Metric Tensor¶

Arrow-reversal transformation for SU(2) representation spaces.

Description¶

The invariant metric \(g^{(j)}\) implements the canonical isomorphism between a representation and its dual. It allows converting an incoming edge (primal space) to an outgoing edge (dual space) and vice versa, while correctly tracking the sign factor required by SU(2) representation theory.

Definition¶

\[g^{(j)}_{m,\,m'} = (-1)^{j - m}\,\delta_{m,\,-m'}\]

This is a \((2j+1) \times (2j+1)\) real orthogonal matrix. It satisfies:

\[g^{(j)} \cdot \left(g^{(j)}\right)^\top = I\]

\[\left(g^{(j)}\right)^2 = (-1)^{2j}\, I\]

For integer spins (\(j \in \mathbb{Z}\)) the metric is symmetric (\(g^\top = g\)). For half-integer spins it is antisymmetric (\(g^\top = -g\)), or equivalently \(g^\top = (-1)^{2j} g\).

Python API¶

The following functions are available in the Yuzuha Python namespace:

Function	Description
`fs_phase_for_spin(spin)`	Frobenius-Schur indicator \((-1)^{2j}\)
`compute_conjugate(spec)`	Canonical basis tensor with all arrows reversed

import yuzuha

jhalf = yuzuha.Spin(1)   # j = 1/2
j1    = yuzuha.Spin(2)   # j = 1

# Frobenius-Schur phase
phase_half = yuzuha.fs_phase_for_spin(jhalf)   # -1  (half-integer)
phase_int  = yuzuha.fs_phase_for_spin(j1)      # +1  (integer)

# Conjugated basis
spec = yuzuha.CGSpec.from_edges([
    yuzuha.Edge.incoming(jhalf),
    yuzuha.Edge.incoming(jhalf),
    yuzuha.Edge.outgoing(j1),
])
basis_conj = yuzuha.compute_conjugate(spec)

Arrow Reversal¶

To convert an incoming edge to an outgoing edge, contract with \(g\):

\[\tilde{T}^{m'}_{\ldots} = \sum_m g^{(j)}_{m, m'}\, T^{\ldots}_{m}\]

To convert an outgoing edge to an incoming edge, contract with \(g^{-1}\):

\[\tilde{T}_{\ldots m'} = \sum_m \left(g^{(j)}\right)^{-1}_{m', m}\, T^{\ldots m}\]

The two operations differ by the Frobenius-Schur phase \((-1)^{2j}\) due to the identity \(g^{-1} = (-1)^{2j} g^\top\).

Frobenius-Schur Indicator¶

The Frobenius-Schur indicator of SU(2) representations is:

\[\text{FS}(j) = (-1)^{2j} = \begin{cases} +1 & j \in \mathbb{Z} \\ -1 & j \in \mathbb{Z} + \tfrac{1}{2} \end{cases}\]

This indicator appears as a phase factor whenever a contracted edge pair has its arrows flipped. In the X-symbol bond inversion scheme, flipping a contracted pair with spin \(j\) multiplies the X-symbol by \((-1)^{2j}\).

Notes¶

The metric \(g^{(j)}\) is not the standard Euclidean metric. It is the unique (up to a global phase) SU(2)-invariant bilinear form on \(V_j\). Using the ordinary identity in place of \(g\) gives incorrect arrow-reversal results.