Partial bosonization

Nuclear Physics B263 (1986) 1-22 © North-Holland Publishing Company

PARTIAL B O S O N I Z A T I O N The formalism of Cheshire cat bag models Sudhir N A D K A R N I

The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark H.B. N I E L S E N

The Niels Bohr Institute, University of Copenhagen and NORDITA, Blegdamsvej 17, DK-2100 Copenhagen ~i, Denmark Received 21 June 1985 By bosonizing fermionic theories on only a part of space-time, one obtains idealized bag models where the physics is independent of the bag radius. Such "Cheshire cat models (CCM's)" were introduced in an earlier paper, where it was suggested that realistic bag models are in fact approximate CCM's. The present paper further explores abelian CCM's in 1 + 1 dimensions. First, the boundary action for CCM's is derived in the lagrangian formalism by demanding invariance under parity and chiral rotations. Next, a quantum reinterpretation of the classical lagrangian bag boundary conditions is used to obtain the bosonic representation of fermions as soliton operators. Last, these soliton operators are used to construct CCM's in the hamiltonian framework. The Cheshire cat criterion (CCC) - independence of the energy spectrum on the bag radius - is presented as the commuting of the energy and momentum operators for the model.

I. Introduction

Cheshire cat models (CCM's) were introduced in an earlier paper [1] as idealized bag models characterized by an unphysical bag wall. The bag radius in such models is an experimentally unmeasurable parameter which has nothing to do with physical quantities such as the confinement radius. This situation comes about because, in CCM's, one matches the fermionic theory inside the bag with an exactly equivalent bosonic theory outside. The two theories are connected through dynamical bag boundary conditions which result from a boson-fermion interaction at the bag surface. Much of the focus of the present paper will be on this boundary term. We will concern ourselves here with the lagrangian and hamiltonian formulations of CCM's. We shall work in 1 + 1 dimensions, where the exact matching of fermionic a.nd bosonic theories is possible. For simplicity, we shall restrict ourselves to abelian theories, i.e. fermions of only one colour and flavour. The non-abelian case is tackled elsewhere [2]. In sect. 2 we derive the boundary lagrangian for CCM's on the basis of fairly general assumptions, and obtain the resulting bag boundary conditions. Classically, these imply the confinement of fermions inside the bag, and thus cannot yield 1

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S. Nadkarni, H.B. Nielsen / Partial bosonization

CCM's. However, we show that a full quantum reinterpretation of these boundary conditions permits fermions to leak out of the bag and emerge as solitons in the boson representation. This is done in sect. 3, where we derive the soliton operator representation of fermions and obtain the commutation relations of these soliton operators with the canonical boson variables. At this stage, we have all the ingredients needed to construct C C M ' s in the hamiltonian framework, which we do in sect. 4. We construct a hamiltonian operator H that yields the correct volume equations of motion as well as all the boundary c o n d i t i o n s - both classical and quantum. We also construct a m o m e n t u m operator P that translates fermions and bosons in their corresponding volume regions, but converts fermions into bosons, and vice versa, across the bag wall. We finally show that H and P commute: this is the mathematical embodiment of the Cheshire cat criterion (CCC), namely, that the energy spectrum of a CCM be independent of the position of the bag wall. Our conclusions are presented in sect. 5. The appendices contain a review of our notation and a discussion of the locality properties of soliton operators.

2. Boundary action for Cheshire cat models

We show in this section that certain assumptions, to be outlined below, lead uniquely to the following boundary action for 1 + 1-p dimensional abelian CCM's: Son =

Io

d E " {½n . ~ e ~'/4~5~ ¢},

n

where d E " ~ d E . n" is an area element, n. being the bag normal.

2.1. G E N E R A L

CONSIDERATIONS

The C C M action has the form S = S i n -}- S o u t + Son ,

where Si. is fermionic and Sout is bosonic, while the boundary action is of the form Son = f

d E " ~.(47, 0, ¢, derivatives).

do n

Defining the boundary lagrangian via n"2C ~ ~ B ,

we have So~=fo. ~ u d X .

S. Nadkarni, H.B. Nielsen / Partial bosonization

The variation of Son is then given by

f 8~B f 6So. = J d~, I ~ 34) +

858B a( n . O4) )

~(n. 04))

+ [similar terms in 4~, 0]} , where we have used the shorthand

858B &Ya ~---t'0-64) 64)

6~wB 8( t" 04))'

etc., t~, being the surface tangent, orthogonal to n~, tv ~ E

n~,.

Note that variations of 4) along the bag surface do not affect n. 04) and thus 84) and 8n. 04) are independent variations. We shall consider here only theories without derivative interactions, so that variations of Sin and Sou, yield surface contributions arising purely from the kinetic pieces, 8~surface ~in+out = (

d,y{_lis~b+lid~lqt~b_(n.

do n

0 4) ) (~4) }

where the surface element d 2 ~' has been chosen to point from "in" to "out". Setting the total surface variation equal to zero, we obtain the boundary equations ½ irtO = a ~ B

65~B n . 04)= 64)' 6LfB 0=

8(normal derivatives) "

In the remainder of this section, we shall derive the form of 5qB by imposing restrictions based on (i) locality, renormalizability and hermiticity, (ii) CCC, (iii) discrete symmetries ~, ~g and 3- and (iv) global chiral invariance.

2.2. L O C A L I T Y ,

RENORMALIZABILITY

AND

HERMITICITY

~B has the dimensions of mass. The only elementary operators with that dimension are the following:

mo,

a.4),

¢;A~,,

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S. Nadkarni, H.B. Nielsen / Partial bosonization

where mo is some mass parameter or collection of mass parameters (mo~ m~o1), m~o2), . . . ) which may occur in the theory as masses or coupling constants, and A is a 2 x 2 matrix-valued dimensionless operator. The only dimensionless elementary operator available is 05. Dimensionless composite operators can be constructed from 05, and from ratios of the above dimensionful operators. To satisfy locality and renormalizability, we shall not allow such ratios in our boundary action. That is to say, ratios such as t~b/t~vs@ or (c9~05)2/rno are excluded from ~B. Our starting form for ~B is then

~cCa= roof(05) + n. c3g(05) + t. Oh(6 ) +1 d~A(05)@, where f, g, h, A are dimensionless functions of the field 05. The third term integrates out in the boundary action and can be omitted. We now parametrize A(05) by decomposing it in terms of the 2 × 2 basis {I, ys, y~} as follows:

A = S + i y s P + V~.y*' P =~(x/s2Sp2+iYsx/~)+(vn.-at.)Y" = p(cos a + iT5 sin a ) + vyt - a e . . n ~y'~ = p e ivy"+ ~(v + ysa) • Hermiticity of ~B is then equivalent to the hermiticity of f, g, p, a, v, a. The requirements of locality, renormalizability and hermiticity thus dictate the following form for the boundary lagrangian:

~B

=

m o f ( 05 ) + n . 0g(05) + ½tT/p( 05) e 'V5'~ (6) 0 +1 ~tt[ v(05) + ysa(05)]O,

where f, g, p, a, v, a are hermitian operators.

2.3. C H E S H I R E CAT C R I T E R I O N

In a true CCM, the bosonic and fermionic degrees of freedom should in no way be independently constrained at the boundary. Instead, there should be full communication of information across the boundary surface: no decoupled boundary conditions should occur. We therefore wish to test the above form of ~B against this requirement. On varying (n. 0)05, we find that g'(05)= 0 at the boundary. This constrains the value of 05, in disagreement with the CCC, and we are obliged to eliminate the term (n. O)g(¢) from ~nNext we vary ~ to obtain the ~b-equation, i~t~/, = [p e'~5~ + t t ( v + ysa)]~b.

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The determinant o f the matrix [p e % ~ + y t ( v - i+ ysa)] should vanish, otherwise the 4~-equation could be inverted to give qJ = 0 at the b o u n d a r y , violating the CCC. We thus set det [ p e i ~ + n ( v - i + ysa)] = p2+ n 2 ( a 2 + 1 + 2iv - v 2) = 0. Since p, a, v are hermitian, this implies that v = 0". We must then have p2=_nZ(l+a

2) .

According to whether n o is timelike, null or spacelike, we have no solution

P =I0,

n2 = + 1 nn 22==0- 1 .

Thus, timelike bag normals are not allowed. (For later convenience, we have chosen the negative square root in defining p; we lose no generality in doing so, since translating a by 7r changes the sign of p). Our b o u n d a r y lagrangian now contains only the functions c~, a and f We now show that c~ and a cannot both vanish. For if they did, then we would have ~Lt'n

' ln2~tb + roof(C) , ~,a =0

so that the b o u n d a r y equations would be decoupled,

n- 36 = m o f ' ( d Q J

a = a =0,

and the Cheshire cat criterion would be violated. The requirement o f full c o m m u n i c a t i o n across the bag wall, with no decoupled b o u n d a r y equations (the "Cheshire cat criterion") thus leads to the following simplified form for ~ . , '~B

=

~n2~x/1 + a2 e %'~ qJ + ½q~a~tYs~b+ m o f ( O ) ,

where a(~b), a(4~), f ( 4 ' ) are hermitian operators and at least one o f a and a is non-vanishing.

2.4. DISCRETE SYMMETRIES: ~', ~, AND 3r

Since o u r b o u n d a r y action knows only about the kinetic parts o f Sin and S .... it is reasonable to d e m a n d that it have the same symmetries. In this subsection, we * If we had generalized .27B in subsect. 2.2 to include fermion-number non-conserving terms (one might wish to do this to provide a possible mechanism for "fermion leakage", see subsect. 3.1) such as [q~*B(~b)q,+h.c.], q, would have been over-determined unless B(~b) vanished. Thus CCM's cannot contain fermion-number violating terms in the boundary action, unless these are present in the fermionic lagrangian to begin with. We have simply chosen to eliminate such terms right from the start, by imposing fermion number conservation, which is usual in most theories.

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focus on the discrete symmetries: p a r i t y ~ , charge c o n j u g a t i o n ~ a n d time reversal J-. We see w h a t restrictions they i m p o s e u p o n ~B b y a s s u m i n g the t r a n s f o r m a t i o n p r o p e r t i e s o f the f e r m i o n field a n d o b t a i n i n g t h o s e for the three f u n c t i o n s a(05), a(05) a n d f(05) w h i c h o c c u r in ~B. The results are the following. (i) Parity: ~wB is ~ - i n v a r i a n t if a n d o n l y if a, a are o d d u n d e r ~ a n d f is even• (ii) Charge conjugation: 5gB is % i n v a r i a n t if a n d only if a, a are o d d u n d e r qg a n d f is even• (iii) Time reversal: 5fB is J - - i n v a r i a n t if a n d only if a, c~, f are all even u n d e r 3.. The a b o v e t r a n s f o r m a t i o n p r o p e r t i e s i m p l y that a (05), a (05) m u s t be o d d functions o f 05, or m u s t vanish• F u r t h e r , since a a n d a Cannot b o t h vanish, as s h o w n earlier, 05 itself m u s t be o d d u n d e r ~ a n d ~ a n d even u n d e r 3-. This in t u r n i m p l i e s that f(05) m u s t be an even f u n c t i o n o f 05. At this p o i n t , we can r e p a r a m e t r i z e o u r l a g r a n g i a n with respect to a(05) b y using identities such as sec 2 0 = l + t a n 2 0

(even-odd),

cosec 2 0 = 1 + cot 2 0

(odd-odd),

c o s h 2 x = 1 + sinh 2 x

(even-odd),

coth 2 x = 1 + cosech 2 x

(odd-odd),

Since a(05) is an o d d f u n c t i o n while ~/1 + a2(05) is even, we clearly n e e d an e v e n - o d d identity. Thus we can set tan 0

sec 0

or

or

,/1-7-Ja~ =

cosh x ,

a =

l °r

sinh x , or

where 0(05), x ( 0 5 ) , . . , are o d d functions o f 05. These e v e n - o d d p a r a m e t r i z a t i o n s are really e q u i v a l e n t within their d o m a i n s o f validity, since we have o n e - t o - o n e corres p o n d e n c e s such as x=sinh

1 tan 0,

0~(-~,+~), x c (-~, +~).

F o r later c o n v e n i e n c e , we shall choose the a n g u l a r p a r a m e t r i z a t i o n . W e thus secure the f o l l o w i n g ~ , qg, J - i n v a r i a n t b o u n d a r y l a g r a n g i a n , ~B =ln2~

sec 0 eWs~qJ +½ q7 t a n O~tys~O+ roof(rig),

where a(05), 0(05) are o d d functions o f 05 while f(05) is an even function• 05 itself is o d d , o d d , even u n d e r ~ , g, 57. [ N o t e that d e m a n d i n g 3~- or % i n v a r i a n c e w o u l d by itself have b e e n sufficient to derive the a b o v e form o f ~B].

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Before imposing any further restrictions on ~B due to symmetry, we show that the function 0(&) is an irrelevant pne, in the sense that the same boundary equations result no matter what the choice of 0. Thus, 0 can be chosen for convenience. We look at the boundary equations resulting from the above form of 5gB. The 0-equation, ½i~b = 6 ~ , simplifies to ipl0 = n 2 ei~/5(c~+o)~.

The &-equation, n. a & = reads

n. O&=½n2fO ' sec 0 tan 0 eiV5'~@+½ n2f0 ' sec 20vtysO +½ nZq~ sec 0 iysa' eiVs'~O+mof'(cb). Using the qJ-equation, this simplifies to n. 0& = ½q~(a + 0)' sec 0rty5 e i r s ° ~ + roof'(&). Since a occurs only in the combination (a + 0), let us reparametrize as follows: a-~a--O,

0~--0.

Our boundary lagrangian then reads 5( B= ½n24J sec 0 e~(~+°)~b - ½f tan 0~tysqJ+ rn0f(4~), and the boundary conditions become iytq, =

n2

eg~('~)q,,

n. 0& =~al '(4,)0~ty50- -~al '(&) tan O(&) On2 ei~(o)4,+ roof'(&) We see that 0 occurs only in the second term of the 4,-equation. However, this term must in fact vanish. This is easily seen by multiplying the 0-equation on the left by 0, taking the hermitian conjugate and adding the two together, to obtain fn 2

ei~"(6)~b = iq~t~ = 0.

Thus our boundary conditions are in fact completely independent of 0, and so we can choose 0 according to our convenience. We of course choose the simplest case 0 = 0. (This also happens to be the only choice consistent with chiral invariance, which we shall next be imposing.) Then the above boundary conditions follow from the simple lagrangian ~ B = ½nZt~ e i~(~)O + roof(&),

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S. Nadkarni, H.B. Nielsen / Partial bosonization

where a(~b) is any odd hermitian function and f(~b) any even hermitian function of the field ~b, which is odd, odd, even under ~, ~, J-.

2.5. C H I R A L S Y M M E T R Y

We shall finally demand that 5~a be invariant under chiral rotation of the fermion fields, even though there may be chiral symmetry-breaking terms such as motkq, in the fermionic lagrangian. Our motivation is that the boundary interaction is so strong (a 6-function lagrangian density) that even chiral symmetry-breaking mass terms do not change it. Thus, our philosophy is that essentially the same ~ a works for all theories. In our boundary lagrangian we make the following UA(1) rotations, tP ~ e i~'° $ ,

5 ~ ~ e '~5° ,

4~ --> (ao ,

where 0 is some real number (not to be confused with O(&) of the previous subsection). Invariance under UA(1) then means that ~ B = 1 t125 eiys[~( ¢~o)+20] O + m o f ( q~o) .

Now the only continuous internal symmetry operation on the boson field ~ is translation by a real number. If the bosonic description is to be truly equivalent to the fermionic one, then this translation must be the bosonic version of chiral rotation. Thus we must have ,~0 = 4 , + a ( 0 ) ,

for some function 6. On the other hand, chiral invariance of ~ a requires that a(0o)+20

= a ( $ ) +27rn,

f((ao) = f ( t b ) + const.

Since 0 is a continuous parameter, we can set n = 0, const = 0, and write ~(4, + 6 ( o ) ) + 2 o = ~ ( ~ ) , f ( $ + 6 ( 0 ) ) =f(d~) •

Thus, f is a constant function of d~ and may be dropped. Setting 4~ = 0 , we have, since a ( 0 ) = 0 , ,~ ( 6 ( o ) ) = - 2 0 ,

i.e. 6(0)=a

1(-20)

and so, o

4~ ---' d)o = ~b+ a - 1 ( - 2 0 ) ,

s. Nadkarni, H.B. Nielsen / Partial bosonization

9

while the condition on a becomes a(&+a

t(-20))-

a(4,) = - 2 0 .

This relation holds for all values o f 0 and 4,, which implies a(4,) = a14,,

cq = c o n s t .

By scaling 4' in the outside action Sout, we can choose the value of cq to be fixed for all theories. With the hindsight of already k n o w i n g the bosonization relations, we choose al to be the canonical value

The b o s o n i c realization o f chiral symmetry is then seen to be the transformation 0

1

Chiral invariance thus restricts our b o u n d a r y Lagrangian to its final form

where n 2 = 0, - 1 ; 4' is odd, odd, even under 3°, ~, ff and 4' ~ 4' - 0/~/~ under UA(1).

2.6. REVIEW OF ASSUMPTIONS

(i)

~ , = ½t~i~O + [non-derivative interactions], &Pout= ½A 2(0,,4,)2 + [non-derivative interactions],

where A2 is a real number, equal to unity for most theories but different from unity for Thirring-like models. (ii) ~B is local, renormalizable and hermitian. (iii) ~LfR does not lead to any d e c o u p l e d b o u n d a r y conditions (CCC). (iv) ~B is invariant u n d e r 30, ~, 3-. (We really only need invariance u n d e r either 30 or c¢ to establish the final form o f ~ . ) (v) ~B is invariant u n d e r global chiral rotations, even t h o u g h this m a y not be a symmetry o f (~in and ~out.

3. S o l i t o n operators

In the previous section, we derived a b o u n d a r y action that led to the following b o u n d a r y equations in the lagrangian formalism: intp

=

n 2

e~'/T~/~+ ~p

n- 04, = x/--~~nysO

(~b-equation), (4,-equation).

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The e - e q u a t i o n is m e r e l y the n o r m a l c o m p o n e n t o f the w e l l - k n o w n axial current b o s o n i z a t i o n relation:

f n. a4, = ,F~ ~yt-/~O

We have seen that the W-equation, classically i n t e r p r e t e d , i m p l i e s qJyt0 = 0. This violates the C C C a n d thus there can be no classical C C M . As we shall discuss later in this section, a f e r m i o n is r e p r e s e n t e d as a soliton (kink) in the b o s o n p i c t u r e [3], so a f e r m i o n a p p r o a c h i n g the b a g wall o f a true C C M s h o u l d , rather t h a n being reflected b a c k a n d m a k i n g qJmh vanish, emerge on the o u t s i d e as a kink. This m e a n s that at the b a g surface, the value o f the b o s o n field s h o u l d shift from one v a c u u m value to another. This is p r e c i s e l y the effect e m b o d i e d in the t a n g e n t i a l c o m p o n e n t o f the a b o v e b o s o n i z a t i o n relations, which describes a n o n - v a n i s h i n g f e r m i o n current in terms o f a fluctuating b o s o n field. We shall n o w show that this " f e r m i o n l e a k a g e " effect is in fact c o n t a i n e d in the q~-equation w h e n q u a n t u m effects are i n c o r p o r a t e d . This will lead us n a t u r a l l y into a d i s c u s s i o n o f M a n d e l s t a m soliton o p e r a t o r s [3], a n d p r o v i d e us with ingredients essential to m a k i n g the t r a n s i t i o n b e t w e e n l a g r a n g i a n a n d h a m i l t o n i a n f o r m u l a t i o n s .

3.1. CONSEQUENCES OF THE d-EQUATION Since we wish to e v e n t u a l l y m a k e c o n t a c t with the c a n o n i c a l f o r m a l i s m , we c h o o s e a static b a g with n o r m a l (n~) = (0, 1). (We t h e r e b y e x c l u d e the special case n 2 = 0.) Differentiating the 0 - e q u a t i o n along the b o u n d a r y , i.e. a l o n g the time direction, we have i n ~ = n2 e;~4rg~v~+O + n 2 0 [e'~ r-4'~z'5¢']~ -

We use the D i r a c e q u a t i o n ,

to e l i m i n a t e ~,, a n d use the qJ-equation again to get rid o f the e x p o n e n t i a l factors*, to be left with 0~0 = - ix/~(0oq5 + ~/5al ~ ) ~-t . We n o w r e g a r d this as an o p e r a t o r e q u a t i o n in a c a n o n i c a l f o r m a l i s m , with O o & ( X ) = ~-(x), a n d c o m p a r e it with the f o l l o w i n g w e l l - k n o w n o p e r a t o r * The appearance of ~' in the final equation means that we have assumed the d-equation to hold under small deformations of the bag wall. More generally, if we regard 0 in the 0-equation as a soliton operator constructed from &, then the d-equation can be presumed to hold throughout spacetime.

S. Nadkarni, H.B. Nielsen/ Partial bosonization

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equivalence* O~tb = - i T r ( j l + ysjo)th.

Since a~h = x/-~jo is the O-equation, we may identify 3 o d~ = ,,/-~rrj l =- ~,f~ ~hY o Y s th

i.e., t- a 8 = , / ~ 4 , Y ~ ' ~ , ,

as we had set out to show. If we formally integrate the differential equation we obtained from the q,-equation, we get x d~:[ ~-(~:) + Ys0'(sc)] } ~b(Xo)6(x) = e x p { -ix/-~ y~o This is precisely the Mandelstam operator for solitons in the boson representation [3]. We shall next look in some more detail at such operators.

3.2. C O N S T R U C T I O N OF SOLITON OPERATORS

We give below a brief and formal discussion on the construction of soliton operators which create and destroy kink states when acting on the bosonic vacuum. Define ~ ( x ) via the relation

too6 = -a16.

In a canonical equal-time formalism, we may solve for ~ as follows: ~(x) = -

ds~ rr(~:) ,

vr(x)-= 6 ( x ) .

c~

If we impose the canonical commutation relation between d~ and rr, [~(x),

~(y)]x,,-~,o =

ia(x-y),

then we obtain the following equal-time commutator between O and 4~: [ 6 ( x ) , ~b(y)].,,,:>. = - i O ( y - x ) . * The regularized version of this operator equivalence is

3t ~ = - ~ iv:{ jt + YsJo, ~ } , as stated by Sommerfield [4] (see Dashen and Frishman [4] for a direct proof). An equivalent statement, proved by Coleman, Gross and Jackiw [4], is that the canonical fermionic stress-energymomentum tensor can be rewritten in the Sugawara [4] form involving only fermionic currents. Via the bosonization relations this tensor is then seen to be precisely the same as the corresponding bosonic one, thus creating a true CCM [1].

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Using the B a k e r - C a m p b e l l - H a u s d o r t t theorem we then have e'~g(x)~b (y) e -'"g(x) = ~b(y) - ozO(x - y ) , i.e., ~ ( x ) generates step-function configurations o f tb. Generalizing to powers o f 6, e'"4;(x)[4~(Y)]" e -'~g(x) = [6(Y) - aO(x - y ) ] " , and more generally, for any function F, -

.

-

ei'~4~(X)F[cb(y)] e -'"4,(x) = F[~b(y) - aO(x - y ) ] . Choosing

F = e i'~(y) and a = T r, we get ei~(x)

ei~(y)

e-irr,~(x) = e i4~(y) e-i~-o(x-y)

i.e., e i,r,~(X) e i4)(y) = _l_ei,C,(y) e i~,(y) { + i

y > x . y
Thus we have the possibility of constructing anticommuting operators from 4~. To actually do this, we use the B a k e r - C a m p b e l l - H a u s d o r t t identity again, in the form eA e u :

e [ A , B] e B e A

and choose

A = ix/-~(~a + qb)(x), B = ivr-~( ~ + ~b)(y) . Then, for x # y, [A, B] = +i~" and e A e B = --e B e A .

Thus e ~'~('~±~) behaves like an anticommuting fermionic operator. The field ~b in 1 + 1 dimensions is an angular variable [5], possessing degenerate chiral v a c u a connected by

We can therefore define solitons/antisolitons by d e m a n d i n g that such solutions satisfy the asymptotic condition (4,(+oo)) = <4,(-oo)) ± ~ .

Thus, to annihilate a soliton or create an antisoliton, an operator ~7 must satisfy [~b(y), r/(x)] =x/'-~O(x-y)~7(x),

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i.e.,

~(x)4)(y)~7-1(x) = (a(y)-~-~ O ( x - y) . From our analysis above, we see that ~(x) oc e~,~'~( x) will do the job. To make the ~-operators anticommute, we must add some multiple of 4~ to the exponent. We have already shown how to do this. We thus see that the following operators

~7=;z

,/2/(1 + i) exp [ iJ--~ (4~ - ~b)]~ ~(1-i) exp[iJ-~(~+ck)]]'

are anticommuting operators which create or destroy kink configurations. (Here,/z is a mass scale and the factors (1 + i) correspond to our definition of the y-matrices.) For more on these operators, the reader is referred to Mandelstam's paper [3]. Some insight into their locality properties may be gained by recasting the above formalism in terms of the variable U---exp (i 4,JT-~qS); see appendix B. Comparing the soliton operators we have just constructed with the results of the previous subsection, we see that the ~b-equation provides us with a new recipe for their construction. (This is useful for more complicated theories, such as non-abelian ones [2], where the form of these operators is not known a priori.) It is then no surprise that the soliton operators satisfy the original, undifferentiated qJ-equation. Namely, it is easily verified that (e ; ' ~ 5 ~ ( x ~)+ i,)~7(x) = 0,

e~0

where we have regularized the ~-equation by point-splitting. In the following section, we shall use soliton operators to construct C C M ' s in a canonical f r a m e w o r h For this purpose, we only need remember their commutators with the boson variables, which are easily derived: [r/(y), qS(x)] = - x / - - ~ O ( y - x ) r l ( y ) , [rl(y), zr(x)] = +V/--~6(x - y ) y s r l ( y ) . 4. H a m i i t o n i a n formulation of Cheshire cat models

The hamiltonian formalism affords an elegant and rigorous demonstration of the Cheshire cat principle, which as we have seen is not easy to show in the lagrangian framework. Anomalous effects can be built in directly by a judicious choice of commutation relations, and thereby all the relevant boundary c o n d i t i o n s - b o t h classical and q u a n t u m - emerge quite naturally. The Cheshire cat problem in the hamiltonian formalism is to find a pair of operators H and P, satisfying certain criteria. First, H must provide the correct volume equations of motion as well as

S. Nadkarni, H.B. Nielsen / Partial bosonization

14

all the necessary boundary conditions. Second, P must generate the usual translations of fermions and bosons in their corresponding volume regions, but should also translate fermions into bosons (and vice versa) at the bag surface. Third, H and P must commute. This last criterion implies that the energy spectrum of the full theory is independent of bag position, or, equivalently, that m o m e n t u m is conserved. We can write H and P in the form dx ~'r'(X) + Io,~tdx,:~out(x)+HB,

H=I.

foot

dx~°ut(x)+ PB'

P = I,,

where ~i .... t and ~i ..... are the usual energy and m o m e n t u m densities, while HB and PB are boundary terms, which must be cleverly chosen to do the job. In the following subsections, we shall first illustrate the Hamiltonian formalism for C C M ' s with two trivial but enlightening examples: bosonizing bosons and fermionizing fermions. We shall then tackle the nontrivial case ofbosonizing massless free fermions.

4.1. BOSONIZING BOSONS We consider a trivial CCM, discussed in [1], with a boson field g having potential energy U(X) placed inside the bag (which for simplicity we take to be open-ended, lying on the interval (-co, 0)), while outside the bag we place a boson field ¢ having potential energy V((/,). The hamiltonian and m o m e n t u m operators for this system are:

H

1 dxE~rr x2 +½X'2+ U ( X ) ] +

.

rl

2 --I

.,2--

V((~)]

+

+ [4, ( 0 + ) - x ( 0 - ) ] x ' ( 0 - ) ,

-p=

;o2,

dxTrxX'+

io

dx rr+¢'+½ [¢ (0+) - X ( 0 - ) ] [ ~'+(0+) + ~ ' x ( 0 - ) ] .

+

The boundary terms were written down essentially by guesswork. While HB follows canonically from the lagrangian formulation, we are as yet unable to similarly derive PB. We have introduced some point-splitting at the bag wall, for later convenience. Usually we will not explicitly indicate such splitting, writing instead 0±-= 0±131, ~ 0, as simply 0. The energy-momentum 2-vector, ( P ' ) ~ ( H , P), generates derivatives via the Heisenberg equations of motion,

a~=iEP~ ]

S. Nadkarni, ll.B. Nielsen / Partial bosonization

15

which are to be c o m p u t e d using the canonical c o m m u t a t i o n relations I x ( x ) , 7rx(y)] = [05(x), 7r+(y)] = iS(x - y ) , all other c o m m u t a t o r s vanishing. The equations of m o t i o n are obtained by c o m m u t i n g fields with the hamiltonian

49(x) =- i[ H, 05(x)] = O(x)Tra(x) , 7i'6(x)---i[H, 7re(x)] = O(x)[05"(x)- V ' ( 6 ) ] + 6 ( x ) [ 6 ' - X ' ] ,

)~(x) =- i[ H, X(x)] = O(-x)Trx(x ) (rx(x) =~ i[ H, 7rx(x)] = O ( - x ) [ x " ( x ) - U'(X)] + 8 ' ( x ) [ 0 5 - X ] The 8-function terms must be eliminated by setting their coefficients equal to zero, else time derivatives would blow up at the bag surface, resulting in undesirably high-energy configurations which could not possibly occur in C C M ' s . Thus, at the bag surface we have

05 = X,

~h' = X'

at b o u n d a r y .

These are precisely the bag b o u n d a r y conditions which we had obtained earlier [1] in the lagrangian formalism. We see here that in the hamiltonian formalism, bag

boundary conditions emerge as vanishing coefficients of 6-functions at the bag surface. To obtain the space derivatives, we must c o m m u t e fields with the m o m e n t u m operator, P1 -= - P - We get

i [ - P , 05(x)] = O(x)05'(x)+½6(x)(ch- X) , i [ - P , rr6 (x)] = O(x)rr~(x)+ l~8(x)(Tr 6 _ 7rx) , i [ - P , X(x)] = 0 ( - x ) x ' ( x ) + ½8(x)(ch - X) , i [ - P , ~'x(x)] = O(-x)~-'x(x ) +½ 8(x)(Tre, - rrx). Since P generates derivatives in the volume regions, it is clearly a g o o d m o m e n t u m operator there. However, we must check whether P translates correctly between the 05 and X variables at the bag surface. We show below that, for example, the c o m m u t a t o r s o f P with 05 and X are consistent with the translation

e iaPx(x-a)ei~'P=~X(x)' (05(x),

x0"

Of course, it is only necessary to show this across the bag wall. By differentiating and then integrating the left-hand side of the above translation, we obtain the identity

e

i~'PX ( - ~ 1e ) eieP

-= X(

1

-i

dx[P, e )

X(-se)

eiXP]

.

s. Nadkarni, H.B. Nielsen / Partial bosonization

16

Breaking up the integral on the right-hand side into two parts, we use the assumed translation property to get

e-~Px(-½e)e~P=X(-½e)-i -i

I/

dx[P,x(x-½e)]

dx[P, 4,(x-le)].

e

We now substitute for the commutators,

- i [ P, X(x - I s ) ] = O(½e - x ) x ' ( x - ½ e ) + ½ 6 ( x -½ e)(4, - X)lo, -i[P, 4,(x-½e)] = O(x-½e)4,'(x-½e)-F½~(x-le)(4,-X)lo, to obtain e-*~Px(-½~) e '~" = x ( - ½ ~ ) + x ( O ) - x ( - ½ ~ )

+½(4, - x)lo + 4,(½ ~) - 4,(0)+½(4, - x ) l o

=4'(½~), as required. Note that the 8-functions in the commutators with P were crucial in obtaining this result. (Although not indicated explicitly, the point-splitting introduced earlier ensures that the arguments of the t~-functions being integrated over vanish well within the limits of integration, rather than at the endpoints.) We note that the sum of the coefficients of g-functions occurring in the commutators with P of corresponding fields must be equal to the difference of those fields. This is the general requirement that the correct momentum operator must satisfy. We shall see this again when we bosonize fermions. Having shown that H provides the correct equations of motion and boundary conditions, and that P is the correct momentum operator, we next demonstrate that the bag wall is of the Cheshire cat type by showing that H and P commute. Indeed, we have

i[H, - P ] =

dx[5-xX' + ~rx)~'] +

fo

dx[Ti~4,'+ ~eO~']

+½(d, - , , / ) ( ~ , + ~-,~)1o+ ½(4, - x)(~-~ + ~~)1o =

dx[(x"-U'(x))X'+TrxTr'x]+

d x [ ( 4 , " - V'(4,)) 4,' + ~-e~Tr~]

oo

+'(~,

- ~-x )(,,, + ~)1o +l (4, - x)[4,"- v'(4,) + x"- u'(x)] Io

= [½x ' ~ - U ( x ) - ½ c t , ' ~ +

v(4,)]1o+½(4,-x)[4,"-

v'(4,)+x"-

U'(x)]lo

= [ v ( 4 , = x ) - U ( x = 4,)] Io,

where we used the boundary conditions in the last step. We must therefore set the potential energy functions equal to obtain a Cheshire cat model; this is the same result we obtained in ref. [1].

S. Nadkarni, ll.B. Nielsen / Partial bosonization 4.2. F E R M I O N I Z I N G

17

FERMIONS

We repeat the exercise of the preceding subsection, for a model with a fermion inside the bag and another, r/, outside. (The boundary term in this case will be useful to us when we bosonize fermions.) For simplicity, we choose our fermions to be free and massless. The energy-momentum operators can be written compactly as

P,

=

dx ½i[tffr, Y5~ll~]q-

;o

d x ~1,.[ n v .-V ~ O l n ] +-~ , [ 4 ~ v . v ~ n1 .

-

-

.]v.v,~] Io,

where the fermions obey the usual equal-time anticommutation relations {6(x), th*(y)} = {rl(x), r/t(y)} = 6(x - y ) . The equations of motion are ~(x) =--i[ H, ~b(x)] = O(-x)YsO'(x)+½~(x)Ys(n - 0 ) , ~(x) =--i[H, r/(x)] = O(x)y5rf(x)+½8(x)ys(rl - ~b), while the space derivatives are given by i [ - P , 6(x)] = O(-x)~b'(x) + l 6(x)(r I _ 6 ) , i [ - P , r/(x)]

=

O(x)rf(x)+½8(x)O?-~b).

Thus the boundary condition is O=rt,

x=O,

while the sum of the coefficients of the &functions in the commutators with P is ( r l - ~), the difference of corresponding fields, which makes P correctly translate O ~ rl at the surface. It is also easily checked that H and P commute. We have -P=

f ~ dx

½i~b*'OxO+ d x l~" n *-O~,n+~(4' 1. *n - n * 4 , ) l o fo °

and so, after simplification, i[H, - P ] = ½i [ ( 0 ' + n ' ) t y s ( r / - ~b)- ( r / - q~)*y5(~O'+ 7')1, which vanishes by the boundary condition.

4.3. B O S O N I Z I N G

FERMIONS

We consider now the simplest nontrivial CCM, with a free massless fermion inside the bag, taken again to be the half-line (-oo, 0), and a free massless boson

S. Nadkarni, H.B. Nielsen / Partial bosonization

18

O outside. The energy and m o m e n t u m operators for this system are (with the point-splitting at the bag wall explicitly shown): H=

d x ' ~[1r,2 + &,2] + HB,

dx ½i[#~y,~l #J] + ~e

-P =

IT

dx ½i[ 6Vo'0,0 ] +

dx[ 7r,~&'] - P . .

Here the fermion and b o s o n variables satisfy the canonical c o m m u t a t i o n relations

[4,(x), ~Ay)] = i a ( x - y ) , {0(X), #~(Y)} = y o S ( X - y ) . We presently do not have a canonical procedure for deriving the correct b o u n d a r y terms HB and PB ; they were discovered by guesswork. It turns out that the b o u n d a r y terms for bosonizing fermions look precisely the same as those for fermionizing fermions, HB=½i[f(

]

1

_ p . : 1 i[ #~(_½ e) Yor/(l e) - ~(½ e ) Y o 0 ( - ½e ) ] , provided we now interpret ~ as a soliton o p e r a t o r constructed from the bosonic variables, and satisfying the following previously stated c o m m u t a t i o n relations with them, [r/(y), qS(x)] = - - ~ - ~ O ( y - x ) r l ( y ) , [~?(y), ~,~(x)] = + , , / ~ 6 ( x - Y ) Y s : q ( Y ) , while a n t i c o m m u t i n g with the fermion tp. With the above choice of HB and PB, we obtain the correct equations of motion and space derivatives. These are recorded below.

Equations of motion : d) (x) = O(x - ½~ ) , ~ (x) + o(½ ~ - x) ( 4 ~ [ 6~,, n + ~ , , 0 ] , ¢'(x) = O(x-l~)~'~(x)+~(x

- ~' ~){ ~-,~- , 2 ~

~To(X):O(x

- ~1) { ¢

le)~)tt(X)_l_~(X

6(x) = O(-x-'~)~,sO'(x)+~(x

[6~,n +

, - , / ~ 1 [6~o,

~/,q']},

+ ~)~o0]}.

+½~)~5{,7- q.}.

~ ( x ) = - O ( - x - ' ~ ) 6 ' ( x ) % , - ½~(x +½~){ ~ - 6 } ~ , L ( x ) = O ( - x - 5' e)yus(x) " + 6(x + 2' e){~' [ 0-% , Y , r / + (TYuY, O] -Jus}.

S. Nadkarni, H.B. Nielsen / Partial bosonization

19

Space derivatives: i [ - P , 4)(x)1 = O ( x - ½ e ) g a ' ( x ) + O(½s - x),,/-F~[ ' ~,v,,n + ,~Voq,1, i [ - P , &'(x)] = O ( x - ½ e ) & " ( x ) + 6 ( x

i[-P, ~(x)]

1

t

~e){6 - x / ~

1

[6yorl + ¢lYo~b]},

8(x-½e){~-,/g~[~v,,~+,~v,~]},

O(x-~e)~(x)

i [ - P , ~ ( x ) ] = O ( - x - ½ e)¢, ' ( x ) + ~ a' ( x + 2 e ) {' n

- qJ},

i t - P, ~ ( x ) ] = O ( - x - '2e) (~'(x) + ½a ( x + ½e){,~ - ~},

i[-P,j.(x)]

= O ( - x - ~ e1) j ~ ,.,( x )

+

] 1 6 ( x +~e){~[~byu~?+flyu~]-j~}.

For the sake of brevity, we have suppressed the arguments of variables at the b o u n d a r y ; fermionic variables ~b, f, j . , etc. are evaluated at x = - 3 s while bosonic variables (b, &', rr,b, etc. are evaluated at x = +½e. Finally, we demonstrate the CCC, i.e. the c o m m u t i n g of H and P. W e ' h a v e I[H,-P]

=

dx[~i(by,,~xtb+>,t~YoOx~b]+

dx[~-+tb' + rr+¢~'] 12e

+ ½i[ $yon - #~o~b - ,~~o ~, + qT~o,)]1o. On using the volume equations of motion, the integrations can be p e r f o r m e d and we get I •

--t

o

1 ,

--t

i[ H, - P ] = ~ t [ O y~( rl - O ) - ( gl - ~)Y, t l " ] - ~( rr:b + O'2) + ~ t [ - rl

'Yl+-{-

~'Yl'?~']

•

Now, the bosonic energy density can be reexpressed in solitonic, i.e. fermionic, form as ( rr~ + (h'z) = ½igly[~r/. We thus obtain i[H, - P ] : ½i[(47' + ~') Y,(~7 - ~b)- ('0 - qT)Yl(~' + rl')], which vanishes on using the b o u n d a r y condition r / = ~b.

5.

Conclusion

By making the effective bosonic theory outside a bag completely equivalent to the fermionic theory inside, and placing an a p p r o p r i a t e interaction term on the b o u n d a r y of the bag, one can turn the latter into an unphysical "Cheshire cat". Working first in the lagrangian formalism, we have shown that the b o u n d a r y term for such Cheshire cat bag models is essentially uniquely determined from requirements of locality, renormalizability and hermiticity, plus invariance under parity (or charge conjugation) and global chiral rotations. The lagrangian b o u n d a r y conditions do not in fact a p p e a r to describe a Cheshire cat model when looked at na]'vely; however the incorporation of q u a n t u m effects yields the bosonization relations

20

s. N a d k a r n i , H . B . N i e l s e n / P a r t i a l b o s o n i z a t i o n

necessary for the success of the model. Going further, we have given a prescription for obtaining the soliton operator representation of fermions in the boson picture, and derived the commutation relations of these operators with the canonical boson variables. With the help of soliton operators, we have been able to construct Cheshire cat models in the hamiltonian formalism, which has the advantage of providing all the boundary conditions-classical as well as a n o m a l o u s - o n the same footing. Moreover, the Cheshire cat criterion - the independence of the energy spectrum on the bag wall position - can be stated in hamiltonian language simply as the commuting of the energy and momentum operators of the system. In a forthcoming paper [2], we generalize these results to non-abelian Cheshire cat models in 1 + 1 dimensions. We are currently working on the extension of these ideas to 3 + 1-dimensional non-abelian theories such as QCD. We thank I. Zahed for discussions during the early phases of these investigations. The work of S.N. was supported by the Danish Research Council.

Appendix A NOTATION We usually use the same symbol to denote both a spacetime point x ~ (x °, x 2) and the corresponding spatial coordinate x -= xl; which one is meant should be clear from the context. Further notational conventions are recorded below: metric:

(~7~..) ~ diag (1, - 1 ) ,

7- matrices:

%, ,

Ys ~

#,, v=O, 1, 7o71 ,

Weyl representation:

(O+)=',eft-mover @=

g'-

\right-mover/'

e-symbol: ep, v

e, ~ v e

--Ev,~ ~

up

--

(~p

e O1 =

--COl

v

=

1 ,

:

e u , e ~ t 3 = -- ~ ) ~ ~qvt3 + ~7~,t3r l w •

S. Nadkarni, H.B. Nielsen / Partial bosonization

21

Appendix B LOCALITY PROPERTIES OF SOLITON OPERATORS It is interesting to recast our formalism directly in terms of the Ua(1) group element U-= exp (i 4 , ~ 4'); it turns out also to be useful for direct c o m p a r i s o n with the non-abelian case discussed in ref. [2]. The bosonic action outside the bag can be rewritten in terms o f U as s(free) _ _ 1 f out 8rr do d2x(U ut while the b o u n d a r y action becomes

10~'U)2'

dXU{½n.fU~"t)}

Son = ( do n

= fo dX, sn 1 2[qJ-t UO+ + O*+U*O-]9 n

with the introduction o f the obvious notation U ~' --- e ' 44a~'5. -= ½(1 + Y5) U + } ( 1 - Ys) U*. The transformation properties o f U follow from those o f ~b:

~: U(x, t)~ U*(-x, t) 9 %~: U(x, t)-~ U*(x, t ) ,

J-: U(x, t)--, U(x, -t), U(x,t)~e 2i°U(x,t),

Ua(1):

($~e'Vs°$).

Finally, the b o u n d a r y equations are ira/, =

n"U loiJS 27ri

rl 2 U ' / , ~ / ; ,

~150

"

From the 4,-equation we obtain the differential equation for soliton operators,

O,I~=-I(u

I a o U - ~ / 5 U IOIU)I/I ,

which m a y be formally integrated to give

4,(x) =exp {-l f ~ d,( U-lOoU+ ~,,U-'O,U)} 4,o. Define the " d u a l " variable

U(x)=-exp{i,/-~ ~(x)}=exp { - ff~od,g U 'OoU} .

22

S. Nadkarni, H.B. Nielsen / Partial bosonization

We then have, in an obvious notation,

4,(x) = 01/2(x) u ~'/2~,(x)~Oo. To understand the effects of q,(x) acting on bosonic states, we first note that

e"/~'('°F[d)(y)] e -''/~4;(x) = F [ 6 ( y ) -,¢'-~ O(x - y ) ] . Choosing F = U(y), we have

[ l]l'/2(x), U(y)] = 0,

x e y.

Thus 01/2(x) is a local operator, having no effect on the bosonic configuration except in the vicinity of x. At first this is surprising, since U~/2(x) is non-local in terms of qS(x), stepping up its value by ~ on one side ofx. However, U[4, + ~ / ~ ] = U[4,], so things are actually consistent. What about the kink that U1/2(x) creates in a 05-configuration? It is quite easy to see that the 4,-kink is in fact described by a complete winding of the U variable. Such a winding cannot be obtained from the original U-configuration by a continuous deformation. To summarize: the effect of ¢_[ra/2(x) on U(x), though local, is topologically non-trivial. It is also easy to understand why the ~0-0perators anticommute. This is due to the square-root factors U±I/2(x), whose sign is a priori ambiguous, and depends on whether or not there is a twist in the U-configuration. Although the effect of U1/2(x) on U is local, its effect on U ~/2 is non-local: there is a sign change on one side of the twist. Choosing F[~5] = U±~/2(y), we obtain the following relations describing the effect of ~/1/2(X) on U ~/2, [ [Jl/2(X), u ± l / 2 ( y ) ] = O,

x < y,

{/[/'/2(x), U±l/2(y)} = O,

x > y,

from which the anticommutativity of the 0-operators easily follows, for x ¢ y. What happens for x = y requires careful analysis and is discussed (albeit in 4Manguage) by Mandelstam [3]. References [1] S. Nadkarni, H.B. Nielsen and I. Zahed, Nucl. Phys. B253 (1985) 308 [2] S. Nadkarni and I. Zahed, Nucl. Phys. B [3] S. Coleman, Phys. Rev. D l l (1975) 2088; S. Mandelstam, Phys. Rev. D l l (1975) 3026 [4] H. Sugawara, Phys. Rev. 170 (1968) 1659; C.M. Sommerfield, Phys. Rev. 176 (1968) 2019; S. Coleman, D. Gross and R. Jackiw, Phys. Rev. 180 (1969) 1359; R. Dashen and Y Frishman, Phys. Rev. D l l (1975) 2781 [5] J. Kogut and L. Susskind, Phys. Rev. Dll (1975) 3594