Fermion Interactions with Constant θμν

Fermion Interactions with Constant θ_μν in Motion Theory

This document explores the derivation of fermion interactions within Motion Theory, focusing on the scenario where the non-commutativity parameters θ_μν are treated as a constant, antisymmetric background tensor. This approach, utilizing the concept of "intrinsic torsion" encoded by θ_μν, aims to make the theory's fundamental interactions more concrete as a basis for understanding fermion properties and for later analysis involving a dynamic θ_μν(x) field.

1. The Modified Spin Connection and Generalized Dirac Equation

The presence of a constant background θ^ρσ modifies the standard spin connection $\omega_{\mu}^{ab}$ as follows:

$$\tilde{\omega}_{\mu}^{ab} = \omega_{\mu}^{ab} + \kappa C_{\mu\rho\sigma}^{ab} \theta^{\rho\sigma}$$

In this expression, $\omega_{\mu}^{ab}$ is the conventional torsion-free spin connection. The second term, $\kappa C_{\mu\rho\sigma}^{ab} \theta^{\rho\sigma}$, represents the torsion-like influence stemming from the constant background field $\theta^{\rho\sigma}$. Here, $\kappa$ acts as a coupling constant, and $C_{\mu\rho\sigma}^{ab}$ is a structural tensor ensuring appropriate Lorentz covariance and index matching for the interaction.

By substituting the modified covariant derivative $\tilde{D}_\mu = \partial_\mu + \frac{1}{4} \tilde{\omega}_{\mu}^{cd} \gamma_c \gamma_d$ into the Dirac equation $(i\gamma^\mu \tilde{D}_\mu - m)\psi = 0$, we obtain the generalized form:

$$i\gamma^\mu \left( \partial_\mu + \frac{1}{4} (\omega_{\mu}^{cd} + \kappa C_{\mu\rho\sigma}^{cd} \theta^{\rho\sigma}) \gamma_c \gamma_d \right) \psi - m\psi = 0$$

This equation can be divided into the standard Dirac component and a new interaction component attributable to $\theta^{\rho\sigma}$:

$$(i\gamma^\mu D_\mu - m)\psi + i\gamma^\mu \left( \frac{\kappa}{4} C_{\mu\rho\sigma}^{cd} \theta^{\rho\sigma} \gamma_c \gamma_d \right) \psi = 0$$

Here, $D_\mu = \partial_\mu + \frac{1}{4} \omega_{\mu}^{cd} \gamma_c \gamma_d$ denotes the standard covariant derivative in a torsion-free curved spacetime. The novel interaction introduced by the constant $\theta^{\rho\sigma}$ background is captured by the second term.

2. General Analysis of Torsion-Induced Interaction Terms

The new interaction term, potentially added to the Lagrangian density, takes the schematic form:

$$\mathcal{L}_{int} \sim \bar{\psi} \left( i\gamma^\mu \left( \frac{\kappa}{4} M_{\mu}^{cd} \right) \gamma_c \gamma_d \right) \psi$$

where $M_{\mu}^{cd} = C_{\mu\rho\sigma}^{cd} \theta^{\rho\sigma}$.

Dependence on $\theta^{\rho\sigma}$: The interaction's strength and nature are directly dictated by the constant tensor $\theta^{\rho\sigma}$.
Connection to Fermion Spin: The term $\gamma_c \gamma_d$ can be expanded as $\eta_{cd} + 2\Sigma_{cd}$, where $\Sigma_{cd} = \frac{i}{4}[\gamma_c, \gamma_d]$ are the Lorentz group generators. This indicates that these new interactions can involve direct couplings to the fermion's spin ($\Sigma_{cd}$), as well as potentially to the fermion's vector current, depending on the structure of $M_{\mu}^{cd}$. The "intrinsic torsion" represented by $\theta^{\rho\sigma}$ can thereby directly influence the spin dynamics of fermions.

3. Specific Interaction Types from Constant θ_μν

The precise form of the interaction depends on how the structural tensor $C_{\mu\rho\sigma}^{ab}$ projects the constant $\theta^{\rho\sigma}$ into terms that couple with fermion bilinears. We focus on Lorentz invariant outcomes.

3.1. Scalar Interaction (Mass Modification)

If the interaction, through a specific form of $C_{\mu\rho\sigma}^{ab}$, effectively reduces to a Lorentz scalar, it can be written as:

$$\mathcal{L}_{int}^{(S)} = \lambda_S (\theta_{\alpha\beta}\theta^{\alpha\beta}) (\bar{\psi}\psi)$$

Here, $\lambda_S$ is a coupling constant, and $\Theta_S = \theta_{\alpha\beta}\theta^{\alpha\beta}$ is a Lorentz scalar invariant constructed from the constant $\theta$ tensor.

Physical Interpretation: This term directly modifies the fermion's mass: $m \rightarrow m' = m - \lambda_S \Theta_S$. The presence of the $\theta^{\rho\sigma}$ background, via its "intensity" $\Theta_S$, shifts the effective mass of fermions. This interaction is Lorentz scalar and does not, by itself, violate parity.

3.2. Pseudo-Scalar Interaction

Alternatively, if the interaction effectively results in a Lorentz pseudo-scalar form:

$$\mathcal{L}_{int}^{(P)} = \lambda_P (\epsilon_{\alpha\beta\rho\sigma}\theta^{\alpha\beta}\theta^{\rho\sigma}) (\bar{\psi}i\gamma_5\psi)$$

Here, $\Theta_P = \epsilon_{\alpha\beta\rho\sigma}\theta^{\alpha\beta}\theta^{\rho\sigma}$ is a Lorentz pseudo-scalar invariant of the $\theta$ tensor.

Physical Interpretation: This term acts as a pseudo-scalar mass. It violates Parity (P) and potentially CP symmetry. Such terms can influence the chiral properties of fermions and might be relevant for phenomena related to chiral symmetry breaking or axion-like physics if $\theta$ were related to such a pseudo-scalar.

3.3. Other Potential Couplings and Lorentz Symmetry

If the interaction term generated via $C_{\mu\rho\sigma}^{ab}\theta^{\rho\sigma}$ retains uncontracted indices from $\theta^{\rho\sigma}$ leading to vector or tensor couplings (e.g., $A_\mu(\theta) \bar{\psi}\gamma^\mu\gamma_5\psi$ or $\Theta_{\alpha\beta} \bar{\psi}\Sigma^{\alpha\beta}\psi$, where $A_\mu(\theta)$ or $\Theta_{\alpha\beta}$ are constructed from the constant $\theta^{\rho\sigma}$), these terms would generally imply an explicit breaking of Lorentz symmetry, as a constant background tensor field defines preferred directions or planes in spacetime. Such terms are actively searched for in Lorentz violation experiments (e.g., within the Standard-Model Extension framework).

4. Impact of Constant θ_μν Interactions on Chirality

We analyze how these interactions affect left-handed ($\psi_L = P_L \psi$) and right-handed ($\psi_R = P_R \psi$) fermions, where $P_{L/R} = (1 \mp \gamma_5)/2$.

Scalar Interactions ($\mathcal{L}_{int}^{(S)} \propto \bar{\psi}\psi$): Since $\bar{\psi}\psi = \overline{\psi_L}\psi_R + \overline{\psi_R}\psi_L$, this term explicitly mixes left-handed and right-handed chiral states, similar to a standard Dirac mass term. It does not inherently treat $\psi_L$ and $\psi_R$ differently in coupling strength but contributes to mass generation, thus breaking chiral symmetry.
Pseudo-Scalar Interactions ($\mathcal{L}_{int}^{(P)} \propto \bar{\psi}i\gamma_5\psi$): Using $\bar{\psi}i\gamma_5\psi = \overline{\psi_R}i\psi_L - \overline{\psi_L}i\psi_R$ (or similar forms), this term also mixes chiral states and acts as a pseudo-scalar mass, violating parity. It does not, by itself, generate chiral gauge couplings.
Axial-Vector Background Couplings (Lorentz-Violating): An interaction like $\mathcal{L}_{int}^{(AV)} = g_A ( \overline{\psi_R} \gamma^\mu \psi_R - \overline{\psi_L} \gamma^\mu \psi_L ) A_\mu(\theta)$, where $A_\mu(\theta)$ is a constant axial vector derived from $\theta$, explicitly couples differently to left- and right-handed fermion currents. This is a direct chiral-sensitive interaction but arises from an external, Lorentz-violating background. It does not generate the chiral *gauge* interactions of the Standard Model.

In summary, while interactions derived from a constant $\theta_{\mu\nu}$ can modify mass terms (mixing chiralities) or introduce chiral-sensitive Lorentz-violating backgrounds, they do not straightforwardly produce the chiral *gauge* couplings characteristic of the Standard Model's weak interactions (where gauge bosons couple exclusively to left-handed fermions). Achieving such chiral gauge couplings likely requires a dynamic $\theta_{\mu\nu}(x)$ and spontaneous symmetry breaking mechanisms, as discussed in other contexts of Motion Theory.

5. Conclusion for the Constant θ_μν Scenario

The assumption of a constant, antisymmetric background $\theta_{\mu\nu}$ within Motion Theory leads to modifications of the Dirac equation via an altered spin connection. The resulting new interaction terms for fermions, if constrained to be Lorentz invariant, primarily manifest as scalar (mass-modifying) or pseudo-scalar (parity-violating mass) couplings. These depend on scalar or pseudo-scalar invariants of $\theta_{\mu\nu}$. While these interactions impact fermion properties and can affect chirality through mass generation, they do not, in this simplified constant-$\theta$ scenario, directly produce the chiral gauge structure of the weak force. More complex, Lorentz-violating interactions are also possible if $\theta_{\mu\nu}$ defines fixed background directions. This analysis lays the groundwork for exploring the richer phenomenology of a fully dynamic $\theta_{\mu\nu}(x)$ field.