Abstract We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics, and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection.
Abstract We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics, and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection.
Trace dynamics is a matrix-valued Lagrangian/Hamiltonian dynamics whose equilibrium statistical mechanics yields quantum theory via the Adler--Millard conserved charge. A persistent conceptual gap is a canonical (Noether--Hamiltonian) definition of spin: since trace dynamics is fundamentally pre-spacetime (in the sense that classical spacetime geometry is emergent), the conventional interpretation of spin as an ``internal'' angular momentum requires a precise identification of the relevant configuration ``angle'' variable and the space in which it lives. Building on earlier phase-amplitude constructions of complexified trace-dynamical variables, we propose a mathematically sharpened definition: \emph{spin is the canonical momentum conjugate to an orientation variable valued in an appropriate spin group} (e.g.\ $\Spin(3,1)$ on an emergent Lorentzian leaf, or $\Spin(3,3)$ at the 6D pre-localisation stage). This reformulation upgrades the abelian phase θ to a nonabelian group element U(τ), with angular velocity Ω=U−1U˙ and intrinsic spin tensor $S=\delta\,\Tr L/\delta\Omega\in\mathfrak{spin}(p,q)$. We show (i) how the earlier θ-momentum definition arises as a restriction to a one-parameter subgroup; (ii) how quantisation of spin follows from the emergent canonical (anti)commutators induced by equipartition of the Adler--Millard charge together with the topology of the true orientation manifold ($\SU(2)$ double cover); (iii) how the Pauli--Lubanski invariant is recovered and how a 6D (3,3)-signature generalisation naturally appears as a 3-form; and (iv) how division-algebra geometry (octonions, split bioctonions) provides a concrete scaffolding for the relevant spin groups, including the $\SO(3,3)\to \SO(3,1)\times \SO(2)$ leaf selection and the $\SU(3)_{\mathrm{geom}}$-induced internal Spinc structure on CP2-type fibres. We also clarify the relation between Poincar\'e-mass as the translation Casimir and ``square-root mass'' charges that can arise from geometric U(1) factors.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
There are two kinds of spin particles in nature, the Boson and the Fermion.Those with integer value spin 0, 1, 2...are called Bosons, with half-integer value spin /2, 3/2, 5/2...are called Fermions.It is well known that everything in the universe is made of Bosons and Fermions.The spin representations of Boson and the Fermion in conventional quantum mechanics are expressed by Hermitian matrices, which are finite dimensional matrices.Are there so-called the Third Kind Of Particles (TKP), for an example, whose spin maybe /3, /4, /5, /6..., which are neither Bosons nor Fermions?This article concerns about the possible math figure of TKP.More detailed material related to derivative process and ideas evolvement of TKP are given.
In this chapter we shall treat two of the most important observables of quantum mechanical systems: angular momentum and spin. We shall argue that angular momentum is the generator of rotations. In quantummechanics, angular momentum is quantized in units of the reduced Planck constant. The quantization of angular momentum is dealt with by finding the angular momentum eigenvalues and eigenstates. We shall then apply the theory of angular momentum to study the hydrogen atom. Spin is an intrinsic angular momentum of subatomic particles and therefore corresponds to an important intrinsic degree of freedom. We shall discuss briefly how spin was discovered and many of its important properties. Finally, we shall also explore here some further subtleties of quantum theory.a When we consider quantum systems in two or three spatial dimensions, we need to take into account also the effect of rotations. In this case, we need to introduce a physical quantity called the angular momentum, that is, themomentum induced by or connected with rotations. In classical physics the angular momentum of a particle about a given origin is defined by [see Fig. 8.1]
Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity; it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.
Objective collapse models are often implemented so that collapse acts only on the fermionic (matter) sector, while bosonic fields do not undergo fundamental collapse. In generalized trace dynamics (GTD), spontaneous localization is expected to arise when the trace Hamiltonian has a significant anti-self-adjoint component. In this note we show, starting from the STM-atom (spacetime-matter atom) trace Lagrangian written in terms of two inequivalent matrix velocities Q˙1 and Q˙2, that the purely bosonic subsector admits a self-adjoint Hamiltonian, whereas the fermionic sector carries an intrinsic anti-self-adjoint contribution. The key structural input is that making the trace Lagrangian bosonic requires insertion of two \emph{unequal} odd-grade Grassmann elements β1=β2. Assuming natural adjoint properties for these elements, we compute the trace Hamiltonian explicitly via trace-derivative canonical momenta (with bosonic and fermionic variations treated separately) and isolate the resulting anti-self-adjoint term. This provides a first-principles mechanism, within GTD, for why only fermionic degrees of freedom act as collapse channels.