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Lee model and the local field theory
Nuclear Physics 12 (1959) 3 0 9 - - 3 1 3 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
LEE M O D E L A N D T H E LOCAL FIELD T H E O R Y I. B I A L Y N I C K I - B I R U L A
Institute /or Theoretical Physics, Warsaw University R e c e i v e d 31 M a r c h 1959 T h e c o m p a r i s o n b e t w e e n t h e L e e m o d e l a n d t h e u s u a l local field t h e o r y is d i s c u s s e d in a c e r t a i n s i m p l e case. T h e conclusion is t h a t t h e difficulties a s s o c i a t e d w i t h t h e ren o r m a l i z a t i o n in t h e Lee m o d e l do n o t a p p e a r in t h e local field t h e o r y .
Abstract:
1. I n t r o d u c t i o n
The Lee model is the only known model of quantum field theory exactly soluble with renormalization of the coupling constant. However, renormalization of the Lee model in terms of usual concepts of the field theory is impossible if the renormalized coupling constant is greater than a critical coupling constant 1,2,3). In the Lee model without cut-off the critical coupling constant vanishes and thus for any value of the observable coupling constant one must change the fundamental concepts of the field theory. Such a situation in the Lee model implies the possibility that also the usual local field theory is inconsistent for similar reasons 4,5). In this paper we want to show that the difficulties which appear in the Lee model are intimately connected with simplifications made in passing from the local field theory to the Lee model. These difficulties disappear when Lee's interaction Hamiltonian is completed by the lacking terms. We give the comparison of the Lee model with the local field theory for a certain simple case. 2. T h e Model of the Local Field T h e o r y
Let us consider a simple model of non-relativistic quantum field theory which describes the interaction of the fixed fermions V and N with bosons v% We assume the Hamfltonian for this model in the form H = Ho+H
~
= mov f d 3xV* (x)V (x) + m0Nf d 8xN* (x)N (x) +½fds~ [~* (x) + (V$ (x)) *+#* $2 (x)]
(1)
--gofd3.W*(x)N (x) + N t (x)V (x)] ¢ (x).
With the notations 309
310
I.
V)(x) =
BIALYNICKI-BIRULA
V(x) N(x) '
V)t(x) = I]Vt(x), Nt(x)II,
(9.)
j (x) = V)*(x)zl V)(x) -----V* (x)N (x) + N t (x) V (x), 1 V m o = -~(mo +too N ), Amo = moV--moN, we can write Hamiltonian (1) in the form H = H O + H 1 = m o f d'xv)* (x)~v (x)-{-½ f d3x[~#(x)+ ( V ¢ ( x ) ) 2 + # ' ¢ ' ( x ) ]
--go f d3x/"(x)• (x) + A m o f d3x~v*(x)vsv)(x).
(3)
We shall treat the SchrSdinger equation with Hamiltonian (3) b y means of the perturbation m e t h o d with respect to the t e r m H1=
ztm0fd3xwt(x),3v)(x).
(4)
It is well known 1, e) t h a t we must find only the physical one-particle states to carry out the renormalization, since the renormalization constants are defined'in terms of the m a t r i x elements between these states. We denote the one-particle states in zeroth, first and second approximation by Jv)°(x)), [v)l(x)) and Iv)*(x)) respectively. We then obtain b y usual procedure 1 _ _ H1
=
(
,
- °-- - m fv)3(x))=N-½ 1 - - P H
)
m H 1 [v)°(x)), H ° - - m (HI--AE)PH---6-~_
(6) where HOlv)O(x)> : mlv)O(x)>, 1
(7) 2
c$(x--x')A E = (~v° (x)IHIIV)° (x')).
(9)
The vect or lV)(x) ) denotes the column with elements IV (x)) and IN (x)) and P denotes the principal value. F r o m (9) we obtain for energy correction A E = TsArno exp(--2go2/g~) = ~sAm.
(10)
Here geir = [2(2~)8]-XSd3kco-S and An, is the observed mass difference in the first approximation. The equation (7) can be solved 7) upon diagonalizing of the Hamiltonian H ° by means of the unitary transformation
If
U = exp ig o d3 i(x)
'
1.
01)
The multiplication b y w-* in (11) means multiplication of Fourier transform
LEE
MODEL
AND
THE
LOCAL
FIELD
THEORY
311
of ~(x). The result of the transformation U is H O = UHoU-1
(127
--
:
The colon denotes the Wick normal product and m=mo--~m=m
go' r d ' k 2(2r¢)s J o / "
o
We can see from (12) that ~*(x)10 > is an eigenvector of the operator H ° belonging to the eigenvalue m. It follows then that the eigenvector of H ° belonging to the eigenvalue m has the form
lvO(x)> = U-IV+(x)IO>.
(13)
The renormalized (observed) coupling constant can be obtained ~) from the equation
(x-x')g~ = ~0.
(14)
The renormalization constants Z2v and Z2N of the fermion fields can be obtained from the equations
O ( x - x ' ) v ' z J =,
(15 7
cS(x--x')%/~ -----.
(16)
Now we find the renormalized coupling constant. From (6), (13) and (15) we have
~(x-x')gn
l
=
rd j(xlP
+ H 1 p HO-- m J
Using (3), (4) and
(n)
HO--m
(17)
we find
g = goN-1 [1-- <0[e'~aoTl~-*~(x)PHo~e-~gon~-"(*)10>] ,
(is)
where Hob = f dakoJ(k)a*(k)a(k). With the abbreviations a(go" ) = <01e't,o~,~-'~(*)pH~,e-~,n~-t~(x)i0 ) 1 r go' l " r d 3 k x d3k~ 1 = . =~z ~.t L2(2~)3j j oj---g" " " oJJ (oJ~+ • • • + ~ . ) 2 '
(19)
(20)
I. BIALYNICKI-BIRULA
312
we obtain finally g ---- go
1-- (Am)*a (gd) 1 + (~m)*a(eo*) ~ g0(1--2(~m)*a(g0*)).
(21)
It Can be shown that the power series (20) is convergent for go* < gma. where g~max> (2at)*e-1" From (15) and (17) we can obtain the renormalization constants Z, v and z,N:
~¢/-~,v ~_ exp(_go,/2gZcr)N_½[l_Ambl (go, ) + (Ar~),b~(go,) ] '
(22)
V/Z, ~ = exp(--go*/2g2=)N-i[l+Ambl(go*)+(Am)*b2(go2)].
(23)
The functions bl (go*)and b, (go*)are also defined by series which are convergent like (20) in a certain neighbourhood of go* = 0. The vertex part renormalization constant Z 1 can be find from the well known Dyson equation
g ~ ZI-I(Z2VZaN)½Z3½go .
(24)
The boson field renormalization constant Z 3 is equal to 1 and we have from
(24) Z 1 = exp (--go2/g~) [1 + ( A m ) ' C (go*)],
(25)
c (go*) --- b?(go*) + ~b,(go*) + a (go*).
(2S)
where
3. T h e L e e M o d e l The Hamiltonian for the Lee model 1) has the form I4 5 = H a + H I L = mov f d a x V t ( x ) V ( x ) + m o s f d a x N t ( x ) N ( x )
+~_fd3.E~,(x)+(v¢(x))'+~,¢,(x)]
(27)
--go f d3x IV* (x)N (x)~ ~+' (x) + i t (x)V (x)$ (-' (x)]. The renormalization constants in this model are 2 1 = 1,
Z 3 = l,
Z 2 N = 1,
go* f dab 1 (28) (Z2v) -1 = 1 + ~ eo (co--Am) 2"
The renormalized coupling constant defined by (25) has the form
g
1 + 2(2~p
o~ (o~---Am)*
(29)
It is well known that the equation (29) is the source of all difficulties in the Lee model. In fact, no real solution of this equation exists for go if g is finite.
LEE MODEL AND THE LOCAL FIELD THEORY
313
4. D i s c u s s i o n
The comparison between the Lee model and the model described in § 2 can be carried out in terms of Feynman diagrams. In the model of local field theory described b y the Hamiltonian (1) there is an infinite number of self energy and vertex part diagrams. The first few diagrams are the following ~
~
(30)
The symmetry between self energy and vertex part diagrams leads to the reduction of the infinite parts of renormalization constants Z 1 and Z2 in the product ZI-~ (Zzv Z2N)½. In the particular case of equal masses ~.s) we have simply the Ward identity Z1-1 Z, -~ 1. In the Lee model this symmetry is disturbed. One diagram appears here as the correction to the self energy but there are no radiation corrections to the vertex part. In the Lee model instead of (30) and (31) we have
~
(32)
The infinite parts of the constants Z 1 and Z2 cannot reduce and this leads to the known difficulties in the renormalization procedure. The author wishes to thank Professors L. Infeld and W. Kr61ikowski for their interest in this work.
References 1) 2) 3) 4) 5)
T. D. Lee, Phys. Rev. 95 (1954) 1329 W. Heisenberg, Nuclear Physics 4 (1957) 532 G. Kall6n and W. Pauli, Mat. Phys. Medd. Dan. Vid. Selsk 30, No. 7 (1955) W. Heisenberg, Revs. Mod. Phys. 29 (1957) 269 H. Umezawa, Q u a n t u m Field Theory (North-Holland Publishing Company, Amsterdam, 1956) p. 354 6) G. K~IMn, Helv. Phys. Acta 25 (1952) 417 7) O. W. Greenberg and S. S. Schweber, Nuovo Cimento 8 (1958) 378 8) Th. W. Ruijgrok, Physica 24 (1958) 185
LEE M O D E L A N D T H E LOCAL FIELD T H E O R Y I. B I A L Y N I C K I - B I R U L A
Institute /or Theoretical Physics, Warsaw University R e c e i v e d 31 M a r c h 1959 T h e c o m p a r i s o n b e t w e e n t h e L e e m o d e l a n d t h e u s u a l local field t h e o r y is d i s c u s s e d in a c e r t a i n s i m p l e case. T h e conclusion is t h a t t h e difficulties a s s o c i a t e d w i t h t h e ren o r m a l i z a t i o n in t h e Lee m o d e l do n o t a p p e a r in t h e local field t h e o r y .
Abstract:
1. I n t r o d u c t i o n
The Lee model is the only known model of quantum field theory exactly soluble with renormalization of the coupling constant. However, renormalization of the Lee model in terms of usual concepts of the field theory is impossible if the renormalized coupling constant is greater than a critical coupling constant 1,2,3). In the Lee model without cut-off the critical coupling constant vanishes and thus for any value of the observable coupling constant one must change the fundamental concepts of the field theory. Such a situation in the Lee model implies the possibility that also the usual local field theory is inconsistent for similar reasons 4,5). In this paper we want to show that the difficulties which appear in the Lee model are intimately connected with simplifications made in passing from the local field theory to the Lee model. These difficulties disappear when Lee's interaction Hamiltonian is completed by the lacking terms. We give the comparison of the Lee model with the local field theory for a certain simple case. 2. T h e Model of the Local Field T h e o r y
Let us consider a simple model of non-relativistic quantum field theory which describes the interaction of the fixed fermions V and N with bosons v% We assume the Hamfltonian for this model in the form H = Ho+H
~
= mov f d 3xV* (x)V (x) + m0Nf d 8xN* (x)N (x) +½fds~ [~* (x) + (V$ (x)) *+#* $2 (x)]
(1)
--gofd3.W*(x)N (x) + N t (x)V (x)] ¢ (x).
With the notations 309
310
I.
V)(x) =
BIALYNICKI-BIRULA
V(x) N(x) '
V)t(x) = I]Vt(x), Nt(x)II,
(9.)
j (x) = V)*(x)zl V)(x) -----V* (x)N (x) + N t (x) V (x), 1 V m o = -~(mo +too N ), Amo = moV--moN, we can write Hamiltonian (1) in the form H = H O + H 1 = m o f d'xv)* (x)~v (x)-{-½ f d3x[~#(x)+ ( V ¢ ( x ) ) 2 + # ' ¢ ' ( x ) ]
--go f d3x/"(x)• (x) + A m o f d3x~v*(x)vsv)(x).
(3)
We shall treat the SchrSdinger equation with Hamiltonian (3) b y means of the perturbation m e t h o d with respect to the t e r m H1=
ztm0fd3xwt(x),3v)(x).
(4)
It is well known 1, e) t h a t we must find only the physical one-particle states to carry out the renormalization, since the renormalization constants are defined'in terms of the m a t r i x elements between these states. We denote the one-particle states in zeroth, first and second approximation by Jv)°(x)), [v)l(x)) and Iv)*(x)) respectively. We then obtain b y usual procedure 1 _ _ H1
=
(
,
- °-- - m fv)3(x))=N-½ 1 - - P H
)
m H 1 [v)°(x)), H ° - - m (HI--AE)PH---6-~_
(6) where HOlv)O(x)> : mlv)O(x)>, 1
(7) 2
c$(x--x')A E = (~v° (x)IHIIV)° (x')).
(9)
The vect or lV)(x) ) denotes the column with elements IV (x)) and IN (x)) and P denotes the principal value. F r o m (9) we obtain for energy correction A E = TsArno exp(--2go2/g~) = ~sAm.
(10)
Here geir = [2(2~)8]-XSd3kco-S and An, is the observed mass difference in the first approximation. The equation (7) can be solved 7) upon diagonalizing of the Hamiltonian H ° by means of the unitary transformation
If
U = exp ig o d3 i(x)
'
1.
01)
The multiplication b y w-* in (11) means multiplication of Fourier transform
LEE
MODEL
AND
THE
LOCAL
FIELD
THEORY
311
of ~(x). The result of the transformation U is H O = UHoU-1
(127
--
:
The colon denotes the Wick normal product and m=mo--~m=m
go' r d ' k 2(2r¢)s J o / "
o
We can see from (12) that ~*(x)10 > is an eigenvector of the operator H ° belonging to the eigenvalue m. It follows then that the eigenvector of H ° belonging to the eigenvalue m has the form
lvO(x)> = U-IV+(x)IO>.
(13)
The renormalized (observed) coupling constant can be obtained ~) from the equation
(x-x')g~ = ~0
(14)
The renormalization constants Z2v and Z2N of the fermion fields can be obtained from the equations
O ( x - x ' ) v ' z J =
(15 7
cS(x--x')%/~ -----
(16)
Now we find the renormalized coupling constant. From (6), (13) and (15) we have
~(x-x')gn
l
=
rd j(xlP
+ H 1 p HO-- m J
Using (3), (4) and
(n)
HO--m
(17)
we find
g = goN-1 [1-- <0[e'~aoTl~-*~(x)PHo~e-~gon~-"(*)10>] ,
(is)
where Hob = f dakoJ(k)a*(k)a(k). With the abbreviations a(go" ) = <01e't,o~,~-'~(*)pH~,e-~,n~-t~(x)i0 ) 1 r go' l " r d 3 k x d3k~ 1 = . =~z ~.t L2(2~)3j j oj---g" " " oJJ (oJ~+ • • • + ~ . ) 2 '
(19)
(20)
I. BIALYNICKI-BIRULA
312
we obtain finally g ---- go
1-- (Am)*a (gd) 1 + (~m)*a(eo*) ~ g0(1--2(~m)*a(g0*)).
(21)
It Can be shown that the power series (20) is convergent for go* < gma. where g~max> (2at)*e-1" From (15) and (17) we can obtain the renormalization constants Z, v and z,N:
~¢/-~,v ~_ exp(_go,/2gZcr)N_½[l_Ambl (go, ) + (Ar~),b~(go,) ] '
(22)
V/Z, ~ = exp(--go*/2g2=)N-i[l+Ambl(go*)+(Am)*b2(go2)].
(23)
The functions bl (go*)and b, (go*)are also defined by series which are convergent like (20) in a certain neighbourhood of go* = 0. The vertex part renormalization constant Z 1 can be find from the well known Dyson equation
g ~ ZI-I(Z2VZaN)½Z3½go .
(24)
The boson field renormalization constant Z 3 is equal to 1 and we have from
(24) Z 1 = exp (--go2/g~) [1 + ( A m ) ' C (go*)],
(25)
c (go*) --- b?(go*) + ~b,(go*) + a (go*).
(2S)
where
3. T h e L e e M o d e l The Hamiltonian for the Lee model 1) has the form I4 5 = H a + H I L = mov f d a x V t ( x ) V ( x ) + m o s f d a x N t ( x ) N ( x )
+~_fd3.E~,(x)+(v¢(x))'+~,¢,(x)]
(27)
--go f d3x IV* (x)N (x)~ ~+' (x) + i t (x)V (x)$ (-' (x)]. The renormalization constants in this model are 2 1 = 1,
Z 3 = l,
Z 2 N = 1,
go* f dab 1 (28) (Z2v) -1 = 1 + ~ eo (co--Am) 2"
The renormalized coupling constant defined by (25) has the form
g
1 + 2(2~p
o~ (o~---Am)*
(29)
It is well known that the equation (29) is the source of all difficulties in the Lee model. In fact, no real solution of this equation exists for go if g is finite.
LEE MODEL AND THE LOCAL FIELD THEORY
313
4. D i s c u s s i o n
The comparison between the Lee model and the model described in § 2 can be carried out in terms of Feynman diagrams. In the model of local field theory described b y the Hamiltonian (1) there is an infinite number of self energy and vertex part diagrams. The first few diagrams are the following ~
~
(30)
The symmetry between self energy and vertex part diagrams leads to the reduction of the infinite parts of renormalization constants Z 1 and Z2 in the product ZI-~ (Zzv Z2N)½. In the particular case of equal masses ~.s) we have simply the Ward identity Z1-1 Z, -~ 1. In the Lee model this symmetry is disturbed. One diagram appears here as the correction to the self energy but there are no radiation corrections to the vertex part. In the Lee model instead of (30) and (31) we have
~
(32)
The infinite parts of the constants Z 1 and Z2 cannot reduce and this leads to the known difficulties in the renormalization procedure. The author wishes to thank Professors L. Infeld and W. Kr61ikowski for their interest in this work.
References 1) 2) 3) 4) 5)
T. D. Lee, Phys. Rev. 95 (1954) 1329 W. Heisenberg, Nuclear Physics 4 (1957) 532 G. Kall6n and W. Pauli, Mat. Phys. Medd. Dan. Vid. Selsk 30, No. 7 (1955) W. Heisenberg, Revs. Mod. Phys. 29 (1957) 269 H. Umezawa, Q u a n t u m Field Theory (North-Holland Publishing Company, Amsterdam, 1956) p. 354 6) G. K~IMn, Helv. Phys. Acta 25 (1952) 417 7) O. W. Greenberg and S. S. Schweber, Nuovo Cimento 8 (1958) 378 8) Th. W. Ruijgrok, Physica 24 (1958) 185