UDC 530.145

MATHEMATICAL PHYSICS

L. Sh. KHODZHAEV

ON THE THEORY OF \(2(2s+1)\)-COMPONENT FREE QUANTIZED SPINOR FIELDS

(Presented by Academician N. N. Bogolyubov, 14 XI 1966)

In the present paper we give a group-theoretic and functional formulation of the formalism of \(2(2s+1)\)-component free quantized spinor fields \((^{1,2})\), with the aim of subsequently studying the covariant structure of the elements of the causal \(S\)-matrix for arbitrary spin.

To define \(2(2s+1)\)-component free spinor fields as operator-valued generalized functions, we introduce into consideration a rigged Hilbert space \((^{3})\) \(D_S \subset H_{L_2} \subset D_S'\), induced by a unitary irreducible representation \([ms]\) of the spinor Poincaré group \(\widetilde P_+^\uparrow\) for a particle of mass \(m\) and spin \(s\).

The states of the system are described by vector-valued generalized functions from the space

\[ D_S'=\bigoplus_{n=0}^{\infty} D^{(n)}_{S(\bar\Omega_n^+)}[ms]_n, \tag{1} \]

i.e., the space of linear continuous functionals in the nuclear space \((^{3})\) of Fock \(D_S\). These vector-valued generalized functions have the form

\[ |\rho(\Phi)\rangle = \sum_{n=0}^{\infty} |\rho^{(sm)n}(\Phi)^{(sm)n}\rangle = \sum_{n=0}^{\infty} \sum_{(s_3)^n} |\rho^{(sm)n}_{(s_3)^n}(\Phi)^{(sm)n}_{(s_3)^n}\rangle \tag{2} \]

for any element \(\Phi=\{\Phi^{(sm)n}_{(s_3)^n}(p)_n\}_0^\infty \in D_S\), with components of the function
\(\Phi^{(sm)n}_{(s_3)^n}(p)\in S(\bar\Omega_n^+)\), where \(S(\bar\Omega_n^+)\) is the space of basic functions, where

\[ \bar\Omega_n^+=\bigotimes_{i=1}^{n}\bar\Omega_{m_i}^+,\qquad \bar\Omega_{m_i}^+= \{p_i\in \widetilde R^4;\; p_i^2=(p_i^0)^2-(\vec p_i)^2=m_i^2,\; p_i^0>0,\; i=1,\ldots,n\}; \]

\[ |\rho^{(sm)n}_{(s_3)^n}(\Phi)^{(sm)n}_{(s_3)^n}\rangle \]

are generalized basis vectors of the canonical system of the continuous representation

\[ [ms]_n=\prod_{i=1}^{n}[m_i s_i] \]

of the spinor Poincaré group \(\widetilde P_+^\uparrow\); \(s_3^i=-s_i,\ldots,s_i,\ i=1,\ldots,n\), are the eigenvalues of the third component of the spin-projection operator. On these generalized basis vectors is spanned the \(n\)-particle subspace \(D^{(n)'}_{S(\bar\Omega_n^+)}[ms]_n\)—a Hilbert space of generalized states. The transformation properties of the generalized vectors of the canonical system with respect to a unitary transformation \(U(a,A)\) of the Poincaré group \(\widetilde P_+^\uparrow\) are determined according to

\[ U(a,A)\,|\rho^{(sm)n}_{(s_3)^n}(\Phi)^{(sm)n}_{(s_3)^n}\rangle = |\rho^{(sm)n}_{(s_3)^n}(U(a,A)\Phi)^{(sm)n}_{(s_3)^n}\rangle, \tag{3} \]

where

\[ (U(a,A)\Phi)^{(sm)n}_{(s_3)^n}(p)_n= \]

\[ = \exp\left[i\sum_{j=1}^{n} a\cdot p_j\right] \sum_{(s_3')^n} \bigotimes_{j=1}^{n} D^{s_j m_j}_{s_j' s_j}\,[R^{-1}(A,p_j)]\, \Phi^{(sm)n}_{(s_3')^n}(A^{-1}p)_n. \tag{4} \]

where
\[ R^{-1}(A,p)=(p\cdot \widetilde{\sigma}/m)^{1/2}A^{-1}(A^{-1}p\cdot\sigma/m)^{1/2}\in SU(2,C); \quad \sigma=(\sigma_0,\vec{\sigma}); \]
\[ \widetilde{\sigma}=(\sigma_0-\vec{\sigma}) \]
are the Pauli matrices;
\[ (p\cdot\sigma/m)^{1/2}=[2m(p^0+m)]^{-1/2} \bigl[(m+p^0)+\vec p\,\vec\sigma\bigr]\in SL(2,C) \]
is the Lorentz transformation to the rest frame, i.e.
\[ (p\cdot\sigma/m)^{1/2}\widetilde p=p;\qquad \widetilde p=(m,0); \]
\(D^{sm}[R^{-1}(A,p)]\) is the usual \((2s+1)\times(2s+1)\)-dimensional matrix representation of the Wigner rotation \((^4)\)
\(R^{-1}(A,p)\in SU(2C)\). In formula (4) it is assumed that
\[ (Ap)^\mu=\Lambda(A)^\mu{}_\nu p^\nu \]
for \(A\in SL(2,C)\) and \(\Lambda(A)\in L_+^\uparrow\).

We note that the multiparticle generalization of the basis vector
\[ \bigl|\rho_{(s_3)n}^{(sm)n}(\Phi)_{(s_3)n}^{(sm)n}\bigr\rangle \]
is obtained on the basis of the Schwartz kernel theorem \((^6)\) by means of the direct product of one-particle generalized basis vectors
\[ \bigl|\rho_i^{s_i m_i}(\Phi)_{s_i}^{s_i m_i}\bigr\rangle \]
for arbitrary
\[ \Phi_{s_i}^{s_i m_i}(p_i)\in D_{s(2m_i)}^{(1)}[m_i s_i],\qquad i=1,\ldots,n, \]
which are common generalized eigenvectors of the operators of 4-momenta \(\widehat P^\mu\) and of the 3rd component of the projection of the spin operator \(\widehat S_3(p)\), defined by the relation
\[ \widehat S_j(p)=\frac{1}{2m}\sum_{\mu,\nu,\sigma,\lambda=0}^{3} \varepsilon_{\mu\nu\sigma\lambda}(p\cdot\widetilde{\sigma}/m)_j^{1/2\,\mu}P^\nu M^{\sigma\lambda}. \tag{5} \]

We now introduce into consideration the operator-valued generalized creation functions
\[ a_{s_3}^{*\,sm}(\widetilde f_{s_3}) \]
and annihilation functions
\[ a_{s_3}^{sm}(\widetilde f_{s_3}) \]
of a particle of mass \(m\) and spin \(s\), and the corresponding operator-valued generalized functions
\[ b_{s_3}^{*\,sm}(\widetilde g_{s_3}) \]
and
\[ b_{s_3}^{sm}(\widetilde g_{s_3}) \]
for an antiparticle, where
\[ \widetilde f_{s_3}(p),\ \widetilde g_{s_3}(p)\in S(\overline{\Omega}_m^+), \]
defined in the canonical system of generalized basis vectors, transforming according to the unitary representations
\[ D^{sm}[R(A,p)] \quad\text{and}\quad D^{sm}[R^{-1}(A,p)] \]
of the spinor Poincaré group \(\widehat P_+^\uparrow\). Analogously one may introduce one-particle operator-valued generalized functions
\[ \widehat a_\alpha^{*\,sm}(\widetilde f^\alpha),\quad \widehat a_\alpha^{sm}(\widetilde f^\alpha),\quad \widehat b_\alpha^{*\,sm}(\widetilde g^\alpha),\quad \widehat b_\alpha^{sm}(\widetilde g^\alpha), \]
\(\alpha=-s,s\), defined in the spinor system of generalized basis vectors, transforming according to the nonunitary finite-dimensional representations
\[ D^{sm}(A),\quad D^{sm}(A)^{-1+},\quad D^{sm}(A)^{-T} \quad\text{and}\quad D^{sm}(A)^* \]
of the group \(SL(2,C)\).

Between these two classes of operator-valued generalized functions the following relations hold. For example \((^5)\):
\[ \sum_\alpha \widehat a_\alpha^{sm}(\widetilde f^\alpha) = \sum_{s_3} a_{s_3}^{sm}(\widetilde g_{s_3}), \tag{6} \]
where
\[ \widetilde g_{s_3}(p)= \sum_\alpha D_{\alpha s_3}^{sm}\bigl[(p\cdot\sigma/m)^{1/2}\bigr]\, \widetilde f^\alpha(p); \]
\[ \sum_\alpha \widehat b_\alpha^{*\,sm}(\widetilde f_-^\alpha) = \sum_{s_3} b_{s_3}^{*\,sm}(\widetilde h_{s_3}), \tag{7} \]
where
\[ \widetilde h_{s_3}(p)= \sum_\alpha D_{\alpha s_3}^{sm}\bigl[(p\cdot\sigma/m)^{1/2}\varepsilon^{-1}\bigr]\, \widetilde f^\alpha(-p), \]
\[ p^0=\sqrt{\vec p^{\,2}+m^2} \]
for arbitrary
\[ \widetilde f^\alpha(p),\quad \widetilde f_-^\alpha(p)=\widetilde f^\alpha(-p)\in S(\overline{\Omega}_m^+). \]

In order to formulate the condition of causality or local commutativity, it is evidently necessary to define quantities with causally independent supports. To this end we introduce into consideration \((2s+1)\)-component spinor fields
\[ \varphi_\alpha^{sm}(x) \quad\text{and}\quad \chi_{sm}^{\dot\alpha}(x), \]
respectively in the \((s,0)\)- and \((0,s)\)-representations, regarded as operator-valued generalized functions, i.e. to each function
\[ f^\alpha(x)\in S(R^4) \]
there is put in corres-

there corresponds the linear operator

\[ \varphi_{\alpha}^{sm}(f^{\alpha})=\sum_{\alpha=-s,s}\varphi_{\alpha}^{sm}(f^{\alpha}) =2\pi\sum_{\alpha=-s,s}\left[\hat a_{\alpha}^{sm}(\widetilde f_{-}^{\alpha})+\hat b_{\alpha}^{*\,sm}(\widetilde f_{\alpha})\right]= \]
\[ =\sqrt{2\pi}\sum_{s_3}\left[a_{s_3}^{sm}(\widetilde g_{s_3})+b_{s_3}^{*\,sm}(\widetilde h_{s_3})\right]\quad \text{in } D_S, \tag{8} \]

where \((\Phi,\varphi_{\alpha}^{sm}(x)\psi)\in S'(R^4)\) for \(\Phi,\psi\in D_S\). The field \(\chi_{sm}^{\dot\alpha}(x)\) has an analogous representation.

The spinor \((2s+1)\)-component fields \(\varphi_{\alpha}^{sm}(x)\) and \(\chi_{sm}^{\dot\alpha}(x)\) satisfy the wave equations

\[ (-i)^{2s}\varphi_{\alpha}(\hat\partial_{s}^{\alpha\dot\alpha}g_{\dot\alpha}) =\chi^{\dot\alpha}(g_{\dot\alpha}),\qquad (-i)^{2s}\chi^{\dot\alpha}(\hat\partial_{\dot\alpha\alpha}f^{\alpha}) =\varphi_{\alpha}(f^{\alpha}), \tag{9} \]

\[ \varphi_{\alpha}((\square-m^2)f^{\alpha})=0,\qquad \chi^{\alpha}((\square-m^2)g_{\dot\alpha})=0 \tag{10} \]

for arbitrary \(f^{\alpha}(x), g_{\dot\alpha}(x)\in S(R^4)\), where

\[ \hat\partial^{ss}= \begin{pmatrix} 0 & \hat\partial_{\alpha\dot\alpha}^{ss}\\ \hat\partial_{s\dot s}^{\dot\alpha\alpha} & 0 \end{pmatrix} = \begin{pmatrix} 0 & D_{\alpha\dot\alpha}^{sm}(\widetilde\sigma\!\cdot\!\partial/m)\\ D_{\alpha\dot\alpha}^{sm}(\sigma\!\cdot\!\partial/m) & 0 \end{pmatrix}. \tag{11} \]

Bearing in mind that any interaction preserving parity must include both the \(\varphi_{\alpha}^{sm}(x)\)-field and the \(\chi_{sm}^{\dot\alpha}(x)\)-field, we introduce for consideration the \(2(2s+1)\)-component spinor fields \(\psi^{sm}(x)\) and \(\overline\psi^{sm}(x)\), regarded as operator-valued generalized functions; that is, to each \(2(2s+1)\)-component regular spinor

\[ F(x)=\left\{F_{\chi}(x)= \begin{pmatrix} g_{\dot\alpha}(x)\\ f^{\alpha}(x) \end{pmatrix}\right\},\qquad \overline F_{\chi}(x)=F_{\chi}^{*}(x) \tag{12} \]

with components \(g_{\dot\alpha}(x), f^{\alpha}(x)\in S(R^4)\) there are assigned linear operators

\[ \psi^{sm}(\overline F)=\sum_{\chi=(\alpha,\dot\alpha)}\psi_{\chi}^{sm}(\overline F_{\chi}) =\sum_{\alpha=-s,s}\left[\varphi_{\alpha}^{sm}(f^{\alpha})+\chi_{sm}^{\dot\alpha}(g_{\dot\alpha})\right], \tag{13} \]

\[ \overline\psi^{sm}(F)=\sum_{\chi=(\alpha,\dot\alpha)}\overline\psi_{\chi}^{sm}(F_{\chi}) =\sum_{\alpha=-s,s}\left[\varphi_{\dot\alpha}^{sm}(f^{\dot\alpha})+\chi_{sm}^{\alpha}(g_{\alpha})\right], \tag{14} \]

defined in \(D_S\), and moreover \((\Phi,\psi^{sm}(x)\psi)\in S'(R^4)\) and \((\Phi,\overline\psi^{sm}(x)\psi)\in S'(R^4)\) for \(\Phi,\psi\in D_S\). These fields satisfy the wave equations

\[ (-i)^{2s}\psi^{sm}(\overline F\hat\partial^{ss})=\psi^{sm}(\overline F),\qquad \psi^{sm}((\square-m^2)\overline F)=0; \tag{15} \]

\[ (i)^{2s}\overline\psi^{sm}(\hat\partial^{*\,sm}F)=\overline\psi^{sm}(F),\qquad \overline\psi^{sm}((\square-m^2)F)=0 \tag{16} \]

and the commutation relations for free fields.

The spinor \(2(2s+1)\)-component fields \(\psi^{sm}(x)\) and \(\overline\psi^{sm}(x)\) transform according to a nonunitary finite-dimensional representation of the spinor Poincaré group \(\widetilde{\mathscr P}_{+}^{\uparrow}\) as follows:

\[ U(a,A)\psi^{sm}(\overline F)U^{-1}(a,A) =\psi^{sm}(U(a,A)\overline F), \tag{17} \]

where

\[ (U(a,A)\overline F)(x)=\{U(a,A)\overline F_{\chi}(x)\}_{\chi=(\alpha,\dot\alpha)} \]

\[ =\left\{\sum_{\chi'}\overline F_{\chi'}\!\left[A^{-1}(x-a)\right]D_{\chi'\chi}^{sm}(A)\right\}; \tag{18} \]

\[ D^{sm}(A)= \begin{pmatrix} D^{sm}(A)^{-1} & 0\\ 0 & D^{sm}(A)^{+} \end{pmatrix} \tag{19} \]

for arbitrary \(\overline F(x)\in S(R^4)\) and \(A\in SL(2,C)\).

The \(P\)-, \(T\)- and \(C\)-invariance of these fields can be expressed by means of the functional relations:

\[ U_P \psi^{sm}(\bar F) U_P^{-1}=\xi_P \psi^{sm}(U_P\bar F), \tag{20} \]

where

\[ (U_P\bar F)(x)=\{U_P\bar F_\chi(x)\}_{\chi=(a,\dot a)} =\left\{\sum_{\chi'}\bar F_{\chi'}(Px)R_{\chi'\chi}\right\}, \tag{21} \]

\[ R=\begin{pmatrix}0&I\\ I&0\end{pmatrix}, \qquad \bar F(x)\in S(R^4) \]

(\(P\)-invariance; for \((\xi_P)^2=1,\ U_P^2=1\));

\[ \bar U_T\psi^{sm}(\bar F)U_T^{-1} =\xi_T\bar\psi^{sm}(\bar U_TF), \tag{22} \]

where

\[ (\bar U_TF)(x)=K^T R^T\bar F^T(Tx)\in S(R^4), \tag{23} \]

\[ K^T= \begin{pmatrix} D^s(\varepsilon)&0\\ 0&D^s(\varepsilon) \end{pmatrix}, \qquad D^s_{s'_3s_3}(\varepsilon)=(-)^{s+s'_3}\delta_{s'_3,-s_3} \]

(\(T\)-invariance; for \(|\xi_T|^2=1,\ U_T^2=(-)^{2s}\));

\[ U_C\psi^{sm}(\bar F)U_C^{-1} =\xi_C\,{}^*\psi^{sm}(U_CF), \tag{24} \]

where

\[ (U_CF)(x)=\{(U_CF)_\chi(x)\}_{\chi=(a,\dot a)} =\left\{\sum_{\chi'} C^T_{\chi'\chi}\bar F^T_{\chi'}(x)\right\}, \tag{25} \]

\[ C= \begin{pmatrix} D^s(\varepsilon)&0\\ 0&(-)^{2s}D^s(\varepsilon) \end{pmatrix} \tag{26} \]

(\(C\)-invariance; for \(|\xi_C|^2=1,\ U_C^2=1\)).

One can define Green’s functions for these fields.

Taking this opportunity, I express my deep gratitude to N. N. Bogolyubov, A. N. Tavkhelidze, Nguyen Van Hieu, I. Todorov, B. V. Medvedev-Vereshchagin and A. V. Efremov for useful discussions and valuable remarks.

Joint Institute
for Nuclear Research

Received
19 IX 1966

CITED LITERATURE

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5. L. Sh. Khodjaev, On the theory of \(2(2S+1)\)-component free quantized spinor fields, I, Preprint, D-2-3010, Joint Inst. Nuclear Research, Dubna, 1966.
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