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On the Question of a Unified Field Theory*
Does the “Dualism” of Waves and Particles Always Exist?
D. I. Blokhintsev
Even at the very earliest stage in the development of quantum mechanics, the equal validity of the corpuscular and wave pictures of the motion of matter was demonstrated. Corpuscular and wave properties thus proved to be only two aspects of one and the same physical essence.
But does this “dualism”** in fact exhaust all possible states of matter?
This question may also be formulated more precisely: does matter always exhibit corpuscular properties?
It is customary to think that modern theory gives an affirmative answer to this question. Namely, every free wave field obeying the laws of quantum theory possesses corpuscular properties, i.e., in interaction it gives up and receives energy \(\varepsilon\) and momentum \(\mathbf p\) in portions:
\[ \varepsilon = \pm \sqrt{m^2 c^4 + c^2 p^2}, \qquad \pm \mathbf p, \tag{1} \]
where \(m\) is the rest mass of the particle, and \(\mathbf p\) is the particle momentum.
The particles themselves—photons, positrons, electrons, mesons, nucleons—from this point of view should be regarded as excited states of the corresponding fields\(^{1,2}\), and therefore, perhaps, it would be appropriate to call them by the general name “poletons.”
It is considerably more difficult to answer the question posed above if one takes into account that the various physical fields in fact inter—
* The articles by D. I. Blokhintsev and Ya. I. Frenkel devoted to the question of a unified field theory are being published in the order of discussion. —Ed.
** This is how the capacity of matter to exhibit, in some relations, corpuscular properties and, in others, wave properties is often called.
interact with one another and, consequently, are not free. It is precisely when this interaction is taken into account that the well-known difficulties of the modern theory arise.
Recently it has become possible to develop formal methods for eliminating infinities, effective at least in electrodynamics (interaction of fields)\(^3\).
However, in what follows we shall be interested not in this aspect of the matter. We wish to draw attention to other consequences of the modern theory of interacting fields, which indicate the existence of field states incompatible with corpuscularity, i.e. leading to a violation of the “dualism” of waves and particles.
To illustrate our assertion concerning the violation of the “dualism” of waves and particles, let us turn to examples. Let us first consider the case of interaction of two scalar fields \(\Psi(x,t)\) and \(\Phi(x,t)\), and assume that the interaction energy of these fields, \(W\), has the especially simple form: \(W = g\Psi\Phi\), where \(g\) is a coupling constant having dimension \(L^{-2}\).* Then the relativistically invariant field equations will have the form:
\[ \Box^{2}\Psi-\frac{m^{2}c^{2}}{h^{2}}\Psi=g\Phi, \tag{2} \]
\[ \Box^{2}\Phi-\frac{M^{2}c^{2}}{h^{2}}\Phi=g\Psi, \tag{2'} \]
where
\[ \Box^{2}=-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}} +\frac{\partial^{2}}{\partial x^{2}} +\frac{\partial^{2}}{\partial y^{2}} +\frac{\partial^{2}}{\partial z^{2}}, \]
and \(m\) and \(M\) are the masses of the particles associated with the free fields, i.e. existing when \(g=0\). It is convenient to consider these equations with the aid of normal coordinates \(q_k\) and \(Q_k\) such that
\[ \Psi(x,t)=\int q_k e^{ikx}\,dk,\qquad \Phi(x,t)=\int Q_k e^{ikx}\,dk. \tag{3} \]
It is not difficult to show that \(q_k\) and \(Q_k\) will satisfy the equations:
\[ \frac{d^{2}q_k}{dt^{2}}+\omega_k^{2}q_k=gc^{2}Q_k,\qquad \frac{d^{2}Q_k}{dt^{2}}+\Omega_k^{2}Q_k=gc^{2}q_k. \tag{4} \]
Here the natural frequencies \(\omega_k\) and \(\Omega_k\) are equal to:
\[ \omega_k=\frac{\varepsilon_k}{h}=\sqrt{m^{2}c^{4}+h^{2}c^{2}k^{2}},\qquad \Omega_k=\frac{E_k}{h}=\sqrt{M^{2}c^{4}+h^{2}c^{2}k^{2}}. \tag{5} \]
Let us now introduce new normal coordinates \(\xi_k\) and \(\eta_k\):
\[ \xi_k=\alpha q_k+\beta Q_k,\qquad \eta_k=\delta q_k+\gamma Q_k. \]
\[ \text{* This example was partially considered by us earlier as well in }{}^{2}. \]
Then, by a suitable choice of \(\alpha, \beta, \gamma, \delta\), one can arrange that \(\xi_k, \eta_k\) will satisfy independent equations:
\[ \frac{d^2 \xi_k}{dt^2}+\omega_k^{\prime\,2}\xi_k=0,\qquad \frac{d^2 \eta_k}{dt^2}+\Omega_k^{\prime\,2}\eta_k=0, \tag{6} \]
where the new frequencies \(\omega'_k\) and \(\Omega'_k\) will be equal to:
\[ \omega'_k=\frac{1}{h}\sqrt{m^{\prime\,2}c^4+h^2c^2k^2},\qquad \Omega'_k=\frac{1}{h}\sqrt{M^{\prime\,2}c^4+h^2c^2k^2}, \tag{7} \]
where \(m'\) and \(M'\) are determined by the rest masses of the particles corresponding to the new fields, for which \(\xi_k\) and \(\eta_k\) are normal oscillations. Simple calculations of these masses \(m'\) and \(M'\) lead to the result:
\[ m^{\prime\,2}=\frac{m^2+M^2}{2}+\sqrt{\frac{(m^2-M^2)^2}{4}+\frac{h^4g^2}{c^4}}, \tag{8} \]
\[ M^{\prime\,2}=\frac{m^2+M^2}{2}-\sqrt{\frac{(m^2-M^2)^2}{4}+\frac{h^4g^2}{c^4}}. \tag{8'} \]
For a small coupling constant \(g\), both masses \(m'\) and \(M'\) are real, and the fields will be “quantized,” i.e., the exchange of energy and momentum will occur “corpuscularly,” in portions, according to (1). Such fields may be regarded as aggregates of “poletons.”
However, for a coupling constant \(g>\dfrac{mc}{h}\dfrac{Mc}{h}\), one of the masses \((M')\) becomes imaginary; the dispersion law for the frequencies (7) takes the form:
\[ \Omega_k^{\prime\,2}=c^2k^2-\left|\frac{M'c^2}{h}\right|^2=c^2(k^2-k_0^2) \tag{7'} \]
so that, for small \(k\), the oscillator equation (6) will be:
\[ \frac{d^2\eta_k}{dt^2}-c^2k_0^2\eta_k=0. \tag{6'} \]
Such an oscillator will not be “quantized,”*) and we arrive at a special state of the field to which no particles correspond.
Let us now consider another example, closer to the schemes of modern theory.**) Namely, suppose that the interaction energy of the fields \(\Psi\) and \(\Phi\) has the form \(W=\dfrac{1}{2}g\Psi\Phi^2\). Such an interaction is nonlinear (of third order with respect to \(\Psi,\Phi\)). Third-order interactions are quite typical in modern theory.
*) The asymptotic behavior of the eigenfunctions, in contrast to the ordinary oscillator, will have the form:
\[ \Psi_\lambda=\frac{1}{\sqrt{\eta}}\,e^{\pm \frac{i}{2}\eta^2\pm i\lambda\lg\eta+\cdots}, \]
\(\lambda\) is a real number proportional to the energy.
**) Concerning the general scheme of modern quantum field theory, see, for example, 4.
A problem with a nonlinear interaction cannot be solved exactly. Usually one seeks a solution in the form of an expansion in powers of the interaction constant \(g\). We shall choose another path—the path of qualitative analysis. Such a method of consideration does not require assumptions about the smallness of \(g\).
We shall base the analysis on Hamilton’s method. Introducing normal coordinates \(q_k\) and \(Q_k\), and the momenta \(\pi_k\) and \(\Pi_k\) conjugate to them, we may write the Hamiltonian function in the form:
\[ H=\frac{1}{2}\int\left(\pi_k^2+\omega_k^2 q_k^2\right)\,dk+ \frac{1}{2}\int\left(\Pi_k^2+\Omega_k^2 Q_k^2\right)\,dk+ \frac{1}{2}g\int\!\!\int q_k Q_k Q_{k-s}\,dk\,ds . \tag{9} \]
The potential energy of our field is, obviously, equal to:
\[ U\{q,Q\}= \]
\[ =\frac{1}{2}\int\left(\omega_k^2 q_k^2+\Omega_k^2 Q_k^2\right)\,dk+ \frac{1}{2}g\int\!\!\int q_k Q_k Q_{k-s}\,dk\,ds \tag{10} \]
and consists of the potential energy of the oscillators plus the energy of their interaction (the latter being the double integral).
Let us examine this energy in more detail and show that it is positive-indefinite.
Assume, for the proof, that:
\[ q_k=X \quad \text{in the interval } k=2s\pm\frac{\Delta}{2}, \qquad q_k=0 \quad \text{outside this interval,} \]
\[ Q_k=Y \quad \text{in the interval } k=s\pm\frac{\Delta}{2}, \qquad Q_k=0 \quad \text{outside it.} \]
Then
\[ 2U=\Delta\left(\omega^2X^2+\Omega^2Y^2+g\Delta XY^2\right), \tag{11} \]
where \(\omega^2,\Omega^2\) are the mean values of \(\omega_{2s}^2,\Omega_s^2\) in the interval \(\Delta\) around \(k=2s\) and \(k=s\), respectively.
If
\[ \Omega^2+g\Delta X<0, \tag{12} \]
then as \(Y\to\pm\infty\), \(U\to-\infty\), and only for small \(X,Y\) (or \(g\cdot\Delta\)) is \(U>0\).
Thus, the indefiniteness of \(U\) has been proved.
The figure shows in more detail the relief of the potential energy. The solid lines in the figure are the lines of constant \(U\); the line \(aa'\) denotes the line \(\Omega^2+g\Delta X=0\). For small \(X,Y\) there is a “crater,” the walls of which rise steeply as \(X,Y\) increase. In this “crater” the motion will be “quantized”—the field will possess corpuscular properties. However, this quantization will be approximate, since to the left of the line \(aa'\) there are two
symmetrically located “abysses” \((U \to -\infty)\), separated from one another by the “mountain” \(cbc'\) and by the barrier \(aa'\) from the “crater” near \(O\).
Just as the water of a mountain lake situated in a deep sinkhole would seep through the rock into a deep ravine, so in our case there will occur an overflow of the field from the “crater” \(O\) into the abyss \((abc)\) and \((a'bc')\) through the potential barrier \((aa')\).
In those same regions where the “abysses” are located, the field is not quantized, just as it is not quantized in the case of an oscillator with negative stiffness (6′). To this state of the field one cannot assign any particles—“poletons.” It is not excluded, of course, that the cubic interaction itself
\[ W=\frac{1}{2}g\Psi\Phi^2 \]
is only an approximation, just as the potential energy of an anharmonic oscillator is an approximation:
\[ \frac{1}{2}m\omega^2x^2+gx^3+\ldots . \]
Fig. 1.
The appearance in the interaction of higher terms, for example of the type
\[ \frac{1}{2}g\Psi^2\Phi^3, \]
could restore the positive definiteness of \(U\). Then the “abysses” could disappear, and in their place slopes or new craters could form.
Especially curious would be the emergence of new craters, since their existence would signify the possibility of small oscillations of the field at new equilibrium positions; and together with this we would return again to quantized states of the field, and hence also to particles, but arising not near the zero field, as usual, but near a field of finite magnitude. The transformation of these particles into ordinary ones is possible by seepage (“tunnel effect”) through the potential barriers separating the craters.
It would now be premature to analyze these higher interactions in greater detail. But from the examples considered above with a quadratic \((g\Psi\Phi)\) interaction and a cubic
\[ \left(\frac{1}{2}g\Psi\Phi^2\right) \]
interaction of fields, it is seen that not every excited state ...
field may be regarded as the equivalent existence of particles—“poletons.”
Apparently there also exist states of fields that cannot be reduced to “poletons,” i.e., wave-corpuscular dualism does not exhaust all conceivable states of matter: there may also be states that cannot be reduced to particles. Whether these states are a purely mathematical possibility or whether they are in fact realized in nature remains an open question.
It is supposed, for example, that the decay of a $\mu$-meson occurs into an electron and two neutrinos. But can it be considered excluded that the neutrino in general is not a “poleton” and does not possess corpuscular properties?
Perhaps more careful studies of meson decay occurring with the participation of neutral particles (“neutrino,” “neutretto”) will give a more definite answer to this question.
References
- Ya. I. Frenkel, UFN, 42, 69 (1950).
- D. I. Blokhintsev, UFN, 42, 76 (1950).
- Ya. A. Smorodinsky, UFN, 39, 325 (1949).
- G. Wentzel, Introduction to the Quantum Theory of Wave Fields, Gostekhizdat, 1940.