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Dirac Theory As a Relativistic Quantum Theory of a Single Electron, and Its Optical-Mechanical Analogy

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Dirac Theory As a Relativistic Quantum Theory of a Single Electron, and Its Optical-Mechanical Analogy Noname manuscript No. (will be inserted by the editor) N. L. Chuprikov Dirac theory as a relativistic quantum theory of a single electron, and its optical-mechanical analogy Received: date / Accepted: date Abstract Dirac’s equation with static external fields is considered as an equation of relativistic quan- tum theory, which describes the quantum dynamics of an electron in four-dimensional space-time. It is shown that the Dirac electron has only one intrinsic degree of freedom – spin. The quantum ensemble of an electron with a given energy consists of two inseparable from each other subensembles of “heavy” and “light” spin-1/2 quasiparticles to have opposite (relative) intrinsic parities. In the case of electri- cal scalar potential nonuniform in the direction of a particle motion, these subensembles behave as analogs of the electrical and magnetic components of the plane monochromatic electromagnetic wave propagating in a dispersive medium, nonuniform in the direction of a wave motion. The role of states with negative energies and the properties of the velocity operator are explained. Keywords Dirac equation one-particle formulation electron’s intrinsic degrees of freedom velocity operator · · · PACS 03.65.-w 03.65.Xp 42.25.Bs 1 Introduction According to the “official” point of view [1–5], the Dirac theory is a field theory that needs to be quantized; while a consistent Dirac theory describing the quantum dynamics of a single electron is, in principle, impossible. But, in our opinion, it is worth returning to this conclusion, especially since in fact there is still no consensus on the nature of the Dirac equation. This is evidenced, for example, by the recent controversy [6, 7] around the Dirac theory in the standard and Foldy-Wouthuysen [8] representations. Note that a successful description of the fine structure of the spectrum of the hydrogen atom was given precisely within the framework of the one-particle Dirac model; and only the Lamb shift requires a multiparticle (field) approach for its explanation. The excellent agreement of the Dirac model with experimental data means that on the scales large enough compared to the Compton wavelength, up to the atomic scales, this one-particle model is quite justified; and the fact that relativistic corrections arXiv:1702.02021v4 [physics.gen-ph] 31 Jul 2020 are small in comparison with the corresponding energy eigenvalues calculated in the framework of the nonrelativistic model suggests that in this problem, the Dirac theory is consistent, in the nonrelativistic limit, with the Pauli theory. The problem is that it is far from easy to see this agreement at the level of their formalisms, since these two theories have different symmetries. The key role here is played by the fact that in the nonrelativistic theory, the operation of spatial inversion does not lead to any new physical results, while in the relativistic theory the inclusion of inversion in the symmetry group requires the simultaneous N. L. Chuprikov Tomsk State Pedagogical University, 634041, Tomsk, Russia E-mail: [email protected] 2 consideration of a pair of spinors (see [4], p. 94). In other words, in the Pauli theory the state of a particle is given by a two-component spinor, while in the Dirac theory this is done with the help of two two-component spinors – the large and small components of the Dirac bispinor in the standard representation, which, unlike the Pauli spinor, obeys the system of four first-order differential equations (DE), rather than two DE of the second order. The consequences of doubling the number of wave functions and equations turned out to be so unusual that all known attempts to give them a consistent physical interpretation within the framework of a one-particle quantum theory did not lead to success. Among these consequences, we single out the following three: (1) the four-component wave function is “numbered”, in addition to the spin, by the (relative) intrinsic parity; but this doubling of quantum numbers creates an unusual situation: on the one hand, since this function describes the states of an electron, both of its quantum “numbers” should be considered as intrinsic degrees of freedom of this particle (see, for example, [2, 9]); on the other hand, this would contradict the Pauli theory that predicts the existence of only one intrinsic degree of freedom for an electron; in addition, one must take into account the fact that the number of intrinsic degrees of freedom of a particle does not depend on the velocity with which it moves; (2) the transition from a two-component wave function to a four-component one leads to the fact that in the Dirac theory, instead of (2 2)-matrix operators of the Pauli theory, there appear (4 4)-matrix operators; among them there are× new operators that have no analogs in the Pauli theory, and× there are some that are known (for example, the velocity operator), but they acquire so unusual properties in the space of four-component wave functions, that the question of their physical interpretation remains open to this day; (3) the new equation has solutions not only with positive, but also with negative energies; this calls into question the possibility of applying the well-known quantum mechanical interpretation of the energy eigenvalues in the Dirac theory. Our goal is to consider all these issues using the example of an electron in an external electric field which is inhomogeneous along the direction of a particle motion. We follow the approach [10], according to which, at a fixed energy, the quantum dynamics of an electron in the four-dimensional space-time is equivalent to the quantum dynamics of the ’heavy’ and the ’light’ quasiparticles in the usual three- dimensional space, which are described, respectively, by the large and small components of the Dirac bispinor with the positive and negative intrinsic parities. This quantity is not the intrinsic degree of freedom of the electron (Section 2). In the case considered, the division into the subensembles of heavy and light quasiparticles reflects the fact that, at a fixed energy and z-component of spin, the quantum dynamics of the electron in four-dimensional space-time is analogous to the propagation of a linearly polarized plane monochromatic electromagnetic wave in a nonuniform medium with dispersion; the subensembles of heavy and light quasiparticles are, respectively, analogs of the electric and magnetic components of this wave (Section 3). In Section 4 it is shown that the general solution of the Dirac equation for a free electron contains only solutions with positive energies: states with negative energies are in principle unavailable for a free electron with a positive energy, since spontaneous transitions of such an electron into the region of negative energies contradict the energy conservation law (Section 5). An explanation of the unusual properties of the velocity operator is given in Section 6. 2 The concepts of ’heavy’ and ’light’ Schr¨odinger quasiparticles in the Dirac theory Let us consider the case when an electron moves along the OZ-axis in an external scalar field φ(z). In the standard representation, the Dirac equation for this quasi-(1 + 1)-dimensional system has the form ∂Ψ(z,t) i¯h = Hˆ Ψ(z,t); Hˆ = cα pˆ + eφ(z)+ βmc2, (1) ∂t D D z z Φ χ where Φ = + and χ = + are, respectively, the large and small components of the bispinor Φ− χ− Φ 0 σz 1 0 Ψ = ; αz = , β = , σz is the Pauli matrix, m is the electron mass, and χ σz 0 0 1 − pˆz = i¯h d/dz. Since the external field does not depend on time, the search for a general solution to this equation− is reduced to solving the stationary Dirac equation HˆDΨ = EΨ. (2) 3 As was shown in [10], this equation can be written in the form of the “Pauli equation” for the large component Φ to be a two-component wave function describing the state of a “heavy Schr¨odinger quasiparticle” with the effective mass M(E,z) = [E + mc2 eφ(z)]/(2c2) and energy ǫ(E)= E mc2, as well as the equality connecting the small component with− the large – − 1 1 pˆ pˆ + eφ(z) Φ = ǫ(E)Φ, χ = σ pˆ Φ. (3) z 2M(E,z) z 2M(E,z)c z z In addition, it can be written in the form of the “Pauli equation” for the small component χ to be a two-component wave function describing the state of the “light quasiparticle” with the effective mass µ(E,z) = [E mc2 eφ(z)]/(2c2) and energy ǫ′(E)= E + mc2, as well as the equality connecting the large component− with− the small – 1 1 pˆ pˆ + eφ(z) χ = ǫ′χ, Φ = σ pˆ χ. (4) z 2µ(E,z) z 2µ(E,z)c z z Both these quasiparticles have the same electrical charge. Thus, in the four-dimensional space-time the quantum dynamics of an electron with a given energy is equivalent to two interconnected quantum dynamics in the ordinary three-dimensional space – the quantum dynamics of the heavy quasiparticle and the quantum dynamics of the light quasiparticle. It is important to emphasize that the Dirac equation (2) is not equivalent to the system of the “Pauli equations” (3) and (4) if they are considered without the equalities connecting the large and small components. A theory, where these components are independent of each other, a priori violates the symmetry of the Dirac equation. The well-known “Dirac theory” in the Foldy-Wouthuysen representation is just such a theory. The transformation underlying this representation differs from the identity transform only when it acts on functions that do not obey the Dirac equation.
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