On the vacuum structure in quantum electrodynamics with massless fermions

Nuclear Phvsics B236 (1984) 109-124 © North-Holland Publishing Company

ON T H E V A C U U M S T R U C T U R E IN Q U A N T U M E L E C T R O D Y N A M I C S WITH MASSLESS FERMIONS V. A. R U B A K O V

Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow, 11 7312, USSR Received 12 August 1983 Extending the analysis of Krasnikov et al. [1] of the two-dimensional massless Q E D we argue that in the four-dimensional massless Q E D (i) the ground state has a 0 structure, (ii) the global chiral group U(I)LEF r X U ( 1 ) R I G H T is broken down t o U(1)CHIRALITY X Z 2 . The implications of this observation are discussed. We also consider an equivalence between free massless fermions and free massless scalars in four dimensions.

1. Introduction

The problem of the vacuum structure is one of the most important problems in modern field theory. The properties of the ground state determine both the particle spectrum and the characteristic features of interactions inherent in the theory. In gauge theories, the structure of the ground state is strongly connected with the requirement of gauge invariance [1-4], which is fundamental in these theories. In particular, in theories with massless fermions the ground state may not be the eigenstate of the operators of fermion number and/or chirality, provided that the unitary operators U[g] implementing some gauge transformations carry these quantum numbers. In this case the perturbation theory vacuum [0) is not gauge invariant, and the true ground state [0) is a linear superposition [1-4], [0)=

~

e-"°(U[g])"[O),

(1.1)

n=--oo

with an indefinite fermion number and/or chirality. In the two-dimensional massless Q E D (the Schwinger model [5]) in the transverse gauge, the operators of the residual gauge transformations have been constructed explicitly [ 1, 4] and they carry fermion number and chirality. It is this fact that leads to the double-0 vacuum structure [6] of the Schwinger model; furthermore, the absence of fermionic excitations in this model [6] can be naturally understood in terms of the absence of fermion number and chirality as quantum numbers labeling the physical states. In four-dimensional non-abelian gauge theories with massless fermions, the operators U[g] also carry fermion number of chirality [2, 3], provided that the corresponding current is anomalous. This fact has been established within the euclidean functional integral approach, the relevant gauge field configurations being 109

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V,A. Rubakov / Vacuum structure in QED

characterized by a non-vanishing winding number [7]. The Schwinger model example shows that the anomalous quantum numbers may not be the only quantum numbers carried by the operators of gauge transformations: indeed, in this model the fermion number is not anomalous. Therefore, the single-0 vacuum structure in fourdimensional gauge theories [2, 3] might not be complete [1]. In this paper we attempt to discuss this problem in the simplest case of fourdimensional quantum electrodynamics with massless fermions. Since the exact solution of this model is not known, we are unable to construct the operators of gauge transformations explicitly. However, for establishing qualitative properties of these operators, such as their fermion number or chirality, the explicit construction might not be obligatory. For qualitative discussion the functional integral approach might turn out to be more adequate. Even in the Schwinger model, the euclidean functional integral technique is not convenient for establishing the double-0 vacuum structure, although chirality (but not fermion number!) breaking can be understood in terms of the functional integral over euclidean gauge fields with non-zero winding number [8, 1]. On the other hand, it is rather convenient to consider both fermion number and chirality breaking within the framework of pseudo-euclidean functional integral formalism [1]. Guided by the Schwinger model example, we prefer to use the pseudo-euclidean technique in this paper also in the four-dimensional case. Surprisingly enough, the analysis of Kranikov et al. [1] can be rather straightforwardly generalized; we argue that the operators implementing some gauge transformations carry non-vanishing fermion number (but not chirality). This implies that the ground state in four-dimensional massless Q E D has a 0 structure (1.1) and that the true vacuum [0) has indefinite fermion number; the physical excitations can be characterized by the only additive quantum number (chirality), so the spectrum of the model is expected to consist either solely of bosons or of bosons and Majorana fermions. Furthermore, since the fermion number is related to the electric charge, we expect that there are no charged states in massless Q E D , in agreement with ref. [9]. The absence of the fermionic excitations in the Schwinger model is known to be connected with the fermion-boson equivalence in two dimensions [10]. To understand the possibility of the transformation of fermions into bosons in four dimensions, we show in this paper that free massless four-dimensional fermions are equivalent to free massless real scalars, the transformation being non-linear and non-local. The only, but crucial, difference in comparison with two dimensions is that fermionic currents are expressed non-locally through the scalar field; this fact makes the bosonization useless for solving four-dimensional theories. We note in passing that other examples of fermion-boson transformation in four dimensions have been considered in connection with the fermion-monopole problem [11, 12] and nonlinear tr-models [13]. This paper is organized as follows. In sect. 2 we recapitulate the functional integral analysis [1] of the fermion number breaking in the Schwinger model. Extending

V.A. Rubakov / Vacuum structure in Q E D

111

this analysis to four dimensions, we present in sect. 3 the arguments in favour of the fermion number breaking in the four-dimensional massless Q E D . We discuss the relevant gauge field configurations, zero fermion modes and fermionic determinants. However, we do not attempt to evaluate the functional integrals over the gauge field fluctuations near these configurations, so we are unable to calculate the f e r m i o n - n u m b e r violating matrix elements. Sect. 4 contains some concluding remarks. The equivalence between free massless fermions and scalars in four dimensions is considered in the appendix. In the whole paper we set the electromagnetic charge equal to 1 in order to simplify the formulas.

2. The fermion number breaking and the functional integral in the Schwinger model The vacuum structure in the Schwinger model has been discussed from the point of view of gauge invariance some time ago [4,1]. In the transverse gauge, a~,A" = 0, the residual gauge transformations are characterized by the gauge functions of the following general form,

~(x)=~+(x°-x')-~ (x°+x'). It is convenient to introduce two topological numbers [1] associated with any residual gauge transformation

1 n~ = -

77"

[~(~)

-

~ ( - ~) ] .

The gauge transformations are implementable if and only if n+ and n are integer; the unitary operators U[a~+1) ] and U [ a <1)] implementing the gauge transformations with n = 0, n+ = 1 and n_ = 1, n+ = 0 respectively carry fermion number 1 and chirality ±1. The requirement of gauge invariance leads to the double-0 vacuum structure [1, 4], [0+, 0 - ) =

E e in+o+-in o (U[og(+l)])n+(u[ol(l)])n n+,n_

jO) '

(2,1)

where [0) is the perturbation theory vacuum. From (2.1) it is clear that both fermion n u m b e r and chirality are broken. The matrix elements of gauge invariant operators over the ground state (2.1) are independent of a particular choice of the gauge functions c~ 1~ [1]. This is not the case for gauge variant operators. The functional integral analysis of the fermion number and chirality breaking in the Schwinger model has been performed by Krasnikov et al. [1]. For reasons that become clear later, we concentrate here on fermion n u m b e r breaking. The relevant gauge function has the topological numbers n+ = n_ = 1; we take for simplicity a ( x ) = ½1r[e (x ° - x 1) + e ( x ° + x l ) ] .

(2.2)

To see that the unitary operator U[c~] implementing this gauge transformation

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V.A. R u b a k o v / Vacuum structure in O E D

carries two units of fermion number, consider the matrix element +


(2.3)

where the subscripts 1 and 2 label upper and lower components of the spinor field. Since the operator ~Ol(X)q~2(x) is gauge variant, we expect that F2 depends on a particular choice of the gauge function a, as well as on the coordinates x" appearing in (2.3). The matrix element (2.3) is given by the standard pseudo-euclidean functional integral in the transverse gauge, but with the following boundary conditions [1], A~, (x) = positive frequency function as x ° ~ - ~ , A t (x) - 0~a (x) = negative frequency function as x ° ~ + ~ ,

(2.4)

qJ(x), q~(x) = positive frequency functions as x ° ~ - 0 0 , e-i~x)q~(x), ei~)q~(x) = negative frequency functions as x ° ~ +c~.

(2.5)

These boundary conditions are the generalizations of Feynman ones for the case of "sandwiching" between the perturbation theory vacuum 10) and its gauge transform, U[a][0). The simplest gauge field configuration obeying (2.4) is fi,, ( x ) = 0~,a+(x),

(2.6)

where c~+(x) = ½i log [(x') 2 - (x°) 2 -

iex°],

(2.7)

is a positive frequency part of the gauge function (2.2). The configuration (2.6) is a "pure gauge" with the complex "gauge function" a+(x); nevertheless, its neighbourhood gives the dominant contribution to the matrix element (2.3) [1]. Note that the appearance of complex gauge fields is not specific to the vacuum structure problem: even in the free theory with real external sources, the saddle-point fields are complex. To see that the matrix element (2.3) is non-vanishing, consider the functional integral over fermions in the external field (2.6). There are two zero modes of the Dirae operator I~ = y" ( 0 - i/i), obeying (2.5), namely

q/')(x) = ei~+~x~(10) ,

~(2)(X)= ei"+(x)(~) .

(2.8)

These zero modes tend to zero as x ° ~ - 0 0 , and their gauge transforms e-i'~x~ q~ 1)'~2)(x)------e-i'~ ~x){(10), ( ~ ) } ,

(2.9)

which are relevant in the distant future because of (2.5), tend to zero as x ° ~ +oo,

V.A. Rubakov / Vacuum structure in QED

113

since the negative frequency part of the gauge function is equal to

a - ( x ) = -½i log [(x') 2 - (x°) .2+ iex°].

(2.10)

On the other hand, the Dirac operator ~i = 3" ( - ~ - ieA), which acts on q~, has no zero modes obeying (2.5) and vanishing as x ° ~ - ~ . According to the general rules [2, 14] the functional integral over fermions for F2(x) does not vanish, and the matrix element (2.3) is proportional to the product of the zero modes [1] Fe(x ) = e zi~'+(x)const.

(2.11)

This result coincides with that obtained within the operator approach [1], so the whole functional integral procedure is justified. To discuss the norms of the zero fermion modes and fermionic determinants, it is convenient to perform the Wick rotation into euclidean space-time. The boundary conditions (2.5) make it impossible to perform the rotation in terms of the original variables [1]. Instead, it is convenient to introduce new variables [1]

A~,(x) c = A~,(x) -a,~a +( x ) , ~C(x) = e-i'~+~x)~(x),

(2,12)

~C(x) = ei~+~x)~(x) ,

(2.13)

which obey the following boundary conditions,

A ~ ( x ) , e-i~c~x)¢C(x), ei"C~x)~C(x) (negative frequency functions as x ° ~ + ~ 5 [ positive frequency functions as x ° ~ - ~ ,

(2. 14)

where aC(x) = O ( x ° ) a - ( x ) - O(-xO)a+(x) .

Note that the transformation (2.12), (2.13) is in fact a "gauge transformation" with a complex "gauge function" a+(x), so its jacobian is equal to 1 [1]. Note also that in the case of an external gauge field (2.6) this transformation reduces the Dirac operator D to the free operator t~ = 3" ~. Since the function a ~_(x) has a correct causal behaviour, the functional integral over A~,, 4,~_, ~ can be Wick rotated. The boundary conditions (2.14) transform into

AE~(XE),e--i'~E(xE)~bE(xE),e+i'~E(xE)~E(xE)~o,

as ]xE[-~ O0,

where the superscript E denotes the euclidean quantities, and aZ(X E) = a c(--ix E, X E) = ½i log [(xE)2].

(2.15)

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V.A. R u b a k o v / Vacuum structure in O E D

The procedure described above leads to the following euclidean versions of the zero modes (2.8),

which are evidently the zero modes of a free euclidean Dirac operator ~E and obey the boundary conditions (2.15). It is worth noting that the functions ~(~,2)= e ~EOE(1'2), which are relevant because of the boundary conditions (2.15), have norms I ~ * ~ d2x E logarithmically divergent in the infrared region. This behaviour is characteristic also of the zero modes associated with chirality breaking in the Schwinger model [8] as well as to the zero modes appearing in the monopole-fermion problem [11, 15]. The functional integral over other fermion modes is equal to some constant [1]. We discuss this point later in connection with the four-dimensional problem. To conclude this section we summarize the main observations made in connection with the fermion number breaking in the Schwinger model: (i) to establish the qualitative properties of the operators of the residual gauge transformations, it is convenient to proceed within the pseudo-euclidean functional integral approach; (ii) complex, pure gauge configurations of the gauge field A , are responsible for the fermion number breaking; (iii) the non-vanishing difference between the numbers of the zero modes of the Dirac operators t~ and /~ in the presence of these gauge field configurations is crucial; (iv) the euclidean counterparts of the zero fermion modes, ~(1,2) have logarithmically divergent norms. In sect. 3 we apply the procedure outlined above to the four-dimensional massless QED. We find that these observations, up to minor modifications, are still valid, so we argue that fermion number breaking occurs also in four dimensions.

3. Fermion number breaking in four-dimensional massless QED In this section we consider a four-dimensional Q E D with one massless fermion field (which is necessarily Dirac because of the requirement of the anomaly cancellation). We are interested in the properties of the unitary operators U [ a ] implementing the gauge transformations. We restrict ourselves to the case of spherically symmetric gauge functions,

a(x)=a(r,t),

r--lxl,

(3.1)

and choose the gauge condition so that the residual gauge transformations obey

(G2 - G 2 )t~(r, • t) =O .

(3.2)

This gauge can be specified by decomposing the gauge field A , ( x , t) over spherical harmonics and then imposing the gauge condition on each term of the decomposition

V.A. Rubakov / Vacuum structure in QED

115

separately. For instance, in the case of spherically symmetric fields, Ao=ao(r,t) ,

A=X-a~(r,t) , r

(3.3)

the gauge condition reads O,ao- ~al = 0.

(3.4)

(Note in passing that (3.4) does not coincide with O,A • = 0 in the case of spherically symmetric fields (3.3).) For the gauge transformation to be non-singular at the origin, the gauge functions should obey the following boundary condition, Ora(r

= 0, t) = 0,

(3.5)

otherwise 0~ = ( x / r ) O r a would have a singularity at r = 0 . The gauge functions obeying (3.2) and (3.5) have the following form, a(r, t ) = f l ( t - r ) + 1 3 ( t + r ) .

(3.6)

Let us try to extend the arguments of the preceeding section. We expect that the non-trivial gauge transformations correspond to the functions 13 with non-vanishing "topological number" /3(+oo)-/3(-oo); for definiteness we concentrate on a particular form (cf. (2.2)) a(r, t) = C [ e ( t - r) + e ( t + r)], (3.7) where the constant C is yet to be specified. To argue that the unitary operator U[a] of this gauge transformation carries fermion number, we consider the matrix element (cf. (2.3)) +

(O[U[a]q~l(X)O~(x)qJ3(x)O4(x)]O) ~ F4(x),

(3.8)

where 10) is a perturbation theory vacuum and subscripts 1 , . . . , 4 are spinor indices. Again, this matrix element is given by the pseudo-euclidean functional integral with the boundary conditions (2.4), (2.5), and the simplest gauge field configuration obeying (2.4) is given by A~ ~ c9~o~+ ,

a +(r, t) =/Clog (r 2 - t 2+ iet).

(3.9)

As in sect. 2, we have to consider the zero modes of the Dirac operators /0 = 3" ( 0 - i,~) and ~ = 3'(-c9-- i/~), supplemented by the boundary conditions (2.5). We find that there exist four zero modes of the operator/~,

/~/{1t ~b(1) = eia+\O / .... ,

~J(4) ~---ei'~÷ ( i ) ,'

(3.10)

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V.A. Rubakov / Vacuum structure in QED

and that the o p e r a t o r / ~ has no zero modes. The Wick rotation into euclidean space is performed exactly in a way described in sect. 2, and the euclidean counterparts of (3.10) are

t~(1)'''(4) (X E) = ei'~e(xE)~E(1),..'.,(4) , where E = (iC/~) log ( r Z + x Ez) and q,~(i) = 6}. In order that these functions have norms I q~+q~d4xEdiverging at most logarithmically at large [xE[, the constant in (3.6) should be chosen as follows, C=*r.

(3.11)

In this case, the euclidean functional integral over fermions which corresponds to the contribution of the gauge field (3.9) to the matrix element (3.8) is

f eI~VEaV4~Ed4xEoE(xE)...~b4E(xE) d~bE dt~E = const • det'

~E .

(3.12)

This equation requires some explanation. First, note that the change of variables (2.12), (2.13), which is to be performed before the Wick rotation, reduces the Dirac operator D to a free one, so the exponential on the left-hand side of (3.12) coincides with the free action. Second, the integration is to be performed over the fermion fields obeying the boundary condition (2.15), so the integral is proportional to the product of the zero modes times the determinant over the non-zero modes, the latter is denoted by det'/I e . Finally, the euclidean forms, 0 E~i) of the zero modes (3.10) are simply constants (cf. (2.16)), so the constant on the right-hand side of (3.12) does not depend on x z. Now we argue that Det' ~E is equal to the determinant Det ~E of a free Dirac operator with standard boundary conditions, so the right-hand side of eq. (3.12) is some non-vanishing constant (on the left-hand side of (3.12), the standard normalization factor (det ~E)-I is implicitly assumed). According to the 't Hooft prescription [16], the evaluation of Det' ~E begins with introducing a spherical box of radius R in euclidean space-time. Then one calculates the eigenvalues An(R) of the operator /rE supplemented by the boundary condition ei'~E(R)~bE(R) = 0 ,

(3.13)

which is a finite-volume counterpart of (2.15). The determinant D e t ' ~ E is formally equal to det'~' = lim [I An(R). R~oe n The same equation is valid for the free determinant Det~ 'E, but the boundary condition (3.13) should be replaced by 0E(R) = 0 .

(3.14)

V.A. R u b a k o v / Vacuum structure in Q E D

117

Since e i'vE(R) is not equal to zero at large but finite R, eqs. (3.13) and (3.14) actually coincide, so the eigenvalues are the same. We conclude that det'/~E =

det ~E,

(3.15)

which is the desired result. Turning back to pseudo-euclidean space-time, we obtain from (3.12) and (3.15) that the contribution of the gauge field configuration (3.9) to the matrix element (3.8) is const

• e 4i'~+(x) .

If we assume, by the analogy to the Schwinger model, that the neighbourhood of (3.9) gives the dominant contribution to (3.8), we would get F4 = c o n s t • e 4i"+. So we argue that F4 # 0, i.e. the operator U[c~] implementing the gauge transformation (3.7), (3.11) carries four units of fermion number and zero chirality. It is worth noting that the above arguments do not depend on the explicit form (3.7) of the gauge function, i.e. they are valid for any gauge function of the form (3.6) provided that /3(o0)-/3(-o0) = 2 r r .

(3.16)

Before turning to the consequences of the above observation, let us discuss the properties of a gauge function obeying (3.6), (3.16). At a distant future (as well as at a distant past) this gauge function varies only near the light cone, and takes different constant values inside and outside it, as shown in fig. 1. There is evidently a topological structure associated with the gauge functions of this type. We believe that any unitary operator implementing a gauge transformation with a gauge function (not necessarily spherically symmetric) having the same structure carries four units of fermion number and zero chirality. In particular, we conjecture that the difference between the numbers of the zero modes of the Dirac operators /~ and ./~ supplemented by the boundary conditions (2.5) is equal to 4, provided that the field A , interpolates between zero and 0 , a (i.e. obeys (2.4)) and the gauge function t~(x)

t: --2~

/

=_. Fig. 1

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V.A. Rubakov / Vacuum structure in OED

has this topological structure. In other words, we expect that there exist some kind of index theorem in pseudo-euclidean space associated with this type of U(1) gauge field configurations which might be analogous to the integrated form of the axial anomaly [17]. We hope to turn to this problem in future. Now we wish to discuss some consequences of the above considerations. Since the operator U [ a ] implementing the gauge transformation (3.6), (3.16) carries fermion number, the gauge invariant ground state has a 0 structure (1.1) and its fermion number is indefinite. Therefore, fermion number cannot label the excitations above this vacuum, so the spectrum of the model cannot consist of massless fermions. On the other hand, U [ a ] does not carry chirality, so chirality can be a good quantum number. We point out the difference between the four-dimensional massless Q E D and the Schwinger model: in the four-dimensional case the gauge function should obey the boundary condition (3.5), so it cannot have a form, say, [e(t+r)-e(t-r)], which is precisely the form of the gauge function relevant to chirality breaking in the Schwinger model (with x 1 substituted for r) [1]. More generally, the future light cones in four and two dimensions separate the space into two and three disconnected regions respectively, so in four dimensions one can imagine only the gauge function structure shown in fig. 1, while in two dimensions another structure (fig. 2) is also possible. In the Schwinger model the operators of the gauge transformations of the latter type are just ones carrying chirality and thus they are responsible for the second vacuum 0 angle. For these reasons, the absence of chirality breaking in four dimensions seems to be natural. There is yet another global symmetry, in the four-dimensional case, which is allowed to be unbroken in spite of the above observations. Since there exist four zero modes (3.10), the operator U [ a ] carries f o u r units of fermion number. This means that states with fermion numbers N r equal to 1, 2 or 3 cannot be gauge transformed into a state with zero fermion number, while a state with N v = 4 can, i.e. fermion number may be conserved mod 4. Since for even (odd) chirality the fermion number is also even (odd), we expect that the initial global group U(1)L × U ( 1 ) a is broken down to U(1)chiraltyXZ2. Of course, there can exist some other effects, not discussed in this paper, which breaks this symmetry further.

=_to eL

,..,

"Z

Fig. 2

V.A. Rubakov / Vacuum structure in QED

119

What would be a spectrum of the four-dimensional massless Q E D if the unbroken global subgroup is indeed U(1) x Z27 Evidently, there would be no massless fermions and the massless fermions initially present in the lagrangian would be transformed into either massive Dirac ones, or Majorana fermions and bosons, or only bosons. The latter two possibilities seem plausible, since the residual Z2 symmetry could hardly be implemented in the former case. On the other hand, the appearance of bosons in the spectrum could be natural, since in four dimensions (as well as in two dimensions) fermions could in a sense be equivalent to bosons. Specifically, we show in the appendix that the free Weyl fermion field is equivalent to a massless real scalar one, so the free massless Dirac field is equivalent to two real free scalars ~1 and q~2. Turning on the electromagnetic interaction might result in the survival of these scalars as physical excitations, the Z2 symmetry being implemented as a transformation interchanging ~1 and q~2. To conclude this section we note that the above arguments apply only to strictly massless fermions. Indeed, adding a fermionic mass term (whatever small it is) would lead to the disappearance of the zero modes (3.10) since the non-singular solutions of the Dirac equation would exponentially grow as r ~ oc. Therefore, we expect that the limit mF-~ 0, c~(/x) fixed, (/x is some normalization point) is discontinuous.

4. Conclusion

In this paper we have argued that the ground state of four-dimensional massless Q E D has a 0 structure associated with the fermion number breaking. Of course, there are several objections to the (quite formal) arguments presented above. First, we have discussed contributions to the functional integral coming from c o m p l e x gauge field configurations. Second, we have not performed the integration over the fluctuations near these configurations, which could in principle result in the suppression of fermion n u m b e r breaking matrix elements. Third, we have used a peculiar gauge, which has not even been completely specified. In fact, the main justification of our arguments comes from the Schwinger model, in which they work well. Can one apply the above arguments to the non-abelian gauge theories with massless fermions? The answer to this question is unclear. At first sight, any gauge group contains an U(1) subgroup, so naively the above arguments can be straightforwardly generalized. However, the self-iteration of the gauge fields can play a very essential role. For example, all field configurations (including zero fermion modes), appearing in the above discussion, are long ranged, and confinement effects can suppress them. In any case, before proceeding to the non-abelian theories it seems necessary to understand better the vacuum structure of the four-dimensional massless Q E D .

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V.A. Rubakov / Vacuum structure in QED

I am deeply indebted to V.A. Matveev and A.N. Tavkhelidze for stimulating interest and discussions. The criticism and helpful comments by my colleagues from the Theoretical Division of the Institute for Nuclear Research of the Academy of Sciences are also acknowledged. I am especially grateful to N.V. Krasnikov, V.A. Kuzmin, M.E. Shaposhnikov, F.V. Tkachov and V.F. Tokarev for numerous discussions.

Appendix FERMION-BOSON EQUIVALENCE IN FOUR DIMENSIONS In this appendix we show that in four dimensions, a free massless left-handed fermionic field is equivalent to a free massless real scalar field, the correspondence being similar to that in two dimensions [10]. Our strategy will be as follows. We first decompose the fermionic field over the operators with fixed angular momentum. Each term in this decomposition is effectively two-dimensional, the two coordinates being r, t. Then we apply an integral transformation to each term, which reduces the equation of motion to the two-dimensional Dirac equation with a specific boundary condition at r = 0. This "two-dimensional" Dirac field is known [12] to be equivalent to the "two-dimensional" free scalar field, also with a boundary condition at the origin. On the other hand, the four-dimensional massless scalar field is also decomposed over the spherical harmonics; each term of the decomposition is equivalent, after another integral transformation, to the two-dimensional free scalar field. The two sets of the two-dimensional scalars turn out to be the same. Since all the manipulations are invertible, this establishes the desired equivalence. The free massless left-handed fermionic field in four dimensions, ~0(x, t), can be decomposed as follows,

~b(x,t)=l r

~u~'k(r,t)X~'k(X--r),

~

(A.1)

n = l , 2 .... /.t = ± 1 k =±l,...,±(n -1 )

X~'k

~rx/r

where are the eigenfunctions of j 2 , J 3 and with the eigenvalues ( n - ½ ) , k, p~ respectively, ~,k are the operators obeying the equations of motion,

(-i O,-irz Or'Jl-~TI)~)n'k= 0 ,

(A.2)

and the equal-time anticommutation relations, +

{v"'k(r,

t), v"'k(r', t)} = ~ ( r -

r'),

(A.3)

V.A. Rubakov / Vacuum structure in QED

121

other anticommutators vanishing, Here we have introduced the notation

Eq. (A.2) should be supplemented with the boundary condition

v~'k(r=O, t ) = 0 ,

(A.4)

which ensures the absence of the singularity at the origin. Note that the Green functions of the operators v n'k, which were explicitly constructed in ref. [11], resemble the boundary condition (A.4). Consider first the case n = 1, k = ½, and introduce the new variables F~_'i and F+'~ in the following way,

( )f0

v+l(r, t) = Or-

F_l(r', t) dr',

v_,(r, t) = -F+l(r, t).

(A.5)

(We omit the superscripts 1, ½here). It is straightforward to verify that in terms of these variables eqs. (A.2), (A.3) read

- i O,F- ir2 OfF = 0 , {F(r, t), F+(r ', t)} = a ( r - r ' ) ,

(1.6)

where F

\F-I/"

Furthermore, the boundary condition (A.4) takes the form F+l(r = 0, t) = 0 ,

F - l ( r = 0, t) < oo.

(A.7)

Therefore, the transformation (1.5) reduces the initial field v ~4 to a free twodimensional fermionic field with boundary conditions (A.7). The F field can be bosonized; the explicit expression being [12] F = :e;~+;~2: (01),

(1.8)

where ~ is a free massless two-dimensional real scalar field obeying

Orq~(r = 0, t) = 0 , ~(r, t) = -

Io

O,~(r', t) dr'.

(A.9) (A.10)

We conclude that the first two terms in (A.1) are equivalent to two fields q~L4(r, t) obeying (A.9).

V.A. Rubakov / Vacuum structurein OED

122

Consider now the case of an arbitrary n >~ 2. T h e integral t r a n s f o r m a t i o n of the form /')+1

(r, t) =

( n) f0 0r-

r '"-1 v'n-leltr,' t) dr' ,

v~l (r, t) = -~_~T 1 (r, t ) ,

(A.11)

is easily checked to reduce eqs. ( 1 . 2 ) - ( 1 . 4 ) to the same equations for ~ with (n - 1) substituted for n. T h e r e f o r e , the t r a n s f o r m a t i o n (A.10) lowers n by one, and the n-fold application of the t r a n s f o r m a t i o n s of this kind reduces v" to a free twodimensional Dirac field with the b o u n d a r y conditions ( 1 . 7 ) . So we find that the initial fermion field (A.1) is equivalent to an infinite set of real massless twodimensional scalars , , k obeying ( 1 . 9 ) . In fact, the bosonization p r o c e d u r e is slightly m o r e complicated, since it is necessary to apply the Klein t r a n s f o r m a t i o n to the right-hand side of eq. (A.8) in o r d e r to ensure the correct a n t i c o m m u t a t i o n relations b e t w e e n F _ 1 and F+~ as well as b e t w e e n F "'k with different n and k [18]. This Klein t r a n s f o r m a t i o n can also be p e r f o r m e d recursively* [18]. L e t us turn to the real massless four-dimensional scalar field q)(x, t). Its d e c o m p o s i tion analogous to ( 1 . 1 ) reads

cl)(x, t) = - 1 ~ r

Ys,s~(x)~s,S~(r, t) ,

(A.12)

3 = 0 , 1 .... J3=0,± 1,...±J

w h e r e Ys, s~ are the standard spherical harmonics and the o p e r a t o r s ~ s's3 satisfy the equations of motion,

(0 2. 0:-t. .J(J;l))~'sS~=O,~ .

(1.13)

the equal time c o m m u t a t i o n relations,

[Ot~s'S3(r,t),

ffs'S3(r', t)] =

-i~(r-

r'),

(A.14)

and the b o u n d a r y conditions ~'sJ3(r = 0, t) = 0 .

(A.15)

W e n o w show that the set {~.s.s3} is equivalent to the set of real massless twodimensional scalar fields {ffs,s~} obeying 0r~S'S3(r = 0, t ) = 0 . * I am indebted to M.S. Serebryakov for discussion of this point.

(A.16)

V.A. R u b a k o v / Vacuum structure in Q E D

123

In the case J = J3 = 0 the field ~.o.oalready obeys the two-dimensional D'Alambert equation, and it is straightforward to verify that the field ~°'°(r, t) = - f '

0rsr°'d(r, t') d t ' ,

J-

(A.17)

oz~

has all the desired properties. For J = 1 the transformation from ~"I'J3 to ~1"J3 reads

( )f0

~'"J3(r, t) = 0 r -

~'J3(r', t) d r ' .

(A.18)

For J i> 2 we apply the integral transformation ~J(r, t ) =

0r-

1

r'J-l~J-2(r ', t) dr',

(A.19)

lowering J by two (i.e. ~ obeys eqs. (A.13)-(A.15) with ( J - 2 ) substituted for J). Therefore, in a finite number of steps ~-J.J3can be reduced either to ffJ=~ or to ~.J=0, so it is also equivalent to the free massless two-dimensional scalar field ~J, J3 obeying (A.16). Therefore, both free Weyl fermions and free real massless scalars are equivalent to infinite sets of real massless two-dimensional scalars, { n.k; n = 1, 2 . . . . ; k = ±½. . . . . +(n -½)} and {&J'J~; J = 0 , 1 , . . . ; J 3 = 0 , +1 . . . . . +J} respectively, defined on a half-plane (r, t) and obeying the boundary conditions (A.9) and (A.16). Since one can easily establish a one-to-one correspondence between these two sets, and since all the above transformations, eqs. (A. 1), (A. 5), (A. 11 ), (A. 12), (A. 17)-(A. 19) can be inverted, the desired equivalence is shown. We note in conclusion that the equivalence formula (A.8) is correct only in a sector with zero fermion number associated with F; however, this does not place any restriction, since one can always proceed in this sector and use the cluster property [18]. Note also, that the above construction does not contradict the spin-statistics theorem, since the fermion field 4J(x, t) and the boson field ~(x, t) have spin ½ and 0 respectively.

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V.A. R u b a k o v / Vacuum structure in Q E D

[8] N.K. Nielsen and B. Schroer, Nucl. Phys. B120 (1977) 62; Phys. Lett. 66B (1977) 373; B. Schroer, Acta Phys. Austr. suppl. 19 (1978) 155; K.D. Rothe and J.A. Swieca, Ann. of Phys. 117 (1979) 382 [9] V.N. Gribov, Nucl. Phys. B206 (1982) 103 [10] S. Coleman, Phys. Rev. D l l (1975) 2088; S. Mandelstam, Phys. Rev. D11 (1975) 3026; A.K. Pogrebkov and V.N. Sushko, Teor. Mat. Fiz. 24 (1975) 425; 26 (1976) 419; G.F. Dell'Antonio, Y. Frishman and D. Zwanziger, Phys. Rev. D6 (1972) 988 [11] V.A. Rubakov, Inst. Nucl. Res. preprint P-0211 (1981); Nucl. Phys. B203 (1982) 311 [12] C.G. Callan, Phys. Rev. D26 (1982) 2058 [13] E. Witten, Princeton preprint (1983) [14] R.D. Peccei and H. Quinn, Nuovo Cim. 41A (1977) 303; N.V. Krasnikov, V.A. Rubakov and V.F. Tokarev, Phys. Lett. 79B (1978) 423; Yad. Fiz. 29 (1979) 1127 [15] V.A. Rubakov, Proc. Int. Seminar on quantum field theory and high-energy physics, Serpukhov, July 1981 [16] G. 't Hooft, Phys. Rev. D14 (1976) 3432 [17] A.S. Schwarz, Phys. Lett. 67B (1977) 172; N.K. Nielsen and B. Schroer, Nucl. Phys. B127 (1977) 493; L.S. Brown, R.D. Carlitz and C. Lee, Phys. Rev. D16 (1977) 417; R. Jackiw and C. Rebbi, Phys. Rev. D16 (1977) 1052; N.H. Christ, Phys. Rev. D21 (1980) 1591 [18] M.B. Halpern, Phys. Rev. D12 (1975) 1684