Ⅱ-6. Particle Physics
RIKEN
Accel. Prog. Rep. 56 (2023)

RIKEN Accel. Prog. Rep. 56 (2023)

QST algebra
Y. Akiba∗1X̂p (εX̂q ) = ∓X̂q (εX̂p ). The reduction operator evaluates a n-fold vee product of PAs as follows

The standard model (SM) successully describes thenature, but it has at least 25 free parameters. Simple formulas with no free parameter for 24 SM parameters have been reported:1) 15 particle masses,
4 Cabbibo-Kobayashi-Maskawa (CKM) quark mixing
parameters, 3 neutrino mixing angles, the ﬁne structure constant, and the strong coupling constant. Moreover, the quarternion-spin-isospin (QST) model thatpredicts these formulas,2) and their implications ofgravity and cosmology3) have also been reported. This
article reports an update of the model.
In the QST model, the Planck time, tpl = 5.3912 ×
10−44 s, is the minimum time period in nature. tpl isa fundamental constant of nature, similar to the speedof light in vacuum c and the Planck constant ℏ. Consequently, the Planck length lpl = ctpl is the minimum distance in nature; therefore, a spacetime point
has a ﬁnite minimum size, and the number of spacetime points in a ﬁnite volume is also ﬁnite. Thus, eachspace-time point can be numbered.
In the QST model, the change in the state from aspacetime point |n⟩ to the subsequent spacetime point|n + 1⟩ is described as() ()∑|n + 1⟩ = 1 + dŜ EA [n] = 1 +δp [n]dŜpEA |n⟩,• If more than one of the X̂p ’s are identical[X̂p1 ∨ · · · ∨ X̂pn ] = 0.
• If X̂pk X̂pk+1 = X̂pk+1 X̂pk for any k (1 ≤ k < n),[X̂p1 ∨ · · · ∨ X̂pn ] = 0.
• If X̂pk X̂pk+1 = −X̂pk+1 X̂pk for all k (1 ≤ k < n),[X̂p1 ∨ · · · ∨ X̂pn ] = X̂p1 · · · X̂pn .
The change of physical state from |n⟩ to |n + N ⟩ isexpressed as|n + N ⟩ = (1 + dŜ EA [n + N − 1])(· · · (1 + dŜ EA [n])|n⟩In the model, it is shown that the product of any two
EAs vanishes, that is, dŜpEA dŜqEA = 0. Thus(∑)1 + dŜ EA [k] = exp(dŜ EA [k]) = exp
δp [k]dŜpEA|n + N ⟩ = exp(n+N −1∑k=n)dS[k] |n⟩This formula represents the physical action from one
spacetime point |n⟩ to another spacetime point |n+N ⟩along with one path. Thus, the physical state |tf ⟩ isthe sum of all of such actions.
(∑)∑expdŜ EA [k] |xi ⟩,|xf ⟩ =∑where dŜ EA [n] =δp [n]dŜpEA is the change in thestate from |n⟩ to |n + 1⟩, and δp [n] = 0 or 1, anddŜpEA (p = 1, · · · , 49) are operators that are denoted
as elementary actions (EAs). This implies that eachspacetime point is associated with a 49-dimentional
vector space that describes the physical action, andan EA is the basis of the vector space. We denote thisvector space as the EA space. Here the coeﬃcient ofdŜpEA is not a real number; however it is limited toeither 0 or 1. This inidicates that the coeﬃcient ﬁeldof the EA space is not real R, but a ﬁnite ﬁeld F2 ={0, 1}.
In the QST model, an EA is a 48-fold product ofoperators that we denote as primordial actions (PAs).

where the ﬁrst sum runs over all possible paths
from|xi ⟩ to |xf ⟩. This equation corresponds to thefollowing path integral formula∫|tf ⟩ =Dϕ exp (idS[ϕ]) |ti ⟩( ∫)∫4=
Dϕ exp i L[ϕ]d x |ti ⟩This indicates that an EA dŜpEA corresponds to a termof Lagrangian as followsdŜpEA = [X̂p1 ∨ X̂p2 ∨ · · · ∨ X̂p48 ],dŜpEA ↔ Lp d4 x

where ∨ is a binary product operator in the model analogous to the wedge product ∧ in the exterior algebra,X̂pk are PAs, and [] represents a “reduction” operator
that evaluates a vee product in term of normal product. A PA is one of the following 64 operators.

In the QST model, all terms of the standard model
Lagrangian are derived as one of the 49 EAs.
A paper describing the QST model is still inprogress.

{I µ σ ν τ a , ετ a , εi, εI [c−a] τ a , εiτ 3 , −εI c , −εi, ε, −ε, i, 1},

Referenceswhere ε is the sign operator. If X̂p X̂q = ±X̂q X̂p then

1) Y. Akiba, RIKEN Accel. Prog. Rep. 54, 69 (2021).
2) Y. Akiba, RIKEN Accel. Prog. Rep. 54, 70 (2021).
3) Y. Akiba, RIKEN Accel. Prog. Rep. 54, 71 (2021).

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