PHYSICAL REVIEW D V O L U M E 1 0 , N U N B E R 1 2 1 5 D E C E M B E R 1 9 7 4
Goldstone and quasi-Goldstone bosoi~s in ~nassless and massive e8ectrodyl-aamics
Kurt Haller Department of Physzcs, LTnz~>erszt$' of Conncctzcut, Storvs, Con?zectzcut OGI ' I
(Received 15 July 1974)
Comment is made on the modifications that are required in the demonstration that Gold- stone bosons exist in electrodynamics in order to accomlnodate the indefinite metric of the underlying space and the infinite range of the theory.
One of the interest ing c a s e s t o which the argu- ment that demons t ra tes the existence of Goldstone bosons h a s been applied is quantum electrodynam- i c s (QED) in the Lorentz gauge.' Although the argument is formally straightforward there a r e mathematical complications that a r i s e i n QED that a r e rooted, on the one hand, i n the fact that the theory must be embedded in an indefinite- m e t r i c space,' and, on the other , in the m a s s l e s s - n e s s of the photon. Because of the dynamical s t ruc ture of Lorentz-gauge QED, and i t s formula- tion i n a n indefinite-metric space , the equation a,A, = 0 , which plays the ro le of the conservation equation i n establishing the existence of the Gold- stone bosons, does not hold a s an operator identi- ty. Because of the mass lessness of the photon, many mat r ix elements that a r e manipulated i n the course of the argument a r e i l l defined, and even the limiting p r o c e s s e s that a r e used t o give them meaning a r e problematical. In a thought provoking review ar t i c le , Bernstein3 cal ls attention to the p rob lems that beset such formal proofs and r e f e r s t o the necessi ty for maneuvering judiciously be- tween the dangers of pedantic r i g o r and inconclu- sive handwaving.
The purpose of th i s note is t o clar i fy the situa- tion i n QED by making use of two devices, The f i r s t of these is a formulation of Lorentz-gauge QED i n which the subsidiary condition is imposed a s a c ~ n s t r a i n t , ~ ~ ' and in which the s t ruc ture of the underlying indefinite-metric space is particu- l a r ly t ransparent . The other device is a formula- tion of mass ive QED' which closely para l le l s Lorentz-gauge QED in i t s u s e of the Gupta-Bleuler method7 and a n indefinite-metric space, a s well a s i n the fact that it genera tes the mass ive "Feyn- man-gauge" photon propagator -id,, (M' +k2 - i ~ ) - ' . Since the theory is unitary, and s ince, moreover , a l l probabilities i n i t a r e positive semidefinite, i t can be used effectively a s a fo rm of "short-range" QED, which i n the M - 0 l imi t approaches Lorentz gauge QED without generating s ingular i t ies in the photon propagator a s the photon m a s s vanishes.'
Since i t s h a r e s a l l of the indefinite-metric fea tures of m a s s l e s s QED, but h a s a finite range, many of the infinite-range ambiguities of $ED can be r e - solved by car ry ing out a l l s t eps in the proof f o r nonvanishing photon m a s s , and allowing the m a s s t o vanish a f te r the argument i s complete.
We will u s e the resu l t s and notation of Refs, 4 and 5 f ree ly and formulate the so-called "clas- sical" argument f o r the existence of Goldstone bosons by making modifications appropriate to the indefinite-metric space. We note that the m a s s - l e s s photon field obeys
o ( ~ , A , ) = O , ( 1
where we have used the y = O f o r m of the general- ized Lorentz-gauge e q ~ a t i o n s . ~ Similarly, the massive photon field obeys
Equations (1) and ( 2 ) do not imply that a,A, = 0 o r a,W, = 0 , but ra ther that
In these equations, p(%) is the charge density e,$(ii)y4$(ii), a d p $ ) is i t s F o u r i e r t rans form. a R ( i ; ) , aQ(E), a$(;), and a$(c) a r e l inear combina- tions of longitudinal and timelike opera tors that annihilate and c rea te photons i n the "allowed" and "forbidden" modes. F o r the m a s s l e s s c a s e they a r e given by
4300 K U R T H A L L E R 18 -
~ $ 6 ) =a:$) = 2-lI2[a,t$) + ia; $11. In the appropriate representat ion, afi(i;)I 0) and a;@) 10) descr ibe the allowed and forbidden z e r o - norm '"host" s ta tes , respective1 y.
F o r the mass ive case , these corresponding modes a r e given by
where N(f2) is the normalization factor
and where k o = / G I f o r the m a s s l e s s photons, and ko = (Af2 + / i; 1 ')'/' f o r the mass ive photons. In the mass ive c a s e the "allowed" and "forbidden" one- photon modes a r e not ze ro-norm s ta tes and a r e given by l inear combinations of a$@) 1 0 ) and a$(;) / 0) . The Green 's function A(%, x,; M ) is given by
ag, X,;AI) = -(2r id3 l d g e i i . j (sink ,x,)/k,
It is convenient t o use a representat ion in which
s tate vec tors , / [) , and opera tors , 0 , a r e t rans - formed according to I 5 ) - I g) = eDI 5 ) and O - O ' = eDOe-D. The f o r m of D differs f o r the m a s s l e s s and massive photons, and is given ex- plicitly in Ref. 4 for the f o r m e r and i n Ref. 6 f o r the l a t t e r cases . The resulting representat ion, i n which the Hamiltonian has the fo rm H and 5, =ilk, 1, p r e s e r v e s a l l dynamical equations and commutation rules . The theory is equally "man- ifestly covariant" i n the new representation; i.e., the d, (and @,) continue to obey the s a m e sub- s idiary condition a f te r a Lorentz t ransformation i n which they t ransform like four-vectors. The advantage of th i s new, so-called "hatted" rep- resentation is that the opera tors 8,d, and ;,w, have the v e r y s imple space-t ime behavior
Although Eqs. (5) and (6) s e e m to be nearly iden- t i ca l , this appearance is somewhat illusory. Whereas n$($) =a:@) i n Eq. (5) and [a&), a$(g1)] = 0 in the m a s s l e s s photon theory, the more compli- cated commutation ru les given in Sec. II of Ref. 6 apply in the massive photon case. Equations (5) and (6) s ta te that $,A, and $,@,, respectively, have vanishing expectation values within the phys- i ca l subspace. We have previously10 called such opera tors "L' operators." Matrix elements of U opera tors , i .e . , ( a l * ~ I P ) , vanish when aQ($)lfi) = 0 o r (al*a$(i;) = 0, fo r a l l G.
We can express the f ie ld variables in the i r hatted representat ion and obtain, i n addition t o Eqs. (5) and (6),
f o r the m a s s l e s s c a s e , and
10 - G O L D S T O N E A N D Q U A S I - G O L D S T O N E B O S O N S I N M A S S L E S S . . . 4301
f o r the massive case. We can now proceed with one f o r m of the argument that demonstrates the existence of Goldstone bosons.
We note that
and that, upon integration over lfi, we obtain the vacuum expectation value (01 [ Q , , ~ , a , ( x ) I l o ) = - i ,
where 6, = l d ~ d , @ ) . Equation (14) provides u s with a spontaneously broken symmetry which we c a n u s e to look for Goldstone bosons. F r o m Eq. (7) we find that 9, is given by
Since the zero-momentum denominator in Eq. ( 1 5 ) makes the mat r ix elements of @, meaningless, $, i s often defined by extending the spat ial inte- g r a l over a sphere of finite radius R and subse- quently letting R- a (as in Ref. 1 ) . This has the effect of replacing the 6 function in Eq. (15) by a finite-width packet that receives contributions f r o m a finite interval of momentum s ta tes cen- t e red about 2 = 0 , and subsequently letting the width vanish. Equation (14) can be rewri t ten a s
Since u p @ ) and a@) commute with the interact ion Hamiltonian they commute not only with the t r a n s v e r s e photon and spinor opera tors at equal t imes , but a l so with the "dressed" opera tors f o r t r a n s v e r s e photons, e lectrons and positrons. The contributing intermediate s ta tes i n Eq. (16) the re - f o r e a r e limited to I a J ( 0 ) ) (aR(0)I f o r the f i r s t t e r m and 1 ag*(0)) ( a Q ( 0 ) / f o r the second. Since
1 a i ( 0 ) ) and ( ~ ~ ( 0 ) 1 a r i s e in different components of the unit operator," the argument s e e m s to un- cover two "Goldstone" boson s ta tes , not one, pro- vided that the limiting p r o c e s s we have used can be t rusted. One of these s ta tes , / a i ( 0 ) ) is for - bidden by the subsidiary condition; the other , I a i ( ~ ) ) , constitutes the longitudinally polarized component of the spin-one system. It r epresen ts a n unobservable part ic le because i t s norm is zero. Both a r e degenerate with the vacuum. Since one expects a single Goldstone boson in the theory,
th i s resul t is puzzling. If we use a different argument , i n which we
and then a s s u m e that
the difficulties prol i ferate . The aR(0) and [the l a t t e r identical to a & ( ~ ) ] , which a r e par t of
but not of 5, i,, a r e coupled to c u r r e n t s in fi, and connect to many different intermediate s tates . We a r e caught in the di lemma that if we tal
4302 K U R T H A L L E R - 10
where C is an a r b i t r a r y constant. Lf we let G- 0 smoothly then th i s approaches C - ix, . Similar fo rmal manipulations of the left-hand s ide of Eq. (17) yield
an equation of v e r y questionable validity. The mat r ix elements of d ,(0) a r e unbounded, the t ime dependence i s paradoxical, and no rel iable con- clusions can be drawn about the intermediate s ta