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Gauge independence and unitarity in the Weinberg unified theory
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
G A U G E I N D E P E N D E N C E AND U N I T A R I T Y IN T H E W E I N B E R G U N I F I E D T H E O R Y * J,P. HSU and E.C.G. SUDARSHAN Center for Particle Theory, The University of Texas at Austin, Austin, Texas 78712. USA Received 10 June 1974 We formulate the Weinbergtheory with the help of the Lagrangianmultiplier formalism.A new fictitious Lag~angianis obtained, which differs from the usual one. We showby "explicitcalculationthat it restores unitarity and gauge independenceof the theory.
During the past two years methods have been developed to deal with the field theory of complicated models of self-interacting vector mesons like the Yang-Mills model and the Weinberg model [1 ]. The non-Abelian gauge invariance of such theories involves a Faddeev--Popov Jacobian which may be associated with a suitable fictitious particle Lagrangian. It has been believed generally that these methods obtain a theory which is renormalizable and unitary, for all values of the three gauge parameters a, ~, ~7which occur in the gauge-fixing terms; there are also formal proofs of this satisfying result. We formulate the Weinberg theory within the Lagrangian multiplier formalism and derive equations satisfied by the unphysical components of the vector fields. We isolate the contributions from the interactions of these components in terms of a fictitious Lagrangian. The fictitious Lagrangian so constructed from the source of the unphysical fields differs from the usual fictitious Lagrangian [2]. It aims at removing the extra amplitudes contributed by unphysical particles, so that the theory is unitary. Explicit calculations are carried out to the twoloop level to verify that it does restore unitarity and gauge independence of the theory. Let us consider the Lagrangian [2, 1] ~W ='QWl +£?W2'
(1)
.l~Wl = ~ [OuI4r~v- av ~ + i e ( ~ A v - W~vA u) -iGcos20(W+Zv - W~vZ )[2- ~[3.Av-avA.-ie(W+Wv--W~vW-)[2
- l l a u Z v - a Z u + i G cos20(W~+Wv- - W~vWu-)12 + [0uS++ (i/v~)G cos0 W+SO+i(-eAu+~G cos20Z )S+ +iMWu+t2 + [~SO - (i/2)GZ SO + (I/x/~)G cos0 Wu'-S + - (i/V~)Msec0Z 12- ~ [S+S- +(SO+ v/V/2)(ff 0 + o / V ~ ) - 02/2] 2,
(2)
where e = - G cos O sin 0; we shall eventually choose ~ = ~ (cf. discussions below). The neutral vector particle Z has a mass M z = M sec 0. Leptons can be introduced in (1) without affecting our fictitious Lagrangian at all (cf. eq. (9) below). Introducing the Lagrange multiplier fields X± , XA, and XZ, (1) can be written as
* Work supported in part by the U.S. Atomic Energy Commission.
349
Volume 51B, number 4
PHYSICS LETTERS
+ ×-(OuW++--~/ S + ) + × + ( b u W - - ~ S - ) }
19 August 1974
+M2{2 X2 + 1~ Xi'+ ~×+X-).
(4)
The Lagrangian (4) leads to field equations for W~, Z , etc and constraint equations for X± , ×z and Xh- For example, we have - Du[DuAv-O Au-ie(Wu+Wu-- W~W-)] + M3,XA + ieW+UW~z -ieW-UW~m - i e ( S - E ~ - S + E v - ) = O, Wrvv= ~ Wry- ~vW+ + ie(W+A v - W~vA u ) - iG cos20(W+Z - WvvZ ) ,
E +" = auS+ + (i/vr2)G cos 0 Wu+S0+ i(-eA u + ~G cos20Z)S + ÷ iMWu+.
(5)
The divergence of the field equations together with the relations from (4) gives rises to the equations for the Lagrange multiplier fields: GMz GM + ([3+M2/rl)Xz+--~ t~Xz - - ~ ( x - S + + x S - ) + iG cos20 Ou(W+X- - W-X +) : O,
l S O : ~--~(~ +ix),
E]XA-iebU(W+X - - W-X +) = O,
(6) (7)
(I3+M2 /~)X +- ieA~uX + + iG cos20 ZU~uX+ MG cos + O 2~ X+(~ - ix)
G cos OM z -2f1 S+Xz + ieW+OuXA --iG cos20 W+aUXz = 0,
(8)
and the conjugate equation of (8). These equations are derived by tedious but straightforward calculations. In these derivations, one sees the fantastic and complete cancellation of the non-renormalizable source terms [3,4]. This renders the theory renorrnalizable as we shall see below. The source terms in these equations determine the interactions of the unphysical particles in the Lagrangian [3,4]. The unphysical particles in the intermediate states of physical processes contribute extra unwanted amplitudes and upset unitarity and gauge independence of the theory. Using the expressions (6)-(8), these extra amplitudes could be expressed in term of a functional deterrninant by using path integral. They must be removed from the theory in order to restore unitarity and gauge independence of the theory [4, 5]. Another way to take care of the functional determinant is to introduce four fictitious complex scalar-fermion fields (D +, De), (/9-, D - ) , (DA, DA), and (D Z, DZ)" We need complex fictitious" fields because (1) involves two unphysical degrees of freedom corresponding to the photon, the neutral vector meson Z, W-, and W+ wRh masses 0, MZ/#I/2, M/~ 1/2 and M/~ 1/2 respectively [5]:. {We may remark that if au W*. . . m (3) and (4) xs. replaced. the gauge mvarlant expressions (~ / 2 . - +- " wAU) W-+ then eq (7) becomes rg×. = 0. Because u of this free field equation obeyed by XA we need not introduce the fictitious fields D A, D A .) The fictitious Lagrangian for these fictitious fermion is ./2ff = --D+([3 +tM2/~)D+ + iefi +(A OUD+ - W+O"DA) +GM S+~+D_ _ iG cos2O D + ( Z aUD+ - W+D~Dz ) 2~7 L _ MG cos 0 ~+D+(qj 2~ - ix) - D-([::] +M2 /~)D - - ieD- (A DUD- - Wu- 3UDh ) + ~ S - D - D z + iGcos 2 0 D - ( Z ua uD - - W - O uDZ) -
2
GM
-
-Dz(I3+M~/B)Dz+-~Dz(D-S
+
MG cos 0 u + + - - - 2 f f - ~ _D- - D - ( 4 + i x ) - - -DA [3DA +eD i--A a ( W ~ D - - W~-D ) +
GMz
-
+D S - ) - - - ~ - - ~ D z D z ' i G c o s
2
-
u
+
ODza ( W ~ D - - W ~ D
+
),
(9)
which could, in fact, be inferred from the eqs. (6), (7), and (8) for the Lagrange multiplier fields. Of course, there 350
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
is an extra factor (-1) for each dosed loop of the fictitious field. The theory is completely defined by (1) and (9) [3,5]. It is renormalizable by standard power counting. If one compares (9) with the usual fictitious Lagrangian [2] (i.e., the Faddeev-Popov fictitious Lagrangian) in the Weinberg theory, one sees that while there is an overall similarity between the two and though some terms are exactly the same, yet many terms are different. These differences show up in the calculations of matrix elements. For example, (9) involves the term
ieDAO"(W+D--Wu-D+)
(10)
as the source for D A field. The corresponding term in the usual fictitious Lagrangian [2] is iG sinO ~AOU(W~-¢+ - Wu~b + - ),
Gsin0 = -e/cos0,
(1!)
where ~+, ~ - , and ~A are complex fictitious scalar-ferrnions. One may simply check unitarity in the one-loop self-energy diagram of the spin 1 boson W+ to see ff our fictitious Lagrangian is correct. It is sufficient to calculate the contribution of two unphysical particles with masses 0 and k/]~l/2: W+(p) -> W+s(p') +'l'(q):
-e2eet3{-MZg~tJ-2q~qtJ+M2(1 -a)q~q~/q2},
(12)
W+(p) ~ S+(p') +7(q):
+e2ee~(-M2g°C+M2(1 -ot)qaq#/q2},
(13)
W+(p) ~D+(p ') +DA(q):
-e2ece#qaq ;~,
(14:
W+(p)~D-(p')+DA(q):
-e2 e, e#qaq~,
(15)
where W+ s is the negative metric spin 0 particle associated with the 4-vector field W/a+ and eu is the polarization of the physical (spin 1)"vector boson W+. The overall minus sign in (12) is due to the negative metric of W2 and those in (14-(15) are due to the factor (-1) associated with the fictitious loop. Ilaese amplitudes (12)-(15) must cancel among themselves as required by unitarity because they cannot be cancelled by anything else. This is indeed true for any a and ~. The usual fictitious Lagrangian leads to the same result. In order to verify unitarity at the two-loop level, we show that there is no net contribution due to two unphysical particles (with masses M[~1/2 and Mz[rl1/2) and one physical particle (with mass 3/) in the intermediate states of the two-loop self-energy diagrams for the neutral vector boson Z with momenta p~ and polarization e~: intermediate states
amplitudes-squared
[¥+(P2)S-(P3)X(P4)
+ [a+b cos20 - c cos20 + d cos 20] Z
W+(P2)S-(P3)Zs(P4 )
- [a+bcos20 sin20 - d sin40] 2
W+(p2) B/s-(p3)Zs(P4 )
+ [a-b cos40 +c[~-d(sin40 + cos20)] 2
W+(p2) Ws-(P3)X(P4) W+(P2)D- (P3)Dz(P4)
+c/rl_dsin20] 2 -[-beos30+(c/rl)cosO-dcos30]2[-bcos30+ccosO-dcos30]2
W+(P2)D+(P3)Dz (p4)
Af
(16)
_ [a_ccos20
--Af
where we have set ~ = 1 for simplicity, ~ is arbitrary and a = - G 2 cos 0(e 1 "p4)(e2
"p3)[[(Pl -P4 )2 -
m2 ] - G2c°s 0 sin20 eI "e2[2,
m 2 = 2Xo2,
b = G 2 cos 0 [ - 2 e I "P2 e2 "P4 -- 2el "P4 e2 "Pl + el "e2(Pl +P2)'P4 ] / ( - 2Pl "P2 +M2)' c --- G2 cos0
sin20~zel.e2[[E(-EPl.P2+M2)],
d-GEcosO(el.P3)(e2.P4)[(-EPl.P3+M2z). 351
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
Z s is the negative metric scalar particle associated with the 4-vector field Z~ and e~ is the polarization vector associated with W+(P2). The amplitudes-squared in (16) must cancel among themselves because they cannot be cancelled by anything else in the theory. A corresponding cancellation must occur for intermediate states such as W-S+×s, W-S+Zs, etc. Indeed, the sum of the amplitudes-squared in (16) is zero for any 77. When one calculates the contribution for the intermediate states W+~-ffz and W+ff+¢ z by using the usual fictitious Lagrangian [2], one obtains the same results. Based on the fictitious Lagrangian (9), we verify that the contributions due to two unphysical particles with massesM/~l/2 andMz/rll/2 in the one-loop self-energy diagram of W- vanish for any ~ and 77.We have also verified unitarity for ~ -~ ~ and arbitrary a by computing the net contribution due to three unphysical particles (with the same massM/~1/2) in the intermediate states of the two-loop self-energy diagrams for the vector boson W+. We note that there is a gauge symmetry embedded in the Lagrangian. It makes the fantastic and complete cancellation of the non.renormalizable source terms possible in the equations for the Lagrangian multiplier fields, just like the cancellation which happens in the exact gauge invariant Lagrangian (without mass terms). This indicates that the dynarriics of interactions interlocked with the exact gauge symmetry is essentially unaltered by the mass terms. In the usual approach [6, 7], the fictitious Lagrangian is obtained by considering the change of gauge conditions under gauge transformations. It alms at compensating the changes of the gauges during the evolution of a physical system, so that the theory remains gauge invariant. On the other hand, the present approach aims at removing the extra unwanted amplitudes contributed by unphysical particles. In view of the above results, we believe that these two different approaches are equivalent although they may lead to apparently different fictitious Lagrangians. In simpler cases, the fictitious Lagrangians obtained by these two different approaches are identical [3, 5]. Either approach restores both unitarity and gauge independence of the theory. This indicates that the dynamical origins for violations of unitarity and gauge independence are the same. We are happy to thank Professor B.W. Lee for his comments on an early version of this paper.
References [1] [2] [3] [4] [5] [6] [7]
352
S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; 26 (1972) 1688. K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. 10D (1972) 2923. See,for example, J.P. Hsu and E.C.G. Sudarshan, Phys. Rev. D9 (1974) 1678. J.P. Hsuand E.C.G. Sudarshan, CPT Preprints 218 and 225 (1974). J.P. Hsu, Phys. Rev. D9 (1974) 1113. G. 't Hooft, Nuclear Phys. B33 (1971) 173. B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3137; D5 (1972) 3155.
PHYSICS LETTERS
19 August 1974
G A U G E I N D E P E N D E N C E AND U N I T A R I T Y IN T H E W E I N B E R G U N I F I E D T H E O R Y * J,P. HSU and E.C.G. SUDARSHAN Center for Particle Theory, The University of Texas at Austin, Austin, Texas 78712. USA Received 10 June 1974 We formulate the Weinbergtheory with the help of the Lagrangianmultiplier formalism.A new fictitious Lag~angianis obtained, which differs from the usual one. We showby "explicitcalculationthat it restores unitarity and gauge independenceof the theory.
During the past two years methods have been developed to deal with the field theory of complicated models of self-interacting vector mesons like the Yang-Mills model and the Weinberg model [1 ]. The non-Abelian gauge invariance of such theories involves a Faddeev--Popov Jacobian which may be associated with a suitable fictitious particle Lagrangian. It has been believed generally that these methods obtain a theory which is renormalizable and unitary, for all values of the three gauge parameters a, ~, ~7which occur in the gauge-fixing terms; there are also formal proofs of this satisfying result. We formulate the Weinberg theory within the Lagrangian multiplier formalism and derive equations satisfied by the unphysical components of the vector fields. We isolate the contributions from the interactions of these components in terms of a fictitious Lagrangian. The fictitious Lagrangian so constructed from the source of the unphysical fields differs from the usual fictitious Lagrangian [2]. It aims at removing the extra amplitudes contributed by unphysical particles, so that the theory is unitary. Explicit calculations are carried out to the twoloop level to verify that it does restore unitarity and gauge independence of the theory. Let us consider the Lagrangian [2, 1] ~W ='QWl +£?W2'
(1)
.l~Wl = ~ [OuI4r~v- av ~ + i e ( ~ A v - W~vA u) -iGcos20(W+Zv - W~vZ )[2- ~[3.Av-avA.-ie(W+Wv--W~vW-)[2
- l l a u Z v - a Z u + i G cos20(W~+Wv- - W~vWu-)12 + [0uS++ (i/v~)G cos0 W+SO+i(-eAu+~G cos20Z )S+ +iMWu+t2 + [~SO - (i/2)GZ SO + (I/x/~)G cos0 Wu'-S + - (i/V~)Msec0Z 12- ~ [S+S- +(SO+ v/V/2)(ff 0 + o / V ~ ) - 02/2] 2,
(2)
where e = - G cos O sin 0; we shall eventually choose ~ = ~ (cf. discussions below). The neutral vector particle Z has a mass M z = M sec 0. Leptons can be introduced in (1) without affecting our fictitious Lagrangian at all (cf. eq. (9) below). Introducing the Lagrange multiplier fields X± , XA, and XZ, (1) can be written as
* Work supported in part by the U.S. Atomic Energy Commission.
349
Volume 51B, number 4
PHYSICS LETTERS
+ ×-(OuW++--~/ S + ) + × + ( b u W - - ~ S - ) }
19 August 1974
+M2{2 X2 + 1~ Xi'+ ~×+X-).
(4)
The Lagrangian (4) leads to field equations for W~, Z , etc and constraint equations for X± , ×z and Xh- For example, we have - Du[DuAv-O Au-ie(Wu+Wu-- W~W-)] + M3,XA + ieW+UW~z -ieW-UW~m - i e ( S - E ~ - S + E v - ) = O, Wrvv= ~ Wry- ~vW+ + ie(W+A v - W~vA u ) - iG cos20(W+Z - WvvZ ) ,
E +" = auS+ + (i/vr2)G cos 0 Wu+S0+ i(-eA u + ~G cos20Z)S + ÷ iMWu+.
(5)
The divergence of the field equations together with the relations from (4) gives rises to the equations for the Lagrange multiplier fields: GMz GM + ([3+M2/rl)Xz+--~ t~Xz - - ~ ( x - S + + x S - ) + iG cos20 Ou(W+X- - W-X +) : O,
l S O : ~--~(~ +ix),
E]XA-iebU(W+X - - W-X +) = O,
(6) (7)
(I3+M2 /~)X +- ieA~uX + + iG cos20 ZU~uX+ MG cos + O 2~ X+(~ - ix)
G cos OM z -2f1 S+Xz + ieW+OuXA --iG cos20 W+aUXz = 0,
(8)
and the conjugate equation of (8). These equations are derived by tedious but straightforward calculations. In these derivations, one sees the fantastic and complete cancellation of the non-renormalizable source terms [3,4]. This renders the theory renorrnalizable as we shall see below. The source terms in these equations determine the interactions of the unphysical particles in the Lagrangian [3,4]. The unphysical particles in the intermediate states of physical processes contribute extra unwanted amplitudes and upset unitarity and gauge independence of the theory. Using the expressions (6)-(8), these extra amplitudes could be expressed in term of a functional deterrninant by using path integral. They must be removed from the theory in order to restore unitarity and gauge independence of the theory [4, 5]. Another way to take care of the functional determinant is to introduce four fictitious complex scalar-fermion fields (D +, De), (/9-, D - ) , (DA, DA), and (D Z, DZ)" We need complex fictitious" fields because (1) involves two unphysical degrees of freedom corresponding to the photon, the neutral vector meson Z, W-, and W+ wRh masses 0, MZ/#I/2, M/~ 1/2 and M/~ 1/2 respectively [5]:. {We may remark that if au W*. . . m (3) and (4) xs. replaced. the gauge mvarlant expressions (~ / 2 . - +- " wAU) W-+ then eq (7) becomes rg×. = 0. Because u of this free field equation obeyed by XA we need not introduce the fictitious fields D A, D A .) The fictitious Lagrangian for these fictitious fermion is ./2ff = --D+([3 +tM2/~)D+ + iefi +(A OUD+ - W+O"DA) +GM S+~+D_ _ iG cos2O D + ( Z aUD+ - W+D~Dz ) 2~7 L _ MG cos 0 ~+D+(qj 2~ - ix) - D-([::] +M2 /~)D - - ieD- (A DUD- - Wu- 3UDh ) + ~ S - D - D z + iGcos 2 0 D - ( Z ua uD - - W - O uDZ) -
2
GM
-
-Dz(I3+M~/B)Dz+-~Dz(D-S
+
MG cos 0 u + + - - - 2 f f - ~ _D- - D - ( 4 + i x ) - - -DA [3DA +eD i--A a ( W ~ D - - W~-D ) +
GMz
-
+D S - ) - - - ~ - - ~ D z D z ' i G c o s
2
-
u
+
ODza ( W ~ D - - W ~ D
+
),
(9)
which could, in fact, be inferred from the eqs. (6), (7), and (8) for the Lagrange multiplier fields. Of course, there 350
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
is an extra factor (-1) for each dosed loop of the fictitious field. The theory is completely defined by (1) and (9) [3,5]. It is renormalizable by standard power counting. If one compares (9) with the usual fictitious Lagrangian [2] (i.e., the Faddeev-Popov fictitious Lagrangian) in the Weinberg theory, one sees that while there is an overall similarity between the two and though some terms are exactly the same, yet many terms are different. These differences show up in the calculations of matrix elements. For example, (9) involves the term
ieDAO"(W+D--Wu-D+)
(10)
as the source for D A field. The corresponding term in the usual fictitious Lagrangian [2] is iG sinO ~AOU(W~-¢+ - Wu~b + - ),
Gsin0 = -e/cos0,
(1!)
where ~+, ~ - , and ~A are complex fictitious scalar-ferrnions. One may simply check unitarity in the one-loop self-energy diagram of the spin 1 boson W+ to see ff our fictitious Lagrangian is correct. It is sufficient to calculate the contribution of two unphysical particles with masses 0 and k/]~l/2: W+(p) -> W+s(p') +'l'(q):
-e2eet3{-MZg~tJ-2q~qtJ+M2(1 -a)q~q~/q2},
(12)
W+(p) ~ S+(p') +7(q):
+e2ee~(-M2g°C+M2(1 -ot)qaq#/q2},
(13)
W+(p) ~D+(p ') +DA(q):
-e2ece#qaq ;~,
(14:
W+(p)~D-(p')+DA(q):
-e2 e, e#qaq~,
(15)
where W+ s is the negative metric spin 0 particle associated with the 4-vector field W/a+ and eu is the polarization of the physical (spin 1)"vector boson W+. The overall minus sign in (12) is due to the negative metric of W2 and those in (14-(15) are due to the factor (-1) associated with the fictitious loop. Ilaese amplitudes (12)-(15) must cancel among themselves as required by unitarity because they cannot be cancelled by anything else. This is indeed true for any a and ~. The usual fictitious Lagrangian leads to the same result. In order to verify unitarity at the two-loop level, we show that there is no net contribution due to two unphysical particles (with masses M[~1/2 and Mz[rl1/2) and one physical particle (with mass 3/) in the intermediate states of the two-loop self-energy diagrams for the neutral vector boson Z with momenta p~ and polarization e~: intermediate states
amplitudes-squared
[¥+(P2)S-(P3)X(P4)
+ [a+b cos20 - c cos20 + d cos 20] Z
W+(P2)S-(P3)Zs(P4 )
- [a+bcos20 sin20 - d sin40] 2
W+(p2) B/s-(p3)Zs(P4 )
+ [a-b cos40 +c[~-d(sin40 + cos20)] 2
W+(p2) Ws-(P3)X(P4) W+(P2)D- (P3)Dz(P4)
+c/rl_dsin20] 2 -[-beos30+(c/rl)cosO-dcos30]2[-bcos30+ccosO-dcos30]2
W+(P2)D+(P3)Dz (p4)
Af
(16)
_ [a_ccos20
--Af
where we have set ~ = 1 for simplicity, ~ is arbitrary and a = - G 2 cos 0(e 1 "p4)(e2
"p3)[[(Pl -P4 )2 -
m2 ] - G2c°s 0 sin20 eI "e2[2,
m 2 = 2Xo2,
b = G 2 cos 0 [ - 2 e I "P2 e2 "P4 -- 2el "P4 e2 "Pl + el "e2(Pl +P2)'P4 ] / ( - 2Pl "P2 +M2)' c --- G2 cos0
sin20~zel.e2[[E(-EPl.P2+M2)],
d-GEcosO(el.P3)(e2.P4)[(-EPl.P3+M2z). 351
Volume 51B, number 4
PHYSICS LETTERS
19 August 1974
Z s is the negative metric scalar particle associated with the 4-vector field Z~ and e~ is the polarization vector associated with W+(P2). The amplitudes-squared in (16) must cancel among themselves because they cannot be cancelled by anything else in the theory. A corresponding cancellation must occur for intermediate states such as W-S+×s, W-S+Zs, etc. Indeed, the sum of the amplitudes-squared in (16) is zero for any 77. When one calculates the contribution for the intermediate states W+~-ffz and W+ff+¢ z by using the usual fictitious Lagrangian [2], one obtains the same results. Based on the fictitious Lagrangian (9), we verify that the contributions due to two unphysical particles with massesM/~l/2 andMz/rll/2 in the one-loop self-energy diagram of W- vanish for any ~ and 77.We have also verified unitarity for ~ -~ ~ and arbitrary a by computing the net contribution due to three unphysical particles (with the same massM/~1/2) in the intermediate states of the two-loop self-energy diagrams for the vector boson W+. We note that there is a gauge symmetry embedded in the Lagrangian. It makes the fantastic and complete cancellation of the non.renormalizable source terms possible in the equations for the Lagrangian multiplier fields, just like the cancellation which happens in the exact gauge invariant Lagrangian (without mass terms). This indicates that the dynarriics of interactions interlocked with the exact gauge symmetry is essentially unaltered by the mass terms. In the usual approach [6, 7], the fictitious Lagrangian is obtained by considering the change of gauge conditions under gauge transformations. It alms at compensating the changes of the gauges during the evolution of a physical system, so that the theory remains gauge invariant. On the other hand, the present approach aims at removing the extra unwanted amplitudes contributed by unphysical particles. In view of the above results, we believe that these two different approaches are equivalent although they may lead to apparently different fictitious Lagrangians. In simpler cases, the fictitious Lagrangians obtained by these two different approaches are identical [3, 5]. Either approach restores both unitarity and gauge independence of the theory. This indicates that the dynamical origins for violations of unitarity and gauge independence are the same. We are happy to thank Professor B.W. Lee for his comments on an early version of this paper.
References [1] [2] [3] [4] [5] [6] [7]
352
S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; 26 (1972) 1688. K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. 10D (1972) 2923. See,for example, J.P. Hsu and E.C.G. Sudarshan, Phys. Rev. D9 (1974) 1678. J.P. Hsuand E.C.G. Sudarshan, CPT Preprints 218 and 225 (1974). J.P. Hsu, Phys. Rev. D9 (1974) 1113. G. 't Hooft, Nuclear Phys. B33 (1971) 173. B.W. Lee and J. Zinn-Justin, Phys. Rev. D5 (1972) 3137; D5 (1972) 3155.