UDC 530.12+531.51

MATHEMATICAL PHYSICS

K. P. STANYUKOVICH

ANALYSIS OF THE DIRAC EQUATION IN RIEMANNIAN SPACE

(Presented by Academician L. I. Sedov on 24 January 1966)

In Riemannian space, for spinors the covariant derivative is defined in the form

\[ (\ );_\mu=\nabla_\mu=\partial_\mu-A_\mu^a I_a, \tag{1} \]

where \(I_a\) is an infinitesimal operator that defines the algebra of the group corresponding to the given field \(A_\mu^a\).

This definition makes it possible to write the momentum operator, applying it to spinor functions, in the form

\[ -\hat p_\mu=i\hbar\nabla_\mu=i\hbar(\partial_\mu-A_\mu^a I_a). \]

Let us now write the Dirac equation in covariant form \((1)\)

\[ L^\mu\partial_\mu\psi+k\psi=0. \tag{2} \]

Here \(L^\mu\) is the “spin” of the particle; \(L^\mu=L^\nu e_\nu^{\mu};\ k=mc/\hbar\). Equation (2) is covariant (invariant) with respect to the group of transformations

\[ \psi'=\psi\exp[\varepsilon_{(x_i)}^a I_a]. \]

In the case of the Lorentz group

\[ \nabla_\mu=\partial_\mu-\tfrac{1}{2}M_{ik}\Delta_{\mu(i,k)}, \tag{3} \]

where \(M_{ik}\) is the generator of the Lorentz group, and

\[ [M_{ik}M_{jl}]=f_{ik;\,jl}^{ps}M_{ps}, \]

where \(f_{ik;\,jl}^{ps}\) is the structural constant of the group.

The Ricci coefficients of parallel transport are expressed through the tetrads as follows:

\[ \Delta_{\mu(i,k)}=e_{(i)}^{\tau}\nabla_\mu e_{\tau(k)}, \quad \text{where } \nabla_\mu e_{\tau(i)}=\partial_\mu e_{\tau(i)}-\Gamma_{\mu\tau}^{\lambda}e_{\lambda(i)}. \]

In this case the parallel transport of a vector \(\xi_i\), if it is written in the orthonormal frame, will have the form

\[ \delta\xi_i=\Delta_{\mu(i,k)}\xi_k\delta n^\mu. \]

The transport of any arbitrary quantity \(\psi_j\) will have the form

\[ \delta\psi^j=\tfrac{1}{2}\Delta_{\mu(i,k)}M_{ik}^{js}\psi_s\delta n^\mu. \]

In the case of the spinor representation of the Lorentz group

\[ M_{ik}=\tfrac{1}{2}[\gamma_i\gamma_k]=s_{ik}. \]

In this case the Dirac equation in curvilinear coordinates takes the form \((^{1})\)

\[ \gamma^i e_i^\mu\left(\partial_\mu-\frac14[\gamma_i\gamma_k]\Delta_{\mu(i,k)}\right)\psi+k\psi=0, \tag{4} \]

where \(e_i^\mu e_{\mu,k}=\delta_{ik},\quad e_i^\mu e_i^\nu=g^{\mu\nu},\quad \frac12[\gamma_i\gamma_k]=s_{ik}=M_{ik}\).

For a more explicit allowance for the gravitational field, let us square the Dirac equation, analogously to how this is done to allow for the electromagnetic field \((^{2})\):

\[ \gamma^\mu\gamma^\nu(\partial_\mu-\tfrac12 s_{ik}\Delta_{\mu(i,k)}) (\partial_\nu-\tfrac12 s_{jl}\Delta_{\nu(j,l)})\psi = \]

\[ = (g^{\nu\mu}-s^{\nu\mu})\nabla_\mu\nabla_\nu\psi = g^{\nu\mu}\nabla_\nu\nabla_\mu\psi-\tfrac12 s^{\mu\nu}[\nabla_\mu\nabla_\nu]\psi = k^2\psi. \tag{5} \]

Since

\[ [\gamma^\mu\gamma^\nu]_+=2g^{\mu\nu},\qquad s^{\mu\nu}=s^{ik}e_i^\mu e_k^\nu,\qquad [\nabla^\mu\nabla^\nu]=M_{ik}R_{\mu\nu(i,k)}, \]

we shall have

\[ g^{\mu\nu}\bigl[(\partial_\mu-\tfrac12 s_{ik}\nabla_{\mu(i,k)}) (\partial_\nu-\tfrac12 s_{jl}\nabla_{\nu(j,l)})\bigr]\psi - \]

\[ -\tfrac14 e^\mu_p e_p^\nu\gamma^\gamma\gamma^p\gamma^j\gamma^l R_{\mu\nu(j,l)}\psi = k^2\psi, \tag{6} \]

where \(R_{\mu\nu(j,l)}\) is the curvature tensor in frames in orthogonal coordinates; it is expressed in terms of the general Ricci curvature tensor \(R^\tau_{\mu\nu\lambda}\) in the following way:

\[ R_{\mu\nu(j,l)}=e_j^\lambda e_\tau^l R^\tau_{\mu\nu\lambda}. \tag{7} \]

Since it is sufficient for us to make only an estimate of the influence of the gravitational field on the spin of the particle (and, moreover, a very small one, without being interested in still smaller details), we shall write

\[ \tfrac14 e^\mu_p e_p^\nu\gamma^\gamma\gamma^p\gamma^j\gamma^l R_{\mu\nu(j,l)} = A^{ik}R_{ik}=R^*, \]

where \(A^{ik}\) is some dimensionless tensor; \(R^*\) is a quantity proportional to the scalar curvature \(R^*=\alpha R\).

Let us now rewrite equation (6) in the form

\[ (g^{\mu\nu}\hat p_\mu\hat p_\nu+m^2c^2+\hbar^2R^*)\psi=0. \tag{8} \]

It is obvious that

\[ \hat p_\mu=-i\hbar(\partial_\mu-\tfrac12 s^{ik}\Delta_{\mu(i,k)}). \tag{9} \]

We shall now clarify the meaning of the correction \(\hbar^2R^*=Am^2c^2\), where \(A\) is some dimensionless scalar, and estimate the value of \(A\). In the case of interaction with its own field we shall have:

\[ R^*=-\alpha\chi T=8\pi G\alpha\rho/c^2, \]

where \(\rho\) is the density of matter producing the gravitational field; hence

\[ A=\frac{\alpha\hbar^2 8\pi G\rho}{m^2c^4} =\frac{\hbar}{r_0^2}\,\frac{\hbar G}{c^3}\,\frac{8\pi\alpha\rho r_0^3}{m^2cr_0}, \tag{10} \]

where \(r_0\) is the characteristic radius of elementary particles, \(r_0\simeq10^{-13}\) cm. The quantity \(\hbar G/c^3=r_\Phi^2\), where \(r_\Phi\simeq10^{-33}\) cm is the characteristic magnitude of the metric fluctuation of the gravitational field. Let us denote the quantity

\[ \hbar\,\frac{\hbar G}{r_0^2c^3} =\hbar\left(\frac{r_\Phi}{r_0}\right)^2 =\hbar_g \tag{11} \]

and call \(\hbar_g\) the “Planck constant” for gravitational interactions \((^{3,4})\).

Thus, \(\hbar_g=\hbar\sigma_g/\sigma_p\), where \(\sigma_g\) and \(\sigma_p\) are the cross sections for gravitational and strong interactions, respectively. Further, since \({}^{4}/_{3}\pi\rho r_0^3=m_p=m\),

then we shall have

\[ A = 6\alpha \hbar_g / m c r_0 \simeq \hbar_g / \hbar = \sigma_g / \sigma_p = (r_\Phi / r_0)^2 \simeq 10^{-40}. \]

In the more general case, if the “radius” of curvature \(r^* \ne r_0\), \(A = r_0 r_\Phi^2 / r^{*3}\). If the effective “radius” of curvature is taken to be \(r^* = r_\Phi = 10^{-33}\) cm (5), then \(A = r_0 / r_\Phi = 10^{20}\).

Since \(m^2 c^2 + \hbar^2 R^* = m^2 c^2(1 + A) = m_0^2 c^2\), where \(m_0\) is the mass “reduced” in the gravitational field, then

\[ m_0 = m \sqrt{1 + A} = m \sqrt{1 + \sigma_g / \sigma_p}. \tag{12} \]

In an external field (in the field of the Metagalaxy) \(R^* = 12a/a^2\), where \(a\) is the radius of curvature of the Metagalaxy; \(A = 12\alpha M_0 r_0^2 \hbar_g / m^2 c a^3\); since \(M_0/a^2 \simeq m/r_0^2\), we have

\[ A = \frac{12\alpha \hbar_g}{mac} \simeq \frac{\hbar_g}{\hbar}\frac{r_0}{a} = \left( \frac{\hbar_g}{\hbar} \right)^2 \simeq 10^{-80}. \]

Here \(M_0 \simeq 10^{80} m\) is the mass of the Metagalaxy. Since the quantity \(a\) varies with time, \(a \sim t\), it follows that \(R^* \sim t^{-2}\), and the quantity \(A\) also decreases with time. Thus, the interaction of particles, or, more precisely, the interaction of particle spin, changes (decreases) with time.

The expression \(m^2 c^2 + \hbar^2 R^*\) in the case of an external field can also be written in the form \(\hbar^2(1/r_0^2 + \text{const}/a^2)\); here it is evident that the quantities \(\hbar c/r_0 = E_p\), \(\hbar c/a \simeq E_g = E_p r_0/a \simeq E_p \cdot 10^{-40}\) characterize, respectively, the energy of the nucleon and of the graviton.

In investigating such small energy corrections and distances of order \(r_0 \cdot 10^{-20} = 10^{-33}\) cm, there is, of course, no complete certainty that the Dirac equation is applicable to such situations. But the fact that these small corrections lead to effects of the gravitational field that are correct in order of magnitude indicates the applicability of the Dirac equation in a gravitational field.

In conclusion the author expresses deep gratitude to L. I. Sedov and G. A. Sokolik for discussing the paper.

Scientific-Research
Institute of Introscopy

Received
14 I 1966

REFERENCES

1. V. A. Fock, D. D. Ivanenko, Phys. Zs., 30, 648 (1929).
2. C. Weber, G. Bete, F. Hoffmann, Mesons and Fields, 1, ch. 5, § 3, IL, 1957, p. 57.
3. D. I. Blokhintsev, Abstracts of the First Soviet Gravitation Conference, Moscow State University Press, 1961.
4. K. P. Stanyukovich, Gravitational Field and Elementary Particles, “Nauka,” pt. II, § 9, 1965.
5. L. D. Landau, Collected Volume: Niels Bohr and the Development of Physics, IL, 1960.