Proposition. Normal ordering adds a matrix-trace central character [fcap-0035]
Proposition. Normal ordering adds a matrix-trace central character [fcap-0035]
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.