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On the self-stress of composite particles
:....... i:i~::iii•.:?i~!i/~!,i•:i! ¸~i~¸ ~,
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ON:THUg: SELI~-STRESS OF COMPOSITE PARTICLES ~. S T R A T H D E E and Y. TAKAHASHT
Dublin in,ititute /or Advanced Studies, Dubtin, Ireland Received 23 May 1958 ,~4,b~tritct: I t i~ pro~ed t h a t the Pals-Epstein formula which shows the close relation t~tween s e l f , s t r e ~ and self-energy holds true even for bound states. This formula is used t o show t h a t t h e self-stress of the composite particle vanishes in the rest frame, ie:0 t h a t t h e binding energy transforms like a mass. This makes it possible for the c o m p o s i t e p a r t i c l e models proposed by many authors in connection with strange l?asticles to join company with the relativistic particles.
1, Introduction The interestin~ question has been raised whether the particles so far . . . . ed are of in elementary or composite nature. The words elementary ~1_b~rv and composite, however, have not been g~ven unambiguous meaning by the present quantum field theory" and are used in a casual sense only. It was, as far as we know, Taketani who first suggested a possible experimental distinco tion between the two kinds of particles a). The mass of a composite particle, calcuJated on field the('~:y, should be given as t h e algebraic sum of the binding energy and the constituent elementary masses. It is therefore important to ensure that the bind;.ng energy so calculated behaves like a mass under Lorentz t:~-ansforn~ations. In other words it must be shown that tile self-stress of the composite particle vanishes in the rest frame ~). This consideration, which was raised by Sakata a few years ago s) becomes quite important in view of the nunlber of composite models proposed in connection with the strange particles 4). In this paper we shall discuss the self-stress of composite partlcl~ " " ' ~..... In order to treat this problem we must, of course, assume the existence of a hound state solution to the eigenv~'ue problem. In section 2 we shall prove, in a formal manner, that the self-s ~ss indeed vanishes. This proof employs a g e n e r ~ e d Pais'Epstein formula which is derived in section 3. Furttier, it. is shown t h a t the formal proof is contradicted by an actual calculation usirtg the Fermi,Yang model (made covariant by Maki 5)). Previousderivations of the Pals,Epstein formula are open to certahi
il:~:
. . . . .
,it, 'STRATI'tDF,F, AND V. TAKAH,ANH/~
:
ables on tile parameters of the system ...... a procedure which justified, leads to the correct result in this ca~e 7). This point wi in section 3. We emphasize that for the treatment considerm ions are of essential importance because the expressed either independer::tlyof the coupling constant or by means of a perturbation exr~ansion. This formula must be proved without any recourse to approximatim~ In the last section we shall discuss briefly the physical significance of the Pals-Epstein folTnula. Here also we indicate, formally, how a theory which lacks divergences would also be free of the self-stress difficulty. 2. The Self-Stress of a Composite Particle
For this discussion we make use of the Pais-Epstein formula for the case of two particles bound together by some interaction. (This formula will be derived in section 3.) If S(0) and E(0)are the self-stress and energy respectively (measured in the rest frame) and K, m are the component particle masses and g the coupling constant (of dimension L ~ in natural units), then
s(o) = l
(
0
V +m
0
0
-,,gug -1
)
(2.1)
It follows from dimensional considerations that E(O) is of the form
E(O) = ~, a,~k~*mJg~,
(2.2.)
i, L k
where the a~jk are dimensionless numbers and the sum is restricted by
1 = i+i--nk.
(2.3)
Tile substitution of (2. ° ) into (2.1), on the other hand, yields S(O) := ~- Z (i+]--nk--i)a~j~'mJg~ i, t, k
so that, by (2.3),
s(o) :
o.
(2.4)
It can be seen, however, that just as for elementary particles, the relation (2.2) does not ahvays ho!d true. This effect is an immediate consequence of the divergences inherent in the present field theory. To illustrate this we cite the example of the composite nucleon-antinucleon model of the pion. The The pion mas~ E(0), as calculated by Maki 5), is given by E
=
(4g)-,},
(2,5)
vhere I0, t~o and J'0 are diverging quantities which, on introducing a c u t o f f omen~gum 2. and putting e _~ )~/~, can be written in the form :
(,..6) In (2.6)the:terms :~ 0(1/0~)have been omitted. It is e ~ d e n t now that t h e dimensionaljustification of (2,2) is no longer valid: the. new parameter ~ h a s dimension L -~ and should be entered also~ In fact a straightforward substitution of (2.5)into (2.1) yieMs
S(O)
~ ~.-~ (J'o)"
4r'g
~x-t--
tn ~¢--F} In =
,s
(,,., ~,
Thus the self*stress of the composite particle fails to vanish because the energy diverges. This situation is exactly the same as for the elementary particle. In other words, the selpstress diHiculty is again absorbed i~n the
set/-energy dilliculty and would not arise in a divergence-]ree theory~ 3. The Generalized Pais-Epstein Formul~ The primary task in this section is the derivation of an expresskm h~r the derivative of an energy eigenvalue with respect to one of the parameters contained in the Lagrangian density, viz.
where .ff'(x) is the Lagrangian density, ~ is an eigenstate of the Hamilt~miam and 2~ is some parameter appearing in the Lagrangian. it must be Nm't h~ mind that the field operators also depend on the 2,'s; hence the suttice,s ~, ~ , on (0.W/02~)¢~ which mean that: the derivative is explicit~ Our purpose n~ght be clarified by a short digression c,n the matter of derivatives. Consider a system. ~4th one degree of freedom characterized by a Hamiltonian H and suppose that only one parameter a p w a r s in ft (e.g:~the 1 ~+ toq~), 2 harmonic oscillator: H = ~(p o~ is the parameteri E~id,~:nih .................. OH/Oo is not well-defined whereas (OH/Oo~)~,is, And, in general (OH/&,,)o :.~: (OH/Oo~)~,,,,where qp and q'p' are two different sets of canonical vad~b!e-.~., It is possible, however, to define a total derivative dH
(OH)
.......i:H.
F,r.~.
which i s independent of the chosen set qp, The diagomd ~ "~°~ ~ (OH/Oa~)q, in the energy representation are then seen to 1~~ m~le~:~,.~:~:~:~~:~ the set qp. In fact ,,E tdH~&o!E .~ (E'iOH,~&~"iE') , (For simple harmonic oscillator 1;;, = (iJ2~o)qp~.)
I t can then be shown that and it. 7/~ ~ - - E ' " ,,~
where na is a time-like future directed (n° > O) unit vector. The metric ( - 1 , + I , + I , +1) is used (~o = t). Natural units {!i = c = 1) are used. The abbreviations a
0 ~ ¢'(~) = C ¢'(=) = ¢ ; ( x ) and
~z,G¢,(x ) = ~ ¢'(x) = 0,4'(=) = ¢/(=) will prove convenient. The system to be considered comprises a number of interacting boson and fermion fields ~'(x) characterized by the Lagrangian density
~(x) = .z',o,(z)+ ~.g, .z',,(x),
(3.U
l
~:here ~o~ is the sum of the free iield densities and the .5f ~,'~are interaction terms. The latter need not be specified i.~ detail except for restrictions on terms of the tFpe (¢~)2, ea~, etc. which must not appear. The particle masses are assumed to appear in £a~o) only. The canonical momentum and Hamiltonian density operators are ,~(,) = - n A a . . ~ / a ¢ / )
= a2/a¢/
~(~)= X ='(z),~,/(z)-.e
(3"°4
and the total Hamiltonian is
H(,) = f., d~(x)ar(,)
(3.~)
0.ItI~) -- o.
(3,4.)
which must satisfy
i(The eigenvalues of H are energies in a coordinate
system
with
The equations of motion are ia,¢,(x)
= [¢'(x), HI =
(3.B)
together with 0 , ~ = 0 where ~ is the state vector. Let 2 represent one of the parameters in the Lagrangian so that H = H(X) and Cr __ cr(~; X), ~r = ~'(2; X). If 2 were replaced by 2 + ~ the Hamiltonian would change and the evolution in time of the system would be altered, The commutation rules (3.5), however, would be unchanged. The latter fact implies the existence of a unitary operator A (r) suct! that ~r(~+~2, X) = A-l(~')~*(2, x)A(z),
xca(t)
(ILT)
and similarly for ~F, where A (z) satisfies iO,.A(z) ---- K(82, ~., ,)A(v)
(3.S)
with K a functional of the ~'(,~.; x), ~'(2; x) to be determined. We cannot in general find a vo such that A(vo) = 1 (as evidenced by the discussion at the beginning of the section) so we put The equations of motion for the 0"(2+62; x) must be examined no'~~. From (3.6), (3.7)and (3.8) i0, ¢'(2+62; x) ----A -1 (,)ia,0'(2; z). A (T) +A=X(z)[¢r(2; x), K(&X,2, r)]A(r) = ~L~, (*T , ,~j~, ,~. x), M(~)~
(3.|0!
where
M(v) -----A-~(r)(1-I(2)+K(d0., L v))A(T}
z4
£3~~1,
and, similarly, i0,~'(~+~2; x) = E~'(2+o).; xL M(r)i~
' .... ':~
It follows also from (3:8) that
From (3.10), (3A2), (3.13) we deduce that M(T) c~m dif,er by n~ n,~r~, t]~ul~ a c-number from :H0+~52). Thus we p u t = :
.:
....
HCa+~2) ---- A-~Ct)(HCa)+K( ~jt, 2, t))A(~).
(3~t~ii
that
.... ~":'~ . ~.:'
In g~meral, :if F ~=~ ~,, ~.....
-~, . . . .
dF__.... lira :--:i.F,~(z,~';-,:r),,, .~(2 .+,.~")z;. ; '
""
"~
...
]
I
~
, '
"
' .la'
V,'~,;
z~*-,)Z,'l~l(x)(aF;aX)#,
" A(r) ~
~o 0~ i,e.
d F(£x)~:~ (OF'ie~. ...-:-i¢.-.-.i:FO.; ..... x; .... F .=-~,(.~.I ,, deK(i,
r')i
(3.1(;)
where we have used A (r.),~.=l..~-ib2(r+j ,. , b : K (/.,r')) .+.
,/,.,:
which follows from (3.8) and , c ' An immediate consequence of {,.I. " 1~;; '~ is f
f,
; ,~
where q~ is an eigenstate of H. (We have used the symbol d/d~ somewhat unconventionally in the alcove work. It impIies that t h e d e p e n d e n c e on ~. of observables throug~ the ¢', .~' is taken into account. If 2 is one of ~:veral parameters then these are regardect, " together ~ t"h x, as fix:ed in the evaluation
of dF/dL) We are now in a position to calculate the d e r i v a t i v e s of the energy eigenvalues. Since the HamiltonLms H(i) and H(/. : Lb;.) are ~-independent we can introduce the two sets (complete) of their respective eigenvectors {~().)} and t/~e(z-rbZ)~. '~.... ~ "'~ The in,:tex ~, standing for t h e eigenvatues :)f a complete commutirtg set of ob:~er~abtes, runs over N)th the d i ~ - e t e and the continuous spectra. These vectors are chosen such t h a t Consider now (G(~')' (H(~'+'~-)-n(;~i)~(~.+~;.)) : = .
dZ
( E d 3 + ~ ; , ) ' E . ( ~ ) ) ( ~ . ( ~ ) , .c2¢b{~.))
-.:..-:~/.(I.:~!~ ....G(;4)(~A~.), Q, /o
oq'~
= --~~-
('r/", (.I.), q~,e()-)) -Jr-(Ee (2) -- E,,, (,i.1) (qS~(2), ~, q~,e(,I.)) + O (&~.2i ,
i t is easily shown that an appropriate choice of phases for the q~(2) will make the diagonal elements of .Q~ vanish. Thus, using (3.18), dE.(2)
-=
where it is understood that the normalizing factor (¢~, ¢~) has been removed from the fight hand side. We shall next make use of the following two formulae wl, ich are derived in the appendix: =
0.;
(a.22)
and
T~,/' -= Z n,gl(O'L~IOg,)*,~,-- Z rn's(0"~ii~m,)tt÷i,
(3.23)
where T~, is the energy-momentum tensor, the m, are the different particle masses and the g~ are coupling constants. The dimension of g~ is given b y n~ (In natural units if [x] = L, then Fc, L~] = L"'.) Assuming that the in, and g~ are the only parameters appearing in ~ , tliel:t E~ wiU be a function of just these variables and (reverting to the u!sagc Oj02 for did2) we have
[~. m, OiOm,- ~. n,g, OlOg,]E<<(.,,g) = j
d~(x)(¢,,,, T~'(x)q~,),
(3.24)
where we have used (3.,1), (3.22) and (3.23). Choosing a coordinate ~ t t , in in which n~ = (1, 0, 0, 0), so that E~ becomes an energy then, for a s:tate q~ with zero momentum in this system,
which .is the generalized Pais-Epste:in fornmla used in section 2~
4. D i s c u s s i o n Tlxis section is devoted to a few remarks which, it is tm wd. ma:F h,~p t:,~ clarify: the ::present theoretical situation with regard to sellost~e:s~:.... I t :was pointed out in the introduction that the u:~e of a Ctii~:~ff par~i,m,,:~.~ in evaltmtin ambiguous integrals lies at the root of the di[iicultv l~l!L~........v
....
Then, h o r n
:+
i
~
y
. . . .
:
dimel~.si,.~nMcon~ideratio~, .......E((:}) =,~
;:aOiOa:-..~.;.2 ,~,~/~m,+ 2,~g~a/ag,~E(o).
(,t,t)
All of t b e ~ parameters except a apf~ar i:n the Hamiltonian ~ that eq, mtions (,3.~l),:~3 o~ and (3,;3) " ' : " ear ~, t~ apphed, i,e,, *}
t~
-=
....
)
,
wher~ a~ is a state, of zero i'l(torl]entull~. Whu~;
5 (0) .%%...... ~aaE (o)/aa. It might parameters would not conclusion,
(4.3)
I~.*thought that a could be regarded as a function of the other (e.g, a = =/m~,with = dimensionless). Then the term aaE(o)/aa ap!~ar in (4.1) and we should conclude that S(0) .--- 0. This ]however, would be incorrect simply because
aE (o)/a~ ~ (¢), aii/a,n~ ¢)),,,~ if a ==~:g/mzl : So the difficulty remains t. ]'tie physical meaning oi self*stress can t:,~made rnore clear b y qualitative considerations. Let us regard a as the radius of a particle and V = ~,~aa as its volume, Tiien (4:3) reads
s(o) -v(oE(~)/~v). V(SE/OV) as the work done in compressing
(~.3')
We can picture the energy into a volume V so that ~.--3EJOV corresponds to the outward pressure at the surface of the particle. A non-vanishing self-stress then implies that the particle is unstable. One should not conclude from this that a particte of finite size cannot be stable. Eq, (4.~') was arrived at under the assumption that E ( 0 ) contained a while, the iiamiltonian did not. If the theory is constructed f r o m the start W%" . . . l & a, view to obtaining a finite self' energy, i.e. if a. is built into the Lagrangian then the argument must be attered. Proceeding again from (,t.l) we have : :
- e(o) = (¢,. ~,,~/a,,-.: Z:,,,. a/o,,,,+ X n,g,~/agi~a¢,)~.:. t See, Ior instance, S Boro-,Stz and W. KoCh,' tel;t)?, [
,)
i
= ( # , j d ' zT,,a(x)~)re,~t = -- E 0 ( ) + 3 S 0 ( )
,a~hence s(o)
This cuVoII gian ?, giving
=
o.
!~!
a r g u m e n t shows t h a t the self-stress problem would disap~:~a~" ~f ila~ c o u l d b e i n t r o d u c e d consistently from the begim:ing m the |,ag~a:~:~=, It also explains the success of the mixed field a n d reg:datc r t~i~ ~:~t'~ a zero self-stress.
Appendix I n o r d e r to verify equation (3.22) we need to k n o w tl> ,.t~:~ ......... ~::.,~,~.~. :.: (0~,,/0,1.) ÷,,. These are easily discovered by int roducing t lv.ed::ri~ a t i~ ~.~si ~,, ~:~:.~:!~ a n d parallel to aft), respectively, O, = nt'O~,
and
0~, = 0 ~ - - ~ ¢ 0 , ,
,~?
T h e n a n y function of O~ can be considered a function ol 0~,,-~, % 0~ ~,~,~~-~,~ p r o b l e m becomes one of examining ( 0 ¢ ~ ; 0 x ) , . and ( ? 0:~, ~ ) ~ , • ...... N':~:~ =
(aC,L/aa)~ ,
lim ( 5 } . ) - l [ A ( z ) a , , ¢ ' ( ~ + 6 2 ; x)
A,-a(r) ...........a , ,
~a.
8A.+0
= O~hm
(a2) L, ( }0 ( ..... ,
.
.
.
.
.
¢~ .'i,:,,~ a"!:
A
~-~0 -
-
O,,~(O0 (,., x)/O).'~,.
== 0.
B u t because [A, arj ~ ~ ~ " 0 it is seen that,. in g~:~u~:~a:,, : (O0;().;
x) :O;,~, , =/: o,
Using the definition (3.2) o f / ~ ()~; x).
aeff.;x)
Za'9.;
'
"~'"
te
we h a v e
'-= .... \ 02/exlmeu Formula
In~naural
C3 . :23 ) c a n
be arriv(~l at b'~~ ,~impt~ units the dimensions are ......
.::~
,,, _
i
!.ff~ ....
1;.4,
~,¢ri L/'. and
fm ~] ~: Lt,:~!~: : ':ur:
:: *
we regard ~ as a function of ~', ~ , m~ and:gv Appl~ng E iers formula we get
,, !i9.~
The equations of motion can be used to write
Substituting (A.7) into (A.6) we get
-
where Fa is some function of the ¢" which can be dropped by a suitable choice of Lagrangian. The canonical energy-momentum tensor T~', is defined Tt' .-_-~-;; , , + H.C.--d~'~.ff'(x) so that a~
T ~(x) ~¢,; ~b/(z)+H.C.--4~(x).
(A.9)
C.m~bining (::~.8) and (A.9) T'a(z) = "
~ 'n' (a-'~--"~ +
('~gl)
(A. ,o~
References 1) M, Taketani, Sorfishiron no Kenkyu 2 (1950) 188 2) A. Pats, Developments ~n the Theory of ~ectrons (hastitute for Advanced Study and PrinceLon University, 1;}48); W. H~tler, The Quantum: T h e o ~ o f Raxtiation, 3rd e& (Oxford :1954~; J. M. Janch and F. Rohrlich~ Theo . . . . . . . . . . . . . . . . . . . . . . {Addls,~,nWesley Publishing C0.£ Cambridge, Mass.i 1955):'; Lhysik 5!1 (Springer, Berlin, !958)- H Umeeawa Quantum FieIdThe~,tv'lNnrth iar~ll~ad Publishir ',~ _ - :, ~ msterda,m; 19o6~ :: , : ,;: ,, ~ ;Jl S~ .: . . . . . o Sakata,: Private Comnmnieation : .... = i : ,: ~. Sakar~; Prog. Thevr" Phys.: 16 f1956)686;: M. G:;G~ldha~r~ 'NFS!:IRe~;: 1t}i i::11956) 433
tg) 1739; Z~ Maki~, ~ i ~ , ]~i:~'~L| ; ~
~
I, 1938)
9) j~[Y~kawa-and H-Umez~wa, Prog. Theor. Phys~ 4 (t9t~)! ....(!950):I22; F. Rohrlich, Phys. Rev. 77 ~i950) 357
~6S }:' "V,:;~...:~ :~.~':'i :~ 6 ~:~,:~ ~
• ii~ I:•,•: ,i ~ ,
,:~,
•>;
•
•
' ,,~
•
ON:THUg: SELI~-STRESS OF COMPOSITE PARTICLES ~. S T R A T H D E E and Y. TAKAHASHT
Dublin in,ititute /or Advanced Studies, Dubtin, Ireland Received 23 May 1958 ,~4,b~tritct: I t i~ pro~ed t h a t the Pals-Epstein formula which shows the close relation t~tween s e l f , s t r e ~ and self-energy holds true even for bound states. This formula is used t o show t h a t t h e self-stress of the composite particle vanishes in the rest frame, ie:0 t h a t t h e binding energy transforms like a mass. This makes it possible for the c o m p o s i t e p a r t i c l e models proposed by many authors in connection with strange l?asticles to join company with the relativistic particles.
1, Introduction The interestin~ question has been raised whether the particles so far . . . . ed are of in elementary or composite nature. The words elementary ~1_b~rv and composite, however, have not been g~ven unambiguous meaning by the present quantum field theory" and are used in a casual sense only. It was, as far as we know, Taketani who first suggested a possible experimental distinco tion between the two kinds of particles a). The mass of a composite particle, calcuJated on field the('~:y, should be given as t h e algebraic sum of the binding energy and the constituent elementary masses. It is therefore important to ensure that the bind;.ng energy so calculated behaves like a mass under Lorentz t:~-ansforn~ations. In other words it must be shown that tile self-stress of the composite particle vanishes in the rest frame ~). This consideration, which was raised by Sakata a few years ago s) becomes quite important in view of the nunlber of composite models proposed in connection with the strange particles 4). In this paper we shall discuss the self-stress of composite partlcl~ " " ' ~..... In order to treat this problem we must, of course, assume the existence of a hound state solution to the eigenv~'ue problem. In section 2 we shall prove, in a formal manner, that the self-s ~ss indeed vanishes. This proof employs a g e n e r ~ e d Pais'Epstein formula which is derived in section 3. Furttier, it. is shown t h a t the formal proof is contradicted by an actual calculation usirtg the Fermi,Yang model (made covariant by Maki 5)). Previousderivations of the Pals,Epstein formula are open to certahi
il:~:
. . . . .
,it, 'STRATI'tDF,F, AND V. TAKAH,ANH/~
:
ables on tile parameters of the system ...... a procedure which justified, leads to the correct result in this ca~e 7). This point wi in section 3. We emphasize that for the treatment considerm ions are of essential importance because the expressed either independer::tlyof the coupling constant or by means of a perturbation exr~ansion. This formula must be proved without any recourse to approximatim~ In the last section we shall discuss briefly the physical significance of the Pals-Epstein folTnula. Here also we indicate, formally, how a theory which lacks divergences would also be free of the self-stress difficulty. 2. The Self-Stress of a Composite Particle
For this discussion we make use of the Pais-Epstein formula for the case of two particles bound together by some interaction. (This formula will be derived in section 3.) If S(0) and E(0)are the self-stress and energy respectively (measured in the rest frame) and K, m are the component particle masses and g the coupling constant (of dimension L ~ in natural units), then
s(o) = l
(
0
V +m
0
0
-,,gug -1
)
(2.1)
It follows from dimensional considerations that E(O) is of the form
E(O) = ~, a,~k~*mJg~,
(2.2.)
i, L k
where the a~jk are dimensionless numbers and the sum is restricted by
1 = i+i--nk.
(2.3)
Tile substitution of (2. ° ) into (2.1), on the other hand, yields S(O) := ~- Z (i+]--nk--i)a~j~'mJg~ i, t, k
so that, by (2.3),
s(o) :
o.
(2.4)
It can be seen, however, that just as for elementary particles, the relation (2.2) does not ahvays ho!d true. This effect is an immediate consequence of the divergences inherent in the present field theory. To illustrate this we cite the example of the composite nucleon-antinucleon model of the pion. The The pion mas~ E(0), as calculated by Maki 5), is given by E
=
(4g)-,},
(2,5)
vhere I0, t~o and J'0 are diverging quantities which, on introducing a c u t o f f omen~gum 2. and putting e _~ )~/~, can be written in the form :
(,..6) In (2.6)the:terms :~ 0(1/0~)have been omitted. It is e ~ d e n t now that t h e dimensionaljustification of (2,2) is no longer valid: the. new parameter ~ h a s dimension L -~ and should be entered also~ In fact a straightforward substitution of (2.5)into (2.1) yieMs
S(O)
~ ~.-~ (J'o)"
4r'g
~x-t--
tn ~¢--F} In =
,s
(,,., ~,
Thus the self*stress of the composite particle fails to vanish because the energy diverges. This situation is exactly the same as for the elementary particle. In other words, the selpstress diHiculty is again absorbed i~n the
set/-energy dilliculty and would not arise in a divergence-]ree theory~ 3. The Generalized Pais-Epstein Formul~ The primary task in this section is the derivation of an expresskm h~r the derivative of an energy eigenvalue with respect to one of the parameters contained in the Lagrangian density, viz.
where .ff'(x) is the Lagrangian density, ~ is an eigenstate of the Hamilt~miam and 2~ is some parameter appearing in the Lagrangian. it must be Nm't h~ mind that the field operators also depend on the 2,'s; hence the suttice,s ~, ~ , on (0.W/02~)¢~ which mean that: the derivative is explicit~ Our purpose n~ght be clarified by a short digression c,n the matter of derivatives. Consider a system. ~4th one degree of freedom characterized by a Hamiltonian H and suppose that only one parameter a p w a r s in ft (e.g:~the 1 ~+ toq~), 2 harmonic oscillator: H = ~(p o~ is the parameteri E~id,~:nih .................. OH/Oo is not well-defined whereas (OH/Oo~)~,is, And, in general (OH/&,,)o :.~: (OH/Oo~)~,,,,where qp and q'p' are two different sets of canonical vad~b!e-.~., It is possible, however, to define a total derivative dH
(OH)
.......i:H.
F,r.~.
which i s independent of the chosen set qp, The diagomd ~ "~°~ ~ (OH/Oa~)q, in the energy representation are then seen to 1~~ m~le~:~,.~:~:~:~~:~ the set qp. In fact ,,E tdH~&o!E .~ (E'iOH,~&~"iE') , (For simple harmonic oscillator 1;;, = (iJ2~o)qp~.)
I t can then be shown that and it. 7/~ ~ - - E ' " ,,~
where na is a time-like future directed (n° > O) unit vector. The metric ( - 1 , + I , + I , +1) is used (~o = t). Natural units {!i = c = 1) are used. The abbreviations a
0 ~ ¢'(~) = C ¢'(=) = ¢ ; ( x ) and
~z,G¢,(x ) = ~ ¢'(x) = 0,4'(=) = ¢/(=) will prove convenient. The system to be considered comprises a number of interacting boson and fermion fields ~'(x) characterized by the Lagrangian density
~(x) = .z',o,(z)+ ~.g, .z',,(x),
(3.U
l
~:here ~o~ is the sum of the free iield densities and the .5f ~,'~are interaction terms. The latter need not be specified i.~ detail except for restrictions on terms of the tFpe (¢~)2, ea~, etc. which must not appear. The particle masses are assumed to appear in £a~o) only. The canonical momentum and Hamiltonian density operators are ,~(,) = - n A a . . ~ / a ¢ / )
= a2/a¢/
~(~)= X ='(z),~,/(z)-.e
(3"°4
and the total Hamiltonian is
H(,) = f., d~(x)ar(,)
(3.~)
0.ItI~) -- o.
(3,4.)
which must satisfy
i(The eigenvalues of H are energies in a coordinate
system
with
The equations of motion are ia,¢,(x)
= [¢'(x), HI =
(3.B)
together with 0 , ~ = 0 where ~ is the state vector. Let 2 represent one of the parameters in the Lagrangian so that H = H(X) and Cr __ cr(~; X), ~r = ~'(2; X). If 2 were replaced by 2 + ~ the Hamiltonian would change and the evolution in time of the system would be altered, The commutation rules (3.5), however, would be unchanged. The latter fact implies the existence of a unitary operator A (r) suct! that ~r(~+~2, X) = A-l(~')~*(2, x)A(z),
xca(t)
(ILT)
and similarly for ~F, where A (z) satisfies iO,.A(z) ---- K(82, ~., ,)A(v)
(3.S)
with K a functional of the ~'(,~.; x), ~'(2; x) to be determined. We cannot in general find a vo such that A(vo) = 1 (as evidenced by the discussion at the beginning of the section) so we put The equations of motion for the 0"(2+62; x) must be examined no'~~. From (3.6), (3.7)and (3.8) i0, ¢'(2+62; x) ----A -1 (,)ia,0'(2; z). A (T) +A=X(z)[¢r(2; x), K(&X,2, r)]A(r) = ~L~, (*T , ,~j~, ,~. x), M(~)~
(3.|0!
where
M(v) -----A-~(r)(1-I(2)+K(d0., L v))A(T}
z4
£3~~1,
and, similarly, i0,~'(~+~2; x) = E~'(2+o).; xL M(r)i~
' .... ':~
It follows also from (3:8) that
From (3.10), (3A2), (3.13) we deduce that M(T) c~m dif,er by n~ n,~r~, t]~ul~ a c-number from :H0+~52). Thus we p u t = :
.:
....
HCa+~2) ---- A-~Ct)(HCa)+K( ~jt, 2, t))A(~).
(3~t~ii
that
.... ~":'~ . ~.:'
In g~meral, :if F ~=~ ~,, ~.....
-~, . . . .
dF__.... lira :--:i.F,~(z,~';-,:r),,, .~(2 .+,.~")z;. ; '
""
"~
...
]
I
~
, '
"
' .la'
V,'~,;
z~*-,)Z,'l~l(x)(aF;aX)#,
" A(r) ~
~o 0~ i,e.
d F(£x)~:~ (OF'ie~. ...-:-i¢.-.-.i:FO.; ..... x; .... F .=-~,(.~.I ,, deK(i,
r')i
(3.1(;)
where we have used A (r.),~.=l..~-ib2(r+j ,. , b : K (/.,r')) .+.
,/,.,:
which follows from (3.8) and , c ' An immediate consequence of {,.I. " 1~;; '~ is f
f,
; ,~
where q~ is an eigenstate of H. (We have used the symbol d/d~ somewhat unconventionally in the alcove work. It impIies that t h e d e p e n d e n c e on ~. of observables throug~ the ¢', .~' is taken into account. If 2 is one of ~:veral parameters then these are regardect, " together ~ t"h x, as fix:ed in the evaluation
of dF/dL) We are now in a position to calculate the d e r i v a t i v e s of the energy eigenvalues. Since the HamiltonLms H(i) and H(/. : Lb;.) are ~-independent we can introduce the two sets (complete) of their respective eigenvectors {~().)} and t/~e(z-rbZ)~. '~.... ~ "'~ The in,:tex ~, standing for t h e eigenvatues :)f a complete commutirtg set of ob:~er~abtes, runs over N)th the d i ~ - e t e and the continuous spectra. These vectors are chosen such t h a t Consider now (G(~')' (H(~'+'~-)-n(;~i)~(~.+~;.)) : = .
dZ
( E d 3 + ~ ; , ) ' E . ( ~ ) ) ( ~ . ( ~ ) , .c2¢b{~.))
-.:..-:~/.(I.:~!~ ....G(;4)(~A~.), Q, /o
oq'~
= --~~-
('r/", (.I.), q~,e()-)) -Jr-(Ee (2) -- E,,, (,i.1) (qS~(2), ~, q~,e(,I.)) + O (&~.2i ,
i t is easily shown that an appropriate choice of phases for the q~(2) will make the diagonal elements of .Q~ vanish. Thus, using (3.18), dE.(2)
-=
where it is understood that the normalizing factor (¢~, ¢~) has been removed from the fight hand side. We shall next make use of the following two formulae wl, ich are derived in the appendix: =
0.;
(a.22)
and
T~,/' -= Z n,gl(O'L~IOg,)*,~,-- Z rn's(0"~ii~m,)tt÷i,
(3.23)
where T~, is the energy-momentum tensor, the m, are the different particle masses and the g~ are coupling constants. The dimension of g~ is given b y n~ (In natural units if [x] = L, then Fc, L~] = L"'.) Assuming that the in, and g~ are the only parameters appearing in ~ , tliel:t E~ wiU be a function of just these variables and (reverting to the u!sagc Oj02 for did2) we have
[~. m, OiOm,- ~. n,g, OlOg,]E<<(.,,g) = j
d~(x)(¢,,,, T~'(x)q~,),
(3.24)
where we have used (3.,1), (3.22) and (3.23). Choosing a coordinate ~ t t , in in which n~ = (1, 0, 0, 0), so that E~ becomes an energy then, for a s:tate q~ with zero momentum in this system,
which .is the generalized Pais-Epste:in fornmla used in section 2~
4. D i s c u s s i o n Tlxis section is devoted to a few remarks which, it is tm wd. ma:F h,~p t:,~ clarify: the ::present theoretical situation with regard to sellost~e:s~:.... I t :was pointed out in the introduction that the u:~e of a Ctii~:~ff par~i,m,,:~.~ in evaltmtin ambiguous integrals lies at the root of the di[iicultv l~l!L~........v
....
Then, h o r n
:+
i
~
y
. . . .
:
dimel~.si,.~nMcon~ideratio~, .......E((:}) =,~
;:aOiOa:-..~.;.2 ,~,~/~m,+ 2,~g~a/ag,~E(o).
(,t,t)
All of t b e ~ parameters except a apf~ar i:n the Hamiltonian ~ that eq, mtions (,3.~l),:~3 o~ and (3,;3) " ' : " ear ~, t~ apphed, i,e,, *}
t~
-=
....
)
,
wher~ a~ is a state, of zero i'l(torl]entull~. Whu~;
5 (0) .%%...... ~aaE (o)/aa. It might parameters would not conclusion,
(4.3)
I~.*thought that a could be regarded as a function of the other (e.g, a = =/m~,with = dimensionless). Then the term aaE(o)/aa ap!~ar in (4.1) and we should conclude that S(0) .--- 0. This ]however, would be incorrect simply because
aE (o)/a~ ~ (¢), aii/a,n~ ¢)),,,~ if a ==~:g/mzl : So the difficulty remains t. ]'tie physical meaning oi self*stress can t:,~made rnore clear b y qualitative considerations. Let us regard a as the radius of a particle and V = ~,~aa as its volume, Tiien (4:3) reads
s(o) -v(oE(~)/~v). V(SE/OV) as the work done in compressing
(~.3')
We can picture the energy into a volume V so that ~.--3EJOV corresponds to the outward pressure at the surface of the particle. A non-vanishing self-stress then implies that the particle is unstable. One should not conclude from this that a particte of finite size cannot be stable. Eq, (4.~') was arrived at under the assumption that E ( 0 ) contained a while, the iiamiltonian did not. If the theory is constructed f r o m the start W%" . . . l & a, view to obtaining a finite self' energy, i.e. if a. is built into the Lagrangian then the argument must be attered. Proceeding again from (,t.l) we have : :
- e(o) = (¢,. ~,,~/a,,-.: Z:,,,. a/o,,,,+ X n,g,~/agi~a¢,)~.:. t See, Ior instance, S Boro-,Stz and W. KoCh,' tel;t)?, [
,)
i
= ( # , j d ' zT,,a(x)~)re,~t = -- E 0 ( ) + 3 S 0 ( )
,a~hence s(o)
This cuVoII gian ?, giving
=
o.
!~!
a r g u m e n t shows t h a t the self-stress problem would disap~:~a~" ~f ila~ c o u l d b e i n t r o d u c e d consistently from the begim:ing m the |,ag~a:~:~=, It also explains the success of the mixed field a n d reg:datc r t~i~ ~:~t'~ a zero self-stress.
Appendix I n o r d e r to verify equation (3.22) we need to k n o w tl> ,.t~:~ ......... ~::.,~,~.~. :.: (0~,,/0,1.) ÷,,. These are easily discovered by int roducing t lv.ed::ri~ a t i~ ~.~si ~,, ~:~:.~:!~ a n d parallel to aft), respectively, O, = nt'O~,
and
0~, = 0 ~ - - ~ ¢ 0 , ,
,~?
T h e n a n y function of O~ can be considered a function ol 0~,,-~, % 0~ ~,~,~~-~,~ p r o b l e m becomes one of examining ( 0 ¢ ~ ; 0 x ) , . and ( ? 0:~, ~ ) ~ , • ...... N':~:~ =
(aC,L/aa)~ ,
lim ( 5 } . ) - l [ A ( z ) a , , ¢ ' ( ~ + 6 2 ; x)
A,-a(r) ...........a , ,
~a.
8A.+0
= O~hm
(a2) L, ( }0 ( ..... ,
.
.
.
.
.
¢~ .'i,:,,~ a"!:
A
~-~0 -
-
O,,~(O0 (,., x)/O).'~,.
== 0.
B u t because [A, arj ~ ~ ~ " 0 it is seen that,. in g~:~u~:~a:,, : (O0;().;
x) :O;,~, , =/: o,
Using the definition (3.2) o f / ~ ()~; x).
aeff.;x)
Za'9.;
'
"~'"
te
we h a v e
'-= .... \ 02/exlmeu Formula
In~naural
C3 . :23 ) c a n
be arriv(~l at b'~~ ,~impt~ units the dimensions are ......
.::~
,,, _
i
!.ff~ ....
1;.4,
~,¢ri L/'. and
fm ~] ~: Lt,:~!~: : ':ur:
:: *
we regard ~ as a function of ~', ~ , m~ and:gv Appl~ng E iers formula we get
,, !i9.~
The equations of motion can be used to write
Substituting (A.7) into (A.6) we get
-
where Fa is some function of the ¢" which can be dropped by a suitable choice of Lagrangian. The canonical energy-momentum tensor T~', is defined Tt' .-_-~-;; , , + H.C.--d~'~.ff'(x) so that a~
T ~(x) ~¢,; ~b/(z)+H.C.--4~(x).
(A.9)
C.m~bining (::~.8) and (A.9) T'a(z) = "
~ 'n' (a-'~--"~ +
('~gl)
(A. ,o~
References 1) M, Taketani, Sorfishiron no Kenkyu 2 (1950) 188 2) A. Pats, Developments ~n the Theory of ~ectrons (hastitute for Advanced Study and PrinceLon University, 1;}48); W. H~tler, The Quantum: T h e o ~ o f Raxtiation, 3rd e& (Oxford :1954~; J. M. Janch and F. Rohrlich~ Theo . . . . . . . . . . . . . . . . . . . . . . {Addls,~,nWesley Publishing C0.£ Cambridge, Mass.i 1955):'; Lhysik 5!1 (Springer, Berlin, !958)- H Umeeawa Quantum FieIdThe~,tv'lNnrth iar~ll~ad Publishir ',~ _ - :, ~ msterda,m; 19o6~ :: , : ,;: ,, ~ ;Jl S~ .: . . . . . o Sakata,: Private Comnmnieation : .... = i : ,: ~. Sakar~; Prog. Thevr" Phys.: 16 f1956)686;: M. G:;G~ldha~r~ 'NFS!:IRe~;: 1t}i i::11956) 433
tg) 1739; Z~ Maki~, ~ i ~ , ]~i:~'~L| ; ~
~
I, 1938)
9) j~[Y~kawa-and H-Umez~wa, Prog. Theor. Phys~ 4 (t9t~)! ....(!950):I22; F. Rohrlich, Phys. Rev. 77 ~i950) 357
~6S }:' "V,:;~...:~ :~.~':'i :~ 6 ~:~,:~ ~