(d) Give each external line a factor
eτp−
for the p− of that line and the τ of the vertex to which it connects.
(e) Read off the vertices from the action as usual.
(f) Integrate
∞ dτ
0
for each τ difference between consecutive (though not necessarily connected) ver-
tices. (Performing just this integration gives the usual old-fashioned perturbation theory in terms of energy denominators [2.1], except that our external-line factors differ off shell in order to reproduce the usual Feynman rules.)
(g) Integrate
∞ dp+ dD−2pi
−∞
(2π)D−1
for each loop.
The use of such methods for strings will be discussed in chapt. 10.
2.6. Covariantized light cone
There is a covariant formalism for any field theory that has the interesting prop-
erty that it can be obtained directly and easily from the light-cone formalism, without any additional gauge-fixing procedure [2.7]. Although this covariant gauge is not as general or convenient as the usual covariant gauges (in particular, it sometimes has additional off-shell infrared divergences), it bears strong relationship to both the lightcone and BRST formalisms, and can be used as a conceptual bridge. The basic idea
of the formalism is: Consider a covariant theory in D dimensions. This is equivalent to a covariant theory in (D + 2) − 2 dimensions, where the notation indicates the addition of 2 extra commuting coordinates (1 space, 1 time) and 2 (real) anticommuting coordinates, with a similar extension of Lorentz indices [2.8]. (A similar use of OSp groups in gauge-fixed theories, but applied to only the Lorentz indices and not the coordinates, appears in [2.9].) This extends the Poincaré group ISO(D−1,1) to a graded analog IOSp(D,2|2). In practice, this means we just take the light-cone transverse indices to be graded, watching out for signs introduced by the corresponding change in
2.6. Covariantized light cone
21
statistics, and replace the Euclidean SO(D-2) metric with the corresponding graded
OSp(D-1,1|2) metric:
i = (a, α) ,
δij → ηij = (ηab, Cαβ) ,
(2.6.1)
where ηab is the usual Lorentz metric and
Cαβ = Cβα = σ2
(2.6.2)
is the Sp(2) metric, which satisfies the useful identity
C
γ
δ
αβ C γδ = δ[α δβ]
→
A[αBβ] = CαβCγδAγBδ .
(2.6.3)
The OSp metric is used to raise and lower graded indices as:
xi = ηijx
i
j
,
xi = xjηji ;
ηikηjk = δj
.
(2.6.4)
The sign conventions are that adjacent indices are contracted with the contravariant (up) index first. The equivalence follows from the fact that, for momentum-space
Feynman graphs, the trees will be the same if we constrain the 2 − 2 extra “ghost”
momenta to vanish on external lines (since they’ll then vanish on internal lines by momentum conservation); and the loops are then the same because, when the momentum integrands are written as gaussians, the determinant factors coming from the 2 extra anticommuting dimensions exactly cancel those from the 2 extra commuting
ones. For example, using the proper-time form (“Schwinger parametrization”) of the
propagators (cf. (2.5.1)),
1
∞
=
dτ e−τ(p2+m2)
,
(2.6.5)
p2 + m2
0
all momentum integrations take the form
1
dD+2p d2p
+papa+pαpα+m2) = dDp e−f(papa+m2)
π
α e−f (2p+p−
π D/2
=
e−fm2
, (2.6.6)
f
where f is a function of the proper-time parameters.
The covariant theory is thus obtained from the light-cone one by the substitution
(p−, p+; pi)
→
(p−, p+; pa, pα) ,
(2.6.7a)
22
2. GENERAL LIGHT CONE
where
p− = pα = 0
(2.6.7b)
on physical states. It’s not necessary to set p+ = 0, since it only appears in the combination p−p+ in OSp(D,2|2)-invariant products. Thus, p+ can be chosen arbitrarily on external lines (but should be nonvanishing due to the appearance of factors of 1/p+).
We now interpret x± and xα as the unphysical coordinates. Vector indices on fields
are treated similarly: Having been reduced to transverse ones by the light-cone formalism, they now become covariant vector indices with 2 additional anticommuting
values ((2.6.1)). For example, in Yang-Mills the vector field becomes the usual vector field plus two anticommuting scalars Aα, corresponding to Faddeev-Popov ghosts.
The graphical rules become:
(a) Assign a τ to each vertex, and order them with respect to τ .
(b) Assign (p+, pa) to each external line, but (p+, pa, pα) to each internal line, all directed toward increasing τ . Enforce conservation of (p+, pa, pα) at each vertex
(with pα = 0 on external lines).
(c) Give each internal line a propagator
1
2
Θ(p
+pαpα+m2)/2p+
+)
e−τ(pa
2p+
for the (p+, pa, pα) of that line and the positive difference τ in the proper time between the ends.
(d) Give each external line a factor
1 .
(e) Read off the vertices from the action as usual.
(f) Integrate
∞ dτ
0
for each τ difference between consecutive (though not necessarily connected) ver-
tices.
(g) Integrate
d2pα
Exercises
23
for each loop (remembering that for any anticommuting variable θ,
dθ 1 = 0,
dθ θ = 1, θ2 = 0).
(h) Integrate
∞
2
dp+
−∞
for each loop.
(i) Integrate
dDp
(2π)D
for each loop.
For theories with only scalars, integrating just (f-h) gives the usual Feynman
graphs (although it may be necessary to add several graphs due to the τ -ordering of non-adjacent vertices). Besides the correspondence of the τ parameters to the usual Schwinger parameters, after integrating out just the anticommuting parameters the
p+ parameters resemble Feynman parameters.
These methods can also be applied to strings (chapt. 10).
Exercises
(1) Find the light-cone formulation of QED. Compare with the Coulomb gauge for-
mulation.
(2) Derive the commutation relations of the conformal group from (2.2.2). Check
that (2.2.4) satisfies them. Evaluate the commutators implicit in (2.2.7) for each
generator.
(3) Find the Lorentz transformation Mab of a vector (consistent with the conventions of (2.2.2)). (Hint: Look at the transformations of x and p.) Find the explicit
form of (2.2.8) for that case. Solve these equations of motion. To what simpler
representation is this equivalent? Study this equivalence with the light-cone anal-
ysis given below (2.2.8). Generalize the analysis to totally antisymmetric tensors
of arbitrary rank.
(4) Repeat problem (3) for the massive case. Looking at the separate SO(D-1,1)
representations contained in the SO(D,1) representations, show that first-order
formalisms in terms of the usual fields have been obtained, and find the corre-
sponding second-order formulations.
24
2. GENERAL LIGHT CONE
(5) Check that the explicit forms of the Poincaré generators given in (2.3.5) satisfy the correct algebra (see problem (2)). Find the explicit transformations acting
on the vector representation of the spin group SO(D-1). Compare with (2.4.1-2).
(6) Derive (2.4.11). Compare that p− with the light-cone hamiltonian which follows
from (2.1.5).
(7) Calculate the 4-point amplitude in φ3 theory with light-cone graphs, and com-
pare with the usual covariant Feynman graph calculation. Calculate the 1-loop
propagator correction in the same theory using the covariantized light-cone rules,
and again compare with ordinary Feynman graphs, paying special attention to
Feynman parameters.
3.1. Gauge invariance and constraints
25
3. GENERAL BRST
3.1. Gauge invariance and constraints
In the previous chapter we saw that a gauge theory can be described either in a
manifestly covariant way by using gauge degrees of freedom, or in a manifestly unitary way (with only physical degrees of freedom) with Poincaré transformations which are nonlinear (in both coordinates and fields). In the gauge-covariant formalism there is a D-dimensional manifest Lorentz covariance, and in the light-cone formalism a D − 2-dimensional one, and in each case a corresponding number of degrees of freedom.
There is also an intermediate formalism, more familiar from nonrelativistic theory: The hamiltonian formalism has a D − 1-dimensional manifest Lorentz covariance (rotations). As in the light-cone formalism, the notational separation of coordinates
into time and space suggests a particular type of gauge condition: temporal (time-
like) gauges, where time-components of gauge fields are set to vanish. In chapt. 5, this formalism will be seen to have a particular advantage for first-quantization of relativistic theories: In the classical mechanics of relativistic theories, the coordinates are treated as functions of a “proper time” so that the usual time coordinate can be treated on an equal footing with the space coordinates. Thus, canonical quantization with respect to this unobservable (proper) “time” coordinate doesn’t destroy manifest Poincaré covariance, so use of a hamiltonian formalism can be advantageous, particularly in deriving BRST transformations, and the corresponding second-quantized
theory, where the proper-time doesn’t appear anyway.
We’ll first consider Yang-Mills, and then generalize to arbitrary gauge theories.
In order to study the temporal gauge, instead of the decomposition (2.1.1) we simply separate into time and spatial components
a = (0, i) ,
A · B = −A0B0 + AiBi .
(3.1.1)
The lagrangian (2.1.5) is then
L = 1F 2
(p
4
ij
− 12 0Ai − [∇i, A0])2 .
(3.1.2)
26
3. GENERAL BRST
The gauge condition
A0 = 0
(3.1.3)
transforms under a gauge transformation with a time derivative: Under an infinites-
imal transformation about A0 = 0,
δA0 ≈ ∂0λ ,
(3.1.4)
so the Faddeev-Popov ghosts are propagating. Furthermore, the gauge transformation
(3.1.4) does not allow the gauge choice (3.1.3) everywhere: For example, if we choose periodic boundary conditions in time (to simplify the argument), then
∞
δ
dx0 A0 ≈ 0 .
(3.1.5)
−∞
A0 can then be fixed by an appropriate initial condition, e.g., A0|x0=0 = 0, but then the corresponding field equation is lost. Therefore, we must impose
δS
0 =
= −[∇
δA
i, F 0i] = −[∇i, p0Ai]
at x0 = 0
(3.1.6)
0
as an initial condition. Another way to understand this is to note that gauge fixing eliminates only degrees of freedom which don’t occur in the lagrangian, and thus
can eliminate only redundant equations of motion: Since [∇i, F 0i] = 0 followed from the gauge-invariant action, the fact that it doesn’t follow after setting A0 = 0 means some piece of A0 can’t truly be gauged away, and so we must compensate by imposing
the equation of motion for that piece. Due to the original gauge invariance, (3.1.6) then holds for all time from the remaining field equations: In the gauge (3.1.3), the lagrangian (3.1.2) becomes
L = 1A
(p
[A
2
i✷ Ai − 12
iAi)2 + [Ai, Aj ]piAj + 1
4
i, Aj ]2
,
(3.1.7)
and the covariant divergence of the implied field equations yields the time derivative of (3.1.6). (This follows from the identity [∇b, [∇a, F ab]] = 0 upon applying the
field equations [∇a, F ia] = 0. In unitary gauges, the corresponding constraint can be derived without time derivatives, and hence is implied by the remaining field equations under suitable boundary conditions.) Equivalently, if we notice that (3.1.4) does not fix the gauge completely, but leaves time-independent gauge transformations, we need to impose a constraint on the initial states to make them gauge invariant. But the
generator of the residual gauge transformations on the remaining fields Ai is
δ
G(xi) = ∇i, i
,
(3.1.8)
δAi
3.1. Gauge invariance and constraints
27
which is the same as the constraint (3.1.6) under canonical quantization of (3.1.7).
Thus, the same operator (1) gives the constraint which must be imposed in addition to the field equations because too much of A0 was dropped, and (2) (its transpose) gives the gauge transformations remaining because they left the gauge-fixing function A0
invariant. The fact that these are identical is not surprising, since in Faddeev-Popov quantization the latter corresponds to the Faddeev-Popov ghost while the former
corresponds to the antighost.
These properties appear very naturally in a hamiltonian formulation: We start
again with the gauge-invariant lagrangian (3.1.2). Since A0 has no time-derivative
terms, we Legendre transform with respect to just .
Ai. The result is
1
SH =
dDx tr L
g2
H
,
LH = .AiΠi − H , H = H0 + A0iG ,
H
2
2
0 = 1 Π
F
,
2
i − 14
ij
G = [∇i, Πi] ,
(3.1.9)
where . = ∂0. As in ordinary nonrelativistic classical mechanics, eliminating the
momentum Πi from the hamiltonian form of the action (first order in time deriva-
tives) by its equation of motion gives back the lagrangian form (second order in time derivatives). Note that A0 appears linearly, as a Lagrange multiplier.
The gauge-invariant hamiltonian formalism of (3.1.9) can be generalized [3.1]:
Consider a lagrangian of the form
L
A
H = .
zM eM (z)πA − H ,
H = H0(z, π) + λiiGi(z, π) ,
(3.1.10)
where z, π, and λ are the variables, representing “coordinates,” covariant “momenta,”
and Lagrange multipliers, respectively. They depend on the time, and also have
indices (which may include continuous indices, such as spatial coordinates). e, which is a function of z, has been introduced to allow for cases with a symmetry (such
as supersymmetry) under which dzM e A
M
(but not dz itself) is covariant, so that π
will be covariant, and thus a more convenient variable in terms of which to express the constraints G. When H0 commutes with G (quantum mechanically, or in terms
of Poisson brackets for a classical treatment), this action has a gauge invariance
generated by G, for which λ is the gauge field:
δ(z, π) = [ζiGi, (z, π)] ,
∂
δ
− λiG
∂t
i
= 0
→
(δλi)Gi = .ζiGi + [λjGj, ζiGi] ,
(3.1.11)
28
3. GENERAL BRST
where the gauge transformation of λ has been determined by the invariance of the
“total” time-derivative d/dt = ∂/∂t + iH. (More generally, if [ζiGi, H0] = fiiGi, then δλi has an extra term −fi.) Using the chain rule ((d/dt) on f(t, qk(t)) equals ∂/∂t +
.
qk(∂/∂qk)) to evaluate the time derivative of G, we find the lagrangian transforms as a total derivative
d
δL
A
H =
(δzM )e
π
dt
M
A − ζ iiGi
,
(3.1.12)
which is the usual transformation law for an action with local symmetry generated
by the current G. When H0 vanishes (as in relativistic mechanics), the special case ζi = ζλi of the transformations of (3.1.11) are τ reparametrizations, generated by the hamiltonian λiGi. In general, after canonical quantization, the wave function satisfies the Schrödinger equation ∂/∂t + iH0 = 0, as well as the constraints G = 0 (and thus
∂/∂t + iH = 0 in any gauge choice for λ). Since [H0, G] = 0, G = 0 at t = 0 implies G = 0 for all t.
In some cases (such as Yang-Mills), the Lorentz covariant form of the action can
be obtained by eliminating all the π’s. A covariant first-order form can generally be obtained by introducing additional auxiliary degrees of freedom which enlarge π to
make it Lorentz covariant. For example, for Yang-Mills we can rewrite (3.1.9) as
L
2
i
2
H = 1 G
F
F
2
0i − G0
0i + 1
4
ij
→
L
2
1 = − 1 G
+ GabF
4
ab
ab
,
(3.1.13)
where G0i = iΠi, and the independent (auxiliary) fields Gab also include Gij, which have been introduced to put 1F 2 into first-order form and thus make the lagrangian 4
ij
manifestly Lorentz covariant. Eliminating Gij by their field equations gives back the hamiltonian form.
Many examples will be given in chapts. 5-6 for relativistic first-quantization,
where H0 vanishes, and thus the Schrödinger equation implies the wave function is
proper-time-independent (i.e., we require H0 = 0 because the proper time is not
physically observable). Here we give an interesting example in D=2 which will also
be useful for strings. Consider a single field A with canonical momentum P and
choose
iG = 1(P + A′)2 ,
(P
4
H0 = 14
− A′)2 ,
(3.1.14)
where ′ is the derivative with respect to the 1 space coordinate (which acts as the index M or i from above). From the algebra of P ± A′, it’s easy to check, at least
at the Poisson bracket level, that the G algebra closes and H0 is invariant. (This
3.2. IGL(1)
29
algebra, with particular boundary conditions, will be important in string theory: See chapt. 8. Note that P + A′ does not form an algebra, so its square must be used.)
The transformation laws (3.1.11) are found to be
←
→
δA = ζ 1 (P + A′) ,
δλ = .ζ
∂
2
− λ 1 ζ .
(3.1.15)
In the gauge λ = 1 the action becomes the usual hamiltonian one for a massless
scalar, but the constraint implies P + A′ = 0, which means that modes propagate
only to the right and not the left. The lagrangian form again results from eliminating P , and after the redefinitions
√
ˆ
1 − λ
1
λ = 2
,
ˆ
ζ =
2
ζ
,
(3.1.16)
1 + λ
1 + λ
we find [3.2]
L = −(∂
ˆ
+A)(∂−A) + 1 λ(∂
2
−A)2
;
←
→
δA = ˆ
ζ∂
ˆ
ˆ
−A
,
δˆ
λ = 2∂+ζ + ˆ
ζ ∂ −λ ;
(3.1.17)
where ∂± are defined as in sect. 2.1.
The gauge fixing (including Faddeev-Popov ghosts) and initial condition can be
described in a very concise way by the BRST method. The basic idea is to construct
a symmetry relating the Faddeev-Popov ghosts to the unphysical modes of the gauge
field. For example, in Yang-Mills only D − 2 Lorentz components of the gauge field
are physical, so the Lorentz-gauge D-component gauge field requires 2 Faddeev-Popov ghosts while the temporal-gauge D − 1-component field requires only 1. The BRST
symmetry rotates the additional gauge-field components into the FP ghosts, and vice versa. Since the FP ghosts are anticommuting, the generator of this symmetry must
be, also.
3.2. IGL(1)
We will find that the methods of Becchi, Rouet, Stora, and Tyutin [3.3] are the
most useful way not only to perform quantization in Lorentz-covariant and general
nonunitary gauges, but also to derive gauge-invariant theories. BRST quantization is a more general way of quantizing gauge theories than either canonical or path-integral (Faddeev-Popov), because it (1) allows more general gauges, (2) gives the Slavnov-Taylor identities (conditions for unitarity) directly (they’re just the Ward identities for BRST invariance), and (3) can separate the gauge-invariant part of a gauge-fixed 30
3. GENERAL BRST
action. It is defined by the conditions: (1) BRST transformations form a global group with a single (abelian) anticommuting generator Q. The group property then implies
Q2 = 0 for closure. (2) Q acts on physical fields as a gauge transformation with the gauge parameter replaced by the (real) ghost. (3) Q on the (real) antighost gives a BRST auxiliary field (necessary for closure of the algebra off shell). Nilpotence of Q
then implies that the auxiliary field is BRST invariant. Physical states are defined to be those which are BRST invariant (modulo null states, which can be expressed
as Q on something) and have vanishing ghost number (the number of ghosts minus
antighosts).
There are two types of BRST formalisms: (1) first-quantized-style BRST, origi-
nally found in string theory [3.4] but also applicable to ordinary field theory, which contains all the field equations as well as the gauge transformations; and (2) second-quantized-style BRST, the original form of BRST, which contains only the gauge
transformations, corresponding in a hamiltonian formalism to those field equations
(constraints) found from varying the time components of the gauge fields. However,
we’ll find (in sect. 4.4) that, after restriction to a certain subset of the fields, BRST1 is equivalent to BRST2. (It’s the BRST variation of the additional fields of BRST1 that leads to the field equations for the physical fields.) The BRST2 transformations were originally found from Yang-Mills