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Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

A. P. Isaev, A. A. Provorov

Abstract

In this paper, we derive characteristic identities for the split Casimir operator of the Lie algebra $so(2r)$ in tensor products of spinor representations of the same and opposite chiralities. Using these identities, we explicitly construct projectors onto invariant subspaces of this operator and compute their traces. The results obtained allow us to derive explicit expressions for the colour factors of ladder Feynman diagrams in gauge theories with gauge group $Spin(2r)$. In addition, we obtain a new form of a solution to the Yang-Baxter equation that is invariant under the action of the Lie algebra $so(2r)$ in spinor representations.

Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

Abstract

In this paper, we derive characteristic identities for the split Casimir operator of the Lie algebra in tensor products of spinor representations of the same and opposite chiralities. Using these identities, we explicitly construct projectors onto invariant subspaces of this operator and compute their traces. The results obtained allow us to derive explicit expressions for the colour factors of ladder Feynman diagrams in gauge theories with gauge group . In addition, we obtain a new form of a solution to the Yang-Baxter equation that is invariant under the action of the Lie algebra in spinor representations.
Paper Structure (14 sections, 11 theorems, 120 equations, 8 figures)

This paper contains 14 sections, 11 theorems, 120 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. The split Casimir operator of a simple Lie algebra
  4. First approach to the derivation of characteristic identities for the split Casimir operator of the Lie algebra $so_{2r}$ in tensor products of spinor representations
  5. Characteristic identity of the split Casimir operator of the Lie algebra $so_{2r}$ in the representation $\rho\otimes \rho$
  6. Characteristic identities of the split Casimir operator of the Lie algebra $so_{2r}$ in the representations $\Delta_{\pm}\otimes \Delta_{\pm}$ and $\Delta_{\pm}\otimes \Delta_{\mp}$
  7. Second approach to the derivation of characteristic identities for the split Casimir operator of the Lie algebra $so_{2r}$ in tensor products of spinor representations
  8. Characteristic identities of the split Casimir operator of the Lie algebra $so_{2r}$ in the representations $\Delta_{\pm}\otimes \Delta_{\pm}$ and $\Delta_{\pm}\otimes \Delta_{\mp}$
  9. Colour factors of ladder Feynman diagrams in a gauge theory with fermions transforming in spinor representations of the gauge group $\mathrm{Spin}(2r)$
  10. Solutions of the Yang--Baxter equation invariant under the action of $so_{2r}$ in spinor representations
  11. Method of solving the Yang--Baxter equation based on properties of Casimir operators
  12. Solutions of the Yang--Baxter equation invariant under the action of the Lie algebra $so_{2r}$ in the representations $\Delta_\pm\otimes \Delta_\pm$
  13. Lie algebra $so_N$ and the Clifford algebra $\mathcal{C}l_N$
  14. Defining representation of $so_N$ and its exterior powers

Key Result

Proposition 1

The split Casimir operator of $so_{2r}$ in the representation $\rho$ satisfies Identity I2 char iden is the characteristic identity of $\widehat{C}_\rho$.

Figures (8)

  • Figure 1: Graphical interpretation of the operator $\widehat{C}_{T\cdot T}$. In the case of a self-dual representation $T$, the horizontal lines should be regarded as unoriented.
  • Figure 2: Feynman rules for the theory with Lagrangian \ref{['Lagrangian']}. If the representations $\Delta_+$ and $\overline{\Delta}_+$ are equivalent, the fermion lines should be treated as unoriented.
  • Figure 3: Graphical interpretation of the operator $\widehat{C}_{++}$. For even $r$, the fermion lines should be regarded as unoriented (see the isomorphisms \ref{['Delta bar(Delta) iso']}).
  • Figure 4: Graphical interpretation of the operator $\widehat{C}_{++}^L$ in terms of a Feynman diagram describing fermion interaction via gluon exchange.
  • Figure 5: Vacuum Feynman diagram obtained by closing the diagram in Fig. \ref{['fermion-fermion scattering']}.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Proposition 1
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 2
  • Proposition 4
  • ...and 15 more