Physica 18D (1986) 274-279 North-Holland, Amsterdam

T H E M A X W E L L , Y A N G - M I L L S A N D E I N S T E I N E Q U A T I O N S A N D C L O S E D P A T H

P A R A L L E L P R O P A G A T I O N

C a r l o s K O Z A M E H a n d E z r a T. N E W M A N Department of Physics and Astronomy, University of Pittsburgh, USA

Two linear partial differential equations (and several non-linear generalizations) for a function H, defined on the unit sphere, will be described and their relationship to gauge fields discussed. The two equations are

and

~/-/+ ~Y= 0 (1)

~H + [H, A-] + ~ Y = 0, (2)

where the differential operators ~ and ~ are given by either

{o io) ~ H = - s i n 0 00 sin0 0-~ (H/sinO),

or (using complex stereographic coordinates (~', 2) given by ~" = eiee cot 8/2)

¢Y= 2p2 ~-~(Y/P),

~H= 2pi ff---~( H/P),

with P = 12(1 + ~'~). The function A-is to be considered as (essentially) an arbitrary function of three variables u, 8, q~ (or u, ~', ~) where u is a particular given function of (0, q~) and four parameters x '~, (a = 0,1, 2, 3), i.e. A-has the form

A-{u(e, ~, x°) , 0, ~) .

In eq. (1), H and Aare scalar valued while in eq. (2) they are matrix (or Lie algebra) valued. The claim now is that eq. (1) is equivalent to the vacuum Maxwell equations and eq. (2) is equivalent to the anti-self-dual

Yang-Mills equations. The x a, which only appear as parameters in the equation, represent the Minkowski space-time points. In both equations the function H(x a, 0, q,) has a simple geometric interpretation as an infinitesimal holonomy operator (or parallel propagator) associated with a particular family of paths, while the function A-( u, q~, 0) is the free characteristic data for the respective fields. In both cases the fields F,~ b and vector potential 7, are easily determined from knowledge of H.

Eqs. (1) and (2) have been generalized to the following cases; a) the full vacuum Yang-Mills equations on Minkowski space; b) the Maxwell and Yang-Mills equations on a given asymptotically flat space-time; c) the asymptotically flat vacuum solutions of the Einstein equations, with the self or anti-self-dual fields as a simple special

case. Only in the maxwell and self-dual Yang-Mills case are the equations linear. In each of the succeeding cases the interactions

between H and its complex conjugate H gets more complicated. Moreover the equations become integro-differential equations which is a manifestation of the non-Huygens nature of the propagation.

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C. Kozameh and E. Newman / Maxwell, Yang- Mills and Einstein equations 275

1. Introduction

The subject of this paper is the reformulation of gauge theories in terms of a non-local variable. By a gauge theory I will mean either Maxwell theory, Yang-Mills theory or Einstein's General Relativ- ity or combinations of them. The Maxwell or Yang-Mills theories could be in either fiat space or in curved space. The non-local variable, which has a simple geometric interpretation, played (in one form or another) an important role in under- standing the SU(2) and general relativistic instan- ton solutions as well as Penrose's non-linear gravi- ton construction. It was the desire to see how this variable could be used in the general cases, which led to this research [1-3].

For reasons of comprehensibifity and simplicity I will confine myself almost completely to the case of vacuum Maxwell equations in real Minkowski space, with just a scattered set of comments about Yang-Mills. General Relativity will essentially be left out completely. We can do this without any great loss because our basic non-local variable has the same meaning in all cases- only the equations governing this variable become more complicated in the other cases.

Rather than starting from the beginning, I would like to begin at the end and work backwards, i.e. I will begin with the equations for the non-local variable, (denoted by H), explain the equations, describe the geometric meaning of H and finally indicate where the equations come from.

2. Equations governing H

We will begin with the following three equa- tions:

y=0, (1)

~H + ¢./T+ [H, A-] = 0, (2)

~H + ~ A + [H, A-] = Nasty functional of H, H,

A, A, (3)

where $ and ~ are linear differential operators (on the unit sphere) given by either

0 ~ H = - s i n 0 ( 00

~ X = - s i n 0 { O + -

0 H sin 0

i O

or (using complex stereographic coordinates ~', 2 given by ~ = ei*cot 0/2)

~ H = 2 O H

a

with P = ~(1 + ~2). The function X is to'be con- sidered as (essentially) an arbitrary function of three variables u, 8, ¢ or (u, ~, 2) where u is a particular given function of 8, ¢ (or ~, 2) and four parameters x a (a=0,1 ,2 ,3) , i . e . . 4has the form X(u(O, ¢,x°),o, ¢).

In (1) H and A are scalar valued while in (2) and (3) they are matrix- or Lie algebra-valued.

The idea is now to seek regular (on the sphere) solutions of (1), (2) or (3) for some range of the parameters x a.

The claim now is that with the above regularity condition, (1) is equivalent to the vacuum Maxwell equations, (2) is equivalent to the anti-self-dual Yang-Mills equations while (3) is equivalent to the full Yang-Mills equations with (2) being a special case of (3) with H = 0 . The x ~ which appear in (1), (2) and (3), only as parameters, represent the Minkowski space-time points. The function ,,T(u, 0, ¢) is the free-characteristic data for the respective fields. In all cases the fields F~b and vector potential "G are easily determined from knowledge of H ( x ~, O, dp). Before giving the mean- ing of H and its relationship to Fab and T~, I would like to say a word about the integration of (1), (2) and (3).

The ~ operator has a simple Green's function K(R, ¢p, 8', d/) so that the solution to (1) can be

276 C. Kozameh and E. Newman/Maxwell , Yang- Mills and Einstein equations

given immediately by

I~( x`,, O, , ) = f f K( O, , , 0', , ') 0A-sin 0' d0 ' dq~',

are respectively referred to as the self and anti- self-dual parts of Fat , . One has that

" + = =lF t,, -iF .

(2) can be integrated exactly for a class of special cases (i.e. choices of A) (e.g. instanton solutions) and also by an iteration series, (3) only by an iteration series. In all these cases the solutions are unique.

If the field is complex with the property that

F`, t ,=0 or E = i B

the field is called self-dual and if

3. Background material

To give geometric meaning to H, we require some background material and notation.

We begin with a brief discussion of the Hodge dualing operation. If we have a Maxwell (or Yang-Mills) field, Fob, then the dual field is

F`,*b ~ 1 r~cd ~ S a b c d 1~

with e being the Levi-Civita symbol, or more simply if

F ~ = 0 or E = - i B

the field is anti-self-dual. This decomposition and the special cases of self and anti-self-dual fields plays an important role later.

A second technical point we want to consider is the definition of a twistor or twistor surface.

We begin with Minkowski space and standard Minkowski coordinates x`,. The light cone from the point x`, can be given by

y a = X a + r l a,

F~b =

then

0, E 1 , E 2 , E 3

- E l , 0, B3, - B 2

-E2, - B 3 , 0, B 1

- E3, BE, - B 1 , 0

0, - B x ' - B 2 ' - B 3 /

B1, 0, E3, - E 2

B2, -E3, O, El ]' B3, E2, - El,

i.e. E ---, - B and B ---, E. Fields can be decom- posed into

r`,t,= +

where

F ~ = ½( F`, b - iFa*b ),

F ~ = ½(F`, b + iF,,;)

where r is an (affine) distance and where l`, is some normalized null vector, la la = 0, given by

1 U = ~ (1, sin 0 cos q~, sin 0 sin q~, cos O),

Vz

o r

l ~ _ m 1 (1 + ~'g, ~ + g, i ( ~ - ~'), - 1 + ~g). 22v~P

As (0, ¢) (or ~, ~) move over the sphere, l`, points in all null directions and thus as r, O, ~ vary all points of the null cone of x ~ are reached.

An extremely important point is now to realize that l`, remains null even when the (0, ~) take on complex values or when ~ is not the complex conjugate of ~, then denoted by ~. (It is at this point that it becomes much more convenient to use stereographic coordinates rather than O, q~.) We can now refer to the complex light cones of the

C. Kozameh and E. Newman~Maxwell, Yang- Mills and Einstein equations 277

point x ~ by letting r, ~, ~ take on independent complex values.

We now consider different portions of this com- plex null cone. We can hold ~ fixed and vary r and ff or hold ff fixed and vary r and ~. In each

case we get a 2-complex dimensional surface in complex Minkowski space. In the first case the two-surface is referred to as a twistor with the collection of two-surfaces of the same type being twistor space. The other type of two-surface is referred to as a dual twistor. We could spend a considerable amount of time studying properties of these twistor surfaces. We however need only the following property: the twistor surfaces are self-dual while the dual twistor surfaces are anti- self-dual. By this we mean that the skew tensor (say l[a~lb] ) formed from a pair of vectors, l a and ~ , tangent to the twistor surface is self-dual while 1[~ m b] (in the dual twistor surface) is anti-self-dual.

It is very convenient to choose m ~= ~1 ~= P3/0~( l ~) and ~ a = ~1 a = pO/O~(la).

The projection of a Maxwell (Y.M.) field into a twistor (or dual twistor) surface picks out the self (or anti-self) dual portions of the field, e.g. Fabl[amb] = F~b )ltam b].

We are now in position to define our non-local variable, H ( x ~, ~, ~). We begin at a point x a and choose the dual twistor surface through x a given by g-constant. On this surface we define the (in- finitesimal) triangle Ax(~, ~) by taking two sides as the null geodesics y~ = x ~ + r/~(~, ~) and y~ = x ~ + rla(~ + d~', ~) for 0 < r _< oo. The triangle is then closed by the connecting vector rm~d~, in the limit as r-* oo. H(x ~, ~,~)d~" is defined as the infinitesimal parallel propagator or infinitesimal holonomy operator around the triangle Ax(~, ~).

H is thus defined by

~ y ~ d x ~

I + H d ~ = e aa

or, via Stokes theorem,

By using the properties of A one has

H ( x ~, ~, ~) = fo~Fab(XC + rU_)l~mbrdr.

H is thus a (weighted) integral of the field, pro- jected into the dual twistor surface, along a null geodesic from x ~ to o0. In a similar fashion, only now using ~x which lies in a twistor surface we obtain the conjugate of H, i.e.

= f0=Fo (x c + rr)t° r dr.

For real fields H and H are complex conjugates of each other but for self-dual fields H 4:0 and H = 0 while for anti-self-dual fields H ~ 0 and H = 0.

The equations (1), (2) or (3) which govern H are obtained in the following fashion. We now con- sider two neighboring dual twistor surfaces pass- ing through x a given by g = constant and g + d~ = const, and, in addition, two neighboring twistor surfaces, also passing through x ~, given by ~ = constant and ~ + d~ = constant. Keeping r real, these four two-surfaces bound a real three dimen- sional region, Vx(~, g), which, if we put a "cap" or base on it at r = o0, has the form of a pyramid with x ~ at the apex or "peak". The basic idea is now to consider the Maxwell or Yang-Mills equa- tions in the region Vx(~, ~) or more precisely to consider the volume integral of the field equations over V x and to this apply Stokes theorem. One thus has, among other terms, the surface integral of Fab o v e r the boundary of Vx. The four sides involving the twistor and dual twistor surfaces can be expressed in terms of H and H, while the " cap " or "base" side can be expressed in terms of the characteristic data A and A given at r = o0.

To illustrate this procedure, we consider the Maxwell equations ~Ta Fab= 0 and ~7[aFbc] = 0 in the form

Hd~=~oa' /adx~= ffaFabdXa A dx b. ~toFL~ = ~t~,Fb~l + i v t a F ~ ] = O.

278 C. Kozameh and E. Newman~Maxwell, Yang-Mills and Einstein equations

Integrating over Vx(~, ~) and applying Stokes the- orem we have

fvx lT[aFhcl d x ~ A d x b A d x c

= foEFab dXa A d x b = O. x

The integrals over the two twistor surfaces vanish because they are self-dual and F~ is anti-self-dual. The integrals over the two dual twistor surfaces yield the differences between H at ~ + d~ and H at ~, i.e. ~H. Finally the integral over the "cap" becomes lim r 2 F ~ m a ~ b which can be expressed as d .4wi th A(u, ~, 5) the characteristic data given at r = oo. (The value of u depends on the apex point x a and ~', g i.e. u = u (x a, ~, 5). We thus have eq. (1). The additional terms in eqs. (2) and (3) arise from the commutator terms in the Yang-Mills equations.

Acknowledgements

We would like to thank the Institute for Theo- retical Physics, University of California, Santa Barbara for the aid, stimulation and hospitality during the period when some of this work was being done.

We also acknowledge the support from a grant from the National Science Foundation.

Appendix

We would like to make several comments about the integration of eq. (2), our version of the (anti) self-dual Yang-Mills equations.

The first thing to note is that if we consider a new variable, G(xa,~ , ~) defined by the regular solution of

~ G = - G A ,

we can consider G to be a that

(A.1)

"potential" for H such

4. Conclusion

To conclude we mention two generalizations of our discussion. Equations (1), (2) and (3) have been generalized to include Maxwell and Yang-Mills in asymptotically fiat curved space- times. The complication in this case is that the twistor and dual twistor surfaces do not globally exist. This yields extra interaction terms between H and H. This point of view has also been ex- tended to general relativity. The same complexities arise there, again due to the lack of global twistor and dual twistor surfaces. However, there is in the G.R. case an important conceptual simplification namely the relevant vector bundle is the cotangent (or tangent) bundle and hence H has two space- time indices. It turns out to be possible to express this H in terms of derivatives of a scalar function. The details are now in the process of being worked out with the special case of the self-dual Einstein equations completely understood.

H = G- Z~G. (A.2)

The compatibility or integrability conditions be- tween (A.1) and (A.2) is just eq. (2) of the text. Eq. (A.1), known as the Spading equation [4], thus can be used to replace (2) as an equation equivalent to the anti-self-dual Yang-Mills equations. Since the operator ~ has a well defined Green's function, K, (A.1) can be rewritten as the integral equation

G ( x , ~ , g ) = I - ~ - I G A = I - f f K G A d S , (A.3)

where dS is the area element on the sphere

dS = 2 7 dT/A d~ / (1 + T/~) 2 (A.4)

and

1 + ~ (A.5) K(~', 5, ~/, ~) = 4~r(f - ~)"

In recent discussions with M. Ablowitz and H.

C. Kozameh and E. Newman/Maxwell, Yang-Mills and Einstein equations 279

Segur we s h o w e d that the a p p r o a c h to (ant i )self-

d u a l Y a n g - M i l l s fields of Beals a n d C o i f m a n (ref.

[5]) is e s sen t i a l ly the same as u s i n g (A.1) or (A.3).

References

[1] The principal reference for this work, in which the ideas and proofs are given in detail, is C. Kozameh and E.T. Newman, Phys. Rev. D 31 (1985).

[2] The application of these ideas to Maxwell theory is in S. Kent, C. Kozameh and E.T. Newman, J. Math. Phys.

[3] The detailed application to general relativity of these ideas has not yet been completed but preliminary results are reported in A Note on Self-Dual Gravitational Fields by C. Kozameh and E.T. Newman, to appear in Foundations of Physics.

[4] The study of self-dual Yang-Mills equations via the Spar- ling Equation is described in E.T. Newman, Phys. Rev. D 22 (1980) 3023.

[5] R. Beals and R.R. Coifmann, Multi-dimensional Scattering and Inverse Scattering, conference paper.