Proposition. Normal ordering adds a matrix-trace central character [fcap-0035]
AGENTDRAFTED
Let \(K\) be a field in which \(2\) is invertible. On
\(V=M_N(K)\) take
\[Q_{\mathrm {tr}}(Y)=\operatorname {Tr}(Y^2),\qquad
B_{Q_{\mathrm {tr}}}(X,Y)=2\operatorname {Tr}(XY),\]
and write \(d_{ij}=\iota _{Q_{\mathrm {tr}}}(E_{ij})\). The Clifford
relation is
\[d_{ij}d_{kl}+d_{kl}d_{ij}
=2\delta _{jk}\delta _{li}.\]
Let \(q(X)\) be the quadratic Clifford lift of the adjoint action
\(Y\mapsto [X,Y]\). The normal-ordered lift used for the CAR module is
\[D(X)=q(X)+\frac N2\operatorname {Tr}(X)1.\]
On a matrix unit it has the concrete form
\[D(E_{ij})=\frac 12\sum _{k=0}^{N-1}d_{ik}d_{kj}.\]
Indeed,
\(d_{ik}d_{kj}=\beta _{Q_{\mathrm {tr}}}(E_{ik},E_{kj})
+\delta _{ij}1\); summing the contraction term over \(k\) gives
\((N/2)\delta _{ij}1\). This derives both the coefficient and its positive
sign.
There are three different actions here. The ordinary exterior
extension of the adjoint action is the derivation
\[\delta _X(Y_1\wedge \cdots \wedge Y_r)
=\sum _sY_1\wedge \cdots \wedge [X,Y_s]\wedge \cdots \wedge Y_r.\]
The quadratic Clifford action is the inner derivation
\([q(X),-]=[D(X),-]\). The CAR representation on the full Clifford
algebra is instead left multiplication,
\[\rho _{\mathrm {CAR}}(X)c=D(X)c,
\qquad \rho _{\mathrm {CAR}}(X)c-cD(X)=[D(X),c].\]
On the central matrix the three formulas read:
\[\operatorname {ad}_I=0,\qquad q(I)=0,
\qquad D(I)=\frac {N^2}{2}1.\]
The left-regular carrier has dimension \(2^{N^2}\). For \(N\geq 1\) it
is not an irreducible exterior spinor carrier; \(N=0\) is the
one-dimensional trivial exception.
The exact identities are formalized at the pinned TauCeti revision.
The
normal-ordered lift,
its
matrix-unit and generator formulas,
and the
left-action comparison
have the field and inverse-two hypotheses stated above. They do not
identify this module with an irreducible spinor representation.
[pressley1986loop, Section 12.1, pp. 230--233]
let the determinant-square-root cover of \(U(W)\) act on
\(\bigwedge ^kW\) by \(u^{-1}\bigwedge ^k(A)\), with \(u^2=\det A\).
Differentiating first along a unitary curve gives
\(d\bigwedge (X)-\frac 12\operatorname {Tr}_W(X)\) for
\(X\in \mathfrak u(W)\); complexifying then gives the corresponding
\(\mathfrak {gl}(W)\) formula. No arbitrary complex matrix is being
presented as the derivative of a unitary curve.
[panyushev2001exterior, Section 2, Example 2.5(3), preprint pp. 8--9]
works over an algebraically closed field of characteristic zero and gives
the ordinary spinor correction as
\(\mathbb k_{-\nu }\otimes \bigwedge W\), where \(\nu \) is half the sum of
the weights with multiplicity. Evaluating \(\nu \) is half the exterior
trace. These two sources agree on
the ordinary minus-half-trace normalization. They do not produce the
positive \((N/2)\operatorname {Tr}\) term above: the trace of
\(\operatorname {ad}_X\) on \(M_N\) is zero, while that term is the
contraction created by normal ordering on the full left-regular
Clifford module.