Goldstone and quasi-Goldstone bosons in massless and massive electrodynamics

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<ul><li><p>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 </p><p>Goldstone and quasi-Goldstone bosoi~s in ~nassless and massive e8ectrodyl-aamics </p><p>Kurt Haller Department of Physzcs, LTnz~&gt;erszt$' of Conncctzcut, Storvs, Con?zectzcut OGI ' I </p><p>(Received 15 July 1974) </p><p>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. </p><p>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. </p><p>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.' </p><p>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 $ED can be r e - solved by car ry ing out a l l s t eps in the proof f o r nonvanishing photon m a s s , and allowing the m a s s t o vanish a f te r the argument i s complete. </p><p>We will u s e the resu l t s and notation of Refs, 4 and 5 f ree ly and formulate the so-called "clas- sical" argument f o r the existence of Goldstone bosons by making modifications appropriate to the indefinite-metric space. We note that the m a s s - l e s s photon field obeys </p><p>o ( ~ , A , ) = O , ( 1 </p><p>where we have used the y = O f o r m of the general- ized Lorentz-gauge e q ~ a t i o n s . ~ Similarly, the massive photon field obeys </p><p>Equations (1) and ( 2 ) do not imply that a,A, = 0 o r a,W, = 0 , but ra ther that </p><p>and that </p><p>In these equations, p(%) is the charge density e,$(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 </p></li><li><p>4300 K U R T H A L L E R 18 - </p><p>and </p><p>~ $ 6 ) =a:$) = 2-lI2[a,t$) + ia; $11. In the appropriate representat ion, afi(i;)I 0) and a;@) 10) descr ibe the allowed and forbidden z e r o - norm '"host" s ta tes , respective1 y. </p><p>F o r the mass ive case , these corresponding modes a r e given by </p><p>and </p><p>where N(f2) is the normalization factor </p><p>and where k o = / G I f o r the m a s s l e s s photons, and ko = (Af2 + / i; 1 ')'/' f o r the mass ive photons. In the mass ive c a s e the "allowed" and "forbidden" one- photon modes a r e not ze ro-norm s ta tes and a r e given by l inear combinations of a$@) 1 0 ) and a$(;) / 0) . The Green 's function A(%, x,; M ) is given by </p><p>ag, X,;AI) = -(2r id3 l d g e i i . j (sink ,x,)/k, </p><p>It is convenient t o use a representat ion in which </p><p>s tate vec tors , / [) , and opera tors , 0 , a r e t rans - formed according to I 5 ) - I g) = eDI 5 ) and O - O ' = eDOe-D. The f o r m of D differs f o r the m a s s l e s s and massive photons, and is given ex- plicitly in Ref. 4 for the f o r m e r and i n Ref. 6 f o r the l a t t e r cases . The resulting representat ion, i n which the Hamiltonian has the fo rm H and 5, =ilk, 1, p r e s e r v e s a l l dynamical equations and commutation rules . The theory is equally "man- ifestly covariant" i n the new representation; i.e., the d, (and @,) continue to obey the s a m e sub- s idiary condition a f te r a Lorentz t ransformation i n which they t ransform like four-vectors. The advantage of th i s new, so-called "hatted" rep- resentation is that the opera tors 8,d, and ;,w, have the v e r y s imple space-t ime behavior </p><p>and </p><p>Although Eqs. (5) and (6) s e e m to be nearly iden- t i ca l , this appearance is somewhat illusory. Whereas n$($) =a:@) i n Eq. (5) and [a&amp;), a$(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. </p><p>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), </p><p>f o r the m a s s l e s s c a s e , and </p></li><li><p>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 </p><p>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. </p><p>We note that </p><p>and that, upon integration over lfi, we obtain the vacuum expectation value (01 [ Q , , ~ , a , ( x ) I l o ) = - i , </p><p>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 </p><p>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 </p><p>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 </p><p>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, </p><p>th i s resul t is puzzling. If we use a different argument , i n which we </p><p>integrate </p><p>and then a s s u m e that </p><p>the difficulties prol i ferate . The aR(0) and [the l a t t e r identical to a &amp; ( ~ ) ] , which a r e par t of </p><p>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</p></li><li><p>4302 K U R T H A L L E R - 10 </p><p>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 </p><p>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 tes in Eq. (18). </p><p>It i s instruct ive to c a r r y out s i m i l a r arguments f o r the massive boson case , where the finiteness of the range is par t of a consistent, unitary theory. In Ref. 6 it i s shown that the v e r y same conditions that make the range finite a l so modify the operator a lgebra and the geometr ic p roper t i es of the in- definite-metric space. The M- 0 and the E- 0 l imi t s of the mass ive photon theory a r e both well defined, but it is important , in the sp t t i a l inte- g r a l s that a r i s e in these proofs , that k - 0 befire .W- 0. Equations (2.8) in Ref. 6 show that when ,W-0 while /;/ is finite, then a$(k)- a:(;) and az(G)- a&amp;(E). But when E - 0 f o r nonvanishing m a s s , a$(O) - -a:(0) and a$(O) - aA(0). The simul- taneous limit c- 0 and M- 0 is not well defined. If we define 4, = Scl: Go(?), then we find that i t is given by </p><p>where q =p(O) and i s the constant total charge of the system. Since a$(O) = -a:(0) and s ince aQ(0) and aB(0) commute with I?,, the t ime dependent Q, is given by </p><p>(19) In contrast to QA, GW i s well defined. F r o m </p><p>( 0 [P,(?).a,D,(x)] 10) = i I f i &amp; ~ ~ , - $ , x ~ ; ~ ) </p><p>we can infer that </p><p>and s ince Q^, a s well a s 5,@,(0) can only connect to the / a i ( 0 ) ) (aQ(0) / component of the unit opera- </p><p>t o r , the only boson whose existence is required by Eq. (20) i s the 1 a i ( 0 ) ) s ta te which is forbidden by the subsidiary condition. Since it has a m a s s dd#O we will cal l i t a "quasi-Goidstone" boson. The explicit representat ions of Q , and a,@, make it easy to verify that Eq. (20) i s cor rec t . </p><p>When the second argument-is applied t o the m a s - sive case , we find that Jdz V . %' = 0 and that Jdii 8,@, is well defined and equal to a , ~ , . F r o m </p><p>we can conclude direct ly that </p><p>(21) and, again that in) (nl must contain 1 a i (0) ) (aQ(0) 1 , which is indeed the only s tate that can contribute to the sum. At th i s point one can le t M- 0 and infer that , i n th i s l imit , the Goldstone boson s ta te I a i ( 0 ) ) is required by the theory. Ambiguities in the argument a r i s e in the m a s s l e s s photon c a s e because the indefinite met r ic of the space and the infinite range conspire t o make the necessary manipulations of questionable validity; but casual- ly conceived limiting p r o c e s s e s do not necessar i ly improve the situation. To resolve the difficulties one requ i res a limiting procedure in which the theory, before the l imit h a s been taken, is con- s is tent , interpretable , and in which al l the math- ematical consequences of receding f rom the limit- ing value of the appropriate parameter have been carefully developed. It i s interesting to speculate what would happen in a m a s s l e s s theory f o r which there may not be a consistently interpretable massive version (for example, some non-Abelian gauge theor ies may be in th i s category). </p><p>Finally it should be noted that the Goldstone bo- son is dynamically detached f rom the other part i - c les and f r o m the spinor cur ren ts in the following sense. When the I a&amp;(g)) s ta tes a r e disallowed a s initial s ta tes , they a r e permanently excluded. However, i f they should be admitted initially they would couple dynamically t o the cur ren ts and interact with the other part ic les in the theory. As h a s been shown i n Ref. 6 , a theory that admits these "quasi-Goldstone" bosons a s initial s ta tes is not probabilistically interpretable . In the case of the Coulomb gauge, the nonlocal equal-time commutation ru les allow al l the d, and the a^, d, t o be U operators , and mat r ix elements of their products can never connect through the Goldstone boson. The commutation ru les and the equations of motion can al l be consistently maintained in a space f rom which the Goldstone boson h a s been excluded.12 </p></li><li><p>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 ... 4303 </p><p>'see, for example, G. S. Guralnik, C. R. Hagen, and T, W. B. Kibble, in Advances i n Particle Phys ics , edited by R. L. Cool. and R. E. Marsh&amp; wile^, New York, 1968), Vol. 11. </p><p>'F. Strocchi, Phys. Rev. D 2, 2334 (1970); 166, 1302 (1968); g, 1479 (1967). </p><p>3 ~ . Bernsteln, Rev. Mod. Phys. 46, 7 (1974). k. Haller and L. F. Landovitz, Phys. Rev. D 2, 1498 </p><p>(1970). 5 ~ . P. Tomczak and I</p></li></ul>