field theory without surface terms

Volume 194, number 3 PHYSICS LETTERS B 13 August 1987

FIELD THEORY WITHOUT SURFACE TERMS

Victor TAPIA International School for Advanced Studies. 1-34014 Trieste, Italy

Received 6 February 1987

It is shown that the surface terms appearing in the variation of the action can always be given account of by means of an extra term in the lagrangian in such a way that there is no need for imposing conditions on the boundary or asymptotic behaviour of the fields for the stationarity of the action.

In field theory the surface terms appear ing in the var ia t ion of the act ion are current ly taken away by fixing the boundary or asymptot ic behav iour of the fields. The previous procedure involves a sufficient condi t ion for the s ta t ionar i ty o f the action. The necessary condi t ion leads to a source te rm at the boundary in the field equations. In this case it is not necessary to fix the bounda ry behav iour o f the fields. This is the correct field theory following f rom the given action. The condi t ions at the bounda ry d imin- ish the number of Cauchy da ta which can be given independent ly , therefore both theories are different.

The field equat ions with the source te rm can be obta ined from a new lagrangian const ructed in terms of the original one but having no surface terms. The hami l ton ian formula t ion o f the theory is then straightforward.

Let us consider the act ion

S = f 2f(0 A, 0~,¢ A) d4x , (1) .Q

where LP is the lagrangian density; 0 A A = 1 . . . . . m, are the fields describing the theory, m being their number; Otj~A=OoA/OxFt; X 'u, f l = 0 , 1, 2, 3, are local coordinates of the space- t ime; i = 1, 2, 3 has the usual meaning of a space-like index.

We assume that the region I2 of the space- t ime over which the theory is formula ted is s imply connected. We assume fur thermore that we can write 1 2 = S ® [h, t2], i.e., we can in t roduce a system of s imply connected space-like surfaces X and a t ransverse t ime-

like vector field over g2; this is equivalent to the split- t ing x ~ = (t, M). For a closed 27 without boundary no addi t iona l assumpt ions are int roduced. If27 is closed w i t h a boundary 027 we assume 27=I®$2, where I = [0, a] and $2 has the topology of 027. For an open space we assume S: to have the topology o f a two- sphere, S 2, and I = [0, oo); in this case we put for- mally the boundary at infinity, a = or. In the last two cases we call r the coordinate on I and r=a the

boundary . The var ia t ion of the act ion is given by

12

+ (2)

where

OL, e OL,¢' 8A~W= dO-- ~ -- D~, O(o~oA ) . (3)

Du = d/dxU; O~ 5e= ni( OLHO( OiO A) ), with ni the outer normal to OX; d.S, and dS2 are the volume and surface elements on 27 and OX, respectively; d.S = dr dS2; ~A =dOA/dt.

The field equat ions are ob ta ined by requiring 8 S = 0 . The first integral in (2) depends on the val- ues o f ~ A in the in ter ior of the interval [tl, t2] while the second one does it at tl and tz. Therefore, both terms must be independent ly zero.

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For the second integral we put

8CpA( t~ ) =80A( t2) --0, (4)

but 50 A is unrestricted and arbitrary in the rest. For the first integral there exist drastic differences

depending on whether the three-space 27 is open or closed with or without boundary. For a closed space without boundary one is certain that no complica- tions could possibly arise since then 027= 0 and the surface term does not appear in (2). For an open or closed space with boundary the presence of the surface integral is unavoidable.

Since the variations 50 m in the first integral are arbitrary what must be zero are the factors multi- plying 60 A.

The sufficient condition is

5A5(=0, (5a)

plus, in the case of an open or closed space with boundary, a condition on the boundary behaviour of the fields

02 5(los = 0 . (5b)

Eqs. (4) then imply that in order to integrate the field equations (5a) a complete set of Cauchy data is provided by the functions CA(t I ) and ~A(t2).

Since for an open or closed space with boundary the Cauchy data, evolving with (5a), must satisfy (5b) they cannot all be given independently. This means that the system is a constrained one. The hamiltonian formalism is then constructed as for a constrained system with (5b) as constraint. The correct hamiltonian is the canonical one plus a linear combination of the constraints

H=Hc+ ~ 0~5(2AdS2, (6) OF

8A 5(+ 20A ~ LeO(r--a) = 0 . (8)

The extra term can be considered as a source or force at the boundary depending, however, on the fields.

As in classical mechanics, when a generalised force is present, one would like to find a lagrangian 5(' giving account of the extra term in (8). Difficulties appear when trying to formulate the hamiltonian formalism since it is not known how to deal with the surface terms, unless of proceeding as in (5) and (6). Therefore, nothing is gained unless the new lagrangian gives no rise to surface terms.

The solution is

4#' =5(--20A ± 5([orO(r-a)O A , (9)

where 0~ = n'O,~ A, with n ~ normal to OX and satis- fying ninz= 1; O(x) is the Heaviside step function

O(x)= l f o r x > 0 ,

=½ f o r x = 0 ,

= 0 f o r x < 0 . (10)

0) 5(las behaves as a constraint, therefore deriva- tives are taken only with respect to r in 0 and 0~. The lagrangian 5(' correctly reproduces the field equations (8) since

5A5(' =~A 5(+20~ 5(fi(r--a). (11 )

Furthermore

0~5( ' [0s - -0 . (12)

Therefore, one can give account of the extra term in the field equations by means of an extra term in the lagrangian.

The energy density is

( ' =~A 05('/0~A 5('

with ,~A some Lagrange multipliers. This gives rise to the surface energy term well known in general rela- tivity [ 1 ].

The necessary condition in order that the first integral be zero can be obtained by rewriting it as

t2

f f [SAS+20A l 5(d(r--a)]80 A drdS2 dt . t l v"

(7)

The field equations are now

=c+O) 5(larO(r-a)O A (13)

such that the energy is

E ' = | ( d S = E . (14) _r-

Furthermore it is a conserved quantity

f ~ - = - ~AOA- 5('dS2 = 0, (15) OX

409


due to (12). Now for the formula t ion of the hami l ton ian for-

malism. We first observe that the canonical momenta are the same:

~.~i = 0 ~ ' / 0 ~ ~ = ~A • (1 6 )

The canonical hami l ton ian is the funct ion appear ing on the r ight-hand side in the def ini t io in o f the energy such that the energy is the numer ica l value o f the hamil tonian. Due to (16) eq, (14) t ranslates to

H'c=Hc . (17)

The fact that H'~ and H in (6) differ is not strange since when in t roducing condi t ions at the boundary one is not dealing with the same theory. The possi- bi l i ty for H'c of reproducing the s tandard results, e.g., the behaviour of the energy for asymptot ica l ly fiat spaces in general relat ivity, is not excluded since it

is evaluated on solutions of (8) while H is on solu- t ions of (5a) subjected to (5b) .

To conclude, for any lagrangian there exists a new lagrangian equivalent to the original one in the sense of giving the same field equat ions but lacking the surface terms. Therefore, we have been able to free field theory f rom the surface terms or, equivalently, f rom the condi t ions on the boundary behav iour of the fields.

Reference

[1] P.A.M. Dirac, Phys. Rev. 114 (1959) 924; R. Arnowitt, S. Deser and C.W. Misner, in: Gravitation: an introduction to current research, ed. L. Witten (Wiley, New York, 1962) p. 227; B.S. DeWin, Phys. Rev. 160 (1967) 1113; T. Regge and C. Teitelboim, Ann. Phys. (NY) 88 (1974) 286.

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field theory without surface terms

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