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Action principle for higher order Lagrangians with an indefinite metric
ANNA13
OF
PHYSICS:
Action
20,
203-218
Principle
(1962)
for
Higher Order Lagrangians Indefinite Metric*
with
an
A consistent action principle with an indefinite metric and a Lngrangian of any finite order requires that the Lagrangian be invariant under an operation called metric conjugation. This in turns implies that (i) an observable must commute with the metric operator, (ii) the S-matrix must. be unitary, and
(iii) transitions between states of opposite metric parity are forbidden. For higher order Lagrangians the energy is still the o&component of the energymomentum tensor 1”‘“. It was necessary to split the derivatives of the field variables into normal and tangential components so that canonical conjugate variables could be defined. I. INTRODUCTION
In his well-known papers, Schwinger (1) developed the quantum action prinriple for a Lagrangian containing only the first order derivatives of the field variables. It is assumedthat. the Hilhert spacehas a positive definite metric. The t,wo obvious generalizations, wit’h which we shall concern ourselves, consist of a quant,um action principle with a Lagrangian containing fin&e order of derivatives of the field variables and a Hilbert space of indefinit)e metric. h consistent action principle demands that the Lagrangian be invariant under metric conjugation. with respect to the positive
The Hcrmitian definite metric,
conjugat,e
of an operator,
defined
is replaced by the adjoint operator, defined wihh respect to the indefinite metric. We assume that t’he indefinit)e metric is nondegenerate. The details of these results are given in Se&ion II. The Lagrangian formalism, developed in Scct,ion III, owes much to t#hepapers of Misra (2)) Chang (3)) and deWet# (4 ) . However, our paper generalizes their works by developing a covariant formalism completely in terms of conjugate yariahles, by int)roducing spin in a symmet,ric way, and by proving the rquivalence of t’he Euler-Lagrange equations and the Heisenberg equations of motion. * Supported
in part by the U. S. .4ir Force Office of Scientific
t Now at Lawrence Radiation Laboratory, fornia. 1 Now on a Resident Research Associateship White Oak, Silver Spring, Maryland. 203
University
of
Research. California,
at the U. 8. Naval
Ordnance
Berkeley, Laboratory,
Cali-
201
BARUT
-4NL)
MMULLES
In Section IV, we t’reat, briefly t,he introduction of the elect,romagnet,ic field in the Lagrangian. Section V contains the summary and conclusions. We shall employ real world coordinates with t(hemetric go0= 1 and y” = - 6”. We introduce the notation E$,= a/a? and a,,(") = a,a,ah . . . (n times) with n = 0 being the unit operat#or.Also, we have a,(.)+(r) = &P~,(s). The derivat,ive is separated into a normal (time increasing) directional derivat#ive (8) and a tangential derivat#ive (a,) in the form
3, = ap -
n,a” = 2
n,8,
so that n,~”
= n,a”
- n,n”S
E 0,
(n,n” = 1).
Here also we have #k, = 82 . . . (k-times),
7x:” = n,n, . . . ( k-times).
In this paper we shall deal with two types of variations. The first is called the “substantial” variation (5) which is defined so that &,cn)(6&) = & ( dr(n$). This variation compares the transformed field and the original field at. t(he same point r, i. e., 6&(s) = $(.r) - 4(x). Th is variation does not involve any coordinate transform. The second is the totmavariation (%$) which is composedadditively of &p at the point zP and the change in +(~-I’) produced by moving from .r, on the surface c t,o .c, + 6x, on u + 6~. We will consider only motions corresponding to a local rigid displacement of the surface (d&r:, + iUs, = 0). This is just the condition that .cP+ 2’ = .c’ + 6s” is an infinitesimal Lorentz transformation. Further, if we limit the displacements to infinitesimal translations plus rotations. 6.r’ = a” + 2’“~~, the field coordinates undergo the linear transformation &‘(.r’)
- &(s)
= ~(Lm$#)b(.r).
Thus, we may write 8c$,= I&& + f&(x) - &‘(r)
= &#B,+ &p(s)G.rp - ~~&s$$“(x).
For higher order derivatives we have &#$A, = &&.)
+ &pdxY
- ~~~~~S:b&~~n~ .
Finally we remark that for he,, linear in .r, , the variation of the derivative is given by
HIGHER
This leads to the important
II.
HILBERT
SPACE
ORDER
LAGRANGIAN
205
THEORY
relation
WITH
INDEFISITE
METRIC
AND
ACTION
PRINCIPLE
We first review very briefly the properties of a Hilbert space (4) with an indefinite metric. For details we refer the reader to the review papers of Pandit (6) and Nagy (7 ) In order to have a physically interpretable theory the Hilbert space ( $1 must cont,ain a subspace (A?‘) of positive definite metric whose vectors describe the physical states of the system (8). A linear operator in the total space (@) is an observable if it acts as a Hermitian operator on the physical st#ates of $1’. h linear operator I? is introduced into Q so that the norm of a vect#or is defined 1)) II ti II = (* I r I #), which can be positive, negative, forms, i. e., reality condition,
(1)
or zero. Since we are int,erested in self-adjoint
(4 I r I+> = (+ I r I ti)
(2)
where the bar denotes complex conjugation, IY must be Hermitian with respect to the positive definite metric. The positive definite metric is assumed to exist,. We furt,her assume that r is unitary (I’+ = r = I?‘) and that the metric is nondegenerate. ?;ondegenerat)e means that t,here does not exist a + # 0 such that (#I rl+) = Oforall4inQ. The nondegeneracy of the metric allows us to introduce a complete orthonormal set of stat#es { fi;). We get, the completBeness relation by expanding a vector 4 in $5 in terms of ($i} . Hence (#; / r j 4) = F
Ci(lc/j
where now 11I/~ 11= Ni = fl.
1 r 1 fiii = T An arbitrary
14) = c N&i
GijCi
Ij #j 11= Ci I/ $i Ij ,
vector can then be written I r 14) 14%)
as (3)
and, Dhus, we have the ident’ity I = c We introduce
a linear operator,
Ni / Si>(#? I r.
0, and define it,s adjoint,
(1 I r I 04) = (otti I r Id.
(41 O’, by
The expectat#ion value of 0 follows from the above relation and is real for selfadjoint operators. The relat#ion between the adjo& and t,he Hermitian conjllgate, O’I.~., defined with respect t,o the positive definite metric is the following: O+ zz r0h.c.r.
(5)
The self-adjointness of 0 does not insure its diagonalizability. However, if r commutes with Oh.c. then a self-adjoint’ operator is diagonalizahle by virtue of its Hermit&y. Consider a nonsingular transformation S which leaves the metric invariant, i. e., leaves Dhenorm of all vect’ors invariant. As usual, we have thah 0’ = SOS-’ and XtS = 1. However, now S-’ = I?Sh.‘. r; S is pseudo-unitary with respect,to the metric I’. Lastly, we int’roduce the concept of metric conjugat’ion as follows. Let) r
=
firi, L
m,
r,l-
=
0,
r loi
=
where n = number of fields in the theory. We define m&c following relations J/i = r&jr;
= +ji # j,
tii'
=
riGiri
10)
(6)
conjugation by the =
ciqi
17)
where ci’s can always be set equal to fl by a suitable renormalization of the +i’s. If ci is equal to plus (minus) one, then fii has positive (negative) metric parity. We then have fi> = c&“‘. The metric operator ri, due to its Hermiticity, can be made diagonal in the representation labeled by the eigenvalues of t,he number operator of t’he ith field N;: n i where n i = (c i) n’. By metric invariance, it is meant that for any operator 0’ = I’Or = 0. Consider now two spacelike surfaces (TVand c2on which the dynamics is known. We introduce on u1 and u2 a complete set, of operators & and & , respectively, with the corresponding set of eigenvalues &’ and &‘. The kansition amplik~de between these two sets is (&
1 r 1duz).
(8)
A variation of this amplitude, assuming I? is not a dynamical variable, is given by 6(&,
I r I &)
= i(41’ul / mw12 / tA2)
= (&‘a1I r 1&2) - (&1 I r I .G2 ).
(9)
The state ) zisi) is infinitesimally close to ( Ei’ai). The reality condition (2) along with (9) leads to 8W13= 6W,n + 6wt~ ) 6W12= -a?%1 ) 8W12= BWi2 .
(10)
HIGHER
Remember
ORDER
LAGRASGIAX
that (t) is the adjoint
operation.
207
THEORY
If
Wu = 1” d4.2- s(.rj, CT”
(11)
then d:(x) must also be self-adjoint. Let 1{i’irL) be related to j (i’g;) by a pseudounitary
transformation
1f,‘lTi) = r: / &Ti) with
I’ = exp (-iF)
= 1 - iF. F must he self-adjoint 6 / (i’ui)
= -iFi
with
the result that
/ tj’ui),
6(,ti~i 1 = (.$i’ui 1iFi. From our transition 8(&l
amplitude,
I r I &z>
we have
= 6( (61 Cl = i(&q
j ) r
I &2>
1 (FlI’
-
+
(5l’Ul
I rq
I &2))
T’Fz) 1&‘uu)
= i(tl'ul / rsw12 I 12'u2) 01
swlz = rFlr - F2. An analogous result is obtained
(12)
for the variat’ion
SO = -i(OF
-
of an operator,
0,
rFr0).
(13)
The important difference from the conventional formalism is t,he appearance of the met’ric operator in the last two relations. If r were to commute with an arbitrary generat,or F, then the only difference between the indefinite case and t,he posit’ive definite case is to replace adjointness by Hermiticity. In the next section we shall prove the following theorem: if a *field theory is yuantized using either commutators or anticommutators and an indejkite metric, then the consistency oj’ yuantization demands the invariance of the Lagrangian under metric conjugation. This, as we will see, leads to generators which must commute with the metric operator. Therefore, relations ( 12) and ( 13) reduce to t,he conventional form. III.
L.AGRASGIAN
A variat,ion
FORMALISM
24ND
of the action can be written
COMMUTATIOX
RELATIONS
as
which involves the effects of varying the field components at each point by 64 and altering the region of integration by a displacement 6.~” of the points on the boundary surfaces.
The variation
of t,he Lagrangian
density of finite order N is
&, 2 = 2 -al: 6,, &,L, . n=O ac#$tn,
i 15)
We define the functions (Iti) which,
due to their symmetry,
obey the relation
where x
ps(“)
E
N-l n+l)
5,
(- 1)n~aX~mlL~Y(n)X(m).
(10)
As an example we obtain for N = 2 and n = 1, 7rp" = w$
(-l)"ax~m~~,~~X(m)
= Lp"
and for N = 3 and 'n = 0 3-1~2
2 = ,% (-i)"axcm~L~X(m' = L" - aiLpA + aXa,LfiXp. In Eq. (18) 6”~ is underst’ood to be symbolic in that the order of operators should not be changed during the processof variation. Further, we assume’that the commutation relations of 6& and the structure of the Lagrangian are such that identical contributions are obtained from terms differing only in the position of A&. Using Gauss’ theorem, and inserting (18) into (14), we obtain t,he EulerLagrange equations and a function ~(a)
go (--I)"
a+) Lfi(") = 0
("0) (21)
1 See ref 1 (a),
p. 917.
HIGHER
so that the relation
ORDER
LAGR.4NGIAX
(14) can be written 6W1? = 5(q)
THEORY
209
as - 5(u‘J.
We can associate the funct’ions F( ni) with the physical infinitesimal generators of Eq. (12) only if 5((ri) commutes wit,h the metric operat.or r. Whether t!he commutator of I? and F is zero or not depends on the commutation relations between r and t,he fields ($,I. From relation (7) we have [I’, $J* = 0. Since r is not a dynamical variable, this implies [I’, Gid.cn~+& = 0. Thus, in (21) if 8oaVi,l>+i commut8cs ( anticommutjes) wit’h I?, then ?r:“(“) must commute (anticommut’e) with r so t,he second term of Eq. (21) commutes with r. Sow the commutation relations of the $” ‘““s with I’ depend on the structure of the Lagrangian density which is bilinear in the field variables and their derivat’ives. From its definition in l&l. ( I!)), we see t’hat, STY’ is proportional to t,he coefficient of &(“P$~ appearing in the Lagrangian. Therefore, the coefficient of &cn$i , which is of course a function of &cl l+j’, must commute (,anticommutes) with I?. Thus the Lagrangian density must be invariant under met,ric conjugation. It, will be shown that t*he conditions (a), (b) and (c) of the previous paper (9) follow from the metric invariance of L. We may rewrit’e (12) and (13) as
and 60 = - i[O, F(a)]-.
(13’)
In addition, the equation
[r, F(u)]- = 0
(22)
tells us that physical observables commute with r since the generators contain the physics of the sysbem. In other words, from is), our generators are also Hermitian operators as is needed (see first paragraph of Section II). We are now in a position t’o study Eq. (21) in detail so as to obtain the canonical commutation relations, the energy-moment8um tensor, etc. In order to make the analogy with classical mechanics, it is necessary to separate dvcn@and iTr’(n) intro canonical and const,raint variables. Thus we wish to write Eq. (21) in terms of the independent variations of the field variables on the surface. Note that once the value of 6& on the surface is known, the variations &,&c~N$ are not completely arbitrary but tZheir variations in t,he normal direction are arbitrary’ (10). We shall further restrict ourselves to surfaces such that a(“&+ tends to zero sufficient,ly rapidly at spatial infinity. Thus for any function gy(.r), we have / da, &(f’“) 2 See ref.
S(a),
p. 4G5.
= /du$j”
= 0,
da, = ,n,,da
(23)
210
BARUT
ASD
MTLLEK
and k 6 N - 1. To prove this let us choose a part,icular where j”” = g” d“(‘-‘)&#J a spacelike surface (n, = (1, 0, 0, 0) j, then jdugd”
= jd3.r(a3’
zz
- n,i3y> = jd3d3”fu
jd32(aOfB
+
diP”
--don)
=
+ a,P” - n, m ay’,
jd3Ndjji
=
jdS'f
r~
0.
This will insure us that the Too component of the stress tensor is the energy density.3 Xow once 60(a”‘+) is known on the surface u, a,( 608’~‘+) is determined; hence only 60( 8(k+1)+) is arbitrary. The second term in (21) can be rewritten as N-l
N-l s
da, c ?T’”(%~,(n~ so cj, = II=”
du, c s
~“~(~‘(rjv + n,#%o
d
II=0
N-1 =
s
da, Go (-1)” + . . . + (-1y
;
~n:k’t3ycn-kl 7rPV(“)$(Q~O $
0 + . . . + (-1)”
; 0
n2”’ ?ypv(“) p
so 4
(24)
1 + j da,9,.B”“,
where
n
0n1
= ,n!/m!(n
- m)!.
Here a,B”” is a complicated expression containing 8(k’&$; its integral da, vanishes by virtue of (23). Collecting terms in (ail), we have
over
where we have defined (25) The generat’or becomes (26) 3 See ref.
4,
p. 551.
HIGHER
ORDER
LAGIZ.4NGI.4N
211
THEORY
We substitute the total variat,ion & for &#I by noting that, when
Hence F(u)
= i‘ da, [gl
[7F%(P+)
- n’k”d,(8(k’c#&l.d (27) + ?.pP”S,a
We define the quantit’y ~&A”- ], (k)bpp4 1 (&I = .2 h with j”;k; = -j*$.
iY”‘+] -
,
+ g(k)vShrgq + ph~q(k)+,
(23)
r\;ot,e also that &;
= ,lJj’::;
- f$)
= -a~(f:;;&r,j
CA”= f@- wrSXvg(k~~tA” + dp&r”
The first term is zero from the antisymmetry of j$; and from the theorem s
da, dhj+” =
s
&A & p.
The final result is
(29) where
The conservation laws arc arrived at from rigid displacements of the coordinate system given by 6x” = up + ~““x, and 8(8’“‘+,) = 0. The displacement generator can be wri-tten as
where
with MAW = -j@’ Since conservation integral invariant, J”“( ul) = Y’(us).
E TXfiZY_ Txvx”.
(32)
laws are associated with variations that leave the action we see that F(u,) = F(az) implies PP( ul) = P”( a?) and This leads to the differential conservation laws d,T”’
= &MA’”
= 0,
(33)
which in turn imply that T’” is symmetric. The conservation of “charge” can be obtained from the required invariance of the self-adjoint (also Hermitian due to metric invariance) Lagrangian under constant phase transformations exp ( f ic%X) where c’ = fl, 0. With &r, = 0 and 8(8’“‘+,) = it”MX8’“‘+, , we have Fax(u) = / cLT,@~ a~k”(ik%sx~(k)~,) k=”
zsxQ(u),
(34)
where we have defined N-l
Q(u)
=
1
da,
(35)
j',
The field equations fix 6: up t’o an arbitrary divergence terms since the EulerLagrange equations are invariant under 6: ---f 6: + d,h" with h" being metric invariant. However, WI2 is not invariant under this subst8itution. This change in W,, can be expressed as an alteration of the basis eigenvectors on (rl and g‘2 . Consider the generator of the infinitesimal transformation, H(a), defined by 10’) = (1 - iH) 10’). Then the new WI2 is obtained from (#(ul) 1 I? / 6" (Us) ) = (O’(U~) 1( r - iI?H? + iH,J?) I O”(uz)). With H = I du,h” we get the result SW12= y’ d4xd, h’“. g2 In particular, if we choose h” = -i( F’(u)
= [ da, (F
x:1;
=r(k)“8(k)+)so that we have
~(?#~)cf~ + h” - Ty’iirp)
or
(36) F’(~)
= - / au, &’
&r(k)p8(k)++ T”‘Gr,) .
Therefore F (a), Eq. (29), corresponds to SW = p6q - HFt and F’(u), Eq. (36), corresponds to 615’ = --6pq - H6t of classical particle mechanics. Defining
HIGHER +(h-I
~
,rLp?T(k)~
ORDER
LAGRANGIAS
21:3
THEORY
and 5?’ E npTBY, we write
(37) F(6x)
= -1
daYi%s, .
E’rom our analogy with particle mechanics, we can identify F(c%‘~‘~) with t#he generator of transformations 8(L)+ + g(k)+ - @@‘+ and ?ick’ -+ iick’, while F( &?lk)) with the generator of 73(k)-+ iick) - %?(” and a(k)+ -+ ack)+. From (13’) we obtain the commutation relat,ions [#“‘4, F( ,j#k’$)]-
= &‘“‘&
[j;@-,, F(@‘“‘+)]-
= 0
[i;(k), F(&+k))]-
= &‘“‘;
[#“‘c$, F(h?(k))]-
= 0.
(38)
At this point one conventionally would use the relation [9, BC]- = [A, B],C B[A, C], and assume that for bosons 67i’k’ and 8ackJ4 commute with all quantities and for fermions they would anticommute. However, we now know that a successful qu:tnGzation can be carried out for bosons whose particle operators anticommute f.o plus one and antiparticle anticommutes to minus one if an indefinite metric is used. The opposite (commutators) is true for fermions. Therefore, we only say t,hat if the independent canonical coordinatIes (8’“‘4) obey bose (fermi) statisttics, then 89(k) and 88(k)+ commute (anticommute ) with all quantities independent of the spin statist*ics connect’ion. We then obtain from Eq. (38) the final result,
[8(k)4e(.rj,+L’),(.r’)]*
= iP6,b6,(~
- s’),
(.v - 2’)’
<
0
(39)
where s
da~(z’)d,(a
- 2’) = f(x).
This relation is the main result of the present formalism. The consistency of the formalism is proved by showing that the Euler-Lagrange equations (20) are equivalent t,o t,he Heisenberg equations which are derivrd using relation (39) and the displacement, generator F(6.r). It is sufficient t’o show that, a(k)+ and jjck) are indeed canonical conjugaks and that Too is the Hamilt80nian density of the syst,eni, i. e., 6H @qj=T
I (k), ’
6H= S(iFk))
-JCk)&
(10)
To prove these relat,ions n-e consider a family of spacelike planes and such that sn = 1 = constant,. In t,his “natural” coordinate system i 1, 0, 0, 0 j . Now from (30) we have
a
frame II, =
C-21) where ~3”~~)meaus that t,he time derivaGve becomes
is taken
k-times.
From
(25)
A(‘)~
e,a”‘k+l),(+, %&)” = z (-]y+k (;) ,a”(,& but in our particular frame of reference il~(,-k)~O(k+l)“(n-k) = ay(~~_k)?yO(k+l),(~l-ki
-771,(n-C~-(n-tinO(k+‘)“(“-t) a = a,lcrt-k,a~Jc't+l) + al(n_k)aO!k+l)i("-k)_ a,l(n--k)TO(rL+l) "(k+l)j(rt-k) = aicr,-k)?r
and Iye obtain
where the index i = 1, 2, 3. Next we evaluate 6H. From it’s definition,
we have
6H =
(43) , or, using Eq. (41), N-l
6H = =
1{
dsx C [&(k)Oa’J(k+1)4 + n(k)u,j(ao(kfl~~)] _ $
k=O N-l k&o
[jQ)“$(k+l)$
+
T(k)Uj(ao(ktl)~)]
-
2 n=U
,Q + $$ ,Qr + . . . P jy(n)~~p(“)
.
HIGHER
Neglecting
t,otal divergences
ORDER
LAGRANGIAN
THEORY
and using Ihe symmetry
213
of L’(“) and +,,cn) , we have
N-1 + . . . + =& &+~“‘a”(k+“~ . (
)I
Comparing
(4s) and (J-l), we obtain the relabions
Finally we use the Euler-Lagrange equat,ions and the definitions arrange the right hand side of the ahovc relat,ions and ohtain
of sCk”, re-
The equations (4.5) are the desired relations in the part’icular frame of reference we are working. Hence we have shown that gik)+ and 73(k)are canonical conjugat’es and that Too is the energy density. IV.
ELECTROMAGNETIC
IXTERACTIOXS
In this se&on WC wish to introduce elwt,romagnetic interactions inO0 our Lagrangian C = G(&cn) , &! ~~1) in a gauge invariant, way. If we add the electromagnet,ic w&or in the usual way ( a, ---f I), = d, + ieL4,and Dp* = d, - ied,) then the Lagrangian C(qQn) ) &I, is obviously
gauge invariant.
--) .e(D,c&
Sow however
D&$+1
the order of the Dp(n)‘s is important
216
BARUT
since they do not commute.
AND
RlULLEN
With this new Lagrangian
the Eq. (20) becomes (46)
where
D,(., = D,,D,,
. . . D,,
and
D (n), = QJL
. . . D&i,
.
The reason for the inverted order for DTn),, in (46) is the following. an arbitrary term in the Lagrangian such as
Consider
D,*D”*$+#“‘. We write II/“” to stand for a function linear in 4 and whose indices (p, Y) insure Lorentz covariance of the Lagrangian. Let us look at t.he 8&t variation of this term
D,*D”“Go&‘“, which we write
as
= ara”6~+~“”
- i”s,$.+A ,,24“#“” - ied,(A.6~+)~p”
- id,a”s~+pt)s,q”.
This last term is equal to
= &+#~~{[a,a,,- e”A”A, + ie(A”a, + ayAp)]@‘“) - ied,[6,&t(,A”J/‘” - ,4”#“‘)] = 6,y$+D”D,~“” + &BP. The B” term is a total divergence which does not contribute to the equations motion. For an arbitrary number of derivatives, we then have
(D,“D,* A S&+ variation relation ( 47 ) .
. ’ . Dn*s,$+))J/‘“~~~~~+(D~ e . . D”Dg,V”“.)
of the Lagrangian V. SUMMARY
will lead to equation
+ d,B”. (46) by virtue
of
(47) of the
AND CONCLUSIONS
For a consistent quantum field theory using either commutation or anticommutation relations and a Hilbert spacewith an indefinite metric, we demand the invariance of the Lagrangian under metric conjugation defined in Section II. This invariance insures the Hermiticity (observability) of the generators F (i.e. P’, J”“, Q, etc.) and of the unitarity of the S-matrix. In other words the
HIGHER
ORDER
LAGRANGIAN
217
THEORY
metric invariance of d: leads to zero commut,ation relations between I’ and F and S. In the previous paper (9) the Hermiticity of ohservables and the unitarity of the S-matrix were listed as condit,ions (b ) and (c) , respectively, for a consistent physical theory. In order to explain electromagnetic and strong vector coupling (meson) intjeractions w&h 2-component spinors, the condition (a) of metric invariance of transition amplitudes was postulated, i. e. cd, WJ’I = ($7 I?) where c+cd must that transitions prove that this Lagrangian. To
= w9(1J/,
r-41,
equal 1 and leads t,o Q = cm and $’ = rqr = c&. This means between states of opposite metric parity are forbidden. We now result is also a consequence of the metric invariance of the see t,his consider the arbitrary transition (91 rol+)
and use the relations +l
=
cjtil.
=
r = n ri, 2
riwi,
ci = fl,
to write
4 = c.+r+r.
ti = dw, Now
we write
the transition (0 I wo+
Remembering
symbolically I 0) =
c+c+(~
as 1 rprrorg
/ 0).
also that r lo)
= I o>,
r” = 1,
we may writme
(ti I ro Id4= W& I or i 4). We have shown
that for physical
observables
0 must commute
with
I?, hence
Therefore c+ = c4 or the states 1 +) and / 4) must have t’he same metric parity for a nonzero physical transition amplitude. The concept of metric invariance was necessa.ry in order to develop an action principle using either commutation or anticommutation relations. A physical interpretation for this action principle, conditions (a ), (b), and (c) staled above, is obtained from the metric invariance of the Lagrangian. The other problems encountered with the action formalism were t,he definitions
218
II.\ Ill:T
AND
MULLI*:N
of canonical conjugates and the interpretat)ion of t,he energy operator. In t,he former, it was necessary t#o split’ t’he fields (a,~~$) into normal (canonical) and t,angential component’s. The expression for the momenta canonical to a(k)+ are no longer simple but deprnd on the tangent,ial derivatives of t(he fields. In t,he derivation of t)he energy-momentum tensor 7”” an additional term appeared in our generator. However, this nddit,ional km, which is a total t#angential dcrivative d,B’“, can be assumed t(o t’end to zero at’ spatial infinity. Thus, in the higher order case the energy again is the T”” component of 7”“. The consistency of our action formalism for a higher order theory lies in the fact that from our definit’ions of canonical conjugatles and energy, we are led to Heisenherg equat,ions of motion equivalent to t!he Euler-Lagrange equations. RECEIVED
: February
28, l!)G2
1. J. SCHWINGER, Phys. Reu. 82, 914 (1951a); ibid. 91, 713 (19511)). d. S. P. MISRA, Im&zn J. Phys. 33, 461 (1959a); ibid. 33, 520 (1959h). 5. T. H. CHANG, Proc. (‘ambridge Ph.il. Sac. 44, 76 (1948). 4. J.S. DEWET, Proc. Cnmbridge PhGl. SOI,. 44, 5-X (1938). 5. E. M. Co~sori, “Introduct,ion to Tensors, Spinors, and Itelativistic Wave Equations,” p. 13. Hafner, New York, 1953. 6. L. Ii. PANDIT, NUOZJO cimento Isupp2. 11, 157 (1959). 7. For a complete bibliography up t,o June 1960 see R. 1,. XAGY, h’trouo c~inrenlo Suppl. 17, 92 (1960). 8. R. ilsc0~1 AXD 13. MINARDI, Xuclrar P&p. 4, 2-Q (1958). 9. A. 0. BARUT AND G. H. MITI.I.EX, Anr~. phys. (iv. E’.) 20, 184-20;! (1962). 10. W. THIRRING, “Principles of Quantum Elect,rodynamics,” p. 36. Academic Press, New York, 1958.
OF
PHYSICS:
Action
20,
203-218
Principle
(1962)
for
Higher Order Lagrangians Indefinite Metric*
with
an
A consistent action principle with an indefinite metric and a Lngrangian of any finite order requires that the Lagrangian be invariant under an operation called metric conjugation. This in turns implies that (i) an observable must commute with the metric operator, (ii) the S-matrix must. be unitary, and
(iii) transitions between states of opposite metric parity are forbidden. For higher order Lagrangians the energy is still the o&component of the energymomentum tensor 1”‘“. It was necessary to split the derivatives of the field variables into normal and tangential components so that canonical conjugate variables could be defined. I. INTRODUCTION
In his well-known papers, Schwinger (1) developed the quantum action prinriple for a Lagrangian containing only the first order derivatives of the field variables. It is assumedthat. the Hilhert spacehas a positive definite metric. The t,wo obvious generalizations, wit’h which we shall concern ourselves, consist of a quant,um action principle with a Lagrangian containing fin&e order of derivatives of the field variables and a Hilbert space of indefinit)e metric. h consistent action principle demands that the Lagrangian be invariant under metric conjugation. with respect to the positive
The Hcrmitian definite metric,
conjugat,e
of an operator,
defined
is replaced by the adjoint operator, defined wihh respect to the indefinite metric. We assume that t’he indefinit)e metric is nondegenerate. The details of these results are given in Se&ion II. The Lagrangian formalism, developed in Scct,ion III, owes much to t#hepapers of Misra (2)) Chang (3)) and deWet# (4 ) . However, our paper generalizes their works by developing a covariant formalism completely in terms of conjugate yariahles, by int)roducing spin in a symmet,ric way, and by proving the rquivalence of t’he Euler-Lagrange equations and the Heisenberg equations of motion. * Supported
in part by the U. S. .4ir Force Office of Scientific
t Now at Lawrence Radiation Laboratory, fornia. 1 Now on a Resident Research Associateship White Oak, Silver Spring, Maryland. 203
University
of
Research. California,
at the U. 8. Naval
Ordnance
Berkeley, Laboratory,
Cali-
201
BARUT
-4NL)
MMULLES
In Section IV, we t’reat, briefly t,he introduction of the elect,romagnet,ic field in the Lagrangian. Section V contains the summary and conclusions. We shall employ real world coordinates with t(hemetric go0= 1 and y” = - 6”. We introduce the notation E$,= a/a? and a,,(") = a,a,ah . . . (n times) with n = 0 being the unit operat#or.Also, we have a,(.)+(r) = &P~,(s). The derivat,ive is separated into a normal (time increasing) directional derivat#ive (8) and a tangential derivat#ive (a,) in the form
3, = ap -
n,a” = 2
n,8,
so that n,~”
= n,a”
- n,n”S
E 0,
(n,n” = 1).
Here also we have #k, = 82 . . . (k-times),
7x:” = n,n, . . . ( k-times).
In this paper we shall deal with two types of variations. The first is called the “substantial” variation (5) which is defined so that &,cn)(6&) = & ( dr(n$). This variation compares the transformed field and the original field at. t(he same point r, i. e., 6&(s) = $(.r) - 4(x). Th is variation does not involve any coordinate transform. The second is the totmavariation (%$) which is composedadditively of &p at the point zP and the change in +(~-I’) produced by moving from .r, on the surface c t,o .c, + 6x, on u + 6~. We will consider only motions corresponding to a local rigid displacement of the surface (d&r:, + iUs, = 0). This is just the condition that .cP+ 2’ = .c’ + 6s” is an infinitesimal Lorentz transformation. Further, if we limit the displacements to infinitesimal translations plus rotations. 6.r’ = a” + 2’“~~, the field coordinates undergo the linear transformation &‘(.r’)
- &(s)
= ~(Lm$#)b(.r).
Thus, we may write 8c$,= I&& + f&(x) - &‘(r)
= &#B,+ &p(s)G.rp - ~~&s$$“(x).
For higher order derivatives we have &#$A, = &&.)
+ &pdxY
- ~~~~~S:b&~~n~ .
Finally we remark that for he,, linear in .r, , the variation of the derivative is given by
HIGHER
This leads to the important
II.
HILBERT
SPACE
ORDER
LAGRANGIAN
205
THEORY
relation
WITH
INDEFISITE
METRIC
AND
ACTION
PRINCIPLE
We first review very briefly the properties of a Hilbert space (4) with an indefinite metric. For details we refer the reader to the review papers of Pandit (6) and Nagy (7 ) In order to have a physically interpretable theory the Hilbert space ( $1 must cont,ain a subspace (A?‘) of positive definite metric whose vectors describe the physical states of the system (8). A linear operator in the total space (@) is an observable if it acts as a Hermitian operator on the physical st#ates of $1’. h linear operator I? is introduced into Q so that the norm of a vect#or is defined 1)) II ti II = (* I r I #), which can be positive, negative, forms, i. e., reality condition,
(1)
or zero. Since we are int,erested in self-adjoint
(4 I r I+> = (+ I r I ti)
(2)
where the bar denotes complex conjugation, IY must be Hermitian with respect to the positive definite metric. The positive definite metric is assumed to exist,. We furt,her assume that r is unitary (I’+ = r = I?‘) and that the metric is nondegenerate. ?;ondegenerat)e means that t,here does not exist a + # 0 such that (#I rl+) = Oforall4inQ. The nondegeneracy of the metric allows us to introduce a complete orthonormal set of stat#es { fi;). We get, the completBeness relation by expanding a vector 4 in $5 in terms of ($i} . Hence (#; / r j 4) = F
Ci(lc/j
where now 11I/~ 11= Ni = fl.
1 r 1 fiii = T An arbitrary
14) = c N&i
GijCi
Ij #j 11= Ci I/ $i Ij ,
vector can then be written I r 14) 14%)
as (3)
and, Dhus, we have the ident’ity I = c We introduce
a linear operator,
Ni / Si>(#? I r.
0, and define it,s adjoint,
(1 I r I 04) = (otti I r Id.
(41 O’, by
The expectat#ion value of 0 follows from the above relation and is real for selfadjoint operators. The relat#ion between the adjo& and t,he Hermitian conjllgate, O’I.~., defined with respect t,o the positive definite metric is the following: O+ zz r0h.c.r.
(5)
The self-adjointness of 0 does not insure its diagonalizability. However, if r commutes with Oh.c. then a self-adjoint’ operator is diagonalizahle by virtue of its Hermit&y. Consider a nonsingular transformation S which leaves the metric invariant, i. e., leaves Dhenorm of all vect’ors invariant. As usual, we have thah 0’ = SOS-’ and XtS = 1. However, now S-’ = I?Sh.‘. r; S is pseudo-unitary with respect,to the metric I’. Lastly, we int’roduce the concept of metric conjugat’ion as follows. Let) r
=
firi, L
m,
r,l-
=
0,
r loi
=
where n = number of fields in the theory. We define m&c following relations J/i = r&jr;
= +ji # j,
tii'
=
riGiri
10)
(6)
conjugation by the =
ciqi
17)
where ci’s can always be set equal to fl by a suitable renormalization of the +i’s. If ci is equal to plus (minus) one, then fii has positive (negative) metric parity. We then have fi> = c&“‘. The metric operator ri, due to its Hermiticity, can be made diagonal in the representation labeled by the eigenvalues of t,he number operator of t’he ith field N;: n i where n i = (c i) n’. By metric invariance, it is meant that for any operator 0’ = I’Or = 0. Consider now two spacelike surfaces (TVand c2on which the dynamics is known. We introduce on u1 and u2 a complete set, of operators & and & , respectively, with the corresponding set of eigenvalues &’ and &‘. The kansition amplik~de between these two sets is (&
1 r 1duz).
(8)
A variation of this amplitude, assuming I? is not a dynamical variable, is given by 6(&,
I r I &)
= i(41’ul / mw12 / tA2)
= (&‘a1I r 1&2) - (&1 I r I .G2 ).
(9)
The state ) zisi) is infinitesimally close to ( Ei’ai). The reality condition (2) along with (9) leads to 8W13= 6W,n + 6wt~ ) 6W12= -a?%1 ) 8W12= BWi2 .
(10)
HIGHER
Remember
ORDER
LAGRASGIAX
that (t) is the adjoint
operation.
207
THEORY
If
Wu = 1” d4.2- s(.rj, CT”
(11)
then d:(x) must also be self-adjoint. Let 1{i’irL) be related to j (i’g;) by a pseudounitary
transformation
1f,‘lTi) = r: / &Ti) with
I’ = exp (-iF)
= 1 - iF. F must he self-adjoint 6 / (i’ui)
= -iFi
with
the result that
/ tj’ui),
6(,ti~i 1 = (.$i’ui 1iFi. From our transition 8(&l
amplitude,
I r I &z>
we have
= 6( (61 Cl = i(&q
j ) r
I &2>
1 (FlI’
-
+
(5l’Ul
I rq
I &2))
T’Fz) 1&‘uu)
= i(tl'ul / rsw12 I 12'u2) 01
swlz = rFlr - F2. An analogous result is obtained
(12)
for the variat’ion
SO = -i(OF
-
of an operator,
0,
rFr0).
(13)
The important difference from the conventional formalism is t,he appearance of the met’ric operator in the last two relations. If r were to commute with an arbitrary generat,or F, then the only difference between the indefinite case and t,he posit’ive definite case is to replace adjointness by Hermiticity. In the next section we shall prove the following theorem: if a *field theory is yuantized using either commutators or anticommutators and an indejkite metric, then the consistency oj’ yuantization demands the invariance of the Lagrangian under metric conjugation. This, as we will see, leads to generators which must commute with the metric operator. Therefore, relations ( 12) and ( 13) reduce to t,he conventional form. III.
L.AGRASGIAN
A variat,ion
FORMALISM
24ND
of the action can be written
COMMUTATIOX
RELATIONS
as
which involves the effects of varying the field components at each point by 64 and altering the region of integration by a displacement 6.~” of the points on the boundary surfaces.
The variation
of t,he Lagrangian
density of finite order N is
&, 2 = 2 -al: 6,, &,L, . n=O ac#$tn,
i 15)
We define the functions (Iti) which,
due to their symmetry,
obey the relation
where x
ps(“)
E
N-l n+l)
5,
(- 1)n~aX~mlL~Y(n)X(m).
(10)
As an example we obtain for N = 2 and n = 1, 7rp" = w$
(-l)"ax~m~~,~~X(m)
= Lp"
and for N = 3 and 'n = 0 3-1~2
2 = ,% (-i)"axcm~L~X(m' = L" - aiLpA + aXa,LfiXp. In Eq. (18) 6”~ is underst’ood to be symbolic in that the order of operators should not be changed during the processof variation. Further, we assume’that the commutation relations of 6& and the structure of the Lagrangian are such that identical contributions are obtained from terms differing only in the position of A&. Using Gauss’ theorem, and inserting (18) into (14), we obtain t,he EulerLagrange equations and a function ~(a)
go (--I)"
a+) Lfi(") = 0
("0) (21)
1 See ref 1 (a),
p. 917.
HIGHER
so that the relation
ORDER
LAGR.4NGIAX
(14) can be written 6W1? = 5(q)
THEORY
209
as - 5(u‘J.
We can associate the funct’ions F( ni) with the physical infinitesimal generators of Eq. (12) only if 5((ri) commutes wit,h the metric operat.or r. Whether t!he commutator of I? and F is zero or not depends on the commutation relations between r and t,he fields ($,I. From relation (7) we have [I’, $J* = 0. Since r is not a dynamical variable, this implies [I’, Gid.cn~+& = 0. Thus, in (21) if 8oaVi,l>+i commut8cs ( anticommutjes) wit’h I?, then ?r:“(“) must commute (anticommut’e) with r so t,he second term of Eq. (21) commutes with r. Sow the commutation relations of the $” ‘““s with I’ depend on the structure of the Lagrangian density which is bilinear in the field variables and their derivat’ives. From its definition in l&l. ( I!)), we see t’hat, STY’ is proportional to t,he coefficient of &(“P$~ appearing in the Lagrangian. Therefore, the coefficient of &cn$i , which is of course a function of &cl l+j’, must commute (,anticommutes) with I?. Thus the Lagrangian density must be invariant under met,ric conjugation. It, will be shown that t*he conditions (a), (b) and (c) of the previous paper (9) follow from the metric invariance of L. We may rewrit’e (12) and (13) as
and 60 = - i[O, F(a)]-.
(13’)
In addition, the equation
[r, F(u)]- = 0
(22)
tells us that physical observables commute with r since the generators contain the physics of the sysbem. In other words, from is), our generators are also Hermitian operators as is needed (see first paragraph of Section II). We are now in a position t’o study Eq. (21) in detail so as to obtain the canonical commutation relations, the energy-moment8um tensor, etc. In order to make the analogy with classical mechanics, it is necessary to separate dvcn@and iTr’(n) intro canonical and const,raint variables. Thus we wish to write Eq. (21) in terms of the independent variations of the field variables on the surface. Note that once the value of 6& on the surface is known, the variations &,&c~N$ are not completely arbitrary but tZheir variations in t,he normal direction are arbitrary’ (10). We shall further restrict ourselves to surfaces such that a(“&+ tends to zero sufficient,ly rapidly at spatial infinity. Thus for any function gy(.r), we have / da, &(f’“) 2 See ref.
S(a),
p. 4G5.
= /du$j”
= 0,
da, = ,n,,da
(23)
210
BARUT
ASD
MTLLEK
and k 6 N - 1. To prove this let us choose a part,icular where j”” = g” d“(‘-‘)&#J a spacelike surface (n, = (1, 0, 0, 0) j, then jdugd”
= jd3.r(a3’
zz
- n,i3y> = jd3d3”fu
jd32(aOfB
+
diP”
--don)
=
+ a,P” - n, m ay’,
jd3Ndjji
=
jdS'f
r~
0.
This will insure us that the Too component of the stress tensor is the energy density.3 Xow once 60(a”‘+) is known on the surface u, a,( 608’~‘+) is determined; hence only 60( 8(k+1)+) is arbitrary. The second term in (21) can be rewritten as N-l
N-l s
da, c ?T’”(%~,(n~ so cj, = II=”
du, c s
~“~(~‘(rjv + n,#%o
d
II=0
N-1 =
s
da, Go (-1)” + . . . + (-1y
;
~n:k’t3ycn-kl 7rPV(“)$(Q~O $
0 + . . . + (-1)”
; 0
n2”’ ?ypv(“) p
so 4
(24)
1 + j da,9,.B”“,
where
n
0n1
= ,n!/m!(n
- m)!.
Here a,B”” is a complicated expression containing 8(k’&$; its integral da, vanishes by virtue of (23). Collecting terms in (ail), we have
over
where we have defined (25) The generat’or becomes (26) 3 See ref.
4,
p. 551.
HIGHER
ORDER
LAGIZ.4NGI.4N
211
THEORY
We substitute the total variat,ion & for &#I by noting that, when
Hence F(u)
= i‘ da, [gl
[7F%(P+)
- n’k”d,(8(k’c#&l.d (27) + ?.pP”S,a
We define the quantit’y ~&A”- ], (k)bpp4 1 (&I = .2 h with j”;k; = -j*$.
iY”‘+] -
,
+ g(k)vShrgq + ph~q(k)+,
(23)
r\;ot,e also that &;
= ,lJj’::;
- f$)
= -a~(f:;;&r,j
CA”= f@- wrSXvg(k~~tA” + dp&r”
The first term is zero from the antisymmetry of j$; and from the theorem s
da, dhj+” =
s
&A & p.
The final result is
(29) where
The conservation laws arc arrived at from rigid displacements of the coordinate system given by 6x” = up + ~““x, and 8(8’“‘+,) = 0. The displacement generator can be wri-tten as
where
with MAW = -j@’ Since conservation integral invariant, J”“( ul) = Y’(us).
E TXfiZY_ Txvx”.
(32)
laws are associated with variations that leave the action we see that F(u,) = F(az) implies PP( ul) = P”( a?) and This leads to the differential conservation laws d,T”’
= &MA’”
= 0,
(33)
which in turn imply that T’” is symmetric. The conservation of “charge” can be obtained from the required invariance of the self-adjoint (also Hermitian due to metric invariance) Lagrangian under constant phase transformations exp ( f ic%X) where c’ = fl, 0. With &r, = 0 and 8(8’“‘+,) = it”MX8’“‘+, , we have Fax(u) = / cLT,@~ a~k”(ik%sx~(k)~,) k=”
zsxQ(u),
(34)
where we have defined N-l
Q(u)
=
1
da,
(35)
j',
The field equations fix 6: up t’o an arbitrary divergence terms since the EulerLagrange equations are invariant under 6: ---f 6: + d,h" with h" being metric invariant. However, WI2 is not invariant under this subst8itution. This change in W,, can be expressed as an alteration of the basis eigenvectors on (rl and g‘2 . Consider the generator of the infinitesimal transformation, H(a), defined by 10’) = (1 - iH) 10’). Then the new WI2 is obtained from (#(ul) 1 I? / 6" (Us) ) = (O’(U~) 1( r - iI?H? + iH,J?) I O”(uz)). With H = I du,h” we get the result SW12= y’ d4xd, h’“. g2 In particular, if we choose h” = -i( F’(u)
= [ da, (F
x:1;
=r(k)“8(k)+)so that we have
~(?#~)cf~ + h” - Ty’iirp)
or
(36) F’(~)
= - / au, &’
&r(k)p8(k)++ T”‘Gr,) .
Therefore F (a), Eq. (29), corresponds to SW = p6q - HFt and F’(u), Eq. (36), corresponds to 615’ = --6pq - H6t of classical particle mechanics. Defining
HIGHER +(h-I
~
,rLp?T(k)~
ORDER
LAGRANGIAS
21:3
THEORY
and 5?’ E npTBY, we write
(37) F(6x)
= -1
daYi%s, .
E’rom our analogy with particle mechanics, we can identify F(c%‘~‘~) with t#he generator of transformations 8(L)+ + g(k)+ - @@‘+ and ?ick’ -+ iick’, while F( &?lk)) with the generator of 73(k)-+ iick) - %?(” and a(k)+ -+ ack)+. From (13’) we obtain the commutation relat,ions [#“‘4, F( ,j#k’$)]-
= &‘“‘&
[j;@-,, F(@‘“‘+)]-
= 0
[i;(k), F(&+k))]-
= &‘“‘;
[#“‘c$, F(h?(k))]-
= 0.
(38)
At this point one conventionally would use the relation [9, BC]- = [A, B],C B[A, C], and assume that for bosons 67i’k’ and 8ackJ4 commute with all quantities and for fermions they would anticommute. However, we now know that a successful qu:tnGzation can be carried out for bosons whose particle operators anticommute f.o plus one and antiparticle anticommutes to minus one if an indefinite metric is used. The opposite (commutators) is true for fermions. Therefore, we only say t,hat if the independent canonical coordinatIes (8’“‘4) obey bose (fermi) statisttics, then 89(k) and 88(k)+ commute (anticommute ) with all quantities independent of the spin statist*ics connect’ion. We then obtain from Eq. (38) the final result,
[8(k)4e(.rj,+L’),(.r’)]*
= iP6,b6,(~
- s’),
(.v - 2’)’
<
0
(39)
where s
da~(z’)d,(a
- 2’) = f(x).
This relation is the main result of the present formalism. The consistency of the formalism is proved by showing that the Euler-Lagrange equations (20) are equivalent t,o t,he Heisenberg equations which are derivrd using relation (39) and the displacement, generator F(6.r). It is sufficient t’o show that, a(k)+ and jjck) are indeed canonical conjugaks and that Too is the Hamilt80nian density of the syst,eni, i. e., 6H @qj=T
I (k), ’
6H= S(iFk))
-JCk)&
(10)
To prove these relat,ions n-e consider a family of spacelike planes and such that sn = 1 = constant,. In t,his “natural” coordinate system i 1, 0, 0, 0 j . Now from (30) we have
a
frame II, =
C-21) where ~3”~~)meaus that t,he time derivaGve becomes
is taken
k-times.
From
(25)
A(‘)~
e,a”‘k+l),(+, %&)” = z (-]y+k (;) ,a”(,& but in our particular frame of reference il~(,-k)~O(k+l)“(n-k) = ay(~~_k)?yO(k+l),(~l-ki
-771,(n-C~-(n-tinO(k+‘)“(“-t) a = a,lcrt-k,a~Jc't+l) + al(n_k)aO!k+l)i("-k)_ a,l(n--k)TO(rL+l) "(k+l)j(rt-k) = aicr,-k)?r
and Iye obtain
where the index i = 1, 2, 3. Next we evaluate 6H. From it’s definition,
we have
6H =
(43) , or, using Eq. (41), N-l
6H = =
1{
dsx C [&(k)Oa’J(k+1)4 + n(k)u,j(ao(kfl~~)] _ $
k=O N-l k&o
[jQ)“$(k+l)$
+
T(k)Uj(ao(ktl)~)]
-
2 n=U
,Q + $$ ,Qr + . . . P jy(n)~~p(“)
.
HIGHER
Neglecting
t,otal divergences
ORDER
LAGRANGIAN
THEORY
and using Ihe symmetry
213
of L’(“) and +,,cn) , we have
N-1 + . . . + =& &+~“‘a”(k+“~ . (
)I
Comparing
(4s) and (J-l), we obtain the relabions
Finally we use the Euler-Lagrange equat,ions and the definitions arrange the right hand side of the ahovc relat,ions and ohtain
of sCk”, re-
The equations (4.5) are the desired relations in the part’icular frame of reference we are working. Hence we have shown that gik)+ and 73(k)are canonical conjugat’es and that Too is the energy density. IV.
ELECTROMAGNETIC
IXTERACTIOXS
In this se&on WC wish to introduce elwt,romagnetic interactions inO0 our Lagrangian C = G(&cn) , &! ~~1) in a gauge invariant, way. If we add the electromagnet,ic w&or in the usual way ( a, ---f I), = d, + ieL4,and Dp* = d, - ied,) then the Lagrangian C(qQn) ) &I, is obviously
gauge invariant.
--) .e(D,c&
Sow however
D&$+1
the order of the Dp(n)‘s is important
216
BARUT
since they do not commute.
AND
RlULLEN
With this new Lagrangian
the Eq. (20) becomes (46)
where
D,(., = D,,D,,
. . . D,,
and
D (n), = QJL
. . . D&i,
.
The reason for the inverted order for DTn),, in (46) is the following. an arbitrary term in the Lagrangian such as
Consider
D,*D”*$+#“‘. We write II/“” to stand for a function linear in 4 and whose indices (p, Y) insure Lorentz covariance of the Lagrangian. Let us look at t.he 8&t variation of this term
D,*D”“Go&‘“, which we write
as
= ara”6~+~“”
- i”s,$.+A ,,24“#“” - ied,(A.6~+)~p”
- id,a”s~+pt)s,q”.
This last term is equal to
= &+#~~{[a,a,,- e”A”A, + ie(A”a, + ayAp)]@‘“) - ied,[6,&t(,A”J/‘” - ,4”#“‘)] = 6,y$+D”D,~“” + &BP. The B” term is a total divergence which does not contribute to the equations motion. For an arbitrary number of derivatives, we then have
(D,“D,* A S&+ variation relation ( 47 ) .
. ’ . Dn*s,$+))J/‘“~~~~~+(D~ e . . D”Dg,V”“.)
of the Lagrangian V. SUMMARY
will lead to equation
+ d,B”. (46) by virtue
of
(47) of the
AND CONCLUSIONS
For a consistent quantum field theory using either commutation or anticommutation relations and a Hilbert spacewith an indefinite metric, we demand the invariance of the Lagrangian under metric conjugation defined in Section II. This invariance insures the Hermiticity (observability) of the generators F (i.e. P’, J”“, Q, etc.) and of the unitarity of the S-matrix. In other words the
HIGHER
ORDER
LAGRANGIAN
217
THEORY
metric invariance of d: leads to zero commut,ation relations between I’ and F and S. In the previous paper (9) the Hermiticity of ohservables and the unitarity of the S-matrix were listed as condit,ions (b ) and (c) , respectively, for a consistent physical theory. In order to explain electromagnetic and strong vector coupling (meson) intjeractions w&h 2-component spinors, the condition (a) of metric invariance of transition amplitudes was postulated, i. e. cd, WJ’I = ($7 I?) where c+cd must that transitions prove that this Lagrangian. To
= w9(1J/,
r-41,
equal 1 and leads t,o Q = cm and $’ = rqr = c&. This means between states of opposite metric parity are forbidden. We now result is also a consequence of the metric invariance of the see t,his consider the arbitrary transition (91 rol+)
and use the relations +l
=
cjtil.
=
r = n ri, 2
riwi,
ci = fl,
to write
4 = c.+r+r.
ti = dw, Now
we write
the transition (0 I wo+
Remembering
symbolically I 0) =
c+c+(~
as 1 rprrorg
/ 0).
also that r lo)
= I o>,
r” = 1,
we may writme
(ti I ro Id4= W& I or i 4). We have shown
that for physical
observables
0 must commute
with
I?, hence
Therefore c+ = c4 or the states 1 +) and / 4) must have t’he same metric parity for a nonzero physical transition amplitude. The concept of metric invariance was necessa.ry in order to develop an action principle using either commutation or anticommutation relations. A physical interpretation for this action principle, conditions (a ), (b), and (c) staled above, is obtained from the metric invariance of the Lagrangian. The other problems encountered with the action formalism were t,he definitions
218
II.\ Ill:T
AND
MULLI*:N
of canonical conjugates and the interpretat)ion of t,he energy operator. In t,he former, it was necessary t#o split’ t’he fields (a,~~$) into normal (canonical) and t,angential component’s. The expression for the momenta canonical to a(k)+ are no longer simple but deprnd on the tangent,ial derivatives of t(he fields. In t,he derivation of t)he energy-momentum tensor 7”” an additional term appeared in our generator. However, this nddit,ional km, which is a total t#angential dcrivative d,B’“, can be assumed t(o t’end to zero at’ spatial infinity. Thus, in the higher order case the energy again is the T”” component of 7”“. The consistency of our action formalism for a higher order theory lies in the fact that from our definit’ions of canonical conjugatles and energy, we are led to Heisenherg equat,ions of motion equivalent to t!he Euler-Lagrange equations. RECEIVED
: February
28, l!)G2
1. J. SCHWINGER, Phys. Reu. 82, 914 (1951a); ibid. 91, 713 (19511)). d. S. P. MISRA, Im&zn J. Phys. 33, 461 (1959a); ibid. 33, 520 (1959h). 5. T. H. CHANG, Proc. (‘ambridge Ph.il. Sac. 44, 76 (1948). 4. J.S. DEWET, Proc. Cnmbridge PhGl. SOI,. 44, 5-X (1938). 5. E. M. Co~sori, “Introduct,ion to Tensors, Spinors, and Itelativistic Wave Equations,” p. 13. Hafner, New York, 1953. 6. L. Ii. PANDIT, NUOZJO cimento Isupp2. 11, 157 (1959). 7. For a complete bibliography up t,o June 1960 see R. 1,. XAGY, h’trouo c~inrenlo Suppl. 17, 92 (1960). 8. R. ilsc0~1 AXD 13. MINARDI, Xuclrar P&p. 4, 2-Q (1958). 9. A. 0. BARUT AND G. H. MITI.I.EX, Anr~. phys. (iv. E’.) 20, 184-20;! (1962). 10. W. THIRRING, “Principles of Quantum Elect,rodynamics,” p. 36. Academic Press, New York, 1958.