Individual Particle Localization per Relativistic de Broglie–Bohm

Abstract

The significance of the de Broglie/Bohm hidden-particle position in the relativistic regime is addressed, seeking connection to the (orthodox) single-particle Newton–Wigner position. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here for free spin-0 particles in 1 + 1 dimensions and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion of a specific Newton–Wigner-localized state is presented in detail. The presence of virtual pairs found is interpreted as the Bohm picture of the spatial extension beyond single point particles proposed in the 1960s as to why space-like hyperplane dependence of the Newton–Wigner wavefunctions may be needed to achieve Lorentz covariance. For spin-1/2 particles the convective current is speculatively utilized for achieving parity with the spin-0 theory. The spin-0 improper quantum potential is generalized to an improper stress tensor for spin-1/2 particles.

Appendices

The previously unpublished research in Appendices A and B was completed in 1988 and 2004, respectively.

Appendix A: Discrete Plane-Wave Superposition

The previously unpublished research in Appendices A and B was completed in 1988 and 2004, respectively.

The integral I, expressed as sums for the superposition of a finite number of discrete plane waves, is particularly useful for computing particle trajectories, which may be determined by calculating contours from the resulting transcendental equation. Equation (13) for the discrete case reads:

$$ \psi [z,t] = \sum\limits_{k'} {\omega_{k'}^{ - 1/2} \phi_{k'} e^{{ik'z - i\omega_{k'} t}} } . $$

(13’)

Equation (10) then becomes, in terms of the expected value $\left\langle {k/\omega_{k} } \right\rangle c^{2}$ of the group velocity:

$$ v/c^{2} = \frac{{ \left\langle {k/\omega_{k} } \right\rangle + \tfrac{1}{2}\sum\limits_{k'' \ne k'} {\sum {(\omega_{k'} \omega_{k''} )^{ - 1/2} \phi_{k'}^{*} \phi_{k} (k'' + k') Exp[i(k'' - k')z - i(\omega_{k''} - \omega_{k'} )t] } } }}{{1 + \tfrac{1}{2}\sum\limits_{k'' \ne k'} {\sum {(\omega_{k'} \omega_{k''} )^{ - 1/2} \phi_{k'}^{*} \phi_{k''} (\omega_{k'} + \omega_{k''} ) Exp[i(k'' - k')z - i(\omega_{k''} - \omega_{k'} )t]} } }}. $$

(10’)

Finally, Eq. (12) translates to:

$$ \begin{aligned} I = \, & i(z - \left\langle {k/\omega_{k} } \right\rangle c^{2} t) \\ & + \;\tfrac{1}{2}\sum\limits_{k'' \ne k'} {\sum {(\omega_{k'} \omega_{k''} )^{ - 1/2} \phi_{k'}^{*} \phi_{k''} (\omega_{k'} + \omega_{k''} ) (k'' - k')^{ - 1} Exp[i(k'' - k')z - i(\omega_{k''} - \omega_{k'} )t]} } . \\ \end{aligned} $$

(12’)

Trajectories z[t] are then easily computed numerically from Eq. (12’) at I = constant, noting that the double sum in Eq. (12’) is pure imaginary.

Figure 6 illustrates trajectories for an ensemble of particles at rest in the mean resulting from a superposition of three plane waves, two of which are ultra-relativistic. Continual creation and annihilation of pairs in the concept of Feynman [8, 9] and Stückelberg [10] of anti-particles as particles moving backwards in time is evident. The particles are virtual in the sense that the pairs are always maximally entangled; any pair created is quickly annihilated. Creation or annihilation of free particles is not possible here where no interaction with other fields is permitted. Note that there exist situations where the anti-particles are not simply apparent, i.e., cannot be removed by proper Lorentz transformation. Diagonals of the figure form the light cone, and the position-axis length is about 0.01 Compton wavelengths.

Fig. 6

Particle trajectories in the mean rest frame corresponding to the superposition of three extraordinary plane waves, two of which are ultra-relativistic. Diagonals of the figure specify the light cone and the vertical axis is of the order of 0.01 Compton wavelength. Note the peculiar virtual multi-particle aspect as a result of an extreme pilot wave’s effect on a single-particle existence

Appendix B: Free Spin-1/2 Particles

2.1 Generalities

The Bohmian spin-1/2 theory is considerably more complicated than for spin-0. With the extra degrees of freedom in the spinor wavefunction, it is not obvious what the representation analogous to Eq. (2) would be, and therefore what sort of Lagrangian or Hamilton–Jacobi equation would result. Nevertheless, by adopting a small part of the scalar field results, a complete free spin-1/2 single-particle theory is possible.

The obvious choice for a Bohmian theory of single spin-1/2 particles is to assume the velocity is given in terms of the conserved current $\bar{\psi }\,\gamma_{\mu } \,\psi$ [13]. This approach would include particle trajectories covering internal aspects. For example, with a superposition of two positive-energy spin-1/2 plane waves of different frequencies with spins aligned and normal to momentum, particles oscillate transversely to both spin and momentum with |v| repeatedly often reaching the speed of light c momentarily while advancing at small constant velocity along the momentum. Many other states’ particles oscillate close to the speed of light. Such states have been described [28] as comprising a set of measure 0 relative to the set of possible states; of course, a measure-0 set depends on the measure, and furthermore some such sets can be significant or interesting.

Here, however, we focus on the conserved convective part of the spin-1/2 current in its Gordon decomposition (regardless of Dirac’s original intent of positive probability densities for solely particles). The internal current associated with zitterbewegung and with the curl of the magnetic moment density is then separated off. This approach with convective current then parallels the above spin-0 theory.

Further, allow for the possibility of an effective rest mass function μ₀[x]. Then Eqs. (3) and (5) with the scalar theory suggest [sidestepping Eq. (2)]:

$$ \mu_{0} u_{\mu } = \frac{\hbar }{{2i\bar{\psi }\psi }}\left[ {\bar{\psi }\frac{\partial \psi }{{\partial x_{\mu } }} - \frac{{\partial \bar{\psi }}}{{\partial x_{\mu } }}\psi } \right], $$

(37)

but where now the spinor $\bar{\psi } = \psi^{\dag } \gamma_{0}$, ψ solves the free Dirac equation, and γ₀ is

$$ \gamma_{0} = \left[ {\begin{array}{*{20}c} +1 & 0 & 0 & 0 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & { - 1} \\ \end{array} } \right] . $$

(38)

Since $u_{\mu } \,u^{\mu } = - c^{2}$, Eq. (37) immediately expresses the effective invariant mass μ₀ in terms of ψ.

$$ (\mu_{0} c)^{2} = - \mu_{0} u_{\mu } \cdot \mu_{0} u^{\mu } . $$

(39)

However, an exact analogue of Eq. (8) is obtained by defining a specific spin tensor in addition to the quantum potential in terms of the Dirac spinors. As with the scalar field, let

$$ \varPhi \equiv - \tfrac{{\hbar^{2} }}{{2m_{0} }}(\bar{\psi }\psi )^{ - 1/2} \Box (\bar{\psi }\psi )^{1/2} . $$

(40)

Furthermore, define an improper tensor $T_{\mu \upsilon }^{spin}$ as:

$$ \begin{aligned} T_{\mu \upsilon }^{spin} \hbar^{ - 2} \equiv & \tfrac{1}{2}(\bar{\psi }\psi )^{ - 1} \left[ {\frac{{\partial \bar{\psi }}}{{\partial x_{\mu } }}\frac{\partial \psi }{{\partial x_{\upsilon } }} + \frac{{\partial \bar{\psi }}}{{\partial x_{\upsilon } }}\frac{\partial \psi }{{\partial x_{\mu } }}} \right] \\ & - \tfrac{1}{2}(\bar{\psi }\psi )^{ - 2} \left[ {\left( {\frac{{\partial \bar{\psi }}}{{\partial x_{\mu } }}\psi } \right)\left( {\bar{\psi }\frac{\partial \psi }{{\partial x_{\upsilon } }}} \right) + \left( {\frac{{\partial \bar{\psi }}}{{\partial x_{\upsilon } }}\psi } \right)\left( {\bar{\psi }\frac{\partial \psi }{{\partial x_{\mu } }}} \right)} \right]. \\ \end{aligned} $$

(41)

The symmetric second-rank tensor field $T_{\mu \upsilon }^{spin}$ has the interesting property that it is invariant under the replacement, $\psi \to f[x] \cdot \psi$, where $f[x]$ is an arbitrary complex-valued function of space–time x. In particular, $T_{\mu \upsilon }^{spin}$ is independent of an overall amplitude or phase, and would vanish if ψ were a scalar.

In terms of Φ and $T_{\mu \upsilon }^{spin}$, Eq. (39) reads:

$$ (\mu_{0} c)^{2} = (m_{0} c)^{2} + 2m_{0} \varPhi + tr[T_{{}}^{spin} ], $$

(42)

similar to Eq. (8) aside from the final term. Equation (42) follows from the fact that the four components of ψ each solve the GKS equation. The proof is entirely straightforward and can be checked using a program capable of symbol manipulation.

The equations of motion can now be obtained from

$$ \mu_{0} d(\mu_{0} u_{\mu } )/d\tau = \mu_{0} u^{\upsilon } \partial (\mu_{0} u_{\mu } )/\partial x_{\upsilon } $$

(43)

by expressing the right-hand side in terms of ψ by using Eq. (37). Remarkably, the resulting equations of motion can be expressed in terms of Φ and $T_{\mu \upsilon }^{spin}$ as simply:

$$ \mu_{0} u^{\upsilon } \partial (\mu_{0} u_{\mu } )/\partial x_{\upsilon } = - \frac{{\partial m_{0} \varPhi }}{{\partial x_{\mu } }} - \frac{1}{{\bar{\psi }\psi }}\partial^{\upsilon } \left[ {\bar{\psi }\psi T_{\mu \upsilon }^{spin} } \right], $$

(44)

similar to Eq. (9), except for stress tensor $\bar{\psi }\psi T_{\mu \upsilon }^{spin}$. The proof of Eq. (44), though straightforward, has an excessive number of steps, but can be checked using a program capable of symbol manipulation.

2.2 Non-relativistic Limit

We now specialize to positive-energy solutions of the Dirac equation by using the Foldy–Wouthuysen representation with 3rd and 4th components of the spinor set equal to zero. An arbitrary such spinor with four real components can be represented in terms of an amplitude A and phase S as

$$ \psi = Ae^{{\tfrac{i}{\hbar }S}} u(\hat{s}) $$

(45)

where $\hat{s}$ is a 3-dimensional x-dependent unit spin pseudovector field, and where $u(\hat{s})$ with u^†u = 1 and s_k = u^†σ_k u is:

$$ u(\hat{s}) = \frac{1}{{\sqrt {2(1 + s_{3} )} }}\left[ {\begin{array}{*{20}c} {1 + s_{3} } \\ {s_{1} + is_{2} } \\ 0 \\ 0 \\ \end{array} } \right] . $$

(46)

Equation (45) is of course the spin-1/2 analogue of the Bohm–de Broglie expression, Eq. (2), with $u(\hat{s})$ to account for the extra two degrees of freedom. In terms of the amplitude A, the quantum potential is again given by Eq. (7). Furthermore, a straightforward calculation indicates that the tensor $T_{\mu \upsilon }^{spin}$ is given simply by:

$$ T_{\mu \upsilon }^{spin} = \tfrac{{\hbar^{2} }}{4}\frac{{\partial s_{l} }}{{\partial x_{\mu } }}\frac{{\partial s_{l} }}{{\partial x_{\upsilon } }} $$

(47)

The non-relativistic limits can now be given. The quantum potential Φ is given by

$$ \varPhi = - \tfrac{{\hbar^{2} }}{{2m_{0} }}A^{ - 1} \Delta A, $$

(48)

where Δ is the Laplacian operator, and the spin tensor $T_{ij}^{spin}$ is approximately:

$$ T_{ij}^{spin} = \tfrac{{\hbar^{2} }}{4}\frac{{\partial s_{l} }}{{\partial x_{i} }}\frac{{\partial s_{l} }}{{\partial x_{j} }}. $$

(49)

Interestingly, this tensor is identical to an earlier result found [11] for a non-relativistic spinning object, although the particle in the theory presented here is not taken as spinning; rather, the pilot wave has a spin component that affects where the particle moves, for example in a Stern–Gerlach experiment.

The ensemble equations of motion become:

$$ \frac{{\partial v_{j} }}{\partial t} + \frac{{\partial v_{j} }}{{\partial x_{i} }}v_{i} = - m_{0}^{ - 2} \left( {\frac{{\partial m_{0} \varPhi }}{{\partial x_{j} }} + \frac{1}{{A^{2} }}\frac{\partial }{{\partial x_{i} }}\left[ {A^{2} T_{ji}^{spin} } \right]} \right), $$

(50)

similar to the Navier–Stokes equation for a viscous fluid, except that here dissipation and anti-dissipation cancel in the mean.

Note that since

$$ \begin{aligned} \int {d^{3} x} A^{2} \frac{\partial }{{\partial x_{j} }}\frac{\Delta A}{A} = \, & - 2\int {d^{3} x} \frac{\partial A}{{\partial x_{j} }}\Delta A \\ = \, & 0 , \\ \end{aligned} $$

(51)

Equations (48) and (50) give the ensemble average 〈dv_j/dt〉 at any fixed time as:

$$ \int {d^{3} x} A^{2} \frac{{dv_{j} }}{dt} = 0. $$

(52)

Furthermore, even in the non-relativistic limit, the ensemble fluid is rotational, with circulation given by:

$$ curl_{k} v = \frac{\hbar }{{4m_{0} }} \varepsilon_{kji} \varepsilon_{lmn} s_{l} \frac{{\partial s_{m} }}{{\partial x_{j} }}\frac{{\partial s_{n} }}{{\partial x_{i} }}. $$

(53)

Particles with spin therefore exhibit richer motion than scalar particles. The tensor component of the force of wave on particle results in a circulatory ensemble fluid.