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Notas de Física

CBPF-NF-037/86LAGRANGIAN PROCEDURES FOR HIGHER ORDER FIELD

EQUATIONS

by

C.G. Boiiini and J.J. Giambiagi

RIO DE JANEIRO1986

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Centro Brasileiro de Pesquisas FísicasArea de PublicaçõesKua Dr. Xavier Sij;aud, 150 - 49 andar22.290 - Riu da Junuiro, RJBRASIL

ISSN 0029 - 3865

CBPF-NF-037/86LAGRANGIAN PROCEDURES FOR HIGHER ORDER FIELD

EQUATIONS

by

C.G. Bollini1'7 and J.J. Giambiagi1'3

'Centro Brasileiro de Pesquisas Físicas - CBPF/CNPqRua Dr. Xavier Sigaud, 15022290 - Rio de Janeiro, RJ - Brasil

20n leave from Depto. de Fisica, Fac. de Ciências Exactas de Univ. Nac. deLa Plata, C.I.C prov. Buenos Aires

*Centro Latino-americano de Física - Brasil

CBPF-NF-037^86

ABSTRACT

We present in a pedagogical way a Lagrangian procedure for

the treatment of higher order field equations. We build the

energy-momentum tensor and the conserved density current. In

particular we discuss the case in. which the derivatives appear

only in the invariant L>'A\embertian operator. We discuss some

examples. We quantize the fields and construct the corresponding

Hamiltonian which is shown not to be positive definite. We give

the rules for the causal propagators.

Key-^ords: Higher order equations; Lagrangian procedures; Field

theory.

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1 INTRODUCTION

**

Ever since the advent of differential equations foi the

description of physical systems, the consideration of higher or

der field equations has# always been present in the mind of phy_

sicists. It is almost impossible to mention all references to

earlier works on the subject, but we would like to point out

that a Lagrangian treatment was already present: in Courant-Hilbart's

book'- J, and the classical retarded Green function was cons-

tructed in reference

Nowadays the subject is acquiring increasing importance due

to the consideration of gravity theories with Lagrangian con-

taining terms quadratic in the curvature tensor *- -• . Further-

more, supersymmetry in higher dimensions leads to higher ordei.

equations, so that dimensionality of space-time could be re-

lated to the order of the field equations'- -I L. -• . Also in this

context,it might turn out to be impossible to get complete

conformai invariance with fields obbeying only second order

wave, equations *- -* .

It seems then convenient to attack thrs problem with cano-

nical methpds, trying to understand and overcome, if possible,

the difficulties one encounters1- J .

It is for these reasons that we here developé a

general formalism for higher order equations starting from a

Lagrangian and building up from it, the canonical tensors. In

this respect we don't pretend to be fully original, but we

rather try to systematize in an easy way, the procedures that

can be followed for the development of the theory.

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For an alternative, more mathematical, viewpoint see re-

ference Q8[] •

We should also mention that it is possible to follow Schwinger's

P9~!Action integral methods'— J the canonical tensors coinciding,

up to divergences, with those obtained here. The equations of

motion are, of course, the same.

Lastly we would like to point out the general appearance

of negative energy states, which should be related to an in-

definite metric in the "Hilbert space" of states, for whose

treatment several references can be given (see for example Q-O]h

It should be pointed out also that the whole atittude and

philosophy regarding the usual S-matrix problem should be

changed in theories of higher derivatives. This is one of the

main problems which has tp be clarified in the near future if

one intends to go ahead with higher order equations. We intend

to treat this problem in a forthcomming paper.

2 CANONICAL TENSORS

We start with a Lagrangian function of a scalar field i>

and of its first m derivatives, in an n-dimensional space-time,

(2.1)

The principle of least action A, allows us to write

A «= I d"x c& , (2.2)

- f dnx« £

After integrating by parts the variation of the derivatives

of the field <fr,(2.3) and (2.4) lead- to the following Euler

«quation t•

+ 3a

+ a-

n

(2.5)

Similarly, we can deduce the generalized Nõther theorem.

1^ + ... \ (2.6)

In particular we construct the energy-momentum tensor

by considering inifinitesimal translation:

, ... etc. (2.7)

Replacing (2.7) in (2.6) we get»

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(2 .8)

where:

• ) •

(2.9)

This tensor i s not necessar i ly symmetric but i t can be sym

ne t r i zed following Be l in fan te ' s procedure"- -I. Anyway, as the

symmetry i s broken by divergence terms, the total energy-momentum vec

tor i s well defined by (2..!»).

>W = J d n^2.10)

When $ i s complex and the Lagra^.Tian i s phase- invar ian t , ;one i s

led from (2.6) to the conserved cur ren t : fô<{>=i£<í> ; 6<|>* =-ie<J>*)

j ° = ic < rt—x — 3. + . . .

. . . - c.cA

(2.11)

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3 FUNCTIONS OF ITERATED D'ALEMPERTIAN

We shall discuss special case in which the Lagrangian

is a function of the derivatives of <J>, only through the in-

variant D1 Aleinbertian operator:

D = naeaa<>6 (3.D

We define

<|>(i0 = D s * (*Co) = *) (3.2)

c£=cG(4,(s)) (s = 0,1/-.-M)

The principle of least action now leads (cf. (2.4), (2.5)),

to the Euler equation:

2 f+D 2 -7-fT+ ... = 0

Or:

M

I s

Similarly, we have the Nother theorem:

I ( 8

(3.4)

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(Conpare with (2.6)). *

From (3.4) we deduce, using (2.7), the

tensor

energy-momentum

f la'-s,t*0 I 3*

(s+t+1) -»an8

-3 U(3.5)

(Compare with (2.9)).

This tensor can again be symmetrized following Ref.

It is easily seen that

3pT"v - o

The conserved current takes now the form

where ch means hermitian conjugate

(3.6)

-ch

(3.7)J

4 EXAMPLES

a) Let us begin with the simple example of the usual Klein-Gor

don equation, treated from the point of view of a higher order

Lagrangian,

«6» - \ ** (4.1)

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from which

giving the equations of. motion

- V2** -§ 0<fr* +O(- | 9*) = 0 (4.3)

For the energy-momentum tensor:

V

and for the Hamiltonian

T S3 » —- A * A «̂ yp m™m •• ^ç mm* ^ <v © A *

- f- ^ 9*G4» - -j 9D9* - P29*9 )

(4.4).

- " T •*

trtiich leads/ up to a divergence, tó

H» |V 9 | 2 * | i | 2 • P 2 U | 2 (4.5)

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The current (3.7Í is here the usual one:" j v =-ie

b) Another example.

- 5 l»V (4.6)

leading to the eq.. of motion

- 0 ie: D G $ -u*$ - 0 (4.7)

and for the energy-momentum tensor

" a 9 d <p — n, 03 (4.o;

The Hamiltonian is given by: • '

• • + 2 p2*2 - (4.9)

Using Fourier development for <f>,

• (x) - ake*iltxi(k) (4.10)

The equation of motion (4.7) implies:

(k* -y")i - 0 (4.11)

I.•»

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4>2(k)ôtk2 + y2) (4.12)

And we can see the appearance of two kind of particles. One

"normal" with k2 «= p2, and another "abnormal", with negative

mass square: k2 = -u2. This second particle is a "tachyon"*- -*.

c) Let us consider a slightly more general example L -J

(4.13)

With Euler equations:

o - ! (raj +m|) , B = mjm2 (4.14)

(D+mj)(GJ+m2) • - 0 • (4.15)

From (3.5):

Taking the Fourier transform of

(4.16)

dke"ikx}(k) ; í*(k) - $(-k) (4.17)

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and using equation (4.15):

(-k2 + ra2)(-k2

Which means that:

• (k) » • 1 (k )ó (k 2 -mj )+ •2(k)«(k2 -m2)

6(k +u . )+6(k - « . ) 6(k +w-)+6(* -w .(k) + 4 > ( k ) ^_ -

(4.18)

Vie2 +mj , w2 = VKZ +m2

We have then:

'2 '(4.19)

total Hamiltonian is obtained now, replacing (4.19) in

(4.16) and integrating over the space variables. -The result i«:

I (4.20)

Note the difference in sign between the contributions of

the partial fields <P^ and <f>2to the total energy. This is a

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general feature of higher order equations (see for example

Ref. £7]). It» quantization can be carried through from (4.20) by

imposing Heinsenberg equations of motion:

|H , (4.21)

Implying for (4.19)

| H ,*l

Wiese commutation relations together with (4.20) give

\2[ -1

< t > ( £ ) , * * ( £ ' )-1 l J m í " n 2

(4.23)

_ 2 _ _ 2m l m 2

With a s ign i f i ca t ive difference in sign. One of the partial

f i e ld s gives an indefinite metric to the Hubert space of par-

t i d o s t a t e s . (See for example Ref. (^lOj).

In view of the relat ions (4 .23) , we can redefine the fields

• j and <f>2 in such a way that (we take m̂ > m*):

mj -(4.24)

Thus we get:

H - j •dkjw^j - » (4.25)

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and

= 6(k-k') ; U,(k),$*(k') 1= -óík-fc') (4.26)

5 MANY MASSIVE STATES

Just to establish the pattern we will first consider the

following equation of motion:

(D+io])(D+m=) ( D+m*)(D + mj) * = 0 (5.1)

which can be written as:

D+ a ) + - 0 (5.2)

where

4a , - I m| ; a « I m|in| ; a

3 i - 1 x 2 x 3

The Lagrangian i s :

(5.4)

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_ A<2) 1 .CD" • + 2 a3*

(5.5)

From (3.5), the Hamiltonian density is seen to be:

/ 3&

* I —n

•<!) . £ (5.6)

Using (5.5) and the equation of motion (5.2), we get:

(5.7)

For the Fourier transform of the field, we write:

«(x) í dke~ikx$(k) ; (5.6)

+ ^(k)6(k2-ro? ) (5.9)

Then, as i n ( 4 . 1 8 ) , (4.19) we gets

i J ^ (5.10)

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From (5.7) and (5.10), the total Hamiltonian can be com

puted:

>3 +(roí-ro0(m2-<)(ra3-< ) •?•• j (5'Assuming m* < m£ < m* < m*, we see that the coefficients of

•?•, and 4»*i>_ are' possitives, while those of <f>*<i>2 and 4»?4>, are

negative (compare with j_13J and Appendix C of reference

How that the pattern is clearly established, we can gene-

ralize the results to any number of massive states.

The equation of motion is:

8 (D+inO* = 0 ' (5.12)

Or equivalently

I a . O J * = I a.«frlj) = 0 (5.13)j-0 3 j-0 J

with '

aM * X '' aM-l " I mj ' aM-2 =

*... au e s A in* ...m, /«../a — n m. (5*14)M-S jj j g o J B l j

In (5.14) the symbol ['means that all the indices are dif

ferent.

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The Lagrangian is

M

j-1 l 2 j-0a.*(j) (5.15)

From (3.5) and (5.15), (5.16) we get

s + t . 1 ( » i * ) < 6 (5.17,st *

The Fourier development of ^(x) leads to the representa-

tion:

f M

«(x) = I dkeikr I ^(•.(k)e-iwJ.t + •*(-k)eiwJt)(5.18)J j-i j J . . J '

where

u. • + V k 2 +m*J Y J

Of course

U)(x)

idkeikr J t i l l (#.(k).-

l-i + •t(-k).i"jt) (5.19)

j-1 ""j

From which, (5.17) takes the form:

(5-20)

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Taking m* < m2 < ... < •£, the sign of the coefficient of

#••. is seen to be (-1)**1.

Heisenberg equation (4.21) implies

(5.21)

Which, together with (5.20) gives:

where

and

S. « sign of JI (mt2-m?) (5.23)

Redefining now the operators ••»

— ^ * Ah —» ••(ic) (5.24)

If* finally obtain the commutation relations:

tt'Q- 8 ^ c< ({-<•) (5.25)

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and the haniltonian:

H - J d k | S . ^

H = J dk I (- (5.26)

6 PROPAGATORS

Looking at (5.12), we see that the propagator should, at

least formally, be given by:

A(x) = F

(-p2+m?)

(6.1)

Where F means Fourier transform.

Of course, the right hand side of (6.1) is not «ell defined,

as it has poles at each m?, so a prescription must be given, for the

effective calculation of F. This prescription is equivalent to

the choice of boundary conditions. The. simplest way to express

quantum causality is by analytic continuation in the coef-

ficients of the metric Ql4j / going from euclidean to hyper-

bolic one.

Explicitly, one starts from an euclidean space:

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In this way we get analytic functions (distributions) of a,

which are continued to a » i +e, generating . the well

known "ie" (See reference |~15]).

Pornula (6.1) will be well defined by the prescription

p1 • p* + ie.

Alternatively we can take the usual definition:

c = <0| T*(x)*tO)|O > (6.2)

Defining now the partial fields:

•,W = Ídki^í^.íkje-^J1 + •i(-k)eiMjt) (6.3)j j j •*

with

M

I *.(x) (6.4)

And using the commutation relations (5.22) we arrive at :

Ac(x) » I <0| T#j(x)^(O)|O>

A (x) « Z -1- AJ J )(X) (6.5)

Where A^J (x) is the usual Feynman propagator for a par-

ticle with mass m..

Taking the Fourier transform of (6.5), we get»

j»l Cj in?-p2~io

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Or, taking into account (5.23):

I jr r 7 r- (6.7)=l H (mj-m2) (m?-P

2-io)

Which is nothing but

In coincidence with (6.1) except taht the prescription "io"

is incorporated in (6.8).

Note that the rule p2 •*• p 2 + io, is equivalent to the usual

one m2 •*• m2 - io, for propagators of the form (6.1) . However

this is not true in general. Take for example equation (4.7)

whose formal Green function is:

-V*

2v»

I t is evident that for the second Fourier transform in (6.9)

the prescriptions do not coincide.

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BIBLIOGIIAPHY

R. Courant, D. Hilbert, Methods of Math. Physics. Vol. II

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L O R- Delbourgo, V.B. Prasad, J. Phys. Gl, (1975) 377

[V) C» G* Bollini, J.J. Giarabiagi. Phys. Rev. D 32 (19B5) 3316

£6 3 S. Fubini: Private communication.

Jjf3 S. W. Hawking: Preprint.Univ. of Cambridge. Depart. of

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V. Aldaya, J. Azcârraga: J. Fhys. A 13 (1981) 2545

T. Kimura: Lett. Nuovo Cim. 5 (1972X 81.

P.R. Rodrigues: J. Math. Phys.ltJ (1977) 1720.

P.R. Rodrigues: "Mecânica em Fibrados dos Jatos de Ordern

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£l23 J* DHar, E.C.G. Sudarshanj Quantum Field Theory of Inter

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£l3] A. Pais. G.E. ühlenbeck: Phys. Rev. 79 (1950) 145

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