Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
The Dirac equation provides a description of spin 1/2 particles, consistent with both the principles of quantum mechanics and of special relativity. Often its presentation to students is based on mathematical propositions that may hide the physical meaning of its contents. Here we show that Dirac spinors provide the quantum description of two unit classical vectors: one whose components are the speed of an elementary particle and the rate of change of its proper time and a second vector which fixes the velocity direction. In this context both the spin degree of freedom and antiparticles can be understood from the rotation symmetry of these unit vectors. Within this approach the Dirac Lagrangian acquires a direct physical meaning as the quantum operator describing the total time-derivative.
We present a geometric interpretation of intrinsic spin in which a point-like particle possesses an internal rotational degree of freedom associated with its proper-time evolution. Rather than identifying spin with the mechanical rotation of an extended body, the proposed framework introduces an internal phase variable parameterizing a compact manifold along the particle's worldline. The resulting internal clock evolves at twice the de Broglie mass-energy frequency, allowing the rest-frame wavefunction to be written as ψ(ϕ)=Ae−iϕ/2. This construction reproduces the characteristic two-state structure, spinorial 4π periodicity, and fundamental SU(2) representation of spin-21 particles. Because the internal phase acts as an evolution parameter rather than a canonical operator, the framework avoids the dimensionality and algebraic obstructions associated with quantizing a spatial angular coordinate. The construction extends consistently to the standard Dirac formalism, preserving the established structure of relativistic quantum mechanics and quantum field theory while providing a geometric interpretation of the origin of intrinsic spin.
This paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R 4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R 3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine's statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by coordinate and momentum spinors. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Quantum mechanics describes interactions using complex wave functions that cannot be observed directly; only upon measurement do they translate into probabilities expressed as the squares of the magnitudes of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed partial derivatives on coordinates. Section 3 presents a version of quantum theory in spinor coordinate space. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. An equation of motion for the electron is proposed that is relativistically invariant. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations are presented, which can serve as a model of the graviton. Section 5 presents Einstein's inhomogeneous equation, which features the Riemann tensor rather than the Ricci tensor, with the energy-momentum tensor of a charged particle having four indices rather than two. In the case where a particle of matter has a charge, another Riemann tensor is added to Einstein’s equation, in addition to the usual Riemann curvature tensor, which describes the additional curvature created for a charged particle by an electromagnetic field.
This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
The foundations of quantum theory are closely tied to a formulation of classical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formulations of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be represented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation [special characters omitted]indicates that the 8-dimensional even subalgebra [special characters omitted] of the Space-time algebra, STA is isomorphic to APS Cl 3, which is isomorphic to complex quaternions [special characters omitted]. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra [special characters omitted]. The algebras Cl1,3 and Cl 3,1 are not isomorphic but their even sub-algebras are[1]. The Klein paradox is resolved in APS by considering Feynman's picture of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern-Gerlach magnet acts on the eigenspinor Λ field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved.