Representation Theory of sl(2, C)

• Representation theory of sl(2, C) is a study of how the simple Lie algebra acts on vector spaces via unitary irreducible representations decomposed into SU(2) and SU(1,1) bases.
• The methodology employs a functional-differential framework and hypergeometric functions to explicitly compute generator matrix elements across discrete and continuous bases.
• Its applications extend to quantum gravity and harmonic analysis, linking geometric quantities to representation theory in both compact and non-compact settings.

The representation theory of $\mathfrak{sl}(2, \mathbb{C})$ concerns the systematic paper of the ways in which this simple complex Lie algebra can act linearly on vector spaces, with analytic continuation to the non-compact group $SL(2, \mathbb{C})$. This theory underpins a wide spectrum of mathematical physics, most notably in harmonic analysis, relativistic quantum mechanics, and quantum gravity, with deep structural links to the representation categories of its real forms $SU(2)$ and $SU(1,1)$. A central theme is the classification and analysis of unitary irreducible representations (irreps), the explicit construction of their bases in different decompositions, and the computation of generator matrix elements, especially in contexts requiring Lorentzian or non-compact structures.

1. Decomposition of Unitary Irreducible Representations

The irreducible unitary representations of $SL(2, \mathbb{C})$ admit several canonical decompositions by restriction to maximal compact and non-compact subgroups. The most studied decompositions are with respect to $SU(2)$ (compact) and $SU(1,1)$ (non-compact), both of which are maximal subgroups distinguished by their role in spacelike and timelike geometry, respectively.

• $SU(2)$ Decomposition: The representation space $\mathcal{H}(\rho, n)$ decomposes as a direct sum over $SU(2)$ irreps (Equation (18)):

$$\mathcal{H}(\rho, n) \cong \bigoplus_{j = n/2}^{\infty} \mathcal{D}^j$$

where $\mathcal{D}^j$ are finite-dimensional $SU(2)$ representations.

• $SU(1,1)$ Decomposition: The same $(\rho, n)$ irrep decomposes into $SU(1,1)$ representations as (Equation (19)):

$$\mathcal{H}(\rho, n) \cong \mathcal{H}_D \oplus \mathcal{H}_C$$

with $\mathcal{H}_D$ comprising (delta-function-normalized) discrete series and $\mathcal{H}_C$ the continuous series, each possibly realized in "J" (discrete eigenvalues) or "K" (continuous eigenvalues) bases.

This structural understanding is crucial both for constructing explicit basis states and for transferring computational techniques (such as integration over $SU(2)$ or $SU(1,1)$ Haar measure) from the well-understood theory of these subgroups to the more intricate representations of $SL(2, \mathbb{C})$ (1007.0937).

2. Basis States: Discrete (J) versus Continuous (K)

The paper distinguishes two non-equivalent bases in which the representation generators are diagonalized:

• Discrete ("J") basis: Eigenstates $|\Psi^{j}_{m}\rangle$ of a compact generator (e.g., $J^3$) are labeled by integer/half-integer $m$ and spin $j$, with a discrete spectrum. These feature naturally in $SU(2)$ decompositions and also in the discrete series part of $SU(1,1)$ decompositions.
• Continuous ("K") basis: Eigenstates $|\Psi_{j, \lambda, \sigma}\rangle$ of a non-compact generator (e.g., $K^1$) are labeled by a continuous eigenvalue $\lambda$ (and possibly a degeneracy index $\sigma$), characteristic of continuous series $SU(1,1)$ representations.

Selection of basis is context-dependent: the $J$ basis is especially suited to models involving spacelike geometry, such as spacelike 2-cells in spin foam models of quantum gravity, while the $K$-basis is indispensable for understanding timelike 2-cells and thus Lorentzian quantum geometry.

3. Explicit Construction: Differential/Functional Framework

The explicit derivation of the generator matrix elements employs a functional-differential approach:

• Functional Realization: Representation spaces are realized in terms of functions on ($\mathbb{C}^2\backslash \{0\}$), $SU(2)$, or $SU(1,1)$, depending on the decomposition, with inner products invoking $SL(2, \mathbb{C})$-invariant measures (Equations (5)-(7)).
• Generators as Differential Operators: Generators act as explicit differential (and multiplicative) operators in the chosen coordinates:
  • For example, in the $SU(2)$ parametrization, the non-compact generator $K$ acts by (Equation (61)):

  $$K = -(\rho/2 + i) \cos\theta - i \sin\theta \frac{\partial}{\partial \theta}$$ - In the $SU(1,1)$ "J" basis (Equation (64)):

  $$K = -\left((\rho/2 + i) \cosh t + i \sinh t \frac{\partial}{\partial t}\right)$$

The action on state functions, constructed from D-functions (Wigner $d$-functions or their $SU(1,1)$ analogues), enables translation to matrix elements by systematically decomposing the resultant function into shifted spin ($j\pm1, j$) contributions.

• Hypergeometric Functions as State Functions: The D-functions (Equation (31)) can be written using hypergeometric functions, e.g.,

$$F^{(j)}_{m m'}(z) = (1-z)^{(m + m')/2} z^{(m - m')/2} {}_2F_1(-j + m, j + m + 1, m - m' + 1; z)$$

Application of the generator operator $O$ (Equation (68)) to these yields the master equation for matrix elements (Equation (69)), with precisely computable coefficients:

$$OF^{(j)}_{m m'} = (\rho/2 + i(j+1))C^{(j+1)}_{m m'} F^{(j+1)}_{m m'} - m A^{(j)}_{m m'} F^{(j)}_{m m'} + (\rho/2 - ij)C^{(j-1)}_{m m'} F^{(j-1)}_{m m'}$$

This equation, valid for all decompositions, systematizes the translation from differential representation to algebraic matrix elements.

4. Matrix Elements: Master Equation and Coefficient Structure

The generator matrix elements in all natural bases are determined by the action of $O$ on the hypergeometric basis, as above.

• The master equation reveals that the action of a generic generator can be written as a sum over states of spin $j-1$, $j$, $j+1$, each term with computable structural coefficients $A$, $C$ [(1007.0937), eqs. (70)-(72)].
• The universality of this equation for both the $SU(2)$ and $SU(1,1)$ decompositions (and for both discrete and continuous series of the latter) manifests the analytic parallelism between the representation structures built on compact and non-compact subgroups.
• In particular, this universal structure renders the computational handling of Lorentzian unitary irreps essentially parallel to the compact case, despite significant analytic differences e.g. unitary structure, measure, normalization.

5. Physical and Geometric Applications

The technical distinction between the discrete ($J$) and continuous ($K$) bases has direct physical counterparts:

• Spin foam models (quantum gravity): The $J$ basis is naturally associated with spacelike faces; the $K$ basis is necessary for representation of timelike faces, as shown by the requirement to have a non-compact spectrum for correct encoding of quantum area/boost degrees of freedom.
• Covariant State Sum Models: Decomposition into irreducible $SU(1,1)$ and $SU(2)$ representations reflects the internal $SL(2, \mathbb{C})$ symmetry of Lorentzian models and underlies the mapping between discrete quantum numbers (spins) and geometric quantities (areas, angles).
• This analytic machinery is foundational for the harmonic analysis employed in such models, allowing one to match invariant inner products, compute amplitudes, and connect physical observables to representation-theoretic quantum numbers.

6. Comparative Perspective: SU(2) versus SU(1,1) Reductions

• The $SU(2)$ reduction offers a conventional, tractable Hilbert space decomposition, but cannot accommodate the representations associated to non-compact degrees of freedom essential in Lorentzian settings.
• The $SU(1,1)$ reduction is critical for a complete picture in Lorentzian signature, despite analytic subtleties associated with continuous spectra, normalization, and labelings.
• The explicit functional/differential framework in all decompositions reveals striking formal similarities; the hypergeometric basis and operator $O$ allow a unified algebraic and analytic treatment [(1007.0937), Table I].
• Unification via the master equation not only simplifies computations but makes clear how the subtleties of non-compact representation theory can be understood by analytic continuation and generalization of compact (angular momentum) constructions.

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In summary, the representation theory of $\mathfrak{sl}(2, \mathbb{C})$ is underpinned by the parallel decomposition into $SU(2)$ and $SU(1,1)$ irreducibles, with explicitly defined basis states and generator actions in both discrete and continuous labelings. The analytic strategy—expressing both state functions and generators in terms of hypergeometric functions and their differential identities—enables explicit, unified computation of matrix elements, ultimately supporting applications in mathematical physics and quantum geometry where the $SL(2, \mathbb{C})$ unitary representation theory is foundational (1007.0937).