We propose a general method for the description of arbitrary single spin-j states transforming according to (j,0)+(0,j) carrier spaces of the Lorentz algebra in terms of Lorentz-tensors for bosons, and tensor-spinors for fermions, and by means of second order Lagrangians. The method allows to avoid the cumbersome matrix calculus and higher \partial^{2j} order wave equations inherent to the Weinberg-Joos approach. We start with reducible Lorentz-tensor (tensor-spinor) representation spaces hosting one sole (j,0)+(0,j) irreducible sector and design there a representation reduction algorithm based on one of the Casimir invariants of the Lorentz algebra. This algorithm allows us to separate neatly the pure spin-j sector of interest from the rest, while preserving the separate Lorentz- and Dirac indexes. However, the Lorentz invariants are momentum independent and do not provide wave equations. Genuine wave equations are obtained by conditioning the Lorentz-tensors under consideration to satisfy the Klein-Gordon equation. In so doing, one always ends up with wave equations and associated Lagrangians that are second order in the momenta. Specifically, a spin-3/2 particle transforming as (3/2,0)+ (0,3/2) is comfortably described by a second order Lagrangian in the basis of the totally antisymmetric Lorentz tensor-spinor of second rank, \Psi_[ \mu\nu]. Moreover, the particle is shown to propagate causally within an electromagnetic background. In our study of (3/2,0)+(0,3/2) as part o

The natural recognition of quantum nonlocality follows from the fact that a quantum wave is spatially extended. The waves of fermions display nonlocality in low energy limit of quantum fields. In this ab initio paper, we propose a complex-geometry model that reveals the effect of nonlocality on the interaction between material particles of spin 12. To make nonlocal properties appropriately involved in a quantum theory, the special unitary group SU(n) and spinor representation D(1∕2,1∕2) of Lorentz group are generalized by making complex spaces—which are spanned by wave functions of quantum particles—curved. The curved spaces are described by the geometry used in general relativity by replacing the real space with complex space and additionally imposing the analytic condition on the space. The field equations for fermions and for bosons are, respectively, associated with geodesic motion equations and with local curvature of the considered space. The equation for fermions can restore almost all the terms of quadratic form of Dirac equation. According to the field equation, it is found that for the U(1) field (generalized quantum electrodynamics), when the electromagnetic fields E⃗ and B⃗ satisfy E⃗2−B⃗2≠0, the bosons will gain masses. In this model, a physical region is empirically defined, which can be characterized by a determinant occurring in boson field equation. Applying the field equation to U(3) field (generalized quantum chromodynamics), the quark-confining property ca

speak of Fermi–Dirac statistics for half-integer-spin particles (fermions) and Bose–Einstein statistics for integer-spin particles (bosons). While lecturing Paul Adrien Maurice Dirac (, dih-RAK; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for both quantum electrodynamics and quantum field theory, coining the former term. He was Lucasian Professor of Mathematics at the University of Cambridge from 1932 to 1969, and a professor of physi Another story told of Dirac is that when he first met the young Richard Feynman at a conference, he said after a long silence, "I have an equation. Do you have one too?" After he presented a lecture at a conference, one colleague raised his hand and said: "I don't understand the equation on the top-right-hand corner of the blackboard". After a long silence, the moderator asked Dirac if he wanted to answer the question, to which Dirac replied: "That was not a question, it was a comment." Dirac was… Dirac discovered the relativistic equation for the electron, which now bears his name. The remarkable notion of an antiparticle to each fermion particle – e.g. the positron as antiparticle to the electron – stems from his equation. He is credited as being the one to create quantum field theory, which underlies all theoretical work on sub-atomic or "elementary" particles today, work that is fundamental to our understanding of the forces of nature, alongside creating quantum electrodynamics and coining the term. He proposed and investigated the concept of a magnetic monopole, an object not yet known empirically, as a means of bringing even greater symmetry to James Clerk Maxwell's equations of electromagnetism. Dirac also coined the terms "fermion" (particles with half-integer spin) and "boson" (particles with whole-integer spin). Throughout his career, Dirac was motivated by the principles of mathematical beauty, with Peter Goddard stating that "Dirac cited mathematical beauty as the ultimate criterion for selecting the way forward in theoretical physics". Dirac was recognised for being mathematically gifted, as during his time in university, academics had affirmed that Dirac had an "ability of the highest order in mathematical physics", with Ebenezer Cunningham stating that Dirac was "quite the most original student I have met in the subject of mathematical physics". Therefore, Dirac was known for his "astounding physical intuition combined with the ability to invent new mathematics to create new physics". During his career, Dirac made numerous important contributions to mathematical subjects, including the Dirac delta function, Dirac algebra and the Dirac operator.

The waves of fermions display nonlocality in low energy limit of quantum fields. In this \QTR{it}{ab initio} paper we propose a complex-geometry model that reveals the affection of nonlocality on the interaction between material particles of spin-1/2. To make nonlocal properties appropriately involved in a quantum theory, the special unitary group SU(n) and spinor representation $D^{(1/2,1/2)}$ of Lorentz group are generalized by making complex spaces--which are spanned by wave functions of quantum particles--curved. The curved spaces are described by the geometry used in General Relativity by replacing the real space with complex space and additionally imposing the analytic condition on the space. The field equations for fermions and for bosons are respectively associated with geodesic motion equations and with local curvature of the considered space. The equation for fermions can restore all the terms of quadratic form of Dirac equation. According to the field equation it is found that, for the U(1) field [generalized Quantum Electrodynamics (QED)], when the electromagnetic fields $\vec E$ and $\vec B$ satisfy $\vec E^2-\vec B^2\neq 0$, the bosons will gain masses. In this model, a physical region is empirically defined, which can be characterized by a determinant occurring in boson field equation. Applying the field equation to U(3) field [generalized Quantum Chromodynamics (QCD)], the quark-confining property can be understood by carrying out the boundary of physical regi

The Momentum-First (M-First) framework posits that gravity's primary role is to modify the kinematic rules of quantum mechanics. We apply this principle to Dirac fermions in stationary spacetimes, deriving a single, unified quantum Hamiltonian from first principles. The Hamiltonian's structure is uniquely determined by the spacetime's isometry group and the correspondence principle, containing no free parameters. In its static limit, it resolves the neutron star shallow heating puzzle via a gravitationally induced contextual mass. In its rotational limit, it provides the theoretical foundation for the anomalous Earth fly-by velocity shifts. The theory further predicts novel, testable phenomena, including a gravitational spin-Hall effect and a gravitational screening of charge. The ability of a single, coherent framework to solve existing anomalies and predict new physics marks it as a compelling paradigm for quantum-gravity interactions.