Second quantization and gauge invariance.

Transcription

1 1 Second quanizaion and gauge invariance. Dan Solomon Rauland-Borg Corporaion Moun Prospec, IL June, 1. Absrac. I is well known ha he single paricle Dirac equaion is gauge invarian. This means ha observable quaniies, such as he curren densiy, are no affeced by a gauge ransformaion. However wha happens when he mehod of second quanizaion is applied o conver a single paricle heory ino a field heory? In his case i will be shown ha he heory is no longer gauge invarian. This will be shown by considering he second quanizaion of a zero mass Dirac field in 1+1 dimensions and examining he change of he curren densiy operaor due o a gauge ransformaion. 1. Inroducion. Quanum field heory is generally assumed o be gauge invarian [1,]. A change in he gauge is a change in he elecromagneic poenial ha doesn produce a change in he elecric or magneic field. Such a change should no produce any change in a physically observably quaniy such as he curren or charge densiy. However i is well known ha when he vacuum curren is calculaed using sandard perurbaion heory he resuls are no gauge invarian. Non-gauge invarian erms appear in he resuls which mus be removed o yield a physically correc resul. For an example of his consider Secion 14. of Greiner e al [] where he soluion for he vacuum polarizaion ensor is given by, where k and sp k k g k k k k g sp k (1.1) are defined in Ref. []. As discussed in [] he firs erm o he righ of he equals sign is gauge invarian however he second erm is no unless

2 sp k is zero. I is shown by Greiner e al ha his is no he case. Therefore his second erm mus be removed in order o ge a physically correc resul. Anoher insance of his ype of problem is provided in Secion 6.4 of Nishijima [] where an expression for he vacuum polarizaion ensor is derived and is shown o include non-gauge invarian erms which mus be dropped form he expression o obain he correc gauge invarian resul. A number of addiional examples and a deailed discussion of his problem is given in Ref [4,5]. The purpose of his paper is o show ha he failure of gauge invariance occurs during he process of second quanizaion of he quanum field. We will do his by examining he effec of a gauge ransformaion on he curren operaor of a second quanized Dirac field in 1+1 dimensions wih zero mass fermions.. Gauge invariance of he single paricle curren densiy. The Dirac equaion for a single massless fermion in 1+1 dimensions in he presence of an exernal elecric poenial,,,, where, and, A z A z, is, 1 z, i H z,,, 1 (.1) H H A z A z (.) H i z 1 where is he Pauli marix wih. 1 The elecric field is given by, A 1 A E z (.) (.4) A gauge ransformaion is a change in he elecric poenial ha does no produce a change in he elecric field. Such a change is given by, A, A A, A z. (.5) 1 1

3 where z, is an arbirary funcion. where z The soluion of (.1) can be easily shown o be,, and can be wrien as, The quaniy W z, where c1 z, and, and, Le, z, W z, z, (.6) is he soluion o he free field Dirac equaion, z, i H z, is given by, ih z, e z W z, (.7). (.8) e ic1 e ic c z saisfy he following differenial equaions, c c c A A1 z 1 1 z, A (.9) (.1) c A A1. (.11) z and z, A1 z (.1) Use his in (.4) o obain E. Therefore (.1) is a gauge ransformaion from zero elecric field. Use (.1) in (.1) and (.11) o obain,,,, c z c z z. (.1) 1 Use his in (.6) along wih (.9) and (.8) o obain, i z, ih The curren densiy for a single fermion is defined by, z, e e z. (.14),,, J z z z. (.15)

4 4 To deermine he impac of a gauge ransformaion on he curren densiy use (.14) ih i z, along wih z, e z e in he above expression for J z, ih ih, o obain, J z e z e z. (.16) We see ha he dependence on he funcion z, does no appear in he above expression. Therefore he gauge ransformaion does no change he curren densiy which proves ha he curren densiy for a single fermion is gauge invarian. This is, of course, a sandard resul.. Second quanizaion. As we have jus shown he curren densiy for a single fermion is gauge invarian. The nex sep is o apply he usual mehods of second quanizaion and o deermine wheher or no he resuling quanum field heory is gauge invarian. We will follow he approach of Ref [6] in which a second quanized formulaion for massless fermions ineracing wih an exernal field in a wo dimensional model was discussed. Le eigenvalues where,, p z, p be he eigenfuncions of he free field Hamilonian wih energy. They saisfy he relaionship,, p z, p, p, p H z z (.1) p 1 1 p e L p 1 p ipz ;, p p (.) and where is he sign of he energy, p is he momenum, and L is he 1 dimensional inegraion volume. We assume periodic boundary condiions so ha he momenum p r L quaniies where r is an ineger. According o he above definiions he, p z are negaive energy saes wih energy, p p and he quaniies, p z are posiive energy saes wih energy, p p. The, p z form an orhonormal basis se and saisfy,

5 5 z where inegraion from L o L, p, p z dz pp (.) is implied. Define he projec operaors P where P projecs ino he negaive energy saes and P projecs ino he posiive energy saes. The projecion operaors are defined by heir acion on a funcion f x : P f, k, k, k Following Ref. [6] define he field operaor, f. (.4) f bp f d P f The operaors bp f and d P f (CAR), b P f, b P g f, P g. (.5) saisfy he canonical anicommuaion relaionships, d P f, d P g f, P g. (.6) wih all oher CARs equal o zero. These operaors ac on a Fock space H. The vacuum sae H is annihilaed by b and d : b P f, d P f. (.7) Consider a uniary operaor V ha acs on he single paricle wave funcion ha V x V x x such. How does his uniary operaor impac he Fock space? If his V saisfies a cerain condiion hen he Fock space will be aced by he operaor V where V acs on he field operaor according o: Vf V f V I is has been shown ha in order for V. (.8) o exis he uniary operaor V mus be Hilber-Schmid [7]. In his case he operaor V is said o be uniary implemenable. 4. Failure of gauge invariance. In Ref [6] a generalized charge operaor Q A is defined, Q A b A b b A d d A b d A d (4.1) n nm m n nm m n nm m m nm n n, m

6 6 where A is a bounded operaor on he Hilber space and where, A, A, A,, A, nm, n, m nm n m, bn b, n, dn d, n If V is a uniary operaor acing on he Hilber space and V quanized operaor hen i is shown in [6] ha, V Q A V QVAV A (4.) is he associaed second (4.) where, A Tr P AP P Tr P AP P (4.4) wih, P. (4.5) V P V A formal expression for he curren densiy operaor smeared over a real-valued funcion f x is given in Ref [8] by, : : J f z z f z dz. (4.6) Using his relaionship as a model and referring o (4.1) we find ha he second quanized curren densiy operaor smeared over f z is given by Q f. The effec of he gauge ransformaion is o ac on he Hilber space wih he uniary operaor V i e. I has been shown ha his operaor is uniary implemenable [8, 9]. Use his in (4.) o show ha he effec of he gauge ransformaion on he curren operaor is, where, i i i i Q f e Q f e Q e fe f. (4.7) i i i i f Tr P fe P e P Tr P fe P e P. (4.8) The gauge ransformaion has aken curren operaor Q f operaor Q f ino he curren densiy. If he curren densiy operaor is gauge invarian hen he quaniy mus disappear from he righ hand side of (4.7).

7 7 i i Consider he erm Qe fe i i. Since e fe f i is eviden ha i i Q e fe Q f. Therefore his par of he expression is independen of. Nex we have o evaluae f. In he Appendix i is shown ha, Use his in (4.7) o obain, 1 d z f dzf z. (4.9) dz 1 d z Q f Q f dzf z. (4.1) dz The las erm in he above expression is, in general, non-zero and is dependen on Therefore he curren densiy operaor is no gauge invarian. 5. Conclusion. z. We have examined he effec of a gauge ransformaion on he curren densiy for zero mass Dirac field in 1+1 dimensions. This problem was moivaed by he fac ha sandard calculaions of he vacuum curren densiy in quanum field heory yield nongauge invarian resuls. I was shown ha for a single fermion he curren densiy is, indeed, gauge invarian. However when he mehod second quanizaion is used o produce a field heory hen he resul is no longer gauge invarian. This, hen, explains why calculaions of he vacuum curren using perurbaion heory do no produce gauge invarian resuls and mus be correced by removing he non-gauge invarian erms. Appendix. In he following we will evaluae f. Use (.4) and (4.5) o obain, This can be rewrien as, Tr P AP P, AV, V. (5.1), p, k, k, p p k Tr P AP P TR A z V z z z V z z z dzdz., k, k, p, p k p (5.) where TR is used o symbolize he race operaion over he spinnor indices only. Use (.) o obain,

8 8 and, 1 L ikzz ikzz, k z, k z e e (5.) k k L ipzz ipzz, p z, p z e e. (5.4) p Nex use A f p and V z 1 1 i z e along wih he above relaionships o obain, where, 1 i z i z (5.5) Tr P AP P dz dzf z e e B z z L i pk zz i pk zz. (5.6) B z z e e k, p Use p n L and k m L o obain, B z z exp i n m z z exp i n m z z n m L L. (5.7) Now ake L and use, in (5.7) o obain, n L L g g d (5.8) n L B z z d d exp i z z exp i z z (5.9) To evaluae his use s and s in he above o obain This yields, s 1 L B z z ds ds exp is z z exp is z z. (5.1) s L B z z sds exp is z z exp is z z. (5.11) This can be furher evaluaed o obain, Use, L d B z z i exp is z z ds dz. (5.1)

9 9 in (5.1) o obain, exp is z z ds z z (5.1) il d B z z z z dz. (5.14) Therefore, i d Tr P AP P dzdzf ze e z z dz This is evaluaed o obain, Similarly i can be shown ha, i z i z. (5.15) z 1 d Tr P AP P dzf z. (5.16) dz Tr P AP P Tr P AP P Use (5.16) and (5.17) along wih (4.4) o obain, which is Eq. (4.9) in he ex.. (5.17) f dzf z 1 d z dz

10 1 References. 1. J. Schwinger. On gauge invariance and vacuum polarizaion. Phys. Rev. 81: (1951).. W. Greiner, B. Muller, and T. Rafelski, Quanum Elecrodynamics of Srong Fields. Springer; Berlin (1985).. K. Nishijima, Fields and Paricles: Field heory and Disperion Relaions, W.A. Benjamin, New York: (1969). 4. D. Solomon, A new look a he problem of gauge invariance in quanum field heory, Phys. Scripa 76: (7). 5. D. Solomon, Gauge invariance and he vacuum sae. Can. J. Phys. 76: (1998). 6. P. Falkenseiner and H. Grosse. Fermions in ineracion wih ime-dependen fields Nucl. Phys. B5: (1988). 7. B. Thaller. The Dirac Equaion. Springer-Verlag; Berlin (1985). 8. A.L. Carey, C.A. Hurs, and D.M. O Brien. Fermion currens in 1+1 dimensions J. Mah. Phys. 4: 1-1 (198). 9. P. Falkenseiner and H. Grosse. Uniary implemenabiliy of gauge ransformaions for he Dirac operaor and he Schwinger erm. J. Mah. Phys. 8: (1987).