NSD in Lattice Gauge Theory

• NSD is a quantum Hamiltonian framework on a lattice that offers a non-perturbative, anomaly-free definition of the Standard Model using chiral SO(10) fields.
• It employs strong SO(10)-invariant interactions on a mirror boundary to gap unwanted fermions, ensuring a finite-dimensional, strictly local Hilbert space.
• The approach reproduces the Standard Model spectrum with unitary evolution and offers a blueprint for constructing other anomaly-free chiral gauge theories.

The term "New Standard Definition" (NSD), as introduced in the context of non-perturbative gauge theories, refers to a Hamiltonian, lattice-based, anomaly-free, fully quantum mechanical definition of the Standard Model of particle physics. This approach realizes the Standard Model as the low-energy sector of a chiral SO(10) lattice gauge theory, providing an explicit construction of its Hilbert space, operator content, and anomaly structure from first principles in a strictly local quantum system. This framework directly addresses the absence of a non-perturbative, operator-level realization of chiral gauge theories and is distinguished by its rigorous management of gauge anomalies via symmetry-protected topological (SPT) phases and mirror sector gapping (Wen, 2013).

1. Construction Overview and Motivation

NSD begins with a cubic 3 + 1‐dimensional spatial lattice of size $L^3$ and continuous time. Each site houses 16 two-component Weyl fermions (the SO(10) spinor representation), and on each link are SO(10) gauge variables and conjugate electric fields as in the Kogut–Susskind Hamiltonian formalism. The framework applies strong, SO(10)‐invariant interactions on a single “mirror” boundary layer to gap out unwanted mirror fermions, preserving the chiral 16 of SO(10) in the bulk. After spontaneous breaking of SO(10) to SU(3)×SU(2)×U(1) via a standard Higgs mechanism, the low-energy sector contains precisely three families of 16 Weyl fermions—the particle content of a modified Standard Model.

The salient features of the NSD approach are:

• Discrete space, continuous time, ensuring a quantum Hamiltonian with well-defined unitary evolution.
• Locality, with each Hamiltonian term $H_X$ acting on only nearby sites/links.
• Initial finite number of degrees of freedom, guaranteeing a finite- (or at most countably infinite-) dimensional Hilbert space for any finite $L$.

2. Explicit Hilbert Space Realization

Each lattice site $i$ is equipped with:

• Fermions: 16 two-component Weyl operators $\psi_{i,\alpha}$, $\alpha=1\ldots16$, transforming as the SO(10) spinor. The canonical algebra is

$$\{\psi_{i,\alpha}, \psi_{j,\beta}^\dag\} = \delta_{ij} \delta_{\alpha\beta},$$

with $SO(10)$ action $\psi_i \to R(g)_{\alpha\beta} \psi_{i,\beta}$.

• Gauge fields: Oriented link $\langle ij \rangle$ carries an SO(10) holonomy $H_X$0 and electric field $H_X$1 ($H_X$2), satisfying

$H_X$3

The local gauge Hilbert space is $H_X$4.

• Higgs sector: Real 10-vector $H_X$5 or other SO(10) Higgs on each site or boundary layer as necessary for symmetry breaking and mirror gapping.

Mirror sector gapping employs strong SO(10)‐invariant Yukawa/Higgs-type interactions of the form

$H_X$6

where $H_X$7, $H_X$8 and $H_X$9 are SO(10) spinor matrices, and $L$0 ensures $L$1 but preserves $L$2 for unbroken SO(10).

3. Lattice Hamiltonian Formulation

The total Hamiltonian is decomposed as

$L$3

with

• Fermion kinetic: $L$4 (tuning $L$5 to achieve a gapless Weyl spectrum in the continuum limit).
• Gauge field dynamics: $L$6 where $L$7 is the oriented plaquette product.
• Mirror fermion gapping: $L$8 as above, only on a boundary layer.
• Higgs-induced symmetry breaking: $L$9, with $i$0 in an SO(10) representation that breaks to $i$1.

The entire system evolves unitarily with a strictly local Hamiltonian, and high-energy cutoffs on $i$2 can render the full Hilbert space finite for each $i$3.

4. Emergence of the Standard Model Spectrum

Under the decomposition $i$4, the 16-dimensional spinor branches as

$i$5

with degrees of freedom: $i$6. Three such copies (families) yield 48 Weyl fermions, which upon SO(10) breaking couple chirally to the inherited $i$7 gauge fields. Thus, the visible sector at low energy is precisely the (modified) chiral Standard Model content.

5. Anomaly and Symmetry Protection Mechanism

Chiral gauge theories are non-trivial to define non-perturbatively due to potential gauge anomalies. In the NSD construction, anomaly cancellation is enforced via two mechanisms:

• Anomaly polynomial: The 6-form $i$8 vanishes for the chiral SO(10) theory when the three SM triangle anomalies cancel family by family.
• SPT phases and mirror sector gapping: The connection between gauge anomalies and SPT orders ensures that a chiral gauge theory admitting a symmetric gapping of its mirror boundary can be realized non-perturbatively if and only if it is anomaly-free. For SO(10), with $i$9 for $\psi_{i,\alpha}$0, the 16 Weyl mirror fermions can be fully gapped, showing all local and global anomalies are absent. This SPT/mirror construction is central in guaranteeing a non-anomalous, fully quantum Hamiltonian definition.

6. Finite-Volume Quantum Theory and Continuum Limit

All degrees of freedom reside on a lattice of finite $\psi_{i,\alpha}$1. For any such $\psi_{i,\alpha}$2, the quantum Hilbert space is a tensor product of finite- (or at most countably infinite-) dimensional local Hilbert spaces. The time evolution is generated by a sum of local Hamiltonian terms, ensuring strict locality and quantum unitarity. By sending $\psi_{i,\alpha}$3 and tuning couplings to the critical regime, the construction reproduces the full chiral SO(10) theory in $\psi_{i,\alpha}$4 dimensions, and upon spontaneous symmetry breaking, the entire Standard Model spectrum.

A plausible implication is that the NSD protocol provides a blueprint for a non-perturbative definition of other anomaly-free chiral gauge theories in the same dimension, subject to the same anomaly cancellation and SPT gapping requirements.

7. Significance and Distinction from Prior Definitions

The NSD as articulated by X.-G. Wen (Wen, 2013) establishes a fully operator-based, Hamiltonian, local, and anomaly-free realization of the Standard Model on a finite lattice:

• No reliance on path integrals or perturbative expansions.
• Unambiguous chiral gauge theory definition at the quantum level, guaranteed by SPT-mirror correspondence.
• Finite-dimensional (or controllably countable) Hilbert space for finite lattice size and explicit mapping to standard field theory in the continuum.

This approach contrasts with traditional formulations, which either fail to non-perturbatively define chiral gauge theories due to mirror doubling, or lack local, anomaly-free Hamiltonian constructions. The NSD concept thus represents a foundational advance in the rigorous quantum definition of the Standard Model and other chiral gauge theories.