---
title: General Standard Model (GSM) in GQFT
url: https://www.emergentmind.com/topics/general-standard-model-gsm
type: topic
---

# General Standard Model (GSM) in GQFT

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 [2508.20128].

## 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 \(\Psi_{-}\leftrightarrow\Psi_{+}\) [2508.20128].

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 [2508.20128].

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 \(\Psi_s\) with \(s=\mp\), the generators are
\[
\Sigma^{ab}=\tfrac{i}{4}[\Gamma^a,\Gamma^b],\qquad
\Sigma^a_{\mp}=\Gamma^a\Gamma_{\mp},\qquad
\Sigma_{\mp}=\pm\tfrac12\,\Gamma_9,
\]
with \(\Gamma_{\mp}=\frac12(1\mp\gamma_9)\). These decompose as
\[
SP(1,3):\{\Sigma^{ab}\},\qquad
W^{1,3}:\{\Sigma^a_{\mp}\},\qquad
SP_c(1,1):\{\Sigma_{\mp}\}.
\]
The non-vanishing commutation relations include
\[
[\Sigma^{ab},\Sigma^{cd}]
= i(\eta^{bc}\Sigma^{ad}-\dots),
\]
\[
[\Sigma^{ab},\Sigma^c_{\mp}]
= i(\eta^{bc}\Sigma^a_{\mp}-\eta^{ac}\Sigma^b_{\mp}),
\]
\[
[\Sigma_{\mp},\Sigma^a_{\mp}]
= i\,\Sigma^a_{\mp},
\qquad
[\Sigma^a_{\mp},\Sigma^b_{\mp}]=0
\]
[2508.20128].

The framework distinguishes global external symmetry in coordinate spacetime from internal spin-fiber symmetry. Coordinate spacetime carries \(PO(1,3)=P^{1,3}\ltimes SO(1,3)\), whereas the \(WS_c(1,3)\) 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 [2508.20128].

A closely related but distinct notation appears in mathematical physics, where \(G_{SM}\) denotes the Standard Model gauge group itself. In an octonionic Spin(9) construction, the subgroup commuting with a certain complex structure \(J_R\) is
\[
C_{Spin(9)}(J_R)\cong [SU(3)\times SU(2)\times U(1)]/\mathbb Z_6,
\]
with Lie algebra \(su(3)\oplus su(2)\oplus u(1)\) [1912.11282]. 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,
\[
l_{L,R}^i=(\nu_{L,R}^i,e_{L,R}^i),\qquad
q_{L,R}^i=(u_{L,R}^i,d_{L,R}^i),
\qquad i=1,2,3,
\]
assembled into \(\Psi_{\mp}^i\). Under \(WS_c(1,3)\),
\[
\Psi_{\mp}^i\to
S(\varpi^{ab})\,S_{\mp}(\varpi^a)\,S_{\mp}(\varpi)\,\Psi_{\mp}^i,
\]
and under the scaling gauge \(GS(1)\),
\[
\Psi_{\mp}^i\to \xi^{3/2}\,\Psi_{\mp}^i
\]
[2508.20128].

The gauge-field content includes the electroweak and strong fields \(B_\mu\), \(W_\mu^i\), and \(A_\mu^\alpha\) as in the Standard Model, together with the spin gauge field \({}_{\mu}^{ab}(x)\in SP(1,3)\), the chirality-boost gauge field \({}_{\mu}^a(x)\in W^{1,3}\), the conformal-spin gauge field \({}_{\mu}(x)\in SP_c(1,1)\), the scaling gauge field \(\mho_\mu(x)\in GS(1)\), and the gravigauge field \(\chi_\mu^{\;a}(x)\) with dual \({}_a^{\;\mu}(x)\) [2508.20128].

Gravigauge spacetime is defined through
\[
\eth_a\equiv {}_a^{\;\mu}\partial_\mu,
\qquad
[\eth_c,\eth_d]
= {}_{cd}^{\;\;a}\,\eth_a,
\]
with antisymmetric structure functions \({}_{cd}^{\;\;a}=-\,{}_{dc}^{\;\;a}\). This is the differential-geometric setting in which the GSM action is written [2508.20128].

Using the spin-frame measure \([\zeta]=dx^0dx^1dx^2dx^3\,\chi\), where \(\chi=\det\chi_\mu^{\;a}\), the action takes the form
\[
S_{GSM}
= \int [\zeta]\bigl[\mathcal{L}_{\rm kin}
+\mathcal{L}_{\rm int}
+\mathcal{L}_{\rm grav}\bigr].
\]
After gauge-fixing conformal-boost and scaling to unit values, the paper gives schematically
\[
S_{GSM}
=\int dx^4\sqrt{-g}\Bigl\{
\tfrac12\sum_{f}\bar f\,\gamma^a\chi_a^{\;\mu} iD_\mu f
-\tfrac14\sum_X F_{\mu\nu}^X F^{X\,\mu\nu}
+(D_\mu H)^\dagger D^\mu H - V(H)
+\dots
+\frac{\chi}{4}\,{}_{aa'}^{\mu\nu\mu'\nu'}\,{}_{\mu\nu}^{a}\,{}_{\mu'\nu'}^{a'}
\Bigr\},
\]
with
\[
g_{\mu\nu}=\chi_\mu^{\;a}\chi_\nu^{\;b}\eta_{ab}
\]
[2508.20128].

The fermionic covariant derivative is
\[
iD_\mu^{(\Psi_s)}
=i\partial_\mu
+g'\,B_\mu\,\Sigma_{Y\,s}^{(\Psi)}
+g\,W_\mu^i\,\Sigma_{L\,s}^i
+g_3\,A_\mu^\alpha T^\alpha
+\tfrac12\,{}_\mu^{ab}\Sigma_{ab}
+{}_\mu^a\Sigma_{a\,s}
+{}_\mu\Sigma_s
+\mho_\mu\Bigl(\tfrac32\Bigr).
\]
The field-strength sector includes the usual electroweak and strong tensors, the spin-gauge curvature
\[
R_{\mu\nu}^{ab}
=\partial_\mu{}_\nu^{ab}-\partial_\nu{}_\mu^{ab}
+{}_\mu^{ac}\,{}_\nu^{cb}-{}_\nu^{ac}\,{}_\mu^{cb},
\]
and the spin-covariant gravigauge field strength
\[
{}_{\mu\nu}^a
=D_\mu\chi_\nu^{\;a}-D_\nu\chi_\mu^{\;a}
\]
[2508.20128].

The scalar sector contains the Standard-Model Higgs doublet \(H\), with
\[
V_H(H)=\tfrac14\lambda_h\,(H^\dagger H-v_h^2)^2,
\]
and three singlets \(\phi_w\), \(\phi_e\), and \(\Phi_\kappa\) associated with W-spin, E-spin, and scaling. The general scaling-gauge invariant potential \(V(\phi_w,\phi_e,\Phi_\kappa)\) is stated not to be fixed by symmetry alone [2508.20128].

## 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 \({}_\mu^{ab}\) with current \(\bar\Psi\{\Sigma_{ab},\gamma^c\}\Psi\); chirality-boost-spin interaction via \({}_\mu^a\) coupling to fermionic bilinears \(\bar\Psi\{\Sigma_{a\,s},\gamma^c\}\Psi\); chiral-conformal-spin interaction via \({}_\mu\) to scalar fermion densities \(\bar\Psi\Psi\); scaling gauge interaction via \(\mho_\mu\) to all fields with scaling weight; and scalar self- and cross-couplings among \(H,\phi_w,\phi_e,\Phi_\kappa\) [2508.20128].

These interactions are not presented as independent phenomenological additions, but as structural consequences of gauging \(WS_c(1,3)\times GS(1)\). 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 \(\chi_\mu^{\;a}\). In the “gravidynamics” picture, \(\chi_\mu^{\;a}\) is a Goldstone-like field of broken \(SP(1,3)\), and it generates the metric through
\[
g_{\mu\nu}=\chi_\mu^{\;a}\chi_\nu^{\;b}\eta_{ab}.
\]
A central claim is that the quadratic term in the gravigauge field strength is equivalent, up to total derivatives, to the Einstein–Hilbert action:
\[
\tfrac14\,\chi\,
{}_{aa'}^{\mu\nu\mu'\nu'}\,{}_{\mu\nu}^a\,{}_{\mu'\nu'}^{a'}
=\chi\,R
-2\,\partial_\mu\bigl(\chi\,{}^{\mu\rho}{}_{a}^{\;\sigma}{}_{\rho\sigma}^a\bigr).
\]
On this basis, gravity is treated as a gauge force of local spin symmetry rather than as a separate classical background theory [2508.20128].

The equation of motion for \(\chi_\mu^{\;a}\) is written as a gauge-type gravitational equation in GQFT,
\[
\nabla_\nu\,{}^{\mu\nu}_a
= {}_a^{\;\mu},
\qquad
{}^{\mu\nu}_a=-{}^{\nu\mu}_a,
\qquad
\nabla_\mu\,{}_a^{\;\mu}=0,
\]
and its projection into coordinate spacetime yields generalized Einstein equations,
\[
R_{\mu\nu}-\tfrac12g_{\mu\nu}R+\gamma_G\,g_{\mu\nu}
=8\pi G_\kappa\,T_{\mu\nu},
\]
together with antisymmetric parts absent in General Relativity [2508.20128].

## 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 \({}_\mu^a\) acquires a mass
\[
m_G\equiv g_4\,\beta_w\,v_w,
\]
is parity-odd under a residual \(Z_2\), and decouples from direct Standard Model currents, which makes it a stable “dark graviton” candidate [2508.20128].

For the inflationary and dark-energy sectors, the nonlinear parametrization of \(\phi_e\) and \(\phi_w\) defines
\[
\phi_e=M_S\sinh\frac{\phi_s}{M_S}\sin\chi_c,\qquad
\phi_w=M_S\sinh\frac{\phi_s}{M_S}\cos\chi_c,
\qquad
M_S=\beta_\kappa M_\kappa.
\]
The proposed scalar potential is split as
\[
V(\phi_s,\chi_c)=V_p(\phi_s)+V_d(\phi_s,\chi_c),
\]
with
\[
V_p(\phi_s)\sim\frac18\lambda_P^4M_S^4\frac{(\sinh^2(\phi_s/M_S)-\epsilon_p^2)^2}
{(1+\lambda_p^{-2}\sinh^2(\phi_s/M_S))^2}.
\]
Inflation is described through the slow-roll parameters
\[
\epsilon_V
=M_\kappa^2
\Bigl(\frac{V_p'}{V_p}\Bigr)^2,
\qquad
\eta_V
=M_\kappa^2\frac{V_p''}{V_p},
\]
which can be made \(\ll 1\) for \(\beta_\kappa\ll 1\) [2508.20128].

At late times, the “dark cosmino” \(\varphi_c\) is stated to sit at the minimum of \(V_d\) and to generate a tiny vacuum energy
\[
\Lambda_D^4\sim\tfrac18\lambda_D^4\Lambda_\kappa^4,
\qquad
\Lambda_\kappa\sim10^{-3}\,{\rm eV},
\]
thereby providing dynamical dark energy [2508.20128].

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 \(U_Y(1)\times SU_L(2)\times SU_C(3)\) by \(WS_c(1,3)\times GS(1)\times Z_2\); Yukawa couplings are Hermitian, so strong-CP is stated to be naturally small; neutrinos become massive without an extra seesaw; and the framework predicts
\[
\sin^2\theta_W=1/4
\]
at unification. It also introduces new gauge bosons associated with \(SP(1,3)\), \(W^{1,3}\), \(SP_c(1,1)\), and scaling, together with new scalar singlets, and it predicts extra gravitational-wave polarizations, specifically spin-0 and spin-1 transverse modes [2508.20128].

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, \(G_{SM}\) can denote the Standard Model gauge group
\[
[SU(3)\times SU(2)\times U(1)]/\mathbb Z_6,
\]
for example as the centralizer of a complex structure in the Spin(9) spinor representation [1912.11282]. 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 [1910.01948; 2107.12630]. In the present context, however, GSM refers specifically to the “General Standard Model” of GQFT [2508.20128].

Source: https://www.emergentmind.com/topics/general-standard-model-gsm