IL ~UOVO CIMENTO
VOL. XIX, ~. 1
1o Gennaio 1961
Field Theories with ((Superconductor** Solutions.
J. GOLDSTONE
CERN
- Geneva
(rieevuto 1'8 Settembre 1960)
- - T h e conditions for the existence of non-perturbative type
~ superconductor ~)solutions of field theories are examined. A non-covariant
canonical transformation method is used to find such solutions for a theory
of a fermion interacting with a pseudoscalar bosom A covariant renormalisable method using Feynman integrals is then given. A (~superconductor ~) solution is found whenever in the normal perturbative-type
solution the boson mass squared is negative and the coupling constants
satisfy certain inequalities. The symmetry properties of such solutions
are examined with the aid of a simple model of self-interacting boson
fields. The solutions have lower symmetry than the Lagrangian, and
contain mass zero bosons.
Summary.
1.
-
Introduction.
This paper reports some work on the possible existence of field theories
with solutions analogous to the Bardeen model of a superconductor. This
possibility has been discussed b y NAMer (1) in a report which presents the
general ideas of the t h e o r y which will not be repeated here. The present work
merely considers models and has no direct physical applications b u t the n a t u r e
of these theories seems worthwhile exploring.
The models considered here all have a boson field in t h e m from the beginning. I t would be more desirable to construct bosons out of fermions and
this t y p e of t h e o r y does contain t h a t possibility (1). The theories of this paper
have the dubious a d v a n t a g e of being renormalisable, which at least allows
one to find simple conditions in finite terms for the existence of (( supercon-
(1) y . ~A~BU: Enrico Fermi Institute for l~uclear Studies, Chicago, Report 60-21.
FIELD
THEORIES
WITH
(( S U P E R C O N D U C T O R
*) S O L U T I O N S
155
ducting )> solutions. I t also a p p e a r s t h a t in fact m a n y features of these solutions can be found in v e r y simple models with only boson fields, in which the
analogy to the B a r d e e n t h e o r y has almost disappeared. I n all these theories
the relation between the boson field and the actual particles is more indirect
t h a n in the usual p e r t u r b a t i o n t y p e solutions of field theory.
2. - N o n - c o v a r i a n t
theory.
The first model has a single fermion interacting with a single pseudoscalar
boson field with the L a g r a n g i a n
L = ~ ( i 7 ~ f ~ -- m~) + ~ I ~1[~ ~x~
~q~ -#~qg)--
go~f175~p~-- - ~-~1~o(P' •
(The last t e r m is necessary to obtain finite results, as in p e r t u r b a t i o n theory.)
The new solutions can be found b y a n o n - c o v a r i a n t calculation which perhaps
m a y show more clearly w h a t is h a p p e n i n g t h a n the c o v a r i a n t t h e o r y which
follows.
L e t ~(x) = (1/~/V) ~ qk exp Ilk.x] and let Pk be the conjugate m o m e n t u m
t
to qk. L e t a~o
, bk, be the creation operators for F e r m i particles of mom e n t u m k spin a and antiparticles of m o m e n t u m - - k , spin - - a respectively. R e t a i n only the m o d e k = 0 of the boson field in the H a m i l t o n i a n .
(The significance of this a p p r o x i m a t i o n appears below.) T h e n
(1)
H
=H~+H
B+H
z,
Sx= ~. Ei(atia~+ b~b~),
i
(k, a
is replaced b y a single symbol i)
HB
1
2
~ 2
= ~CPo + ~oqo),
H, =
go
qo Z
+ b,a ) +
~o
W h e n H~ is t r e a t e d as a p e r t u r b a t i o n , its only finite effects are to alter the
boson mass and to s c a t t e r fermion pairs of zero t o t a l m o m e n t u m . These effects
can be calculated e x a c t l y (when V--> vo) b y writing a~bi
+ ~= c~
+ and treating
cit as a boson creation operator (~). The H a m i l t o n i a n becomes
H'= ~ 2E,e~c~+ ~1 (po~ + #2q2) + ~go qo ~ (c~+ c~).
(~) N. N. BOGOLIU•OV:
7, 51 (1958).
2urn. Eksp. Teor. Fiz., 34, 73 (1958); Soy. Phys. J.E.T.P.,
156
5. GOLDSTON:E
The (1/V)q~o t e r m s has no finite effects. Let
e t, + e,
V~E,,
-
q';
i\/~(c
~, -
e,) = p , ,
then
H , = ~ (P~ q- K~q°~)-[- E, 12 (p~ -F 4E,, q,)
, -F ~go
qo ~ v / ~ q ,
-- ~ E,.
H ' represents a set of coupled oscillators and is easily diagonalized. The frequencies of the normal modes are ogo, o9~, given b y the roots of the equation
go~~
4Ei
I n the limit as V - + co, this becomes
( 2 = ) ~ j 4 ~ - o9~
(2~) j
/~ + ~
}
+ ,~E~',4.~,-- o9~) •
The first two terms in the integral diverge. This procedure corresponds exuctly
to the covuriant procedure of ealculuting the poles of the boson p r o p a g a t o r
.
.
k6o
.
.
.
.
.
.
D(Ic, o9) = ~o~ _ kS _ la~ - - II(k, 09)'
kw
Fig. 1.
and including i n / / only the lowest polarization p a r t shown in Fig. 1.
This comparison shows t h a t the two divergent terms can be absorbed into
the renormalized mass and coupling constant. The renormalization is carried
out at the point k, ---- 0 instead of as usual at the one-boson pole of D. Thus if
/7(0, o9) = A + Bo92 + g~//l(og~),
then
Z
D(O, o9) =
o9 2 - -
z=
1
i-~
'
ffl --
gilll(o~)
'
/z~ _ f f ~ _ A
-2'
g~
g~o
=5"
E q u a t i o n (2) becomes
/t~ - - o9~
2g 1~ 4/"
dsk
= (2~)~ o9 J 4 E ~ : 4 ~ - -
= F(o9~).
0)2)
FIELD
THEORIES
~ V I T H (( S U P E R C O N D U C T O R
)) S O L U T I O N S
157
The isolated r o o t of this equation, ~Oo,
2 is f o u n d as s h o w n in Fig. 2.
W h e n # l2> 0, ~oo is t h e square of the p h y s i c a l b o s o n mass. W h e n # 2~ 4 m ~,
this r o o t disappears. This is
t h e ease w h e n t h e b o s o n can
d e c a y into a f e r m i o n pair.
There is a l w a y s a n o t h e r r o o t
2
for m2 large a n d negative. This
"\
I
c o r r e s p o n d s to the w e l l - k n o w n
\
\
2
\
2
(~g h o s t )> difficulties ~nd will
"x~ol./
I
\
be i g n o r e d here. W h e n /zl2 < 0
[
092
\
4m 2
(but n o t too large), there is
\
a n e g a t i v e r o o t for ~o~. This
#~
is usually t a k e n to m e a n s i m p l y
t h a t t h e t h e o r y w i t h td~ < 0
does n o t exist. H e r e it is t a k e n
Fig. 2.
to i n d i c a t e t h a t the a p p r o x i m a tion used is w r o n g a n d t h a t t h e H a m i l t o n i a n (1) m u s t be i n v e s t i g a t e d f u r t h e r .
This is done b y a series of canonical t r a n s f o r m a t i o n s b a s e d on the idea t h a t
in the v a c u u m s t a t e the e x p e c t a t i o n v a l u e of t h e b o s o n field is n o t zero. L e t
,
A
qo = qo + - v ' > ,
go
A is a p a r a n l e t e r to be fixed later.
t h e ferinion field
ai :
po = p'o,
Also m a k e a B o g o l i u b o v t r a n s f o r m a t i o n on
Oi
COS 2- g i - -
bi = sin
0~
2 - ~ Z i¢ 4.
sin
Oi
^t
~- fl~,
0_
cos ~ ~i
Z
A
tg 0i = - - •
Ei
Then
H =Ho+H~+H2+H~,
~°=--Z(v~E~+~--Fi)+i A•
~v+
HI = qo V ~ #8 go +
2o A ~ go
A
~v,
2~ A4
}
158
ft. GOLDSTONE
1
'o
1
,i /
~
As
go
,,
~. ~
x
Ho= f4-vq° + ~ c ' ~ q °
,~
go
,
+ c-~q°~ v
A
,
Ei
~ ,
(~, o~, + fl~ ~,) "
N o w choose A so t h a t H1 = 0. Ho is just a c o n s t a n t p r o p o r t i o n a l to V a n d
H3 has no finite effects. This leaves H2, which can be diagonalised b y t h e
m e t h o d used a b o v e for the original H a m i l t o n i a n (1).
One solution of H~ = 0 is A = 0. This gives the original a p p r o x i m a t i o n ,
which is no use when # 1o < 0. O t h e r solutions are given b y
1 A 2 g~~
1
f'~+~°~=V
~E~+A0=(2
(3)
where G is a finite function.
~
~
3It
/~;
2~
+ ggG(A2)
'
Let
(2~)U E ~ = ~ '
6g{ f d 3 k
21
~o + (~y~)~j~ - ~ ,
~
g]
go=~
2
Pl,
tt~ Z can be identified as the lowest order p e r t u r b a t i o n theoretic valueg.
of t h e renormalized boson maSSy four-boson interaction c o n s t a n t a n d w a v e
function renormalization, arising f r o m the g r a p h s in
x\
,
Fig. 1 and 3.
/
"\
As before, the renormalizations are carried out a t
/ /
>
k = 0. E q u a t i o n (3) becomes
(4)
/
1.
A2
~I+~Z~ ~ =
g~G(A~),
\
/
N
Thus an equation for A 2 is obtained which is finite in
t e r m s of constants which would be the r e n o r m a l i z e d
p a r a m e t e r s of the t h e o r y in the o r d i n a r y solution.
I t will be shown below t h a t when #1~< 0, equation (4) has solutions for a
certain r a n g e of 21, and t h a t t h e n there does exist a real boson.
/
N
Fig. 3.
FIELD
THEORIES
WITH
(( S U P E R C O N D U C T O R )) S O L U T I O N S
15,9
3. - Covariant t h e o r y .
A first approach to a covariant t h e o r y can be made b y calculating the
fermion Green's function in a self-consistent field approximation. In perturbation t h e o r y the term represented b y Fig. 4 vanishes
b y reflection invariance.
However, suppose it gives a contribution 75d to
the fermion self-energy. Then
1
S(p) =
y , p ' - - m - - 75X1 '
Fig. 4.
A _
and evaluating Fig. 4 with this value of S gives
1 g! ( d t p T r ~ 5
1
#~o (27~)4iJ
y~p~-- m -- 75A --
4g] 1 f f l 4
Zl
tt2o ( 2 ~ ) q
~ p2 _ m 2 _ zl~,
which is the same as equation (3) without the 2 term, and again has the perturbation solution gl = 0 and possibly other solutions.
A general covariant t h e o r y can be found b y using the F e y n m a n integral
technique. This gives an explicit formula for the fermion Green's function
S(x', x ) = fS(x', x; qD) exp [-- iW(ss)] exp [ifL~(~s)d*x] 8~
f e x p [-- iW(T)] exp [ifL~(~) d'x] ~cp
Here S(x'xcf) is the fermion Green's function calculated in an external boson
field ~0 (and without interacting bosons). E x p [-- iW(~)] is the v a c u u m - v a c u u m
S-matrix amplitude in an external field and LM(~) is the boson part of the
Lagrangian. The integration are carried out over all fields ~(x, t).
Let
1
q~(xt) = D X cfk exp [ikx]
where k is now a 4-vector and ~2 a large space-time volume. To obtain the
B a r d e e n - t y p e solutions, first do all the integrations except t h a t over To (this
has k - - ~o = 0) and p u t ~e = Q7~ (Z finite). Then
S(x', x) -- j'S(x', x; Z) exp [--izQ_F(Z)] d Z
.fe p [- iQF(Z)]dz
F(z)=w(x)+2z +~z',
]~0
J . GOLDSTON~
is the one particle Green's function calculated in a constant external field
~(x)----Z and including all the interacting boson degrees of freedom except
k = co = 0. E x p [ - - i Q w ] is the v a c u u m - v a c u u m a m p l i t u d e calculated in the
s a m e way. The idea is to look for s t a t i o n a r y points of F(Z ) other th~n Z = 0.
I f F'(Z~)----0, t h e n in the limit ~2-~ co, S ( x ' x ) = S(x'x)5 ).
- - i f 2 w ( z ) is given b y the sum of all connected v a c u u m diagrams. I t is
e a s y to see t h a t
Z2w(z) = Z V,.
,=o
2n!
where V~. is the p e r t u r b a t i o n t h e o r y value of the
e x t e r n a l m o m e n t a zero. F o r e x a m p l e V~----//(0).
V2.---- -2.v(~)]7""~where v / Z is the boson w a v e function
is finite p r o v i d e d the t e r m s #~ and ~o are absorbed
fore, the renormalizations are carried out a t k ~--0.
Z = v'ZZ(%
then
/v(Z(r)) ----
2n-boson v e r t e x with all
I n p e r t u r b a t i o n theory,
renormalization, ~nd _v(r)
~
into V~ a n d V4. As beThus if
2n.
;U ~"
an expression f r o m which all the divergences h a v e been removed. I t also
follows t h a t if F'(Z(~")) = 0, the new values
for
the v e r t e x p a r a m e t e r s V~ are given b y
\%
\
3"
V'.-
F" ~"
/
I n particular the new boson mass is given b y
N
/
I
/
/
%
Fig. 5.
F ' ( Z ) is in fact
d i a g r a m s with one
covered b y p u t t i n g
This gives
,,%
(This is really the mass operator at k----0,
not the mass.) Thus one condition for the
existence of these a b n o r m a l solutions is t h a t
F(Z ) should h a v e a m i n i m u m at some nonzero value of Z.
easier to evaluate t h a n F. I t is g i v e n b y the sum of all
external boson line. The previous a p p r o x i m a t i o n s are regoz = A and including only the diagrams in Fig. 5.
F(Z) --= ff0Z2÷
2.o Z3
6-
4g°~ f_
Z
+ ( 2 ~ ) 4 i J P ~ - m ~ - g~oz2 d 4 p .
FIELD
TtIEOR1ES
WITH
(( S U P E R C O N D U C T O R
161
~> S O L U T I O N S
~-[enee
F,(Z(,))
A~2
2 (,)+ 21 Z(r~
]["
~i
_ g (r)~4
~12:(r)4
d4p.
Thus A is given b y
=-~+~
gl
6gl
--~
1+
~
log 1 +
m2
m=J
Le~
4~r2#~
2 ~ 2 ).1 _
m~g2 = A ,
3
B,
g{
-Z]- 2 = x .
m~
Then
A ÷ B x = (l ÷ x) log ( l + x) - - x = h(x) .
The new boson m a s s is given b y
~
= F'(z) = ~x{B
--
h'(x)}.
I t can be seen f r o m Fig. 6 t h a t there will be
roots with/~-~ >. 0 only when /~ . 0 and B > Bcrit ,
where B¢~i~ is given b y h ' ( x ) = B . Thus the abn o r m a l solutions exist when
B>>O,
O>A>,--(e'~--I--B).
/
(x)
X
Fig. 6.
This calculation is exact in the limit g~--~ 0 keeping # J2 g l2, 21/g4 and glx (r)
finite. Thus in at least one case a solution of the required t y p e can be established as plausibly us the more usual p e r t u r b a t i o n t h e o r y solutions.
4. - S y m m e t r y
properties
and a simple
model.
I t is now necessary to discuss the principal peculiar feature of this t y p e
of solution. The original L a g r a n g i a n h a d a reflection s y m m e t r y , t~rom this
it follows t h a t F(•) m u s t be an even function. Thus is Z--Z1 is one solution
of F'(Zl ) = 0, ;/l------X1 is another. B y choosing one solution, the reflection
s y m m e t r y is effectively destroyed. I t is possible to m a k e a v e r y simple m o d e l
which shows this kind of behaviour, a n d also d e m o n s t r a t e s t h a t so long as
there is a boson field in the t h e o r y to s t a r t with, the essential features of t h e
a b n o r m a l solutions h a v e v e r y little to do with fermion pairs.
11
- II
Nuovo
Cimento.
162
J. GOLDSTON]~
Consider the t h e o r y of a single neutrM pseudoscalar boson interacting with
itself,
L = 2 \~x/~ ~x~
g~9~] - - ~ 90~"
Normally this t h e o r y is quantized b y letting each mode of oscillation of the
classical field correspond to a q u a n t u m oscillator whose q u a n t u m n u m b e r
gives the n u m b e r of particles. W h e n
tt02 < 0, this approach will not work.
However, if ~3 > 0, the function
~q
+~
,
is as shown in Fig. 7.
The classical equations
Fig. 7.
([[J~ + g~)90 + -6 90 = 0 ,
now have solutions 90=-4-~¢/~6/x~/X o corresponding to the minima of this
curve. Infinitesimal oscillations round one of these minima obey the equation
(E]~ - 2/,0) 890 = o.
These can now be quantized to represent particles of mass ~--2#2o.
simply done b y making the transformation 9 0 - 90'+ X
This is
6/~
23
Then
.5 = ~ \~x# ~x, + 2~:o90 '5 - - - ~ 90', - -
90,3 + -5 ).-o "
This new Lagrangian can be treated b y the canonical methods.
I n any state with a finite n u m b e r of particles, the expectation value of 90
is infinitesimally different from the v a c u u m expectation value. Thus the eigenstates corresponding to oscillations round 90= Z a r e all orthogonM to the
usual states corresponding to oscillations round 90----0, and also to the eigenstates round 90 = Z. This means t h a t the theory has two v a c u u m states,
w i t h a complete set of particle states built on each vacuum, b u t t h a t there
i.s a superselection rule between these two sets so t h a t it is only necessary to
FIELI)
TIIEORIES
a,V I T I [
q SUPERCONDUCTOR
163
~) S O L U T I O N S
consider one of t h e m . The s y m m e t r y cp ~ - - q ~ has disappeared. Of course
it can be restored b y introducing linear combinations of states in the two sets
b u t because of the superselection rule this is a highly artificial procedure.
Now consider the ease when the s y m m e t r y group of the L a g r a n g i a n is continuous instead of discrete. A simple e x a m p l e is t h a t of a complex boson
field, ~ = (fz'~-rip,,)/~/2
L-
~P* ~P
~x~ ~x:'
2 ,
~o
t~ocp~v- - ~ @*q~)~.
The s y m m e t r y is p -~ exp [i~]p. The canonical t r a n s f o r m a t i o n is ~s ----P'q-Z
IX l~ The phase of Z is not determined.
Z real the new L a g r a n g i a n is
2o
Fixing it destroys t h e s y m m e t r y .
With
)
z = 2 \~x~, ~~x ÷ 2#~p'? +
The particle corresponding to the p~ field has zero mass. This is true even
when tile interaction is included, and is tile new w a y the original s y m m e t r y
expresses itself.
A simple picture can be m a d e for this t h e o r y b y thinking of the two dimensional v e c t o r p at each point of space. I n the v a c u u m state the vectors h a v e
m a g n i t u d e Z and are all lined up (apart f r o m the q u a n t u m fluctuations). The
massive particles q/1 correspond to oscillations in the direction of Z. The massless particles p~ correspond to ~ spin-wave ~>excitations in which only the direction of ~v m a k e s infinitesimal oscillations. The mass m u s t be zero, because
when all the q~(x) r o t a t e in phase there is no gain in energy because of the
symmetry.
This time there are infinitely m a n y v a c u u m states. A state can be specified b y giving the phase of Z and then the n u m b e r s of particles in the two
different oscillation modes. There is now ~ superseleetion rule on the phase
of Z. States with a definite charge can only be constructed artificially by
superposing states with different phases.
5.
-
Conclusion.
This result is completely general. W h e n e v e r the original L a g r a n g i a n has
a continuous s y m m e t r y group, the new solutions h a v e a reduced s y m m e t r y
a n d contain massless bosons. One consequence is t h a t this kind of t h e o r y
164
J. GOLDSTONE
cannot be applied to a vector particle without losing Lorentz invariance. A
m e t h o d of l o s i n g s y m m e t r y is of c o u r s e h i g h l y d e s i r a b l e i n e l e m e n t a r y p a r t i c l e t h e o r y b u t t h e s e t h e o r i e s will n o t do t h i s w i t h o u t i n t r o d u c i n g n o n - e x i s t e n t
m a s s l e s s b o s o n s (unless d i s c r e t e s y m m e t r y g r o u p s c a n b e u s e d ) . SKYWAY, (3)
h a s h o p e d t h a t one set of fields c o u l d h a v e e x c i t u t i o n s b o t h of t h e u s u a l t y p e
a n d of t h e (~ s p i n - w a v e ~ type~ t h u s for e x a m p l e o b t a i n i n g t h e ~ - m e s o n s as
c o l l e c t i v e o s c i l l a t i o n s of t h e f o u r K - m e s o n fields~ b u t t h i s d o e s n o t s e e m p o s s i b l e i n t h i s t y p e of t h e o r y . T h u s if a n y u s e is t o b e m a d e of t h e s e s o l u t i o n s
s o m e t h i n g m o r e c o m p l i c a t e d t h a n t h e s i m p l e m o d e l s c o n s i d e r e d in t h i s p a p e r
will b e n e c e s s a r y .
(3) T. H. R. S K Y R ~ : P r o c . R o y . Soc., A 252, 236 (1959).
RIASSU~TO
(*)
Si esaminano le condizioni per l'esistenza di soluzioni (~superconduttrici )~ del tipo
non p e r t u r b a t i v o delle teorie di campo. Si usa un metodo non covariante di trasformazioni canoniche per trovare queste soluzioni per la teoria di un fermione interagente
con un bosone pseudoscalare. Si espone poi un metodo covariante rinormalizzabile
usando integrali di Feynman. Si trova un soluzione (~superconduttrice )) ogniqualvolta
nella soluzione normale del tipo perturbativo ]a massa bosonica elevata al q u a d r a t o
negativa e le costanti di accoppiamento soddisfano ad alcune ineguaglianze. Si esaminano le propriet~ di simmetria di tall soluzioni con l'aiuto di un semplice modello
di campi di bosoni auto-interagenti. Le soluzioni hanno simmetria inferiore al lagrangiano, e contengono bosoni di massa zero.
(*) Traguzioree a cura della Redazione.