General Standard Model (GSM) in GQFT

• The General Standard Model (GSM) is a unified gauge-theoretic framework that extends the Standard Model by incorporating gravity and cosmology through additional symmetry groups.
• It introduces new gauge and scalar sectors derived from the intrinsic properties of leptons and quarks, offering novel predictions including massive neutrinos and extra gravitational-wave modes.
• GSM presents a representation-theoretic approach to unify particle interactions, gravitational dynamics, and cosmological phenomena, potentially explaining dark matter, dark energy, and inflation.

The General Standard Model (GSM) is a recently proposed framework within Gravitational Quantum Field Theory (GQFT) that is intended to unify particle physics and cosmology in a single gauge-theoretic construction. It is formulated from first principles based exclusively on the intrinsic properties of leptons and quarks, and it enlarges the conventional Standard Model symmetry $U_Y(1)\times SU_L(2)\times SU_C(3)$ to

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$

with

$WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$

In this formulation, the electromagnetic, weak, strong, and gravitational interactions, together with the Higgs scalar interaction, are incorporated in a common structure, and the framework further introduces new gauge and scalar sectors (Wu, 26 Aug 2025).

1. Foundational construction from fermionic degrees of freedom

The starting point of the GSM is the set of sixteen two-component Weyl fermions of leptons and quarks, including right-handed neutrinos, in each family. These are assembled into two equivalent sixteen-component chiral spinor representations,

$$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$

which reveal a discrete chiral-duality $Z_2$ exchanging $\Psi_{-}\leftrightarrow\Psi_{+}$ (Wu, 26 Aug 2025).

The paper states the physical principle as “physics is governed by intrinsic properties of leptons and quarks.” On that basis, the GSM does not begin from a purely geometric reformulation of gravity or from an abstract enlargement of the Standard Model gauge group; instead, it derives its extended structure from the representation content of the fermionic sector itself. Localizing $WS_c(1,3)\times GS(1)$ introduces new gauge fields, including a spin-gauge field and the gravigauge field $\chi_\mu^{\;a}$, the latter being identified as the object through which gravity is unified with the other interactions (Wu, 26 Aug 2025).

This suggests that the GSM is designed as a representation-theoretic extension of the Standard Model rather than as a minimal phenomenological modification. A plausible implication is that the framework treats spacetime and internal dynamics as more tightly coupled than in ordinary Yang–Mills plus General Relativity formulations.

2. Enlarged gauge symmetry and the role of $WS_c(1,3)$

The enlarged internal gauge structure is organized around the conformal inhomogeneous spin group $WS_c(1,3)$. In a chiral sector $$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$0 with $$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$1, the generators are

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$2

with $$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$3. These decompose as

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$4

The non-vanishing commutation relations include

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$5

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$6

$$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$7

(Wu, 26 Aug 2025).

The framework distinguishes global external symmetry in coordinate spacetime from internal spin-fiber symmetry. Coordinate spacetime carries $$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$8, whereas the $$\hat G_{GSM}=U_Y(1)\times SU_L(2)\times SU_C(3)\times WS_c(1,3)\times GS(1)\times Z_2,$$9 symmetries act internally in spin-fiber space. This separation is central to the GSM’s claim that gravity can be reformulated as a gauge interaction without collapsing the distinction between coordinate transformations and intrinsic gauge transformations (Wu, 26 Aug 2025).

A closely related but distinct notation appears in mathematical physics, where $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$0 denotes the Standard Model gauge group itself. In an octonionic Spin(9) construction, the subgroup commuting with a certain complex structure $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$1 is

$WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$2

with Lie algebra $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$3 (Krasnov, 2019). That result concerns the ordinary Standard Model gauge group rather than the “General Standard Model” of GQFT, but it is relevant because the acronym “GSM” is used in both contexts.

3. Field content, gravigauge spacetime, and dynamical structure

The fermionic matter fields are the three families of leptons and quarks,

$WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$4

assembled into $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$5. Under $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$6,

$WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$7

and under the scaling gauge $WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$8,

$WS_c(1,3)=SP(1,3)\rtimes W^{1,3}\rtimes SP_c(1,1).$9

(Wu, 26 Aug 2025).

The gauge-field content includes the electroweak and strong fields $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$0, $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$1, and $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$2 as in the Standard Model, together with the spin gauge field $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$3, the chirality-boost gauge field $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$4, the conformal-spin gauge field $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$5, the scaling gauge field $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$6, and the gravigauge field $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$7 with dual $$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$8 (Wu, 26 Aug 2025).

Gravigauge spacetime is defined through

$$\Psi_{-}^i \equiv \begin{pmatrix}\Psi_L^i\ \Psi_R^i\end{pmatrix}, \qquad \Psi_{+}^i \equiv \begin{pmatrix}\Psi_R^i\ \Psi_L^i\end{pmatrix}, \qquad i=1,2,3,$$9

with antisymmetric structure functions $Z_2$0. This is the differential-geometric setting in which the GSM action is written (Wu, 26 Aug 2025).

Using the spin-frame measure $Z_2$1, where $Z_2$2, the action takes the form

$Z_2$3

After gauge-fixing conformal-boost and scaling to unit values, the paper gives schematically

$Z_2$4

with

$Z_2$5

(Wu, 26 Aug 2025).

The fermionic covariant derivative is

$Z_2$6

The field-strength sector includes the usual electroweak and strong tensors, the spin-gauge curvature

$Z_2$7

and the spin-covariant gravigauge field strength

$Z_2$8

(Wu, 26 Aug 2025).

The scalar sector contains the Standard-Model Higgs doublet $Z_2$9, with

$\Psi_{-}\leftrightarrow\Psi_{+}$0

and three singlets $\Psi_{-}\leftrightarrow\Psi_{+}$1, $\Psi_{-}\leftrightarrow\Psi_{+}$2, and $\Psi_{-}\leftrightarrow\Psi_{+}$3 associated with W-spin, E-spin, and scaling. The general scaling-gauge invariant potential $\Psi_{-}\leftrightarrow\Psi_{+}$4 is stated not to be fixed by symmetry alone (Wu, 26 Aug 2025).

4. Novel interactions and gauge-theoretic gravity

From the expanded covariant derivative and commutators, the GSM contains, in addition to Standard Model forces, several new interaction types. The paper lists: spin-gauge interaction via $\Psi_{-}\leftrightarrow\Psi_{+}$5 with current $\Psi_{-}\leftrightarrow\Psi_{+}$6; chirality-boost-spin interaction via $\Psi_{-}\leftrightarrow\Psi_{+}$7 coupling to fermionic bilinears $\Psi_{-}\leftrightarrow\Psi_{+}$8; chiral-conformal-spin interaction via $\Psi_{-}\leftrightarrow\Psi_{+}$9 to scalar fermion densities $WS_c(1,3)\times GS(1)$0; scaling gauge interaction via $WS_c(1,3)\times GS(1)$1 to all fields with scaling weight; and scalar self- and cross-couplings among $WS_c(1,3)\times GS(1)$2 (Wu, 26 Aug 2025).

These interactions are not presented as independent phenomenological additions, but as structural consequences of gauging $WS_c(1,3)\times GS(1)$3. In that sense, the new bosonic and scalar sectors are not optional appendages but parts of the defining symmetry content. This suggests that the GSM seeks a unified origin for both known and new interactions at the level of gauge principle and representation theory.

The gravitational sector is formulated through the gravigauge field $WS_c(1,3)\times GS(1)$4. In the “gravidynamics” picture, $WS_c(1,3)\times GS(1)$5 is a Goldstone-like field of broken $WS_c(1,3)\times GS(1)$6, and it generates the metric through

$WS_c(1,3)\times GS(1)$7

A central claim is that the quadratic term in the gravigauge field strength is equivalent, up to total derivatives, to the Einstein–Hilbert action: $WS_c(1,3)\times GS(1)$8 On this basis, gravity is treated as a gauge force of local spin symmetry rather than as a separate classical background theory (Wu, 26 Aug 2025).

The equation of motion for $WS_c(1,3)\times GS(1)$9 is written as a gauge-type gravitational equation in GQFT,

$\chi_\mu^{\;a}$0

and its projection into coordinate spacetime yields generalized Einstein equations,

$\chi_\mu^{\;a}$1

together with antisymmetric parts absent in General Relativity (Wu, 26 Aug 2025).

5. Dark sector, inflation, and cosmological interpretation

The GSM explicitly ties its enlarged gauge and scalar sectors to dark matter, dark energy, and inflation. In the dark-matter sector, the chirality-boost gauge boson $\chi_\mu^{\;a}$2 acquires a mass

$\chi_\mu^{\;a}$3

is parity-odd under a residual $\chi_\mu^{\;a}$4, and decouples from direct Standard Model currents, which makes it a stable “dark graviton” candidate (Wu, 26 Aug 2025).

For the inflationary and dark-energy sectors, the nonlinear parametrization of $\chi_\mu^{\;a}$5 and $\chi_\mu^{\;a}$6 defines

$\chi_\mu^{\;a}$7

The proposed scalar potential is split as

$\chi_\mu^{\;a}$8

with

$\chi_\mu^{\;a}$9

Inflation is described through the slow-roll parameters

$WS_c(1,3)$0

which can be made $WS_c(1,3)$1 for $WS_c(1,3)$2 (Wu, 26 Aug 2025).

At late times, the “dark cosmino” $WS_c(1,3)$3 is stated to sit at the minimum of $WS_c(1,3)$4 and to generate a tiny vacuum energy

$WS_c(1,3)$5

thereby providing dynamical dark energy (Wu, 26 Aug 2025).

A plausible implication is that the GSM treats cosmology not as an effective afterthought but as a direct consequence of the same gauge-scalar structure that organizes the particle sector. In the language of the paper, the framework is meant to provide a unified description of both fundamental interactions and cosmic evolution.

6. Relation to the Standard Model, predicted departures, and terminological scope

Relative to the Standard Model, the GSM makes several explicit structural and phenomenological claims. It extends $WS_c(1,3)$6 by $WS_c(1,3)$7; Yukawa couplings are Hermitian, so strong-CP is stated to be naturally small; neutrinos become massive without an extra seesaw; and the framework predicts

$WS_c(1,3)$8

at unification. It also introduces new gauge bosons associated with $WS_c(1,3)$9, $WS_c(1,3)$0, $WS_c(1,3)$1, and scaling, together with new scalar singlets, and it predicts extra gravitational-wave polarizations, specifically spin-0 and spin-1 transverse modes (Wu, 26 Aug 2025).

These are presented as predictions and theoretical consequences of the framework rather than as experimentally established results. The paper’s scope is therefore broader than that of an ordinary beyond-the-Standard-Model extension: it proposes a simultaneous reformulation of gauge structure, gravitation, and cosmology. This suggests an ambitious unification program whose principal contribution, at present, is theoretical architecture.

The acronym “GSM” also requires careful disambiguation. In mathematical-physics usage, $WS_c(1,3)$2 can denote the Standard Model gauge group

$WS_c(1,3)$3

for example as the centralizer of a complex structure in the Spin(9) spinor representation (Krasnov, 2019). In communications and signal-processing literature, “GSM” commonly denotes generalized spatial modulation, including DNN-based GSM signal detection and low-complexity improved-throughput generalized spatial modulation schemes (Shamasundar et al., 2019, An et al., 2021). In the present context, however, GSM refers specifically to the “General Standard Model” of GQFT (Wu, 26 Aug 2025).