Octonionic Formulation of the Fully Symmetric Maxwell's Equations in 3+ 1 Dimensions

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2 Octonions

The octonionic algebra O is an eight dimensional algebra over the ﬁeld of the real numbers R. Its elements can be represented as

O = e0y0 + e1y1 + e2y2 + e3y3 + e4y4 + e5y5 + e6y6 + e7y7 7 = e0y0 + eaya , (1) a=1 X where yk (k =0, 1,..., 7) are real numbers, ea (a =1, 2,..., 7) are imaginary octonion units and e0 is the multiplicative unit element. The set of octets (e0, e1, e2, e3, e4, e5, e6, e7) is known as the octonion basis and its elements satisfy the following multiplication rules:

e0 = 1 , i.e., e0ea = eae0 = ea

eaeb = −δabe0 + fabcec, (a, b, c = 1, 2,..., 7) , (2) where δab is the usual Kr¨onecker symbol and the structure constants fabc are completely antisym- metric, take the value 1 for the following combinations:

fabc = +1 ∀(abc) = (123), (471), (257), (165), (624), (543), (736) , (3) and vanish otherwise. The octonion basis elements satisfy the following additional relations:

[ea, eb] = 2 fabc ec ,

{ea, eb} = −2 δabe0 ,

ea( eb ec) 6= (ea eb ) ec , (4) where the brackets [ , ] and { , } denote, as usual, the commutator and the anti-commutator, respectively.

Notice that the sub-algebra generated by (e0, e7) is isomorphic to the algebra C of the complex numbers while the sub-algebra generated by (e0, e1, e2, e3) is isomorphic to the algebra H of quaternions.

3 Representation of octonions in terms of 4×4 matrix

The algebra of octonions is homeomorphic to a four-dimensional algebra over the complex numbers. In order to see this, consider that the algebra of the complex numbers is isomorphic to the sub- 2 algebra of O generated by the elements (e0, e7), since (e7) = −e0. Now, any octonion O = e0y0 + e1y1 + e2y2 + e3y3 + e4y4 + e5y5 + e6y6 + e7y7 ∈ O can be written as

O = (y0 + e7y7) e0 + (y1 + e7y4) e1 + (y2 + e7y5) e2 + (y3 + e7y6) e3 . (5)

2 Denoting

Y0 = y0 + e7y7 ,

Yj = yj + e7yj+3 , ∀j =1, 2, 3 , (6) equation (5) becomes

O = Y0e0 + Y1e1 + Y2e2 + Y3e3 . (7)

Equation (7) represents octonions in terms of elements of the sub-algebra of quaternions (the sub- algebra generated by (e0, e1, e2, e3), with “coeﬃcients” in the sub-algebra (e0, e7), isomorphic to the algebra of complex numbers. It is well known that the algebra of quaternions is isomorphic to the algebra of Pauli matrices through the identiﬁcation

e0 = 1 and ej = −iσj , ∀j = 1, 2, 3, (8) where the unit matrix 1 and the Pauli matrices σj are given, as usual, by

1 0 0 1 0 −i 1 0 1 = , σ1 = , σ2 = , σ3 = . (9) " 0 1 # " 1 0 # " i 0 # " 0 −1 #

In order to obtain a representation of H in terms of real matrices one has to proceed in the following 0 1 way. Deﬁne J = . Since J 2 = −1, we may replace the imaginary number i by J in (8), " −1 0 # through the following identiﬁcations:

e0 = 1 ⊗ 1, e1 = −J ⊗ σ1, e2 = 1 ⊗ (−iσ2) = −1 ⊗ J, and e2 = −J ⊗ σ3.

This provides following identiﬁcation of the elements of the quaternionic basis with real 4 × 4 matrices:

1 0 0 −J 0 −1 −J 0 e0 = , e1 = , e2 = , e3 = . 0 1! −J 0 ! 1 0 ! 0 J!

The algebra of real 4 × 4 matrices so obtained is isomorphic to the quaternionic algebra H. With these identiﬁcations we can represent the algebra of octonions in terms of an algebra of 4 × 4 matrices whose entries are elements of the sub-algebra generated by (e0, e7). Denoting by π(O) the representative of an octonion O given as (7), we have:

Y0 −Y3 −Y2 −Y1  Y3 Y0 Y1 −Y2  π(O) = . (10)  Y2 −Y1 Y0 Y3     Y1 Y2 −Y3 Y0      Identifying the entries Yk, k = 0,..., 3, above, with complex numbers, eq. (10) provides an homomorphism (not an isomorphism) between O and an algebra of 4×4 complex matrices. Equation (10) suggests, in addition, a convenient deﬁnition of derivative operations, as we will see below.

3 4 Fields and a derivative operator

4 Let us now consider Yk, k = 0,..., 3, as functions Yk(t, x1, x2, x3) deﬁned in R taking values on the sub-algebra of O generated by (e0, e7), i.e., taking values on the e0–e7 plane. Denote by ∂t partial derivative with respect to t and by ∂k the partial derivative with respect to xk, k =1, 2, 3, and deﬁne a derivative operator ∂ in terms of quaternion basis elements as equation (8),

∂ = ∂te0 + ∂1e1 + ∂2e2 + ∂3e3 . (11)

By analogy with (10), the derivative ∂ can be written in terms of 4 × 4 matrix as

∂t −∂3 −∂2 −∂1

∂ ∂t ∂ −∂ ∂ =  3 1 2  . (12)  ∂2 −∂1 ∂t ∂3     ∂1 ∂2 −∂3 ∂t      4 Consider now a field of octonions R ∋ (t, x1, x2, x3) 7→ O(t, x1, x2, x3) ∈ O in the representation (10). The matrix representation (12) of the derivative ∂ suggests to define the derivative ∂π(O) of a octonionic field O by taking the matrix product of (12) with (10):

∂ Y − ∂ Y − ∂ Y − ∂ Y −∂ Y ∂ Y − ∂ Y − ∂ Y −∂ Y − ∂ Y − ∂ Y ∂ Y −∂ Y − ∂ Y ∂ Y − ∂ Y  t 0 1 1 2 2 3 3 3 0 + 2 1 1 2 t 3 2 0 3 1 t 2 + 1 3 1 0 t 1 + 3 2 2 3  ∂ Y − ∂ Y + ∂ Y + ∂tY ∂tY − ∂ Y − ∂ Y − ∂ Y ∂ Y + ∂tY − ∂ Y + ∂ Y −∂ Y − ∂ Y − ∂tY + ∂ Y ∂π(O) =  3 0 2 1 1 2 3 0 1 1 2 2 3 3 1 0 1 3 2 2 3 2 0 3 1 2 1 3  .  − − − − − − − −   ∂2Y0 + ∂3Y1 + ∂tY2 ∂1y3 ∂1Y0 ∂tY1 + ∂3Y2 ∂2Y3 ∂tY0 ∂1Y1 ∂2Y2 ∂3Y3 ∂3Y0 ∂2Y1 + ∂1Y2 + ∂tY3   − − − − − − − −   ∂1Y0 + ∂tY1 ∂3Y2 + ∂2Y3 ∂2Y0 + ∂3Y1 + ∂tY2 ∂1Y3 ∂3Y0 + ∂2Y1 ∂1Y2 ∂tY3 ∂tY0 ∂1Y1 ∂2Y2 ∂3Y3  (13)

The matrix elements in equation (13) are analogous to the corresponding matrix elements in equation (10), that has a compact form as equation (7). This suggests to deﬁne a derivative operator on an octonionic ﬁeld O(t, x1, x2, x3) by imposing π ∂O = ∂π(O). One gets, ∂O = [∂tY0 − ∂1Y1 − ∂2Y2 − ∂3Y3] e0 + [∂1Y0 + ∂tY1 + ∂2Y3 − ∂3Y2] e1+

[∂2Y0 + ∂tY2 + ∂3Y1 − ∂1Y3] e2 + [∂3Y0 + ∂tY3 + ∂1Y2 − ∂2Y1] e3 ∈ O , (14) or, in a more compact form,

−→ −→ ∂O =[∂tY0 − ∂j Yj ]e0 + ∂j Y0 + ∂tYj + ∇× Y ej , (15) j −→ where Y is the 3-vector with components (Y1, Y2, Y3). Above we used Einstein’s summation convention (the summation over j being from 1 to 3). This expression will be used to describe the fully symmetric Maxwell’s equations in the next section.

5 Generalised Maxwell’s equations generated by octonions

Until now we considered the functions Yk, k =0,..., 3, as functions of t and of space coordinates xj , j =1, 2, 3. The e0–e7 plane in the octonions algebra O is a sub-algebra isomorph to the algebra of the complex numbers C. Our next step is to consider t as an octonionic variable t 7→ e0t and to perform an analytic continuation of the functions Yk, to this complex plane, by performing a sort of Wick rotation e0t 7→ e7x0 (with x0 ∈ R). Formally, by this procedure of analytic continuation of

4 the functions Yk t, ~x remain functions taking values on the e0–e7 plane, i.e., we may, as before, write (0) (1) Yk e7x0, ~x = e0Yk x0, ~x + e7Yk x0, ~x , (0) (1) with Yk and Yk being real function analogous to the real and imaginary parts of Yk.

By this procedure, the time derivatives ∂t are replaced in (14) by −e7∂x0 ≡−e7∂0. Hence, (15) becomes

(0) (1) (0) (1) ∂O = −e7∂0 Y0 + e7Y0 − ∂j Yj + e7Yj e0 h i −−→ −−→ (0) (1) (0) (1) −→ (0) (1) + ∂j Y0 + e7Y0 − e7∂0 Yj + e7Yj + ∇× Y + e7Y ej " j # −−→ (1) (0) (0) (1) −→ (0) = e0 ∂0Y0 − ∂j Yj + ∂j Y0 + ∂0Yj + ∇× Y ej " j ! # −−→ (0) (1) (1) (0) −→ (1) +e7 −∂0Y0 − ∂j Yj + ∂j Y0 − ∂0Yj + ∇× Y ej . (16) " j ! # Now, we change our notations by writing −−→ −−→ −→ (0) (1) −→ −→ Y = Y + e7Y ≡ E + e7 H, (17) (0) (1) ∂0Y0 = ∂0Y0 + e7∂0Y0 ≡ −ρm + e7ρe , (18) (0) (1) ∂j Y0 = ∂j Y0 + e7∂j Y0 ≡ jm j − e7 je j , ∀j =1, 2, 3 . (19) With this new notation, eq. (16) becomes

−→ −→ ∂O = e0 ρe − ∂j Ej + (jm)j + ∂0Hj + ∇× E ej j −→ −→ +e7 ρm − ∂j Hj + − je − ∂0Ej + ∇× H ej . (20) j j By imposing ∂O = 0, the diﬀerent components of equation (20) provide eight equations corresponding to the coeﬃcients of e0, e7, ej and e7ej , j = 1, 2, 3. These equations can be more properly written in vector form:

−→ −→ ∇ · E = ρe , (21) −→ −→ ∇ · H = ρm , (22) −→ −→ −→ ∂H −→ ∇× E = − − j , (23) ∂t m −→ −→ −→ ∂ E −→ ∇× H = + j . (24) ∂t e

As we now recognise, equations (21)–(24) are the fully symmetric Maxwell’s equations and allow for the possibility of magnetic charges and currents, analogous to electric charges and currents. It is a well known fact that equations (21)–(24) are unchanged under the so-called duality transformations,

5 deﬁned, for θ ∈ [0, 2π), by −→ −→ −→ −→ ρe ρe je je E E 7→ R(θ) , −→ 7→ R(θ) −→ and −→ 7→ R(θ) −→ ρm! ρm! jm! jm! H ! H !

cos θ − sin θ where R(θ) := . We should also recall that, using arguments from quantum sin θ cos θ ! mechanics, Dirac [16] argued that the existence of an isolated magnetic charge would imply the quantisation of electric charge. From (21)–(24) we easily obtain the so-called continuity equations for the electric and magnetic −→ −→ −→ −→ charge densities and currents: ∂0ρe + ∇ · je = 0 and ∂0ρm + ∇ · jm = 0. By using (18) and (19), (0) (1) these continuity equations imply, as one easily sees, Yo = 0 and Yo = 0, or simply Y0 = 0, where is the D’Alembertian operator.

6 Conclusion

We have considered a representation of octonions as a four dimensional algebra over complex numbers. After a suitable deﬁnition of a derivative operation over ﬁelds of octonions in 3+1 dimensions we derived the fully symmetric Maxwell’s equations (i.e., with electric and magnetic charges and currents) as a single equation.

Acknowledgment. One of the authors (P.) wishes to thank FAPESP for ﬁnancial support.

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