UDC 530.12

PHYSICS

K. P. STANYUKOVICH

SPHERICAL WAVE OF A SCALAR PLANKEON

(Presented by Academician L. I. Sedov, 23 IV 1969)

The basic equation for studying a spherical wave corresponding to a “scalar” plankeon has the form (^1)

\[ \left(g^{ik}\hat p_i \hat p_k + m^2c^2 + \hbar^2 R/6\right)\psi = 0. \tag{1} \]

The interval corresponding to the “Einstein universe” is

\[ -ds^2 = -c^2 d\tau^2 + \frac{dr^2}{1-r^2/a^2} + r^2 d\Omega^2 . \tag{2} \]

In this case

\[ R = 6/a^2 . \tag{3} \]

Since for the scalar wave function

\[ g^{ik}\hat p_i\hat p_k = -\hbar^2 g^{ik}\nabla_i\nabla_k = -\hbar^2 g^{ik}\left( \frac{\partial^2}{\partial x^i \partial x^k} -\Gamma^m_{ik}\frac{\partial}{\partial x^m} \right), \]

equation (1) takes the form

\[ \left[ g^{ik}\left( \frac{\partial^2}{\partial x^i\partial x^k} -\Gamma^m_{ik}\frac{\partial}{\partial x^m} \right) -\left(\frac{m^2c^2}{\hbar^2}+\frac{1}{a^2}\right) \right]\psi = 0. \tag{4} \]

For the metric (2), the nonzero Christoffel symbols have the form

\[ \Gamma^1_{11}= \frac{r}{a^2(1-r^2/a^2)},\qquad \Gamma^1_{22}=r\left(1-\frac{r^2}{a^2}\right),\qquad \Gamma^1_{33}=r\sin^2\theta\left(1-\frac{r^2}{a^2}\right), \]

\[ \Gamma^2_{12}=\Gamma^3_{13}=1/r,\qquad \Gamma^2_{33}=-\sin\theta\cos\theta,\qquad \Gamma^3_{23}=\operatorname{ctg}\theta . \]

Now (4) can be written in explicit form

\[ \left[ -\frac{\partial^2}{c^2\partial \tau^2} +\left(1-\frac{r^2}{a^2}\right) \left(\frac{\partial^2}{\partial r^2} +\frac{2}{r}\frac{\partial}{\partial r}\right) -\frac{r}{a^2}\frac{\partial}{\partial r} +\frac{1}{r^2} \left[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \right] -\left(\frac{m^2c^2}{\hbar^2}+\frac{1}{a^2}\right) \right]\psi = 0. \tag{5} \]

Since

\[ \frac{\partial^2\psi}{\partial r^2} +\frac{2}{r}\frac{\partial\psi}{\partial r} = \frac{1}{r}\frac{\partial^2(r\psi)}{\partial r^2}, \qquad r\frac{\partial\psi}{\partial r} = \frac{\partial(r\psi)}{\partial r}-\psi, \]

then (5) takes the form

\[ \left[ -\frac{\partial^2(r\psi)}{c^2\partial \tau^2} +\left(1-\frac{r^2}{a^2}\right) \frac{\partial^2(r\psi)}{\partial r^2} -\frac{r}{a^2}\frac{\partial(r\psi)}{\partial r} +\frac{r\psi}{a^2} +\frac{1}{r^2} \left[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial(r\psi)}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2(r\psi)}{\partial\varphi^2} \right] -\left(\frac{m^2c^2}{\hbar^2}+\frac{1}{a^2}\right)r\psi \right]=0, \]

or

\[ \left[ -\frac{\partial^2}{c^2\partial \tau^2} +\left(1-\frac{r^2}{a^2}\right)\frac{\partial^2}{\partial r^2} -\frac{r}{a^2}\frac{\partial}{\partial r} +\frac{1}{r^2} \left( \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \right) -\frac{m^2c^2}{\hbar^2} \right]\varphi = 0, \tag{6} \]

where \(\varphi=\psi r\).

Set \(r=a\sin\chi\); then (6) takes the form

\[ \left[ -\frac{\partial^2}{c^2\partial \tau^2} +\frac{1}{a^2}\frac{\partial^2}{\partial \chi^2} +\frac{1}{a^2\sin^2\chi} \left( \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\right) +\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} \right) -\frac{m^2c^2}{\hbar^2} \right]\varphi=0. \tag{7} \]

The operator

\[ \frac{1}{\sin\theta} \left[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \left(\sin\theta\frac{\partial}{\partial\theta}\right) \right] +\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\varphi^2} =-\hat l^{\,2}=-l(l+1), \]

where \(l=0,1,2,\ldots\) is the angular momentum (orbital quantum number).

Therefore (7) takes the form:

\[ \left[ -\frac{\partial^2}{c^2\partial \tau^2} +\frac{1}{a^2}\frac{\partial^2}{\partial\chi^2} -\left( \frac{m^2c^2}{\hbar^2} +\frac{l(l+1)}{a^2\sin^2\chi} \right) \right]\varphi=0. \tag{8} \]

Let

\[ \varphi=A(\chi)e^{-\frac{i}{\hbar}E\tau}\mathcal{Y}_{lm}(\theta;\varphi), \tag{9} \]

where

\[ \mathcal{Y}_m=\theta_{lm}e^{im\varphi}; \tag{10} \]

\[ \theta_{lm}= \frac{(-1)^{l+m}}{2^l l!} \sqrt{\frac{(2l+1)(l-m)!}{2(l+m)!}}\, \sin^m\theta\, \frac{d^{\,l+m}\sin^{2l}\theta}{(d\cos\theta)^{l+m}}, \]

\(m=0,\pm1,\pm2,\ldots\) is the magnetic quantum number.

Substituting (9) into (8), we obtain

\[ \left[ \left(\frac{E^2}{c^2}-m^2c^2\right)\frac{a^2}{\hbar^2} -\frac{l(l+1)}{\sin^2\chi} \right]A+A_{\chi\chi}=0. \tag{11} \]

Denote

\[ (E^2/c^2-m^2c^2)a^2/\hbar^2=\xi^2 \tag{12} \]

and instead of \(l(l+1)\) introduce \(n(n-1)=l(l+1)\), whence \(n_1=l+1,\ n_2=-l\); then (11) assumes the standard form

\[ \sin^2\chi A_{\chi\chi}+\left[\xi^2\sin^2\chi-n(n-1)\right]A=0. \tag{13} \]

The general solution of equation (13), as is known, has the form \((^2)\)

\[ A=\sin^n\chi \left(\frac{1}{\sin\chi}D\right)^n \left(\bar A_1 e^{-i\xi\chi}+\bar A_2 e^{i\xi\chi}\right), \tag{14} \]

where

\[ \left(\frac{1}{\sin\chi}D\right)^n = \frac{1}{\sin\chi}\frac{d}{d\chi} \left[ \frac{1}{\sin\chi}\frac{d}{d\chi} \left( \frac{1}{\sin\chi}\frac{d}{d\chi}(\cdots) \right) \right]. \tag{15} \]

Thus we find that

\[ \psi= \frac{\bar A_1}{r}\theta_{lm}e^{-\frac{i}{\hbar}E\tau+im\varphi} \sin^{l+1}\chi \left(\frac{1}{\sin\chi}D\right)^{l+1} e^{-i\xi\chi} = \frac{\varphi}{r}, \]

\[ \psi^{*}= \frac{\bar A_2}{r}\theta_{lm}^{*}e^{\frac{i}{\hbar}E\tau-im\varphi} \sin^{-l}\chi \left(\frac{1}{\sin\chi}D\right)^{-l} e^{-i\xi\chi} = \frac{\varphi^{*}}{r}, \tag{16} \]

where \(\xi=pa/\hbar,\ p=\sqrt{E^2/c^2-m^2c^2}\) is the momentum.

For \(l=0,\ m=0\ (n=0)\) we shall have

\[ \psi=\frac{A_1}{r}e^{-\frac{i}{\hbar}(E\tau+pa\chi)},\qquad \psi^{*}=\frac{A_2}{r}e^{\frac{i}{\hbar}(E\tau+pa\chi)}. \tag{17} \]

The trace of the energy–momentum tensor corresponds to the given wave equation (1) and has the form \((^3)\)

\[ -T=\frac{m^2c^2}{\hbar^2}\langle \psi\psi^{*}\rangle. \tag{18} \]

If we choose \(A_1A_2=\beta\dfrac{m^2c^3a^2}{\hbar}\), where \(\beta=\mathrm{const}\) is determined below, then \(\nabla\psi\nabla\psi^*\) will have the dimension of energy density (as it should). In this case

\[ \langle\psi\psi^*\rangle =\frac{8\pi\beta m^2c^3a^2}{\pi^2a^3\hbar} \int_0^a \frac{dr}{\sqrt{1-r^2/a^2}} =\frac{8m^2c^3\beta}{\pi\hbar}\chi\Bigg|_0^{\pi/2} =\frac{4\beta m^2c^3}{\hbar}; \tag{19} \]

\[ -T=4\beta m^4c^5/\hbar^3; \tag{20} \]

\[ R=6/a^2=-\chi T=\frac{32\pi\beta m^4c}{\hbar^3}G. \tag{21} \]

If we define \(-T=2\varepsilon\) (with \(3p+\varepsilon=0\)), which holds for the Einstein space with metric (2), then we find

\[ -T=3mc^2/2\pi a^3. \tag{22} \]

From (20) and (22) we have \(mca=\hbar(3/8\pi\beta)^{1/3}\). Put \(\beta=3/8\pi\); then

\[ mca=\hbar. \tag{23} \]

From (21) and (23) we have

\[ m=\sqrt{\frac{c\hbar}{2G}},\qquad a=L=\sqrt{\frac{2G\hbar}{c^3}}=2Gm/c^2=r_g, \tag{24} \]

i.e., for \(m\) and \(a\) we obtain the Planck values (the values of the mass and size of the planckeon), with \(a=L=r_g\)—the gravitational radius, as must be the case for an Einstein universe. At the same time, the result obtained checks the calculations performed.

Denote

\[ \frac{1}{\hbar}[E\tau+pa\chi] =\frac{1}{\hbar}\left[E\tau+pa\arcsin\frac{r}{a}\right] =\alpha, \]

then (17), for \(m=0,\ l=0\ (n=0)\), takes the form

\[ \psi=\frac{A_1}{r}e^{-i\alpha}. \tag{25} \]

The de Broglie wave will propagate according to the law:

\[ E\tau+pa\arcsin\frac{r}{a}=\alpha\hbar, \]

or

\[ r=a\sin\frac{E\tau-\alpha\hbar}{pa} =a\sin\frac{E(\tau-\tau_0)}{pa} =a\sin\left[\frac{E}{pc}\frac{c(\tau-\tau_0)}{a}\right]. \tag{26} \]

Hence the phase velocity is

\[ \frac{V_{\mathrm{ph}}}{c} = \sqrt{1+\frac{m^2c^2}{p^2}}\, \cos\left[ \sqrt{1+\frac{m^2c^2}{p^2}}\, \frac{c(\tau-\tau_0)}{a} \right], \]

the group velocity is

\[ \frac{V_{\mathrm{gr}}}{c} = \frac{1}{\sqrt{1+m^2c^2/p^2}}. \]

Expression (26) shows that the de Broglie wave corresponding to the planckeon is localized in the region \(0\le r\le a=r_g=L\) and that the wave packet, held by its own gravitational field, is stable and does not spread.

For values \(c\tau/a\ll 1\),

\[ r=c(\tau-\tau_0)E/pc =E(\tau-\tau_0)/p =c(\tau-\tau_0)\sqrt{1+m^2c^2/p^2}, \tag{27} \]

i.e., for \(a\to\infty\), when gravity is switched off, the usual-

…situation of spreading of the wave packet, since as \(\tau \to \infty\), \(r \to \infty\). The conclusion regarding the stability of the wave packet in its own gravitational field of a definite energy is fundamental, although quite expected. This conclusion is significant in studying the structure of elementary particles in their own gravitational field.

Earlier M. A. Markov \({}^{6}\) proposed that the maximon (a particle equivalent to the planckeon) is described by a particle-like solution of the Einstein and Dirac equations in the form of a limiting state of a wave packet whose energy is gravitationally closed in a region of size \(L\).

Received
22 IV 1969

CITED LITERATURE

\({}^{1}\) E. A. Tagirov, P. A. Chernikov, Preprint R2-3777, Dubna, 1968. \({}^{2}\) E. Kamke, Handbook of Ordinary Differential Equations, Part 3, Vol. I, IL, 1950, p. 666 (2.424). \({}^{3}\) K. A. Bronnikov, V. N. Melnikov, K. P. Stanyukovich, Preprint ITF, 88–89, Kiev, 1968. \({}^{4}\) K. P. Stanyukovich, DAN, 168, No. 4 (1966). \({}^{5}\) K. P. Stanyukovich, V. G. Lapchinsky, Abstracts of the V International Conference on Gravitation, 1968. \({}^{6}\) M. A. Markov, ZhETF, 51, No. 9 (1966).