Two questions on gauge fields

Foundations of Physics, Vol. 24. No. 5, 1994

Two Questions on the Geometry of Gauge Fields

N. C. A. da Costa,~ F. A. Doria, 2 A. F. Furtado-do-Amaral, 3 and J. A. de Barros 4

Received October 9, 1991; revised July 7, 1992

We first show that a theorem by Cartan that generalizes the Frobenius integrability theorem allows us (given certain conditions) to obtain noncurvature solutions for the differential Bianchi conditions and for higher-degree similar relations. We then prove that there is no algorithmic procedure to determine, for a reasonable restricted algebra of functions on spacetime, whether a given connection form satisfies the preceding conditions. A parallel result gives a version of G6del's first incompleteness theorem within an (axiomatized) theory of gauge fields.

1. I N T R O D U C T I O N

The present paper deals with two interlocked questions in theoretical and mathematical physics. We are interested in a given phenomenon; we for- malize it in a reasonable way, and state a result that the describes some "nice" necessary and sufficient conditions for that phenomenon to happen. Then, given a specific example, can we actually check whether our example satisfies the prescribed conditions?

The geometrical question we deal with here started out of a simple and yet interesting problem in the mathematics of gauge fields, the problem of the "Bianchi condition copies." That problem can be stated as follows:

t Institute for Advanced Studies, University of S~o Paulo, Av. Prof. Luciano Gualberto, trav. J, 374, 05655-010 S~o Paulo SP Brazil.

2 Center for the Study of Mathematical Theories of Communication, School of Communica- tions, Federal University of Rio de Janeiro, Av. Pasteur, 250. 22295-900 Rio de Janeiro RJ Brazil.

3 Institute of Physics, Federal University of Rio de Janeiro, 21949-900 Rio de Janeiro RJ Brazil.

4 Brazilian Center for Physical Research, R. Dr. Xavier Sigaud, 150. 22290-000 Rio de Janeiro RJ Brazil.

783

0015-9018/94/0500-0783507.00/0 �9 1994 Plenum Publishing Corporation

784 da Costa et M.

closed differential forms on manifolds can be locally represented with the help of the Poincar6 lemma in a well-behaved way as the exterior derivative of lower-degree forms. Closed covariant differential forms appear in several problems in general relativity and gauge field theories. Such are the differential Bianchi identities for linear or gauge curvature tensors or stability conditions for connection forms under the action of a gauge transformation. Penrose ~2-~1 has used the differential Bianchi identities as the actual spinorial sourceless gravitational equations; more recently one can ask the following question within the theory of chiral gauge anomaliest3"28~: if we are given an arbitrary, non-gauge-field, tensorial 2-form that happens to satisfy the differential Bianchi identities with respect to a previously given conection form ~, can we develop an analogue of the theory of anomalies for that form? Which would be the corresponding physical interpretations? Which are the minimum requirements (if any) to be imposed on such a non-gauge-field form, so that it represents a physically sensible object in the context of gauge theory?

In order to handle those problems, we must at first look for a classification of the forms that satisfy the differential Bianchi identities.

If we use tensorial notation and a downstairs coordinate system on a A principal fiber bundle, and if 0Ej ...... pl is a covariantly closed differential

form (A, B .... denote gauge indices while /~, v .... are coordinate indices as usual), then it satisfies

A A B = 0 (1) a[~,0,,p .... ] + 7B[~O,,~ . . . . . ]

(The ~ , are local components for the connection form, and [ . . . ] denotes skew symmetrization in the enclosed indices.)

A If the 0[~,.1 are local components of a curvature form, then they can be expressed as some sort of covariant derivative of the connection form on an adequate principal bundle with the help of the Cartan structure equations. But is that representation always available for any arbitrary coveriantly closed differential forms, in the case where such forms do exist?

We show in Sec. 2 of the present paper that there is a whole continuum of different noncurvature solutions for Eq. (1), given certain integrability conditions obtained with the help of a theorem by Cartan that generalizes the Fr~Sbenius integrability theorem; the integrability conditions we get are seen to be related to degeneracies in the corresponding gauge potentials.

With the conditions for the existence of noncurvature solutions for Bianchi-like equations available, we then turn to the second question: can we algorithmically detect whether an arbitrary connection form ),g~, satisfies those integrability conditions? We prove in Sec. 3 that even for a restricted

Geometry of Gauge Fields 785

algebra ~ of smooth functions over spacetime that can be explicitly represented by finite expressions in a first-order language for gauge field theory (as in Ref. 5), there is no general algorithm to test whether a given connection form has that kind of degeneracy. Moreover, if we axiomatize gauge field theory within a reasonable axiomatic system, ~s 7.9) we can show that there both exist a degenerate and a nondegenerate connection which are such that their degeneracy (or nondegeneracy) cannot be proved within the given axiomatization for gauge fields. (See also Ref. 26 on those questions.)

The present paper is dedicated to the memory of Professor Carlos Marcio do Amaral (1922-1988), teacher and friend, who introduced two of the authors (FAD and AFFA) to the beautiful world of Dirac-like equations and Clifford algebras in physics.

2. A D E G E N E R A C Y IN THE BIANCHI IDENTITIES

Our notation in the present Section is standard and follows Ref. 21. In particular, ix denotes the interior product w.r.t, the vector field X; L~' x is the Lie derivative w.r.t.X. Lowercase Greek letters will in general denote differential forms on prescribed manifolds, and we suppose our objects to have as much smoothness as required by our results.

2.1. Main Tools

Let V be a real finite-dimensional vectorspace; let V* be its dual, the space of all real-valued linear forms over V. Let A(V*) be the exterior algebra of V. We suppose given a pairing

( ' " l ' " ) : Vx V* ~ R (2)

linear in both factors, where ~ denotes the reals. For a linear subspace U_~ V, we define U ~ V* to be the (linear subspace) of all forms that annihilate U:

( u I u ~ (3)

An ideal I ~ A(V*) is the linear subspace such that given f le A(V*) and cte/ , /3 A ~teI (an ideal "incorporates" new objects via the exterior product). A set F c I is a generator for I if every element /3 of I can be written as /3=etA T, c~eA(V*) and ) ,eF. The ideal I is said to be homogeneous if FcAr(V), r ~ < n = d i m V, fixed. For a review of these notions see, 25, Chap. I. We state:

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786 da Costa et al.

Proposition 2.1. U c V be a subspace. Then I has generators in A(U ~ if and only if

Vet ~ L ixcr ~ I

for all x e U.

Let I ~ A ( V * ) be a homogeneous ideal, and let

(4)

Proof See Ref. 25, p. 28. []

Remark 2.2. Two different ideals may determine the same U ~ Let us be given R 4, and four linearly independent 1-forms eti, i = 1 ..... 4, on R 4. One can, for instance, check that the ideal I generated by et~ A et3, et~ A et4, and et~ A etz_et3 A et4 determines the same U ~ as the ideal I ' generated by etl A et2, etl A (3{3 etl A et4, and et3A 0(4. Yet, I and I ' are different (see Ref. 25, p. 27). []

From now on, let M be an n-dimensional Ck-differentiable real manifold, k>n; let TM and T*M be the corresponding tangent and cotangent bundles, and let A(M) be the exterior bundle over M. The preceding definitions extend to M and its bundles. Given a homogeneous ideal I~_A(M), we put

D / = {Xe TM: Vet ~ I, ixet e I} (5)

For our purposes here, we say that I is nonsingular if and only if dim I is constant over a regular open set A ~ M [A is regular open if and only if A = interior(closure A)]. Our main tools are a consequence of the two versions of the Frobenius Theorem ([25] , p. 132 and [16], p. 97). Let D ~_ TM be a p-dimensional subbundle, p ~< n. We say that D is integrable if D = TN, the tangent bundle of N, where N _ M is an immersed submanifold. Then,

Proposition 2.3 (Frobenius' Theorem, First Version). D ~ TM is integrable if and only if D is a Lie algebra, that is, if and only if for X, YeD, [X, Y]eD.

Proof See the references. []

Form D~ T ' M , the set of all differential forms that vanish on D, and get the ideal I(D ~ generated by D o in A(M). If d is the exterior derivative on A(M), we say that I(D ~ is a differential ideal if and only if dl(D ~ ~ l(D~ Then,

Proposition 2.4 (Frobenius' Theorem, Second Version). D is integrable if and only if I(D ~ is a differential ideal.


P r o o f See the references. []

On a local chart, if the r = n - p linearly independent forms o9i~ D O are the idears generators, the differential condition for the ideal is equivalent to the existence of l-forms 0~ such that

do) i = 0.i. A r j, i, j = 1, 2 .... r (6)

Corollary 2.5. D is integrable over a local chart U ~ M if and only if there are functions u~,/~J, and u = (u~) invertible over U such that

oY= UI" d~ j (7)

P r o o f See Ref. 16, p. 97. []

We notice that Eq. (6) is a kind of vanishing covariant differential condition.

Up to this point, everything is pretty standard. We now state and prove a result by Cartan that extends the Fr6benius theorem to a more general situation:

Proposition 2.6 (Cartan). If I~_ A ( M ) is nonsingular, homogeneous, and differential, then D 1 is integrable.

Also,

P r o o f We need a lemma:

Lemma 2.7. I f ~ A ( M ) ,

i tx , v ] o t = d ( [ i x , i v ] ~ ) + i x d i v ~ - i v d i x o t

P r o o f We have

i t x. v]~ = -~xi v~ - i v ~ x ~

(8)

. ~ x ~ = d i x u + i x du

Then,

i t x ' v]~ = d(ixiyCX) - d( iyixOt) + i x d iv ~ - i y di x ot []

Now back to the proof of the theorem. Notice that D t is a linear space (as a consequence of the linearity of ix) . Let now X 1 ..... Xp span D t. Then, from Eq. (5), i x k l c L k = 1 ..... p. For ceEL

ic x,. xj] cr = d( [ i xi, i xj] r ) + ix, di xj ~ - i xj di x~ ot

788 da Costa et aL

Clearly the Lie product [X b, Ark] e D +, as each term in the sum above belongs to L []

To conclude, let now F~_Ar(M), r fixed, be a set of generators for I(F), and let the r-forms of, i = 1 ..... p span F. Then,

Lemma 2.8. The homogeneous nonsingular ideal I(F) is differential, that is, dI(F) c I(F), if and only if there are differentiable 1-forms 0j such that

do)i= O} A O)J (9)

Proof Sufficiency: For an c~ �9 A(M), write ~ = ~ ^ co( Then,

As we have

we get

d~=do~ A o ) i + ( - - l)deg~t 0~ A do) i

do) i = Oj A O)J

d~=dot A o)i+ ( _ l)dCg~ A O~ ^ 09Jel

Our reasoning extends to arbitrary ~ by iinearity. Thus, sufficiency is proved.

For necessity, as dI c L given ~ ~--- ~ A (s d~ = d0[ A (2) i --~

( - -1 ) acg '~A do)( As dr we must have ct ^ do)i~I. There are two alternatives here: either ct e I or not. If ct e / , d~ �9 I trivially. If not, do) i �9 L Thus, do) i must be a linear combination of products of a 1-form with the ideal's generator, that is,

do)i= 0~ A o)J (10) []

2.2. Noncurvature Solutions for the Differential Bianchi Identities

We are given the following geometrical setting for our main problem in the present subsection: let M be a real n-dimensional differentiable manifold, with a Ck-structure, k > n (M will be our basis manifold). Let G be a finite-dimensional semisimple Lie group. W form the principal bundle P(M, G), with G as the fiber and bundle group, and x: P(M, G) --* M as the canonical projection. We will also suppose several associated vector bundles over M. We endow P(M, G) with a connection; thus, we are given


a Ck-decomposition TP = Y/ ~ .Jcd of P(M, G)'s tangent bundle into a vertical space ~r and a horizontal space ~fr the connection form 7 annihilates the horizontal space, 7 ( ~ ) = 0 . We will consider tensorial forms on P(M, G). In our case, a L(G)-valued tensorial form on P(M, G), where L(G) is the group's Lie algebra, is an r-form ct on P(M, G), r<~n, with values in L(G), and such that, given any vectorfields X, Y,... on the bundles,

~(X, Y,...)=7(hX, hY,...) (11)

where hX denotes X's component w.r.t, the horizontal space ~)~. Now we have: a differential Bianchi condition on an r-form as in Eq. ( 11 ) is given by

dcz(hX, hr,.. .) = 0 (12)

where d is the exterior derivative on P(M, G). We look for noncurvature solutions for Eq. (12). Locally, given a basis r U for the Lie algebra L(G), any L(G)-valued form can be written as ~(X,...)= ~(X, . . . )~ , , and Eq. (12) becomes

d~"= -7~ A ~b (13)

which is a condition similar to the one condition expressed by Eq. (6) where, however, ct" has now an arbitrary degree.

Our first problem can now be formulated as follows:

1. Are there connection forms 7 so that we can find at least a tensorial form ~ which cannot be obtained out of 7's curvature form over P(M, G) and which, however, still satisfies a Bianchi- like differential condition for 7, as the one in (13)?

2. If so, which properties do characterize those 7 and ~?

We must at first clarify what we mean by "a form that cannot be obtained out of 7's curvature." Let c t ( - ) denote a linear functional defined over the space Jr of all curvature forms on P(M, G) with values in the space of L(G)-valued tensorial forms on the bundle. We note the covariant exterior derivative operator with reference to y by D(7). Then,

Definition 2.9. We say that ~(12) is curvature-related if and only if O ~ J / i d e n t i c a l l y implies that D ( 7 ) a ( 1 2 ) - 0 . []

Remark 2.10. We can take a closer look at the set of curvature- related forms.

Let 0(7) be a curvature form on P(M, G) derived from the connection form 7, and let B(f2) be the linear algebra over R generated by ~2 and its

790 da Costa et aL

contractions, products, Lie-algebra products, and the like. Let K ( I2 ) c B(I2) be the restriction of B to a linear algebra of Lie-algebra-valued tensorial forms on P(M, G). An arbitrary element ~ of K is thus given by some linear functional ~(~). Obtain K ~ { ~ K : D(y)ct =0} . Now let y' be a different connection form on P, and let ~ ' be the corresponding curvature form. In general, even if D(7)~(g2)=0, we may have D(y')ct(g2'):~0. (Just think about a trivial case, then consider a nontrivial one.) However, there will be some functionals ~ such that the covariant exterior derivative D~-= 0, for any y and ~2. As an example, ~(g2)=kg2, k~R. Write

A(O) = {~ ~ K~ D(~/) ~(g2')--- 0, any y'}

A simple (but cumbersome) calculation shows that A(I2) = {kg2: k ~ • }; it suffices to examine all possible forms for the functionals ct, compute the corresponding covariant exterior derivativeds, and check for conditions that make them equal to zero; if nontrivial, we discard that ct. []

We then say:

Definition 2.11. A Lie-algebra-valued form at on P is nontrivial if and only if ~ ~ 0 and ct is not curvature-related. []

We are now going to answer the above two questions simultaneously. Let us at first suppose that there exists a tensorial r-form ~ on a

bundle P(M, G) endowed with a connection form y such that ~t satisfies Eq.(12) over a trivializing open domain 7t-I(U)=UxGcP(M,G), UcM. Let I(ct) be the ideal in A(P) generated by the 0t". Let us also suppose that the bundle's group G coincides with the connection's Ambrose-Singer holonomy group, generated by the curvature form 12 associated to ~.(21)

Then,

Lemma 2.12. If there are such ct", then l(ct) is a nonsingular homogeneous differential ideal in A(P).

Proof Immediate. The differential condition is a consequence of Eq. (13) together with Lemma 2.8. []

Lemma 2.13. If there are such 0t ~, then for I(~), ~e ~ D t.

Proof If fl is any form on P(M, G), given a generator ct ~ for I(~), if X is a vertical vector on P(M, G), we have

i x ( f l A O t a ) = i x f l A O t a + ( - - l ) d e g ~ " f l A i x C t a = i x f l A C t a (14)

which ensures [together with (5)] that any vertical X~D ~. []


Remark 2.14. As the preceding result is also trivially valid when ---0, we conclude that D t will always contain at least the vertical vector

fields ~e. []

Now notice that as D'___ ~V" O)3r we can ask about o,U = D t - ~r. We first define:

Definition 2.15.

1. I(0t) is trivial if and only if )~r = ~ .

2. I(ct) is nontrivial if and only if ~ r ~ . []

We must now show that there are nontrivial l(a), and relate them to our goal:

Proposition equivalent:

1.

2.

.

2.16 (Main Result I). The following conditions are

l(a) is nontrivial.

3s is horizontal, integrable, and ' ) , restricted to the integral manifold of o,~ is fiat.

There is a nontrivial, noncurvature related tensorial L(G)-valued r-form ct on P(M, G) that satisfies the differential Bianchi identities with respect to y.

Proof We will first show that Condition 1 implies Condition 2. Suppose that . ~ is nonempty. It is obviously a horizontal distribution. As I(~) is homogeneous, nonsingular, and differential, Jg is integrable. Let K be its integral submanifold in P. Then the restriction of y to K, y K, is a fiat partial connection, a~

Now we show that Condition 2 implies Condition 3. Since )'K is fiat, and as we are over a trivializing domain, we can write for a given Lie algebra basis, for 1-forms 0" that determine K through the Frobenius theorem:

dO"= - ~ ^ 0 b (15)

There is a gauge transformation g that sends 7K over 0, and the 0" are 0 = g ~ v , the g = (g~,) here in the adjoint representation, so mapped on the -b _b,~

that we have dOa= O. Call that gauge the zero gauge for )'K. Let A(O) be the real exterior algebra generated by the O ". Notice that (at the zero gauge) if ~ ~ A(O), then d~ = 0, due to linearity. If ~ e A(O), for several a, is a linear independent homogeneous set of r-forms from A(O), where

792 da Costa e t al.

r < dim(K), we then form 0~ = ~"-?a an L(G)-valued tensorial r-form, so that we have, at the zero gauge, dR = 0.

If we go back to the original gauge of Eq. (15), we get

do:"= - -~ ^ o:b (16)

We therefore reach Condition 3. From Condition 3 to Condition 1: immediate. []

We have therefore answered Question 1 above: we will have noncurvature solutions of arbitrary degrees for the Bianchi differential conditions precisely for the nontrivial I(o:), or when the connection form 7 splits into

=y~ 0)7 ~ and 70 is a fiat partial connection form that determines an integrable ~ , in the sense of Kamber and Tondeur/2~ Actually we will have a whole continuum of such noncurvature solutions within the annihilator of oU. It is also immediately apparent that such noncurvature, Bianchi-satisfying objects, are not curvature-related.

For examples of those degenerate connections that imply our phenomenon of Bianchi copies, see Refs. 12, 13, and 20.

Remark 2.17. If o: is a tensorial l-form, the space spanned by the o:" annihilates [in the sense of Eq. (3)] " / / @ ~ . Therefore, Proposition 2.16 allows us to assert that (locally at least) there is a gauge transformation that sends each o:a onto an exact form, that is, there is a (possibly local) gauge map g: P(M, G)I U ~ P ( M , G)I U, where U is open, U c M , so that

dfl=g-lo:g, fl a function (17)

The same clearly applies to r-forms, r > l , that satisfy Eq.(12). Moreover, they will all be expressed as (locally) exact L(G)-valued forms in the same gauge, which is the one where the flat portion of the connection goes to zero. []

The previous Remark answers our Question 2. To emphasize it:

Corollary 2.18. If I(o:) is nontrivial, then there is a gauge where (locally at least) we can write o:"= dfl", for all indices a. []

The counterpart of that result goes as follows:

Corollary 2.19. If I(ct) is trivial, then l(o:) = I(I2). []

Remark 2.20. In the case where there are no nontrivial I(o:), y has no flat portion, and the connection and curvature presents no degeneracies. See on that the discussion in Ref. 12. []


Let d be the manifold of all connection forms on P(M, G) with its usual Fr6chet topology. 1~21 Then:

Corollary 2.21. The set d ~ ~r of Bianchi-copied connection forms is of first category in ~/.

Proof. See Ref. 12. []

Remark 2.22. If we weaken our irreducibility condition for G, the preceding results allow us to include another situation where we have noncurvature solutions for the Bianchi identities: let G be a gauge group, let H c G be a proper subgroup, and let H ' c G-H. Let yn be a L(H)-valued connection form. If ~ is L(H')-valued, and (locally at least) ~ = d/L then trivially satisfies the Bianchi identities with respect to yH; however, we are in that case dealing with a reducible bundle group.

However, from the physical point of view, the noncurvature ~ thus obtained does not interact with the field given by 1'. []

For more on the possible physical interpretation of that degeneracy, a question that has been raised in the Introduction, see Sec. 4.

3. G O D E L I N C O M P L E T E N E S S IN G A U G E FIELD T H E O R Y

The second question that we have asked in the Introduction deals with a problem that will require some concepts from mathematical logic; we will talk about "formal systems," "formal languages," "first-order theories and languages," "algorithmic procedures," "models," and "the standard model for arithmetic." These concepts can be found in Ref. 24. However, we will only require an intuitive grasp of those concepts in what follows, as our main results here are straightforward consequences of Proposition2.16 together with Richardson's Theorem, stated in Proposition 3.1. The present section is a direct application to gauge field theories of a general undecidability and incompleteness theorem introduced elsewhere Refs. 6-8.

We have proved in the previous section a formal property about gauge fields. Now, if we are given a specific situation, can we examine the actual field we are handling in our theoretical constructs in order to see whether it satisfies that property? Can we ahvays check whether a given gauge field satisfies the degeneracy conditions spelled out in Proposition 2.16?

We can be still more specific: particular physical fields are usually explicitly written out (locally at least) with the help of the following algebra of m-variable functions ~[x~,..., x m] defined on an open domain in ~'", for an arbitrary m:


- - The algebra of partial functions (that is, not necessarily defined over the whole real line), ~r which includes m-variable real ele- mentary functions: polynomials, sines, cosines (and the number 7r); base e exponential; the absolute value function Ixl, which can appear through the expression Ixl= + x / ~ ; real constants restricted to the rational numbers Q to which we add 7t and e; and everything closed under finite sums, products, and function composition.

- - Rational functions: if f , g belong to the preceding algebra, we extend that algebra to include objects of the form fig.

- - Derivatives and quadratures: derivatives and functions of the form

q)(x) = f ( y ) dy

plus the corresponding proper and improper definite integrals, where x takes (in the proper case) real values a = f ( q ) , where q ~ Q and f is in the algebra.

- - The sign function:

, l (x ) = x / I x l , x ~ 0

r/(x)= 0, x = 0

Everything is then again closed under finite sums, products, and function composition. We abbreviate that algebra as ~m. Notice that it is a countable algebra of piecewise smooth functions which includes several smooth subalgebras.

Let us now suppose that we have formalized gauge field theory within a first-order theory T (5"6'1~ whose (denumerable) language is L~-with the help of a Suppes predicate. ~4"271 Moreover, we suppose that L~r~L~Ar, where L~r is the first-order language of arithmetic, t'-41 The actual form taken by that axiomatization will not really matter; we can take T to be, for instance, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), as in Ref. 5; for our purposes we only require that T include arithmetic and that T be arithmetically consistent, that is, T as a model M, and the standard model N c M for arithmetic is a model for arithmetic in T. (6)

We need two more concepts from the theory of formal systems: let be an arbitrary sentence in the theory T. If we can algorithmically decide whether any such ~ is a theorem of T, then T is a decidable theory. If not, T is an undecidable theory.

If, for every sentence ~ of T, either ~ or its negation is a theorem of


T, then T is a complete theory. If there is a ~ such that neither ~ nor its negation is a theorem of T, then T is incomplete.

The incompleteness of (formalized) arithmetic was proved by Kurt G6del in 1931 in a famous paper, t~8~ Our chief result in the present section is a direct consequence of G6del's First Incompleteness Theorem through Hilbert's Tenth Problem. t6)

Let ~ be an arbitrary algebra of functions. The set E ( ~ ) c L~-will be a set o f expressions for the functions in ~ if, given any f e ~ , there is at least one E ( f ) ~ E ( ~ ) that represents f Since L~- is a denumerable language, E ( ~ ) is denumerable. We also require that E ( ~ ) be decidable within L~-, that is, given a finite sequence of letters ~ in the alphabet of L~-, we must be able to algorithmically check whether ~ is in E ( ~ ) or not.

(For example, if we accept that derivatives are to be taken in the sense of distribution theory/9~ we have a finite expression in E ( ~ ) ~ for q,

rt(x) = d Ixl

so that we will not have to add a new symbol to represent that function to ther set of expressions for our algebra.)

We will derive our undecidability and incompleteness result here from a theorem that stems from a result by Richardson~6"8"z91:

Proposition 3.1. If T is arithmetically consistent, then:

1. Undecidability. For each natural number m we can explicitly and algorithmically construct the expression for a continuous, piecewise smooth function B,,(x) such that:

- - B , . ( x ) >i O.

- - Either there is a real x so that B.,(x) > O, or for all real x, Bo,(x) = O.

- - However, there is no general algorithm to check, for an arbitrary value of m, which option holds.

2. Incompleteness. There is an expression for a real-defined and real-valued continuous and piecewise smooth function B ( x ) ~ s l ~ such that it is true in the model M that B ( x ) = O, for all real x, but such that this fact cannot be derived from the axioms of T. []

That result leads to the following~6~:

Corollary 3.2. If T is arithmetically consistent, then we can explicitly


and algorithmically construct the expression for a smooth real-defined and real-valued function O,,,(x) so that:

1. For all real x, O,,,(x)= 0 if and only if there is no real x such that B,,,(x) > O.

2. For all real x, O, , (x )= 1 if and only if there is a real x such that Bm(x)>0.

Proof See Ref. 6. The succession of definite integrals,

f+OT., . --X" K(m) = B,,,(x) e dx

has finite values, for each m. We then write

O,,,(x) = O(m) = q(K, ,) (18) []

Corollary 3.3. If T is arithmetically consistent, then there is no general algorithm to decide, for an arbitrary m and given a countable, algorithmically generated set of expressions for smooth functions (re(X), whether

for all real x.

Proof Put

( , , (x ) = O,,(x)

~m(X) = O,,,(X) + B,,,(X)

where the B,, are given in Corollary 3.1. []

Thus, while we can explicitly and algorithmically obtain an expression for 0 .... we cannot in general decide whether an arbitrary expression for smooth functions in our formal language is such an "on-off" function.

3.1. Undecidability and Incompleteness in Gauge Field Theory

We can now prove our main theorem in the present section:

Proposition 3.4 (Main Result II). If T is arithmetically consistent then:

!. Given a principal bundle P(M, G), there is an expression for a connection form ~, of which it is true in the model M that the


.

.

associated ideal l(ct) is trivial, but such that this fact cannot be proved from the axioms of T.

Given a principal bundle P(M, G), there is an expression for connection form 2 of which it is true in M that l(ct) is nontrivial, but again such that this fact cannot be proved from the axioms of T.

There is a denumerable family of expressions for connection forms .... m a natural number, such that, for an arbitrary value of m,

there is no general algorithm to decide whether the corresponding ideal Im(Ct) is trivial or not.

Proof We are going to use the "incompleteness" portion of Proposi- tion 3.1, plus Corollary 3.2 and Eq. (18). We fix a given coordinate system on a local trivialization of the bundle given by U x G, and let in that coordinate system y0 be a connection form such that the corresponding ideal is trivial, and let 20 be another connection form on the same bundle such that the corresponding ideal is not trivial. Then form locally, as in Corollary 2.16,

y = ( 1 - B ( x Z ) ) ~ ' ~ ~

2 = B ( x l) yo+ (1 - -B(x l ) ) 2 0

where x I is a local coordinate on M, and B is given in Proposition 3.1. Then, clearly, V satisfies Assertion 1 in the Proposition, and 2 satisfies Assertion 2. For Assertion 3 we take,

y,, = (1 - 0,,) ~,o + 0,,2o

We are done. []

Remark 3.5. A parallel result can be obtained if we choose 7o to be irreducible while 2 ~ is reducible. Therefore, as an example, the characteriza- tion given by Ref. 17, pp. 54 and 55, for a reducible connection cannot be algorithmically implemented. []

4. DISCUSSION AND C O N C L U S I O N S

Thus, we have answered our interlocked questions with two different kinds of answers. The first question deals with a problem that can be


reduced to a "neat" geometrical formulation, and can be solved with the help of two classical integrability conditions: noncurvature solutions for the Bianchi differential conditions are seen to exist whenever the connection form on the bundle has a fiat portion. This is equivalent to the existence of a nontrivial integrable distribution (in the geometrical sense) in the horizontal bundle over P(M, G). If required, we can offer specific examples t12~ for those connections with a fiat portion.

We can even consider examples within particular physical theories for that kind of degeneracy, as we noticed that the vacuum Einstein equations ~22~ were shown to be equivalent in a spinorial formalism to the Bianchi identities. While unicity is ensured by the imposition that the linear connection on spacetime be a metric Christoffel connection, we see that even here we can find degenerate cases (e.g., plane gravitational waves) where there may be noncurvature objects that satisfy the Bianchi identities. The same applies to gauge fields in a more relaxed setting, as no metric condition exists in that case. For some exploration of those degeneracies which imply an example of classical symmetry breaking without a Higgs particle, see Refs. 1 and 2; for further examples, see Ref. 15.

On the other hand, the classification of the noncurvature forms that satisfy a differential Bianchi condition is obviously dependent on the basis manifold's De Rham cohomology, since the noncurvature Bianchi copies arise out of tensorial forms that are closed "downstairs," that is, over M or over a topologically nontrivial open set U in M. They do not geometrically behave as curvature forms, and as such it does not seems to make sense to try to develop, as we asked at the beginning, a sort of "parallel" anomaly theory for them. Their physical meaning is assuredly closer to their relation to the De Rham cohomology for M . I11't41

The second answer is of a different sort. We can provide examples of connection forms with the degeneracies described in Proposition 2.16, but given an arbitrary expression for a connection form on an open set U in M, an expression with only rational coefficients plus irrationals like n and e, square roots, sines, cosines and exponentials, derivatives and integrals - - t ha t is to say, an expression with the usual "terms" in the language of theoretical physics, we cannot in general algorithmically check whether the corresponding connection form is degenerate or not. Moreover, gauge field theory is seen to be incomplete in the metamathematical sense of the word, and clearly that result can be generalized to many other situations in physicst6); that appears to be a major obstacle in the actual business of computing results in physics.

To do physics we must compute---calculate something at the end of our theorizing. But what will happen if such a computation cannot be done at all?


5. ACKNOWLEDGMENTS

The authors wish to thank Fapesp, CNPq/Brazil, FINEP, the Fulbright Program, the Institute for Advanced Studies at the University of S~o Paulo, the Center for the Study of Mathematical Theories of Com- munication at Rio's Federal University, the Brazilian Center for Physical Research, and the Institute of Physics, again at Rio's Federal University, for partial support of the present work.

This work was completed while F.A.D. was a Senior Fulbright Scholar at Stanford University for the 1989-1990 term. F.A.D. wishes to thank Pro- fessor Patrick Suppes for his hospitality at the Institute for Mathematical Studies in the Social Sciences, Stanford University; he also acknowledges detailed criticisms and a conversation with Professor Morris Hirsch on related results presented in Refs. 6 and 7. Finally, interaction between the four authors was smoothly ensured by the BITNET system, to whose operators we are deeply indebted.

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Two questions on gauge fields

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