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c 2001 International Press (cid:13) Adv. Theor. Math. Phys. 1 (2001) 165–209 2 Partitioning of electric and 0 0 2 n magnetic energy in SU(2) a J 2 2 gauge theory 1 v 0 5 0 1 Clifford H. Taubes 1 0 2 0 / h Department of Mathematics p Harvard University - h Cambridge, MA 02138 t a [email protected] m : v i X Introduction r a It has been known for over twenty years that the Coulomb solution in Maxwellian electro/magneto-statics for a point charge at the origin in 3-space is unstable when embedded in a non-abelian gauge theory with large coupling constant. A version of this phenomenon was first seen by Mandula [Ma] and other manifestations of it were described in some detail by Sikivie and Weiss [SW1]-[SW3], Jackiw, Jacobs and Rebbi [JKR], Jackiw and Rossi [JR] and in Jackiw’s lecture notes, [J]. 1 Supported in part by the National Science Foundation e-print archive: http://xxx.lanl.gov/xxxx 166 C.H. TAUBES Theinstabilityarisesmoreorlessforthefollowingreason: TheMaxwell- ian Coulomb solution has no magnetic field. However, in a non-abelian gauge theory, static electric and magnetic fields interact; and so it is energetically favorable at large coupling to have a magnetic field that suppresses the coulomb like behavior of the electric field. This insta- bility has nothing to do with the singular nature of a point charge distribution; it occurs, for example, when the charge is uniformly dis- tributed over a ball. This said, the Quixotic purpose of this article is to shed light on the dependence of the minimal energy on the coupling constant for the afore mentioned uniform charge distribution. Of particular interest are the relative contributions to this minimal energy from the electric and magnetic fields. Theorem 1, below, reports the results on these questions. Of further interest are the functional forms of the electric and mag- neticfieldsforanenergyminimizer. However, astheirdetailedbehavior resists analysis, the study here concentrates instead on the behavior of these fields for an energy critical point that minimizes energy under a restrictive hypothesis. Even so, the energy of this critical point is very close to the minimum energy and may well equal the minimum energy. In any event, the long range behavior of the electric and magnetic po- tentials for this other critical point are described below in some detail with Theorem 2 summarizing most of the salient features. Here is the background for the story: Let su(2) denote the Lie algebra of SU(2), thus, the vector space of 2 2 complex valued, anti- × hermitian and traceless matrices. Now, let A denote an su(2) valued 1-form, and let B (dA+A A) denote its ‘magnetic field’. Here, A ≡ ∗ ∧ is the Hodge star isomorphism from 2-forms to 1-forms; for example ∗ (dx1 dx2) = dx3. Introduce a norm on su(2) by the requirement that ∗ ∧ its square to send τ su(2) to τ 2 2 trace (τ2); then use the latter ∈ | | ≡ − with the Euclidean norm on T R3 to defined the norms of su(2) valued ∗ 1-forms. This done, the interest here is with the vector space, , of su(2) A valued 1-forms A which vanish where r 1, are smooth where r > 1, ≤ have locally square integrable first derivatives and are such that B 2 A | | is integrable over R3. Of special interest are the su(2)-valued 1-forms that minimize the energy functional, : (0, ), whose definition E A → ∞ follows. PARTITIONING of ELECTRIC and MAGNETIC ENERGY 167 To define , it is necessary to first digress with an introduction to E the covariant derivative, , defined by A . This derivative sends A ∇ ∈ A an su(2) valued function Ψ to the su(2) valued 1-form Ψ = dΨ + A ∇ AΨ ΨA. The covariant derivative of Ψ in the direction of a vector v is − obtained by contracting v with Ψ. The covariant derivative defines A ∇ the covariant Laplacian, , which is the sum of the second powers of A ∇ the covariant derivatives in each of the 3 coordinate directions. Now, let τ1 su(2) denote a fixed element with norm 1 and let ρ ∈ denote the su(2) valued function that equals 3(4π) 1τ1 on the unit ball − in R3 and vanishes on the complement of this ball. Then each A ∈ A a determines a unique su(2) valued function, Ψ , that obeys A Ψ = ρ (1) A A −∇ and is such that both Ψ and Ψ 3 square integrable over R3. A A A |∇ | | | Standard calculus of variation techniques can be used to argue both for the existence and uniqueness of Ψ and to study its dependence on A. A Given the preceding, specify a positive number g, the coupling constant; and then define by the rule E (A) 2 1g 2 B 2 +2 1g2 Ψ 2. (2) − − A − A A E ≡ | | |∇ | Z Z Here, and below the notation is such that the integral sign with no indication of domain or measure denotes integration over R3 with the Euclidean volume element. A critical point of is, by definition, an su(2) valued 1-form A with E the property that d (A+ta) = 0 (3) t=0 dtE | for all a . A critical point A is called a local minimizer of if ∈ A E (A+ta) (A) (4) E ≥ E for every a and, given a, for all t in some neighborhood of 0 R. A ∈ global minimizer of obeys (4) for all a and t R. E ∈ A ∈ With regards to the choice of ball radius 1 and charge normaliza- tion ρ = 1 in the definitions of and , note that changes of either | | A E can be absorbed by a combination of rescaling and g. Indeed, if the R E 168 C.H. TAUBES corresponding variational problem is posed in the ball of radius R, with charge density ρsuch that ρ = q, andwith coupling constant g, then | | the corresponding energy infimum is equal to the product of q1/2R 1 − R times the energy infimum for the R = 1, q = 1 and coupling constant gq1/2 version of the functional in (2). In any event, this rescaling property justifies the focus here on the use of the unit ball and ρ = 1 | | charge in the definitions of and . A E R By the way, the 1-forms in are required to vanish inside the unit A ball for three reasons of which pay small homage to the underlying physics. First, high energy particle physics experiments observe that the non-abelian magnetic fields associated to the subnuclear forces are small at short distances. Second, physics would, in any event, give the charge density ρ as a functional of some other field (say a spinor with values in C2), and the latter would, perforce, obey an equation that also involved the 1-form A. Because such an auxiliary equation is not considered here, a formulation that allows A to vary in the ball lacks a certain logical consistency. Finally, allow A to vary where ρ = 6 0 and there is no invariant notion of charge density to allow energy comparisons. The latter point underlies statements in some of the afore-mentioned references to the effect that the Coulomb solution is unstable for all values of the coupling constant. Note that this variational setting enjoys a restrictive sort of gauge invariance. Indeed, if A is in and h : R3 SU(2) is the identity A → matrix on the unit ball, then the ‘gauge equivalent’ 1-form h 1Ah + − h 1dh is also in and (A) = e(h 1Ah+h 1dh). − − − A E As a parenthetical remark, note that the variational setting here can be formulated in a completely gauge invariant manner. To do so, first change the definition of so that the requirement that A A vanish in the ball is replaced by the requirement that B vanish in A the ball. This done, make the following requirements on ρ: First, it should vanish outside the ball and its norm should equal 3/(4π) inside. Second, require that ρ = 0 where ρ = 0. This done, then the A ∇ 6 energy can be considered to be a function of pairs (A,ρ), subject to E the preceding constraints. As such, this energy is gauge invariant since (h 1Ah+h 1dh,h 1ρh) and (A,ρ) have the same energy for all smooth − − − h : R3 SU(2). → The author knows the explicit form of only one critical point of PARTITIONING of ELECTRIC and MAGNETIC ENERGY 169 (up to gauge equivalence), this being the gauge transforms of the E ‘coulomb’ solution which has A = 0 with the corresponding Ψ given A in terms of the radial coordinate r on R3 as Ψ = (8π) 1(3 r2)τ1 where r 1. (5) A=0 − • − ≤ Ψ = (4π) 1r 1τ1 where r 1. A=0 − − • ≥ The energy, (0), of the Coulomb solution is equal to 3g2(40π) 1. − E With the preceding understood, consider: Theorem 1. For each g > 0 the corresponding version of the functional achieves its global minimum on . When g (√2π)1/2 this E A ≤ minimum is attained only on the Coulomb solution and its gauge transforms. However, when g > (6π)1/2 then the Coulomb solution is no longer the minimizer of . Moreover, there exists a g-independent con- E stant c such that when g > (6π)1/2, then g2(40π) 1 +c 1g inf g2(40π) 1 +cg. (6) − − − ≤ E ≤ A In addition, if, for such g, the corresponding version of achieves its E infimum at A , then ∈ A g2(20π) 1 Ψ 2. (7) • − ≤ r 1|∇A A| ≤ c 1g g2 R Ψ 2 cg. • − ≤ r 1|∇A A| ≤ ≥ c 1g g 2R B 2 and g 2 B 2 cg. • − ≤ − 1 r 1+c/g| A| − 1 r| A| ≤ ≤ ≤ ≤ R R Ψ < Ψ everywhere, and Ψ cg 1/2r 1 where r 2. A A=0 A − − • | | | | | | ≤ ≥ Remark that the factor g2(40π) 1 appears in (6) and in the first − point of (7) because A = 0 in the unit ball and (40π) 1 is the minimum − over the set of solutions in the unit ball to the equation - ∆ Ψ = ρ of A=0 the functional that sends Ψ to Ψ 2. By the way, note in the r 1|∇A=0 | second point of (7) that a substa≤ntial fraction of the magnetic energy R is contributed from the shell of thickness (g 1) that surrounds the − O unit ball. Note also that the results in Theorem 1 do not change in any 170 C.H. TAUBES substantial way if the charge distribution ρ in (1) is allowed to vary in the ball as ρ τ1 with ρ positive with integral 1. 0 0 As remarked above, the author has little to say about the detailed point to point behavior of the fields (A,Ψ ) when A is an absolute A minimizer of a large g version of . On the otherhand, quite a bit can E besaidaboutthese fields fortheabsolute minimizer of ’srestriction to E a certain subset, . Here, A when A is gauge equivalent to 0 0 A ⊂ A ∈ A A = αsinθdϕτ2 where α is a function on R3, θ and ϕ are the standard 0 spherical coordinates and τ2 su(2) has unit length and is orthogonal ∈ to τ1. In this regard, the critical points of ’s restriction to are also 0 E A unrestricted critical points of . This understood, the next theorem E describes the minimizers of ’s restriction to . 0 E A Theorem 2. For each g > 0, there is a function α on R3, unique up to multiplication by 1, and charcterized by the fact that the global ± minimizers of ’s restriction to are all gauge equivalent to A 0 E A ≡ αsinθdϕτ2. In this regard, α = 0 when g (6π)1/2, but not so when ≤ g > (6π)1/2. In addition, there is a constant, c 1, with the following ≥ significance: For g > (6π)1/2, the energy (A) obeys (6) and A with its E corresponding Ψ obeys (7). Moreover, A Ψ = Ψτ1 with Ψ a positive function on R3. A • α = f sinθ with f > 0 where r > 1. • | | At points where r c, • ≥ a) Ψ √2g 2e g1/4/cr 1. − − − ≤ b) α = fr 1sinθ with f g1/2/c. − | | ≥ At points where r c, • ≫ a) Ψ = √2g 2(e +m )r 1 where e is a positive constant less − 0 Ψ − 0 than e g1/4/c and m cr 1. − Ψ − | | ≤ b) α = (c0 +mα)r−[(9−8e20)1/2−1]sinθ where c0 is a positive constant and m cr 1. α − | | ≤ Ofparticular interest totheauthor withregardstoTheorem 2isthe fact that the electric potential, Ψ, although Coulomb like at large distance, has an effective charge √ 2g 2e that is absolutely microscopic − 0 − PARTITIONING of ELECTRIC and MAGNETIC ENERGY 171 at large g. Meanwhile, the magnetic potential, α, falls to zero rather like the field of a dipole charge with dipole moment of size (g1/2). O As with the statement of Theorem 1, the conclusions of Theorem 2 are not changed substantially if the charge distribution ρ in (1) is allowed to vary in the ball as ρ τ1 with ρ positive with integral 1. 0 0 This said, only the ρ = 3(4π) 1 case is discussed below. 0 − The remainder of this article is occupied with the proofs of Theo- rems 1 and 2. In this regard, the assertions of Theorem 2 are proved first as they are used in part to prove Theorem 1. In particular, the assertions of Theorem 2 are proved as various propositions and lemmas in Sections a-j below. These are all summarized in Section k to tie up the argument for Theorem 2. Section k then ends with the proof of Theorem 1. Before starting, note that the author benefitted at an early stage of this project from conversations with R. Scott. a) A reformulation of the variational problem on 0 A The variational problem given by ’s restriction to the set of su(2)- 0 E A valued 1-forms gauge equivalent to αsinθdϕτ2 with α a function on R3 can be reformulated as follows: Let g 0 be a constant, and, for now, let ρ be a function with ≥ compact support where the distance, r, to the origin is less than 1 and with integral equal to 1. Now consider an equation for a C0 function α of the first two of the spherical coordinates (r,θ) on R3 which is given as follows: First, introduce the unique, C1 solution ψ ψ[α] on R3 to ≡ the equations ∆ψ +r 2α2ψ = ρ where r > 1. (8) − • − lim = 0. r • →∞ 172 C.H. TAUBES Here, ψ is a function of (r,θ) too, and ∆ = r 2(r2( ) ) +(sinθ) 1(sinθ( ) ) − r r − θ θ · · is the standard Laplacian. Then, with ψ understood, require that α = 0 where r 1. (9) • ≤ α r 2((sinθ) 1(sinθα) ) g4ψ2α = 0 where r > 1. rr − − θ θ • − − − r 2α is square integrable on R3. − • Here, and also below when ambiguities are unlikely, the partial derivative of a function, f, with respect to r is written as f and with respect r to θ as f . When ambiguity can arise, these derivatives are written θ below as ∂ f and ∂ f, respectively. r θ Of special interest are those functions α for that are absolute minimizers of the functional (α) 2 1g 2 r 2(α2 +r 2(sinθ) 2(sinθα)2) E0 ≡ − − − r − − θ Z +2 1g2 ( ψ 2 +r 2α2ψ2). (10) − − |∇ | Z Note that (α) = (αsinθdϕτ2) with as in (2), so is the restric- 0 0 E E E E tion of to those A of the form αsinθdϕτ2. E ∈ A Note that the introduction of the 1-form a αsinθdϕ allows the ≡ r 1 parts of (8) and (9) to be rewritten as ≥ ∆ψ + a 2ψ = 0, (11) • − | | ∆a g4 ψ 2a = 0. • − − | | In terms of the 1-form a, the energy becomes 0 E = 2 1g 2 a 2 +2 1g2 ( ψ 2 + a 2 ψ 2). (12) 0 − − − E |∇ | |∇ | | | | | Z Z PARTITIONING of ELECTRIC and MAGNETIC ENERGY 173 Note that in (10) defines a bonafide functional on the space, , 0 0 E C of continuous functions on R3 that vanish on the unit ball, are invariant under rotations about the axis defined by the spherical angle ϕ and are such that r 2(α2 +r 2(sinθ) 2(sinθα)2) has finite integral. Moreover, − r − − θ techniques from the calculus of variations establish that achieves its 0 E infimimum on . In addition, standard elliptic regularity techniques 0 C (as found, for example, in Chapter 6 of [Mo]) justify the assertion that any critical point of on cC is smooth where r > 1. 0 0 E b) Some simple inequalities The subsequent analysis exploits some basic observations that concern and the solutions to (8) and (9). In this regard, the first observation 0 E is obtained by multiplying both sides of the top equation in (11) by ψ and then integrating the result over R3. A subsequent integration by parts then finds that ( ψ 2 + a 2 ψ 2) = ψρ. (13) |∇ | | | | | Z Z The second observation follows by contracting both sides of the bottom equation in (11) with the 1-form a and then integrating the result over R3. This done, an integration by parts finds g 2 a 2 = g2 a 2 ψ 2. (14) − |∇ | | | | | Z Z The third observation concerns a pair ψ and ψ where these functions ′ solve the versions of the second that are defined by a corresponding pair, a and a. This understood, note the following: ′ If a a everywhere with the inequality strict somewhere, ′ | | ≥ | | then ψ < ψ everywhere. (15) ′ Indeed, this follows from the maximum principle because the top equation in (8) implies that ∆(ψ ψ )+( a2 a 2)ψ + a 2(ψ ψ ) = 0. (16) ′ ′ ′ ′ − − | |−| | | | − In particular, this equation precludes a non-negative maximum for ψ − ψ . ′ 174 C.H. TAUBES The observation in (15) implies that for any a, the corresponding ψ is sandwiched as ψ < ψ ψ , (17) D Coul ≤ where ψ is the solution to ∆σ = ρ on R3 which decays to zero as Coul − r , while ψ is the solution to ∆σ = ρ in the ball where r 1 D → ∞ − ≤ which vanishes at r = 1. For example, in the case where ρ = ρ = 0 3(4π) 1 where r 1, − ≤ (8π) 1(1 r2) < ψ (8π) 1(3 r2) where r 1. (18) − − • − ≤ − ≤ 0 < ψ (4π) 1r 1 where r 1. − − • ≤ ≥ By the way, note that (13) and (18) imply that (40π) 1 < 2 1 ( ψ 2 + a 2 ψ 2) 3(20π) 1 (19) − − − |∇ | | | | | ≤ Z in the case where ρ is the constant 3(4π) 1. − c) The coulomb solution The Coulomb solution to (8) and (9) has α = 0 and ψ ψ given by 0 ≡ ψ (x) = (4π) 1 x ( ) 1ρ. (20) Coul − − | − · | Z For example, when ρ has the constant value 3(4π) 1 in the unit ball − and is zero outside, then ψ is equal to (8π) 1(3 r2) in the ball and 0 − − (4π) 1r 1 outside. As the following lemma attests, the Coulomb so- − − lution, α 0, is the minimizer of when g is small and not the 0 ≡ E minimizer when g is large. Lemma 3. When g g , then α = 0 is the absolute minimizer of 0 0 ≤ E on , and when g > g , it is not. In the case where ρ = 3(4π) 1 inside 0 0 − C the ball and zero outside, g = (6π)1/2. 0 The proof of this lemma exploits two fundamental inequalities that are also used elsewhere in the paper: (21)