aa r X i v : . [ h e p - t h ] F e b Electromagnetic knots from de Sitter space
Olaf Lechtenfeld
Institut f¨ur Theoretische Physik and Riemann Center for Geometry and PhysicsLeibniz Universit¨at Hannover, Appelstrasse 2, 30167 Hannover, Germany
Abstract
We find all analytic SU(2) Yang–Mills solutions on de Sitter space by reducing the field equationsto Newton’s equation for a particle in a particular 3d potential and solving the latter in a specialcase. In contrast, Maxwell’s equations on de Sitter space can be solved in generality, by separatingthem in hysperspherical coordinates. Employing a well-known conformal map between (half of)de Sitter space and (the future half of) Minkowski space, the Maxwell solutions are mappedto a complete basis of rational electromagnetic knot configurations. We discuss some of theirproperties and illustrate the construction method with two nontrivial examples given by rationalfunctions of increasing complexity. The material is partly based on [1, 2].
Talk presented at the RDP online workshop ”Recent Advances in Mathematical Physics” - Regio2020,06 December 2020, to appear in PoS(Regio2020)011.
Description of de Sitter space
Four-dimensional de Sitter space is a one-sheeted hyperboloid (of radius ℓ ) in R , ∋ { Z , Z , . . . , Z } given by − Z + Z + Z + Z + Z = ℓ . (1)Constant Z slices are 3-spheres of varying radius, yielding a parametrization of dS ∋ { τ, ω A } as Z = − ℓ cot τ and Z A = ℓ sin τ ω A for A = 1 , . . . , with τ ∈ I := (0 , π ) and ω A ω A = 1 . (2)The details of the embedding ω A : ( χ, θ, φ ) ∋ S ֒ → R are irrelevant. The Minkowski metric d s = − d Z + d Z + d Z + d Z + d Z (3)induces on dS the metric d s = ℓ sin τ (cid:0) − d τ + dΩ (cid:1) with dΩ for S , (4)showing that dS is conformally equivalent to a finite cylinder I × S . We wish to find solutions to the Yang–Mills (and Maxwell) equations on de Sitter space. Due totheir conformal invariance in four spacetime dimensions, we may also study the problem on the finiteMinkowskian cylinder
I × S .The gauge potential taking values in a Lie algebra g can always be chosen as A = X a ( τ, ω ) e a on I × S (5)where X a ∈ g , and { e a , a = 1 , , } is a basis of left-invariant one-forms on S ≃ SU (2) , with d e a + ε abc e b ∧ e c = 0 and e a e a = dΩ . (6)There is no d τ component because we picked the temporal gauge A τ = 0 . In terms of the S coordinates ( a, i, j, k = 1 , , and B, C = 1 , , , ) these one-forms can be constructed as e a = − η aBC ω B d ω C where η ijk = ε ijk and η ij = − η i j = δ ij . (7)Dual to the e a are the left-invariant vector fields R a = − η aBC ω B ∂∂ω C ⇒ [ R a , R b ] = 2 ε abc R c (8)generating the right multiplication on SU(2), so that an arbitrary function Φ on S obeys dΦ( ω ) = e a R a Φ( ω ) . (9)1he full SO(4) isometry group of S is generated by left-invariant R a and right-invariant L a .In this language, the gauge field two-form becomes ( ˙ X a ≡ dd τ X a ) F = F τa e τ ∧ e a + F bc e b ∧ e c = ˙ X a e τ ∧ e a + (cid:0) R [ b X c ] − ε abc X a + [ X b , X c ] (cid:1) e b ∧ e c , (10)where we define R [ b X c ] = R b X c − R c X b , and the Yang–Mills Lagrangian reads L = t r F µν F µν = − t r F τa F τa + t r F ab F ab = − t r (cid:8) ˙ X a ˙ X a − X a X a + ε abc X a D [ b X c ] − ( D [ a X b ] )( D [ a X b ] ) (cid:9) (11)with the short-hand D a := R a + X a . The Yang–Mills equations then take the form ¨ X a = − X a + 2 ε abc R b X c + R b R [ b X a ] + 3 ε abc [ X b , X c ]+ 2[ X b , R b X a ] − [ X b , R a X b ] − [ X a , R b X b ] − (cid:2) X b , [ X a , X b ] (cid:3) (12)with the Gauss law R a ˙ X a + [ X a , ˙ X a ] = 0 . (13) The simplest Yang–Mills solutions are most symmetric. To obtain them, let us impose SO(4) sym-metry by setting X a ( τ, ω ) = X a ( τ ) . The Yang–Mills equations then become ordinary matrix differ-ential equations [3, 4, 5], ¨ X a = − X a + 3 ε abc [ X b , X c ] − (cid:2) X b , [ X a , X b ] (cid:3) and [ X a , ˙ X a ] = 0 . (14)These three coupled ordinary differential equations for the three matrix functions X a ( τ ) are still toocomplicated. However, for the gauge group SU(2), these equations admit some analytic solutions. Solet us choose a spin- j representation of g = su (2) and introduce the three SU(2) generators T a , [ T b , T c ] = 2 ε abc T a and t r ( T a T b ) = − C ( j ) δ ab for C ( j ) = j ( j +1)(2 j +1) . (15)A simple ansatz for the matrices X a is X = Ψ T , X = Ψ T , X = Ψ T with Ψ a = Ψ a ( τ ) ∈ R . (16)The resulting simplification of Yang–Mills Lagrangian density, L = 4 C ( j ) (cid:8) ˙Ψ a ˙Ψ a − (Ψ − Ψ Ψ ) − (Ψ − Ψ Ψ ) − (Ψ − Ψ Ψ ) (cid:9) , (17)suggests an interpretation of { Ψ a } as the coordinates of a Newtonian particle in R moving in apotential V (Ψ) = (Ψ − Ψ Ψ ) + (Ψ − Ψ Ψ ) + (Ψ − Ψ Ψ ) . (18)2igure 1: Contours of the Newtonian potential in (18).The only analytic nonabelian solutions come from Ψ = Ψ = Ψ =: Ψ with ¨Ψ = 16 Ψ (Ψ − − , (19)leading to elliptic functions Ψ( τ ) , except for the special cases Ψ( τ ) = 0 or (the vacuum), Ψ( τ ) = (the sphaleron), and the bounce solution in the double-well potential. The corresponding gaugepotential takes the simple form A = Ψ( τ ) g − d g for g : S −→ SU (2) , (20)and the SU(2) color electric and magnetic fields are E a = F τa = ˙Ψ T a and B a = ε abc F bc = 2 Ψ (Ψ − T a . (21)Their total de Sitter energy and action is finite and proportional to double-well energy. These analyticYang–Mills configurations are related to Minkowski-space solutions found in the seventies [6, 7, 8](for a review from this period, see [9]). Their stability, however, has been analyzed only recently [10].3 All Maxwell solutions on de Sitter space
The other analytic solutions to (12) and (13) are abelian, i.e. excite only a single direction in isospinspace. In this case we can drop the matrix valuedness and treat the X a as real functions. Dropping allcommutator terms, the Yang–Mills equations (12) turn into the linear Mawell equations, ¨ X a = ( R − X a + 2 ε abc R b X c (22)where R ≡ R b R b is the laplacian on S , and we refined the temporal gauge to the Coulomb gauge A τ = 0 and R a X a = 0 , (23)which takes care of the Gauss law.The coupled wave equations (22) may be completely solved by separation of variables. Seekingfactorized complex basis solutions X a ( τ, ω ) = Z a ( ω ) e iΩ τ , (24)one learns that the frequency Ω only depends on the SO(4) spin j ∈ N , − R Z ja ( ω ) = 2 j (2 j +2) Z ja ( ω ) ⇒ (cid:0) (Ω j ) − j +1) (cid:1)(cid:0) (Ω j ) − j (cid:1) = 0 , (25)where the second factor appears only for j ≥ . The basis solutions Z ja to the linear system come intwo types and carry two further labels m and n [1]:• type I : j ≥ , m = − j, . . . , + j , n = − j − , . . . , j +1 , Ω j = ± j +1) Z j ; m,n + = p ( j − n )( j − n +1) / Y j ; m,n +1 Z j ; m,n = p ( j − n +1)( j + n +1) Y j ; m,n Z j ; m,n − = − p ( j + n )( j + n +1) / Y j ; m,n − (26)• type II : j ≥ , m = − j, . . . , + j , n = − j +1 , . . . , j − , Ω j = ± jZ j ; m,n + = − p ( j + n )( j + n +1) / Y j ; m,n +1 Z j ; m,n = p ( j + n )( j − n ) Y j ; m,n Z j ; m,n − = p ( j − n )( j − n +1) / Y j ; m,n − (27)where Z ± = ( Z ± i Z ) / √ , and the hyperspherical harmonics Y j ; m,n ( ω ) with m, n = − j, − j +1 , . . . , + j and j = 0 , , , . . . (28)are characterized by − R Y j ; m,n = j ( j +1) Y j ; m,n and i2 R Y j ; m,n = n Y j ; m,n . (29) Z a ( ω ) is not to be confused with the ambient-space coordinates Z A . The label m is the eigenvalue of i2 L . A = X a ( τ, ω ) e a is a linear combination with X a ( τ, ω ) = X jmn n c I j ; m,n Z j ; m,na I ( ω ) e j +1) i τ + c II j ; m,n Z j ; m,na II ( ω ) e j i τ + c.c. o . (30)Each complex solution yields two real ones (real part and imaginary part). We count j +1)(2 j +3) real type-I solutions and j +1)(2 j − real type-II solutions ( j ≥ ), which add up to j +1) solutions for j> and 6 solutions for j =0 , as it should. Constant solutions ( Ω = 0 ) are not allowed;the simplest ones are j =0 type I or j =1 type II. The most general j =0 configuration is X ( j =0) a = n c , − √ (cid:16) − i0 (cid:17) + c , (cid:16) (cid:17) − c , +1 1 √ (cid:16) (cid:17)o e τ + c.c. . (31)The parity inversion, which interchanges left and right invariance, relates spin j type I solutions withspin j +1 type II solutions, swopping labels m and n . Finally, electromagnetic duality is realized byshifting | Ω j | τ by ± π , which produces from a solution A a dual solution A D . We shall now see thatthis basis of Maxwell solutions relates to so-called electromagnetic knots in Minkowski space. The Z + Z < half of dS is also conformally related to future Minkowski space R , ∋ { t, x, y, z } , Z = t − r − ℓ t , Z = ℓ xt , Z = ℓ yt , Z = ℓ zt , Z = r − t − ℓ t with x, y, z ∈ R and r = x + y + z but t ∈ R + , (32)since t ∈ [0 , ∞ ] corresponds to Z ∈ [ −∞ , ∞ ] but Z + Z < . In these Minkowski coordinates, d s = ℓ t (cid:0) − d t + d x + d y + d z (cid:1) . (33)One may cover the entire R , by gluing a second dS copy and using the patch Z + Z > .We shall employ the direct relation between the cylinder and Minkowski coordinates, cot τ = r − t + ℓ ℓ t , ω = γ xℓ , ω = γ yℓ , ω = γ zℓ , ω = γ r − t − ℓ ℓ , (34)with the convenient abbreviation γ = 2 ℓ p ℓ t + ( r − t + ℓ ) . (35)Since t = −∞ , , ∞ corresponds to τ = − π, , π , the cylinder gets doubled to I × S , andfull Minkowski space is covered by the cylinder patch ω ≤ cos τ . The cylinder time τ is a regularsmooth function of ( t, x, y, z ) , but more useful will be exp(2i τ ) = (cid:2) ( ℓ + i t ) + r (cid:3) ℓ t + ( r − t + ℓ ) . (36)5igure 2: An illustration of the map between a cylinder I× S and Minkowski space R , . TheMinkowski coordinates cover the shaded area. Its boundary is given by the curve ω = cos τ . Eachpoint is a two-sphere spanned by ω , , , which is mapped to a sphere of constant r and t .A slightly lengthy computation yields the Minkowski-coordinate expressions for the one-forms [1], e = e µ d x µ = γ ℓ (cid:16) ( t + r + ℓ ) d t − t x k d x k (cid:17) ,e a = e aµ d x µ = γ ℓ (cid:16) t x a d t − (cid:0) ( t − r + ℓ ) δ ak + x a x k + ℓ ε ajk x j (cid:1) d x k (cid:17) , (37)with the notation ( x i ) = ( x, y, z ) and ( x µ ) = ( x , x i ) = ( t, x, y, z ) . (38)Due to the conformal invariance of the Maxwell equations, our oscillatory solutions on the cylin-der I× S may be transferred to a basis of Maxwell solutions on Minkowski space (with certainfall-off properties). To accomplish this task, we only have to effect the coordinate change from ( τ, ω ) ∼ ( τ, χ, θ, φ ) to x ≡ ( x µ ) = ( t, x, y, z ) ∼ ( t, r, θ, φ ) , (39)so that A = X a ( τ ( x ) , ω ( x )) e a ( x ) = A µ ( x ) d x µ yielding A µ ( x ) with A t = 0 , (40) d A = ˙ X a e ∧ e a − ε abc X a e b ∧ e c = F µν d x µ ∧ d x ν yielding F µν ( x ) . (41)From this, we obtain electric and magnetic fields E i = F i and B i = ε ijk F jk . For the com-putation it is helpful to recognize that exp(2i τ ) is a rational function of t and r . It follows that allphysical quantities (and the gauge potential) are rational functions of the Minkowski coordinates! The S angular coordinates ( θ, φ ) on both sides can be identified. The map ( τ, χ ) ( t, r ) realizes the Penrosediagram of Minkowski space [2]. All knot solutions on Minkowski space
As we shall see below, the simplest ( j =0 ) solutions neatly reproduces the celebrated Hopf-Ra˜nadaelectromagnetic knot [11, 12]. From our construction, some general features of all knot solutions canbe inferred.Firstly, at spatial infinity (for t fixed) all field strengths decay like r − , but they fall off only as ( t ± r ) − along the light-cone. Hence, the asymptotic energy flow is concentrated on past and futurenull infinity and peaks on the light-cone of the spacetime origin. Secondly, the “knot basis” forms acomplete set of finite-action configurations. Of course, it does not contain plane waves. Thirdly, theobvious conserved (in Minkowski time) quantities are helicity and energy, h = Z R (cid:0) A ∧ F + A D ∧ F D (cid:1) and E = Z R d x (cid:0) ~E + ~B (cid:1) , (42)where the spatial integration is done at fixed t . Their common scale is determined by the amplitudeof the solution, but their ratio is fixed for the basis configurations. Both quantities are best computedin the “sphere frame” at t = τ = 0 , F = E a e a ∧ e + B a ε abc e b ∧ e c . (43)Let us focus on type I solutions of a fixed spin j and suppress these indices. For those one finds E a = − iΩ X mn c m,n Z m,na e iΩ τ + c.c. and B a = − Ω X mn c m,n Z m,na e iΩ τ + c.c. , (44)which yields (cid:0) E a E a + B a B a (cid:1) = 2Ω (cid:12)(cid:12)P m,n c m,n Z m,na ( ω ) (cid:12)(cid:12) . (45)The Minkowski energy at t =0 is easily pulled back to the cylinder frame and evaluated by exploitingthe orthogonality properties of the hyperspherical harmonics [2], E = ℓ Z S d Ω (1 − ω ) (cid:0) E a E a + B a B a (cid:1) = ℓ (2 j +1) Ω X m,n | c m,n | . (46)A similar computation produces an expression for the helicity. It turns out that single-spin solutions(of both types) have a universal energy-to-helicity ratio E/h = | Ω | /ℓ .Fourthly, so-called null fields are easily characterized, ~E − ~B = 0 = ~E · ~B ⇔ ( ~E ± i ~B ) = 0 ⇔ X a ( E a ± i B a ) = 0 . (47)For fixed spin j and type I we infer from above that E a + i B a = −
2i Ω X mn c m,n Z m,na ( ω ) e iΩ τ (no c.c.!) , (48)hence in such a sector we have [2] F µν null ⇔ X a (cid:16)X mn c m,n Z m,na ( ω ) (cid:17) = 0 . (49)7iven the known form of the functions Z m,na ( ω ) we can expand this expression in hyperspherical har-monics and arrive at (4 j +1)(4 j +2)(4 j +3) homogeneous quadratic equations for (2 j +1)(2 j +3) complex parameters c m,n . This system is vastly overdetermined, but only j +6 j +1 equations areindependent, and thus we are still left with j +2 free complex parameters for the solution manifold,which is explicitly parametrized as follows [2], c m,n ( w, ~z ) = q ( j +2 j +1 − n ) w j +1 − n j +2 e π i k m j +1 − n j +2 z m with w ∈ C ∗ and ~z ≡ { z m } ∈ C j +1 (50)and a choice of j +1 integers k m ∈ { , , . . . , j +1 } (one of which can be absorbed into z m ). Giventhat the overall scale of the solutions is irrelevant, the null fields form a complete-intersection pro-jective variety of complex dimension j +1 inside C P (2 j +1)(2 j +3) − . The simplest example occursfor spin j =0 , where the single null-field relation c , = 2 c , − c , defines a generic rank-3 quadric in C P or, alternatively, a cone over C P lying in C . We close with two concrete examples. First, the j =0 case represents SO(4)-symmetric Maxwellsolutions in de Sitter space, meaning X a ( τ, ω ) = X a ( τ ) thus R a X b = 0 and trivializing (22) to ¨ X a = − X a ⇒ X a ( τ ) = ξ a cos (cid:0) τ − τ a ) (cid:1) , (51)which describes an ellipse in R . We may always choose a frame where ξ = 0 and τ = 0 . Theoverall amplitude is irrelevant as all equations are linear, and solutions can be superposed at will.Specializing to ξ = ξ = − and τ = π ⇔ c , − = c , = 0 and c , ∈ i R , (52)one has a null configuration with components X ( τ ) = − sin 2 τ , X ( τ ) = − cos 2 τ , X ( τ ) = 0 . (53)The result of a short computation yields ~E + i ~B = ℓ (cid:0) ( t − i ℓ ) − r (cid:1) ( x − i y ) − ( t − i ℓ − z ) i( x − i y ) + i( t − i ℓ − z ) − x − i y ) ( t − i ℓ − z ) . (54)This is the announced Hopf–Ra˜nada electromagnetic knot [11, 12]. Our approach also yields itsgauge potential.Second, let us take the real part of the ( j ; m, n ) = (1; 0 , type I basis solution. Combining e τ +e − τ = 2 cos 4 τ and expressing Y ,⋆ from (26) in terms of ω A , we get X ± = − √ π ( ω ± i ω )( ω ± i ω ) cos 4 τ and X = − √ π ( ω + ω − ω − ω ) cos 4 τ . (55) These are the generic solutions. There also exist special solutions with c m,n = 0 for | n | 6 = j +1 . Every solution X a ( τ ) spontaneously breaks the SO(4) invariance by the choice of integration constants ( ξ a , τ a ) . ℓ =1 ) ( E +i B ) x = − t − i) − x − y − z ) ×× n y + 3i ty − xz + 2 t y + 2i txz − x y − y + 4 yz + 4i t y − t xz − tx y − ty + 4i tyz + 10 x z + 10 xy z − xz + 2(i txz + x y + y + yz )( − t + x + y + z ) + (i ty − xz )( − t + x + y + z ) o , ( E +i B ) y = 2i(( t − i) − x − y − z ) ×× n x + 3i tx + yz + 2 t x − tyz − x − xy + 4 xz + 4i t x + 6 t yz − tx − txy + 4i txz − x yz − y z + 2 yz + 2( − i tyz + x + xy + xz )( − t + x + y + z ) + (i tx + yz )( − t + x + y + z ) o , (56) ( E +i B ) z = i(( t − i) − x − y − z ) ×× n t + t − x − y + 3 z + 4i t − tx − ty + 4i tz − t − t x − t y − t z + 11 x + 22 x y + 10 x z + 11 y − y z + 3 z + 2i t ( t − x − y − z )( t − x − y − z ) − ( t + x + y − z )( − t + x + y + z ) o . Figures 3 and 4 below show t =0 energy density level surfaces and a particular magnetic field line.Figure 3: Energy density level surfaces at t =0 for the (1; 0 , solution above.9igure 4: A particular magnetic field line for the (1; 0 , solution above. • Rational electromagnetic fields with nontrivial topology have been investigated since 1989• We introduced a new construction method based on two insights: – the simplicity of solving Maxwell’s equations on a temporal cylinder over a three-sphere – the conformal equivalence of a cylinder patch { τ, ω } to Minkowski space { x } ≡ { t, ~x } • The gauge potential is transferred via A = X ν ( τ, ω ) e ν = X ν ( τ ( x ) , ω ( x )) e νµ ( x ) d x µ • Only finite-time τ ∈ ( − π, + π ) dynamics is required on the cylinder• Our solutions have finite energy and action, by construction• Energy and helicity are easily computed, null fields can be fully characterized• A complete basis was constructed for sufficiently fast spatially and temporally decaying fields• The non-Abelian extension couples different j components of X a and will be harder to treat• The method may be useful for numerics of Yang–Mills dynamics in Minkowski space Acknowledgments
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