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.
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.
Paper P5 of a ten part series P1-P10 on Interaction, Cohesive Emergence, and Fundamental Constants in Generalised Trace Dynamics P5 — Statistical state, conserved charge and inter-atom exchange This document writes down, choice by choice and with the source of each choice, the parent dynamical system
M0 from which the next stage of the programme is to derive, rather than postulate, the interaction responsible for cohesive emergence.
M0 is the seven-matrix model of the companion paper (P2) placed in the setting the programme's own papers prescribe: atoms as diagonal blocks of one master matrix with the off-diagonal blocks as the only coupling (the master extension of GTD, which the finite-measure test shows is an extension and not a rewriting), the harmonic trap identified --- on P1's chosen adjoint branch --- with the covariant-velocity term of the gauge audit, and the state taken to be the Adler--Millard canonical ensemble with its matrix multiplier
< conjugate to the conserved charge $\Ct$. Exact consequences of that state are then derived in
M0. (i) The multiplier is a $\U(N)$ angular velocity: integrating the bosonic momenta turns the ensemble weight $e^{-\Tr\lt\Ct}$ into a centrifugal configuration-space term $-\tfrac{1}{2\beta^2}\sum_A\Tr[\lt,Q_A]^2$, and the bosonic charge is $\langle\Ct_B\rangle=\beta^{-1}\langle\mathbb I(Q)\rangle\lt$ with
I(Q) the inertia operator of P2 --- ``charge = inertia
× angular velocity''; the fermionic charge is the fermions'
u(N) charge operator and enters the fermion determinant as a matrix-valued chemical potential. (ii) For $\lt=\lambda\ieff$ Adler's uniform-
ℏ assumption is exact by Schur's lemma, with $\hbar=\tfrac{\lambda}{\beta N}\langle\sum_A\Tr[\ieff,Q_A]^2\rangle$, the angular velocity times the moment of inertia about $\ieff$; the ensemble exists only for
2λ/β<m, with or without the quartic. (iii) If $\ieff$ distinguishes two atoms, the strings between them are softened by exactly
(2λ/β)2 in frequency squared and the block-separating orientation is a saddle of the effective potential along the $\U(N)$ orbit at every separation and every
λ>0: the state's first effect is on the orientation of the condensate relative to $\ieff$, and its effect on the shape is a Stage 1 calculation. On symmetry, the multiplier breaks $\U(N)$ to the stabiliser of
< and nothing on the octonionic index: in
M0 the GTD state cannot break
G2 explicitly at any order. The working hypothesis that $\ieff$ is an imaginary octonionic unit is examined under both of its readings: as the Adler--Millard multiplier on an octonionic module (
N=8n) it is admissible and has $\SU(3)$ stabiliser in the module's
G2, but P2's action does not tie that
G2 to the octonionic index and no $\U(8n)$-invariant term can; as a multiplier for the symmetry that does act on the octonionic index it breaks
G2 only to $\U(2)$ or a torus. Colour $\SU(3)$ is also the stabiliser of a vector-type condensate, the collinear commuting configuration, and the centrifugal term prefers centrally symmetric condensates over the regular simplex that the fermion sea of P3 selects, so the colour-versus-geometry competition in the state is two-sided. Stage 1 is four calculations on existing machinery, each with a kill criterion. Five choices remain the author's; defaults are stated.
Standard quantum theory presupposes an external time parameter whose operational meaning requiresa classical clock, hence a regime dominated by macroscopic classical bodies. The theory thereforeuses its own classical limit to give operational meaning to time. We ask what plays the role ofphysical clock-time in a pre-quantum, pre-spacetime reformulation, taking Adler's trace dynamics asthe dynamical substrate and a generalised trace-dynamics (GTD) action on a split-bioctonionic arenaas the unifying kinematics. The proposal is that physical time is the modular(Tomita--Takesaki/Connes) flow of the equilibrium state of the uncollapsed degrees of freedom. Thelogical chain is explicit: (i)~the bare action carries Adler's external ordering parameter
τ;(ii)~the trace-dynamics canonical state, constrained by the conserved Adler--Millard charge$\Ct$, is postulated to define a faithful normal KMS state for the physical
τ-flow on thespacetime-sector algebra; (iii)~the KMS--modular equivalence then identifies the
τ-flow withthe modular flow of that state, up to the inverse-temperature rescaling; (iv)~modular time becomes ageometric clock-time in known geometric modular-flow situations, including theBisognano--Wichmann/Rindler case and the ideal de~Sitter static patch. The inverse temperature usedfor the cosmological reservoir,
β=2π/HΛ in units
ℏ=c=kB=1, is fixed byidentifying the trace-dynamics bath with the de~Sitter horizon; it is an explicit physicalassumption rather than a consequence of Adler's statistical mechanics. The construction isadditionally phase-dependent: the split-biquaternionic spacetime sector that hosts the modularflow exists only after the first
E8→SU(3)×E6 branching, so in the exact
E8×ωE8 phase
τ is a pure ordering parameter and clock-time is undefined.Throughout we distinguishbetween general theorems of modular theory and the open question whether their hypotheses aresatisfied by the GTD operator algebra. Every load-bearing statement carries an epistemic-status tag:\tagD~derived, \tagI~inherited, \tagC~conditional, \tagO~open.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator
L^z=ℏ(x∂y∂−y∂x∂) is not that it is the operatorfor the
z-component
Lz of the angular momentum
L, but rather the expression of the operator for the angular momentum
L when it is aligned with the
z-axis. Hence what we are used to note (erroneously) as
L^z is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.