数理解析研究所講究録
第 822 巻 1993 年 56-6156

Fermion Fock space on$S^{3}$

Tosiaki KORI
$S^{3}$

上のフェルミオン・フォック空間の構成
郡 敏昭 (早大・理工)

1. Preliminaries
a Here we give a brief r\’esume of [K1] to fix the notations. Let $M=$, and $E\simeq S^{3}$ be the equator. Let;andresp.
) be the spinor bundle (resp. of positive chirality and ofnegative chirality) on $M$ . The inner product of two spinors. We denote by
is defined by $<\phi(z),$andswitches
Clifford mutiplication of the radial vector field on M.
. Let $H$ (resp.
. Transition for spinors is given by$H^{*})$ be the space of square integrable spinors on $E$ of positive (resp. negative.
chirality. From the definition $<\phi,$ $\psi>=0$ for all $\phi\in H$ and$c^{2}U_{v}\hat{C}^{2}\simeq S^{4}$$S^{+}$$S$$w=v(z)=- \frac{\overline{z}}{|z|^{2}}$$($$S^{-}$$\phi,$$\varphi\in\Gamma(S^{\pm})$$\varphi(z)>=\phi_{1}(z)\overline{\varphi}_{1}(z)+\phi_{2}(z)\overline{\varphi}_{2}(z)$$n$$S^{-}$$\gamma_{0}$$S^{+}$$\gamma_{0}$$\hat{\varphi}(v(z))=-(\gamma_{0}\varphi)(z)$$\psi\in H^{*}$$)$The Dirac operator is of the form$\mathcal{D}=(\begin{array}{ll}0 D^{\uparrow}D 0\end{array});D$: $\Gamma(S^{+})arrow\Gamma(S^{-})$ .

Let
be Hamiltonian on $E$ . We have the radial decomposition of Diracoperator:$D=\gamma_{0}(n-P)$ ,.
$\mathcal{P}$$D\dagger=(n+\mathcal{P})\gamma_{0}$The eigenvalues ofwith multiplicity $(r+1)(r+2)$$r=0,1,2,$. A complete orthonormal system of eigenspinors in $H$ ;was given explicit forms in [K1];$\mathcal{P}are\pm(r+\frac{3}{2}),$$\cdots$$\{\phi_{k,r-k}^{q}, \pi_{q}^{r-k,k}\}_{r,q,k}$$\mathcal{P}\phi_{k,r-k}^{q}=(r+\frac{3}{2})\phi_{k,r-k}^{q}$$\mathcal{P}\pi_{q}^{r-k,k}=-(r+\frac{3}{2})\pi_{q}^{r-k,k}$.

TypesH by$A_{M}$

S-TEX

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$\phi_{k,r-k}^{q}=(\frac{q!k!(r-k)!}{(r+1-q)!})^{-\frac{1}{2}}(\begin{array}{l}q2^{-q+l}h_{k,r-k}^{q-l}h_{k,r-k}^{q}-2^{-q}\end{array})$$\pi_{q}^{r-k,k}=(\frac{q!k!(r-k)!}{(r+1-q)!})^{-\frac{1}{2}}(\begin{array}{ll}2^{-q} \wedge h_{q}^{r-k+1,k}2^{-q} \wedge h_{q}^{r-k,k+l}\end{array})$,where$k(z_{1}, z_{2})=(- \overline{z}_{2}\frac{\partial}{\partial z_{1}}+\overline{z}_{1}\frac{\partial}{\partial z_{2}})^{q}(z_{1}^{k}z_{2}^{r-} )$$2^{-q}h_{k,r}^{q}$
一$2^{-q} \hat{h}_{q}^{r-k,k}(z_{1}, z_{2})=(\overline{z}_{2}\frac{\partial}{\theta\overline{z}_{1}}-z_{1}\frac{\partial}{\partial z_{2}})^{q}(\overline{z}_{1}^{k}z_{2}^{r-k})$

Let $H+(resp. H_{-})$ be the subspace of $H$ spanned by. We put$\pi_{q}^{r-k,k})$,.
$\phi_{k,r-k}^{q}’ s$(resp.

$H_{\pm}^{*}=\gamma_{0}H\pm\cdot$

For a triplet $\lambda=\{\pm r;k,p\},$ $0\leq r,$ $0\leq k\leq r,$ $0\leq p\leq r+1$ , we put$-\lambda=\{\mp r, r-k, r+1-p\}$ . Lexicographic order for the triplets $\lambda=\{s, p, k\}$is defined by
if either $(i)s\geq s’$ , or $(ii)s=s’,$ $k\geq k’$ , or (iii) $s=s’$ ,$k=k’$ and $p\geq p’$ . Hence. The smallest positive is
implies$o+=( \frac{3}{2},0,0)$ while the largest negative is $0_{-}=(-\frac{3}{2},0,1)$ . Let
denotethe triplet at the p-th place after $0+ifp$ is non-negative (resp. at the p-thplace before
if is negative).
(resp.
andWe denote by) the set of all triplets (resp.
$\lambda\geq 0+and\lambda\leq 0_{-})$ . We put alsofor.
A subset of is called Maya diagram if bothandarefinite set. The integer $\chi(S)=\#(Z_{\geq 0}\cap S)-\#(\mathcal{Z}<0\cap S^{c})$ is called chargeof . For each Maya diagram with $\chi(S)=p$ there corresponds a uniquesuch that (1) $s(\nu)=\nu$ for sufficiently
increasing function :small
and (2) Image(. The degree of a Maya diagramis thenumber $d(S)= \sum_{\nu}(s(\nu)-\nu)$ .
$b$$\lambda\geq\lambda’$$-\lambda\leq-\lambda’$$\lambda\geq\lambda’$$\alpha(p)$$o_{-}$$p$$\mathcal{Z}$$\lambda$$\mathcal{Z}<0$$\mathcal{Z}_{\geq 0}$$\mathcal{Z}_{\leq\alpha}=\{\beta\in Z;\beta\leq\alpha\}$$S$$\mathcal{Z}$$S\cap Z_{\geq 0}$$S$$\alpha\in \mathcal{Z}$$S^{c}\cap \mathcal{Z}<0$

$S$

$s$

$lJ$

$Z<\alpha(p)arrow \mathcal{Z}$$s\overline{)}=S$

$S$

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2 Extensions and duality
a Let $R=\{z\in C^{2}; |z|<1\}$ and $\hat{R}=\{w\in\hat{C}^{2}; |w|<1\}$ . Let$N(R)=$ { $\phi\in\Gamma(R,$ $S^{+});\phi$ has$\mathcal{N}^{\dagger}(R)=${$\psi\in\Gamma(R,$ $S^{-}),$$\phi$

has$L^{2}$-boundary value on $|z|=1,$ $D\phi=0$ },$L^{2}$-boundary value on $|z|=1,$$D^{\uparrow}\psi=0$}.

andare defined similarly.
We have proved in [K1] :$\mathcal{N}(\hat{R})$$\mathcal{N}^{\uparrow}(\hat{R})$$Th$ 伽 Or e 家 1.

(1)
(2)
where

$H_{+}\cong \mathcal{N}(R)$,$H_{-}\cong \mathcal{N}(\hat{R})$$H_{-}^{*}\cong \mathcal{N}^{\uparrow}(R)_{0}$$0$,$H_{+}^{*}\cong \mathcal{N}\dagger(\hat{R})_{0}$,in dicates that the spinors in brace are$0$

at $0\in C^{2}$ or at$\hat{0}\in\hat{C}^{2}$.

For instance, the isomorphism
is given as follows:
such thatLet. We shall show that there is afor $|w|=1$ , where. Let$H+be$ the eigenfunction expansion.
Put. The expression on
becomes$H_{+}^{*}arrow \mathcal{N}\dagger(\hat{C}^{2})_{0}$$\hat{\Psi}\in \mathcal{N}\dagger(\hat{R})_{0}$$\psi=\gamma_{0}\phi\in H_{+}^{*}$$\hat{\Psi}(w)=\hat{\psi}(w)$$\wedge(v(z))=-\overline{\gamma_{0}\psi}(z)$$\phi=\sum_{\lambda>0}a_{\lambda}\phi_{\lambda}\in$$\hat{C}^{2}$$\Phi(z)=\sum a_{\lambda}|z|^{-(\lambda-\frac{3}{2})}(\frac{2}{1+|z|^{2}})^{\frac{3}{2}}\phi_{\lambda}(\frac{z}{|z|})$$\hat{\Phi}(w)=\sum a_{\lambda}|w|^{(\lambda+\frac{3}{2})^{\wedge}}(\frac{2}{1+|w|^{2}})^{\frac{3}{2}}\phi_{\lambda}(\frac{w}{|w|})$ノ$\wedge(v(z))=-\overline{\gamma_{0}\Phi(z)}.\hat{\Phi}$$\mathcal{N}^{\uparrow}(\hat{R})_{0}$andis valued infor $|w|=1$ .

$\triangle^{-}$,. We can verify that$\hat{\Psi}=\overline{\gamma}_{0}\hat{\Phi}\in$$\hat{\Psi}(w)=\hat{\psi}(w)$

We define a pairing of $H$ and$H^{*}$

by$( \psi|\phi)=\int_{E}<\phi,$ $\gamma_{0}\psi>\sigma(dz)$for $\phi\in H$ and$\psi\in H^{*}$.

Theorem 1 and Stokes’ theorem yield that $H\pm andH_{\mp}^{*}$ are annihilated mutually by this pairing. On the other hand, $H\pm andH_{\pm}^{*}$ are respectively induality. This is proved by Hahn-Banach’s extension theorem.
A coupling between
andis defined by$\mathcal{N}(R)$$\mathcal{N}^{\uparrow}(\hat{R})_{0}$$- \int_{E}\Phi(z)\cdot\hat{\Psi}(\nu(z))\sigma(dz)=\int_{E}<\Phi,$ $\gamma_{0}\Psi>\sigma(dz)$,. Also the coupling ofandandis defined by the same integral.
The duality between $H\pm andH_{\pm}^{*}$ in the above and Theorem 1 prove thefollowing:for$\Phi\in \mathcal{N}(R)$$\Phi\in \mathcal{N}\dagger(R)_{0}$$\hat{\Psi}\in N\dagger(\hat{R})_{0}$$\Psi\in \mathcal{N}(\hat{R})$59
Theor em 2.
(1) The dual of(2) The dual of$\mathcal{N}(R)$$\mathcal{N}(\hat{R})$is isomorphic to
is isomorphic to$\mathcal{N}^{\uparrow}(\hat{R})_{0}$$N^{\uparrow}(R)_{0}$.

.

3 Fockspace on $E$a Let$e_{\lambda}=\{\begin{array}{l}\phi_{k^{l}r-k}^{p}\in H+if\lambda\geq o+\pi_{p}^{r-k,k}\in H_{-}if\lambda\leq o_{-}\end{array}$

We define the conjugation byand
if $\lambda<0$ . We have$e^{*\lambda}=\gamma_{0}e_{-\lambda}$$\lambda\geq 0$$e^{*\lambda}\in H_{+}^{*}$. It follows that$(e^{*\lambda}|e_{\mu})=\delta_{-\lambda,\mu}$if. In particular$e^{*\lambda}\in H_{-}^{*}$$(e^{*0+}|e_{o-})=1$ .

For a Maya-diagram we put $e_{S}=\wedge e_{\lambda}=e_{\max S}\wedge\cdots$ , the wedge being.
taken on decreasing order. We denote in prticular
The Fock space of charge and total Fock space are introduced as follows:$S$$|\alpha>=e_{\mathcal{Z}_{\alpha-}}=e_{\alpha}\wedge\cdots$$p$$\mathcal{F}_{p}=\Pi_{\{S;\chi(S)=p\}}Ce_{S}$$\mathcal{F}=\oplus_{p}\mathcal{F}_{p}$.

is given a filtration by the degree of Maya-diagramm introduced in setion
with a complete vector space topology.
1 and this filtration endows
For a Maya-diagram we put, the wedge.
being taken on decreasing order. We denote
The dual Fock space is defined as a direct sum with discrete topology:$\mathcal{F}_{p}$$\mathcal{F}_{p}$$S$$e_{S}^{*}= \bigwedge_{-\mu\in S}e^{*\mu}=\cdots\wedge e^{*-\max S}$$<\alpha|=e_{\mathcal{Z}_{\alpha-}}^{*}=\cdots\wedge e^{*-\alpha}$$\mathcal{F}^{*}=\bigoplus_{s}Ce_{S}^{*}$.

The coupling $(|)$ of $H\pm andH_{\pm}^{*}$ extends to give a coupling between. In particular we have.
. We have$\mathcal{F}^{*}$$(e_{S}^{*}|e_{S’})=\delta_{S,S’}$Differentiation$D_{\alpha}$

by$\alpha\in H$$<\alpha|\beta>=\delta_{\alpha,\beta}$

is defined on $H$ by$D_{\alpha} \phi=(e^{*-\alpha}|\phi)=\int_{E}<\phi,$ $\alpha>d\sigma$

It is extended to$\mathcal{F}$for $\phi\in H$ .

by the rule

$D_{\alpha}(\phi\wedge\psi)=D_{\alpha}\phi\wedge\psi+(-1)^{\deg\phi}\phi\Lambda D_{\alpha}\psi$$\mathcal{F}$

and

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for
acts on from the left as an inner derivation.
.
byWe also define the differentiation on$\psi\in \mathcal{F}$$\phi,$$\mathcal{F}$$D_{\alpha}$$H^{*}$$D_{\alpha}^{*}\phi^{*}=(\phi^{*}|e_{\alpha})$by
It is extended to.
acts on$\psi^{*}\in \mathcal{F}^{*}$$D_{\alpha}^{*}$$\mathcal{F}^{*}$$\phi^{*}\in H^{*}$.

$a_{\nu}=D_{\nu}$,$a_{\nu}^{\uparrow}$$a_{\nu},$$a_{\nu}\dagger=e_{\nu}\wedge$$a_{\nu}=\wedge e^{*-\nu}$forfrom the right.

We define the following actions

$b$

for

$D_{\alpha}^{*}(\phi^{*}\wedge\phi^{*})=\phi^{*}\wedge D_{\alpha}^{*}\psi^{*}+(-1)^{\deg\psi^{*}}D_{\alpha}^{*}\phi^{*}\wedge\psi^{*}$$\mathcal{F}^{*}$$\phi^{*},$

,

,

on

$\mathcal{F}$and$\mathcal{F}^{*}:$,
right action onleft action on$a_{\nu}\dagger=D_{\nu}^{*}$$\mathcal{F}$$\mathcal{F}^{*}$,

where exterior multiplications should be arranged in order. We have then therelations,$\{a_{\lambda}, a_{\nu}\}=0$$\{a_{\lambda}^{\dagger}, a_{\nu}^{\uparrow}\}=0$$\{a_{\lambda}^{\dagger}, a_{\nu}\}=\{a_{\lambda}, a_{\nu}\dagger\}=\delta_{\lambda,\nu}$Hencealgebra .

generate a Clifford algebra
acts on and.

$\{a_{\nu}, a_{\nu}\dagger\}$$Pr$$\mathcal{A}$$\mathcal{F}$$\mathcal{A}$.

, which is called fermion operator$\mathcal{F}^{*}$

oposit ion 3.
(1)$a_{\nu}|\alpha>=0$

for$\nu>\alpha$$<\alpha|a_{\nu}=0$

for$\nu\leq\alpha$$a_{\nu}\dagger|\alpha>=0$.

$<\alpha|a_{\nu}\dagger=0$

for$\nu\leq\alpha$

for$\nu>\alpha$.

(2)$(e_{S}^{*}a_{\alpha}|e_{S’})=(e_{S}^{*}|a_{\alpha}e_{S’})$$(e_{S}^{*}|a_{\alpha}^{\uparrow}e_{S’})=(e_{S}^{*}a_{\alpha}^{\uparrow}|e_{S’})$$c$

We shall introduce the following field operators of fermion:$\varphi+(z)=\sum_{\nu\geq 0_{+}}\phi_{\nu}(z)a_{\nu}$$\varphi_{-}(w)=\sum_{\nu\leq 0-}\pi_{\nu}\wedge(w)a_{\nu}$$\varphi^{\underline{\dagger}}(z)=\sum_{\nu\geq 0_{+}}\overline{{}^{t}\phi_{\nu}(z)}a_{\nu}\dagger$$\varphi_{+}^{\dagger}(w)=\sum_{\nu\leq 0-}t\wedge\bigwedge_{\nu}_{\pi}(w)a_{\nu}^{\dagger}$.

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From the above proposition we have;$Pro$$\varphi+(z)|0_{-}>=0$,$\varphi_{+}^{\dagger}(z)|0_{-}>=0$$<0_{-}|\varphi_{-}(z)=0$,$<0_{-}|\varphi^{\underline{\dagger}}(z)=0$.

暇科何 ion 4.
$t$$< \varphi^{\dagger}(x)\varphi(y)>=<0_{-}|\varphi\dagger(x)\cdot\varphi(y)|0_{-}>=\sum_{r}\sum_{q=0}^{r+1}\frac{r+1}{q!}h_{r+1-q,q}^{q}(A, B)$$< \varphi(x)\varphi^{\dagger}(y)>=<0_{-}|\varphi(x)\cdot\varphi\dagger(y)|0_{-}>=\sum\sum\frac{r+2}{q!}h_{r-q,q}^{q}(C, D)r+1$$rq=0$$<\varphi(x)\varphi(y)>=<\varphi^{\dagger}(x)\varphi^{\dagger}(y)>=0$

where$A=\overline{x}_{1}y_{1}+x_{2}\overline{y}_{2}$$B=\overline{x}_{1}y_{2}-x_{2}\overline{y}_{1}$$C=x_{1}\overline{y}_{1}+x_{2}\overline{y}_{2}$$D=x_{1}y_{2}-x_{2}y_{1}$Ref[K]$K\alpha i$$S^{3}$.

$\propto$, T., Dirac $\sqrt at\alpha s\alpha 1S^{4}$ and[K] $K\alpha i,T.,$$E$.

伽寡 $C$ 伽歌$\alpha\rceil S^{3}$.

$|n$)$\iota ffi\dot{s}\alpha\rceil\gamma We\eta sW\dot{\Re}n\sigma s\alpha 1S^{4}.$.

nite$dim\alpha\iota\dot{s}\alpha lalGraae\eta an|an\alpha\rceil$