have a magnetic dipole moment and a charge radius (Darwin term) [7]. The non-minimal ... Abraham-Minkowski dilemma [9]. Mean-field Theory ... Lorentz force.
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Fully relativistic kinetic equation for spin- particles 1
J. Zamanian1, R. Ekman1, and F. A. Asenjo2 Department of Physics, Ume˚ a University, Ume˚ a, Sweden; 2 Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Iba´n˜ez, Santiago, Chile
Introduction
Non-minimal Coupling and Hidden Momentum
Quantum relativity is highly non-trivial but increased understanding of the subject is important not only for basic theory, but also for applications to dense, hot plasmas, such as white dwarfs, laser-generated plasmas, e.g. in icf, and possibly in strongly magnetized environments where the dipole energy is significant. Starting from the Dirac equation, we have developed a fully relativistic kinetic model for spin 12 particles, extending previous semi-relativistic work [1]. The key step is to separate positive and negative energy states of the Dirac equation to order (~k/m)2 where k is the typical wavenumber of the potentials. This corresponds to eliminating pair creation processes and allows us to treat strongly relativistic plasmas without having to use full qed. Our main results are: • Momentum-dependent transformation leads to non-minimal coupling and “hidden momentum”. • A scalar Vlasov-like kinetic equation is found and coupled to Maxwell’s equations in mean-field theory. • We show that the system fulfills an energy conservation law with Poynting vector E × H, but there is no clear distinction between particle and field momentum. In future work we aim to apply the theory in the linear and weakly nonlinear regimes. We also aim to prove further properties of the theory, for example, investigating its relation to the AbrahamMinkowski dilemma in more detail.
Foldy-Wouthuysen Transformation The Dirac Hamiltonian coupled to a gauge potential ˆ = βm + α · (ˆ H p − qA) + qφ contains odd terms that mix upper and lower spinor components. We use a Foldy-Wouthuysen transformation [2, 3] that produces a representation where odd components of observables are suppressed by a power of ~k/m where k is the typical wavenumber of the fields. Explicitly, nm o 3 µ mc µ 1 B B ˆ ˆ × E − E × π)] ˆ p [σ · (π ,σ · B + p H = ˆ + qφ − (1) 2 ˆ + 2ˆ(ˆ + mc2) 2ˆ(ˆ + mc2) √ ˆ =p ˆ − qA, ˆ = m2 + π ˆ 2. for the upper components, with π This Foldy-Wouthuysen transformation is applicable when the typical scale length is long compared to the Compton wavelength and time variations are slow compared to the Compton frequency.
In the Dirac representation the interacting Hamiltonian is related to the free Hamiltonian through the minimal coupling ) H 7→ H − qφ pµ 7→ pµ − qAµ p 7→ p − qA representing a point-like particle interacting with the gauge potential at a single point. On the other hand, eigenstates of the Foldy-Wouthuysen position operator possess multipole moments – we have a non-minimal coupling of a particle that interacts with derivatives of the gauge potential. Thus the electron can have a magnetic dipole moment and a charge radius (Darwin term) [7]. The non-minimal coupling is effected by the substitution pµ 7→ pµ − qAµ + µ˜ µνσρSν Fσρ where Sν is the spin 4-vector and Fσρ is the field tensor, with the free Hamiltonian p H = β m2 + p2 yielding the Hamiltonian (1) after expressing it in lab frame quantities. Because of the non-minimal coupling, in the Foldy-Wouthuysen representation, the relation between the velocity operator, as given by the Heisenberg equation of motion, and the momentum also includes the fields and the spin, ˆ 1 µ m π × E 1 B ˆ˙ = ˆ + ∇p ,π B+ ·σ x 2 ˆ ˆ ˆ + m and the function on phase space in Weyl correspondence is p m p × E v = + µB ∇ p B− · 3s (3) +m The semi-relativistic case has been studied extensively under the heading of “hidden momentum” (reviewed, e.g., in [8]). It is important for the conservation of energy and momentum and thus relevant in the context of the Abraham-Minkowski dilemma [9].
Mean-field Theory Gauge Invariant Wigner Function To obtain a kinetic theory we utilize two transformations. Firstly, a Wigner transformation Z 3 R iq d z i z·p ~ γ A(s,t)·ds z z ~ e Wαβ (x, p, t) = e ραβ x + , x − , t 3 | {z } (2π~) 2 2 Wilson line factor
where ραβ is the density matrix with spinor indices, is applied giving a gauge-invariant function [4, 5]. Secondly, we find a scalar quasi-distribution function f through the spin transformation [6] 1 f (x, p, s, t) = Tr [(1 + s · σ)W (x, p)] , 4π defined on the phase space (x, p, s) where s is a unit vector representing the spin. ˆ π, ˆ x ˆ , σ) is given by putting π ˆ and x ˆ in totally symmetric The expectation value of an operator O( (Weyl) order and taking a moment of f , R 3 2 ˆ x ˆ , σ)ρ] = d pd s O(p, x, 3s)f hOi := Tr[O(π, (2)
In mean-field theory, we use expectation values as sources for the classical Maxwell equations. However, it is not which operators give the charge R obvious and current densities. We add − HfR d2sd3p d4x as an interaction term to the free electromagnetic Lagrangian 12 (E 2 − B 2) d4x. The Euler-Lagrange equations give the sources as moments of f according to (2), 3µB m ∇ · E = |{z} hqi +∇ · s×p | ( + m) {z } ρ free
−P
D E ∂E µB m 3µB m ∇×B= + |{z} hqvi +∇ × 3s −∂t s×p ∂t ( + m) | {z } jfree
M
The required continuity equation ∂tρfree + ∇ · jfree = 0 can be derived from the evolution equation for f indicating the consistency of the model. As a further consistency check, our model has the correct semi-relativistic limit [1, 10].
Kinetic Equation ˆ This gives our main result, the Vlasov-like kinetic equation ˆ H]. The dynamics of f is found from the von Neumann equation for the density matrix, ∂tρˆ = ~i [ρ, i h i h m p×E 2µB m p×E ˜ ˜ 0 = ∂tf + V · ∇xf + q E + V × B ·∇pf + µB ∇x B− · (s + ∇s) · ∇pf + s× B− · ∇s f (4) | {z } +m ~ +m Lorentz force p×E p 1 + µB m∇p (B + +m ) · (s + ∇s) Thomas factor Magnetic dipole force Spin torque R R From the kinetic equation one can derive the continuity ∂t f + ∇ · vf = 0 as mentioned above, and the energy conservation law 1 2 B B p×E 2 ∂t (E + B ) + − 3µB m · s + ∇ · E × H + + µB m3s · − v =0 2 ( + m) where E × H, one of the possible forms of the Poynting vector in a medium, appears. However, one cannot say that E × B is the “true” electromagnetic momentum density – the hidden momentum (3) precludes a unique separation of particle and field momentum [11].
[1] F. A. Asenjo, J. Zamanian, M. Marklund, G. Brodin, and P. Johansson, New. J. Phys. 14, 073042 (2012). [2] L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). [3] A. J. Silenko, Phys. Rev. A 77, 012116 (2008). [4] O. T. Serimaa, J. Javanainen, and S. Varro´, Phys. Rev. A 33, 2913 (1986).
[5] R. L. Stratonovich, Sov. Phys. D 1, 414–418 (1956). [9] D. J. Griffiths, Am. J. Phys. 80, 7–18 (2012). [6] J. Zamanian, M. Marklund, and G. Brodin, New. J. Phys. 12, 043019 [10] A. Dixit, Y. Hinschberger, J. Zamanian, G. Manfredi, and P.-A. (2010). Hervieux, Phys. Rev. A 88, 032117 (2013). [7] L. L. Foldy, Phys. Rev. 87, 688–693 (1952). [11] S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010). [8] D. Babson, S. P. Reynolds, R. Bjorkquist, and D. J. Griffiths, Am. J. Phys. 77, 826–833 (2009).
