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UDC 539.01 + 519.413/47
MATHEMATICAL PHYSICS
N. P. KONOPLEVA, G. A. SOKOLIK
THE GROUP OF MOTIONS OF A LINE
AND THE GEOMETRIC THEORY OF FIELDS
(Presented by Academician L. I. Sedov, January 24, 1967)
- As is known, gauge fields \(A_\mu^{a}\) are introduced on the basis of the condition of covariance of equations of the Gelfand–Yaglom type \((^{1})\):
\[ L^\mu \nabla_\mu \psi + \hat{\lambda}\psi = 0 \tag{1} \]
with respect to localized gauge transformations:
\(\delta\psi = \varepsilon^{a}(x) I_a \psi\) (\(I_a\) are the generators of a representation of some group).
The covariant derivative \(\nabla_\mu\) is defined by the commutation rule:
\[ [\nabla_\mu \delta] = I_a \delta A_{\mu}^{a}, \]
where
\[ \delta A_\mu^{a} = \varepsilon^{b} f_{bc}^{a} A_\mu^{c} + \partial_\mu \varepsilon^{a}. \tag{2} \]
The geometric theory of the potentials \(A_\mu^{a}\) \((^{2,3})\), according to which \(A_\mu^{a}\) are regarded as coefficients of an affine connection of an internal space, presupposes a certain single-valued process of raising parametric indices, with the generators \(I_a\) and \(I^{a}\) related by the relation:
\[ [I^{a} I_b] = f_{b}^{ac} I_c. \tag{3} \]
- Relation (3) may be regarded as the structure formula of the given group only in the case when there exists a Casimir operator
\[ H = g^{ab} I_a I_b,\quad \text{where } g^{ab} = f_m^{al} f_l^{bm}, \quad \text{and} \quad [I^{a} I^{b}] = f_c^{ab} I^c . \]
From \([H I_a] = 0\) it follows \((^{4})\) that
\[
(g^{ab} f_{al}^{c} + g^{lc} f_{al}^{b}) I_c I_b = 0,
\]
and hence \(f_a^{bc} = g^{bl} f_{al}^{c}\).
But from \(f_a^{bc} = g^{bl} f_{al}^{c}\), obviously, it follows that \(f_a^{ac} = 0\). In other words, \(I_a\) and \(I^{a}\) form a basis of the algebra of a single structure under the condition that this structure is given by \(f_{abc}=f_{[abc]}\), i.e., by constants antisymmetric in all indices. In papers \((^{2,3})\) we essentially restricted ourselves to groups with structure \(f_{[abc]}\), i.e., locally compact ones \((^{4})\).
- Let us admit consideration of an equation of a more general type
\[ L^{a} X_a \psi + \hat{\lambda}\psi = 0, \tag{4} \]
where
\[
X_a = \xi_a^{i}\frac{\partial}{\partial x^{i}}
\]
transforms according to the regular representation of the group
\[ \delta X_a = - \varepsilon^{b} f_{ba}^{c} X_c . \]
The covariance condition (4) has the form:
\[ [L_b^{a} I]=f_b^{a}{}_c L^c . \tag{5} \]
(5), obviously, passes into (3) for groups of structure \(f_{[abc]}\).
-
One of the simplest examples of a group of a more general type (\(f_b^{a}{}_c\) are not antisymmetric in all indices) is the one-dimensional model of the Poincaré group of structure \([X_1 X_2]=-X_2\), isomorphic to the group of motions of the line \(I\) (5).
-
Let us investigate the representation \(I\). We find the generators of the regular representation \(I\) in differential form. According to (5), introducing the regular representation on the set of left cosets with respect to the Abelian subgroup
\[ u(t)= \begin{pmatrix} e^t & 0\\ 0 & e^{-t} \end{pmatrix}, \]
we obtain
\[ X_{\omega} f = \left[ \frac{df}{dt}(u\omega(t)) \right]_{t=0} = \frac{\partial f}{\partial \varepsilon^a}\varepsilon^{a'}(0). \]
Starting from the fundamental representation \(I\)
\[ S(\varepsilon,\eta)= \begin{pmatrix} e^\eta & 0\\ \dfrac{\varepsilon}{\eta}\operatorname{sh}\eta & e^{-\eta} \end{pmatrix}, \]
where \(\varepsilon,\eta\) are parameters of \(I\); \(\eta=\eta_1+\eta_2\);
\[ \frac{\varepsilon}{\eta}\operatorname{sh}\eta = e^{\eta_2}\frac{\operatorname{sh}\eta_1}{\eta_1}\varepsilon_1 + e^{-\eta_1}\frac{\operatorname{sh}\eta_2}{\eta_2}\varepsilon_2, \]
we obtain
\[ X_1= \frac{1}{2} \left[ \frac{\partial}{\partial \eta} + \frac{\varepsilon}{\eta} \left( 1-\frac{e^{-\eta}\eta}{\operatorname{sh}\eta} \right) \frac{\partial}{\partial \varepsilon} \right]; \qquad X_2= \frac{e^{-\eta}}{\operatorname{sh}\eta} \frac{\partial}{\partial \varepsilon}. \tag{6} \]
Then from
\[ \frac{\partial \psi_{mn}(\eta)}{\partial \eta} + \frac{m}{\eta} \left( 1-\frac{e^{-\eta}\eta}{\operatorname{sh}\eta} \right) \psi_{mn}(\eta) = n\psi_{mn}(\eta), \]
where \(f_{mn}(\varepsilon,\eta)=\varepsilon^m\psi_{mn}(\eta)\), it follows that
\[ S_{nn'}^{(p,k)}(\varepsilon,\eta) = \sum_{m=0}^{p} \sqrt{\binom{n}{m}}\, \varepsilon^m \eta^{-m}\operatorname{sh}^{m}\eta\, e^{(2n-m)\eta}\delta_{n-m,n'}; \tag{7} \]
\[ 0\le m\le p;\qquad k-p\le n\le k;\qquad \psi_{mn}(\eta)\sim \operatorname{sh}^{m}\eta^{-m}e^{(2n-m)\eta}. \]
In this case
\[ X_2 f_{mn}(\varepsilon,\eta)=\sqrt{mn}\,f_{m-1,n-1}(\varepsilon,\eta) \]
(\(X_2\) shifts the columns \(S_{nn'}^{(p,k)}\) to the right).
The dimension \(s(p,k)\) is specified, as usual, by the character of \(I\):
\[ \chi(\eta)=e^{(2k-p)\eta}\frac{\operatorname{sh}(p+1)\eta}{\operatorname{sh}\eta}, \qquad s(p,k)=\chi(0)=p+1. \tag{8} \]
From this follows the addition formula for \(I\):
\[ T_g f_{mn}(\varepsilon_2\eta_2) = \sum_{k=0}^{m} \sum_{p=0}^{m-k} g_{mn} \binom{m}{k} \binom{m-k}{p} e^{-2p\eta_1} \operatorname{ch}^{k}\eta\, f_{kn}(\varepsilon'\eta')\,f_{m-k,n-k}(\varepsilon,\eta). \]
The number \(k\) is not an independent weight, since
\[ S_{nn'}^{(p,k)}=e^{2(k-p)}S_{nn'}^{(p,p)}. \]
The generators of the representation of weight \(p\)
\[ M_1 e_n=ne_n,\qquad M_2 e_n=\sqrt{n}\,e_{n-1} \tag{9} \]
follow from the expressions
\[ X_1 S_{nn'}^{(p,k)}(\varepsilon,\eta) = nS_{nn'}^{(p,k)}(\varepsilon,\eta), \]
\[ X_2 S_{n+1,n'}^{(p,k)}(\varepsilon,\eta) = \sqrt{nm}\,S_{n,n'}^{(p,k)}(\varepsilon,\eta). \]
It is easy to see that \(f_{n,m}(\varepsilon,\eta)\) form a basis of the irreducible representation \(S^{(p,k)}\).
- In the particular case, from (7) we obtain the infinite-dimensional representation
\[ S_{nn'}^{(\infty)} = \sum_{m=0}^{\infty} \binom{n}{m}^{1/2} \left(\frac{\varepsilon}{\eta}\operatorname{sh}\eta\right)^m e^{(2n-m)\eta}\delta_{n-m,n'}, \tag{10} \]
\[ S_{nn'}^{(p,k)} \to S_{nn'}^{(\infty)} \qquad (p\to\infty). \]
From (9) follows (3)
\[ S_{nn'}^{(\infty)} = \exp(\eta M_{1})\exp\left[ \left(\frac{\varepsilon}{\eta}e^{\eta}\operatorname{sh}\eta\right)M_{2} \right]. \]
It turns out that relation (5) is realized only in the case of the infinite-dimensional representation (10).
Indeed, in case \(I\)
\[ [L^{1}M_{1}]=[L^{1}M_{2}]=0;\qquad [M_{1}L^{2}]=L^{2}, \]
\[ [L^{2}M_{2}]=L^{1};\qquad [M_{1}M_{2}]=-M_{2}. \tag{11} \]
From Schur’s lemma \((^{5,6})\) it follows that \(L^{1}=\lambda E\), but then (11) reduces to Bose statistics:
\[ [A^{+}A^{-}]=-E;\qquad M_{1}=A^{+}A;\qquad M_{2}=A^{-};\qquad L'=-E;\qquad L^{2}=A^{+}, \tag{12} \]
where \(A^{+}=\sqrt{n+1}\,\delta_{n+1,n'}\), \(A^{-}=\sqrt{n}\,\delta_{n-1,n'}\) \((n,n'=0,1,2,\ldots)\).
Then the covariance conditions (5) for the infinite system of coupled equations are written as
\[ (\delta_{nn'}X_{1}+\sqrt{n+1}\,\delta_{n+1,n'}X_{2})\psi_{n'}+\lambda\psi_{n}=0 \qquad (n,n'=0,1,2,\ldots), \]
where \(X_{1}=\tfrac12(x_{1}\partial/\partial x_{1}-x_{2}\partial/\partial x_{2})\), \(X_{2}=x_{2}\partial/\partial x_{1}\) (\(x_{1},x_{2}\) are the variables of the space of the fundamental representation \(I:S(1,\tfrac12)\)); with respect to \(I\) it is realized only on the infinite-dimensional representation \(S_{a}^{(\infty)}\) (12).
- Let us investigate the invariants \(I\): \(\omega=\langle e_{\mu}\mid L^{a}\mid e_{\nu}\rangle\) \((a=1,2)\). In the case \(S(1,\tfrac12)\), the invariant \(I\) is given by the convolution \(\omega_{\mu\nu}=\langle e_{\mu}\mid c\mid e_{\nu}\rangle\), where
\[ c= \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}, \qquad e= \begin{pmatrix} e_{1}\\ e_{2} \end{pmatrix} \]
belongs to the representation \(S(1,\tfrac12)\) given by (7):
\[ S(\varepsilon,\eta)= \begin{pmatrix} e & 0\\ \dfrac{\varepsilon}{\eta}\operatorname{sh}\eta & e^{-\eta} \end{pmatrix}. \]
Then the invariant convolution is
\[ \langle e_{\mu}\mid S^{+}cS\mid e_{\nu}\rangle = \langle e_{\mu}\mid c\mid e_{\nu}\rangle. \]
In this sense \(S(1,\tfrac12)\) are indeed analogous to spinors. In order that invariant convolutions also exist in the case of representations of arbitrary weight \(S^{(p,k)}\), let us note that \(S^{(p,k)}\) is contained among the irreducible representations into which the direct product of \(p\) spinors decomposes:
\[ S(1,\tfrac12)\times\cdots\times S(1,\tfrac12). \]
Thus, in case \(I\) the device called the (6) folded direct product is also valid.
It can be shown that \(S^{(p+p',\,k+k')}\) is always contained in the decomposition \(S^{(p,k)}\times S^{(p',k')}\), and hence, in the general case, the invariant of \(S^{(p,k)}\) has the form:
\[ \omega_{\mu\nu} = \left\langle e_{\mu}\left| \underbrace{c\times\cdots\times c}_{p} \right|e_{\nu}\right\rangle. \]
- Let us carry out the decomposition for the product of two conjugate spinors. Obviously, in the general case \(S^{(p,k)}\) the formulas obtained extend by complete induction:
\[ \begin{aligned} U S(1,1/2)\times [S^{-1}(1,1/2)]^{t}U^{-1} &= U \begin{pmatrix} e^{\eta} & 0\\[2mm] \dfrac{\varepsilon}{\eta}\operatorname{sh}\eta & e^{-\eta} \end{pmatrix} \times \begin{pmatrix} e^{-\eta} & -\dfrac{\varepsilon}{\eta}\operatorname{sh}\eta\\[2mm] 0 & e^{\eta} \end{pmatrix} U^{-1} \\ &= \begin{vmatrix} e^{2\eta} & 0 & 0 & 0\\[1mm] -\sqrt{2}\,\dfrac{\varepsilon}{\eta}e^{\eta}\operatorname{sh}\eta & 1 & 0 & 0\\[2mm] -\varepsilon^{2}\dfrac{\operatorname{sh}^{2}\eta}{\eta^{2}} & \sqrt{2}e^{-\eta}\dfrac{\varepsilon}{\eta}\operatorname{sh}\eta & e^{-2\eta} & 0\\[2mm] 0 & 0 & 0 & 1 \end{vmatrix} = \begin{bmatrix} S^{(2,1)} & 0\\ 0 & f_{00} \end{bmatrix}. \end{aligned} \]
The authors express their gratitude to Prof. K. P. Stanyukovich for advice and interest in the work.
Scientific Research Institute
of Introscopy
Received
29 XII 1966
CITED LITERATURE
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\(^{3}\) G. A. Sokolik, N. P. Konopleva, DAN, 154, 310 (1964).
\(^{4}\) M. Hamermesh, Group Theory and its Applications to Physical Problems, 1962.
\(^{5}\) N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, “Nauka,” 1965, pp. 146–152.
\(^{6}\) F. Murnaghan, The Theory of Group Representations, IL, 1950.
\(^{7}\) R. Utiyama, Phys. Rev., 101, 1597 (1956).
\(^{8}\) G. A. Sokolik, N. P. Konopleva, DAN, 169, No. 3 (1966).