Sugawara Model and Goldstone Bosons

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<ul><li><p>1660 R A B I N D E R N . M A D A N </p><p>for v = 1 and zeroth order; conditions is </p><p>and </p><p>are 1=0 and I = 1 projections on Eq. ( 5 3 , and finally the 1=0 projection on Eq. (5.6) is </p><p>The amplitude-phase version of the regular solution of Ecl. (5.23) has been evaluated in (5.17). Equation (5.24) by the Green's-function technique yields E1=O and +00=0 by the normalization demand on $. The series solution of Eq. (5.25) satisfying our boundary </p><p>Thus Ez is computed from (5.12), and one gets the usual result that Ez= - (9/4)e2. </p><p>I n the preceding we have demonstrated that the amplitude-phase technique otiers a direct and a practical method for evaluating the second-order energy shift. To solve the higher-order perturbation equations, we would like to refer to the rapidly convergent variation- iteration approach of Hirschfelder and collaborators.12 </p><p>ACKNOWLEDGMENTS </p><p>My thanks are due to Professor R. Blankenbecler, Professor L. F. Cook, Professor R. L. Gluckstern, and Professor A. R. Swift for critical comnients on this work. </p><p>l2 J. 0. Hirschfelder, W. B. Brown, and S. T. Epstein, Recent Adzances i n Quantum Chemistry (Academic Press Inc., New York, 1964), Vol. 1, p. 255. </p><p>P H Y S I C A I , R E V I E W D V O L U M E 1 , N U M B E R 6 </p><p>Sugawara Model and Goldstone Bosons Y. FREUNDLICH "ED D. L U R I ~ </p><p>Llepartment of Physics, Technion-Israel Institnte of Teclznology, I la i fe , Israel (Received 28 July 1969) </p><p>We find that neutral and S U ( 2 ) [and consequently S U (2 ) X S U ( 2 ) ] Sugawara current theories are co~npletely eq~~ivalent to canonical 1,agrangian field theories of massless scalar fields. A Sugawara current theory is obtained by eliminating all the massive fields in a theory of spontaneous symmetry breakdown and retaining the Goldstone bosons only. The reverse is also true in that all Abelian or SLT(2) Sugawara theories are necessarily equivalent to these canonical representations, so that for these groups "Sugawara physics" reduces simply to "Goldstone boson physics." </p><p>A NUMBER of recent papers1 have dealt with the suggestion first put forward by Sugawara2 that a dynamical theory could be formulated entirely in terms of currents. In this approach, the currents are re- garded as the fundamental dynamical variables and the theory is defined by stipulating the equal-time commu- tation algebra together with the explicit expression of the stress-energy tensor in terms of the currents. The Hilbert space must then be constructed as a representa- tion of the commutation algebra, whereas the time development of the system is determined by the Heisen- berg equations of motion. It was hoped that this theory would constitute an alternative to old-fashioned canoni- cal Lagrangian field theories. However, in this paper we show that the S U ( 2 ) [and consequently the SD'(2) X </p><p>* Present address: Deuartment of Theoretical Phvsics. Hebrew University, Jerusalem, fsrael. </p><p>1 R. F. Dashen and D. H. Sharp, Phys. Rev. 165, 1857 (1968); D. H . Sharp, ibid. 165, 1867 (1968); C. G. Callen, R. F. Dashen, and D. H. Sharp, ibid. 165, 1883 (1968). </p><p>2 H. Sugawara, Phys. Rev. 170, 1659 (1968). </p><p>S U ( 2 ) ] Sugawara current theory is completely eqz~ivalent to a canonical Lagrangian theory of massless scalar particles obtained by eliminating the massive fields in a model of spontaneous s) mmetry breakdown while retaining the Goldstone3 bosons only. In other words, "Sugawara physics" for S U ( 2 ) X S G ( 2 ) is simply "Goldstone boson physics." b , ~ e begin by illustrating our procedure for the case*:of a single neutral Sugawara current where the mathe- matics is simpler. We then proceed to extend our treat- ment to the S U ( 2 ) Sugawara current theory. Consider a model of spontaneous symmetry breakdown in which the symmetry-breaking scalar fields are elementary dy- namical variables. A model of this type, previously con- sidered by H i g g ~ , ~ is defined by the Lagrangian </p><p>8 = - a,sta,s+ V(S+S) , (1) J. Goldstone, Nuovo Cimento 19, 154 (1961); J. Goldstone, </p><p>A. Salam, and S. Weinberg, Phys. Rev. 127, 965 (1962). P. W. Higgs, Phys. Rev. Letters 13, 508 (1964); Phys. Rev. </p><p>145, 1156 (1966). </p></li><li><p>1 S U G A \ I 7 A R A M O D E L A N D G O L D S T O N E B O S O N S 1661 </p><p>which is invariant under the transformation </p><p>together with the polar decomposition </p><p>and the broken-symmetry condition (p)o= v. Rewriting the Lagrangian (1) in terms of the polar variables (3) and the physical field p l=p+v , we have </p><p>Clearly p' represents the field operator of a massive par- ticle and 0 the field operator of a massless scalar particle, i.e., the Goldstone boson. </p><p>We now show how the Sugawara current theor&gt;- arises within the context of this model. The stress-energy tensor may be expressed in terms of the massive field operator and the current associated with the symmetry. From the Lagrangian (I), we obtain the stress-energy tensor </p><p>e,,= [a,sta,s+ a,staps+6,,6] ( 5 ) and the current </p><p>associated with the transformation (2). In terms of the polar variables (3 ) , these become </p><p>Furthermore, the canonical commutation relations imply the current-current commutation rules </p><p>At the present stage our model is a "mixed" Sugawara theory in the sense that the field operator p2 appears in place of the usual Sugawara constant and the O,, con- tains additional terms relating to the massive field p. T o reduce our model to a '(pure" Sugawara theory it suffices to eliminate the massive field via the constraint p2= v2 =C, where C is the Sugawara constant. The stress- energy tensor (7) thus goes over into </p><p>obtained from (4) by retaining the Goldstone boson only. Moreover, the Sugawara current (8) is associated with the symmetry transformation 0 -+ e+a. </p><p>An SU(2) Sugawara current theory can be derived from a theory of spontaneous symmetry breakdown in an analogous manner. Starting with a complex Lorentz scalar and isospinor theory described by the Lagrangian </p><p>which is invariant under the U(2) gauge transformation5 </p><p>we mabe the polar decolnposition </p><p>where 2 is a constant complex unit isospinor, and impose the broken-symmetry condition (p)o= 7. The Lagran- gian then takes the form </p><p>where 0,. T= -ie-a. ~a,ei@ T (18) </p><p>and p=p'+?I. The Oa fields correspond to the Goldstone bosons. The SU(2) currents associated with the trans- formation (15) are </p><p>which in terms of the polar variables becomes </p><p>The stress-energy tensor </p><p>may then be written as </p><p>while the canonical commutation relations imply the current-current commutation rules </p><p>and the relevant commutation relation (10) becomes [ j~"(~ , t ) , j jb(~ ' , t ) ]=oO (214 [jo(x,t), j,(x,'t)]= --iCdi6(2)(x-x'). (12) </p><p>We eliminate the massive field via the constraint ap2 These are the defining equations of the Sugawara cur- =iv2=C, Equations (20) and (21) are then the defining rent theory for a single neutral current. We have arrived equations of the Sugawara current theory. T~ sum up, a t a canonical representation of the neutral Sugawara our canonical representation of the ~ ~ ( 2 1 sugawara theory based on the Lagrangian </p><p>a! runs from 0 to 3, r0 is the unit 2x2 matrix, and are the C= - i~d,08,0 (13) usual isotopic-spin matrices. </p></li><li><p>1662 Y . F R E U N D L I C H A N D D . L U R I E 1 </p><p>theory consists of interacting massless bosons and is Bardalrci and Halpernqave shown that this is the most defined by the Lagrangian6 general solution to Eq. (27a). Moreover, Eq. (27b) is the </p><p>Lagrangian equation of motion for the massless scalar g= -2CB i0 i P P (22) fields. To establish complete equivalence it must still be </p><p>obtained from (17) by retaining the Goldstone bosons shown that the current algebra implies the canonical </p><p>only. ~h~ sugaMirara currents (19) are associated with commutation relations for the e5 fields when the j,a are the symmetry transformation given by Eq. (28). This is done as f ~ l l o w s . ~ From Eqs. </p><p>(21) and (28) one derives the commutation relations ei8. r -+ ei$#, ~ ~ i &amp; T 0 3 ) [joa(X,t),eiB(~',t).~ </p><p>The Lagrangian equations of motion are 1 </p><p>= -3T~ei%- Ta(2)(x -x') d,jPa= 0 . aei% . r </p><p>The crucial point is that any neutral or SU(2) =3iD,b@bc-l-- 8 ( 2 ) ( ~ -xl) aec , (29) Sugawara current theory is necessarily equivalent to </p><p>a canonical representation obtained in this manner. T o where Dab and O,b are defined by see this for the single neutral-current theory, we apply the Heisenberg equations of motion to derive the e - ~ o . ~ ~e u i e .7 - - -D ,a (o )~~ , equations </p><p>(30) </p><p>a,j,-d,j,=O, (24a) a -;e-in.~-eir9.~= O U n b . r, </p><p>d,j,= 0 dBU (31) </p><p>(24b) </p><p>by stra'ghtforward application of the commutation rela- Use of (30) and (31) allows us to write the fourth compo- tions. From (24a) we infer that j, can be written in the nents of the currents (28) as form joa= -+DabObc-l~~, j,= ca,e, (32) </p><p>a'= 4COab00b. and from (24b) it follows that (33) </p><p>The T" correspond to the canonically conjugate mo- o e = o . (2Sb) menta of the 8. as derived from the Lagrangian (22). </p><p>The identification (25a) reduces the first of the Sugawara Substituting (32) into (29) yields equations of motion to an identity, while the current [ ~ ~ ( ~ , t ) , 0 ~ ( x ' , t ) ] = - i6ab6(Z)(~-~ ' ) . (34) conservation law (24b) is equivalent to the Lagrangian equation of motion (25b). Finally, the current-current The "seemingly" noncanonical representation of the commutation relation (12) implies that Sugawara theory obtained by Bardakci, Frishman, and </p><p>HalpernS would appear to contradict the above equiva- [Co(x,t),O(xl,t)]= -iS(")(x-X') . (26) lence theorems. This representation is based on a par- </p><p>a similar lmanner we prove that any s ~ ' ( 2 ) tic~llar zero-mass limit of the massive Yang-Mills SLlgawara current theory ln~lst necessarilJ- be equivalent theory. However, i t can be shown7 that in this limit the to the canonical model previously obtained. Consider Yang-Mills theory actually reduces to the canonical first the equations of motion for the S1;(2) theory: theory of nlassless scalar fields which we have considered </p><p>in this paper. The reason for this reduction is that the d,j,. T- dvj,. T = ( i / 2~) [ jp . T , j v . T I , (27a) massive yang-Mills fields can be decomposed into a set </p><p>of transverse vector fields and massless scalar field^.^ 8, jP5= 0 . (27b) The limit considered by Bardakci ef ol. consists precisely </p><p>Equation (27a) is reduced to an identity by setting in retaining the longitudinal modes while discarding the transverse ones. </p><p>-- j,= 2Cieisd,e-in. (28) Y. Freundlich, Phys. Rev. (to be published). 6This canonical representation of the Sugawara theory wasalso K. Bardakci, Y. Prishman, and 91. B. Halpern, Phys. Rev. </p><p>found by Bardalrci and I-Ialpern using completely different 170, 1353 (1968). methods: K. Bardakci and 11. B. Halpern, Phys. Rev. 172, 1541 R. Finkelstein and L. Staunton, Ann. Phys. (N. Y . ) 54, 97 (1968). (1969). </p></li></ul>