31.4 Complex structures, U(d) SO(2 d) and the spinor representation

31.4 Complex structures, U(d) SO(2 d) and the spinor representation

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · Lie algebra · Lie algebra representation · complexification

The construction of the spinor representation given here has used a specific choice of the ${\theta }_j,{\overline{\theta }}_j$ (see equations 31.2) and the fermionic annihilation and creation operators. This corresponds to a standard choice of complex structure $J_0$ , which appears in a manner closely parallel to that of the Bargmann-Fock case of section 26.1. The diference here is that, for the analogous construction of spinors, the complex structure J must be chosen so as to preserve not an antisymmetric bilinear form Ω, but the inner product, and one has

$$(J(\cdot ),J(\cdot ))=(\cdot ,\cdot )$$

We will here restrict to the case of $(\cdot ,\cdot )$ positive definite, and unlike in the bosonic case, no additional positivity condition on $J$ will then be required.

J splits the complexification of the real dual phase space $V^{\ast }=\gamma ={\mathbf{R}}^{2d}$ with its coordinates ${\xi }_j$ into a d dimensional complex vector space on which $J=+i$ and a conjugate complex vector space on which $J=-i$ . As in the bosonic case one has

$$\mathcal{V}\otimes \mathbf{C}={\mathcal{V}}_J^{+}\oplus {\mathcal{V}}_J^{-}$$

and quantization of vectors in ${\mathcal{V}}_J^{+}$ gives linear combinations of creation operators, while vectors in ${\mathcal{V}}_J^{-}$ are taken to linear combinations of annihilation operators. The choice of $J$ is reflected in the existence of a distinguished direction $|0{⟩}_F$ in the spinor space $S={\mathcal{F}}_d^{+}$ which is determined (up to phase) by the condition that it is annihilated by all linear combinations of annihilation operators.

The choice of J also picks out a subgroup $U(d)\subset SO(2d)$ of those orthogonal transformations that commute with $J$. Just as in the bosonic case, two diferent representations of the Lie algebra $\mathfrak{u}(d)$ of $U(d)$ are used:

• The restriction to $\mathfrak{u}(d)\subset \mathfrak{so}(2d)$ of the spinor representation described above. This exponentiates to give a representation not of $U(d)$ , but of a double cover of $U(d)$ that is a subgroup of $Spin(2d)$

• By normal ordering operators, one shifts the spinor representation of $\mathfrak{u}(d)$ by a constant and gets a representation that exponentiates to a true representation of $U(d)$ This representation is reducible, with irreducible components the ${\mathring{\Lambda }}^k({\mathbf{C}}^d)$ for $k=0,1,\dots ,d.$

In both cases the representation of $\mathfrak{u}(d)$ is constructed using quadratic combinations of annihilation and creation operators involving one annihilation operator and one creation operator, operators which annihilate $|0{⟩}_F$ . Non-zero pairs of two creation operators act non-trivially on $|0{⟩}_F.$ , corresponding to the fact that elements of $SO(2d)$ not in the $U(d)$ subgroup take $|0{⟩}_F$ to a diferent state in the spinor representation.

Given any group element

$$g_0=e^A\subset U(d)$$

acting on the fermionic dual phase space preserving J and the inner product, we can use exactly the same method as in theorems 25.1 and 25.2 to construct its action on the fermionic state space by the second of the above representations. For A a skew-adjoint matrix we have a fermionic moment map

$$A\in \mathfrak{u}(d)\to {\mu }_A=\sum _{j,k}{\theta }_j{A}_{jk}{\overline{\theta }}_k$$

satisfying

$$\{{\mu }_A,{\mu }_{A^{\prime }}{\}}_{+}={\mu }_{[A,A^{\prime }]}$$

and

$$\{{\mu }_A,\mathbf{\theta }{\}}_{+}=A^T\mathbf{\theta },\{{\mu }_A,\overline{\mathbf{\theta }}{\}}_{+}=\overline{A^T}\overline{\mathbf{\theta }}=-A\overline{\mathbf{\theta }}\text{\hspace{1em}(31.5)}$$

The Lie algebra representation operators are the

$$U_A^{\prime }=\sum _{j,k}{a}_{Fj}^{†}{A}_{jk}{a}_{Fk}$$

which satisfy (see theorem 27.1)

$$[U_A^{\prime },U_{A^{\prime }}^{\prime }]=U_{[A,A^{\prime }]}$$

and

$$[U_A^{\prime },{\mathbf{a}}_F^{†}]=A^T{\mathbf{a}}_F^{†},[U_A^{\prime },{\mathbf{a}}_F]=\overline{A^T}{\mathbf{a}}_F$$

Exponentiating these $\mathrm{gi}$ ves the intertwining operators, which act on the annihilation and creation operators as

$$U_{e^A}{\mathbf{a}}_F^{†}(U_{e^A}{)}^{-1}=e^{A^T}{\mathbf{a}}_F^{†},U_{e^A}{\mathbf{a}}_F(U_{e^A}{)}^{-1}=e^{\overline{A^T}}{\mathbf{a}}_F$$

For the simplest example, consider the $U(1)\subset U(d)\subset SO(2d)$ that acts by

$${\theta }_j\to e^{i\varphi }{\theta }_j,{\overline{\theta }}_j\to e^{-i\varphi }{\overline{\theta }}_j$$

corresponding to $A=i\varphi 1$ . The moment map will be

$${\mu }_A=-\varphi h$$

where

$$h=-i\sum _{j=1}^d{\theta }_j{\overline{\theta }}_j$$

is the Hamiltonian for the classical fermionic oscillator. Quantizing h (see equation 31.4) will give (−$i$) times the Hamiltonian operator

$$-iH=-\frac{i}{2}\sum _{j=1}^d(a_{Fj}^{†}{a}_{Fj}-a_{Fj}{a}_{Fj}^{†})=-i\sum _{j=1}^d\left(a_{Fj}^{†}{a}_{Fj}-\frac{1}{2}\right)$$

and a Lie algebra representation of $\mathfrak{u}(1)$ with half-integral eigenvalues $(\pm i\frac{1}{2})$ Exponentiation will give a representation of a double cover of $U(1)\subset U(d)$

Quantizing h instead using normal ordering gives

$$:-iH:=-i\sum _{j=1}^d{a}_{Fj}^{†}{a}_{Fj}$$

and a true representation of $U(1)\subset U(d)$ , with

$$U_A^{\prime }=i\varphi \sum _{j=1}^d{a}_{Fj}^{†}{a}_{Fj}$$

satisfying

$$[U_A^{\prime },{\mathbf{a}}_F^{†}]=i\varphi {\mathbf{a}}_F^{†},[U_A^{\prime },{\mathbf{a}}_F]=-i\varphi {\mathbf{a}}_F$$

Exponentiating, the action on annihilation and creation operators is

$$e^{-i\varphi {\sum }_{j=1}^d{a}_F{†}_j{a}_{F}j}{\mathbf{a}}_F^{†}{e}^{i\varphi {\sum }_{j=1}^d{a}_F{†}_j{a}_{F}j}=e^{i\varphi }{\mathbf{a}}_F^{†}$$ $$e^{-i\varphi {\sum }_{j=1}^d{a}_F{†}_j{a}_{F}j}{\mathbf{a}}_F{e}^{i\varphi {\sum }_{j=1}^d{a}_F{†}_j{a}_{F}j}=e^{-i\varphi }{\mathbf{a}}_F$$

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