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Particle-like solutions in field theory
ANNALS
OF
PHYSICS:
19,
219-233
(1962)
Particle-Like
Solutions EUGESE
Brandeis
University, CERS,
in Field
P.
Theory
GROSS
Waltham, Massachusetts* and Geneva, Switzerlanrlt
Anumherof quantumfield theories have the propert,y that the field equations, st,udied as classical equations of motion, have particle-like solutions. These are spatially localized exact solut,ions, with a definite value of angular momentum. The role of these solutions in a fully quantum theory is the subject of investigation. We examine, in particular, the theory of a nonrelativistic scalar nucleon, coupled by a local interaction to a scalar meson field. The classical solution (whose spatial extent is a nucleon Bohr radius) is identified with a selfconsistent field approximation to the one-nucleon state. However, quantum corrections to t,his approximation yield the same divergent self-energy as perturbation theory. In this example, the particle-like solutions have nothing to do with the ultraviolet divergences, but are relevant to the treatment of the finit,e part of the theory for strong coupling. The separation of the infinite part is trivial and is accomplished hy a canonical transformation. The particle solutions still exist for the finite part for strong coupling. I. 1STROI)UC:TIOS
X large number of papers have appeared, in which the field equatjions of a cluantum field t,heory are discussed c~lassicslly. The field operators arc trcakd as ordinary
frmctions
of
spacac
and
time.
Specifically,
\vc
jvill
hr
conrerncd
with
t8heorics Involving t#wo fields coupled in a conventional way. For cxamplc, one may cwkdrr a scalar meson field caoapled to t,he elwtromagnrt~ic field or t,he Dirac field coaupled to a meson field. In some of t,hc theories, particle-like csac*t solutions
of
the
nonlinear
classical
fiicxld
cxcluntions
have
hcxrn
found
(1).
These
solutions are characterixcd by a t,imc constant angular momentum, as ~~11 as by other constants of the motion, stwh as rhargc. They t,hlls have some simple spat,ial sgmmctry. The fisrd values of the constants act as normalization c*onditions
on t hc field
amplitndcs.
The
solllt8ions
ar(‘
particle-l&c
in the
scnsc
that
the
* I’crmanent address. Work supported in part by Oficc of Scientific ltese:irch, IT. S. Air Force. t National Science Senior Post Doctoral Fellow (lOGO-1961). In this paper all vectors are writ,ten in lightface type when they occur as subscripts or superscripts. 219
220
GROSS
mutually associated fields are large in a small spatial region and fall off rapidly as one leaves this region. The stress energy t,rnsor is well behaved everywhere, and the t,otal energy is finit’e. Furthermore, some of t’he solutions are st,able with respect to small perturbations. Rosen and E’inkeMein have discussed some general features of such solutions for fields with Lorcntz invariant Lagrangians. In particular, if “particles” are thought of as being defined in terms of integrals of various field propert#ies over regions of high field strrngt,h, the correct laws of moGon are obt.ained as consequences of the field equations. Thus these conveAona1 theories, when t,reat,ed classically, have very attractive propert,ies from the point of view of a field theory of mat,ter (2). In addition the effectively nonlinear nature of the t,heories implies that there may be several solutions for a given set’ of const,ant,s of motion. This offers hope of interpreting some unstable particles as states of different “principal quantum number,” and perhaps rven of reducing t,hr mimer of kinematical attribut,cs “put into” a t’heory. It is unfortunat,ely t’rue that in this work on classical field t,hcories, no field has yet been produced in which the excit’ed stnt#es bear a promising relation t,o known elementary part,icles. Apart from t,his fact, there has been very lit,tle analysis of the relevance of t,he classical part,icle-like solutions t,o t,he quantum t.heory. There is no indicat’ion of particle-like solutions when the same theories are treated by current’ (cssentially perturbation theorct,ic) methods of cluantum field theory. In fact,, the quantum approach leads to divcrgcnt results for self-energies and other physical yuantit,ies. One may wonder whether t#hc particle-like solAons of the classical theory survive in any form in a complete quantum theory. Furthermore, one may ask whether t,he nonperturbative solut,ions of classical t,hcory affect the divergence problem of t,hc quantum t#heory in any way. The present not’e is devoted to an explorat,ion of these questions in a preliminary way. We shall take a very conventional approach and interpret the cslassicaal particle solutions as a definite npproximst8ion to the one-particle state of the quantum theory-essentially a self-consistent field approximation. We shall not t,ry to use t)he classical theory as a starting point for some radical alteration of t#he quantum formalism. The part,icular t#heory t,o be cxamiwd in detail is one considered by Heber (~3) . It involves a complex scalar-nonrelatJivist’ic “ii~iclcon” field $, and a real s(*al:u meson field 4. The Hamilt*onian is
(1.1)
Here fi = I, c = 1, and :-: means that the zero-point, energy is to be suMractc,d. The fields are coupled by a local point int,eraction. While this problem is not
PARTICLE-LIIiE
SOLUTIONS
IN
FIELD
THEORY
221
relativistic, the features presented by t,he point int,eraction and nucleon recoil are still of general interest. Fully relativistic theories involve mainly complicat.ions (and fundamental differences), connec%ed with t’he proper description of the vacuum. The organization of this paper is as follows. In Section II the features of the particle-like solutions are briefly reviewed. The classical solution is connectfed t’o a self-wnsistent, field description of the one-nucleon state of the quantized field theory. This descript,ion yields a finite self-energy for t,hc one-nucleon stat)e. Hut, quant,um perturbation theory leads t’o a logarithmic divergence for t#hc selfenergy. In Sect,ion III, t#he remark that the two calculat,ions correspond to treating different, physical contributions tJo the self-energy is csploit’cd. The self-energy is recomputed by a variational met,hod which includes both effect’s. It is shown t,hat, the quantum corrections to t,he part#icle-like solution again give a divergent self-energy. For t,his example, t,he part,iclc solut,ion represents a caalculation of t,he finite part of the self-energy ( which can he unambiguously separated from the infinite part,), appropriat#e t,o strong coupling. The conclusion is that t#he finit,e cnlassical rsolutZion has nothing t,o do wit#h t,he quantum divergence. This arises from virt,ual emission and absorption of quanta with wavclcngths much short,cr t.han t,hc c*haract,erist,ic txtcnt, of t.he particle. In Scct,ion IV we c~alculate t#he effective mass of the nucleon. This illust~rates horn the classical and pertlwbation approsimations lead t,o different, estimat#es of a finite quantitJy. Section V summarizes our conclusions and notes problems recluiring furt,her invrst.igation. II.
PARTICLE-LIKE
The field equat,ions
SOLUTIONS
corresponding
ANI)
QLTANTIJM
SELF-ENERC:Y
to H arc
An exwt solution having radial symmetry is of the form + = j(r)eeCEt. The nucleon density $J+# is time const,ant and spherically symmetric. The associat,ed ~Iwson field is -g[z-z’l 2 -~~ ++lj4 x’ ) &r’, (2.2) +=$ s lx--‘1 and is also spherically symmetric. t,hr nonlinear eigenvalue equat,ion
The
part’irlc
solution
is
found
Upon
when
eliminating
one
uses
4(x),
the
we find that $ sat’isfies
normalizing
condition
222
GROSS
J $+$ d’.r = 1. One can calculate E and $ by inserting (2.2) into the Hamilt#onian and varying H for normalized G’s, While numerical calculat,ion is required to find t’he detailed behavior of $, it, is easy to est#imatethe characteristic size of t,he part’& when p = 0. One considers the class of normalized functions I/X = X3’*h(Xr) for some h(x) with J h’(x) d”~ = 1. The best,value of t#hcscaling paramct.er X is found by minimizing the totmaenergy with respect t,o X. The result, is E = &
1
cl -
hg”cz ;
Cl =
c* sE/&A
s
(Vh)’
d3x
= s ,x-y, dz&/ h”(x)h2(y)
3
= 0 yields X = (Q/Q) (g”M) . Thus
E = -f ” g2(Mg2) 1 The extent of t,he particle l/X is of t,he order of a nucleon Bohr radius fi’/Mg”. A more detailed study of the existence of the solut)ion has been made by Meyer (4). His results can be most easily expressed in terms of the Fourier components b, of $(x), which are subject to t,he normalization condit,ion c, b,*b, = 1. Meyer proves t)hat t’here exists a set of b, that gives a minimum of H. Furthermore 1b, 1fails t’o zero faster than any power of p as tends t#oinfinity. Physically the high momentum components behave this way becausethe nucleonic matter cont,ained in a small spatial region t’ends to zero as the region shrinks. For p = 0, the scale argument, shows that’ t’he energy is always negative, and is therefore lower than the energy that corresponds to spreading t,he nucleon out over all spaceuniformly. The latt,cr situat#ionalso corresponds to an exact solut,ion of the classical equat’ions, namely $ = 1/da, 4 = 0. I:or p # 0 a minimum coupling strength g2is required before a particle-like solut’ion is stable. These are the main features of the simplest’particle solut8ion.In general, there are other classical solut’ions of higher energy, both with zero and with finit’e field angular momentum. One would like t,o idrnt#ify these solutions with mstable particles. We will not, study these solutions here sinre our main concern is to st#udy t#heconnection het,ween t#hr classical and quantImn theories. We will rest,rict, our caonsidcrat,ionsto t,he lowest energy solution. This connection is not necessarily unique, and one might possibly use the particle solution in ways different, from the one outlined here. We want the classical solut,ion to correspond t’o a single particle. So, we look at the Hamiltonian for a nonrelat’ivistic scalar particle int#eract,ingwit’h a meson field (2.6)
PARTICLE-LIKE
SOLUTIOSS
IX
d)(x)= c
FIELD
223
THEORY
fp’“(al,+ &)P ikr
(2Wk
Ii?
(Ui+e-ikr
(2.7) -akP
Lkr
)
The Hamiltonian (1.1) apptars when (2.6) is subjected to srcond quantizntion wit,h # and ++ treat#cd as operators. Conversely, (2.6) is the effertive Hamiltonian in t’he one-nucleon subspare of ( 1 .I), when ( I. 1 ) is considered as a quant’um field theory. The question of Bose or Fermi statistics for the field does not, arise for the one nucleon &ate in a nonrelativist’ic t’heory. Xow, 1:he Hamiltonian (‘2.0) has been &died in great, dct#ail in the continuum idealization of the t$heory of the polaron. That case corresponds to I; - y/‘/c, wi; = W. In fart, Landau and I’ekar (5)) on intuit#ive grounds, umntct~ed with the idea of scylf-trapping, proposed a semiclassical approximation t,o (2.6). This cons&s of a,ssigning a wave function x(y) to t,hc particle and st’udying t,he meson equations of motion calassicaally. The procedure is precisely to study (1.1) as a fully calassicaal problem. Pekar has shown (6) that the entirely quant,um approsimation that, yields the same results as t#he semiclassical theory is t,o assume the state vector of the system !P to be a product q = x(q)+. Here @ depends only on the meson variables. The electron wave function x(y) is normalized to 1 and x and + are to he det)ermined self-consistently by independent’ variations of the clncrgy. I’or the ground stat#r it, is found t,hat
where
+0 is the vac*uum
st,at#e a&
= 0 and
This corresponds t’o a static shift, of the field oscillators in the presence of t.he matt#cr &r&y X*X. x(y) in turn is t#he wave fun4on of a particle moving in a potential well creat#ed by t,he field oscillat’ors. As applied t)a our case of a point, interaction, I’k = g(2wl;Q)-““, this self-consist,ent’ field approach to thr particle rcxcoil yields a finite self-energy for t#he ground state. If onr accepts t.his interpretat,ion of the classical theory, it hccnmes intSerest,ing to look at’ other approximations to (2.6) in order to find the range of applicabilit’y of t,he rlassical theory. Second order pert’urhnt’ion theory yields the energy,
1g=: -c which
/ T’kI2 k Wk +
diverges
(k”/2M)
logarit’hmirally
=
-6
A/
t’o minus
log
(1
infinit,y.
+
&)
for
/*
Th(x same result,
=
0,
(2.10)
is 0Mained
224 variationally, from uses a state vect,or
GROSS
the inkrmediatc
coupling
approximation.
This
theory
q = es1 @$l!~
(‘7) (2.11)
where s, =
c
ali+/3kr’k” -
(‘.(‘.
13, = - Trg*//( a/; + k2;‘2d/)
(3.12)
Since minus infinity is the lower bound, it is t)he correct answer. Thus one seems forced to the conclusion that, the self-consist’ent field approach is simply not relcvant when t,here is a point interact’ion. The only possible escape from t,his conclusion might, be t,o claim t’hat the class of state vectors used in intermediate wupling theory is too broad, and that one must’ work x-it,h wave functions x(q) normalizable to unit,y, and localized. In Scct8ion III WC will show t#hat# t,his is not. true. Still, our conclusion will not, bc the ut#ter irrelevance of tjhe particle-l& solutions. The reason is that t,he t,wo self-energies wrrrspond to very different physical effects. In the self-consistent8 fitld approach, one does not take into account1 the virtual emission of quant8a by t’he particle, t,ogrthcr with t(hc associated precise recoil. Indeed, if we carry out pcrturbntion corrections to the self-consistent, field, the due to very short wave logarit,hmic dircrgcnce re-emerges. Thi s dirergencc, fluctuations, is essentially indeprndcntS of the smooth partirk of ext,ent fi’, .1/y”. Point8 interact,ions ran give dircrgrnces, even in a nonrelativistic t,heory. We t,hcrefore conrludc that, the particlr-like solutions do not, represent a way to solve thr problrm of ultraviolrt8 divergrnrra ---as onr might first hopcx. On thr contrary, t,hry have Utle to do with that problem. Instc,nd, thry can be wsrd to suggest approsimat8ions to thr finite part of a qnantrtm throry for strong coupling.
Then,
+ C.C.+ C wk atfak + C wk 1fk 1’ > + c
(3.3)
(I’, Cik*+ ,fk-*ct~r )ak + C.C.E Ho’ + Hi’
The lowest eigenfunction of Ho’ is x( ~)a,, We can use the complete set of eigenfwict’ions of HO’ to calcu1at.e the prrturbation corrections arising from H,‘.
PARTICLE-LIKE
SOL~~TIONS IN FIELD THEORY
x?;,
One finds the logarit,hmic divergence of perturbation t,heory. This procedure has the advant’age of being completely general, in tjhr sense that it can he applied to all Hamiltonians, so that, one shows t,hat the esisttw~e of a particle-like solution does not affect, the ultraviolet divergewe. In the present problem, however, it is possible to st,rengthcn the argument by using the variation principle to gain a lower hound to the energy at. every &age. WI: make use of a method ( 8) of studying t#htl recoil prohltm for all wupling st,rcngth:: which is based on a trial state vector incorporatSing both physical contjrihlli,ions t,o t,hc self-wcrgy. For the ground stat#cl the form of the st,atc v&or is * = &(I@,, , s = c apcpn*(q) - c.(‘. (3.3) rpk( q) dclpends dircc%ly on q and also on ar-cragw taken with X( ‘I). I~ormally, it. reduces to the intcrmrdiate coupling t,rial vcrtor when (pL. = &c 1kq, x = l/d% and to that self-consistent field approwh when pk. is indcpcwdtwt8 of q and equ:d to -I’,” ^ (‘itq ) x 1” & @ = ~wI;-~- J ()nc is, however, not rest’rictcd t’o cGther (Jf t,hcsc choiws. In fact fw ~1 given X(U) ( the flmct ions cpk( q) arc’ wmplct,c~ly dctermincd t)y the variation principle. I,et, IIS defiw :I potenM riqj = i - I ?:I[) (V'x x) :w:rwiatwl with my x(y). I,rt, xlrlnll 11) and t312mlw thr cigenfwwtions and c~igcwval~ws of the linear operator - (v' ,‘,Af) + I’(q). ‘I’hcl entities
arc Green’s flmc%ions for the problwl. Thaw, with rcspwt, to cpk*( q) for givr>n XCc/) yields (Ok(q) = -T7i* G,,.(q, q’) ohcys t#he intcgrd c’,(q, whew
q’)
= G”(q
(&I’ IS the Grwn’s
i
frmc%ional
\-aviation
of t,he energy
rJ’““G, ( q, q’ ) tl”q’
( :i.5 )
clqu:ltioll -
q’) +
func*tion
.I
Gk” ( q -
for frw
q” ) 1T( q” ) Gh (1q” ) q’ ) cl:‘*”
( 3.6 )
part~iclw
(3.7) The aI)prosinlat’ioll E,
=
-&
t.o the clnergy of thcl gro~md
st,atr is
/ x*V2x d"q
(3.8)
This expression can be varied with respect’ to x to find the optimum function. We note that the formal st’ructure of this expression is close to t’hat of second order perturbation t’heory. Yet it expresses the self-trapping aspect as well, because of the dependence on x(y) . The same approximation to the ground state energy was found by Haken (9) from the point, of view of the path integral variat,ion principle of Feynman. One can now prove t#he divergence of EQ for any admissible x. This hinges on the fact that the second term diverges logarit,hmically when GA is replaced by Gk”. For this term is t#hen
Consider very large values of ( k (, such that the entire contribution to the sum over g is accounted for, i.e., xa :Y 0 for u > K. Take / Ic 1 > K. Then the expression in question is
The first term diverges logarit,hmically. Furt,hermore, if we solve the integral F
equat’ion for G, by iteration,
) vk j2 j” e”““x*(y)(G~I~G~)x(y’)e-“kq’
d”q d3q’
the term (3.11)
is finite, because the t’erm in brackets becomes l/k4 for large k. All further t’erms in the series converge. Hence, except for the first tsrm, all terms converge individually. Using t#he result’s of Tit,chmarsh ( 10)) one can establish the convergence of the Pntire series for suitable x(q). The argument is more direct’ if we note that to obtain a finite energy, Gk would have to be such t#hat
s
eik4x*(rl)Gk(q,
approaches
q’)x(~‘)e-ib”’
c13qd’q’
(3.12)
zero fast,er t,han 1/k” for Ii very large. Rut t#hen s
eiPqX*(q)(Gk”
VG&(q’)
Cipq d3q d’q
(3.13)
approaches faster than l/1;’ and gives a finite contribution. However G, = Gko+ C,“VGk and Gko gives an infinite contribution. Hence we have a contradiction, and must, conclude that, Ck gives an infinke result’ for t,he self energy. The divergence in t.his t,heory is of a trivial nature. It may be removed by
PARTICLE-LIKE
SOLUTIONS
IK
FIELD
THEORY
227
subtracting the term EJ #+# d”.r from the original Hamilt,onian. This is an energy E, divergent and coupling constant dependent, for each nucleon. We may isolate the divergent, part explicitly by performing a canonical transformation. With CT = es & = _ (2WkW’” ___________ (3.14) Wk + (k2/2m)
pk = Pk p-y
s = Cak qq.* - cc, we have
(3.15) The transformed
Hamilton
CFH7F-’ := E + &
is
- & c k
(ak &* k.p eikq + ak+ pk e-ik.q k.p) (3.16) ak+ k/jk e-ikq + c.c$ - F !$!$
+ C wk ak+ ak + L& (F The divergent. constant
is -1 E
=
_
C
8?&k k
‘Ok
f2)
+
(3.17)
(k’/%\~)
St#art#ing from a statme of zero total momentum (l,l’dG)%O , where a0 is a vacuum (a& = 0) , the terms linear in ak and a&+ of t’he transformed Hamiltonian give no contributSion t#o second order pertBurbation theory. The pert8urbed Aate vector is
1 Gkz = [(k” + /2)/zdz] This leads to t#he convergent
+ (‘d, + ‘dl)
(3.19)
self-energy
E? = - 5
(&)’
/ 13kI2 I82 IL GkZ
(3.50)
All the higher order t,erms in the self-energy converge, as may be shown by a diagrammatic analysis. Basic to t,he point of view we are following is t#he fact that, t’he transformed Hamiltonian, neglecting the divergent, constant R, also has classical particlelike solutions. In the quantm~l version we look for prodwt state vect.01
228
GROSS
\E = #(q)+. coordinate,
Taking
the partial
expectat,ion
value
with
respect
to the nucleon
ak+ alOk PI* k.1 e--l where Ik = 1 #*k. p ei”“# d3y,
(3.22)
Pk = I ‘# * epikqJ, d3q
This presents us with a normal mode problem and with the task of determining a shifted center of oscillation for the uk . The shift can be induced by the unit#ary transformation UP = e”“, &
= c
ak++Ck -
Making use of the &uct,ure Then cI; is det’ermined by
Thus
=
+(q)
the particle exp
The state vector of the original theory this \k. This energy of the self-trapped
(3.23)
t ‘2al;c’~1 = ak + ck
of the quadratic
t,he state vect,or describing \k
a&*,
[C
(ak+
form,
we find ck real and ck = c-~ .
solut#ion -
is (3.25)
ak-)ck]@O
(wit(h a counter state is
term),
is es operat#ing
on
(3.26) where $ is to bc determined HO as to make & a minimum. To put this expression in a simpler form, we measure Compton wavelength fi/AIc of the heavy part,icle. With
lengths
in units
of the
13.27)
/” #’ d3q = /x’(E)
d3,t = 1,
Pa4RTICLE-LIKE
SOLUTIONS
IN
FIELD
229
THEORY
we find for F = 0,
g=!!c 2
(3.28)
where (3.29) The crit,erion for self-t(rapping is now changed. In particular, for the original Hamiltonian, a self-trapped state was possible at, all coupling strengths, when ~1= 0. Now the coupling g2 must be sufficient,ly large. With l.he methods me have presented, there is no continuous transition bet,wcen the state vertors and energies of perturbation theory and t#hose of the self-consistent, field approach. However, as in the case of polaron t,heory, it, is possible to construct’ state vectors which make a continuous transition, and which have an admixture of self-consistent field at all coupling strengths. A careful st,udy of the t’heory would he important to find out to what extent the self-c*onsistentI field feat)ures of the excited state spectrum are germane at weak and int,crmediatr c+oupling st’rengths. However, the significant, point for t,he present considerations is that’ the divcrgrnt part of the theory can be isolated, and that, t,he ctlassical part’icle soluGon remains relevant for the finite part of the theory for strong coupling. IGnally, we isolate the divergent part in the caomplete quantum theory. The theory is made finite by subtracting E j” fi+# d3x, or an energy E per nucleon. To prodlIce such a t,erm explicitly, one transforms the original Hamiltonian by [T = (>iT,
1’ = j yc+(xh(
x - y)7r( y)+(x)
d”y d”lJ
(3.30)
Then r’+(x)r’-’
= esp - i /’ X(x - y)+(y)
d+J #(x) (3.31 )
734x)r’
= 4(x)
+ j X(x - y)++(y)+(y)
d”y
This transformation is the dressing t8ransformat,ion used by Greenberg and Srhwcber (II). Here, however, we arc treating recoil and the dressing is not caompletcb. The choice of X(x - y) is &. = t/;jy&&. . Then t,he transform in thcx
single nucleon divergent part
UH/l-’
s&space is the same as Ey. (3.14)) of the dressing. We have explicitly
and so serves to isolate
the
= Ho + E /- $+ $ d3x + / {gS(x - y) + (111- Li>
.X(x - y) + &
/j/f
+(x’
+(x>+++(y) : dy)dy’)
d”x d3y -
j(x)X(x
- y) $ (y) d3x d3g
: 2 (x - y’) 3t (x - y’)++(x)+(x)
~+(x)~+(x’)~(x)~(x’)
ss
s
1
- y)h(x - y) d3y +’
gX(x
e(x’
- x’)
+ g
- y) e(x
2 s ay
d3x d3y d31/’ (8.32 )
- y)
d3xd3x’
aY
In this expression all the terms except the last one contain a one-nucleon contribution and are direct analogues of terms in Eq. (3.16). The expression j(r) is t,he nucleon current density ( 1/2M) (++V$ - V$+#) . The expression X(Y) r( y’) : involves two meson operators and is ordered so that t’he zero-point contribution Ck (wk/2ci)e”-” has been taken out’ and used in the det#ermination of X(x - y) . The last term is a direct, two-nucleon int,eraction and depends in sign on t,he boson or fermion character of the nucleons. The transformation is chosen t,o remove divergences, so t(he linear t,erms in the meson operators produce finite self-energy cont8rihut.ions. IV.
STATES
OF
NONZERO
MOMENTUM
In this section we briefly discuss the t.heory of the effective mass of a slowly moving particle. This prolrides an example of how t’hc quantum approximation based on a particle-like classica solution yields a point, of view complementary to that of perturbation t,heory, in the analysis of t,he finite part of a field theory. Similar result,s would he obt,ainrd for ot#her problems such as scattering, optic&al absorption, et#c. l~ollowing ref. 8, we work w&h stat,e vcct,ors of the form 4 = PXP (C In t,he intermcdiatc
coupling $(*)
and f is an eigenfunction
aktcpi,(qj - c.c.j+(q)*o
(4.1)
approximation = L&
(py
lpe = pli epik,y
of the trotal momentunl P = d/S. +
c
PO, wit,h eigenvalue
k 1 & 1’
(4.2) P. (4.3)
P.iRTICLE-LIKE
SOLUTIONS
IN
FIELD
THEORY
231
The value of /z?,is (4.4) The energy associated
with the Hamiltonian
(2.6) is
k [(k”/2M)
+ wk - k-k]
(4.5)
The difference bet’ween this energy and the ground state energy is finite, even without the infinite count,er term. For slowly moving particles the effective mass is defined by writing the energy in the form
i.e., by eliminat’ing
2 bet’ween Eels. (3.3))
(4.4), and (4.5). We find
In the semiclassical approach, the state vector is again of the form (4.1). However, (Do is a constant, -4/i , independent of q. #(q) can be written as Ij/o(q, z+?*, where #o( q, A) is a real, localized function. h can be interpretNed as t)he expectation value of the velocity of the partic:lr. We find
(4.7)
l’or small velocities, I+$~may be t,aken to be the spherically symmet,rica wave function of Section II, i.r., the deformation of the wave packet may br neglected. Then
Thus t,hr t,wo theories give different, results for t,he finite effective mass, applicable for different, ranges of coupling st)rength. A t,hrory of the t,ransition, within the framework of the st,ate vert)or C-l.1) , can be drvrloprd for t#he Hamilt’onian (2.6) as in ref. 8. Let, us now glance at the effective mass calwlation for t’hc transformed Hamilt,onian. Equat8ion (L3.16) was arrived at by subt8ra&ng t)hr self-energy for zero
232
.i
QIZOSS
momentum. In contrast to Eq. (3.18)) t,he perturhat,ion stat,e vector for nonzero momentum states conbins a t#crm wit,h single creation operators acting on the vacuum. The state vector (4.1) is cwnvert~rd into a variational ansatz for (3.17) by means of t#he t,ransform (3.14). It also contains selected multimeson terms. For slowly moving particles, the srlf-consistent field approximation applied t,o Ey. (3.16) yields resul& similar to Ey. (3.17). One has only t’o replace ITi;pk by
- pk*Ik/nf.
V. CONCLUSIONS This discussion of a simple theory of t,wo int,cracting fields indicates the irrelevance of t#he classical part#irle solutions to the ultfraviolctS divergence problem. In a sense, our analysis merely illustjrtltes the accepted view t,hat, the quant~um theory brings in divergence t,rouhles cnt,irely independent, of the underlying classical t#heory. The prohlcm t,rtat,ed happens to he one where the classical t,hcory is sat,isfactory, at least for stat(cs involving a few nurleons. At8 the same t,ime, in this example the part’icle-like solutions arc relevant, t#o the analysis of the finite part, of the theory. Of course, for many Hamiltonians, t(herc is no indicat’ion of particle solutions, and onr considorat,ions are not, gcrmanc. In addition t,here are casrs where t’he classical theory is meaningless, i.e., t’he particle collapses and t,hc energy t#ends t,o minus infinit’y. For example, Pckar has shown i 72) t’hat, the s.c.f. treatment of a nonrclativistic part,iclc wit,h dcrivat,ive collplin, u to a nwson field leads t,o such a result. The self-energy divergence in thr theory st’udi4 here is of a l.ery trivial nat,urc, so that one must be cautious in generalizing t#hc: results. The infir& energy is not, conurcted with an infinite mass, as would he Pxpect8ed in a relativistic field t,heory. It would he int,crwCing to invrst’igat,c mhct,hrr the part,irlc solutions have a similar utilit#y for a rcnormalizable rclatSi14stic theory. If t,hey do, this might, encourage one tjn t.ry to cxt,end t#he scope of rrnormulization t#heory. However, in a relat,ivistic t,heory, t#hr definition of t)hc vacuum presents a problem. Tht model studied here hccomts snpcr~ondll~t,iIl, (7 for the many I:ermion case. It, seems t,haf a rclat8ivixt8ic t,htory adrnit8ting one-n~lcleon particlc-like sohtt ions generally shows a tBendcncy t,o hnvc an anomalous vawum state. Surprisingly enough, for simple models having anomalous varlla, such us t,hose considered hy Goldst,one ( IS), one finds parMe solut,ions even when one works from t,he anomalous \Tacuum. These quest8ions require furt’her st#udy. Rut it seems calcar that all of these considerations can only hope to uncover further unexpected structure in wrrrnt, field t,hcorirs, without’ coping n-it’h the problem of ultSraviolet divergences.
.~('ECN(l\I.I,EI,O~~ZlENT
Thr author is indeljted to t,ions and critic:tl comrrrents. ICE(*EIVI’:I)
: I~f~lmlary
23,
Dr.
J. Bell
for
nxtny
stimulating
and
clarifying
discus-
1 !W l~J~F~xI~?I~:KS
I. N. I
/‘hi/s.
/I’cr. 55.94
(1939);
K.J.
I~INI~EI.sTE:IN,R.J~~ ~,EvIER,AsI)~II.J<~ ELSTEIN, (:. FRONSUAI,, BSI) I'. h-.4(
FINKELSTEIN, I)ERMAs,~~~.s.~?~~. s. Whys. h'w. 103,
II'w. 76, IOi!l (1!)49); 1t.J. 83,3% (1951);1t.J.FrN~1571 (1956); A. HouRIET..\;~/-
P'hys.
r/ear l’hys. 4, 408 (1957). ibitl.39,l (1913);ibid.40, ~(~~~~);H.WEYL,“SJ):L~L', ? ~~.~lrti,.l~rn.I’h~sik37,511(1~~1~); Time and M:tttcr,“ p. L’O(i. 1)over. Yew York. 1950; W. Pal-~1, “Theory of Itelativit,y.” l’crgsrnon Press, I,ontlon, 1958. J. (:. HEF:ER, Z. Physik 144, 39 (195(i);Ann. Physik 17, 1W (195(i); ibid. 16, 43 (19551. 4. Ii. MEYER. .Iun. f'hysik 17, 109 (195(i). 5. I,. 1). J,a~u.nc ANI) S. PEKAR, ./. f'hys. ~'.,~.S.f<. 18, 419 (1948). “Urrt,e~su~hurr~t,rl utwr dir 15lrktroncllt.heo~i(! tier Krist:~lle.” Akademisclrr ti. s. J'EtihR, \.erlag, Berlin, 19.S. 7. >I. C:~.R.~RI, Phil. :lfcry. 44, 3% Cl%:<); T. 1). JA?.E! E'. 15. I,uw, AXU 1). PISES, Phys. f&v. 90, 297 (1953); 1;. V. '~YABI~~~ov, Zhu~. Elispll. i Tewrt. Pia. 26, 688 (1953). 8. 14:. I’. (:ROSS, .i/m. I’hys. (.\‘I-) 8, i8 (1959). 9. H. ~~AKI.:s, z. I'hysik 147, .1"3 (1!)57). “15igenfunction b;spnnsions.” 10. I<;. C'. 'I~T(~Hs~~\RsII, Oxford Univ. Press, 19%. 11. 0. \V. I.;REENRERG ASU S. S. SVHWEBER, .~UWO cimento 8, 378 (1958). 12.
$.
1. ~'EKAR,
13. .J. ( ~OI.IWONE,
&mict Phvs.-JETP .1'urwo ci,uu~rI~,
2, I(i:!
C,l!k%i).
19, 154 (1961).
OF
PHYSICS:
19,
219-233
(1962)
Particle-Like
Solutions EUGESE
Brandeis
University, CERS,
in Field
P.
Theory
GROSS
Waltham, Massachusetts* and Geneva, Switzerlanrlt
Anumherof quantumfield theories have the propert,y that the field equations, st,udied as classical equations of motion, have particle-like solutions. These are spatially localized exact solut,ions, with a definite value of angular momentum. The role of these solutions in a fully quantum theory is the subject of investigation. We examine, in particular, the theory of a nonrelativistic scalar nucleon, coupled by a local interaction to a scalar meson field. The classical solution (whose spatial extent is a nucleon Bohr radius) is identified with a selfconsistent field approximation to the one-nucleon state. However, quantum corrections to t,his approximation yield the same divergent self-energy as perturbation theory. In this example, the particle-like solutions have nothing to do with the ultraviolet divergences, but are relevant to the treatment of the finit,e part of the theory for strong coupling. The separation of the infinite part is trivial and is accomplished hy a canonical transformation. The particle solutions still exist for the finite part for strong coupling. I. 1STROI)UC:TIOS
X large number of papers have appeared, in which the field equatjions of a cluantum field t,heory are discussed c~lassicslly. The field operators arc trcakd as ordinary
frmctions
of
spacac
and
time.
Specifically,
\vc
jvill
hr
conrerncd
with
t8heorics Involving t#wo fields coupled in a conventional way. For cxamplc, one may cwkdrr a scalar meson field caoapled to t,he elwtromagnrt~ic field or t,he Dirac field coaupled to a meson field. In some of t,hc theories, particle-like csac*t solutions
of
the
nonlinear
classical
fiicxld
cxcluntions
have
hcxrn
found
(1).
These
solutions are characterixcd by a t,imc constant angular momentum, as ~~11 as by other constants of the motion, stwh as rhargc. They t,hlls have some simple spat,ial sgmmctry. The fisrd values of the constants act as normalization c*onditions
on t hc field
amplitndcs.
The
solllt8ions
ar(‘
particle-l&c
in the
scnsc
that
the
* I’crmanent address. Work supported in part by Oficc of Scientific ltese:irch, IT. S. Air Force. t National Science Senior Post Doctoral Fellow (lOGO-1961). In this paper all vectors are writ,ten in lightface type when they occur as subscripts or superscripts. 219
220
GROSS
mutually associated fields are large in a small spatial region and fall off rapidly as one leaves this region. The stress energy t,rnsor is well behaved everywhere, and the t,otal energy is finit’e. Furthermore, some of t’he solutions are st,able with respect to small perturbations. Rosen and E’inkeMein have discussed some general features of such solutions for fields with Lorcntz invariant Lagrangians. In particular, if “particles” are thought of as being defined in terms of integrals of various field propert#ies over regions of high field strrngt,h, the correct laws of moGon are obt.ained as consequences of the field equations. Thus these conveAona1 theories, when t,reat,ed classically, have very attractive propert,ies from the point of view of a field theory of mat,ter (2). In addition the effectively nonlinear nature of the t,heories implies that there may be several solutions for a given set’ of const,ant,s of motion. This offers hope of interpreting some unstable particles as states of different “principal quantum number,” and perhaps rven of reducing t,hr mimer of kinematical attribut,cs “put into” a t’heory. It is unfortunat,ely t’rue that in this work on classical field t,hcories, no field has yet been produced in which the excit’ed stnt#es bear a promising relation t,o known elementary part,icles. Apart from t,his fact, there has been very lit,tle analysis of the relevance of t,he classical part,icle-like solutions t,o t,he quantum t.heory. There is no indicat’ion of particle-like solutions when the same theories are treated by current’ (cssentially perturbation theorct,ic) methods of cluantum field theory. In fact,, the quantum approach leads to divcrgcnt results for self-energies and other physical yuantit,ies. One may wonder whether t#hc particle-like solAons of the classical theory survive in any form in a complete quantum theory. Furthermore, one may ask whether t,he nonperturbative solut,ions of classical t,hcory affect the divergence problem of t,hc quantum t#heory in any way. The present not’e is devoted to an explorat,ion of these questions in a preliminary way. We shall take a very conventional approach and interpret the cslassicaal particle solutions as a definite npproximst8ion to the one-particle state of the quantum theory-essentially a self-consistent field approximation. We shall not t,ry to use t)he classical theory as a starting point for some radical alteration of t#he quantum formalism. The part,icular t#heory t,o be cxamiwd in detail is one considered by Heber (~3) . It involves a complex scalar-nonrelatJivist’ic “ii~iclcon” field $, and a real s(*al:u meson field 4. The Hamilt*onian is
(1.1)
Here fi = I, c = 1, and :-: means that the zero-point, energy is to be suMractc,d. The fields are coupled by a local point int,eraction. While this problem is not
PARTICLE-LIIiE
SOLUTIONS
IN
FIELD
THEORY
221
relativistic, the features presented by t,he point int,eraction and nucleon recoil are still of general interest. Fully relativistic theories involve mainly complicat.ions (and fundamental differences), connec%ed with t’he proper description of the vacuum. The organization of this paper is as follows. In Section II the features of the particle-like solutions are briefly reviewed. The classical solution is connectfed t’o a self-wnsistent, field description of the one-nucleon state of the quantized field theory. This descript,ion yields a finite self-energy for t,hc one-nucleon stat)e. Hut, quant,um perturbation theory leads t’o a logarithmic divergence for t#hc selfenergy. In Sect,ion III, t#he remark that the two calculat,ions correspond to treating different, physical contributions tJo the self-energy is csploit’cd. The self-energy is recomputed by a variational met,hod which includes both effect’s. It is shown t,hat, the quantum corrections to t,he part#icle-like solution again give a divergent self-energy. For t,his example, t,he part,iclc solut,ion represents a caalculation of t,he finite part of the self-energy ( which can he unambiguously separated from the infinite part,), appropriat#e t,o strong coupling. The conclusion is that t#he finit,e cnlassical rsolutZion has nothing t,o do wit#h t,he quantum divergence. This arises from virt,ual emission and absorption of quanta with wavclcngths much short,cr t.han t,hc c*haract,erist,ic txtcnt, of t.he particle. In Scct,ion IV we c~alculate t#he effective mass of the nucleon. This illust~rates horn the classical and pertlwbation approsimations lead t,o different, estimat#es of a finite quantitJy. Section V summarizes our conclusions and notes problems recluiring furt,her invrst.igation. II.
PARTICLE-LIKE
The field equat,ions
SOLUTIONS
corresponding
ANI)
QLTANTIJM
SELF-ENERC:Y
to H arc
An exwt solution having radial symmetry is of the form + = j(r)eeCEt. The nucleon density $J+# is time const,ant and spherically symmetric. The associat,ed ~Iwson field is -g[z-z’l 2 -~~ ++lj4 x’ ) &r’, (2.2) +=$ s lx--‘1 and is also spherically symmetric. t,hr nonlinear eigenvalue equat,ion
The
part’irlc
solution
is
found
Upon
when
eliminating
one
uses
4(x),
the
we find that $ sat’isfies
normalizing
condition
222
GROSS
J $+$ d’.r = 1. One can calculate E and $ by inserting (2.2) into the Hamilt#onian and varying H for normalized G’s, While numerical calculat,ion is required to find t’he detailed behavior of $, it, is easy to est#imatethe characteristic size of t,he part’& when p = 0. One considers the class of normalized functions I/X = X3’*h(Xr) for some h(x) with J h’(x) d”~ = 1. The best,value of t#hcscaling paramct.er X is found by minimizing the totmaenergy with respect t,o X. The result, is E = &
1
cl -
hg”cz ;
Cl =
c* sE/&A
s
(Vh)’
d3x
= s ,x-y, dz&/ h”(x)h2(y)
3
= 0 yields X = (Q/Q) (g”M) . Thus
E = -f ” g2(Mg2) 1 The extent of t,he particle l/X is of t,he order of a nucleon Bohr radius fi’/Mg”. A more detailed study of the existence of the solut)ion has been made by Meyer (4). His results can be most easily expressed in terms of the Fourier components b, of $(x), which are subject to t,he normalization condit,ion c, b,*b, = 1. Meyer proves t)hat t’here exists a set of b, that gives a minimum of H. Furthermore 1b, 1fails t’o zero faster than any power of p as tends t#oinfinity. Physically the high momentum components behave this way becausethe nucleonic matter cont,ained in a small spatial region t’ends to zero as the region shrinks. For p = 0, the scale argument, shows that’ t’he energy is always negative, and is therefore lower than the energy that corresponds to spreading t,he nucleon out over all spaceuniformly. The latt,cr situat#ionalso corresponds to an exact solut,ion of the classical equat’ions, namely $ = 1/da, 4 = 0. I:or p # 0 a minimum coupling strength g2is required before a particle-like solut’ion is stable. These are the main features of the simplest’particle solut8ion.In general, there are other classical solut’ions of higher energy, both with zero and with finit’e field angular momentum. One would like t,o idrnt#ify these solutions with mstable particles. We will not, study these solutions here sinre our main concern is to st#udy t#heconnection het,ween t#hr classical and quantImn theories. We will rest,rict, our caonsidcrat,ionsto t,he lowest energy solution. This connection is not necessarily unique, and one might possibly use the particle solution in ways different, from the one outlined here. We want the classical solut,ion to correspond t’o a single particle. So, we look at the Hamiltonian for a nonrelat’ivistic scalar particle int#eract,ingwit’h a meson field (2.6)
PARTICLE-LIKE
SOLUTIOSS
IX
d)(x)= c
FIELD
223
THEORY
fp’“(al,+ &)P ikr
(2Wk
Ii?
(Ui+e-ikr
(2.7) -akP
Lkr
)
The Hamiltonian (1.1) apptars when (2.6) is subjected to srcond quantizntion wit,h # and ++ treat#cd as operators. Conversely, (2.6) is the effertive Hamiltonian in t’he one-nucleon subspare of ( 1 .I), when ( I. 1 ) is considered as a quant’um field theory. The question of Bose or Fermi statistics for the field does not, arise for the one nucleon &ate in a nonrelativist’ic t’heory. Xow, 1:he Hamiltonian (‘2.0) has been &died in great, dct#ail in the continuum idealization of the t$heory of the polaron. That case corresponds to I; - y/‘/c, wi; = W. In fart, Landau and I’ekar (5)) on intuit#ive grounds, umntct~ed with the idea of scylf-trapping, proposed a semiclassical approximation t,o (2.6). This cons&s of a,ssigning a wave function x(y) to t,hc particle and st’udying t,he meson equations of motion calassicaally. The procedure is precisely to study (1.1) as a fully calassicaal problem. Pekar has shown (6) that the entirely quant,um approsimation that, yields the same results as t#he semiclassical theory is t,o assume the state vector of the system !P to be a product q = x(q)+. Here @ depends only on the meson variables. The electron wave function x(y) is normalized to 1 and x and + are to he det)ermined self-consistently by independent’ variations of the clncrgy. I’or the ground stat#r it, is found t,hat
where
+0 is the vac*uum
st,at#e a&
= 0 and
This corresponds t’o a static shift, of the field oscillators in the presence of t.he matt#cr &r&y X*X. x(y) in turn is t#he wave fun4on of a particle moving in a potential well creat#ed by t,he field oscillat’ors. As applied t)a our case of a point, interaction, I’k = g(2wl;Q)-““, this self-consist,ent’ field approach to thr particle rcxcoil yields a finite self-energy for t#he ground state. If onr accepts t.his interpretat,ion of the classical theory, it hccnmes intSerest,ing to look at’ other approximations to (2.6) in order to find the range of applicabilit’y of t,he rlassical theory. Second order pert’urhnt’ion theory yields the energy,
1g=: -c which
/ T’kI2 k Wk +
diverges
(k”/2M)
logarit’hmirally
=
-6
A/
t’o minus
log
(1
infinit,y.
+
&)
for
/*
Th(x same result,
=
0,
(2.10)
is 0Mained
224 variationally, from uses a state vect,or
GROSS
the inkrmediatc
coupling
approximation.
This
theory
q = es1 @$l!~
(‘7) (2.11)
where s, =
c
ali+/3kr’k” -
(‘.(‘.
13, = - Trg*//( a/; + k2;‘2d/)
(3.12)
Since minus infinity is the lower bound, it is t)he correct answer. Thus one seems forced to the conclusion that, the self-consist’ent field approach is simply not relcvant when t,here is a point interact’ion. The only possible escape from t,his conclusion might, be t,o claim t’hat the class of state vectors used in intermediate wupling theory is too broad, and that one must’ work x-it,h wave functions x(q) normalizable to unit,y, and localized. In Scct8ion III WC will show t#hat# t,his is not. true. Still, our conclusion will not, bc the ut#ter irrelevance of tjhe particle-l& solutions. The reason is that t,he t,wo self-energies wrrrspond to very different physical effects. In the self-consistent8 fitld approach, one does not take into account1 the virtual emission of quant8a by t’he particle, t,ogrthcr with t(hc associated precise recoil. Indeed, if we carry out pcrturbntion corrections to the self-consistent, field, the due to very short wave logarit,hmic dircrgcnce re-emerges. Thi s dirergencc, fluctuations, is essentially indeprndcntS of the smooth partirk of ext,ent fi’, .1/y”. Point8 interact,ions ran give dircrgrnces, even in a nonrelativistic t,heory. We t,hcrefore conrludc that, the particlr-like solutions do not, represent a way to solve thr problrm of ultraviolrt8 divergrnrra ---as onr might first hopcx. On thr contrary, t,hry have Utle to do with that problem. Instc,nd, thry can be wsrd to suggest approsimat8ions to thr finite part of a qnantrtm throry for strong coupling.
Then,
+ C.C.+ C wk atfak + C wk 1fk 1’ > + c
(3.3)
(I’, Cik*+ ,fk-*ct~r )ak + C.C.E Ho’ + Hi’
The lowest eigenfunction of Ho’ is x( ~)a,, We can use the complete set of eigenfwict’ions of HO’ to calcu1at.e the prrturbation corrections arising from H,‘.
PARTICLE-LIKE
SOL~~TIONS IN FIELD THEORY
x?;,
One finds the logarit,hmic divergence of perturbation t,heory. This procedure has the advant’age of being completely general, in tjhr sense that it can he applied to all Hamiltonians, so that, one shows t,hat the esisttw~e of a particle-like solution does not affect, the ultraviolet divergewe. In the present problem, however, it is possible to st,rengthcn the argument by using the variation principle to gain a lower hound to the energy at. every &age. WI: make use of a method ( 8) of studying t#htl recoil prohltm for all wupling st,rcngth:: which is based on a trial state vector incorporatSing both physical contjrihlli,ions t,o t,hc self-wcrgy. For the ground stat#cl the form of the st,atc v&or is * = &(I@,, , s = c apcpn*(q) - c.(‘. (3.3) rpk( q) dclpends dircc%ly on q and also on ar-cragw taken with X( ‘I). I~ormally, it. reduces to the intcrmrdiate coupling t,rial vcrtor when (pL. = &c 1kq, x = l/d% and to that self-consistent field approwh when pk. is indcpcwdtwt8 of q and equ:d to -I’,” ^ (‘itq ) x 1” & @ = ~wI;-~- J ()nc is, however, not rest’rictcd t’o cGther (Jf t,hcsc choiws. In fact fw ~1 given X(U) ( the flmct ions cpk( q) arc’ wmplct,c~ly dctermincd t)y the variation principle. I,et, IIS defiw :I potenM riqj = i - I ?:I[) (V'x x) :w:rwiatwl with my x(y). I,rt, xlrlnll 11) and t312mlw thr cigenfwwtions and c~igcwval~ws of the linear operator - (v' ,‘,Af) + I’(q). ‘I’hcl entities
arc Green’s flmc%ions for the problwl. Thaw, with rcspwt, to cpk*( q) for givr>n XCc/) yields (Ok(q) = -T7i* G,,.(q, q’) ohcys t#he intcgrd c’,(q, whew
q’)
= G”(q
(&I’ IS the Grwn’s
i
frmc%ional
\-aviation
of t,he energy
rJ’““G, ( q, q’ ) tl”q’
( :i.5 )
clqu:ltioll -
q’) +
func*tion
.I
Gk” ( q -
for frw
q” ) 1T( q” ) Gh (1q” ) q’ ) cl:‘*”
( 3.6 )
part~iclw
(3.7) The aI)prosinlat’ioll E,
=
-&
t.o the clnergy of thcl gro~md
st,atr is
/ x*V2x d"q
(3.8)
This expression can be varied with respect’ to x to find the optimum function. We note that the formal st’ructure of this expression is close to t’hat of second order perturbation t’heory. Yet it expresses the self-trapping aspect as well, because of the dependence on x(y) . The same approximation to the ground state energy was found by Haken (9) from the point, of view of the path integral variat,ion principle of Feynman. One can now prove t#he divergence of EQ for any admissible x. This hinges on the fact that the second term diverges logarit,hmically when GA is replaced by Gk”. For this term is t#hen
Consider very large values of ( k (, such that the entire contribution to the sum over g is accounted for, i.e., xa :Y 0 for u > K. Take / Ic 1 > K. Then the expression in question is
The first term diverges logarit,hmically. Furt,hermore, if we solve the integral F
equat’ion for G, by iteration,
) vk j2 j” e”““x*(y)(G~I~G~)x(y’)e-“kq’
d”q d3q’
the term (3.11)
is finite, because the t’erm in brackets becomes l/k4 for large k. All further t’erms in the series converge. Hence, except for the first tsrm, all terms converge individually. Using t#he result’s of Tit,chmarsh ( 10)) one can establish the convergence of the Pntire series for suitable x(q). The argument is more direct’ if we note that to obtain a finite energy, Gk would have to be such t#hat
s
eik4x*(rl)Gk(q,
approaches
q’)x(~‘)e-ib”’
c13qd’q’
(3.12)
zero fast,er t,han 1/k” for Ii very large. Rut t#hen s
eiPqX*(q)(Gk”
VG&(q’)
Cipq d3q d’q
(3.13)
approaches faster than l/1;’ and gives a finite contribution. However G, = Gko+ C,“VGk and Gko gives an infinite contribution. Hence we have a contradiction, and must, conclude that, Ck gives an infinke result’ for t,he self energy. The divergence in t.his t,heory is of a trivial nature. It may be removed by
PARTICLE-LIKE
SOLUTIONS
IK
FIELD
THEORY
227
subtracting the term EJ #+# d”.r from the original Hamilt,onian. This is an energy E, divergent and coupling constant dependent, for each nucleon. We may isolate the divergent, part explicitly by performing a canonical transformation. With CT = es & = _ (2WkW’” ___________ (3.14) Wk + (k2/2m)
pk = Pk p-y
s = Cak qq.* - cc, we have
(3.15) The transformed
Hamilton
CFH7F-’ := E + &
is
- & c k
(ak &* k.p eikq + ak+ pk e-ik.q k.p) (3.16) ak+ k/jk e-ikq + c.c$ - F !$!$
+ C wk ak+ ak + L& (F The divergent. constant
is -1 E
=
_
C
8?&k k
‘Ok
f2)
+
(3.17)
(k’/%\~)
St#art#ing from a statme of zero total momentum (l,l’dG)%O , where a0 is a vacuum (a& = 0) , the terms linear in ak and a&+ of t’he transformed Hamiltonian give no contributSion t#o second order pertBurbation theory. The pert8urbed Aate vector is
1 Gkz = [(k” + /2)/zdz] This leads to t#he convergent
+ (‘d, + ‘dl)
(3.19)
self-energy
E? = - 5
(&)’
/ 13kI2 I82 IL GkZ
(3.50)
All the higher order t,erms in the self-energy converge, as may be shown by a diagrammatic analysis. Basic to t,he point of view we are following is t#he fact that, t’he transformed Hamiltonian, neglecting the divergent, constant R, also has classical particlelike solutions. In the quantm~l version we look for prodwt state vect.01
228
GROSS
\E = #(q)+. coordinate,
Taking
the partial
expectat,ion
value
with
respect
to the nucleon
ak+ alOk PI* k.1 e--l where Ik = 1 #*k. p ei”“# d3y,
(3.22)
Pk = I ‘# * epikqJ, d3q
This presents us with a normal mode problem and with the task of determining a shifted center of oscillation for the uk . The shift can be induced by the unit#ary transformation UP = e”“, &
= c
ak++Ck -
Making use of the &uct,ure Then cI; is det’ermined by
Thus
=
+(q)
the particle exp
The state vector of the original theory this \k. This energy of the self-trapped
(3.23)
t ‘2al;c’~1 = ak + ck
of the quadratic
t,he state vect,or describing \k
a&*,
[C
(ak+
form,
we find ck real and ck = c-~ .
solut#ion -
is (3.25)
ak-)ck]@O
(wit(h a counter state is
term),
is es operat#ing
on
(3.26) where $ is to bc determined HO as to make & a minimum. To put this expression in a simpler form, we measure Compton wavelength fi/AIc of the heavy part,icle. With
lengths
in units
of the
13.27)
/” #’ d3q = /x’(E)
d3,t = 1,
Pa4RTICLE-LIKE
SOLUTIONS
IN
FIELD
229
THEORY
we find for F = 0,
g=!!c 2
(3.28)
where (3.29) The crit,erion for self-t(rapping is now changed. In particular, for the original Hamiltonian, a self-trapped state was possible at, all coupling strengths, when ~1= 0. Now the coupling g2 must be sufficient,ly large. With l.he methods me have presented, there is no continuous transition bet,wcen the state vertors and energies of perturbation theory and t#hose of the self-consistent, field approach. However, as in the case of polaron t,heory, it, is possible to construct’ state vectors which make a continuous transition, and which have an admixture of self-consistent field at all coupling strengths. A careful st,udy of the t’heory would he important to find out to what extent the self-c*onsistentI field feat)ures of the excited state spectrum are germane at weak and int,crmediatr c+oupling st’rengths. However, the significant, point for t,he present considerations is that’ the divcrgrnt part of the theory can be isolated, and that, t,he ctlassical part’icle soluGon remains relevant for the finite part of the theory for strong coupling. IGnally, we isolate the divergent part in the caomplete quantum theory. The theory is made finite by subtracting E j” fi+# d3x, or an energy E per nucleon. To prodlIce such a t,erm explicitly, one transforms the original Hamiltonian by [T = (>iT,
1’ = j yc+(xh(
x - y)7r( y)+(x)
d”y d”lJ
(3.30)
Then r’+(x)r’-’
= esp - i /’ X(x - y)+(y)
d+J #(x) (3.31 )
734x)r’
= 4(x)
+ j X(x - y)++(y)+(y)
d”y
This transformation is the dressing t8ransformat,ion used by Greenberg and Srhwcber (II). Here, however, we arc treating recoil and the dressing is not caompletcb. The choice of X(x - y) is &. = t/;jy&&. . Then t,he transform in thcx
single nucleon divergent part
UH/l-’
s&space is the same as Ey. (3.14)) of the dressing. We have explicitly
and so serves to isolate
the
= Ho + E /- $+ $ d3x + / {gS(x - y) + (111- Li>
.X(x - y) + &
/j/f
+(x’
+(x>+++(y) : dy)dy’)
d”x d3y -
j(x)X(x
- y) $ (y) d3x d3g
: 2 (x - y’) 3t (x - y’)++(x)+(x)
~+(x)~+(x’)~(x)~(x’)
ss
s
1
- y)h(x - y) d3y +’
gX(x
e(x’
- x’)
+ g
- y) e(x
2 s ay
d3x d3y d31/’ (8.32 )
- y)
d3xd3x’
aY
In this expression all the terms except the last one contain a one-nucleon contribution and are direct analogues of terms in Eq. (3.16). The expression j(r) is t,he nucleon current density ( 1/2M) (++V$ - V$+#) . The expression X(Y) r( y’) : involves two meson operators and is ordered so that t’he zero-point contribution Ck (wk/2ci)e”-” has been taken out’ and used in the det#ermination of X(x - y) . The last term is a direct, two-nucleon int,eraction and depends in sign on t,he boson or fermion character of the nucleons. The transformation is chosen t,o remove divergences, so t(he linear t,erms in the meson operators produce finite self-energy cont8rihut.ions. IV.
STATES
OF
NONZERO
MOMENTUM
In this section we briefly discuss the t.heory of the effective mass of a slowly moving particle. This prolrides an example of how t’hc quantum approximation based on a particle-like classica solution yields a point, of view complementary to that of perturbation t,heory, in the analysis of t,he finite part of a field theory. Similar result,s would he obt,ainrd for ot#her problems such as scattering, optic&al absorption, et#c. l~ollowing ref. 8, we work w&h stat,e vcct,ors of the form 4 = PXP (C In t,he intermcdiatc
coupling $(*)
and f is an eigenfunction
aktcpi,(qj - c.c.j+(q)*o
(4.1)
approximation = L&
(py
lpe = pli epik,y
of the trotal momentunl P = d/S. +
c
PO, wit,h eigenvalue
k 1 & 1’
(4.2) P. (4.3)
P.iRTICLE-LIKE
SOLUTIONS
IN
FIELD
THEORY
231
The value of /z?,is (4.4) The energy associated
with the Hamiltonian
(2.6) is
k [(k”/2M)
+ wk - k-k]
(4.5)
The difference bet’ween this energy and the ground state energy is finite, even without the infinite count,er term. For slowly moving particles the effective mass is defined by writing the energy in the form
i.e., by eliminat’ing
2 bet’ween Eels. (3.3))
(4.4), and (4.5). We find
In the semiclassical approach, the state vector is again of the form (4.1). However, (Do is a constant, -4/i , independent of q. #(q) can be written as Ij/o(q, z+?*, where #o( q, A) is a real, localized function. h can be interpretNed as t)he expectation value of the velocity of the partic:lr. We find
(4.7)
l’or small velocities, I+$~may be t,aken to be the spherically symmet,rica wave function of Section II, i.r., the deformation of the wave packet may br neglected. Then
Thus t,hr t,wo theories give different, results for t,he finite effective mass, applicable for different, ranges of coupling st)rength. A t,hrory of the t,ransition, within the framework of the st,ate vert)or C-l.1) , can be drvrloprd for t#he Hamilt’onian (2.6) as in ref. 8. Let, us now glance at the effective mass calwlation for t’hc transformed Hamilt,onian. Equat8ion (L3.16) was arrived at by subt8ra&ng t)hr self-energy for zero
232
.i
QIZOSS
momentum. In contrast to Eq. (3.18)) t,he perturhat,ion stat,e vector for nonzero momentum states conbins a t#crm wit,h single creation operators acting on the vacuum. The state vector (4.1) is cwnvert~rd into a variational ansatz for (3.17) by means of t#he t,ransform (3.14). It also contains selected multimeson terms. For slowly moving particles, the srlf-consistent field approximation applied t,o Ey. (3.16) yields resul& similar to Ey. (3.17). One has only t’o replace ITi;pk by
- pk*Ik/nf.
V. CONCLUSIONS This discussion of a simple theory of t,wo int,cracting fields indicates the irrelevance of t#he classical part#irle solutions to the ultfraviolctS divergence problem. In a sense, our analysis merely illustjrtltes the accepted view t,hat, the quant~um theory brings in divergence t,rouhles cnt,irely independent, of the underlying classical t#heory. The prohlcm t,rtat,ed happens to he one where the classical t,hcory is sat,isfactory, at least for stat(cs involving a few nurleons. At8 the same t,ime, in this example the part’icle-like solutions arc relevant, t#o the analysis of the finite part, of the theory. Of course, for many Hamiltonians, t(herc is no indicat’ion of particle solutions, and onr considorat,ions are not, gcrmanc. In addition t,here are casrs where t’he classical theory is meaningless, i.e., t’he particle collapses and t,hc energy t#ends t,o minus infinit’y. For example, Pckar has shown i 72) t’hat, the s.c.f. treatment of a nonrclativistic part,iclc wit,h dcrivat,ive collplin, u to a nwson field leads t,o such a result. The self-energy divergence in thr theory st’udi4 here is of a l.ery trivial nat,urc, so that one must be cautious in generalizing t#hc: results. The infir& energy is not, conurcted with an infinite mass, as would he Pxpect8ed in a relativistic field t,heory. It would he int,crwCing to invrst’igat,c mhct,hrr the part,irlc solutions have a similar utilit#y for a rcnormalizable rclatSi14stic theory. If t,hey do, this might, encourage one tjn t.ry to cxt,end t#he scope of rrnormulization t#heory. However, in a relat,ivistic t,heory, t#hr definition of t)hc vacuum presents a problem. Tht model studied here hccomts snpcr~ondll~t,iIl, (7 for the many I:ermion case. It, seems t,haf a rclat8ivixt8ic t,htory adrnit8ting one-n~lcleon particlc-like sohtt ions generally shows a tBendcncy t,o hnvc an anomalous vawum state. Surprisingly enough, for simple models having anomalous varlla, such us t,hose considered hy Goldst,one ( IS), one finds parMe solut,ions even when one works from t,he anomalous \Tacuum. These quest8ions require furt’her st#udy. Rut it seems calcar that all of these considerations can only hope to uncover further unexpected structure in wrrrnt, field t,hcorirs, without’ coping n-it’h the problem of ultSraviolet divergences.
.~('ECN(l\I.I,EI,O~~ZlENT
Thr author is indeljted to t,ions and critic:tl comrrrents. ICE(*EIVI’:I)
: I~f~lmlary
23,
Dr.
J. Bell
for
nxtny
stimulating
and
clarifying
discus-
1 !W l~J~F~xI~?I~:KS
I. N. I
/‘hi/s.
/I’cr. 55.94
(1939);
K.J.
I~INI~EI.sTE:IN,R.J~~ ~,EvIER,AsI)~II.J<~ ELSTEIN, (:. FRONSUAI,, BSI) I'. h-.4(
FINKELSTEIN, I)ERMAs,~~~.s.~?~~. s. Whys. h'w. 103,
II'w. 76, IOi!l (1!)49); 1t.J. 83,3% (1951);1t.J.FrN~1571 (1956); A. HouRIET..\;~/-
P'hys.
r/ear l’hys. 4, 408 (1957). ibitl.39,l (1913);ibid.40, ~(~~~~);H.WEYL,“SJ):L~L', ? ~~.~lrti,.l~rn.I’h~sik37,511(1~~1~); Time and M:tttcr,“ p. L’O(i. 1)over. Yew York. 1950; W. Pal-~1, “Theory of Itelativit,y.” l’crgsrnon Press, I,ontlon, 1958. J. (:. HEF:ER, Z. Physik 144, 39 (195(i);Ann. Physik 17, 1W (195(i); ibid. 16, 43 (19551. 4. Ii. MEYER. .Iun. f'hysik 17, 109 (195(i). 5. I,. 1). J,a~u.nc ANI) S. PEKAR, ./. f'hys. ~'.,~.S.f<. 18, 419 (1948). “Urrt,e~su~hurr~t,rl utwr dir 15lrktroncllt.heo~i(! tier Krist:~lle.” Akademisclrr ti. s. J'EtihR, \.erlag, Berlin, 19.S. 7. >I. C:~.R.~RI, Phil. :lfcry. 44, 3% Cl%:<); T. 1). JA?.E! E'. 15. I,uw, AXU 1). PISES, Phys. f&v. 90, 297 (1953); 1;. V. '~YABI~~~ov, Zhu~. Elispll. i Tewrt. Pia. 26, 688 (1953). 8. 14:. I’. (:ROSS, .i/m. I’hys. (.\‘I-) 8, i8 (1959). 9. H. ~~AKI.:s, z. I'hysik 147, .1"3 (1!)57). “15igenfunction b;spnnsions.” 10. I<;. C'. 'I~T(~Hs~~\RsII, Oxford Univ. Press, 19%. 11. 0. \V. I.;REENRERG ASU S. S. SVHWEBER, .~UWO cimento 8, 378 (1958). 12.
$.
1. ~'EKAR,
13. .J. ( ~OI.IWONE,
&mict Phvs.-JETP .1'urwo ci,uu~rI~,
2, I(i:!
C,l!k%i).
19, 154 (1961).