- Email: [email protected]
Carathéodory's canonical theory for fields and the problem of quark confinement
Volume 70B, number 2
PHYSICS LETTERS
CARATHI~ODORY'S
26 September 1977
CANONICAL THEORY FOR FIELDS
AND THE PROBLEM OF QUARK CONFINEMENT H.A. KASTRUP Instttut fur Theorettsche Phystk, R WTH Aachen, 51 Aachen, FR Germany
Received 4 July 1977 It Is proposed that Carath6odory's canomcal theory for fields is important for classical and quantlzed interacting fields m particle physics, especmlly gauge theories It gives tr (FIavFI~V)= 0 as a condltton for a possible simultaneous confinement of quarks and gluons. In 1929 Carath~odory wrote a paper [1] on the canonical theory (canonical momenta, Hamilton-function, Legendre transformation, Hamllton-Jacobi equation etc.) for field equations. It was the same year in which Heisenberg and Pauli published [2] the first part of their work on the canonical quantizatlon of fields. Their definition of the canonical momenta - which is the only one which has been employed in quantum field theory in the last 50 years - corresponds to the socalled "De Donder-Weyl" canonical theory [3,4] for fields. The theories of Carath6odory and De Donder-Weyl are equivalent (classically) only if the number m of independent variables xU is equal to 1, with an arbitrary number n o f dependent variables (as in mechanics), or, lfn = 1 and m arbitrary (like a field with one degree of freedom). In fact, an a series of very interesting papers [5, 6], the mathematician Lepage gave a characterlzatmn of all possible canonical theories among which Carath~odory's and that of De Donder-Weyl are special cases. However, it became clear from the work of Boerner [7] Lepage [5, 6], E. Holder [8], Dedecker [9] and others 4: that CaratModory's theory is unique among all those possible canonical formalism and that only tt (and not the De Donder-Weyl theory!) could provide a natural generalization of the classical picture: a family ("field") of 1-dimensional trajectories (rays) accompanied by n-dimensional transversal wave frontsl I would like to suggest that Carath~odory's theory is important for the better understanding - or even the solution - of many current problems in particle 4: More justice to the mathematical and physical hterature wall be done In ref [10].
physics, especially the following ones: Why and how are the quarks and gluons so strongly bound, why are magnetic charges confined and where are their traces, what is the dynamical origin of the infrared problem m gauge theories, what is the nature o f phase transitions, how do we have to incorporate gravity into particle physics etc.? All these questions have to do with long range properties of the dynamics, and one can hope to understand at least some aspects of them already on the classical level. Technically, surface terms in integrals are important in those problems and here the Hamilton-Jacob1 theory with its "wave fronts", now transversal to 4-dimensional extremals E 4, becomes relevant. In this short communication I can only summarize a very few of those aspects m Carath~odory's theory which seem to be interesting for particle physics. A more systematic and detailed discussion will be given in ref. [10] and elsewhere. The nlathematlcal language to be used in the following is Cartan's calculus of differential forms [ 1 1 - 1 3 ] , including his theory of partial differential equations [14, 15]. Only In this language can Lepage's penetrating analysis be understood. Let me answer one obvious question at the beginning: for free, noninteracting fields (and perturbation theories in terms o f them) all those different canonical theories can be considered equivalent, even if they describe particles with, e.g. spin degrees of freedom. The reason is that all those degrees of freedom are dynamically decoupled and can be treated independently. Thus, in the case of free fields, we have essentially the case n = 1, for which all canonical theories are equivalent. Next, let me briefly recall those features of classical 195
Volume 70B, number 2
PHYSICS LETTERS
mechanics, the generalizations of which are important for fields: A Lagranglan L = L(t, ql, vl),/= 1,.., n, defines a 1-form co = L d t which belongs to an equivalence class of 1-forms g2 (t, q, o) on R 2n+l m the following sense: if ql = ql(t), vl = ?l/(t) and ( d / d t ) × (~L/Oo/) = ~L/Oql, then it will "annihilate" [12] the form co/ = dq] - vldt. Thus, as far as the extremals are concerned, the two forms co and f2 = co+hjcol are equwalent. The coefficients hl are fixed by the following requirement• the H a m i l t o n - J a c o b i theory considers not just one extremal, but a family o f them (with special coherence properties), so that in a certain region G n+l ~ R n+l one trajectory passes through each point (t, q ) E G n+l [16]. In that case the velocities 01 form a "slope field" d = d~l(t,q). The H a m i l t o n - J a c o b i theory deals with those slope fields for which co becomes a closed form, i.e. dco = 0. Extending this property to ~(t, q, v) requires d ~ = 0 (mod (cot'}) and fixes hI to be h / = ~L/~vi==-pl. We therefore get
f~(t, q, v) = L dt + plcol = - H dt + p]dq/,
(1)
with H = p l y ~ - L . Thus, the Legendre transformatxon ol -+ PI' L ~ H is implemented by a change o f basis 6ol ~ dql in the space of 1-forms! Suppose it is regular, Then the canonical Hamilton equations of motions are obtained [1 1, 17, 18] from i ( X ) d ~ = 0, where i ( X ) means interior multiplication [12] by an arbitrary vector field
X = T(t, q, p)~t + Ql(t, q, p)~/aql+pl(t,q,p)~/~)pl (interior multiplication by a vector field X makes a (k - 1)-form out of a k-form X by defining ( i ( x ) x ) ( x 1. . . . x k _ 1) = x ( x , x 1 ..., & - l ) , where X~, ~ = 1 . . . . k - 1, are vector fields, recall further that by definition dq](3/3q k) = 61k etc.). In addition, the natural boundary conditions at the endpoints ta, c~= 1, 2, o f the action integral are gtven by i(X)gZt=ta = -HTa+pjQla - 0, a = 1,2, that xs to say, the tangent vector Ta3t + Q13] is "annihilated" by the cotangent vector - H d t + Pl dqJ and therefore belongs to the "assocmted system" [12] of ~ . Next, consider a slope field vl= 491(t,q). It induces a m o m e n t u m slope field Pl = q;1(t, q). The form ~2 then becomes ~2(t, q,p = ~(t, q)) = ~2(t, q) = - H d t + O/dq/. We know that d ~ = 0 for 6ol = 0, that is if ql(t) is a solutmn of the equations ~I = 491(t,q). Porecard's lemma [12] then implies that locally ~2 = 196
26 September 1977
dS(t, q). Combined with eq. (1) this gwes the Hamilt o n - J a c o b i equation 0ts +
= 0,
Pl --
s
(2)
Eq. (2) expresses one essentzal aspect of the H a m i l t o n Jacob1 theory. If S(t, q) is a solution of the H . - J . equation, then at each point (t, q) the canonical "covector" ( - H , p/) is normal to the n-dim, surface S(t, q) = o = const., the tangent vectors to these wave fronts belong to the associated system of ~ and the form co = L d t becomes locally exact: L (t, q, v = (~(t,q)) d t = dS, meaning that the action integral over co depends only on the values of q (t) etc. at the endpoints t,~, c~ = 1,2. The generalization of the following propertzes will be especially important later on: In general the directions (1, o]= ep/(t,q)) o f the extremais will not be tangent to the wave fronts, because (1, v/).(~tS, a/S) '= bt S + vialS = L , and in general L 4: 0. However, if there are extremals for which L = 0 m a certain region F C R n+l , then wave fronts and extremals are tangent and we have a singular "focal region" or "caustic" (it suffices that L = const, for a certain motion, because a suitable normalization of the total energy again gives L = 0). In such singular regions the difference between trajectories (particles) and wave fronts breaks down! (Notice that L = 0 means dS = 0, i.e. S has a critical value !) Example: Harmonic oscillator in 3 dimensions. Here the solution with L = 0 is the circular motion for which : = 0, the effective potential 12/(m2r 2) br2/2 is minimal and for which energy E and angular m o m e n t u m l are no longer xndependent but E = col ("Regge trajectory"), co = (b/m) 1/2. The existence of "critical" curves E = E(1) in phase space which are associated with such focal regions, is a general phenomenon [ 19]. Such curves separate qualitatively different types ("phases") o f motions. For the case, where the number m o f independent variables is larger than 1, we take the Minkowska space M 4 with x 2 = (x0) 2 - (x) 2 as an example. Instead o f q2 and ol we have fa, a = 1 .... n, and oa~,t~ = 0 .... 3. The Lagrangian L (x, f, v) depends on 4+ n + 4n variables and defines the form co = L d x 0 A ..A d x 3 (the wedge means exterior multiplication). Because the extremals E 4 annihilate coa = d f a _ oadxta, the form co is equivalent to
Volume 70B, number 2
PHYSICS LETTERS
g2(x, f v) = L dx 0 ^ ... ^ dx 3 + hUacOa ^ d 3 Nu + 'gl "'abhlav('°aAa)bAd2S# u
+ terms with higher powers of coa , where d 3 £ u = (1/3!) euvpo dxV ^ dxO ^ d x ° , 2d2S/av = eu~oa d x P ^ d x 'r , and the coefficients haU/~etc. are completely antlsymmetric in their upper and lower in&ces, respectively. The crucial point now ~s this. the requirement of the existence of a Hamilton-Jacob1 theory, namely d~2 = 0(mod {o3a}), determines only that ha~ is equal to rraU-= O L I O S , but leaves the other coefficients undetermined! The Legendre transformation is again performed by expressing g2 in terms of ( d r a, dxU) instead of (coa, dxU). The canonical Hamilton functmn H (not to be confused with the total energy pO) is defined as the resulting coefficient of dx 0 ^ dx 1 A dx 2 ^ dx 3, and the canonical momenta pau as the coefficients of d f a ^ d 3 Z u. For each different choice ofth~coefficients haul, etc. we therefore get a different canonical theory. If they all vanish, then ft0= - H d x 0 A . .
dx3+ n~adffA d3E u ,
with H = v$ w~a- L. This is the DeDonder-Weyl case, where the canonical momenta are gwen by Zra u. The canonical equations of motion have the form
df dx u
Orru a
d
0F/
dx u
Of a
However, setting all those higher terms equal to 0 is at the expense of the rank of ~20 (the rank of a differential form is the minimal number of 1-forms in the (4+n)-dim space of cotangentials dxU, d f a, in terms of which it can be expressed [12]). For, if we define a u =L d x u + wart au = - T u 1) dxV+ rr~ad f a , tl
(canonical energy m o m e n t u m tensor), then ~0 = aUAd32;u, meaning that in general the rank of g20 is 8 (only i f n = l, it is 4, because the rank of a k-form in k+ 1 variables is always k [ 12]). The transversal wave fronts, now described by 4 functions SU(x, f ) , are obtained from the basic relatmn ~ 0 = dSU^d3Nu,
26 September 1977
giving the De Donder-Weyl Hamllton-Jacobl equation O, S U + H = O, nua = OaSU" The rank of~20 being 8 implies that the dimenston of the transversal surfaces SU(x, f ) = ou = const, is maximally (n -4)-dimensional. This can cause problems at the boundary of the considered region G n+4 and its 3-dim. intersection with the extremals E4! Those defects are no longer there m Carath6odory's theory, essentially defined by ~ c = ( 1 / L 3 ) aO ^ a l ^ a 2 A a 3 ,
(3)
which has the minimal rank 4! Here, the change of basis coa ~ d f a (generalized Legendre transformation) gwes the canonical quantmes H=-L-3ITUvl,
p U a = - L - 3 T U ~ , n va ,
(4)
where [TU~,l = det (TUv), iPu v : algebraic complement o f T u. (Actually, Carath6odory's original generalized Legendre transformation was defined by v au ~ttag= -pUa/H, L -+ F = - 111t. This is probably one of the reasons, why it did not become popular. Its meaning was clarified by E. Holder [8].) If one defines the tensor R a b = vagTr~ - 8~L, then H and pa~ can also be expressed as H= (-z)l-nlRabl
, pUa = ( - z ) l - n R b a r r ~ .
(5)
The condition for the Legendre transformation to be regular is
0p$ -
×
KH4(n-1)ITUul-nL3
o2t
b
1
t
10% Ovv
s0
(6)
with K - % tflu - L. If it exists, the canomcal equations of motions are d f a Vtav = H OH --, dxU ~pVa
dPa H OH -Vu v = --, dx~ Of a
(7)
where VUu = pUa OH/OpVa - 6UvG , G = p~aaH/Op~a - H, and KVU~, = H T U v . The Hamflton-Jacobi equation is obtained from ~ c = d S ° ^ dS1 ^ dS2 ^ dS3, which gwes 10~SUl + H ( x , f , p ) = 0 ,
(Sa)
PUa(X, f ) = (OS)U o O a S ° ,
(8b)
with (0S)U o : algebraic complement of OuSO ; OaSO = 197
Volume 70B, number 2
PHYSICS LETTERS
OSU/Of a. Eq. (8a) is one partial differential equation for four unknown functions SU(x, f). Thus, one can choose three of them conveniently, say S ], ] = 1,2, 3. They have, however, to satisfy the transversality conditlons pOO sJ+ HOaSI = 0 (which follows from eq. a ,o o o (8b)), l f p .u = pau, H = H are determined by a certain a o e x t r e m a l ? (x). This is achieved by setting [7] o]
o
SI(x, f ) = xl _P~a (fa _ f a ( } ) ) , H
(9)
where x J and fa are the "running" variables of Sl(x, f). (The freedom in the choice of S1 at fixed x 0 corresponds to the freedom of prescribing initial conditions f o r f a and pau at fixed x°.) Actually, the choice (9) is part of an important canonical transformation, discovered by E. Holder [8], which has the unique property of transforming the canonical partial differential equations (7) at each fixed space point x into the ordinary canonical equations of mechanics I The deeper geometrical reasons for this extraordinary property were analyzed by Lepage [6], and further discussed by Van Hove [20]. A canonical transformation is a transformation xU ~ )cU(x, f' P),fa ~ PUa~ Dua(x, f' P), for which g2c(j, f,/~) = ~2c(X, f, p) , h"= Ld)?"+ d)aT)a~, preserving the rank of ~2c, and which has the property that the equations coa = 0 imply the equations dra = d f a - fi~ dfcU = O, a = 1 , . , n. Consider the special canonical transformation xU ~ You= ~U(x, f), fa ~ fa = fa. It follows that
b~aIOp~v[ = pOaOp~u + HO a~U , 121100~v[ = H. Thus, if we take )?0 = x 0 ' 21 = el = $1 (x, f), eq. (9), we get/30 = p0,/~aj = 0 , / t = H . Because d:~i/dxU = 5 i pla ~ / H , we have [d~?~/dxN = TOo/H. (The symbol d/dxU means differentiation of a function F(x, f(x)), whereas 8 u means differentiation of F(x, f ) with.¢ a ~0 ~a ^ j = 0, fixed). It follows from eq. (4) that rr~a0 _- Pa, i?0] = 0, T/v= - 6 1 / ~ , / t = T00, Implying the equations
dx 0
Off'
0 0 S 0 + H = 0,
dx 0
0/)O'
(10)
/~a0= 0 a S 0 .
In all these relations the fixed space point e = (o 1, 02, 03) appears as a parameter. The E. Holder transformation is a generalization of a Lorentz transformation to 198
26 September 1977
the rest frame of a particle. It may have far reaching consequences for the quantizatlon of interaction fields. In the E. Holder frame one has only one choice for the quantlzation, because for fixed ~ everything is structurally the same as in mechanics. This suggests the quantlzation rule [p0(x0 = 0, ~), fb (0)1 = (1/i)6ab6(~). Taking into account that/3a0 = p0, f a = f a , 6(e ) = (1/[ det/dxl)6(x) = H/TOo 6(X) this would imply
[pO(x° = O, x),fb(o)] = (1/i)(H/TO)~ ~(x),
(11)
i.e. we get a "renormalizatlon" factor in the original frame! More about this in ref. [10]. Let me illustrate some of the above general properties by a few examples: 1. Classical electrodynamics with external sources /U(x). We take the Lagrangian
L =-¼FuvFuv-V,
V=-jUA u, v c~u = A a,, =OuA c~.
Whereas the conventional Legendre transformation for this Lagranglan with its gauge lnvariant kinetical part as singular, we here get from rra u = - F a u
H= -¼FUVFuv+ V+ (*FuvFV~)2/16L,
(12a)
821, 1 (FUo~FV _ FU[~FVa) OA~,~OAV,, L = L-16(*FUVFvv/4)IO
(12b)
X [4L2+ 4FU~Fu~L + 3(*FuvFUV)2/16 ] × [4L 4 - 3 (*FvvFUV)2L2 -(*FvvFUV)4/44 ] . Therefore, together with eq. (6), we conclude that the regularity of the Legendre transformation depends on the non-vanishing of the various mvarlants *FvvFUV ( F v v - : e v v a o F a a ) , L, tT~"l and K. For plane waves it becomes singular! If *FvvFUV--/: O, the Hamilton function H becomes singular for solutions with L = 0 in a certain region. In this case the canonical m o m e n t a become singular, too: one has a region in field phase space, where the normal dynamics breaks down! This phenomenon becomes even more spectacular, if one considers 2. Gauge fields coupled to fermions: L = - ¼ tr (FuvFUV) + -¢ lqsDuTUqs + I~/qs,
Ou = 8 u _ lgAau ta " For a solution of the field equations one has L =
(12)
Volume 70B, number 2
PHYSICS LETTERS
- ¼ tr (Fur FUr) (taking mto account 0u(~TU ¢ ) = 0). Thus, for a solution of the field equations the quantities H, Pfermlon, u u Pgauge field are all proportional to the reverse of the invanant tr (FuvFUV)! Immedmte consequences, ff in an abehan gauge theory one does perturbatlon theory with respect to a free photon fie/d, then one sits right on top of the singularity FuvFt~v=O, for both, the gauge and the fermion fields. This seems to be a key to the infrared problem! Furthermore, ff in any gauge theory we have a solution o f the coupled equations with tr (FuvFUV) = 0, we can have a breakdown of the normal dynamics, for fermlons (quarks) and gauge fields (gluons) at the same time! This can, for instance, happen, ff *Fur = +FVV (m a certain Lorentz frame.) Thus, our "confining" condition "magnehc energy equals electric energy everywhere in M 4'' (or in a subregion) is connected with a solution, which seems to be closely related to the mstanton solutions [21,22] in euclidean space. Of course, one has to prove, that such solutions do exist in M 4, too. The interpretation of solutions with L = 0 as describing "focal" regions ("caustics") in the phase space for fields is obtained as follows. At each point (x, f ) in R 4+n a basis for the tangent vectors o f an extremal 0 ., 83u' vul,'" , V u ) . I t f •u r t h e r E 4 1 s g t v e n b y e ( u ) --( 8 u, follows from eq. (8b) that a basis o f the space tangent to the surface defined by SU(x, f ) is given by e(a ) = n These two sets are linearly (--uOa.... , --u a3 . .81 .... 8 a). independent if their determinant does not vanish. It has the value L4/ITUv[ [23]. Agam, soluUons with L = 0 or [Tuvl = 0 gwe rise to singularities. One might wonder why solutions with L = 0 are dynamically so special, because usually one does not worry about adding a total dwergence to L ! However, this is in general no longer allowed, because it will change the rank o f ~2 and therefore the (canonical) surface properties associated with it. The important lesson to be learnt from Caratheodory, Lepage and others Is this: a canomcal theory for fields is mainly determined by the algebrmc structure of the basic differential form ~ , especially by its rank! One final remark as to gravity: As L appears in the denominator o f H and pa~, we in principal have to know the basic Lagrangian o f the world, in order to have the proper time development o f / / , f a and pUa! The inclusion of the gravitational part Lg of L will be particularly important i f L = 0 without Lg, because then gravity will provide - a s far as I can see - the only
26 September 1977
possibtlity to stabihze the dynamical singularities in those focal regions. The inclusion o f Lg will, of course, also matter at short distances. I had many stimulating and critical discussions with J. Jersak, M. Kiera, R. Maciejko, M. Magg, M. Rmke and G. Roepstorff. I am also grateful to J. Kijowski for very interesting discussions about his and his colleague's work [24] on the canonical structure of classical fields. Part of this work was done while 1 was at CERN. I thank Prof. J. Prentkl very much for his kind hospitahty.
References [1] C Carath6odory, Acta SCL Math. (Szeged) 4 (1929) 193, reprinted m C. Carath6odory, Gesamm Math. Schriften, Bd 1 (C H Beck, Munchen, 1954) p. 401 12] W Heisenberg und W. Pauh, Z. Physlk 56 (1929) 1. [3] Th. De Donder, C.R Acad. ScL Paris 156 (1913) 868, Thdorle mvariantwe du calcul des variations, nouv. ed. (Gauthlers-Vdlars, Pans, 1935). [4] H Weyl, Ann. Math. 36 (1935) 607. [5] Th. Lepage, Acad. Roy. Belg. Bull CI. SCL (5e S6r.) 22 22 (1936) 716 and 1036. [6] Th. Lepage, Acad. Roy. Belg. Bull C1. SCL (5e S6r.) 27 (1941) 27, 28 (1942) 73 and 247. [7] H. Boerner, Math. Ann (Berhn) 122 (1936) 187, Math. Z 46 (1940) 720 [8] E. Holder, Jber. Deutsch. Math-Vereln (Stuttgart) 49 (1939) 162. [9] P. Dedecker, Calcul des variations, formes dlff6rentlelles et champs geod6slques, Proc. of the Colloque Intern de G6omdtne Dlff~rentielle, Strasbourg 1953 (CNRS, Pans, 1953) p 17, On the generahzatlon of symplectlc geometry to multiple integrals m the claculus of variations, Proc. of the Symposmm on Differential Geometry cal Methods m Mathemahcal Physics, Bonn, 1975, ed. K. Bleuler (Springer, Lecture Notes m Math. 570, Berhn, 1977) p. 395 110] H.A Kastrup, On Carath6odory's canonical theory for fields and ItS applications to particle physics, RWTH Aachen Lecture Notes, to be published. [11] E. Cartan, Leqons sur les mvarlants mt6graux (Hermann, Paris, 1922). [12] C. Godblllon, G6om6trle diff6rentlelle et m~chamque analytlque (Hermann, Paris 1969) [13] Ch. W. Mlsner, K.S Thorne and J A. Wheeler, Grawtatmn (W.H Freeman, San Francisco, 1973). [14] E Cartan, Les systbmes dlff~rentlels ext6rieurs et leurs apphcatlons g6om6trlques (Hermann, Pans, 1945). [15] J. Dieudonn6, Treatise on analysis, vol. IV (Academlc Press, New York and London, 9174) ch. 18. 199
Volume 70B, number 2
PHYSICS LETTERS
[ 16 ] C. Carath6odory, Vanationsrechnung und partlelle Differentlalgleichungen erster Ordnung (B.G. Teubner, Leipzig und Berhn 1935); Engl. transl.: C. Carath~odory, Calculus of variations and partial differential equations (Holden-Day, San Francisco, 1965). [171 A. Lichnerowicz, Bull. ScL Math. (2e S~r.) 52 (1946) 82. [18] J. Klein, Ann. Inst. Fourier 12 (1962) 1. [19] L.A. Pars, A treatise on analytical dynamics (Hememann, London, 1965) ch. 17.
200
26 September 1977
[20] L. Van Hove, Acad. Roy. Belg. Bull. CI. ScL (5e S& ) 31 (1945) 625. [21] A.A. Belavm, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkm, Phys. Lett. 59B (1975) 85. [22] G. 't Hooft, Phys. Rev. DI4 (1976) 3432. [23] H. Boerner, Jber. Deutsch. Math-Verem. (Stuttgart) 56 (1953) 31. [24] J. Kljowski and W. Szczyrba, Comm. Math. Phys. 46 (1976) 183; and references thereto.
PHYSICS LETTERS
CARATHI~ODORY'S
26 September 1977
CANONICAL THEORY FOR FIELDS
AND THE PROBLEM OF QUARK CONFINEMENT H.A. KASTRUP Instttut fur Theorettsche Phystk, R WTH Aachen, 51 Aachen, FR Germany
Received 4 July 1977 It Is proposed that Carath6odory's canomcal theory for fields is important for classical and quantlzed interacting fields m particle physics, especmlly gauge theories It gives tr (FIavFI~V)= 0 as a condltton for a possible simultaneous confinement of quarks and gluons. In 1929 Carath~odory wrote a paper [1] on the canonical theory (canonical momenta, Hamilton-function, Legendre transformation, Hamllton-Jacobi equation etc.) for field equations. It was the same year in which Heisenberg and Pauli published [2] the first part of their work on the canonical quantizatlon of fields. Their definition of the canonical momenta - which is the only one which has been employed in quantum field theory in the last 50 years - corresponds to the socalled "De Donder-Weyl" canonical theory [3,4] for fields. The theories of Carath6odory and De Donder-Weyl are equivalent (classically) only if the number m of independent variables xU is equal to 1, with an arbitrary number n o f dependent variables (as in mechanics), or, lfn = 1 and m arbitrary (like a field with one degree of freedom). In fact, an a series of very interesting papers [5, 6], the mathematician Lepage gave a characterlzatmn of all possible canonical theories among which Carath~odory's and that of De Donder-Weyl are special cases. However, it became clear from the work of Boerner [7] Lepage [5, 6], E. Holder [8], Dedecker [9] and others 4: that CaratModory's theory is unique among all those possible canonical formalism and that only tt (and not the De Donder-Weyl theory!) could provide a natural generalization of the classical picture: a family ("field") of 1-dimensional trajectories (rays) accompanied by n-dimensional transversal wave frontsl I would like to suggest that Carath~odory's theory is important for the better understanding - or even the solution - of many current problems in particle 4: More justice to the mathematical and physical hterature wall be done In ref [10].
physics, especially the following ones: Why and how are the quarks and gluons so strongly bound, why are magnetic charges confined and where are their traces, what is the dynamical origin of the infrared problem m gauge theories, what is the nature o f phase transitions, how do we have to incorporate gravity into particle physics etc.? All these questions have to do with long range properties of the dynamics, and one can hope to understand at least some aspects of them already on the classical level. Technically, surface terms in integrals are important in those problems and here the Hamilton-Jacob1 theory with its "wave fronts", now transversal to 4-dimensional extremals E 4, becomes relevant. In this short communication I can only summarize a very few of those aspects m Carath~odory's theory which seem to be interesting for particle physics. A more systematic and detailed discussion will be given in ref. [10] and elsewhere. The nlathematlcal language to be used in the following is Cartan's calculus of differential forms [ 1 1 - 1 3 ] , including his theory of partial differential equations [14, 15]. Only In this language can Lepage's penetrating analysis be understood. Let me answer one obvious question at the beginning: for free, noninteracting fields (and perturbation theories in terms o f them) all those different canonical theories can be considered equivalent, even if they describe particles with, e.g. spin degrees of freedom. The reason is that all those degrees of freedom are dynamically decoupled and can be treated independently. Thus, in the case of free fields, we have essentially the case n = 1, for which all canonical theories are equivalent. Next, let me briefly recall those features of classical 195
Volume 70B, number 2
PHYSICS LETTERS
mechanics, the generalizations of which are important for fields: A Lagranglan L = L(t, ql, vl),/= 1,.., n, defines a 1-form co = L d t which belongs to an equivalence class of 1-forms g2 (t, q, o) on R 2n+l m the following sense: if ql = ql(t), vl = ?l/(t) and ( d / d t ) × (~L/Oo/) = ~L/Oql, then it will "annihilate" [12] the form co/ = dq] - vldt. Thus, as far as the extremals are concerned, the two forms co and f2 = co+hjcol are equwalent. The coefficients hl are fixed by the following requirement• the H a m i l t o n - J a c o b i theory considers not just one extremal, but a family o f them (with special coherence properties), so that in a certain region G n+l ~ R n+l one trajectory passes through each point (t, q ) E G n+l [16]. In that case the velocities 01 form a "slope field" d = d~l(t,q). The H a m i l t o n - J a c o b i theory deals with those slope fields for which co becomes a closed form, i.e. dco = 0. Extending this property to ~(t, q, v) requires d ~ = 0 (mod (cot'}) and fixes hI to be h / = ~L/~vi==-pl. We therefore get
f~(t, q, v) = L dt + plcol = - H dt + p]dq/,
(1)
with H = p l y ~ - L . Thus, the Legendre transformatxon ol -+ PI' L ~ H is implemented by a change o f basis 6ol ~ dql in the space of 1-forms! Suppose it is regular, Then the canonical Hamilton equations of motions are obtained [1 1, 17, 18] from i ( X ) d ~ = 0, where i ( X ) means interior multiplication [12] by an arbitrary vector field
X = T(t, q, p)~t + Ql(t, q, p)~/aql+pl(t,q,p)~/~)pl (interior multiplication by a vector field X makes a (k - 1)-form out of a k-form X by defining ( i ( x ) x ) ( x 1. . . . x k _ 1) = x ( x , x 1 ..., & - l ) , where X~, ~ = 1 . . . . k - 1, are vector fields, recall further that by definition dq](3/3q k) = 61k etc.). In addition, the natural boundary conditions at the endpoints ta, c~= 1, 2, o f the action integral are gtven by i(X)gZt=ta = -HTa+pjQla - 0, a = 1,2, that xs to say, the tangent vector Ta3t + Q13] is "annihilated" by the cotangent vector - H d t + Pl dqJ and therefore belongs to the "assocmted system" [12] of ~ . Next, consider a slope field vl= 491(t,q). It induces a m o m e n t u m slope field Pl = q;1(t, q). The form ~2 then becomes ~2(t, q,p = ~(t, q)) = ~2(t, q) = - H d t + O/dq/. We know that d ~ = 0 for 6ol = 0, that is if ql(t) is a solutmn of the equations ~I = 491(t,q). Porecard's lemma [12] then implies that locally ~2 = 196
26 September 1977
dS(t, q). Combined with eq. (1) this gwes the Hamilt o n - J a c o b i equation 0ts +
= 0,
Pl --
s
(2)
Eq. (2) expresses one essentzal aspect of the H a m i l t o n Jacob1 theory. If S(t, q) is a solution of the H . - J . equation, then at each point (t, q) the canonical "covector" ( - H , p/) is normal to the n-dim, surface S(t, q) = o = const., the tangent vectors to these wave fronts belong to the associated system of ~ and the form co = L d t becomes locally exact: L (t, q, v = (~(t,q)) d t = dS, meaning that the action integral over co depends only on the values of q (t) etc. at the endpoints t,~, c~ = 1,2. The generalization of the following propertzes will be especially important later on: In general the directions (1, o]= ep/(t,q)) o f the extremais will not be tangent to the wave fronts, because (1, v/).(~tS, a/S) '= bt S + vialS = L , and in general L 4: 0. However, if there are extremals for which L = 0 m a certain region F C R n+l , then wave fronts and extremals are tangent and we have a singular "focal region" or "caustic" (it suffices that L = const, for a certain motion, because a suitable normalization of the total energy again gives L = 0). In such singular regions the difference between trajectories (particles) and wave fronts breaks down! (Notice that L = 0 means dS = 0, i.e. S has a critical value !) Example: Harmonic oscillator in 3 dimensions. Here the solution with L = 0 is the circular motion for which : = 0, the effective potential 12/(m2r 2) br2/2 is minimal and for which energy E and angular m o m e n t u m l are no longer xndependent but E = col ("Regge trajectory"), co = (b/m) 1/2. The existence of "critical" curves E = E(1) in phase space which are associated with such focal regions, is a general phenomenon [ 19]. Such curves separate qualitatively different types ("phases") o f motions. For the case, where the number m o f independent variables is larger than 1, we take the Minkowska space M 4 with x 2 = (x0) 2 - (x) 2 as an example. Instead o f q2 and ol we have fa, a = 1 .... n, and oa~,t~ = 0 .... 3. The Lagrangian L (x, f, v) depends on 4+ n + 4n variables and defines the form co = L d x 0 A ..A d x 3 (the wedge means exterior multiplication). Because the extremals E 4 annihilate coa = d f a _ oadxta, the form co is equivalent to
Volume 70B, number 2
PHYSICS LETTERS
g2(x, f v) = L dx 0 ^ ... ^ dx 3 + hUacOa ^ d 3 Nu + 'gl "'abhlav('°aAa)bAd2S# u
+ terms with higher powers of coa , where d 3 £ u = (1/3!) euvpo dxV ^ dxO ^ d x ° , 2d2S/av = eu~oa d x P ^ d x 'r , and the coefficients haU/~etc. are completely antlsymmetric in their upper and lower in&ces, respectively. The crucial point now ~s this. the requirement of the existence of a Hamilton-Jacob1 theory, namely d~2 = 0(mod {o3a}), determines only that ha~ is equal to rraU-= O L I O S , but leaves the other coefficients undetermined! The Legendre transformation is again performed by expressing g2 in terms of ( d r a, dxU) instead of (coa, dxU). The canonical Hamilton functmn H (not to be confused with the total energy pO) is defined as the resulting coefficient of dx 0 ^ dx 1 A dx 2 ^ dx 3, and the canonical momenta pau as the coefficients of d f a ^ d 3 Z u. For each different choice ofth~coefficients haul, etc. we therefore get a different canonical theory. If they all vanish, then ft0= - H d x 0 A . .
dx3+ n~adffA d3E u ,
with H = v$ w~a- L. This is the DeDonder-Weyl case, where the canonical momenta are gwen by Zra u. The canonical equations of motion have the form
df dx u
Orru a
d
0F/
dx u
Of a
However, setting all those higher terms equal to 0 is at the expense of the rank of ~20 (the rank of a differential form is the minimal number of 1-forms in the (4+n)-dim space of cotangentials dxU, d f a, in terms of which it can be expressed [12]). For, if we define a u =L d x u + wart au = - T u 1) dxV+ rr~ad f a , tl
(canonical energy m o m e n t u m tensor), then ~0 = aUAd32;u, meaning that in general the rank of g20 is 8 (only i f n = l, it is 4, because the rank of a k-form in k+ 1 variables is always k [ 12]). The transversal wave fronts, now described by 4 functions SU(x, f ) , are obtained from the basic relatmn ~ 0 = dSU^d3Nu,
26 September 1977
giving the De Donder-Weyl Hamllton-Jacobl equation O, S U + H = O, nua = OaSU" The rank of~20 being 8 implies that the dimenston of the transversal surfaces SU(x, f ) = ou = const, is maximally (n -4)-dimensional. This can cause problems at the boundary of the considered region G n+4 and its 3-dim. intersection with the extremals E4! Those defects are no longer there m Carath6odory's theory, essentially defined by ~ c = ( 1 / L 3 ) aO ^ a l ^ a 2 A a 3 ,
(3)
which has the minimal rank 4! Here, the change of basis coa ~ d f a (generalized Legendre transformation) gwes the canonical quantmes H=-L-3ITUvl,
p U a = - L - 3 T U ~ , n va ,
(4)
where [TU~,l = det (TUv), iPu v : algebraic complement o f T u. (Actually, Carath6odory's original generalized Legendre transformation was defined by v au ~ttag= -pUa/H, L -+ F = - 111t. This is probably one of the reasons, why it did not become popular. Its meaning was clarified by E. Holder [8].) If one defines the tensor R a b = vagTr~ - 8~L, then H and pa~ can also be expressed as H= (-z)l-nlRabl
, pUa = ( - z ) l - n R b a r r ~ .
(5)
The condition for the Legendre transformation to be regular is
0p$ -
×
KH4(n-1)ITUul-nL3
o2t
b
1
t
10% Ovv
s0
(6)
with K - % tflu - L. If it exists, the canomcal equations of motions are d f a Vtav = H OH --, dxU ~pVa
dPa H OH -Vu v = --, dx~ Of a
(7)
where VUu = pUa OH/OpVa - 6UvG , G = p~aaH/Op~a - H, and KVU~, = H T U v . The Hamflton-Jacobi equation is obtained from ~ c = d S ° ^ dS1 ^ dS2 ^ dS3, which gwes 10~SUl + H ( x , f , p ) = 0 ,
(Sa)
PUa(X, f ) = (OS)U o O a S ° ,
(8b)
with (0S)U o : algebraic complement of OuSO ; OaSO = 197
Volume 70B, number 2
PHYSICS LETTERS
OSU/Of a. Eq. (8a) is one partial differential equation for four unknown functions SU(x, f). Thus, one can choose three of them conveniently, say S ], ] = 1,2, 3. They have, however, to satisfy the transversality conditlons pOO sJ+ HOaSI = 0 (which follows from eq. a ,o o o (8b)), l f p .u = pau, H = H are determined by a certain a o e x t r e m a l ? (x). This is achieved by setting [7] o]
o
SI(x, f ) = xl _P~a (fa _ f a ( } ) ) , H
(9)
where x J and fa are the "running" variables of Sl(x, f). (The freedom in the choice of S1 at fixed x 0 corresponds to the freedom of prescribing initial conditions f o r f a and pau at fixed x°.) Actually, the choice (9) is part of an important canonical transformation, discovered by E. Holder [8], which has the unique property of transforming the canonical partial differential equations (7) at each fixed space point x into the ordinary canonical equations of mechanics I The deeper geometrical reasons for this extraordinary property were analyzed by Lepage [6], and further discussed by Van Hove [20]. A canonical transformation is a transformation xU ~ )cU(x, f' P),fa ~ PUa~ Dua(x, f' P), for which g2c(j, f,/~) = ~2c(X, f, p) , h"= Ld)?"+ d)aT)a~, preserving the rank of ~2c, and which has the property that the equations coa = 0 imply the equations dra = d f a - fi~ dfcU = O, a = 1 , . , n. Consider the special canonical transformation xU ~ You= ~U(x, f), fa ~ fa = fa. It follows that
b~aIOp~v[ = pOaOp~u + HO a~U , 121100~v[ = H. Thus, if we take )?0 = x 0 ' 21 = el = $1 (x, f), eq. (9), we get/30 = p0,/~aj = 0 , / t = H . Because d:~i/dxU = 5 i pla ~ / H , we have [d~?~/dxN = TOo/H. (The symbol d/dxU means differentiation of a function F(x, f(x)), whereas 8 u means differentiation of F(x, f ) with.¢ a ~0 ~a ^ j = 0, fixed). It follows from eq. (4) that rr~a0 _- Pa, i?0] = 0, T/v= - 6 1 / ~ , / t = T00, Implying the equations
dx 0
Off'
0 0 S 0 + H = 0,
dx 0
0/)O'
(10)
/~a0= 0 a S 0 .
In all these relations the fixed space point e = (o 1, 02, 03) appears as a parameter. The E. Holder transformation is a generalization of a Lorentz transformation to 198
26 September 1977
the rest frame of a particle. It may have far reaching consequences for the quantizatlon of interaction fields. In the E. Holder frame one has only one choice for the quantlzation, because for fixed ~ everything is structurally the same as in mechanics. This suggests the quantlzation rule [p0(x0 = 0, ~), fb (0)1 = (1/i)6ab6(~). Taking into account that/3a0 = p0, f a = f a , 6(e ) = (1/[ det/dxl)6(x) = H/TOo 6(X) this would imply
[pO(x° = O, x),fb(o)] = (1/i)(H/TO)~ ~(x),
(11)
i.e. we get a "renormalizatlon" factor in the original frame! More about this in ref. [10]. Let me illustrate some of the above general properties by a few examples: 1. Classical electrodynamics with external sources /U(x). We take the Lagrangian
L =-¼FuvFuv-V,
V=-jUA u, v c~u = A a,, =OuA c~.
Whereas the conventional Legendre transformation for this Lagranglan with its gauge lnvariant kinetical part as singular, we here get from rra u = - F a u
H= -¼FUVFuv+ V+ (*FuvFV~)2/16L,
(12a)
821, 1 (FUo~FV _ FU[~FVa) OA~,~OAV,, L = L-16(*FUVFvv/4)IO
(12b)
X [4L2+ 4FU~Fu~L + 3(*FuvFUV)2/16 ] × [4L 4 - 3 (*FvvFUV)2L2 -(*FvvFUV)4/44 ] . Therefore, together with eq. (6), we conclude that the regularity of the Legendre transformation depends on the non-vanishing of the various mvarlants *FvvFUV ( F v v - : e v v a o F a a ) , L, tT~"l and K. For plane waves it becomes singular! If *FvvFUV--/: O, the Hamilton function H becomes singular for solutions with L = 0 in a certain region. In this case the canonical m o m e n t a become singular, too: one has a region in field phase space, where the normal dynamics breaks down! This phenomenon becomes even more spectacular, if one considers 2. Gauge fields coupled to fermions: L = - ¼ tr (FuvFUV) + -¢ lqsDuTUqs + I~/qs,
Ou = 8 u _ lgAau ta " For a solution of the field equations one has L =
(12)
Volume 70B, number 2
PHYSICS LETTERS
- ¼ tr (Fur FUr) (taking mto account 0u(~TU ¢ ) = 0). Thus, for a solution of the field equations the quantities H, Pfermlon, u u Pgauge field are all proportional to the reverse of the invanant tr (FuvFUV)! Immedmte consequences, ff in an abehan gauge theory one does perturbatlon theory with respect to a free photon fie/d, then one sits right on top of the singularity FuvFt~v=O, for both, the gauge and the fermion fields. This seems to be a key to the infrared problem! Furthermore, ff in any gauge theory we have a solution o f the coupled equations with tr (FuvFUV) = 0, we can have a breakdown of the normal dynamics, for fermlons (quarks) and gauge fields (gluons) at the same time! This can, for instance, happen, ff *Fur = +FVV (m a certain Lorentz frame.) Thus, our "confining" condition "magnehc energy equals electric energy everywhere in M 4'' (or in a subregion) is connected with a solution, which seems to be closely related to the mstanton solutions [21,22] in euclidean space. Of course, one has to prove, that such solutions do exist in M 4, too. The interpretation of solutions with L = 0 as describing "focal" regions ("caustics") in the phase space for fields is obtained as follows. At each point (x, f ) in R 4+n a basis for the tangent vectors o f an extremal 0 ., 83u' vul,'" , V u ) . I t f •u r t h e r E 4 1 s g t v e n b y e ( u ) --( 8 u, follows from eq. (8b) that a basis o f the space tangent to the surface defined by SU(x, f ) is given by e(a ) = n These two sets are linearly (--uOa.... , --u a3 . .81 .... 8 a). independent if their determinant does not vanish. It has the value L4/ITUv[ [23]. Agam, soluUons with L = 0 or [Tuvl = 0 gwe rise to singularities. One might wonder why solutions with L = 0 are dynamically so special, because usually one does not worry about adding a total dwergence to L ! However, this is in general no longer allowed, because it will change the rank o f ~2 and therefore the (canonical) surface properties associated with it. The important lesson to be learnt from Caratheodory, Lepage and others Is this: a canomcal theory for fields is mainly determined by the algebrmc structure of the basic differential form ~ , especially by its rank! One final remark as to gravity: As L appears in the denominator o f H and pa~, we in principal have to know the basic Lagrangian o f the world, in order to have the proper time development o f / / , f a and pUa! The inclusion of the gravitational part Lg of L will be particularly important i f L = 0 without Lg, because then gravity will provide - a s far as I can see - the only
26 September 1977
possibtlity to stabihze the dynamical singularities in those focal regions. The inclusion o f Lg will, of course, also matter at short distances. I had many stimulating and critical discussions with J. Jersak, M. Kiera, R. Maciejko, M. Magg, M. Rmke and G. Roepstorff. I am also grateful to J. Kijowski for very interesting discussions about his and his colleague's work [24] on the canonical structure of classical fields. Part of this work was done while 1 was at CERN. I thank Prof. J. Prentkl very much for his kind hospitahty.
References [1] C Carath6odory, Acta SCL Math. (Szeged) 4 (1929) 193, reprinted m C. Carath6odory, Gesamm Math. Schriften, Bd 1 (C H Beck, Munchen, 1954) p. 401 12] W Heisenberg und W. Pauh, Z. Physlk 56 (1929) 1. [3] Th. De Donder, C.R Acad. ScL Paris 156 (1913) 868, Thdorle mvariantwe du calcul des variations, nouv. ed. (Gauthlers-Vdlars, Pans, 1935). [4] H Weyl, Ann. Math. 36 (1935) 607. [5] Th. Lepage, Acad. Roy. Belg. Bull CI. SCL (5e S6r.) 22 22 (1936) 716 and 1036. [6] Th. Lepage, Acad. Roy. Belg. Bull C1. SCL (5e S6r.) 27 (1941) 27, 28 (1942) 73 and 247. [7] H. Boerner, Math. Ann (Berhn) 122 (1936) 187, Math. Z 46 (1940) 720 [8] E. Holder, Jber. Deutsch. Math-Vereln (Stuttgart) 49 (1939) 162. [9] P. Dedecker, Calcul des variations, formes dlff6rentlelles et champs geod6slques, Proc. of the Colloque Intern de G6omdtne Dlff~rentielle, Strasbourg 1953 (CNRS, Pans, 1953) p 17, On the generahzatlon of symplectlc geometry to multiple integrals m the claculus of variations, Proc. of the Symposmm on Differential Geometry cal Methods m Mathemahcal Physics, Bonn, 1975, ed. K. Bleuler (Springer, Lecture Notes m Math. 570, Berhn, 1977) p. 395 110] H.A Kastrup, On Carath6odory's canonical theory for fields and ItS applications to particle physics, RWTH Aachen Lecture Notes, to be published. [11] E. Cartan, Leqons sur les mvarlants mt6graux (Hermann, Paris, 1922). [12] C. Godblllon, G6om6trle diff6rentlelle et m~chamque analytlque (Hermann, Paris 1969) [13] Ch. W. Mlsner, K.S Thorne and J A. Wheeler, Grawtatmn (W.H Freeman, San Francisco, 1973). [14] E Cartan, Les systbmes dlff~rentlels ext6rieurs et leurs apphcatlons g6om6trlques (Hermann, Pans, 1945). [15] J. Dieudonn6, Treatise on analysis, vol. IV (Academlc Press, New York and London, 9174) ch. 18. 199
Volume 70B, number 2
PHYSICS LETTERS
[ 16 ] C. Carath6odory, Vanationsrechnung und partlelle Differentlalgleichungen erster Ordnung (B.G. Teubner, Leipzig und Berhn 1935); Engl. transl.: C. Carath~odory, Calculus of variations and partial differential equations (Holden-Day, San Francisco, 1965). [171 A. Lichnerowicz, Bull. ScL Math. (2e S~r.) 52 (1946) 82. [18] J. Klein, Ann. Inst. Fourier 12 (1962) 1. [19] L.A. Pars, A treatise on analytical dynamics (Hememann, London, 1965) ch. 17.
200
26 September 1977
[20] L. Van Hove, Acad. Roy. Belg. Bull. CI. ScL (5e S& ) 31 (1945) 625. [21] A.A. Belavm, A.M. Polyakov, A.S. Schwartz and Yu.S. Tyupkm, Phys. Lett. 59B (1975) 85. [22] G. 't Hooft, Phys. Rev. DI4 (1976) 3432. [23] H. Boerner, Jber. Deutsch. Math-Verem. (Stuttgart) 56 (1953) 31. [24] J. Kljowski and W. Szczyrba, Comm. Math. Phys. 46 (1976) 183; and references thereto.