UDC 512.86:519.46

MATHEMATICS

S. S. SANNIKOV

ON EXTRACTING A SQUARE ROOT FROM ANTICOMMUTING SPINORS

(Presented by Academician A. I. Mal'tsev on 10 III 1966)

1. There is an elegant algebraic device which, for the group \(SO(n)\) preserving the nondegenerate symmetric bilinear form
\[ (x,y)=x_1y_1+x_2y_2+\cdots+x_ny_n, \]
allows one to construct its covering group.* This device goes back to Dirac \((^1)\) (it is sometimes called extracting the square root from a vector \((^2)\)) and consists in associating with the vector \(x=(x_1,\ldots,x_n)\) an element \(\hat{x}=x_i\gamma_i\) such that
\[ \hat{x}^2=(x_i\gamma_i)^2=x_1^2+x_2^2+\cdots+x_n^2 . \]

Here the \(\gamma_i\) must satisfy the relations
\[ \gamma_i^2=1,\qquad \gamma_i\gamma_k+\gamma_k\gamma_i=0\quad (i\ne k). \]

The elements \(\hat{x}=x_i\gamma_i\) form an \(n\)-dimensional subspace \(\hat{X}_n\) of the Clifford algebra \(C_n\) of dimension \(2^n\), \(n\) even (or \(2^{n-1}\), \(n\) odd). The automorphisms in \(C_n\) that preserve the subspace \(\hat{X}_n\) invariant,
\[ S\hat{x}S^{-1}=\hat{x}' \tag{1} \]
and the scalar square \(\hat{x}^2=(\hat{x}')^2\), where \(x'=Ox\), \(O\in SO(n)\), form the group \(\operatorname{Spin}(n)\).

Theorem 1. The group \(\operatorname{Spin}(n)\) is locally isomorphic to the group \(SO(n)\) and covers it twice.

Relation (1) defines a projective representation of the group \(SO(n)\). The elements \(S\in\operatorname{Spin}(n)\), as matrices \(2^{n/2}\times 2^{n/2}\), \(n\) even (or \(2^{(n-1)/2}\times 2^{(n-1)/2}\), \(n\) odd), realize a two-valued representation of the group \(SO(n)\) in a \(2^{n/2}\)- (or \(2^{(n-1)/2}\)-) dimensional spinor space \((^2)\).

2. A group exhibiting a close analogy with \(SO(n)\) is the symplectic group \(\operatorname{Sp}(2m)\) of transformations of a \(2m\)-dimensional space \(\Phi^{2m}\) preserving the skew-symmetric bilinear form \([\varphi,\chi]\), \(\varphi,\chi\in\Phi^{2m}\). In canonical form
\[ [\varphi,\chi]= \sum_{\substack{\alpha,\beta=1,2\\ i,j=1,\ldots,m}} \delta_{ij}\varepsilon_{\alpha\beta}\varphi_i^\alpha\chi_j^\beta , \tag{2} \]
where
\[ \varepsilon_{\alpha\beta}= \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}, \]
\(\delta_{ij}\) is the Kronecker symbol.

We wish to transfer the construction described above to the group \(\operatorname{Sp}(2m)\).**

* We also have in mind the case of an indefinite metric form
\[ (x,y)=x_1y_1+\cdots+x_py_p-x_{p+1}y_{p+1}-\cdots-x_ny_n \]
(the group \(O(p,n-p)\), \(p\le n/2\)).

** In the case when the group \(\operatorname{Spin}(n)\subset \operatorname{Sp}(2^{n/2})\) \(\operatorname{Sp}(2^{(n-1)/2})\) and preserves a skew-symmetric bilinear form, we shall call the construction under consideration extraction of the square root from a spinor. It is known \((^3)\) that, for even \(n\), the skew-symmetric form exists if the numbers \(n(n-2)/8\) or \(n(n+2)/8\) are odd; for odd \(n\), if \((n-1)(n-3)/8\) or \((n^2-1)/8\) are odd numbers.

Remark. In the case of a skew-symmetric form, extraction of the square root is possible if \(\Phi^{2m}\) is defined over a Grassmann algebra*, i.e., \(\Phi^{2m}\) is a \(2m\)-dimensional subspace of the Grassmann algebra \(G_{2m}\), generated by the basis elements \(\varphi_i^\alpha\) \((\alpha=1,2;\ i=1,\ldots,m)\): \(\varphi_i^\alpha\varphi_j^\beta+\varphi_j^\beta\varphi_i^\alpha=0\)**.

In this case the quadratic form \([\varphi,\varphi]\ne 0\). To each \(\varphi=(\varphi_1^1,\varphi_2^2,\ldots,\varphi_m^1,\varphi_m^2)\) we assign the element

\[ \hat{\varphi}=\sum_{\substack{\alpha=1,2\\ i=1,\ldots,m}}\varphi_i^\alpha a_\alpha^i \tag{3} \]

such that

\[ \hat{\varphi}^{\,2}=[\varphi,\varphi]=2(\varphi_1^1\varphi_1^2+\ldots+\varphi_m^1\varphi_m^2). \tag{4} \]

The elements \(a_\alpha^i\) must satisfy the relations

\[ a_\alpha^i a_\beta^j-a_\beta^j a_\alpha^i=2\varepsilon_{\alpha\beta}\delta_{ij}. \tag{5} \]

Lemma. The \(a_\alpha^i\) generate an algebra isomorphic to the algebra \((qp)_m\) of the \(m\)-dimensional coordinate and momentum operators of quantum mechanics.

Put \(a_1^i=\sqrt{-2}\,p_i,\ a_2^i=\sqrt{2}\,q_i\). Then from (5) it follows that

\[ q_i p_j-p_j q_i=\sqrt{-1}\delta_{ij},\qquad q_i q_j-q_j q_i=p_i p_j-p_j p_i=0. \tag{5'} \]

Denote by \(\Phi_{2m}\subset(q,p)_m\) the \(2m\)-dimensional subspace generated by monomials of the form \(\hat{\varphi}=\varphi_i^\alpha a_\alpha^i\). In \((q,p)_m\) consider automorphisms preserving the invariant subspace \(\Phi_{2m}\),

\[ T\hat{\varphi}T^{-1}=\hat{\varphi}', \tag{6} \]

where \(\varphi'=S\varphi,\ S\in \operatorname{Sp}(2m)\). These automorphisms preserve the symplectic square: \(\hat{\varphi}^{\,2}=(\hat{\varphi}')^2\).

In the case of the group \(\operatorname{Sp}(2m)\), the analogue of Theorem 1 is

Theorem 2. The automorphisms (6) form a group locally isomorphic to the group \(\operatorname{Sp}(2m)\).

Proof. Consider (6) in infinitesimal form. We have \(S=1+I_\xi\theta_\xi,\ T=1+L_\xi\theta_\xi\), where \(\theta_\xi\) are real parameters of an infinitesimal transformation from the group \(\operatorname{Sp}(2m)\), and \(I_\xi,\ L_\xi\) are infinitesimal operators. The elements \((I_\xi)^\alpha_{ik\beta}\) satisfy the relations

\[ (I_\xi)^\alpha_{ik\beta}\varepsilon_{\alpha\gamma} +\varepsilon_{\beta\lambda}(I_\xi)^\lambda_{ki\gamma}=0, \]

and, moreover,

\[ I_\xi I_\eta-I_\eta I_\xi=C^\zeta_{\xi\eta}I_\zeta, \tag{7} \]

where \(C^\zeta_{\xi\eta}\) are the structure constants of the Lie algebra of the group \(\operatorname{Sp}(2m)\). Equation (6) now gives

\[ L_\xi a_\alpha^i-a_\alpha^i L_\xi=(I_\xi)^\beta_{ik\alpha}a_\beta^k. \tag{8} \]

From (8) for \(L_\xi\) we obtain

\[ L_\xi=\frac14 (I_\xi)^\beta_{ik\alpha}a_\beta^k a_i^\alpha \varepsilon^{\alpha\gamma},\qquad \varepsilon^{\alpha\gamma}\varepsilon_{\gamma\beta}=\delta^\alpha_\beta, \tag{9} \]

where \(\delta^\alpha_\beta\) is the Kronecker symbol. Using the permutation relations for \(a_\alpha^i\) (5), we establish that

\[ L_\xi L_\eta-L_\eta L_\xi=C^\zeta_{\xi\eta}L_\zeta \tag{10} \]

with the \(C^\zeta_{\xi\eta}\) from (7). Thus, the local isomorphism of \(T\) and \(\operatorname{Sp}(2m)\) is established. The finite transformations \(T\) are expressed through infinitesimal operators in the form of formal series:

\[ T=\sum_{n=0}^{\infty}\frac{1}{n!}(L_\xi\theta_\xi)^n=\exp(L_\xi\theta_\xi). \]

With such \(T\), formula (6) defines a projective representation of the group \(\operatorname{Sp}(2m)\). The \(T\) do not always exist as affine transformations in Hilbert space.

* Recall that in the case of a symmetric form, extraction of the square root is possible in any (Euclidean) field (see (2), p. 366).

** The scheme developed is not abstract. Spinors of quantum field theory are quantities of precisely this (algebraic) nature.

Theorem 3. The operators \(T\) define a unitary representation in Hilbert space if the numbers \((I_\xi)^1_{ik1}, (I_\xi)^2_{ik2}\) are real, and \((I_\xi)^1_{ik2}, (I_\xi)^2_{ik1}\) are imaginary.

The proof follows from definition (9). Indeed, the operators \(q_i, p_i\) (5′) can be realized as Hermitian operators in Hilbert space (Stone–von Neumann theorem). Then, under the formulated conditions, the operators

\[ H_\xi=-\sqrt{-1}L_\xi=\frac{1}{2}\left[(I_\xi)^1_{ik1}p_kq_i-(I_\xi)^2_{ik2}q_kp_i-\right. \]
\[ \left.-\sqrt{-1}(I_\xi)^1_{ik2}p_ip_k-\sqrt{-1}(I_\xi)^2_{ik1}q_iq_k\right] \]

are Hermitian, and, consequently, the operators \(T=\exp(\sqrt{-1}H_\xi\theta_\xi)\) are unitary. (We note that \(H_\xi\) have the form of the dynamical operators of an \(m\)-dimensional oscillator.)

For the simply connected group \(\operatorname{Sp}(2m,C)\) the conditions of the theorem are not satisfied. They may hold for some noncompact nonsimply connected subgroup of the group \(\operatorname{Sp}(2m,C)\). In particular, the conditions of the theorem are satisfied for the real symplectic group \(\operatorname{Sp}(2m,R)\). In this case the operators \(T\) form a group which doubly covers the group \(\operatorname{Sp}(2m,R)\).* In Hilbert space the operators \(T\) define a two-valued representation of the group \(\operatorname{Sp}(2m,R)\). In the realization \(q_i=x_i,\ p_i=\dfrac{1}{\sqrt{-1}}\dfrac{\partial}{\partial x_i}\) \((-\infty<x_i<\infty)\), the representation in the class \(L_2(m)\) of quadratically integrable functions \(\int f|x_1,\ldots,x_m|^2 dx_1\ldots dx_m<\infty\) is defined by the formula

\[ T_s f(x)=T_s f(x)T_s^{-1}(T_s\cdot 1)=f(T_sxT_s^{-1})\chi(x;s)=f(xs)\chi(x;s) \]
\[ s\in \operatorname{Sp}(2m,R), \tag{11} \]

where \(\chi(x;s)=T_s\cdot 1\), and

\[ (xs)_i=s^2_{ij2}x_j+s^1_{ij2}\frac{\partial}{\partial x_j}. \]

The group condition leads to the following equation for \(\chi(x;s)\):

\[ T_{s_2}\chi(x;s_1)=\chi(xs_2;s_1)\chi(x;s_2)=\chi(x;s_2s_1) \tag{12} \]

with boundary condition \(\chi(x;1)=1\).

In (4) such representations of the group \(\operatorname{Sp}(2,R)\) are constructed explicitly (one-dimensional oscillator).

1. Our main result consists in the fact that the operation of extracting the square root is solvable at least for doubly connected symplectic groups.** We can transfer this operation also to simply connected symplectic groups, for example \(\operatorname{Sp}(2m,C)\). Then extraction of the root leads to infinite-dimensional spaces of a more general type than Hilbert spaces. Indeed, now the operators \(\sqrt{-1}L_\xi\) are not Hermitian, and, consequently, \(T\) are not unitary.

The representation can be obtained from (11), (12) by complexifying \(s\in\operatorname{Sp}(2m,R)\), as a result of which we obtain the group \(\operatorname{Sp}(2m,C)\), and \(x_i\) (now the transformations (6) are complex) also become complex. In this case the functions \(f(x)\) (\(x_i\) complex) become elements of an infinite-dimensional space with a strongly indefinite metric. For the group \(U\operatorname{Sp}(2)\) this question was partially discussed in (5).

Physical-Technical Institute
Academy of Sciences of the Ukrainian SSR

Received
12 II 1966

CITED LITERATURE

1. P. A. M. Dirac, Proc. Roy. Soc., A117, 610 (1928).
2. H. Weyl, Classical Groups, IL, 1947.
3. P. K. Rashevskii, UMN, 10, no. 2 (1955).
4. S. S. Sannikov, ZhETF, 49, no. 6 (1965).
5. S. S. Sannikov, Ukr. Phys. Zh., 10, no. 6 (1965); Yadern. Fiz., 2, no. 3 (1965).

* The group space of the group \(T\) is a \((4m^2-1)\)-dimensional manifold in Hilbert space.

** For orthogonal groups the extraction of the root is always possible (and leads to finite-dimensional spaces), since all orthogonal groups are at least doubly connected.