Worldline quantization of field theory, effective actions and L∞ structure
Worldline quantization of field theory, effective actions and
L. Bonora 0 1 2 5
M. Cvitan 0 1 2 3
P. Dominis Prester 0 1 2 4
S. Giaccari 0 1 2 3
M. Pauliˇsi´c 0 1 2 4
T. Sˇtemberga 0 1 2 3
0 Radmile Matejˇci ́c 2 , 51000 Rijeka , Croatia
1 Bijeniˇcka cesta 32 , 10000 Zagreb , Croatia
2 Via Bonomea 265 , 34136 Trieste , Italy
3 Department of Physics, Faculty of Science, University of Zagreb
4 Department of Physics, University of Rijeka
5 International School for Advanced Studies (SISSA) and INFN , Sezione di Trieste
We formulate the worldline quantization (a.k.a. deformation quantization) of a massive fermion model coupled to external higher spin sources. We use the relations obtained in this way to show that its regularized effective action is endowed with an L∞ symmetry. The same result holds also for a massive scalar model.
Higher Spin Gravity; Non-Commutative Geometry; Models of Quantum Grav-
1 Introduction 2 Worldline quantization of a fermion model 3
L∞ structure in higher spin theory
L∞ symmetry of higher spin effective actions
Proof of the L∞ relations
Relation L3L1 + L2L2 + L1L3 = 0, degree 0
In the latter this is tied to the short distance behavior and has to do with the finite string
size. So it is related to the mild form of non-locality in string theory. In general, what is
the right amount of non-locality? All these are very general questions for which answers
are not yet available. For the time being we have to content ourselves with the taxonomy
of higher spin models.
Recently we have revisited and generalized a method based on effective actions to
determine the classical dynamics of higher spin fields, [6–8]. The basic idea is to exploit the
one-loop effective actions of elementary free field theories coupled via conserved currents to
external higher spin sources, in order to extract information about the (classical) dynamics
– 1 –
of the latter. We focused on massive scalar and Dirac fermion models, but, no doubt, the
same method can be applied to other elementary fields. In the cited papers we computed the
two-point correlators of conserved currents, which allowed us to reconstruct the quadratic
effective action for the higher spin fields coupled to the currents. We were able to show
that such effective actions are built out of the Fronsdal differential operators [10, 11],
appropriate for those higher spin fields, in the general non-local form discussed in [12, 13].
The method we used in [6–8] is the standard perturbative approach based on Feynman
diagrams. This method is ultra-tested and very effective for two-point correlators. For
instance, as we have seen in , it preserves gauge and diff-invariance (it respects the
relevant Ward identities). We have no reason to doubt that this will be the case also for
higher order correlators, in particular for the crucial three-point ones. But the burden to
guess what the gauge transformations beyond the lowest level are is left to us. In this
regard there exists an alternative quantization method which can come to our help, the
worldline quantization method,1 which we wish to discuss in this paper.
The worldline quantization of field theory is based on the Weyl quantization of a
particle in quantum mechanics, where the coordinates in the phase space are replaced by position
and momentum operator and observables are endowed with a suitable operator ordering. In
order to achieve second quantization one, roughly speaking, replaces the field dependence
on the position and the field derivatives by the corresponding position and momentum
operators, respectively, and relies on the Weyl quantization for the latter. The effective
action is then defined. The important thing is that this procedure comes with a bonus, the
precise form of the gauge symmetry. This has a remarkable consequence, as we will show
in this paper: without doing explicit calculations, it is possible to establish the symmetry
of the full (not only the local part of) effective action and demonstrate its L∞ symmetry.
The latter is a symmetry that characterizes many (classical) field theories, including closed
string field theory (a good introduction to L∞ algebras and field theory is ).
In section 2 we will carry out the worldline quantization of free Dirac fermions coupled
to external sources (the case of a scalar field has already been worked out in ) and derive
heuristic rules, similar to the Feynman diagrams, to compute amplitudes. In section 3 we
will uncover the L∞ structure of the corresponding effective action. Section 4 is devoted
to a summary and discussion of our results.
Worldline quantization of a fermion model
Fermion linearly coupled to higher spin fields
Let us consider a free fermion theory
ddx ψ(iγ ·∂ − m)ψ,
1The literature on the worldline quantization is large. Here we refer in particular to the calculation
of effective actions via the worldline quantization in relation to higher spin theories, [15–17]. The first
elaboration of this method is probably in , to which many others followed, see for instance [18–27].
– 2 –
coupled to external sources. We second-quantize it using the Weyl quantization method
for a particle worldline. The full action is expressed as an expectation value of operators
S = hψ| − γ ·(Pb − Hb ) − m|ψi
Here Pbµ is the momentum operator whose symbol is the classical momentum pµ . Hb is an
operator whose symbol is h(x, p), where
hµ (x, p) =
X 1 µ 1...µ n (x) pµ 1 . . . pµ n
s = n + 1 is the spin and the tensors are assumed to be symmetric. We recall that a
quantum operator Ob can be represented with a symbol O(x, p) through the Weyl map
(2π)d (2π)d O(x, p) eik·(x−Xb)−iy·(p−Pb)
where Xb is the position operator.
Next we insert this into the r.h.s. of (2.2), where we also insert two completenesses
R ddx|xihx|, and make the identification ψ(x) = hx|ψi. Expressing S in terms of symbols
= S0 +
= S0 + X∞ Z
The HS currents are on-shell conserved in the free theory (2.1)
which is a consequence of invariance of S0[ψ] on global (rigid) transformations
Jµ (1) = ψγµ ψ
Jµ (2)1 =
∂(µ 1 ψγµ )ψ − ψγ(µ ∂µ 1)ψ
µ 1···µ s−1 = 0
δnψ(x) = −
(−i)n+1 εµ(n1)···µ n ∂µ 1 . . . ∂µ n ψ(x)
– 3 –
We see that the symmetric tensor field hµ 1...µ n is linearly coupled to the HS (higher spin)
µ 1...µ s−1 (x) =
(s − 1)! ∂z(µ 1
∂zµ s−1 ψ x +
γµ )ψ x −
For instance, for s = 1 and s = 2 one obtains
We shall next show that for the full action (2.5) this extends to the local symmetry.
The consequence is that the currents are still conserved, with the HS covariant derivative
substituting ordinary derivative in (2.9).
Notice that these currents are conserved even without symmetrizing µ with the other
indices. But in the sequel we will suppose that they are symmetric.
The action (2.2) is trivially invariant under the operation
where Gb = −γ ·(Pb − Hb ) − m. So it is invariant under
Writing Ob = e−iEb we easily find the infinitesimal version.
Symb [Hb µ , Eb] = [hµ (x, p) ∗, ε(x, p)]
Dx∗µ = ∂xµ − i[hµ (x, p) ∗, ]
δεψ˜(x, p) = iε(x, p) ∗ ψ˜(x, p)
– 4 –
Let the symbol of Eb be ε(x, p), then the symbol of [iγ ·Pb, Eb] is
An easy way to make this explicit is to use the fact that the symbol of the product of two
operators is given by the Moyal product of the symbols. Thus
Symb [γ ·Pb, Eb] = [γ ·p ∗, ε(x, p)] = γ ·p e− 2i ∂x·∂p ε(x, p) − ε(x, p) e 2i ∂x·∂p γ ·p
where [a ∗, b] ≡ a ∗ b − b ∗ a. Therefore, in terms of symbols,
δεhµ (x, p) = ∂xµ ε(x, p) − i[hµ (x, p) ∗, ε(x, p)] ≡ Dx∗µ ε(x, p)
where we introduced the covariant derivative defined by
metry HS symmetry.
product of symbols
This will be referred to hereafter as HS transformation, and the corresponding
symThe transformations of ψ are somewhat different. They can also be expressed as Moyal
provided we use the partial Fourier transform
ψ˜(x, p) =
ddy ψ x −
and finally we antitransform back the result. Alternatively we can proceed as follows. We
ddx′ddy′ ε(x′, p) hx|eik·(x′−Xb)−iy′·(p−Pb)|ψi
ddx′ddy′ ε(x′, p′) eik·(x′−x)−iy′·phx|eiy′Pb|ψie− 2i y′·k
Next we insert a momentum completeness R ddq|qihq| to evaluate hx|eiy′Pb|ψi and
subsequently a coordinate completeness to evaluate hq|ψi using the standard relation hx|pi =
eip·x. Then we produce two delta functions by integrating over k and q. In this way we get
rid of two coordinate integrations. Finally we arrive at
δεψ(x) = ihx|Eb|ψi = i
ddz ε x + , p e−ip·z ψ(x + z)
= i X∞ Z
(−i∂z)n · ε(n) x +
ψ(x + z)
= iε(0)(x) ψ(x) + εµ(1)(x) ∂µ ψ(x) +
εµ(ν2) ∂µ ∂ν ψ + ∂µ εµ(ν2) ∂ν ψ +
2 ∂µ εµ(1)(x) ψ(x)
4 ∂µ ∂ν εµ(ν2) ψ (x) + . . .
(−i∂z)n · ε(n) x +
ψ(x + z)
where a dot denote the contraction of upper and lower indices. The first method leads to
the same result.
the interacting classical action (2.5)
Now we want to understand the conservation law ensuing from the HS symmetry of
0 = δεS[ψ, h] =
δψ(x) δεψ(x) + δεψ(x)
δhµ (x, p) δεhµ (x, p)
Now we evaluate this expression on the classical solution, in which case the first two terms
vanish (remember that h is the background field). We are left with
Using (2.18), partially integrating and using the following property of the Moyal product
ddp a(x, p)[b(x, p) ∗, c(x, p)] =
ddp [a(x, p) ∗, b(x, p)] c(x, p)
ddp ε(x, p) Dx∗µ Jµ (x, p)
From this follows the conservation law in the classical interacting theory
It is not hard to shaw that for hµ (x, p) = 0 this becomes equivalent to (2.10).
Using the ∗-Jacobi identity (it holds also for the Moyal product, because it is
associative) one can easily get
(δε2 δε1 − δε1 δε2 ) hµ (x, p) = i (∂x[ε1 ∗, ε2](x, p) − i[hµ (x, p) ∗, [ε1 ∗, ε2](x, p)]])
= i Dx∗µ [ε1 ∗, ε2](x, p)
We see that the HS ε-transform is of the Lie algebra type.
Perturbative expansion of the effective action
In this subsection we work out (heuristic) rules, similar to the Feynman ones, to compute
n-point amplitudes in the above fermion model. The purpose is to reproduce formulas
similar to those of  for the scalar case. We would like to point out, however, that this
is not strictly necessary: the good old Feynman rules are anyhow a valid alternative.
We start from the representation of the effective action as trace-logarithm of a
and use a well-known mathematical formula to regularize it
where ǫ is an infrared regulator. The crucial factor is therefore
W [h] = N Tr[ln Gb ]
K[g|t] ≡ Trhe−tGbi = Trhet(γ·(Pb−Hb )+m)i ,
K[g|t] = emt Z
(2π)d trhp|etγ·(Pb−Hb )|pi
known as the heat kernel, where g is the symbol of Gb. The trace Tr includes both an
integration over the momenta and tr, the trace over the gamma matrices,
Next we expand
etγ·(Pb−Hb ) = et γ·Pb X (−1)n
where γ ·Hb (τ ) = e−τ γ·Pbγ ·Hb eτ γ·Pb.
dτn γ ·Hb (τ1) γ ·Hb (τ2) . . . γ ·Hb (τn)
hp|γ ·Hb (τ )|qi = e−τ γ·php|γ ·Hb |qi eτ γ·q
– 6 –
Using a formula analogous to (2.22) for Hb and inserting completenesses one finds
hp|γ ·Hb |qi =
(2π)d (2π)d γ ·h(x, p′)hp|eik·(x−Xb)−iy·(p′−Pb)|qi
ddx γ ·h(x, ∂u)ei(q−p)·x+u· p+2q
Using this we can write
Tr he−tGb i
= emt X (−1)n
hp|γ ·Hb (τ )|qi =
ddx e−τ γ·p γ ·h(x, ∂u) eτ γ·q ei(q−p)·x+u· p+2q
×tr et γ·pn hpn|γ ·Hb (τ1)|p1ihp1|γ ·Hb (τ2)|p2i . . . hpn−1|γ ·Hb (τn)|pni
= emt X (−1)n
Y ddxi (2π)d
where the double brackets means integration of the xi and derivation with respect to the
ui. In turn K(n)µ...µ (t) can be written more explicitly as
Kµ 1...µ n (x1, u1, . . . , xn, un|t) = etmZ
Y ddpj eipj· xj−xj+1−i uj+1+uj
Ke µ 1...µ n (p1, . . . , pn|t)
Now, the nested integral can be rewritten in the following way
Replacing this inside (2.40) we get
Ke µ 1...µ n (p1, . . . , pn|t) =
(−1)n Z ∞ dω eiωt Z ∞
dσ1 . . .
where ω′ = ω − iǫ. ǫ in the exponents allows us to perform the integrals,2 the result being
Ke µ 1...µ n (p1, . . . , pn|t) =
(p/1 − iω′)2
2This is evident with the Majorana representation of the gamma matrices, because in such a case the
term γ ·p in the exponent is purely imaginary, the gamma matrices being imaginary. This term therefore
gives rise to oscillatory contributions, much like the iω term.
– 8 –
We remark that (p/−iω′)2 = ∂(iω) p/−1iω′ . This allows us, via integration by parts, to
Ke µ 1...µ n (p1, . . . , pn|t)=
n −∞ 2π
t Z ∞ dω eiωt tr γµ 1
which has the same form as Ke µ 1...µ n (p1, . . . , pn|t) with all the p/i replaced by p/i + m:
We can also include the factor etm in (2.39) in a new kernel Ke µ 1...µ n (p1, . . . , pn|m, t)
Kµ 1...µ n (x1, u1, . . . , xn, un|t) =
Ke µ 1...µ n (p1, . . . , pn|m, t) =
Y eipj· xj−xj+1−i uj+1+uj
(p/n + m − iω′)2
(p/1 + m − iω′)2
p/ + m − iω′
n −∞ 2π
t Z ∞ dω eiωt tr γµ 1
Integrating further as in the scalar model case, , is not possible at this stage because
of the gamma matrices. One has to proceed first to evaluate the trace over the latter.
Using (2.37) we can write the regularized effective action as
– 9 –
Z ∞ dt emt X∞ Z
Y d xi (2π)d 0
Y eipj·(xj−xj+1) hµ 1 x1, p1 + pn
. . . hµ n xn, pn−1 + pn
The general formula for the effective action is W [h] =
where we have discarded the constant 0-point contribution, as we will do hereafter. The
effective action can be calculated by various methods, of which (2.49) is a particular example.
In the latter case the amplitudes are given by
µ 1,...,µ n (x1, p1,. . ., xn, pn, ǫ) = −N
n! Z ∞
ddqi Z ∞ dω eiωt
/qn + m − iω′
q1 + qn
. . . δ pn −
qn−1 + qn
We stress once more, however, that the regularized effective action (2.50) may not be
derived only via (2.51), that is via the procedure of section 2.2. It could as well be obtained
by means of the ordinary Feynman diagrams.
This amplitude has cyclic symmetry. When saturated with the corresponding h’s, as
in (2.50), it gives the level n effective action. Here we would like to investigate some general
consequences of the invariance of the general effective action under the HS symmetry,
codified by eq. (2.18), assuming for the W(n) the same cyclic symmetry as (2.51). The
invariance of the effective action under (2.18) is expressed as3
0 = δεW [h]
In order to expose the L∞ structure we need the equations of motion (EoM). Here we
can talk of generalized equations of motion. They are obtained by varying W [h, ǫ] with
respect to hµ (x, p):
3Hereafter we assume that the HS symmetry is not anomalous and that there is a regularization procedure
leading to a HS invariant effective action. The question of whether the particular effective action (2.49)
satisfies (2.52) requires an explicit calculation of (2.51) and is left to future work.
δhµ (x, p)
W [h] = 0
Then, expanding in p, we obtain the generalized EoM’s for the components hµ 1...µ n (x).
The most general EoM is therefore where
Fµ (x, p) = 0
Fµ (x, p) ≡
Integrating by parts (2.52) and using (2.26) we obtain the off-shell equation
Dx∗µ Fµ (x, p) ≡ ∂xµ Fµ (x, p) − i[hµ (x, p) ∗, Fµ (x, p)] = 0
Taking the variation of this equation with respect to (2.18) we get
0 = δε(Dx∗µ Fµ (x, p)) = Dx∗µ (δεFµ (x, p)) − i[Dx∗µ ε ∗, Fµ (x, p)]
From (2.56) and (2.57) one can deduce
δεFµ (x, p) = i[ε(x, p) ∗, Fµ (x, p)]
A final remark for this section. Using standard regularizations one obtains that in
general the effective action contains term linear in HS fields, which gives constant
contribution to EoM’s of even-spin HS fields of the form c(s, ǫ) (ηµ )s/2, where c(s, ǫ) are scheme
dependent coefficients which need to be renormalized. As this term is a generalization
of the lowest-order contribution of the cosmological constant term expanded around flat
spacetime, we shall call the part of the effective action that contains the full linear term
and is invariant on HS transformations (2.18), generalized cosmological constant term. In
the next section we shall assume that this term is removed from the effective action.
L∞ structure in higher spin theory
L∞ symmetry of higher spin effective actions
In this section we will uncover the L∞ symmetry of the W [h]. To this end we use the general
transformation properties derived in the previous subsection, notably eqs. (2.54), (2.58),
beside (2.18). We will also introduce a simplification, which is required by the classical
form of the L∞ symmetry. The expansion of the effective action (2.50) is in essence an
expansion around a flat background. As a flat background is not a solution when the
generalized cosmological constant term is present, consistency requires that we take this
term out of an effective action (or, in other words, renormalize the cosmological constant
to zero). This will be assumed from now on. Technically, this means that we now assume
that the sum in (2.50) starts from n = 2, and the sum in (2.55) starts from n = 1, while
all other relations from subsection 2.4 are the same.
To start with let us recall that an L∞ structure characterizes closed string field theory.4
This fact first appeared in [29, 30], see also , as a particular case of a general
mathematical structure called strongly homotopic algebras (or SH algebras), see the introduction
for physicists [32, 33]. It became later evident that this kind of structure characterizes
not only closed string field, but other field theories as well , in particular gauge field
theories [35–37], Chern-Simons theories, Einstein gravity and double field theory . For
other more recent applications, see [38–40].
For the strongly homotopic algebra L∞ we closely follow the notation and definitions
of . L∞ is determined by a set of vector spaces Xi, i = . . . , 1, 0, −1, . . ., with degree
i and multilinear maps (products) among them Lj , j = 1, 2, . . ., with degree dj = j − 2,
satisfying the following quadratic identities:
(−1)i(j−1) X(−1)σǫ(σ; x) Lj (Li(xσ(1), . . . , xσ(i)), xσ(i+1), . . . , xσ(n)) = 0
In this formula σ denotes a permutation of the entries so that σ(1) < . . . σ(i) and σ(i + 1) <
. . . σ(n), and ǫ(σ; x) is the Koszul sign. To define it consider an algebra with product
xi ∧ xj = (−1)xixj xj ∧ xi, where xi is the degree of xi; then ǫ(σ; x) is defined by the relation
x1 ∧ x2 ∧ . . . ∧ xn = ǫ(σ; x) xσ(1) ∧ xσ(2) ∧ . . . ∧ xσ(n)
In our case, due to the structure of the effective action and the equation of motion, we
will need only three spaces X0, X−1, X−2 and the complex
X0 −L→1 X−1 −L→1 X−2 −L→1 0
The degree assignment is as follows: ε ∈ X0, hµ ∈ X−1 and Fµ ∈ X−2.
The properties of the mappings Li under permutation are defined in . For instance
L2(x1, x2) = −(−1)x1x2 L2(x2, x1)
Ln(xσ(1), xσ(2), . . . , xσ(n)) = (−1)σǫ(σ; x)Ln(x1, x2, . . . , xn)
It is worth noting that if all the xi’s are odd (−1)σǫ(σ; x) = 1.
The product Li are defined as follows. We first define the maps ℓi
Therefore, in our case,
δεh = ℓ1(ε) + ℓ2(ε, h) −
2 ℓ3(ε, h, h) −
3! ℓ4(ε, h, h, h) + . . .
ℓ1(ε)µ = ∂xµ ε(x, p)
ℓ2(ε, h)µ = −i[hµ (x, p) ∗, ε(x, p)] = −ℓ2(h, ε)µ
ℓj (ε, h, . . . , h)µ = 0 ,
j ≥ 3
For these entries, i.e. ε, (ε, h), (ε, h, h), . . . we set Li = ℓi.
4Open string field theory is instead characterized by an A∞ structure, see  and references therein.
From the above we can extract L2(ε, ε) ≡ ℓ2(ε, ε). We have
(δε1δε2 − δε2δε1) hµ = δε1 (ℓ1(ε2) + ℓ2(ε2, h)) − δε2 (ℓ1(ε1) + ℓ2(ε1, h))
= δε1 (ℓ2(ε2, h)) − δε2 (ℓ2(ε1, h))
= ℓ2(ε2, δε1h) − ℓ2(ε1, δε2h) = ℓ2(ε2, ℓ1(ε1)) − ℓ2(ε1, ℓ1(ε2)) + O(h)
Now, the L∞ relation (3.1) involving L1 and L2 is
L1(L2(x1, x2)) = L2(L1(x1), x2) − (−1)x1x2L2(L1(x2), x1)
for two generic elements of x1, x2 of degree x1, x2, respectively. If we wish to satisfy it we
By comparing this with (2.29) we obtain
0 = L1(L3(x1, x2, x3))
The next step is to determine L3. It must satisfy, in particular, the L∞ relation
+L3(L1(x1), x2, x3) + (−1)x1L3(x1, L1(x2), x3) + (−1)x1+x2L3(x1, x2, L1(x3))
+L2(L2(x1, x2), x3) + (−1)(x1+x2)x3L2(L2(x3, x1), x2) + (−1)(x2+x3)x1L2(L2(x2, x3), x1)
We define first the ℓi with only h entries. They are given by the generalized EoM:
Let us write Fµ , (2.54) in compact form as
F = ℓ1(h) − 2 ℓ2(h, h) −
3! ℓ3(h, h, h) + . . .
Fµ = X 1
cyclic symmetry. But in order to verify the L∞ relations we have to know these products
for different entries. Following  we define, for instance,
2L2(h1, h2) = ℓ2(h1 + h2, h1 + h2) − ℓ2(h1, h1) − ℓ2(h2, h2)
L2(h1, h2) =
(ℓ2(h1, h2) + ℓ2(h2, h1))
L3(h1, h2, h3) =
(ℓ3(h1, h2, h3) + perm(h1, h2, h3))
In general, when we have a non-symmetric n-linear function fn of the variable h we can
generate a symmetric function Fn linearly dependent on each of n variables h1, . . . , hn
through the following process
We shall define Ln(h1, . . . , hn) by using this formula: replace Fn with Ln and fn with ℓn,
the latter being given by (3.15).
defining the L∞ algebra of the HS effective action are
We shall see that beside Ln(h1, . . . , hn), (3.7) and (3.11) the only nonvanishing objects
which is equivalent to
+(−1)n−khfn(h1 + . . . + hk) + · · · + fn(hn−k+1 + . . . + hn)
+(−1)n−1hfn(h1) + . . . + fn(hn)
In the rest of this section we shall prove that Ln defined in this way generate an L∞
L2(ε, E) = i[ε ∗, E]
where E represents Fµ or any of its homogeneous pieces.
Proof of the L∞ relations
Relation L21 = 0, degree -2
Now let us verify the remaining L∞ relations. The first is L21 ≡ ℓ21 = 0.5
Let us start from ℓ1(ℓ1(ε)). We recall that ℓ1(ε) = ∂xε(x, p) and belongs to X−1. Now
ℓ1(h) = hhWµ(2), hii
Replacing h with ∂xε(x, p) corresponds to taking the variation of the lowest order in h of Fµ
with respect to h, i.e. with respect to (2.18). On the other hand the variation of Fµ is given
by (2.58) and is linear in Fµ . Therefore, since ℓ1(∂xε(x, p)) is order 0 in h it must vanish. In
fact it does, which corresponds to the gauge invariance of the EoM to the lowest order in h.
Next let us consider ℓ1(ℓ1(h)). It has degree -3, so it is necessarily 0 since X−3 = 0.
5We remark that if the generalized cosmological constant term (see end of section 2.4 and beginning
of section 3) is non-vanishing, then ℓ12 6= 0. In this case an enlarged version of L∞, called curved L∞, is
necessary. We thank J. Stasheff for this piece of information. We will not explore this possibility here.
where we used (3.18). More explicitly (3.23) writes
− iℓ1([h ∗, ε])µ =
ℓ2(∂xε, h) + ℓ2(h, ∂xε)
+ L2(ε, hhWµ(2), hii)
ihhWµ(ν2) , [hν ∗, ε])ii =
hhWµ(ν3λ) , ∂xν ε hλii + hhWµ(ν3λ) , hν ∂xλεii − L2(ε, hhWµ(2), hii) (3.25)
To understand this relation one must unfold (2.58). On one side we have
XhhWµ(n+1.1..)µ i...µ n , hµ 1 . . . ∂xµi ε . . . hµ n ii
−i XhhWµ(n+1.1..)µ i...µ n , hµ 1 . . . [hµ i ∗, ε] . . . hµ n ii
Next, we know ℓ2(ε1, ε2), ℓ2(ε, h) and ℓ2(h1, h2), and we have to verify L1L2 = L2L1. The
latter is written explicitly in (3.9) and takes the form
ℓ1(ℓ2(ε, h)) = L2(ℓ1(ε), h)+L2(ε, ℓ1(h))
ℓ2(ℓ1(ε), h) + ℓ2(h, ℓ1(ε)) +L2(ε, ℓ1(h))
i[ε ∗, Fµ ] = i X 1
n + 1
On the other side
The two must be equal order by order in h. Thus we have
L2(ε, F (1)) ≡ ℓ2(ε, F (1)) = i[ε ∗, F (1)]
ℓ2(ε, F ) = i[ε ∗, F ]
i[ε ∗, hhWµ(n+1) , h⊗nii] =
XhhWµ(n+1.2..)µ i...µ n+1 , hµ 1 . . . ∂xµi ε . . . hµ n+1 ii
This is a not too disguised form of the Ward identity for the symmetry (2.18). Setting
n = 1 gives precisely (3.25) provided
L2(ε, hhWµ(2), hii) = i[ε ∗, hhWµ(2) , hii]
The quantity F (1) = hhWµ(2) , hii is the lowest order piece of the EoM (of degree -2),
see (3.14). So we can say
The next relation to be verified is
L1(L2(h1, h2)) = L2(L1(h1), h2) − L2(h1, L1(h2))
The entries of L2 on the r.h.s. have degree -3, so they must vanish. On the other hand
L2(h1, h2) on the l.h.s. has degree -2, and is mapped to degree -3 by L1. So it is consistent
to equate both sides to 0. In particular we can set L2(F (1), h) = 0 (and, more generally,
L2(X−2, h) = 0).
First we should evaluate L3(ε1, ε2, ε3). Its degree is 1, therefore it exits the complex. Is it
consistent to set it to 0? The relevant L∞ relation is
0 = ℓ1(L3(x1, x2, x3))
+L3(ℓ1(x1), x2, x3) + (−1)x1L3(x1, ℓ1(x2), x3) + (−1)x1+x2L3(x1, x2, ℓ1(x3))
+L2(L2(x1, x2), x3) + (−1)(x1+x2)x3L2(L2(x3, x1), x2) + (−1)(x2+x3)x1L2(L2(x2, x3), x1)
In our case the second line equals ∂xL3(ε1, ε2, ε3). Thus if we set L3(ε1, ε2, ε3) = 0, the first
two lines vanish. Using (3.11), we see that the third line is nothing but the ∗-Jacobi identity.
Arguing the same way and using the next L∞ relation, which involves L4, one can
show that L4(ε1, ε2, ε3, ε4) = 0, etc.
From (3.7) we also know that L3(ε, h1, h2) ≡ ℓ3(ε, h1, h2) = 0. Following  we will
set also L3(ε1, ε2, h) = 0, L3(ε1, ε2, F (1)) = 0. Therefore
L3(ε1, ε2, ε3) = 0, L3(ε, h1, h2) = 0, L3(ε1, ε2, h) = 0, L3(ε1, ε2, F (1)) = 0
Let us consider next the entries ε1, ε2, h. The terms of the first two lines in (3.12)
vanish due to (3.34). The last line is
ℓ2(ℓ2(ε1, ε2), h) + ℓ2(ℓ2(h, ε1), ε2) + ℓ2(ℓ2(ε2, h), ε1)
= [hµ ∗, [ε1 ∗, ε2]] − [[hµ ∗, ε1] ∗, ε2] + [[hµ ∗, ε2] ∗, ε1]
which vanishes due to ∗-Jacobi identity.
because of (3.34). The rest is
Now we consider the entries ε, h1, h2. Plugging them into (3.12), the first line vanishes
ℓ2(ℓ2(ε, h1), h2) + ℓ2(h2, ℓ2(ε, h1)) − ℓ2(ℓ2(h2, ε), h1)
− ℓ2(h1, ℓ(h2, ε)) + ℓ2(ℓ2(h1, h2), ε) + ℓ2(ℓ2(h2, h1), ε)
ℓ3(ℓ1(ε), h1, h2) + perm3
+L3(ε, ℓ1(h1), h2) − L3(ε, h1, ℓ1(h2))
where perm3 means the permutation of the three entries of ℓ3. Writing down explicitly the
first line, it takes the form
(ℓ3(ℓ1(ε), h1, h2) + perm3) = −
hhWµ(ν4λ)ρ , ∂xνε h1λ h2ρii + perm3
The last two lines of (3.36) give
ℓ2(ℓ2(ε, h1), h2) + ℓ2(h2, ℓ2(ε, h1)) − ℓ2(ℓ2(h2, ε), h1) − ℓ2(h1, ℓ2(h2, ε)) + ℓ2(ℓ2(h1, h2), ε)
+hhWµ(ν3λ) , h1λ[hν2 ∗, ε]ii+[ε ∗, hhWµ(ν3λ) , hν1 h2λii] + [ε ∗, hhWµ(ν3λ) , h2λ hν1ii] (3.38)
Summing the rhs’s of (3.37) and (3.38) one gets, apart from the second line, (3.36) expressed
in terms of the expressions appearing in the r.h.s. of (3.28) with entries h1, h2, instead of
one single h.
Now let us consider (3.28) for n = 2, i.e.
i[ε ∗, hhWµ(ν3λ) , hν hλii] =
1 hhWµ(ν4λ)ρ , ∂xνεhλhρ + hν∂xλεhρ + hν hλ∂xρεii
−i hhWµ(ν3λ) , [hν ∗, ε]hλ + hν[hλ ∗, ε]ii.
which vanishes because of the ∗-Jacobi identity.
This can be read as 1 3
−i[ε ∗, ℓ2(h, h)] = −
ℓ3(∂xε, h, h) + ℓ3(h, ∂xε, h) + ℓ3(h, h, ∂xε)
+iℓ2(h, [h ∗, ε]) + iℓ2([h ∗, ε], h)
Now we consider the same equation obtained by replacing h with h1 + h2 according to the
symmetrization procedure in (3.17). We get in this way the symmetrized equation
−i[ε ∗, ℓ2(h1, h2)] − i[ε ∗, ℓ2(h2, h1)]
ℓ3(∂xε, h1, h2) + ℓ3(∂xε, h2, h1) + ℓ3(h1, ∂xε, h2)
+ℓ3(h2, ∂xε, h1) + ℓ3(h1, h2, ∂xε) + ℓ3(h2, h1, ∂xε)
+iℓ2(h1, [h2 ∗, ε]) + iℓ2(h2, [h1 ∗, ε]) + iℓ2([h1 ∗, ε], h2) + iℓ2([h2 ∗, ε], h1)
This is the same as the sum of the first, third and fourth lines of (3.36), or, alternatively,
the sum of the rhs’s of (3.37) and (3.38).
Thus (3.36) is satisfied if the two remaining terms in the second line vanish. They
are all of the type L3(ε, h, F (1)) and we can assume that such types of terms vanish. So,
beside (3.34) we have
L3(ε, h, E) = −L3(ε, E, h) = 0
where E represent Fµ or anything in X−2.
The relation with entries ε1, ε2 and E is nontrivial and has to be verified. Consider
again (3.12) with entries ε1, ε2 and E. Due to (3.34), (3.42) the relation (3.12) reduces to
the last line:
ℓ2(ℓ2(ε1, ε2), E) + ℓ2(ℓ2(E, ε1), ε2) + ℓ2(ℓ2(ε2, E), ε1)
= iℓ2([ε1 ∗, ε2], E) + iℓ2([E ∗, ε1], ε2) + iℓ2([ε2 ∗, E], ε1)
= +[E ∗, [ε1 ∗, ε2]]−[[E ∗, ε1] ∗, ε2]−[[ε2 ∗, E] ∗, ε1]
The L∞ relation to be proved at degree 1 is 0 = −
ℓ2(ε, ℓ3(h1, h2, h3)) + perm3
L1(L4(x1, x2, x3, x4))
−L2(L3(x1, x2, x3), x4) + (−1)x3x4L2(L3(x1, x2, x4), x3)
+(−1)(1+x1)x2L2(x2, L3(x1, x3, x4)) − (−1)x1L2(x1, L3(x2, x3, x4))
+L3(L2(x1, x2), x3, x4) + (−1)1+x2x3L3(L2(x1, x3), x2, x4)
+(−1)x4(x2+x3)L3(L2(x1, x4), x2, x3)
−L3(x1, L2(x2, x3), x4) + (−1)x3x4L3(x1, L2(x2, x4), x3) + L3(x1, x2, L2(x3, x4))
−L4(L1(x1), x2, x3, x4) − (−1)x1L4(x1, L1(x2), x3, x4)
−(−1)x1+x2L4(x1, x2, L1(x3), x4) − (−1)x1+x2+x4L4(x1, x2, x3, L1(x4)) = 0
L4(ε1, ε2, ε3, ε4) = 0, L4(ε1, ε2, ε3, h) = 0, L4(ε1, ε2, h1, h2) = 0, L4(ε, h1, h2, h3) = 0
The first and second equality have positive degree, so they must vanish. The fourth has
been proven above, see (3.7). The other is an ansatz to be checked by consistency.
The relation (3.44) with four ε entries has already been commented. The same relation
with three ε entries and one h is also trivial as a consequence of (3.34) and (3.45). The same
happens in the case of two ε entries and two h, as a consequence again of (3.34) and (3.45).
Now let us consider the case of one ε and three h’s. Plugging them into (3.44) here is
what we get in terms of ℓi’s (only the nonzero terms are written down)
ℓ3(ℓ2(ε, h1), h2, h3) + ℓ3(ℓ2(ε, h2), h1, h3) + ℓ3(ℓ2(ε, h3), h1, h2) + perm3
ℓ4(ℓ1(ε), h1, h2, h3) + perm4
−L4(ε, ℓ1(h1), h2, h3) + L4(ε, h1, ℓ1(h2), h3)−L4(ε, h1, h2, ℓ1(h3))
where perm3, perm4 refer to the permutations of the ℓ3, ℓ4 entries, respectively.
Disregarding for the moment the last line, which is of type L4(ε, E, h, h), this equation becomes
[ε ∗, hhWµ(ν4λ)ρ , hν1h2λh3ρii] + perm(h1, h2, h3)
+hhWµ(ν4λ)ρ , [hν1 ∗, ε]h2λh3ρii + perm([h1 ∗, ε], h2, h3)
+hhWµ(ν4λ)ρ , [hν2 ∗, ε]h1λh3ρii + perm([h2 ∗, ε], h1, h3)
+hhWµ(ν4λ)ρ , [hν3 ∗, ε]h1λh2ρii + perm([h3 ∗, ε], h1, h2)
hhWµ(ν5λ)ρσ , ∂xνε h1λh2ρh3σii + perm(∂xε, h1, h2, h3)
For comparison let us go back to (3.28) with n = 3. It writes
i[ε ∗, hhWµ(ν4λ)ρ , hν hλhρii]
hhWµ(ν5λ)ρσ , ∂xν εhλhρhσ + hν ∂xλεhρhσ + hν hλ∂xρεhσ + hν hλhρ∂xσεii
− ihhWµ(ν4λ)ρ , [hν ∗, ε]hλhρ + hν [hλ ∗, ε]hρ + hν hλ[hρ ∗, ε]ii
If now we transform the l.h.s. of this equation to a trilinear function of h1, h2, h3 according
to the recipe (3.20), we obtain precisely eq.(3.47). As a consequence we are forced to set
Considering the entries ε, ε, E, h in (3.44) one can show that
L4(ε, ε, E, h) = 0
for consistency. Using this and evaluating (3.44) with entries ε, ε, h, h, one can see that the
third ansatz in (3.45) is justified.
Relation L1Ln + . . . ± LnL1 = 0, degree n − 3
The general L∞ relation is (3.1). As the n = 4 example shows, for n ≥ 4 it is consistent to
set the values of Ln to zero except when all the entries have degree -1. Schematically, out
of (3.1), the only nontrivial relation is
Written in explicit form in terms of ℓn, it is
L4(ε, E, h, h) = L4(ε, h, E, h) = L4(ε, h, h, E) = 0
(n − 1)!
ℓ2(ε, ℓn−1(h1, . . . , hn−1)) + permn−1
ℓn−1(ℓ2(ε, h1), h2, . . . , hn−1) + ℓn−1(ℓ2(ε, h2), h1, . . . , hn−1) + . . .
+ℓn−1(ℓ2(ε, hn−1), h1, . . . , hn−2) + permn−1
ℓn(ℓ1(ε), h1, . . . , hn−1) + permn
In order to obtain this it is essential to remark that, for entries of degree -1, the factor
(−1)σǫ(σ; x) in (3.1) is 1.
Using now the definition (3.15) and simplifying, (3.52) becomes
−i [ε ∗, hhWµ(νn)1...νn−1 , hν11 . . . hνnn−−11 ii] + permn−1
+i hhWµ(νn)1...νn−1 , [ε ∗, hν11 ]hν22 . . . hνnn−−11 ii + W(n)
µν 1...νn−1 , [ε ∗, hν21 ]hν12 . . . hνnn−−11 ii
+ . . . + W(n)
µν 1...νn−1 , [ε ∗, hν1
n−1]hν12 . . . hνnn−−21 ii + permn−1
hhWµ(νn+1.1..)νn , ∂xν1 ε hν12 hν23 . . . hνnn−1ii + permn
where permn−1 means the permutations of h1, . . . , hn−1, and permn means the
permutations of h1, . . . , hn−1 and ∂xε.
Now, from (3.28) we get
1 XhhWµ(n+1.1..)µ i...µ n , hµ 1 . . . ∂xµi ε . . . hµ n ii = 0
If now we transform the l.h.s. of this equation to a multilinear function of h1, . . . , hn−1
according to the recipe (3.20), we obtain precisely (3.53). This completes the proof of the
n-th L∞ relation.
In this paper we have carried out the worldline quantization of a Dirac fermion field coupled
to external sources. In particular, we have determined the formula for the effective action,
by expanding it in a perturbative series, and determined the generalized equations of
motion. This has allowed us, in the second part of the paper, to show that this set up of
the theory accommodates an L∞ algebra. We remark that this applies to the full effective
action, i.e. not only to its local part, but also to its non-local part.
Although we do not give here an explicit proof, the same symmetry characterizes also
the effective action obtained by integrating out a scalar field coupled to the same external
sources. The proof in the scalar case is actually easier, because the corresponding W(n)’s
come out automatically symmetric (for the basic formulas, see ).
An L∞ symmetry is different from the familiar Lie algebra symmetry in that the
equation of motion plays an essential role, in other words the symmetry is dynamical (for
an early formulation in this sense, see ). The full implications of this (more general)
symmetry are not yet clear. It characterizes a large class of perturbative field theories ,
but certainly not all. For instance, it is not present in the open string field theory `a la
Witten, where it is replaced by an A∞ algebra. The classification of field and string theories
on the basis of such homotopic-like algebra symmetries is under way. For the time being
we intend to use it as a basic working tool in our attempt to generate higher spin theories
by integrating out matter fields.
In what concerns us here, the L∞ algebra symmetry is a symmetry of the equation
of motion of the effective action resulting from integrating out the matter fields. The L∞
symmetry descends from the Ward identities of the current correlators of the matter model.
These Ward identities for current correlators imply the higher spin symmetry of effective
action. Therefore one can say that the L∞ symmetry is the source of the HS symmetry of
the effective action. We recall again that the local part of this action is to be identified,
in our approach, with the classical HS action. This has been proved so far only at the
quadratic level. That it is true at the interacting levels is the bet of our program.
Another character of our paper is the worldline quantization. Let us repeat (see
introduction) that it is not imperative to use the worldline formalism. As we have done
in previous papers, one could use the traditional quantization and compute the effective
action by means of Feynman diagrams. The problem with this approach is that we do
not have a way to fix a priori the form of the currents and the form of the symmetry
transformations except by trial and error (a method that becomes rapidly unsustainable
for increasing spin). The worldline quantization grants both at the same time. In this
resides the importance of the worldline quantization.
The way we interpret the L∞ relations among correlators is very similar to the usual
Ward identities for an ordinary gauge symmetry (we have already pointed out above this
parallelism): these relations must hold for both the classical and quantum theory, they are
the relevant defining relations. Their possible breakdown is analogous to the appearances
of anomalies in ordinary gauge Ward identities. It is interesting that possible obstructions
to constructing higher spin theories in our scheme might be identified with such anomalies.6
Finally another consideration: while so far L∞ algebras have been discussed mostly
in relation to classical (first quantized, in the string field theory case) actions, as we have
remarked above, our L∞7 structure characterizes the full effective field action (including
its non-local part). This is perhaps in keeping with what was noticed in [7, 8]: the effective
action for a single higher spin field, at least at the quadratic order, is characterized by a
unique Fronsdal differential operator inflected in various non-local forms. In any case it is
reassuring to find such a symmetry in the one-loop effective actions obtained by integrating
out matter fields. Our idea of using this method to generate higher spin field theories is
perhaps not groundless.
We would like to thank Jim Stasheff for reading the manuscript and for several suggestions.
This research has been supported by the Croatian Science Foundation under the project
No. 8946 and by the University of Rijeka under the research support No. 126.96.36.199.05.
M. P. would like to thank SISSA (Trieste) for support under the Visiting PhD Students
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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