated. Even the kinematic theory poses considerable difficulties: locality, antimatter, negative energy, and charge are already systematically linked in ways that cannot simply be paraphrased. A further complication is that in the midst of this structure we have what is usually called the Newton-Wigner representation (or, in the case of the Dirac theory, the Foldy-Wouthuysen representation), in which fields and states are no longer covariantly described and where the “local” self-adjoint quantities (NW-local observables) do not obey microcausality, but only satisfy equal-time commutators with respect to a particular inertial frame. If we pass to this represenation, we obtain at a stroke the basic structure of non-relativistic quantum field theory; it is not too hard to descend from that to a many-particle mechanics, and to recover the usual definitions of localization.