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Covariantly second-quantized string
Volume 142B, number 4
PHYSICS LETTERS
26 July 1984
COVARIANTLY SECOND-QUANTIZED STRING W. SIEGEL ~r Department of Physics, University of California, Berkeley, CA 94720, USA Received 20 April 1984
The interacting relativistic string is second-quantized in a manifestly Lorentz-covariant way. Tree graphs are essentially the same as in previous formalisms, but loops contain ghost strings. The method can also be applied to ordinary field theory, and gives a new method of gauge fixing.
1. Introduction. The relativistic string (see, e.g. refs. [1,2] and references therein) is a prime candidate for a renormalizable theory of quantum gravity. It also provides a model for theories of the graviton as a bound state, since the closed string (which contains the graviton in its massless sector) can be considered as a bound state of the open string in some string theories. However, all known string theories suffer from (at least) two deficiencies: (1) an unphysical dimension of spacetime (2, 10, or 26), and (2) the lack of a manifestly covariant formalism. The former problem (i.e., the lack of four-dimensional models) is probably a consequence of the latter, since a covariant second-quantized formalism is almost a necessity for model building, especially for nonperturbative considerations. (Consider the alternatives of first-quantization or light-cone second-quantization as applied to even the simpler case of quantum chromodynamics.) A manifestly covariant formalism would also be useful for studying hadronic strings which arise from quantum chromodynamics (see e.g., ref. [3] ), as well as providing a covariant off-shell formalism for studying the bound-state mechanism described above. {The apparent inconsistency [4] of such a mechanism can be rephrased as follows: The e n e r g y - m o m e n t u m tensor of the underlying classical theory, which contains no graviton, satisfies the conservation equation ~.T = 0, whereas the quantum theory satisfies the generally-coWork supported in part by National Science Foundation Grant PHY8118547. 276
variant conservation equation 0 = V ' T 4: ~ "T. This can be interpreted as implying an anomaly in the classical conservation law 0" T = 0 (though not necessarily in V ' T = 0, as in ref. [5] ).} In this paper a manifestly covariant second-quantization procedure is provided for the string. The basic procedure, also applicable to ordinary field theory, is simple: Quantize in a method analogous to old-fashioned perturbation theory in the light-cone formalism (see e.g., refs. [6,7] ), but set the light-cone "energy" P - (P_+ = 2-1/2 (Pl +P0)) equal to zero for physical states, replace p+ with a new scalar (but conserved) quantity a, and replace the "transverse" momentum Pi with the covariant momentum plus two anticommuting "ghost" momenta. Symbolically, a, o, ~) , ( P - ' P+ ; Pi) ~ (~,-~o,;p 1
E = 0 = 0 = 0 on physical states.
(1)
Furthermore, the anticommuting momenta are introduced in such a way that the SO(D) Lorentz symmetry in D dimensions, with SO(D - 2) manifest, is enlarged to SO(D + 212) [8], with SO(DI2) (and in particular the subgroup SO(D)) manifest. We will verify the statement [9] that the effect of the two ghost dimensions in loops is to act as negative dimensions: (D + 2) - 2 = D, so the usual D-dimensional covariant Feynman integrals are obtained. In fact, the a coordinate acts as Feynman-like parameters, in an analogous (but more covariant) fashion as p+ in the light-cone formalism [ 6 ] Setting the "energy" to zero corresponds to making the 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 142B, number 4
PHYSICS LETTERS
canonically conjugate "proper-time" coordinate into a Schwinger parameter [10]. More simply, it has the following effect on SO(D + 2[2)-covariant products:
2p_p+ +Pi2 _ + E a + Pa2 + ~ 0 _ + Pa2 '
(2)
where in the last step we have used the condition E = 0 = 0 = 0 on external lines. As a result, all Green functions for physical states are independent of a: We can thus choose a on each external line to be an arbitrary (but nonvanishing) constant. Thus, in ordinary field theory we simply introduce two new commuting spacetime coordinates plus two new anticommuting spacetime coordinates. They cancel in loops, and physical states are defined to be independent of them. For strings, things are a hit subtler: In particular, tire variable a becomes identified with the length of the string. Also, the anticominuting coordinates now contain modes which generate Faddeev Popov ghosts, corresponding to the higher-spin fields generated by the modes of the commuting coordinates.
2. Particles. We first apply our method to scalar field theory, in order to compare with the usual covariant formalism, as well as the light-cone formalism. Before giving the explicit perturbation theory rules, we describe the corresponding rules for the light-cone formalism [6,7]. The basic idea is to work in momentum space for (p+, Pi), but in coordinate space for p _ , in terms of its canonically conjugate variable 7. The Feynman propagator then becomes j
dp_
exp(
ip 7)(2p+p_ +p? + ie) -1
=-iO(p+r)(2lp+l) -1 exp(irp2/2p+),
(3)
where 0(u) = 1 for u/> 0, 0 otherwise. We now draw all graphs so that the vertices are ordered with respect to 7. Thus, we can always consider 7 as being positive. Finally, we Wick rotate the proper time r ~ i7. After introducing the appropriate external line factors (for transforming 7 back to p on external lines), the Feynman rules become : (a) Assign a r to each vertex, and r-order them. (b) Assign (p_, p+, Pi) to each external line, but only (p+, Pi) to internal lines, all directed toward increasing 7; enforce conservation o f ( p + , Pi)at each vertex, and total conservation o f p .
26 July 1984
(c) Each internal line gets a propagator
[O(p+)/2p+] exp(-Tp2i /2P+) for the (p+, Pi)associated with the line and the positive difference r in the proper time between the ends. (d) Each external line gets a factor exp(+Tp_) for the incoming p _ associated with the line and the r associated with the vertex to which it is attached. (e) Vertices are the real constants read from the action. (f) Integrate
fat 0 for each proper-time difference r between consecutive (though not necessarily connected) vertices. (g) Integrate dp+ d D -
2pi
f (21-1 for each loop. Step (f) produces a result identical to performing the ( l o o p ) p _ integration in the usual covariant Feynman graphs. (Dependence on the remaining r cancels due to r-translation invariance, i.e., p_ conservation. Its integration would produce a delta function for p _ conservation.) This step also produces the usual (light-cone) old-fashioned perturbation theory rules in terms of energy denominators. (We differ from the standard form, however, in that our external lines introduced in step (d) are chosen to give off-shell resuits identical to Feynman graphs, whereas the usual results may differ by terms which vanish on shell.) Using the rule (1), we obtain the following new covariant rules: (a) Assign a 7 to each vertex, and r-order them. (b) Assign (a, Pa) to each external line, but (a, Pa, 0 0) to internal lines, all directed toward increasing 7; enforce conservation of (a, Pa, O, O) at each vertex (external 0 = 0 = 0). (c) Each internal line gets a propagator [0(a)/a] e x p [ - r ( p 2 + for the
O0)/a]
(a, Pa, O, O) associated with the line and the 277
Volume 142B, number 4
PHYSICS LETTERS
positive difference r in the proper time between the ends. (d) Each external line gets a factor 1. (e) Vertices are the real constants read from the action. (f) Integrate oo
f dT 0 for each proper-time difference r between consecutive (though not necessarily connected) vertices. (g) Integrate
fag dO for each loop (fdO 1 = 0, fdO 0 = 1 ; 0 2 = 0). (h) Integrate
lfdD+2p
=
26 July 1984
dO dO e x p [ - f ( p 2
f d O p e x p ( - f p 2)
=
+
00)]
(rr/f) D/2 ,
(4)
where f is a function of the Schwinger parameters used to exponentiate the propagators. Thus, the SO(D + 212)covariant set of Feynman rules gives results identical to both the usual SO(D)-covariant Feynman rules and the new SO(DI2)-covariant ("old-fashioned") rules listed above. Therefore, our new rules are equivalent to the usual Feynman rules. Formally, our rules are implemented in a secondquantized action by the replacements
~(x a) --, ~(r, ~, xa, 0, g), [] -+ I-](D+4 ) =
--0~
()/()7 + [] --00 ,
oo
f
--oo
for each loop. (i) Integrate
f dDx -+ f
dr doe dDx
dO dO ,
(5a)
as well as the subsidiary conditions (5b)
f dDp for each loop. Steps ( f ) - ( h ) produce a result identical to the unintegrated Feynman graph expressions (though, as in the light-cone case, several graphs may need to be added to form a single Feynman graph, due to the r-ordering of disconnected vertices). This can be proven as follows: In analogy with the light-cone case, these rules are equivalent to a set of SO(D + 212) covariant rules, where r is replaced by its canonically conjugate variable E. We then have (D + 4)-dimensional Feynman rules, except that we set E = 0 = 0 = 0 on external lines, and loop integrations are fdD+2p dO dO instead of fdDp. However, the latter integrals are identical: By the usual tricks (1/p 2 = f ~ dX exp(-Xp2), symmetric integration, etc.), all momentum-space integrations become gaussian. Equivalence then follows from the identity
on physical states described by a wave functional ft. The method can be generalized to gauge theories: We first choose a light-cone gauge A+ = 0, then implement (1), and perform a similar operation on Lorentz indices: A _ is replaced with a new field A r (or it can be eliminated before applying (1), as usual in lightcone t'ormulations);Ai is replaced with (Aa, Ao, Aft). This procedure produces a Feynman-like gauge-fixing term, as well as Faddeev-Popov-like ghosts A o , A ft. The proof of equivalence is the same as for scalar fields, relating the new rules in a covariant gauge to those for ordinary Feynman graphs in a light-cone gauge via an intermediate SO(D + 212) (but light-cone gauge) theory. However, note that the analog of (5a) introduces a-dependence into the interaction terms. In addition to (5b), we also have A o = Aft = 0 on physical states, as well as the usual transversality conditions on A a.
3. Strings. Compared to the usual Feynman rules, our new Feynman rules differ mainly by introducing Feynman-like parameters automatically. (They also 278
Volume 142B, number 4
PHYSICS LETTERS
may give automatically a type of "background-covariant" gauge fixing when applied to gauge theories, analogous to the approach of ref. [11].) Compared to light-cone old-fashioned perturbation theory, they differ mainly by their manifest covariance. The latter is important for string theories, which do not have covariant Feynman graph rules for loops (noncovariant projection operators must be inserted). The light-cone second-quantized string has the following free action [12], in terms of a real string field qb[r, p+, xi(o)] (o parametrizes the distance along the length of the string): oo
SO= f
oo
dp+ f*xi½q~(-2p+)(a/ar+H)*,
d~" f
_ ~
oo
2~¢e/p+
H=f
do ~ [-27ro~'(8/Sxi)2+(1/2TrO/)(axi/30)2]. (6)
0
(c~' is a mass parameter.) Interactions are given by vertices which are delta functionals in xi(o) (or the canonically conjugate Pi(O)), corresponding to splitting and joining of strings. Applying (1) our covariant generalization is to introduce a string field @[~-, c~, xa(o), 0(o), 0(o)[ (we now use 0 as corresponding to x, rather than the canonical conjugate appearing in (1) and (5)), with free action oo
SO= f
H=f
dad Cl)xaC~)Oc~O~cb(-~)(O/O'r+H)c~,
dr f ¢z¢
--
e¢
do ~ {-27ro2[(6/6xa) 2 + (6/6t7) 6/60]
0
+ (1/27rc~')[(Oxa/3o) 2 + (aS/ao) ao/ao] }.
(7)
The vertices are obtained by substituting 6(x i) -->6(x a)
X 6(0)6(0). The conditions for physical states are now E = 0(o) = 0(o) = 0, as well as the usual Virasoro conditions, involving the non-zero modes of (6/6x) 2 + (3x/3o) 2 and (3x/Do)'(8/Sx). The latter conditions might be gauge-fixing conditions for o-reparametrization invariance [13], for which the non-zero modes of 0(o), 0(o) are the Faddeev Popov ghosts. (The global remnant of this invariance is c~-independence.) Some evidence for the latter assertion is given by the fact that the first-quantized ghosts [3,14] have a lagrangian of the form ~ ++(3/3~ + 3/3o)c+
26 July 1984
+ ~ --(O/Or - 3/30)c_, a first-order form of the lagrangian 0(32/3r 2 - 32/3o2)0; the latter (with the same kinetic operator as x) leads to terms as in (7). The equivalence of the covariant formalism to the light-cone one, at least for S-matrices with only external tachyons (the string's lowest mode) is proven as follows: Consider a Lorentz frame where some components of the external momenta vanish on each external line. Then the corresponding x's contribute internally in the same way as the O's, except for statistics. Specifically, in tree graphs these x's and the O's contribute not at all, and in one-loop graphs they contribute only to the measure. Thus, the fact that there are two extra x's and two extra O's has a totally cancelling effect (i.e., the O's are irrelevant in trees, and at one loop they simply restore the exponent of the wellknown measure factor 1/f(w) [1,2] to be 26 - 2 = 24). The equivalence in arbitrary Lorentz frames follows by the Lorentz invariance (manifest or not) of the two formalisms. Equivalence for arbitrary external lines should follow by analogy to ordinary field theory. However, since here the corresponding SO(D)- and SO(D + 212)covariant formulations do not exist, we compare directly the light-cone and SO(DI 2) formulations. If we restrict external momenta to have k_ = 0, then the analysis of loop integrals is similar to the covariant case in field theory: We again have an identification of the form of (4) but we now equate gaussian integrals for f el) Dp(o) c/) 0(o) @ 0 (o) and fQ) D - 2p(o)" (k_ has been set to zero because E is already zero in the SO(DI2) case.) As above, generalization to nonzero k follows by Lorentz invariance. However, the equivalence should be checked explicitly, since formal Lorentz invariance or freedom from ghosts in string theories is sometimes violated by the subtleties of summation over an infinite number of modes. With this new formalism, truly covariant calculations are possible using either the operator formalism [1,2] or the interacting string picture [1 ]. I thank Korkut Bardakci and Stanley Mandelstam for discussions, and Marty Halpern for bringing ref. [9] to my attention.
R eferen ces [1] S. Mandelstam, Phys. Rep. 13 (1974) 259. 279
Volume 142B, number 4 [2] [3] [4] [5]
PHYSICS LETTERS
J.H. Schwarz, Phys. Rep. 89 (1982) 223. A.M. Polyakov, Phys. Lett. 103B (1981) 207. S. Weinberg and E. Witten, Phys. Lett. 96B (1980) 59. L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1984) 269. [6] S. Weinberg, Phys, Rev. 150 (1966) 1313. [7] J.B. Kogut and D.E. Soper, Phys. Rev. D1 (1970) 2901. [8] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189.
280
[9] [10] [11] [ 12]
26 July 1984
G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. J. Schwinger, Phys. Rev. 82 (1951) 664. G. Parisi and Y. Wu, Sci. Sinica 24 (1981) 483. M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110, 1823. [13] O. Hara, Prog. Theor. Phys. 46 (1971) 1549. [14] M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443.
PHYSICS LETTERS
26 July 1984
COVARIANTLY SECOND-QUANTIZED STRING W. SIEGEL ~r Department of Physics, University of California, Berkeley, CA 94720, USA Received 20 April 1984
The interacting relativistic string is second-quantized in a manifestly Lorentz-covariant way. Tree graphs are essentially the same as in previous formalisms, but loops contain ghost strings. The method can also be applied to ordinary field theory, and gives a new method of gauge fixing.
1. Introduction. The relativistic string (see, e.g. refs. [1,2] and references therein) is a prime candidate for a renormalizable theory of quantum gravity. It also provides a model for theories of the graviton as a bound state, since the closed string (which contains the graviton in its massless sector) can be considered as a bound state of the open string in some string theories. However, all known string theories suffer from (at least) two deficiencies: (1) an unphysical dimension of spacetime (2, 10, or 26), and (2) the lack of a manifestly covariant formalism. The former problem (i.e., the lack of four-dimensional models) is probably a consequence of the latter, since a covariant second-quantized formalism is almost a necessity for model building, especially for nonperturbative considerations. (Consider the alternatives of first-quantization or light-cone second-quantization as applied to even the simpler case of quantum chromodynamics.) A manifestly covariant formalism would also be useful for studying hadronic strings which arise from quantum chromodynamics (see e.g., ref. [3] ), as well as providing a covariant off-shell formalism for studying the bound-state mechanism described above. {The apparent inconsistency [4] of such a mechanism can be rephrased as follows: The e n e r g y - m o m e n t u m tensor of the underlying classical theory, which contains no graviton, satisfies the conservation equation ~.T = 0, whereas the quantum theory satisfies the generally-coWork supported in part by National Science Foundation Grant PHY8118547. 276
variant conservation equation 0 = V ' T 4: ~ "T. This can be interpreted as implying an anomaly in the classical conservation law 0" T = 0 (though not necessarily in V ' T = 0, as in ref. [5] ).} In this paper a manifestly covariant second-quantization procedure is provided for the string. The basic procedure, also applicable to ordinary field theory, is simple: Quantize in a method analogous to old-fashioned perturbation theory in the light-cone formalism (see e.g., refs. [6,7] ), but set the light-cone "energy" P - (P_+ = 2-1/2 (Pl +P0)) equal to zero for physical states, replace p+ with a new scalar (but conserved) quantity a, and replace the "transverse" momentum Pi with the covariant momentum plus two anticommuting "ghost" momenta. Symbolically, a, o, ~) , ( P - ' P+ ; Pi) ~ (~,-~o,;p 1
E = 0 = 0 = 0 on physical states.
(1)
Furthermore, the anticommuting momenta are introduced in such a way that the SO(D) Lorentz symmetry in D dimensions, with SO(D - 2) manifest, is enlarged to SO(D + 212) [8], with SO(DI2) (and in particular the subgroup SO(D)) manifest. We will verify the statement [9] that the effect of the two ghost dimensions in loops is to act as negative dimensions: (D + 2) - 2 = D, so the usual D-dimensional covariant Feynman integrals are obtained. In fact, the a coordinate acts as Feynman-like parameters, in an analogous (but more covariant) fashion as p+ in the light-cone formalism [ 6 ] Setting the "energy" to zero corresponds to making the 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 142B, number 4
PHYSICS LETTERS
canonically conjugate "proper-time" coordinate into a Schwinger parameter [10]. More simply, it has the following effect on SO(D + 2[2)-covariant products:
2p_p+ +Pi2 _ + E a + Pa2 + ~ 0 _ + Pa2 '
(2)
where in the last step we have used the condition E = 0 = 0 = 0 on external lines. As a result, all Green functions for physical states are independent of a: We can thus choose a on each external line to be an arbitrary (but nonvanishing) constant. Thus, in ordinary field theory we simply introduce two new commuting spacetime coordinates plus two new anticommuting spacetime coordinates. They cancel in loops, and physical states are defined to be independent of them. For strings, things are a hit subtler: In particular, tire variable a becomes identified with the length of the string. Also, the anticominuting coordinates now contain modes which generate Faddeev Popov ghosts, corresponding to the higher-spin fields generated by the modes of the commuting coordinates.
2. Particles. We first apply our method to scalar field theory, in order to compare with the usual covariant formalism, as well as the light-cone formalism. Before giving the explicit perturbation theory rules, we describe the corresponding rules for the light-cone formalism [6,7]. The basic idea is to work in momentum space for (p+, Pi), but in coordinate space for p _ , in terms of its canonically conjugate variable 7. The Feynman propagator then becomes j
dp_
exp(
ip 7)(2p+p_ +p? + ie) -1
=-iO(p+r)(2lp+l) -1 exp(irp2/2p+),
(3)
where 0(u) = 1 for u/> 0, 0 otherwise. We now draw all graphs so that the vertices are ordered with respect to 7. Thus, we can always consider 7 as being positive. Finally, we Wick rotate the proper time r ~ i7. After introducing the appropriate external line factors (for transforming 7 back to p on external lines), the Feynman rules become : (a) Assign a r to each vertex, and r-order them. (b) Assign (p_, p+, Pi) to each external line, but only (p+, Pi) to internal lines, all directed toward increasing 7; enforce conservation o f ( p + , Pi)at each vertex, and total conservation o f p .
26 July 1984
(c) Each internal line gets a propagator
[O(p+)/2p+] exp(-Tp2i /2P+) for the (p+, Pi)associated with the line and the positive difference r in the proper time between the ends. (d) Each external line gets a factor exp(+Tp_) for the incoming p _ associated with the line and the r associated with the vertex to which it is attached. (e) Vertices are the real constants read from the action. (f) Integrate
fat 0 for each proper-time difference r between consecutive (though not necessarily connected) vertices. (g) Integrate dp+ d D -
2pi
f (21-1 for each loop. Step (f) produces a result identical to performing the ( l o o p ) p _ integration in the usual covariant Feynman graphs. (Dependence on the remaining r cancels due to r-translation invariance, i.e., p_ conservation. Its integration would produce a delta function for p _ conservation.) This step also produces the usual (light-cone) old-fashioned perturbation theory rules in terms of energy denominators. (We differ from the standard form, however, in that our external lines introduced in step (d) are chosen to give off-shell resuits identical to Feynman graphs, whereas the usual results may differ by terms which vanish on shell.) Using the rule (1), we obtain the following new covariant rules: (a) Assign a 7 to each vertex, and r-order them. (b) Assign (a, Pa) to each external line, but (a, Pa, 0 0) to internal lines, all directed toward increasing 7; enforce conservation of (a, Pa, O, O) at each vertex (external 0 = 0 = 0). (c) Each internal line gets a propagator [0(a)/a] e x p [ - r ( p 2 + for the
O0)/a]
(a, Pa, O, O) associated with the line and the 277
Volume 142B, number 4
PHYSICS LETTERS
positive difference r in the proper time between the ends. (d) Each external line gets a factor 1. (e) Vertices are the real constants read from the action. (f) Integrate oo
f dT 0 for each proper-time difference r between consecutive (though not necessarily connected) vertices. (g) Integrate
fag dO for each loop (fdO 1 = 0, fdO 0 = 1 ; 0 2 = 0). (h) Integrate
lfdD+2p
=
26 July 1984
dO dO e x p [ - f ( p 2
f d O p e x p ( - f p 2)
=
+
00)]
(rr/f) D/2 ,
(4)
where f is a function of the Schwinger parameters used to exponentiate the propagators. Thus, the SO(D + 212)covariant set of Feynman rules gives results identical to both the usual SO(D)-covariant Feynman rules and the new SO(DI2)-covariant ("old-fashioned") rules listed above. Therefore, our new rules are equivalent to the usual Feynman rules. Formally, our rules are implemented in a secondquantized action by the replacements
~(x a) --, ~(r, ~, xa, 0, g), [] -+ I-](D+4 ) =
--0~
()/()7 + [] --00 ,
oo
f
--oo
for each loop. (i) Integrate
f dDx -+ f
dr doe dDx
dO dO ,
(5a)
as well as the subsidiary conditions (5b)
f dDp for each loop. Steps ( f ) - ( h ) produce a result identical to the unintegrated Feynman graph expressions (though, as in the light-cone case, several graphs may need to be added to form a single Feynman graph, due to the r-ordering of disconnected vertices). This can be proven as follows: In analogy with the light-cone case, these rules are equivalent to a set of SO(D + 212) covariant rules, where r is replaced by its canonically conjugate variable E. We then have (D + 4)-dimensional Feynman rules, except that we set E = 0 = 0 = 0 on external lines, and loop integrations are fdD+2p dO dO instead of fdDp. However, the latter integrals are identical: By the usual tricks (1/p 2 = f ~ dX exp(-Xp2), symmetric integration, etc.), all momentum-space integrations become gaussian. Equivalence then follows from the identity
on physical states described by a wave functional ft. The method can be generalized to gauge theories: We first choose a light-cone gauge A+ = 0, then implement (1), and perform a similar operation on Lorentz indices: A _ is replaced with a new field A r (or it can be eliminated before applying (1), as usual in lightcone t'ormulations);Ai is replaced with (Aa, Ao, Aft). This procedure produces a Feynman-like gauge-fixing term, as well as Faddeev-Popov-like ghosts A o , A ft. The proof of equivalence is the same as for scalar fields, relating the new rules in a covariant gauge to those for ordinary Feynman graphs in a light-cone gauge via an intermediate SO(D + 212) (but light-cone gauge) theory. However, note that the analog of (5a) introduces a-dependence into the interaction terms. In addition to (5b), we also have A o = Aft = 0 on physical states, as well as the usual transversality conditions on A a.
3. Strings. Compared to the usual Feynman rules, our new Feynman rules differ mainly by introducing Feynman-like parameters automatically. (They also 278
Volume 142B, number 4
PHYSICS LETTERS
may give automatically a type of "background-covariant" gauge fixing when applied to gauge theories, analogous to the approach of ref. [11].) Compared to light-cone old-fashioned perturbation theory, they differ mainly by their manifest covariance. The latter is important for string theories, which do not have covariant Feynman graph rules for loops (noncovariant projection operators must be inserted). The light-cone second-quantized string has the following free action [12], in terms of a real string field qb[r, p+, xi(o)] (o parametrizes the distance along the length of the string): oo
SO= f
oo
dp+ f*xi½q~(-2p+)(a/ar+H)*,
d~" f
_ ~
oo
2~¢e/p+
H=f
do ~ [-27ro~'(8/Sxi)2+(1/2TrO/)(axi/30)2]. (6)
0
(c~' is a mass parameter.) Interactions are given by vertices which are delta functionals in xi(o) (or the canonically conjugate Pi(O)), corresponding to splitting and joining of strings. Applying (1) our covariant generalization is to introduce a string field @[~-, c~, xa(o), 0(o), 0(o)[ (we now use 0 as corresponding to x, rather than the canonical conjugate appearing in (1) and (5)), with free action oo
SO= f
H=f
dad Cl)xaC~)Oc~O~cb(-~)(O/O'r+H)c~,
dr f ¢z¢
--
e¢
do ~ {-27ro2[(6/6xa) 2 + (6/6t7) 6/60]
0
+ (1/27rc~')[(Oxa/3o) 2 + (aS/ao) ao/ao] }.
(7)
The vertices are obtained by substituting 6(x i) -->6(x a)
X 6(0)6(0). The conditions for physical states are now E = 0(o) = 0(o) = 0, as well as the usual Virasoro conditions, involving the non-zero modes of (6/6x) 2 + (3x/3o) 2 and (3x/Do)'(8/Sx). The latter conditions might be gauge-fixing conditions for o-reparametrization invariance [13], for which the non-zero modes of 0(o), 0(o) are the Faddeev Popov ghosts. (The global remnant of this invariance is c~-independence.) Some evidence for the latter assertion is given by the fact that the first-quantized ghosts [3,14] have a lagrangian of the form ~ ++(3/3~ + 3/3o)c+
26 July 1984
+ ~ --(O/Or - 3/30)c_, a first-order form of the lagrangian 0(32/3r 2 - 32/3o2)0; the latter (with the same kinetic operator as x) leads to terms as in (7). The equivalence of the covariant formalism to the light-cone one, at least for S-matrices with only external tachyons (the string's lowest mode) is proven as follows: Consider a Lorentz frame where some components of the external momenta vanish on each external line. Then the corresponding x's contribute internally in the same way as the O's, except for statistics. Specifically, in tree graphs these x's and the O's contribute not at all, and in one-loop graphs they contribute only to the measure. Thus, the fact that there are two extra x's and two extra O's has a totally cancelling effect (i.e., the O's are irrelevant in trees, and at one loop they simply restore the exponent of the wellknown measure factor 1/f(w) [1,2] to be 26 - 2 = 24). The equivalence in arbitrary Lorentz frames follows by the Lorentz invariance (manifest or not) of the two formalisms. Equivalence for arbitrary external lines should follow by analogy to ordinary field theory. However, since here the corresponding SO(D)- and SO(D + 212)covariant formulations do not exist, we compare directly the light-cone and SO(DI 2) formulations. If we restrict external momenta to have k_ = 0, then the analysis of loop integrals is similar to the covariant case in field theory: We again have an identification of the form of (4) but we now equate gaussian integrals for f el) Dp(o) c/) 0(o) @ 0 (o) and fQ) D - 2p(o)" (k_ has been set to zero because E is already zero in the SO(DI2) case.) As above, generalization to nonzero k follows by Lorentz invariance. However, the equivalence should be checked explicitly, since formal Lorentz invariance or freedom from ghosts in string theories is sometimes violated by the subtleties of summation over an infinite number of modes. With this new formalism, truly covariant calculations are possible using either the operator formalism [1,2] or the interacting string picture [1 ]. I thank Korkut Bardakci and Stanley Mandelstam for discussions, and Marty Halpern for bringing ref. [9] to my attention.
R eferen ces [1] S. Mandelstam, Phys. Rep. 13 (1974) 259. 279
Volume 142B, number 4 [2] [3] [4] [5]
PHYSICS LETTERS
J.H. Schwarz, Phys. Rep. 89 (1982) 223. A.M. Polyakov, Phys. Lett. 103B (1981) 207. S. Weinberg and E. Witten, Phys. Lett. 96B (1980) 59. L. Alvarez-Gaum~ and E. Witten, Nucl. Phys. B234 (1984) 269. [6] S. Weinberg, Phys, Rev. 150 (1966) 1313. [7] J.B. Kogut and D.E. Soper, Phys. Rev. D1 (1970) 2901. [8] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189.
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[9] [10] [11] [ 12]
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G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 744. J. Schwinger, Phys. Rev. 82 (1951) 664. G. Parisi and Y. Wu, Sci. Sinica 24 (1981) 483. M. Kaku and K. Kikkawa, Phys. Rev. D 10 (1974) 1110, 1823. [13] O. Hara, Prog. Theor. Phys. 46 (1971) 1549. [14] M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443.