lagrangian procedures for higher order field
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ISSN 0029-3865
CBPFCENTRO BRASILEIRO DE PESQUISAS FÍSICAS
Notas de Física
CBPF-NF-037/86LAGRANGIAN PROCEDURES FOR HIGHER ORDER FIELD
EQUATIONS
by
C.G. Boiiini and J.J. Giambiagi
RIO DE JANEIRO1986
NOTAS DE PtSICA ê uma pre-publicaçao de trabalhooriginal em Pinica
NOTAS UK KlSICA la a preprint of origina I uorka unpubl ia lied in Pliyuics
Pedidos de copias desta publicação devem ser envUdoa aos autores ou ã:
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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.
CBPF-NF-037/86
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.
CBPF-NF-037/86
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)
CBPP-NF-037/86-8-
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) - $(-k) (4.17)
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and using equation (4.15):
(-k2 + ra2)(-k2
Which means that:
• (k) » • 1 (k )ó (k 2 -mj )+ •2(k)«(k2 -m2)
6(k +u . )+6(k - « . ) 6(k +w-)+6(* -w .(k) + 4 > ( k ) ^_ -
(4.18)
Vie2 +mj , w2 = VKZ +m2
We have then:
'2 '(4.19)
total Hamiltonian is obtained now, replacing (4.19) in
(4.16) and integrating over the space variables. -The result i«:
I (4.20)
Note the difference in sign between the contributions of
the partial fields <P^ and <f>2to the total energy. This is a
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general feature of higher order equations (see for example
Ref. £7]). It» quantization can be carried through from (4.20) by
imposing Heinsenberg equations of motion:
|H , (4.21)
Implying for (4.19)
| H ,*l
Wiese commutation relations together with (4.20) give
\2[ -1
< t > ( £ ) , * * ( £ ' )-1 l J m í " n 2
(4.23)
_ 2 _ _ 2m l m 2
With a s ign i f i ca t ive difference in sign. One of the partial
f i e ld s gives an indefinite metric to the Hubert space of par-
t i d o s t a t e s . (See for example Ref. (^lOj).
In view of the relat ions (4 .23) , we can redefine the fields
• j and <f>2 in such a way that (we take m̂ > m*):
mj -(4.24)
Thus we get:
H - j •dkjw^j - » (4.25)
CBPF-HF-037/86
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and
= 6(k-k') ; U,(k),$*(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)
CBPF-MF-037/86
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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)
CIPF-RF-037/86- 1 7 -
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:
dPF-MF-037/86
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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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£6 3 S. Fubini: Private communication.
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