ISSN 0030-400X, Optics and Spectroscopy, 2006, Vol. 100, No. 6, pp. 906–909. © Pleiades Publishing, Inc., 2006. Original Russian Text © N.R. Sadykov, 2006, published in Optika i Spektroskopiya, 2006, Vol. 100, No. 6, pp. 982–985.
PHYSICAL OPTICS
Maxwell’s Equations in the Majorana Representation in Chiral and Gyrotropic Media N. R. Sadykov All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center, Snezhinsk, Chelyabinsk oblast, 456770 Russia Received May 6, 2005; in final form, January 23, 2006
Abstract—Maxwell’s equations in the Majorana representation are generalized to the case of chiral and gyrotropic media. A relation between the dynamic variables and the parameters of chiral and gyrotropic media is found. An expression for the current density 4-vector is obtained for the media considered. PACS numbers: 42.25.B DOI: 10.1134/S0030400X06060154
In [1], where the results of [2] were generalized, it was shown that, if in the Maxwell equations in the Majorana representation ([3], p. 80), which coincide in form with the Dirac equations [4] for a particle with spin s = 1/2 Pˆ 0 ψ + aˆ Pˆ ψ = 0,
momentum 4-vector Pˆ µ = ( Pˆ 0 , –P) = ((i/c)∂/∂t, i—); and ψ is a bispinor consisting of 3D spinors ⎛ ⎞ aˆ = ⎜ – sˆ 0 ⎟ , ⎝ 0 sˆ ⎠
(1)
the transformations Pˆ 0
Pˆ 0 ( AIˆ + ia 0 αˆ 0 + a 5 αˆ 5 + a 6 αˆ 6 ), Pˆ
Pˆ ,
⎛ 0 –i 0 ⎜ sˆ 3 = ⎜ i 0 0 ⎜ ⎝ 0 0 0
(2) ⎛ ⎞ αˆ 0 = ⎜ 0 – I ⎟ , ⎝ I 0 ⎠
are made, then the equations describing a chiral medium are obtained: ε = A + a5 ,
µ = A – a5 ,
x+h E = ------------- , 2
D = εE + κ 1 H,
(3)
x–h H = ------------, 2i 2
where A, a0, a5, and a6 are constants; Λ1 = A2 – a 0 – 2
2
⎞ ⎟ ⎟, ⎟ ⎠
⎛ ⎞ αˆ 5 = ⎜ 0 I ⎟ , ⎝ I 0⎠
⎛ 0 0 i ⎞ ⎜ ⎟ sˆ 2 = ⎜ 0 0 0 ⎟ , ⎜ ⎟ ⎝ –i 0 0 ⎠
(4) ⎛ ⎞ αˆ 6 = ⎜ I 0 ⎟ , ⎝ 0 –I ⎠
⎛ ⎞ ψ = ⎜ x ⎟. ⎝ h⎠
κ = – ( a 0 + ia 6 ),
κ 1 = – ( a 0 – ia 6 ), B = µH + κE,
⎛ 00 0 ⎞ ⎜ ⎟ sˆ 1 = ⎜ 0 0 – i ⎟ , ⎜ ⎟ ⎝ 0 i 0 ⎠
a 5 – a 6 ; the operators αˆ 0 , αˆ 5 , and αˆ 6 are defined in (4); H and E are the vectors of magnetic and electric fields in Maxwell’s equations; and B and D are the vectors of magnetic induction and dielectric displacement, respectively. In (2), ˆI is the unit operator; the dynamic variables aˆ = ( αˆ 1 , αˆ 2 , αˆ 3 ) and αˆ 0 are constructed from 3 × 3 matrices; Pˆ 0 = (i/c)∂/∂t and Pˆ = –i— are, respectively, the temporal and spatial parts of the
It should be noted that x and h are bispinors of dimensionality three and form a second-rank bispinor (the second-rank bispinors consist of primed and unprimed indices ξαβ and ηα' β' (see [5], p. 95)). In this case the number of unknown components coincides with the number of unknowns of an antisymmetric second-rank tensor; therefore, there exists a relation between these two cases (both quantities form equivalent irreducible representations of the extended Lorentz group). The bispinors x and h transform independently with respect to the proper Lorentz group, which, in turn, means that the transformed bispinors are determined to within a phase factor; i.e., x and h are arbitrary complex vectors. This suggests that the quantities H and E entering Eqs. (3) are, in the general case, complex.
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MAXWELL’S EQUATIONS IN THE MAJORANA REPRESENTATION
The dynamic variables aˆ in (4) have a form that, in the case of electrons, corresponds to the spinor representation ([5], p. 102). It is easy to see that both the Pauli matrices and the matrices sˆ β in (2), where β = 1, 2, 3, coincide to within unitary transformations with the matrices composed of the matrix elements of the vector spin operator (sx)nm, (sy)nm, and (sz)nm (see [6], p. 232). The presence of two spinors of dimensionality 2J + 1 in the wave function is a consequence of the fact that the wave function of a quantum system with a fixed J should include two bispinors with different intrinsic parities. Upon inversion of the coordinate system, the spinors x and h are interchanged ([3], p. 28). The bispinors in the Majorana equation (1) are precisely such bispinors. By analogy with the dynamic variables in the Dirac equation, the variable α5 in (4) plays the role of the operator αˆ 0 in the Dirac equation [5]: ⎛ i 0 5 αˆ 5 = γ γ = – -----e λµγρ αˆ λ αˆ µ αˆ γ αˆ ρ = ⎜ 0 I 16 ⎝ I 0 λ , µ, γ , ρ
∑
⎞ ⎟. ⎠
It should be noted that the reality of the coefficients A, a0, a5, and a6 in (2) follows from the Hermitian property of the operator ( ˆI + ia0 αˆ 0 + a5α5 + a6 αˆ 6), which, in turn, indicates the absence of dissipation processes in a chiral medium. If the values of the coefficients A, a0, a5, and a6 are complex, the hermiticity of the operator ( ˆI + ia0 αˆ 0 + a5α5 + a6 αˆ 6 ) is violated and we obtain a case of a dissipative medium; however, the transformation used in the study allows us to consider this case as well. It is easy to verify that, for a dissipative system, the source function appears in the “continuity equation.” Based on the ideas of [1, 2], we will consider in this paper transformations that lead to the anisotropic variant of a chiral medium and to the case of a gyrotropic medium. Let us consider the transformation Pˆ 0
Pˆ 0 ( AIˆ + b 1 aˆ + ib 2 αˆ 5 aˆ ),
Pˆ
Pˆ ,
(5)
where b1 and b2 are real vectors. From (1) and (5) we obtain equations for the spinors x and h sˆ Pˆ x = Pˆ 0 ( AIˆ – b 1 sˆ )x + iP 0 b 2 sˆ h,
(6)
sˆ Pˆ h = – Pˆ 0 ( AIˆ + b 1 sˆ )h + iP 0 b 2 sˆ x.
sˆ i x = – ie j e ijk ξ k ,
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∂h ∂x ∂x curl x = iA -------- + b 1 × -------- – ib 2 × -------- , c∂t c∂t c∂t ∂h ∂x ∂h curl h = – iA -------- + b 1 × -------- – ib 2 × -------- , c∂t c∂t c∂t
No. 6
x+h E = ------------- , 2
x–h H = -----------2i
(8)
introduced within the Majorana classical electrodynamics [7, 8] will be rewritten as ∂ ∂E curl E = – -------- AH + ( b 1 – ib 2 ) × -------- , c∂t c∂t ∂H ∂ curl H = -------- AE + ( b 1 + ib 2 ) × -------- . c∂t c∂t
(9)
It is emphasized that the quantities x = E + iH and h = E – iH (characterizing the circular polarization of a photon, or, in the language of quantum electrodynamics, the photon helicity [8–13]) are the wave functions of the spin states of a photon in the Z representation ([14], p. 42) 1 F 1 = – ------- ( e x + ie y ), 2
1 F 2 = ------- ( e x – ie y ), 2
F3 = ez ,
where F1 and F2 correspond in the Z representation to the states with sz = ±1 and F3 corresponds to the state with sz = 0. It should be noted that the wave functions Fα, where α = 1, 2, 3, appear in different formulas in investigating the mechanisms of excitation of a nucleus by a photon. From the above example, it follows that the Majorana equations, despite their purely classical origin, describe an undeniably quantum particle with spin—the photon. One can see that Eqs. (9) describe a chiral medium; real values of vectors b1 and b2 correspond to the case of a nondissipative medium, while complex values of vectors b1 and b2 correspond to an absorbing medium. In the general case of chiral media, the quantities κ and κ1 in (3) are tensor parameters of the magnetoelectric coupling [15]. It follows from (3) and (9) that the vectors of magnetic induction and dielectric displacement can be written in the operator form D i = εE i + ( Kˆ 1 H ) i ,
( Kˆ 1 ) ik = κ 1 δ ik + e ijk ( b 1 + ib 2 ) j . 2006
(7)
where symbol × denotes the vector product of vector quantities. Equations (7) in the new variables
Kˆ ik = κδ ik – e ijk ( b 1 – ib 2 ) j ,
sˆ i h = – ie j e ijk η k
OPTICS AND SPECTROSCOPY
taken into account become
B i = µH i + ( Kˆ E ) i ,
Equations (6) with the relationships ([3], p. 81)
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Now let us consider the transformations of the form Pˆ 0 ( AIˆ + c 1 α 0 aˆ + c 2 α 6 aˆ ),
Pˆ 0
Pˆ
Pˆ .
(11)
By analogy with (6), transformations (11) yield equations for the spinors x and h sˆ Pˆ x = Pˆ 0 ( AIˆ – c 2 sˆ )x – c 1 sˆ h, sˆ Pˆ h = – Pˆ 0 ( AIˆ – c 2 sˆ )h + P 0 c 1 sˆ x.
(12)
After a manipulation similar to that made in deriving Eqs. (9), we obtain the following equations from (12): ∂ ∂E curl E = – -------- AH + i ( c 2 – c 1 ) × -------- , c∂t c∂t ∂H ∂ curl H = -------- AE – i ( c 1 + c 2 ) × -------- . c∂t c∂t
(13)
One can see that Eqs. (13) describe a gyrotropic medium; real values of vectors c1 and c2 correspond to the case of a nondissipative medium, while complex values of vectors c1 and c2 correspond to an absorbing medium. Let us consider the continuity equation. From (1), (5), and (11), we obtain Pˆ 0 ( AIˆ + b 1 aˆ + ib 2 αˆ 5 aˆ + c 1 αˆ 0 aˆ + c 2 αˆ 6 aˆ )ψ + aˆ Pˆ ψ = 0, +
( Pˆ 0 ψ ) ( AIˆ + b 1 aˆ + ib 2 αˆ 5 aˆ + c 1 αˆ 0 aˆ + c 2 αˆ 6 aˆ )
(14)
+ + ( Pˆ ψ )aˆ = 0.
Equations (14) can be combined to give + + + + Pˆ 0 ( ψ ψ + b 1 ψ aˆ ψ + ib 2 ψ αˆ 5 aˆ ψ + c 1 ψ αˆ 0 aˆ ψ + + + c 2 ψ αˆ 6 aˆ ψ ) + P ( ψ aˆ ψ ) = 0.
(15)
Equality (15) has the form of the continuity equation ∇µ j µ = 0, where the quantity µ + + + j = ( ψ ψ + b 1 ψ aˆ ψ + ib 2 ψ αˆ 5 aˆ ψ + + + + c 1 ψ αˆ 0 aˆ ψ + c 2 ψ αˆ 6 aˆ ψ, ψ aˆ ψ ) = 0
(16)
represents the current density 4-vector of a photon for a chiral medium. In conclusion, we will consider two of the four remaining Maxwell equations. (Because of space limitations, we omit here a detailed consideration of this problem.) Let us show that Eqs. (3) and the two Maxwell equations for —B and —D in a homogeneous medium (A, a0, a5, and a6 are constants) give two equations for the bispinors x and h that coincide with the equations in vacuum ([3] (p. 80) or [7]) —x = 0,
—h = 0.
(17)
In view of (3), it follows from (17) that ( A – a 5 – ia 0 – a 6 )—x = ( A – a 5 + ia 0 – a 6 )—h,
(18) ( A + a 5 + ia 0 + a 6 )—x = – ( A + a 5 – ia 0 – a 6 )—h. Based on Eqs. (18), we obtain Λ1—x = 0 and Λ1—h = 0, which is equivalent to (17). It should be noted that the transformations considered in [1] lead to the case of a chiral medium (3). In the literature, the quantities κ and κ1 are typically purely complex [16]; i.e., a0 = 0 in (3). This is explained by the fact that, in (2), the quantity a0 corresponds to the operator αˆ 0 , which is equivalent to the operator αˆ 5 in the Dirac equation. The operator αˆ 5 in the Dirac equation characterizes the odd-parity effects [17]. Since these effects are exotic and very small in magnitude, they are ignored in most cases; i.e., it is assumed that a0 = 0 in (3). Thus, in this paper, the results of [1] were generalized to the case of chiral media (3), whose tensor parameters are characterized by the antisymmetric second-rank tensor, and the transformations leading to the case of a gyrotropic medium were considered. Note that we considered here a small class of processes that are studied in detail in optics and electrodynamics of continua. Moreover, a simplified model was used even for the chiral and gyrotropic media: the media were assumed to be homogeneous and nonlinear effects, dispersion, etc., were omitted. In our opinion, the results obtained will be useful in describing the evolution of a spin particle with halfinteger spin and nonzero mass. Some effects considered in the Majorana representation (the optical [12] and inverse optical [13] Magnus effects) were generalized to the case of a spin particle with half-integer spin and nonzero mass (the counterpart of the optical Magnus effect [18] and the counterpart of the inverse Magnus effect [19]). REFERENCES 1. N. R. Sadykov, Opt. Spektrosk. 98 (4), 644 (2005) [Opt. Spectrosc. 98 (4), 590 (2005)]. 2. N. R. Sadykov, Opt. Spektrosk. 97 (2), 325 (2004) [Opt. Spectrosc. 97 (2), 305 (2004)]. 3. A. I. Akhiezer and V. B. Berestetskiœ, Quantum Electrodynamics, 4th ed. (Nauka, Moscow, 1981; Wiley, New York, 1974). 4. P. A. M. Dirac, Lectures on Quantum Field Theory (Belfer Gradual School of Science, Yeshina Univ., New York, 1966; Nauka, Moscow, 1990). 5. V. B. Berestetskii, E. M. Lifshitz (Lifshits), and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Nauka, Moscow, 1980; Pergamon Press, Oxford, 1982). 6. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory, 4th ed. (Nauka, Moscow, 1989; Pergamon, New York, 1977). OPTICS AND SPECTROSCOPY
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Translated by V. Bulychev
