Spacetime path formalism for massive particles of any spin

Abstract

Earlier work presented a spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parameterized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology.The present paper extends the foundational spacetime path formalism to include massive, nonscalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincaré invariance, leading to a formulation for the nonscalar propagator in terms of a path integral over the Poincaré group.Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well known generalized Foldy-Wouthuysen transformation.

Consistent Histories of Systems and Measurements in Spacetime

“…III. The formalism presented here can also be extended to particles of non-zero spin [26], but, for simplicity, this will not explicitly be considered here, since the introduction of spin indices does not fundamentally affect the points to be made in this paper. However, note that the introduction of reverse particles in Sec.…”

“…. , N. (When including Fermions, one needs to, of course, antisymmetrize rather than symmetrize the products [26]. )…”

“…Sec-tion III addresses these issues through a modified spacetime formalism. The grounding of this formalism in the spacetime path approach is discussed in the appendix, making the connection to previous work in [24][25][26].…”

“…In [24] I presented an approach, based on a relativistic spacetime path formalism [25,26], in which entire coarse-grained histories of the universe decohere for all time. Each coarsegrained history of all of spacetime is represented by a cosmological state that is constrained by correlations introduced by the measurement-like processes that occur within the history of the universe.…”

Avoiding Haag’s Theorem with Parameterized Quantum Field Theory

“…Extension of this approach to a second-quantized QFT has been somewhat more limited, focusing largely on application to quantum electrodynamics [35][36][37][38][39][40][41]. The formalism I will use here is derived from related earlier work of mine on a foundational parameterized formalism for QFT and scattering [42][43][44][45]. This formalism was developed previously in terms of spacetime paths, but, in the present work, it is presented entirely in field-theoretic mathematical language, without the use of spacetime path integrals (though the concept of paths still remains helpful for intuitive motivation).…”