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arxiv: 2512.09008 · v2 · submitted 2025-12-09 · ✦ hep-th · quant-ph

A new way to unify all fermion and boson fields, including gravity

N.S. Manko\v{c} Bor\v{s}tnik This is my paper

Pith reviewed 2026-05-17 00:06 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords field unificationhigher dimensionsgamma operatorsfermion familiesboson spectrumgravityLagrangian density
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The pith

In 13+1 dimensions, basis vectors from gamma operators unify all observed fermions and bosons including gravity when angular momenta are restricted to 4D spacetime.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in higher dimensions of the form d=2(2n+1), such as d=13+1, the internal spaces of fields can be described using basis vectors constructed as superpositions of odd and even products of gamma^a operators. These basis vectors provide a single framework for all known fermions—quarks, leptons, antiquarks and antileptons appearing in families—and for bosons including gravitons, photons, weak bosons, gluons and scalars. The unification holds only under the condition that fields carry non-zero angular momentum exclusively in ordinary 3+1-dimensional space-time. This leads to equal numbers of internal states for fermions and bosons and fixes the form of their interaction Lagrangian.

Core claim

The description of the internal spaces of fermion and boson fields with basis vectors, which are the superposition of odd and even products of the operators γ^a, offers in d=2(2n+1)-dimensions, such as d=(13+1), a unified picture of all so far observed fermions (quarks, leptons, antiquarks and antileptons that appear in families) and bosons (gravitons, photons, weak bosons, gluons and scalars), under the condition that all fields have non-zero angular momenta only in the d=(3+1), SO(3,1), of ordinary space-time.

What carries the argument

Basis vectors, defined as superpositions of odd and even products of the γ^a operators, that encode the internal degrees of freedom for both fermion and boson fields.

If this is right

  • Bosons appear in two orthogonal groups and carry a spatial index α, with μ=(0,1,2,3) for vectors and tensors and σ≥5 for scalars.
  • In any d=2(2n+1) space the total number of internal fermion states across all families equals the total number of internal boson states.
  • The mutual interactions of the massless fields determine the explicit form of the Lagrangian density for both fermions and bosons.
  • The construction yields concrete illustrations of the basis vectors and their properties in d=13+1 and d=5+1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The matching state counts between fermions and bosons may point to a structural reason for the observed particle spectrum without invoking additional symmetries.
  • The angular-momentum restriction offers a concrete mechanism for reducing the higher-dimensional theory to the observed 4D physics.
  • Open problems listed in the paper, such as deriving masses and couplings, could be addressed by assigning specific basis-vector combinations to known particles.

Load-bearing premise

That all fields are required to carry non-zero angular momentum exclusively in ordinary 4D space-time while living in higher-dimensional internal spaces, imposed without independent derivation from the gamma-operator algebra.

What would settle it

A calculation or observation showing a field state whose description requires non-zero angular-momentum components in the extra dimensions beyond 3+1, or a mismatch between the predicted and observed particle spectrum when the angular-momentum restriction is enforced.

read the original abstract

The description of the internal spaces of fermion and boson fields with "basis vectors", which are the superposition of odd and even products of the operators $\gamma^a$, offers in $d=2(2n+1)$-dimensions, such as $d=(13+1)$, a unified picture of all so far observed fermions (quarks, leptons, antiquarks and antileptons that appear in families) and bosons (gravitons, photons, weak bosons, gluons and scalars), under the condition that all fields have non-zero angular momenta only in the $d =(3+1)$, $SO(3,1)$, of ordinary space-time. Bosons, which also carry the spatial index $\alpha$ (which is for tensors and vectors $\mu =(0,1,2,3)$ and for scalars $\sigma \ge 5$) appear in two orthogonal groups. In any $d=2(2n+1)$-dimensional space the number of internal states of fermions in all families and their Hermitian conjugate partners is equal to the number of internal states of boson states. The article presents general properties of massless fermion and boson fields and their mutual interactions in this theory, which determine the Lagrangian density of both fields and their interactions. It particularly illustrates "basis vectors" and their properties in $d=(13+1)$ and $d=(5+1)$. The article presents new results and discusses open problems in this theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a unification of all observed fermion fields (quarks, leptons, and their conjugates in families) and boson fields (gravitons, photons, weak bosons, gluons, and scalars, including gravity) in d=2(2n+1) dimensions such as d=14, using basis vectors constructed as superpositions of odd and even products of gamma^a operators. Under the explicit condition that all fields carry non-zero angular momentum exclusively in ordinary 4D spacetime (SO(3,1)), the framework claims equal numbers of internal states for fermions and bosons, with bosons appearing in two orthogonal groups; it outlines general properties of massless fields, their interactions, and the resulting Lagrangian density, with illustrations in d=14 and d=6.

Significance. If the gamma-operator construction could be shown to derive the 4D angular-momentum restriction algebraically and to reproduce the observed particle spectrum and couplings without external inputs, the equal-state-count property would constitute a non-trivial structural result for higher-dimensional unification. The presentation of general interaction properties and a Lagrangian offers a concrete starting point for further development, though these remain at the level of conceptual outline rather than explicit verification.

major comments (3)
  1. [Abstract] Abstract: the central claim that the basis vectors furnish a unified picture of all observed fermions and bosons with equal internal-state counts rests on the imposed restriction that angular momenta vanish for a,b>3; no derivation from the Clifford algebra or gamma-operator properties is supplied to show that this restriction follows necessarily rather than being added by hand to match phenomenology.
  2. [Abstract] Abstract and d=14 illustration: the equality between the number of fermion internal states (families plus Hermitian conjugates) and boson states is asserted after imposing the 4D-only angular-momentum condition, yet no explicit counting, table of basis-vector multiplicities, or algebraic closure check is provided to confirm the equality holds independently of the restriction.
  3. [Abstract] Abstract: the manuscript states that mutual interactions determine the Lagrangian density, but supplies neither the explicit form of the Lagrangian nor a derivation showing how the two orthogonal boson groups and their couplings to fermions emerge from the basis-vector algebra.
minor comments (2)
  1. [d=(5+1) illustration] Notation for the basis vectors (superpositions of odd/even gamma products) and the spatial index alpha for bosons could be clarified with an explicit example in d=6 to improve readability.
  2. [Boson groups] The distinction between the two orthogonal boson groups is mentioned but not accompanied by a concrete matrix representation or commutation relations that would allow independent verification.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the detailed review and valuable suggestions. We address the major comments point by point below, clarifying the assumptions and structure of our framework while indicating revisions to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the basis vectors furnish a unified picture of all observed fermions and bosons with equal internal-state counts rests on the imposed restriction that angular momenta vanish for a,b>3; no derivation from the Clifford algebra or gamma-operator properties is supplied to show that this restriction follows necessarily rather than being added by hand to match phenomenology.

    Authors: The restriction to non-zero angular momenta exclusively in the ordinary 4D spacetime (SO(3,1)) is indeed a foundational assumption of the proposed unification, chosen to align with the observed dimensionality and symmetries of our universe. The Clifford algebra in higher dimensions permits additional rotational degrees of freedom, but the construction of the basis vectors is designed such that only the 4D components contribute to the physical angular momentum of the fields. We do not derive this restriction purely algebraically from the gamma-operator properties alone, as it incorporates a physical input regarding the effective dimensionality. In the revised manuscript, we will explicitly state this assumption in the abstract and introduction, along with a brief discussion of its motivation. revision: yes

  2. Referee: [Abstract] Abstract and d=14 illustration: the equality between the number of fermion internal states (families plus Hermitian conjugates) and boson states is asserted after imposing the 4D-only angular-momentum condition, yet no explicit counting, table of basis-vector multiplicities, or algebraic closure check is provided to confirm the equality holds independently of the restriction.

    Authors: The equality of the number of internal states for fermions (across families and their Hermitian conjugates) and bosons is a general feature of the construction in any d = 2(2n+1) dimensions, arising from the balanced structure of odd and even products of the gamma^a operators under the specified conditions. While the abstract asserts this property, the full manuscript details the basis vector construction in d=14, from which the state counting can be verified by enumerating the allowed superpositions. To make this more transparent and independent of the restriction, we will add an explicit table summarizing the multiplicities of fermion and boson states in the d=14 case, along with a note on the algebraic origin of the equality. revision: yes

  3. Referee: [Abstract] Abstract: the manuscript states that mutual interactions determine the Lagrangian density, but supplies neither the explicit form of the Lagrangian nor a derivation showing how the two orthogonal boson groups and their couplings to fermions emerge from the basis-vector algebra.

    Authors: The manuscript outlines how the mutual interactions between the fermion and boson fields, as encoded in the basis vector algebra, determine the structure of the Lagrangian density. The two orthogonal groups of bosons emerge from the distinct classes of even and odd gamma-operator products in the superpositions. However, the presentation remains at the level of general properties and conceptual derivation rather than providing the fully expanded explicit Lagrangian or step-by-step algebraic computation of all couplings. We agree that including a more detailed derivation would strengthen the work. In the revision, we will expand the relevant section to include the explicit form of the Lagrangian density and a clearer derivation of the boson groups and their couplings from the algebra. revision: partial

standing simulated objections not resolved
  • The algebraic derivation of the 4D angular-momentum restriction as a necessary consequence of the Clifford algebra, without phenomenological input.

Circularity Check

1 steps flagged

The non-zero angular momentum restriction to SO(3,1) is imposed by hand to obtain the observed spectrum and equal state counts

specific steps
  1. self definitional [Abstract]
    "offers in d=2(2n+1)-dimensions, such as d=(13+1), a unified picture of all so far observed fermions (quarks, leptons, antiquarks and antileptons that appear in families) and bosons (gravitons, photons, weak bosons, gluons and scalars), under the condition that all fields have non-zero angular momenta only in the d =(3+1), SO(3,1), of ordinary space-time. ... In any d=2(2n+1)-dimensional space the number of internal states of fermions in all families and their Hermitian conjugate partners is equal to the number of internal states of boson states."

    The unified picture and the fermion-boson state-count equality are asserted only after adding the global condition that angular momenta vanish outside SO(3,1). The basis vectors are defined as superpositions of odd/even gamma^a products, but nothing in that definition forces the restriction; it is imposed externally to recover the observed spectrum and the equality, so the claimed unification reduces to the input condition by construction.

full rationale

The paper claims a unified picture of fermions and bosons from the gamma^a basis vectors in d=2(2n+1) dimensions. However, this unification and the equality between fermion family states (plus conjugates) and boson states are stated to hold only under the explicit additional condition that all fields carry non-zero angular momenta exclusively in the ordinary d=(3+1) coordinates. The Clifford algebra itself does not enforce vanishing L^{ab} for a,b>3; the restriction is added to match observations and produce the two orthogonal boson groups. This makes the central counting equality and particle separation an input assumption rather than an output of the algebra, matching the self-definitional pattern. No independent derivation of the restriction from the gamma operators is shown in the abstract or described derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction rests on the Clifford algebra generated by gamma^a operators in d=14, the ad-hoc restriction that angular momentum is non-zero only in 4D, and the postulate that bosons appear in two orthogonal groups. No new particles are invented, but the equal fermion-boson state count is asserted without derivation from lower-dimensional physics.

free parameters (1)
  • dimension d=14
    Chosen as 2(2n+1) to produce the observed fermion families; no derivation from a more fundamental principle is given.
axioms (2)
  • standard math Clifford algebra generated by gamma^a operators defines the internal spaces
    Invoked throughout the abstract as the source of basis vectors.
  • ad hoc to paper All fields carry non-zero angular momentum only in d=(3+1)
    Stated as a condition required for the unification to match observations.

pith-pipeline@v0.9.0 · 5564 in / 1648 out tokens · 38071 ms · 2026-05-17T00:06:32.325358+00:00 · methodology

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Reference graph

Works this paper leans on

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