31.3 Spinors and the Bargmann-Fock construction

31.3 Spinors and the Bargmann-Fock construction

Concept links · terms present in this machine draft; source roles are unverified: vector space · complex inner product · Lie algebra · unitary representation · irreducible representation · group action

While the fermionic analog of the Schr¨odinger construction does not give a unitary representation of the spin group, it turns out that the fermionic analog of the Bargmann-Fock construction does, on the fermionic oscillator state space discussed in chapter 27. This will work for the case of a positive definite symmetric bilinear form $(\cdot ,\cdot )$ . Note though that this will only work for fermionic phase spaces ${\mathbf{R}}^n$ with $n$ even, since a complex structure on the phase space is needed.

The corresponding pseudo-classical system will be the classical fermionic oscillator studied in section 30.3.2. Recall that this uses a choice of complex structure $J$ on the fermionic phase space ${\mathbf{R}}^{2d}$ , with the standard choice $J=J_0$ coming from the relations

$${\theta }_j=\frac{1}{\sqrt{2}}({\xi }_{2j-1}-i{\xi }_{2j}),{\overline{\theta }}_j=\frac{1}{\sqrt{2}}({\xi }_{2j-1}+i{\xi }_{2j})\text{\hspace{1em}(31.2)}$$

for $j=1,\dots ,d$ between real and complex coordinates. Here $(\cdot ,\cdot )$ is positivedefinite, and the ${\xi }_j$ are coordinates with respect to an orthonormal basis, so we have the standard relation $\{{\xi }_j,{\xi }_k{\}}_{+}={\delta }_{jk}$ and the ${\theta }_j,{\overline{\theta }}_j$ satisfy

$$\{{\theta }_j,{\theta }_k{\}}_{+}=\{{\overline{\theta }}_j,{\overline{\theta }}_k{\}}_{+}=0,\{{\overline{\theta }}_j,{\theta }_k{\}}_{+}={\delta }_{jk}$$

In the bosonic case (see equation 26.7) extending the Poisson bracket from $\mathcal{M}$ to $\mathcal{M}\otimes \mathbf{C}$ by complex linearity gave an indefinite Hermitian form on $\mathcal{M}\otimes \mathbf{C}$

$$⟨\cdot ,\cdot ⟩=i\{\bar{\cdot },\cdot \}=i\Omega (\bar{\cdot },\cdot )$$

positive definite on ${\mathcal{M}}_J^{+}$ for positive J. In the fermionic case we can extend the fermionic Poisson bracket from $\nu$ to ${\nu }_{\otimes }\mathbf{C}$ by complex linearity, getting a Hermitian form on ${\nu }_{\otimes }\mathbf{C}$

$$⟨\cdot ,\cdot ⟩=\{\bar{\cdot },\cdot {\}}_{+}=(\bar{\cdot },\cdot )$$

This is positive definite on ${\mathcal{V}}_J^{+}$ (and also on ${\gamma }_J^{-})$ if the initial symmetric bilinear form was positive.

To quantize this system we need to find operators ${\Gamma }^{+}({\theta }_j)$ and ${\Gamma }^{+}({\overline{\theta }}_j)$ that satisfy

$$\begin{array}{rl}[{\Gamma }^{+}({\theta }_j),{\Gamma }^{+}({\theta }_k){]}_{+}& =[{\Gamma }^{+}({\overline{\theta }}_j),{\Gamma }^{+}({\overline{\theta }}_k){]}_{+}=0\\ {}_{+}& ={\delta }_{jk}1\end{array}$$

but these are just the CAR satisfied by fermionic annihilation and creation operators. We can choose

$${\Gamma }^{+}({\theta }_j)=a_{Fj}^{†},{\Gamma }^{+}({\overline{\theta }}_j)=a$$

and realize these operators as

$$a_{Fj}=\frac{\partial }{\partial {\theta }_j},a_{Fj}^{†}=\text{ multiplication by }{\theta }_j$$

on the state space ${\Lambda }^{\ast }{\mathbf{C}}^d$ of polynomials in the anticommuting variables ${\theta }_j$ . This is a complex vector space of dimension $2^d$ , isomorphic with the state space ${\mathcal{H}}_F$ of the fermionic oscillator in $d$ degrees of freedom, with the isomorphism given by

$$\begin{array}{c}1\leftrightarrow |0{⟩}_F\\ {\theta }_j\leftrightarrow a_{Fj}^{†}|0{⟩}_F\\ {\theta }_j{\theta }_k\leftrightarrow a_{Fj}^{†}{a}_{Fk}^{†}|0⟩\\ \dots \\ {\theta }_1\dots {\theta }_d\leftrightarrow a_{F1}^{†}{a}_{F2}^{†}\dots a_{Fd}^{†}|0{⟩}_F\end{array}$$

where the indices $j,k,\dots$ . take values $1,2,\dots ,d$ and satisfy $j<k<\cdots$

If one defines a Hermitian inner product $⟨\cdot ,\cdot ⟩$ on ${\mathcal{H}}_F$ by taking these basis elements to be orthonormal, the operators ${\mathbf{a}}_{Fj}$ and $a_{Fj}^{†}$ will be adjoints with respect to this inner product. This same inner product can also be defined using fermionic integration by analogy with the Bargmann-Fock definition in the bosonic case as

$$⟨f_1({\theta }_1,\dots ,{\theta }_d),f_2({\theta }_1,\dots ,{\theta }_d)⟩=\int e^{-{\sum }_{j=1}^d{\theta }_j{\overline{\theta }}_j}\overline{f_1}{f}_{2}d\overline{\theta }d{\theta }_1\dots d{\overline{\theta }}_{d}d{\theta }_d\text{\hspace{1em}(31.3)}$$

where $f_1$ and $f_2$ are complex linear combinations of the powers of the anticommuting variables ${\theta }_j$ . For the details of the construction of this inner product, see chapter 7.2 of [91] or chapters 7.5 and 7.6 of [110]. We will denote this state space as ${\mathcal{F}}_d^{+}$ and refer to it as the fermionic Fock space. Since it is finite dimensional, there is no need for a completion as in the bosonic case.

The quantization using fermionic annihilation and creation operators given here provides an explicit realization of a representation of the Cliford algebra Clif(2d, $R$) on the complex vector space ${\mathcal{F}}_d^{+}$ . The generators of the Cliford algebra are identified as operators on ${\mathcal{F}}_d^{+}$ by

$${\gamma }_{2j-1}=\sqrt{2}{\Gamma }^{+}({\xi }_{2j-1})=\sqrt{2}{\Gamma }^{+}\left(\frac{1}{\sqrt{2}}({\theta }_j+{\overline{\theta }}_j)\right)=a_{Fj}+a_{Fj}^{†}$$ $${\gamma }_{2j}=\sqrt{2}{\Gamma }^{+}({\xi }_{2j})=\sqrt{2}{\Gamma }^{+}\left(\frac{i}{\sqrt{2}}({\theta }_j-{\overline{\theta }}_j)\right)=i(a_{Fj}^{†}-a_{Fj})$$

Quantization of the pseudo-classical fermionic oscillator Hamiltonian h of section 30.3.2 gives

$${\Gamma }^{+}(h)={\Gamma }^{+}\left(-\frac{i}{2}\sum _{j=1}^d({\theta }_j{\overline{\theta }}_j-{\overline{\theta }}_j{\theta }_j)\right)=-\frac{i}{2}\sum _{j=1}^d(a_{Fj}^{†}{a}_{Fj}-a_{Fj}{a}_{Fj}^{†})=-iH\text{\hspace{1em}(31.4)}$$

where $H$ is the Hamiltonian operator for the fermionic oscillator used in chapter 27.

Taking quadratic combinations of the operators ${\gamma }_j$ provides a representation of the Lie algebra $s$ $\mathfrak{o}(2d)=\mathfrak{spin}(2d)$ . This representation exponentiates to a representation up to sign of the group $SO(2d)$ , and a true representation of its double cover $Spin(2d)$ . The representation that we have constructed here on the fermionic oscillator state space ${\mathcal{F}}_d^{+}$ is called the spinor representation of $Spin(2d)$ , and we will sometimes denote ${\mathcal{F}}_d^{+}$ with this group action as S.

In the bosonic case, $\mathcal{H}={\mathcal{F}}_d$ is an irreducible representation of the Heisenberg group, but as a representation of $\mathcal{M}p(2d,\mathbf{R})$ , it has two irreducible components, corresponding to even and odd polynomials. The fermionic analog is that ${\mathcal{F}}_d^{+}$ is irreducible under the action of the Cliford algebra $\mathrm{Cliff}(2d,\mathbf{C})$ . One way to show this is to show that $\mathrm{Cliff}(2d,\mathbf{C})$ is isomorphic to the matrix algebra $M(2^d,\mathbf{C})$ and its action on ${\mathcal{H}}_F={\mathbf{C}}^{2^d}$ is isomorphic to the action of matrices on column vectors.

While ${\mathcal{F}}_d^{+}$ is irreducible as a representation of the Cliford algebra, it is the sum of two irreducible representations of $Spin(2d)$ , the so-called “half-spinor” representations. $Spin(2d)$ is generated by quadratic combinations of the Cliford algebra generators, so these will preserve the subspaces

$$S_{+}=\mathrm{span}\{|0{⟩}_F,a_{Fj}^{†}{a}_{Fk}^{†}|0{⟩}_F,\dots \}\subset S={\mathcal{F}}_d^{+}$$

and

$$S_{-}=\mathrm{span}\{a_{Fj}^{†}|0{⟩}_F,a_{Fj}^{†}{a}_{Fk}^{†}{a}_{Fl}^{†}|0{⟩}_F,\dots \}\subset S={\mathcal{F}}_d^{+}$$

corresponding to the action of an even or odd number of creation operators on $|0{⟩}_F$ . This is because quadratic combinations of the $a_{Fj},a_{F_j^{†}}$ preserve the parity of the number of creation operators used to get an element of $S$ by action on $|0{⟩}_F.$

Previous section: 31.2 The Schr¨odinger representation for fermions: ghosts · Next section: 31.4 Complex structures, U(d) SO(2 d) and the spinor representation · 中文版本

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