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On the canonical structure of gauge theories
Volume 81 B, number 1
PIIYSICS LETTERS
29 January 1979
ON THE CANONICAL STRUCTURE OF GAUGE THEORIES H.A. KASTRUP Institut fiir Theoretische Physik, RWTH Aachen, 51 Aachen, FR German), Received 7 November 1978
The usefulness of the general canonical theory for fields, as developed by Carathdodory, De Dondcr, Lepage and others, is illustrated in the case of abelian gauge fields.
In a previous communication [1 ] I pointed out the importance of the general canonical framework for field theories as developed by Carathdodory, De Donder, Weyl, Boerner, Lepage, E. H61der, Dedecker and others * 1. Despite its richness and power, only a very small fraction has been used by physicists, due to utter preoccupation with free fields and perturbation theory in terms of them! In this note I want to point out that the formalism has interesting applications to gauge fields, in addition to those mentioned in ref. [1]. Following Cartan [ 2 - 5 ] , it is very helpful to interpret the s o l u t i o n s f a ( x ) , a = 1 .... , n, x = (x 1 , ..., x m) of field equations in geometrical terms: The n functions f a ( x ) of m variables xU, la = 1, ..., m , describe a m-dimensional submanifold F m C R m+n in a (m + n)dimensional space. The different equations are to be interpreted as conditions on the tangent vectors o f the submanifolds F m . Locally there exist coordinate systems for F m in which the tangent spaces in a neighbourhood o f p = (x, z) G F m are spanned by the vector fields x ~ ( p ) = a~ + v ~ ( p ) a a,
a. = alax.,
a a = alaza.
According to Cartan it is preferable to pass from the vector fields X u ( p ) to those (dual) differential forms u.~a(p) = dz a - v~dxU which "annihilate" the vector fields X u : ~ a ( x u ) = 0 for all a and tt! The forms coa span an ideal I{~o a) in the algebra o f exterior forms in R m +n. Frobenius' integrability criterium • 1 The detailed refs. of their work are given in ref. [ 1].
for the "differential system" S = { X u } , namely that X u , X v E S implies [X u, Xv] E S for their commutator, is equivalent to the property that the forms dw a belong to the ideal spanned by the forms w a [6]. The use o f the forms w a is at the heart o f Lepage's general formulation of the canonical framework for fields, which contains De Donder's and Carathdodory's theories as special cases. Up to now physicists have used only De Donders's canonical framework (without naming him), which - I think - is a great loss, as examples show [7]. In the following I want to give additional illustrations how gauge theories fit very naturally into the more general framework and how, for instance, "anomalies" [8] are special examples! In order to be more specific, I shall discuss field theories in Minkowski space, however, most o f the results hold in any dimension >~ 2, and for euclidean spaces, too. I use the metric x 2 = (x0) 2 - x 2. A field theory may be defined by giving the lagrangian form w = L(x, z a, v~) d x 0 ^ ... A d x 3 , /a = 0, ..., 3, a = 1, ..., n, where L is a function on G 4+n X R 4n , G 4+n c R 4+n. One is interested in such 4- (or less) dimensional submanifolds E 4 C G 4+n for which z a = f a ( x ) , v u ~ u f a ( x ) and which make the integral fco stationary, subject to certain boundary conditions. The crucial observation of Lepage is the following: As far as the "extremals" E 4 are concerned the form ~o is only one representative o f a class of forms ~ defined in G 4+n X R 4n , because the forms o.,"a vanish on E 4, where z a = f a ( x ) , o~ = a u f a ( x ) . A general repre29
Volume 81B, number 1
PHYSICS LETTERS
sentative is ~2(x, z, v) = L(x, z, o) dx 0 ^ ... ^ dx 3 +h~acoa A d3X u +(l[4)hVa~coa^ cob ^ d2Su v +(l[3!)h#i;abc coa A cob ^ coc ^ dx#i + (l/4!)habeeco a A cob ^ coc ^ tOe, 3! d3Z#i = e#ivoodXV ^ d x ° ^ d x °, 2d2S#iv = e#ivoodx° ^ d x °. The coefficients h#i,a h#ivab= - ha~ = -h~ba etc. in general can be arbitrary functions of x, z and va. The #i a Hamilton-Jacobi condition d ~ = 0 mod (co } determines only the coefficients haU:haU = OL/Ov u =- nau. This has far reaching consequences for the definition of the canonical momenta [ 1] : The Legendre transformation is defined as a change of bases: co a ~ dz a = coa + v~dx#i, dxU --->d x u in the space of 1-forms in G 4+n, the canonical momenta paU are identified as the resulting coefficient o f dz a ^ d 3 Z#i in Q and the invariant Hamilton-function H ( x , z, p) is the resulting negative coefficient of dx 0 ^ ... A dx 3 ! If all coefficients ha~ etc. vanish, then pau = rrau, as usual. However, suppose that some h ~ :# 0, but that all higher terms vanish. Then =-HdxOA
... A d x 3 +pau dz a ^ d 3 Z u
+ "~lha b#ivd;,aAdzb .
d2Suv,
H = TrUv a - L - hUa~vao b, a #i o #i v
(1) #i
Pa =W~a
_ ht.W vb ab v"
If the functions h ~ are independent of vu, a then the condition for the Legendre transformation to be regular is det [(02 L /Ooauoob ) - haUff] 4= 0
(2)
Before we apply this to gauge theories let me add some general remarks: An important special case occurs, if the 4-form O = ( 1 / 4 ) h ~ dz a ^ dz b ^ dZsuv is closed, dO = 0, on G 4+n. This happens if the coefficients are constants. We then have, e.g., O = d(¼hUa~zadzb ^ dZSuv), i.e., by Stoke's theorem, an. integral f ® can be transformed into a surface integral. The discovery of such "anomalous" terms goes back to Clebsch (!) [9] and they have been discussed by 30
29 January 1979
several authors in connection with the so-called "second variation" [ 1 0 - 1 4 ] . Another way of expressing the above properties is the following [13] : Def'me the a a differential operator d/dx#i = Ou + v~Oa + v~O/Ov v, v #iV a = v Va#i , o n G 4+n X R .4 n , t h e n , because o f h ~ . uu #ira #iv = - h av~ , one has habV#iv vo = hab d(z a o vo ) / d x #i . If, in addition, d(ha~)/dx#i = 0 (which is the case if the h ~ are constants), then hVa~v~vb is a total divergence: we can define a new, equivalent lagrangian, L* = L t,#iv.a,,b which gives the same canonical momenta - ,abV#it,v, pav as in eq. (1), but the old Euler-Lagrange equations. Let us apply these general properties to gauge theories. In order to stress the essential points I confine myself to the abelian case. However, most results hold in nonabelian theories, too (in that case the coefficients ha~ contain the structure constants of the internal Lie groups). We now have z ~ = AC'(x), a = O, 1,2, 3, ~ = O#IA 'x. A simple choice for hUt'a¢is Xeuva#, where euvat 3 is completely antisymmetric and e0123 = +1, ~, = const. We obtain hUjoOuAaOvAO =Xem,,,t30 #iA ~ 0 vA ~ = - 2 Xt *F#ivFW,
(3) •
1 -
L~,oo
Fur =2e#ivoo "
,
F v = OuA v - OvA#i,
which is just the well-known "anomaly" [8]. Since #iv the h~t 3 are constants, our general discussion above gives the usual expression for the anomaly as a total divergence. Another interesting application is the following: The requirement that the coefficients hUa~have to be antisymmetric in their upper and lower indices, respectively, can be fulfiUed by taking for hUa~ the components R#iVc,¢ of a riemannian curvature tensor! Let us take the special case of constant curvature. We then have =
- g¢ ga)"
(4)
If we start with a lagrangian which has for the electromagnetic field the kinetic term Lena = - ( 1 ~4)Fur FUr, then eqs. (1) give the canonical momenta pUa = - F # i
+ X(gUag; - g~gV)OvA~
= - F U a - XbaA#i + Xg~OvA v, #i = 3Xa#iA#i. P#i Whereas the Legendre transformation a#iA ~ ~ pg is
Volume 81B, n u m b e r 1
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29 January 1979
For the "dynamical" momenta pO, j = 1,2, 3, and p0 we therefore have
The most beautiful canonical theory for fields is that of Carath6odory! In a sense [1], it is the only unique extension of the canonical structures in mechanics (including the Hamilton-Jacobi theory for wave fronts, transversal to the extremals), despite its algebraic complexities! It is defined by the only 4form f2 c which has the smallest rank possible, namely 4:
pO = OoA/'
~c =(ILL3) aO ^ al ^ a2 ^ a 3 ,
singular for 2, = 0, this is not so, if e.g. ?, = 1. In that case we get
pUa : -3UA 3u
ACt =
+gUavAS, et
(5)
1 apV
-Pu -3g~
v"
pO = 3.A/'
(6)
i.e. p0 depends only on the spatial components A/. (Notice that we have not yet imposed any gauge condition!) In the case of a free electromagnetic field, where A 0 = - A - 1 3 0 3 j A / (dividing by differential operators is defined by Fourier transformation), the quantization [p0(x), Ak(y)] x o_. ] - y o = - i 6 k s ( x - y ) is equivalent to the 2-point function
a u = L dx ts + w a rraU = _ T
~v
dxv + n~dfa a
(7)
T u P = vaTr u _ 6UL. t) a I) Here, all coefficients hUa~..... habce are determined! The Legendre transformation coa ~ dfa yields
Pau = _L-3TUvna,
H = - L - 3 1 T U v l (8a),
(8b)
(01AU(x)A~(y){O)
= r" p ; ,
=i(gUV
nUaV +nV3 v) n" 3
n=(nU)=const.,
D (+)(x - y),
n.n=+l.
D<+)(x): ~2~-~f d4k O ( k o l 6 ( k 2 ) e - i k x . According to Strocchi and Wightman [15] the theory defined by the above 2-point function is unitary equivalent to the Gupta-Bleuler gauge: In the notation of their proposition 2.2• we have fur = -nu k J n "k with fuPfvo = 0, since k 2 = 0. We still have to impose a gauge condition in order to comply with the two physical degrees of freedom. This can be done, for instance, by taking the axial gauge A 3 = 0. To the best of my knowledge the above quantization of the electromagnetic field has not yet been used• Using the ideal I (coa}, one can give a very nice interpretation o f gauge invariance: The forms co'~ = dz '~ - ogOt dx /./ are generally defined on G 4+4 Let f(x) be a smooth function o f x E M 4 alone. Then each ~c, is invariant under the transformation z c~-~ z c' + 3~'f(x), v~ ~ v~ + 3 u 3%f(x), that is to say, these "gauge" transformations form a very simple automorphism of the ideal I {co'~}. Notice that this automorphism exists already for nonvanishing w a, i.e. not only on the extremals, and that the form ~ is gauge invariant if ~o gu = L dx 0 ^ ... ^ d x 3 and the coefficients ha0, ... have this property, even if the individual canonical momenta pau are gauge dependent!
where ITUvl = det(TU~), TVu : algebraic complement of TU p • Notice that the Hamilton function H i s essentially the characteristic polynomial of the matrix (v~n~) - or the matrix (vaTr~), see ref. [1] - , with the lagrangian L as variable, which, in turn, is a function of of! That is to say, Carath6odory's canonical theory a v makes use of all invariants of the matrix (vuTra) - or a p (odrb) - , not just its trace, like in the case of De Donder-Weyl. By doing so, it is able to realize the minimal rank 4 of f2 c which is so important for the integrability properties of the transversal wave fronts [1]! Before I come back to electrodynamics let me add a few remarks concerning the quantization procedure in Carath6odory's theory, as indicated briefly in ref. [1 ] : The basic Hamilton-Jacobi relation of G 4÷n, where Pff = ~Ua(x, z), ~e(X,Z) = L - 3 a 0 ^ ... ^ a 3 = - H - 3 b O ^ ... ^ b 3 = d S 0 ^ . . . AdS 3,
S u=Su(x,z),
b u = _ H d x u +PUadza , yields the equations
I3vSUl + H ( x , z , p = ~(x, z)) = 0,
(9a)
pPa OoS u + HaaSU = 0,
(9b) 31
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i.e. we have one partial differential equation (9a) for four functions SU(x, z). So we can choose three o f them, say Si, ] = 1, 2, 3, freely, provided the "transversality" relations (9b) and the integrability conditions following from the requirement that vg(x, z = f(x)) = bufa(x) on E 4 are fulfilled• The functions S](x, z) are essential for the canonical H61der transformation [1]:x 0~0 =x 0,x]~.~/=ol=sl(x,z),z a~i a = z a,,which casts the dynamical equations at (x 0, a )
into mechanical form, because/~0 = if0 = p0,/~/a = 0, ~al = 0. The basic commutators [/30 (x 0 = 0, n), f b ( 0 ) ] = (l/i)fib 8(~) in the I tSlder frame take the form
TO 0 [paO(xO = O'x)'fb(o)l = (1/i)Hfba 6(x)'
det(pUv)/H3 = ~6 (* FuvF~)2 /L 3,
(10)
(11)
= [(E 2 - B2) 2 + 4 ( E ' B ) 2 - 4(]aAa) 2] ]4L. Naively one would expect only the first two terms, but the additional third term is typical for Carath6odory's theory: it involves all available invariants! 32
If the external current Ju vanishes, then H is gauge invariant. We obtain the same expression for H if we replace the determinant of the (gauge dependent) canonical energy-momentum tensor by the determinant of of the (gauge invariant) symmetrical energy-momentum tensor 0 uv = -FUC'F~ +-~gUVFe,¢3F e'~. In order to find H a s a function o f A ~ and p~, we have to express the two invariants o f ( F w ) in terms of the pav. Defining p~V = pro, +_pVU, where pW = gUapV, we obtain the relations
*pu~pU~/H = *FuvFUV [ ~g(* FuvFUV)2 - 4/aAaL ] /L 3,
in the original frame. Since [13],fb] = 0, because/3 ] = 0, one expects [p~(x 0 = O,x~,fb(o)] = 0 in general, and therefore TO.. [pl(x O, x), f b (0)] = 0 Combining 0 this• with eq. (10)j anda rraH= TOupUa (see" eqs. (8b)) we obtain, at least formally, the usual quantization rule [Tr0(x 0 = O, x),fb(o)] = (1/i)5abS(x)! Thus, our quantization procedure in the H61der frame is formally compatible with the standard one! However, beyond this formal relationship, important things may happen: H or L, or both, might vanish or have singularities [1,7], the above products of operator-valued distributions are iU-def'med etc.! Only specific examples can tell. Let us return to electrodynamics with the lagrangian L = (-1/4)FtwFUV +]aA a and let us suppose, for a moment, that the two invariants F u v F ~ = - 2 ( E 2 - B 2) and * F ~ F *w = 4 ( E ' B ) 2 do not vanish. It is then possible and convenient to perform the kegendre transformation in a system where E and B are parallel and in which only F01 and F23 are non-vanishing [16,17]. In this system one can choose the gauge A 0 = 0, A 3 = 0 (the relation F = dA, A = A u dxU holds in any reference frame because the operator " d " commutes with mappings [6]). In this way we obtain the Hamilton function [1 ]
H = - z1F u v F uv - jcxA a + (*FuvFUV)2/l 6L,
29 January 1979
det (p +)/H 2 = ( ¼)6 (* FuvFUV)6 /L 6 ; (P-)uv(P- )tw/H = (*Fur F ~ ) 2 /4 L 2 for ]a = 0 only. They give
H= [(p_)uv(p_) u~] 3 /4(*p~pUV)2 for/a = 0, and
H 5/3 det l/6(p+)[detl/Z(p+) --¢1 . PuvP p v ] = 4jaAa det(pvU), otherwise. (The last expression, which uses only the determinants of the matrices (p) and (p+), is quite likely not the most convenient ones!) Like in ref. [1 ] we see again that the possibility of performing the Carathdodory-Legendre transformation hinges on the non-vanishing of *FuvFUV and L. For plane waves it does not exist. The associated singularities are central for the "infrared" problem [1 ] !
References [1] H.A. Kastrup, Phys• Lett. 70B (1977) 195. [2] E. Cartan, Lecons sur les invariants int~graux (Hermann, Paris, 1922), Les syst6mes diff~rentiels ext&ieurs et leurs applications g~om~triques (Hermann, Paris, 1945). [3] E. Kahler, Einfllhrung in die Theorie der Systeme yon Differentialgleichungen (Chelsea, New York, 1949). [41 M. Kuranishi, Amer. J. Math. 79 (1957) 1. [5] J. Dieudonn~, Treatise on analysis, Vol. IV (Academic Press, New York and London, 1974) Ch. 18. [6] E~g. F. Warner, Foundations of differentiable manifolds and Lie groups (Scott, Foresman and Co., Glenview, 1971). [7] H.A. Kastrup, Phys. Lett. 77B (1978) 203; 78B (1978) 433.
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[8] S.L. Adler, Phys. Rev. 177 (1969) 2426; Perturbation theory anomalies, in: Lectures on elementary particles and quantum field theory, Vol. 1, ed. St. Deser, M. Grisaru and H. Pendleton (M.I.T. Press, Cambridge, MA, 1970) p. 1, these lectures contain further references. [9] A. Clebsch, J. Reine Angew. Mat. 56 (1859) 122. [10] M. Hadamard, Bull. Soc. Math. France 30 (1902) 253, 33 (1905) 73. [11] F.J. Terpstra, Math. Ann. 116 (1938) 166. [12] M.R. Hestenes and E.J. McShane, Trans. Am. Mat. Soc. 47 (1940) 501.
29 January 1979
[13] R. Debever, Bull. Soc. Roy. Sci. Liege 10 (1941) 584. [14] L. Van Hove, Sur le signe de la variation seconde des int~grales multiples ~ plusieurs fonctions inconnues. Acad. Roy. Belg. Cl. Sci., Mere. Coll. in-8 o, t. 24, fasc. 5 (Bruxelles, 1949). [15] F. Strocchi and A.S. Wightman, J. Math. Phys. 15 (1974) 2198, 17 (1976) 1930. [16] Ch.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (W.H. Freeman, San Francisco, 1973) § 20.6. [ 17[ H.P. Robertson and Th.W. Noonan, Relativity and cosmology (W.B. Saunders, Philadelphia, 1969) Ch. 4.3.
33
PIIYSICS LETTERS
29 January 1979
ON THE CANONICAL STRUCTURE OF GAUGE THEORIES H.A. KASTRUP Institut fiir Theoretische Physik, RWTH Aachen, 51 Aachen, FR German), Received 7 November 1978
The usefulness of the general canonical theory for fields, as developed by Carathdodory, De Dondcr, Lepage and others, is illustrated in the case of abelian gauge fields.
In a previous communication [1 ] I pointed out the importance of the general canonical framework for field theories as developed by Carathdodory, De Donder, Weyl, Boerner, Lepage, E. H61der, Dedecker and others * 1. Despite its richness and power, only a very small fraction has been used by physicists, due to utter preoccupation with free fields and perturbation theory in terms of them! In this note I want to point out that the formalism has interesting applications to gauge fields, in addition to those mentioned in ref. [1]. Following Cartan [ 2 - 5 ] , it is very helpful to interpret the s o l u t i o n s f a ( x ) , a = 1 .... , n, x = (x 1 , ..., x m) of field equations in geometrical terms: The n functions f a ( x ) of m variables xU, la = 1, ..., m , describe a m-dimensional submanifold F m C R m+n in a (m + n)dimensional space. The different equations are to be interpreted as conditions on the tangent vectors o f the submanifolds F m . Locally there exist coordinate systems for F m in which the tangent spaces in a neighbourhood o f p = (x, z) G F m are spanned by the vector fields x ~ ( p ) = a~ + v ~ ( p ) a a,
a. = alax.,
a a = alaza.
According to Cartan it is preferable to pass from the vector fields X u ( p ) to those (dual) differential forms u.~a(p) = dz a - v~dxU which "annihilate" the vector fields X u : ~ a ( x u ) = 0 for all a and tt! The forms coa span an ideal I{~o a) in the algebra o f exterior forms in R m +n. Frobenius' integrability criterium • 1 The detailed refs. of their work are given in ref. [ 1].
for the "differential system" S = { X u } , namely that X u , X v E S implies [X u, Xv] E S for their commutator, is equivalent to the property that the forms dw a belong to the ideal spanned by the forms w a [6]. The use o f the forms w a is at the heart o f Lepage's general formulation of the canonical framework for fields, which contains De Donder's and Carathdodory's theories as special cases. Up to now physicists have used only De Donders's canonical framework (without naming him), which - I think - is a great loss, as examples show [7]. In the following I want to give additional illustrations how gauge theories fit very naturally into the more general framework and how, for instance, "anomalies" [8] are special examples! In order to be more specific, I shall discuss field theories in Minkowski space, however, most o f the results hold in any dimension >~ 2, and for euclidean spaces, too. I use the metric x 2 = (x0) 2 - x 2. A field theory may be defined by giving the lagrangian form w = L(x, z a, v~) d x 0 ^ ... A d x 3 , /a = 0, ..., 3, a = 1, ..., n, where L is a function on G 4+n X R 4n , G 4+n c R 4+n. One is interested in such 4- (or less) dimensional submanifolds E 4 C G 4+n for which z a = f a ( x ) , v u ~ u f a ( x ) and which make the integral fco stationary, subject to certain boundary conditions. The crucial observation of Lepage is the following: As far as the "extremals" E 4 are concerned the form ~o is only one representative o f a class of forms ~ defined in G 4+n X R 4n , because the forms o.,"a vanish on E 4, where z a = f a ( x ) , o~ = a u f a ( x ) . A general repre29
Volume 81B, number 1
PHYSICS LETTERS
sentative is ~2(x, z, v) = L(x, z, o) dx 0 ^ ... ^ dx 3 +h~acoa A d3X u +(l[4)hVa~coa^ cob ^ d2Su v +(l[3!)h#i;abc coa A cob ^ coc ^ dx#i + (l/4!)habeeco a A cob ^ coc ^ tOe, 3! d3Z#i = e#ivoodXV ^ d x ° ^ d x °, 2d2S#iv = e#ivoodx° ^ d x °. The coefficients h#i,a h#ivab= - ha~ = -h~ba etc. in general can be arbitrary functions of x, z and va. The #i a Hamilton-Jacobi condition d ~ = 0 mod (co } determines only the coefficients haU:haU = OL/Ov u =- nau. This has far reaching consequences for the definition of the canonical momenta [ 1] : The Legendre transformation is defined as a change of bases: co a ~ dz a = coa + v~dx#i, dxU --->d x u in the space of 1-forms in G 4+n, the canonical momenta paU are identified as the resulting coefficient o f dz a ^ d 3 Z#i in Q and the invariant Hamilton-function H ( x , z, p) is the resulting negative coefficient of dx 0 ^ ... A dx 3 ! If all coefficients ha~ etc. vanish, then pau = rrau, as usual. However, suppose that some h ~ :# 0, but that all higher terms vanish. Then =-HdxOA
... A d x 3 +pau dz a ^ d 3 Z u
+ "~lha b#ivd;,aAdzb .
d2Suv,
H = TrUv a - L - hUa~vao b, a #i o #i v
(1) #i
Pa =W~a
_ ht.W vb ab v"
If the functions h ~ are independent of vu, a then the condition for the Legendre transformation to be regular is det [(02 L /Ooauoob ) - haUff] 4= 0
(2)
Before we apply this to gauge theories let me add some general remarks: An important special case occurs, if the 4-form O = ( 1 / 4 ) h ~ dz a ^ dz b ^ dZsuv is closed, dO = 0, on G 4+n. This happens if the coefficients are constants. We then have, e.g., O = d(¼hUa~zadzb ^ dZSuv), i.e., by Stoke's theorem, an. integral f ® can be transformed into a surface integral. The discovery of such "anomalous" terms goes back to Clebsch (!) [9] and they have been discussed by 30
29 January 1979
several authors in connection with the so-called "second variation" [ 1 0 - 1 4 ] . Another way of expressing the above properties is the following [13] : Def'me the a a differential operator d/dx#i = Ou + v~Oa + v~O/Ov v, v #iV a = v Va#i , o n G 4+n X R .4 n , t h e n , because o f h ~ . uu #ira #iv = - h av~ , one has habV#iv vo = hab d(z a o vo ) / d x #i . If, in addition, d(ha~)/dx#i = 0 (which is the case if the h ~ are constants), then hVa~v~vb is a total divergence: we can define a new, equivalent lagrangian, L* = L t,#iv.a,,b which gives the same canonical momenta - ,abV#it,v, pav as in eq. (1), but the old Euler-Lagrange equations. Let us apply these general properties to gauge theories. In order to stress the essential points I confine myself to the abelian case. However, most results hold in nonabelian theories, too (in that case the coefficients ha~ contain the structure constants of the internal Lie groups). We now have z ~ = AC'(x), a = O, 1,2, 3, ~ = O#IA 'x. A simple choice for hUt'a¢is Xeuva#, where euvat 3 is completely antisymmetric and e0123 = +1, ~, = const. We obtain hUjoOuAaOvAO =Xem,,,t30 #iA ~ 0 vA ~ = - 2 Xt *F#ivFW,
(3) •
1 -
L~,oo
Fur =2e#ivoo "
,
F v = OuA v - OvA#i,
which is just the well-known "anomaly" [8]. Since #iv the h~t 3 are constants, our general discussion above gives the usual expression for the anomaly as a total divergence. Another interesting application is the following: The requirement that the coefficients hUa~have to be antisymmetric in their upper and lower indices, respectively, can be fulfiUed by taking for hUa~ the components R#iVc,¢ of a riemannian curvature tensor! Let us take the special case of constant curvature. We then have =
- g¢ ga)"
(4)
If we start with a lagrangian which has for the electromagnetic field the kinetic term Lena = - ( 1 ~4)Fur FUr, then eqs. (1) give the canonical momenta pUa = - F # i
+ X(gUag; - g~gV)OvA~
= - F U a - XbaA#i + Xg~OvA v, #i = 3Xa#iA#i. P#i Whereas the Legendre transformation a#iA ~ ~ pg is
Volume 81B, n u m b e r 1
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29 January 1979
For the "dynamical" momenta pO, j = 1,2, 3, and p0 we therefore have
The most beautiful canonical theory for fields is that of Carath6odory! In a sense [1], it is the only unique extension of the canonical structures in mechanics (including the Hamilton-Jacobi theory for wave fronts, transversal to the extremals), despite its algebraic complexities! It is defined by the only 4form f2 c which has the smallest rank possible, namely 4:
pO = OoA/'
~c =(ILL3) aO ^ al ^ a2 ^ a 3 ,
singular for 2, = 0, this is not so, if e.g. ?, = 1. In that case we get
pUa : -3UA 3u
ACt =
+gUavAS, et
(5)
1 apV
-Pu -3g~
v"
pO = 3.A/'
(6)
i.e. p0 depends only on the spatial components A/. (Notice that we have not yet imposed any gauge condition!) In the case of a free electromagnetic field, where A 0 = - A - 1 3 0 3 j A / (dividing by differential operators is defined by Fourier transformation), the quantization [p0(x), Ak(y)] x o_. ] - y o = - i 6 k s ( x - y ) is equivalent to the 2-point function
a u = L dx ts + w a rraU = _ T
~v
dxv + n~dfa a
(7)
T u P = vaTr u _ 6UL. t) a I) Here, all coefficients hUa~..... habce are determined! The Legendre transformation coa ~ dfa yields
Pau = _L-3TUvna,
H = - L - 3 1 T U v l (8a),
(8b)
(01AU(x)A~(y){O)
= r" p ; ,
=i(gUV
nUaV +nV3 v) n" 3
n=(nU)=const.,
D (+)(x - y),
n.n=+l.
D<+)(x): ~2~-~f d4k O ( k o l 6 ( k 2 ) e - i k x . According to Strocchi and Wightman [15] the theory defined by the above 2-point function is unitary equivalent to the Gupta-Bleuler gauge: In the notation of their proposition 2.2• we have fur = -nu k J n "k with fuPfvo = 0, since k 2 = 0. We still have to impose a gauge condition in order to comply with the two physical degrees of freedom. This can be done, for instance, by taking the axial gauge A 3 = 0. To the best of my knowledge the above quantization of the electromagnetic field has not yet been used• Using the ideal I (coa}, one can give a very nice interpretation o f gauge invariance: The forms co'~ = dz '~ - ogOt dx /./ are generally defined on G 4+4 Let f(x) be a smooth function o f x E M 4 alone. Then each ~c, is invariant under the transformation z c~-~ z c' + 3~'f(x), v~ ~ v~ + 3 u 3%f(x), that is to say, these "gauge" transformations form a very simple automorphism of the ideal I {co'~}. Notice that this automorphism exists already for nonvanishing w a, i.e. not only on the extremals, and that the form ~ is gauge invariant if ~o gu = L dx 0 ^ ... ^ d x 3 and the coefficients ha0, ... have this property, even if the individual canonical momenta pau are gauge dependent!
where ITUvl = det(TU~), TVu : algebraic complement of TU p • Notice that the Hamilton function H i s essentially the characteristic polynomial of the matrix (v~n~) - or the matrix (vaTr~), see ref. [1] - , with the lagrangian L as variable, which, in turn, is a function of of! That is to say, Carath6odory's canonical theory a v makes use of all invariants of the matrix (vuTra) - or a p (odrb) - , not just its trace, like in the case of De Donder-Weyl. By doing so, it is able to realize the minimal rank 4 of f2 c which is so important for the integrability properties of the transversal wave fronts [1]! Before I come back to electrodynamics let me add a few remarks concerning the quantization procedure in Carath6odory's theory, as indicated briefly in ref. [1 ] : The basic Hamilton-Jacobi relation of G 4÷n, where Pff = ~Ua(x, z), ~e(X,Z) = L - 3 a 0 ^ ... ^ a 3 = - H - 3 b O ^ ... ^ b 3 = d S 0 ^ . . . AdS 3,
S u=Su(x,z),
b u = _ H d x u +PUadza , yields the equations
I3vSUl + H ( x , z , p = ~(x, z)) = 0,
(9a)
pPa OoS u + HaaSU = 0,
(9b) 31
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i.e. we have one partial differential equation (9a) for four functions SU(x, z). So we can choose three o f them, say Si, ] = 1, 2, 3, freely, provided the "transversality" relations (9b) and the integrability conditions following from the requirement that vg(x, z = f(x)) = bufa(x) on E 4 are fulfilled• The functions S](x, z) are essential for the canonical H61der transformation [1]:x 0~0 =x 0,x]~.~/=ol=sl(x,z),z a~i a = z a,,which casts the dynamical equations at (x 0, a )
into mechanical form, because/~0 = if0 = p0,/~/a = 0, ~al = 0. The basic commutators [/30 (x 0 = 0, n), f b ( 0 ) ] = (l/i)fib 8(~) in the I tSlder frame take the form
TO 0 [paO(xO = O'x)'fb(o)l = (1/i)Hfba 6(x)'
det(pUv)/H3 = ~6 (* FuvF~)2 /L 3,
(10)
(11)
= [(E 2 - B2) 2 + 4 ( E ' B ) 2 - 4(]aAa) 2] ]4L. Naively one would expect only the first two terms, but the additional third term is typical for Carath6odory's theory: it involves all available invariants! 32
If the external current Ju vanishes, then H is gauge invariant. We obtain the same expression for H if we replace the determinant of the (gauge dependent) canonical energy-momentum tensor by the determinant of of the (gauge invariant) symmetrical energy-momentum tensor 0 uv = -FUC'F~ +-~gUVFe,¢3F e'~. In order to find H a s a function o f A ~ and p~, we have to express the two invariants o f ( F w ) in terms of the pav. Defining p~V = pro, +_pVU, where pW = gUapV, we obtain the relations
*pu~pU~/H = *FuvFUV [ ~g(* FuvFUV)2 - 4/aAaL ] /L 3,
in the original frame. Since [13],fb] = 0, because/3 ] = 0, one expects [p~(x 0 = O,x~,fb(o)] = 0 in general, and therefore TO.. [pl(x O, x), f b (0)] = 0 Combining 0 this• with eq. (10)j anda rraH= TOupUa (see" eqs. (8b)) we obtain, at least formally, the usual quantization rule [Tr0(x 0 = O, x),fb(o)] = (1/i)5abS(x)! Thus, our quantization procedure in the H61der frame is formally compatible with the standard one! However, beyond this formal relationship, important things may happen: H or L, or both, might vanish or have singularities [1,7], the above products of operator-valued distributions are iU-def'med etc.! Only specific examples can tell. Let us return to electrodynamics with the lagrangian L = (-1/4)FtwFUV +]aA a and let us suppose, for a moment, that the two invariants F u v F ~ = - 2 ( E 2 - B 2) and * F ~ F *w = 4 ( E ' B ) 2 do not vanish. It is then possible and convenient to perform the kegendre transformation in a system where E and B are parallel and in which only F01 and F23 are non-vanishing [16,17]. In this system one can choose the gauge A 0 = 0, A 3 = 0 (the relation F = dA, A = A u dxU holds in any reference frame because the operator " d " commutes with mappings [6]). In this way we obtain the Hamilton function [1 ]
H = - z1F u v F uv - jcxA a + (*FuvFUV)2/l 6L,
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det (p +)/H 2 = ( ¼)6 (* FuvFUV)6 /L 6 ; (P-)uv(P- )tw/H = (*Fur F ~ ) 2 /4 L 2 for ]a = 0 only. They give
H= [(p_)uv(p_) u~] 3 /4(*p~pUV)2 for/a = 0, and
H 5/3 det l/6(p+)[detl/Z(p+) --¢1 . PuvP p v ] = 4jaAa det(pvU), otherwise. (The last expression, which uses only the determinants of the matrices (p) and (p+), is quite likely not the most convenient ones!) Like in ref. [1 ] we see again that the possibility of performing the Carathdodory-Legendre transformation hinges on the non-vanishing of *FuvFUV and L. For plane waves it does not exist. The associated singularities are central for the "infrared" problem [1 ] !
References [1] H.A. Kastrup, Phys• Lett. 70B (1977) 195. [2] E. Cartan, Lecons sur les invariants int~graux (Hermann, Paris, 1922), Les syst6mes diff~rentiels ext&ieurs et leurs applications g~om~triques (Hermann, Paris, 1945). [3] E. Kahler, Einfllhrung in die Theorie der Systeme yon Differentialgleichungen (Chelsea, New York, 1949). [41 M. Kuranishi, Amer. J. Math. 79 (1957) 1. [5] J. Dieudonn~, Treatise on analysis, Vol. IV (Academic Press, New York and London, 1974) Ch. 18. [6] E~g. F. Warner, Foundations of differentiable manifolds and Lie groups (Scott, Foresman and Co., Glenview, 1971). [7] H.A. Kastrup, Phys. Lett. 77B (1978) 203; 78B (1978) 433.
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[8] S.L. Adler, Phys. Rev. 177 (1969) 2426; Perturbation theory anomalies, in: Lectures on elementary particles and quantum field theory, Vol. 1, ed. St. Deser, M. Grisaru and H. Pendleton (M.I.T. Press, Cambridge, MA, 1970) p. 1, these lectures contain further references. [9] A. Clebsch, J. Reine Angew. Mat. 56 (1859) 122. [10] M. Hadamard, Bull. Soc. Math. France 30 (1902) 253, 33 (1905) 73. [11] F.J. Terpstra, Math. Ann. 116 (1938) 166. [12] M.R. Hestenes and E.J. McShane, Trans. Am. Mat. Soc. 47 (1940) 501.
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[13] R. Debever, Bull. Soc. Roy. Sci. Liege 10 (1941) 584. [14] L. Van Hove, Sur le signe de la variation seconde des int~grales multiples ~ plusieurs fonctions inconnues. Acad. Roy. Belg. Cl. Sci., Mere. Coll. in-8 o, t. 24, fasc. 5 (Bruxelles, 1949). [15] F. Strocchi and A.S. Wightman, J. Math. Phys. 15 (1974) 2198, 17 (1976) 1930. [16] Ch.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (W.H. Freeman, San Francisco, 1973) § 20.6. [ 17[ H.P. Robertson and Th.W. Noonan, Relativity and cosmology (W.B. Saunders, Philadelphia, 1969) Ch. 4.3.
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