YITP-07-14
Braided quantum field theories
and their symmetries
Yuya Sasai ∗ and Naoki Sasakura †
Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan
Abstract
Braided quantum field theories proposed by Oeckl can provide a framework for
defining quantum field theories having Hopf algebra symmetries. In quantum field the-
ories, symmetries lead to non-perturbative relations among correlation functions. We
discuss Hopf algebra symmetries and such relations in braided quantum field theories.
We give the four algebraic conditions between Hopf algebra symmetries and braided
quantum field theories, which are required for the relations to hold. As concrete exam-
ples, we apply our discussions to the Poincaré symmetries of two examples of noncom-
mutative field theories. One is the effective quantum field theory of three-dimensional
quantum gravity coupled with spinless particles given by Freidel and Livine, and the
other is noncommutative field theory on Moyal plane. We also comment on quantum
field theory on κ-Minkowski spacetime.
∗ e-mail: sasai@yukawa.kyoto-u.ac.jp
† e-mail: sasakura@yukawa.kyoto-u.ac.jp
http://arxiv.org/abs/0704.0822v6
1 Introduction
Symmetry is one of the most important notions in quantum field theory. In many examples, it
is useful in investigating properties of quantum field theories non-perturbatively, is a guiding
principle in constructing field theories for various purposes such as grand unification, or
gives powerful methods in finding exact solutions. It also plays important roles in actual
renormalization procedures. Therefore it should be interesting to study symmetries also in
noncommutative field theories [1, 2, 3, 4, 5], which may result from some quantum gravity
effects [6].
A difficulty in the study in this direction is the apparent violation of basic symme-
tries such as Poincaré symmetry in the noncommutativity of spacetime. For example, the
Moyal plane [xµ, xν ] = iθµν is translational invariant, but is not Lorentz or rotational
invariant. Another example is the three-dimensional spacetime with noncommutativity
[xi, xj] = iκǫijkxk (i, j, k = 1, 2, 3) [7, 8, 9, 10] with a noncommutativity parameter κ. This
noncommutative spacetime is Lorentz-invariant, but is not invariant under the translational
transformation xi → xi + ai with c-number ai. In fact, a naive construction of noncom-
mutative quantum field theory on this spacetime leads to rather disastrous violations of
energy-momentum conservation [10]: the violations coming from the non-planar diagrams
do not vanish in the commutative limit κ→ 0 as in the UV/IR mixing phenomena [11].
In recent years, however, there has been interesting conceptual progress in understanding
symmetries in noncommutative field theories: the symmetry transformations in noncommu-
tative spacetime are not the usual Lie-algebraic type, but should be generalized to have
Hopf algebraic structures. The Moyal plane was pointed out to be invariant under the
twisted Poincaré transformation in [12, 13, 14] and under the twisted diffeomorphism in
[15, 16, 17, 18]. There have been various proposals to implement the twisted Poincaré in-
variance in quantum field theories [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. As for the
noncommutative spacetime with [xi, xj ] = iκǫijkxk, a noncommutative quantum field theory
was derived as the effective field theory of three-dimensional quantum gravity with matters
[31]. Its essential difference from the naive construction mentioned above is the nontrivial
braiding for each crossing in non-planar Feynman diagrams. With this braiding, there ex-
ists a kind of conserved energy-momentum in the amplitudes, and the energy-momentum
operators have Hopf algebraic structures.
Our aim of this paper is to systematically understand these Hopf algebraic symmetries
and their consequences in noncommutative field theories in the framework of braided quan-
tum field theories proposed by Oeckl [34]. In the usual quantum field theories, symmetries
give non-perturbative relations among correlation functions. We will see that such relations
have natural extensions to the Hopf algebraic symmetries in braided quantum field theories,
and will obtain the four conditions for the relations to hold. These conditions should be
interpreted as the criteria of the symmetries in braided quantum field theories.
This paper is organized as follows. In the following section, we review braided quantum
field theory. This review part follows faithfully the original paper [34], but figures are more
extensively used in the proofs and the explanations to make this paper self-contained and
intuitively understandable. We start with braided category and braided Hopf algebra. Then
correlation functions of braided quantum field theory are represented in terms of them.
Finally braided Feynman rules are given.
In Section 3, we first review the axioms of action1 of an algebra on vector spaces. Then
we consider the relations among correlation functions in braided quantum field theory. We
find that four algebraic conditions are required for the relations to hold. Then, as concrete
examples, we discuss whether the noncommutative field theories mentioned above have the
Poincaré symmetry by checking the four conditions. In the former case, we find that the
twisted Poincaré symmetry is implemented only after the introduction of a non-trivial braid-
ing factor, which agrees with the previous proposal in [21, 35]. In the latter case, we find
that the theory has a kind of translational symmetry, which is different from the usual one
by multi-field contributions. We also give some examples of such relations among correlation
functions and the implications.
The final section is devoted to summary and comments. We comment on quantum
field theory on κ-Minkowski spacetime whose noncommutativity of coordinates is [x0, xj] =
xj (j = 1, 2, 3) [36].
2 Review of braided quantum field theory
2.1 Braided categories and braided Hopf algebras
First of all, we review braided categories and braided Hopf algebras [34, 37]. Braided cate-
gories are composed of an object X , which is a vector space, a dual object X∗, which is a
dual vector space, and morphisms
ev : X∗ ⊗X → k (evaluation), (1)
coev : k → X ⊗X∗ (coevaluation), (2)
where k is a c-number. The composition of the two morphisms in an obvious way makes the
identity. Then the braided categories have also an invertible morphism
ψV,W : V ⊗W → W ⊗ V (braiding), (3)
where V,W are any pair of vector spaces. Generally the inverse of braiding is not equal to
the braiding itself.
The braiding is required to be compatible with the tensor product such that
ψU,V⊗W = (id⊗ ψU,W ) ◦ (ψU,V ⊗ id),
ψU⊗V,W = (ψU,W ⊗ id) ◦ (id⊗ ψV,W ). (4)
1We use the italic symbol to distinguish it from the action S.
Figure 1: The evaluation, coevaluation, braiding and its inverse.
Then the braiding is also required to be intersectional under any morphisms in a Hopf
algebra. For example,
ψZ,W (Q⊗ id) = (id⊗Q)ψV,W for any Q : V → Z,
ψV,Z(id⊗Q) = (Q⊗ id)ψV,W for any Q : W → Z, (5)
where Z is a vector space.
We can represent these axioms in pictorial ways [38]. We write the morphisms, ev, coev,
ψ, downwards as in Figure 1. Thus the axioms (4) are represented as in Figure 2, and the
axioms (5) are represented as in Figure 3.
Next we consider the polynomials of X ,
X̂ :=
Xn, with X0 := 1 and Xn := X ⊗ · · · ⊗X︸ ︷︷ ︸
n times
, (6)
where 1 is the trivial one-dimensional space. X̂ naturally has the structure of a braided
Hopf algebra via
· (product) : X̂⊗̂X̂ → X̂, (7)
η (unit) : k → X̂ ; η(1) = 1, (8)
∆ (coproduct) : X̂ → X̂⊗̂X̂ ; ∆φ = φ⊗̂1+ 1⊗̂φ, and ∆(1) = 1⊗̂1, (9)
ǫ (counit) : X̂ → k ; ǫ(φ) = 0, and ǫ(1) = 1, (10)
S (antipode) : X̂ → X̂ ; Sφ = −φ, and S(1) = 1, (11)
where φ ∈ X . The tensor product ⊗ is the same as the usual product of Xs, while the new
tensor product ⊗̂ is the tensor product of X̂s. The coproduct ∆, counit ǫ, antipode S of the
Figure 2: The axioms of braiding (4).
Figure 3: The axioms of braiding (5).
Figure 4: The axioms of coproduct, counit, antipode for products.
products of Xs are defined inductively by
∆ ◦ · = (·⊗̂·) ◦ (id⊗̂ψ⊗̂id) ◦ (∆⊗̂∆), (12)
ǫ ◦ · = · ◦ (ǫ⊗̂ǫ), (13)
S ◦ · = · ◦ ψ ◦ (S⊗̂S). (14)
These axioms are diagrammatically represented in Figure 4.
2.2 Braided quantum field theory
Next we represent braided quantum field theory [34] in terms of the braided category and
the braided Hopf algebra. We take the vector space X as the space of a field φ(x), where
x denotes a general index for independent modes of the field. Thus X̂ is the space of
polynomials of the fields such as φ(x1)φ(x2) · · ·φ(xn), and 1 correspond to the constant field
of unit. We also take the dual vector space X∗ as the space of differentials δ/δφ(x). We take
the evaluation and the coevaluation as follows,
δφ(x)
⊗ φ(x′) → δ(x− x′), (15)
coev : 1 →
φ(x)⊗ δ
δφ(x)
, (16)
Figure 5: The differentials on X̂.
where the distribution and the integration should symbolically be understood, and their
detailed forms, which may contain non-trivial measures, depend on each case.
The differential on X̂ is defined by
diff := (êv ⊗ id) ◦ (id⊗∆); X∗ ⊗ X̂ → X̂, (17)
where
êv|X∗⊗Xn =
ev for n = 1,
0 for n 6= 1.
Diagrammatically this is given by Figure 5.
To see whether the map diff gives really the differential of products, let us compute the
differential of φ(x)φ(y) as a simple example using the definition (17). This becomes
δφ(x′)
⊗ φ(x)φ(y)
= (êv ⊗ id) ◦ (id⊗∆)
δφ(x′)
⊗ φ(x)φ(y)
= (êv ⊗ id) ◦
δφ(x′)
⊗∆(φ(x)φ(y))
= (êv ⊗ id) ◦
δφ(x′)
⊗ (φ(x)φ(y)⊗̂1
+ φ(x)⊗̂φ(y) + ψ(φ(x)⊗̂φ(y)) + 1⊗̂φ(x)φ(y))
= δ(x′ − x)⊗ φ(y) + (êv ⊗ id) ◦
δφ(x′)
⊗ ψ(φ(x)⊗̂φ(y))
where we have used the axiom (12) in deriving the third line. If the braiding is trivial, we
find that the differential (17) satisfies the usual Leibniz rule.
Figure 6: Diagram of ψn,m.
Generally we find a braided Leibniz rule
∂(αβ) = ∂(α)β + ψ−1(∂ ⊗ α)(β) (20)
∂(α) = (ev ⊗ idn−1)(∂ ⊗ [n]ψα), (21)
where ∂ ∈ X∗, α, β ∈ X̂ , and we have used a simplified notation
∂(α) := diff(∂ ⊗ α). (22)
Here n is the degree of α, and [n]ψ is called a braided integer defined by
[n]ψ := id
n + ψ ⊗ idn−2 + · · ·+ ψn−2,1 ⊗ id + ψn−1,1, (23)
where ψn,m is a braiding morphism given in Figure 6.
The proofs of the formula (20), (21) are in Appendix A.
Now we define a Gaussian integration, which defines the path integral. The definition is
given by ∫
∂(αw) := 0 for ∂ ∈ X∗, α ∈ X̂, (24)
where w ∈ X̂ is a Gaussian weight. In field theory, w is the exponential of the free part of
the action, e−S0 .
In order to obtain a formula for correlation functions, we define a morphism γ : X∗ → X
such that
∂(w) := −γ(∂)w. (25)
This morphism is assumed to be commutative with the braiding as in (5). If w = e−S0 ,
γ(∂) = ∂(S0). In field theory, this is the kinetic part of the action, or the inverse of the
propagator.
Starting from (24), we can represent correlation functions of a free field theory in terms of
the braided category and the braided Hopf algebra. This is the analog of the Wick theorem
in braided quantum field theory. The definition of the free n point correlation function is
given by
Z(0)n (α) :=
, (26)
where the degree of α is n. Algebraically, this is given by
2 = ev ◦ (γ−1 ⊗ id) ◦ ψ, (27)
2n = (Z
n ◦ [2n− 1]′ψ!!, (28)
2n−1 = 0, (29)
where
[2n− 1]′ψ!! := ([1]′ψ ⊗ id2n−1) ◦ ([3]′ψ ⊗ id2n−3) ◦ · · · ◦ ([2n− 1]′ψ ⊗ id), (30)
ψ := id
n + idn−2 ⊗ ψ−1 + · · ·+ ψ−11,n−1
= ψ−11,n−1 ◦ [n]ψ. (31)
The proofs of (27), (28), (29) are in Appendix B.
Next we consider correlation functions with the existence of an interaction. For S =
S0 + λSint, a correlation function is perturbatively given by
Zn(α) =
αe−S∫
α(1− λSint + · · · )e−S0∫
(1− λSint + · · · )e−S0
, (32)
where α ∈ Xn. Introducing a morphism Sint : k → Xk, where k is the degree of Sint, the
correlation function is algebraically given by
n − λZ(0)n+k ◦ (id
n ⊗ Sint) + 12λ
n+2k ◦ (id
n ⊗ Sint ⊗ Sint) + · · ·
1− λZ(0)
◦ Sint + 12λ2Z
2k ◦ (Sint ⊗ Sint) + · · ·
. (33)
Acting Zn on α ∈ Xn, we obtain the correlation function (32). One can obviously extend
Sint to include various interaction terms.
2.3 Braided Feynman rules
From the results in the preceding subsection, a correlation function can be represented by
summation of diagrams obeying the following rules below.
Figure 7: Propagator (left) and vertex (right).
Figure 8: The braiding ψ (left) and its inverse ψ−1 (right).
• An n-point function Zn is a morphism Xn → k. Thus a Feynman diagram starts with
n strands at the top and must be closed at the bottom.
• The propagator Z(0)2 : X ⊗X → k is represented by the left of Figure 7, which is the
abbreviation of Figure 9.
• The interaction vertex Sint : k → Xk is represented by the right of Figure 7. Generally
the order of the strands is noncommutative.
• The two kinds of crossings, which are represented in Figure 8, correspond to the braid-
ing and its inverse.
• Any Feynman diagram is built out of propagators, vertices, and crossings, and is closed
at the bottom.
3 Symmetries in braided quantum field theory
In this section, we discuss symmetries in braided quantum field theory. In order to represent
symmetry transformations on fields, we review general description of an action in Section
Figure 9: The propagator, which is abbreviated in the left figure of Figure 7.
3.1. In Section 3.2, we study relations among correlation functions. We find four conditions
for such relations to follow from a symmetry algebra. In Section 3.3 and 3.4, we treat two
examples of (braided) noncommutative field theories and discuss their Poincaré symmetries.
3.1 General description of an action
We review an action of a general Hopf algebra on vector spaces in a mathematical language
[37, 39].
An action αV is a map αV : A⊗ V → V , where A is an arbitrary Hopf algebra and V is
a vector space (in our case, A is a symmetry algebra, and V = X or X∗). We will denote
the coproduct and the counit of the Hopf algebra2 by ∆′ and ǫ′ to distinguish them from
those of the braided Hopf algebra of fields in Section 2. We do not write all the axioms of
an action, but our important axioms are the following.
• αV satisfies the following condition.
αV ◦ (· ⊗ id) = αV ◦ (id⊗ αV ), (34)
where the equality acts on A ⊗ A ⊗ V . This means that αV ((a · b) ⊗ V ) = αV (a ⊗
(αV (b⊗ V ))), where a, b ∈ A. In short we can write this as
(a · b) ⊲ V = a ⊲ (b ⊲ V ). (35)
• An action on 1, which is in a vector space, is defined by
αV (a⊗ 1) = ǫ′(a)1, (36)
where ǫ′(a) is the counit of an algebra a ∈ A.
• An action on a tensor product of vector spaces V,W is defined by
αV⊗W (a) := ((αV ⊗ αW ) ◦∆′)(a) =
αV (a
(1))⊗ αW (ai(2)), a ∈ A, (37)
where ∆′(a) =
ai(1) ⊗ ai(2) is the coproduct of the Hopf algebra A. In the case of a
usual Lie-algebraic transformation, its coproduct is given by ∆′(a) = a ⊗ 1 + 1 ⊗ a,
where 1 is in A. This gives the usual Leibnitz rule.
• Since a Hopf algebra has the coassociativity that
((∆′ ⊗ id) ◦∆′)(a) = ((id⊗∆′) ◦∆′)(a), (38)
2We omit the antipode.
the action on a tensor product of vector spaces, which is obtained by the multiple
operations of ∆′ on a, is actually unique. An important consequence is that one can
divide the action on a tensor product of vector spaces as
a⊲(V1 ⊗ · · · ⊗ Vk−1 ⊗ Vk ⊗ · · · ⊗ Vn) =∑
ai(1) ⊲ (V1 ⊗ · · · ⊗ Vk−1)⊗ ai(2) ⊲ (Vk ⊗ · · · ⊗ Vn) (39)
for any k.
3.2 Symmetry relations among correlation functions and their al-
gebraic descriptions
The expression of the correlation functions (33) is perturbative in interactions, but is a full
order algebraic description. Therefore we can discuss the symmetry of the theory and the
implied relations among correlation functions by using this expression. We may even expect
that the relations will hold non-perturbatively.
In usual quantum field theory, if a field theory has a certain symmetry, there is a relation
among the correlation functions in the form,
〈φ(x1) · · · δaφ(xi) · · ·φ(xn)〉 = 0, (40)
where δaφ(x) is a variation of a field under a transformation a, on the assumption that the
path integral measure and the action are invariant under the transformation.
If the coproduct of a symmetry algebra is not the usual Lie-algebraic type and thus the
Leibniz rule is deformed, the relation will generally have the form,
c(bi)a 〈φ(x1) · · · δbφ(xi) · · ·φ(xn)〉
+c(bi)(cj)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · ·φ(xn)〉
+c(bi)(cj)(dk)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · · δdφ(xk) · · ·φ(xn)〉
+ · · · = 0, (41)
where c···a are some coefficients. Its essential difference from (40) is the multi-field contribu-
tions. In our algebraic language, the relation can be written as
Zn(a ⊲ χ) = ǫ
′(a)Zn(χ), for a ∈ A, χ ∈ Xn. (42)
This is equivalent to Figure 10 in our diagrammatic representation. Then we consider what
an algebraic structure is required for (42) to hold for any a and χ, i.e. the theory is invariant
under the Hopf algebra transformation A.
Figure 10: A relation among correlation functions in the diagrammatic representation. n
is the number of external legs, k is the order of the interaction, and p is the order of the
perturbation. n+ kp is even.
Let us write the coproduct of an element a ∈ A as
∆′(a) =
f s ⊗ gs, (43)
where f s, gs ∈ A. Since the coproduct must satisfy the Hopf algebra axiom [37],
(ǫ′ ⊗ id)∆′(a) = (id⊗ ǫ′)∆′(a) = a, (44)
f s, gs must satisfy
ǫ′(f s)⊗ gs =
f s ⊗ ǫ′(gs) = a. (45)
For all the relations among correlation functions to hold, we find the following four
conditions for any action a ∈ A.
• (Condition 1) Sint must satisfy
a ⊲ Sint = ǫ
′(a)Sint. (46)
• (Condition 2) The braiding ψ is an intertwining operator. That is
ψ(a ⊲ (V ⊗W )) = a ⊲ ψ(V ⊗W ). (47)
• (Condition 3) γ−1 and a are commutative,
a ⊲ (γ−1(V )) = γ−1(a ⊲ V ). (48)
• (Condition 4) Under an action a, the evaluation map follows
ev(a ⊲ (X∗ ⊗X)) = ǫ′(a)ev(X∗ ⊗X). (49)
Condition 1 to 4 are diagrammatically represented in Figure 11. It is clear that, when the
algebra A is generated from a finite number of its independent elements, it is enough for
these generators to satisfy these conditions.
Condition 1 is the requirement of the symmetry at the classical level for the interaction.
We can extend this condition to
(a ⊲ Xn)⊗ Spint = a ⊲ (Xn ⊗ S
int). (50)
The proof is the following. From a coproduct (43) and its coassociativity (39), the right
hand side of (50) is equal to
(f s ⊲ (Xn ⊗ Sp−1int ))⊗ gs ⊲ Sint (51)
Since Condition 1 implies
gs ⊲ Sint = ǫ
′(gs)Sint, (52)
(51) becomes
(f s ⊲ (Xn ⊗ Sp−1int ))⊗ ǫ′(gs)Sint
= a ⊲ (Xn ⊗ Sp−1int )⊗ Sint, (53)
where we have used (45). Iterating this procedure, we obtain the left-hand side of (50).
Condition 2,3,4 can also be extended to
[n + kp− 1]ψ!! ◦ (a ⊲ Xn+kp) = a ⊲ [n+ kp− 1]ψ!! Xn+kp, (54)
(γ−1 ⊗ id)
2 ◦ (a ⊲ Xn+kp) = a ⊲ (γ−1 ⊗ id)
2 Xn+kp, (55)
2 (a ⊲ (X∗ ⊗X)
2 ) = ǫ′(a) ev
2 (X∗ ⊗X)
2 . (56)
We can find that these extended conditions (50), (54), (55), (56) can be represented as
in Figure 12. In the diagrammatic language, the relation among correlation functions holds
if an action can pass downwards through a Feynman diagram and satisfies (36).
3.3 Symmetries of the effective noncommutative field theory of
three-dimensional quantum gravity coupled with scalar parti-
In this subsection, we discuss the Poincaré symmetry of the effective noncommutative field
theory of three-dimensional quantum gravity coupled with scalar particles, which was ob-
tained in [31] by studying the Ponzano-Regge model [40] coupled with spinless particles. The
Figure 11: Conditions 1,2,3, and 4.
Figure 12: A relation among correlation functions is satisfied if the four conditions (46),
(47), (48), (49) are satisfied.
symmetries of this theory is also known as DSU(2), which was discussed in [32, 33]. We first
review the field theory [10, 31].
Let φ(x) be a scalar field on a three-dimensional space x = (x1, x2, x3). Its Fourier
transformation is given by
φ(x) =
dgφ̃(g)e
tr(Xg), (57)
where κ is a constant, X = ixiσi, and g = P 0−iκP iσi ∈ SO(3)3 with Pauli matrices σi. Here∫
dg is the Haar measure of SO(3) and P 0 = ±
1− κ2P iPi by definition. In the following
discussions, we will only deal with the Euclidean case, but the Lorentzian case can also be
treated in a similar manner by replacing SO(3) with SL(2, R).
The definition of the star product is given by
tr(Xg1) ⋆ e
tr(Xg2) := e
tr(Xg1g2). (58)
Differentiating both hands sides of (58) with respect to P i1 := P
i(g1) and P
2 := P
j(g2) and
then taking the limit P i1, P
2 → 0, one finds the SO(3) Lie-algebraic space-time noncommu-
tativity [7, 8, 9],
[xi, xj]⋆ = 2iκǫ
ijkxk. (59)
For example, the action4 of a φ3 theory is
(∂iφ ⋆ ∂iφ)(x)−
M2(φ ⋆ φ)(x) +
(φ ⋆ φ ⋆ φ)(x)
, (60)
where M2 = sin
. Its momentum representation is
P 2(g)−M2
φ̃(g)φ̃(g−1)
dg1dg2dg3δ(g1g2g3)φ̃(g1)φ̃(g2)φ̃(g3), (61)
from which it is straightforward to read the Feynman rules.
Some quantum properties of this scalar field theory were analyzed in [10]. As can be
seen from (59), the naive translational symmetry is violated. In fact, the violation is rather
disastrous. There exists a kind of conserved energy-momentum in the amplitudes of the
tree and the planar loop diagrams, but this energy-momentum is not conserved in the non-
planar loop diagrams. Moreover, the violation of the energy-momentum conservation does
not vanish in the commutative limit κ → 0 due to a mechanism similar to the UV/IR
phenomena [11].
3The identification g ∼ −g is implicitly assumed.
4Since in the Ponzano-Regge model the definition of the weight of partition function is eiS despite of
Euclidean theory, the sign of the mass term is not the usual one.
In the effective field theory of quantum gravity coupled with spinless particles, however,
the Feynman rules contain also a non-trivial braiding rule for each crossing, which comes
from a flatness condition in a graph of intersecting particles [31]. This can be incorporated
as a braiding between the scalar fields,
ψ(φ̃(g1)φ̃(g2)) = φ̃(g2)φ̃(g
2 g1g2), (62)
in the braided quantum field theory.
From the direct analysis of the Feynman graphs with this braiding rule, one can easily find
that the energy-momentum mentioned above is conserved also in the non-planar diagrams.
This suggests the existence of a translational symmetry in the quantum field theory. In
the sequel, we will discuss the embedding of this field theory into the framework of braided
quantum field theory, and will check the four conditions for its translational and rotational
symmetries.
We use the momentum representation, and take X as the space of φ̃(g) and X∗ as that
δφ̃(g)
. We take the braided Hopf algebra of the fields as follows,
∆ : φ̃(g) → φ̃(g)⊗̂1 + 1⊗̂φ̃(g), (63)
ǫ : φ̃(g) → 0, (64)
S : φ̃(g) → −φ̃(g), (65)
ψ : φ̃(g1)⊗ φ̃(g2) → φ̃(g2)⊗ φ̃(g−12 g1g2). (66)
The evaluation and coevaluation maps are given by
δφ̃(g)
⊗ φ̃(g′) → δ(g−1g′), (67)
coev : 1 →
dgφ̃(g)⊗ δ
δφ̃(g)
. (68)
From γ(∂) = ∂S0 = (P
2(g)−m2)φ̃(g−1),
γ−1(φ̃(g)) =
P 2(g−1)−m2
δφ̃(g−1)
. (69)
From the algebraic consistencies in Figure 13, the braidings between X and X∗ and the
braiding between X∗s are determined to be
δφ̃(g1)
⊗ φ̃(g2)
= φ̃(g2)⊗
δφ̃(g−12 g1g2)
, (70)
φ̃(g1)⊗
δφ̃(g2)
δφ̃(g2)
⊗ φ̃(g2g1g−12 ), (71)
δφ̃(g1)
δφ̃(g2)
δφ̃(g2)
δφ̃(g2g1g
. (72)
Figure 13: The algebraic consistency conditions of coevaluation map and X , X∗.
In this derivation, we have used the invariance of the Haar measure d(g−1g′g) = dg′.
Now we consider a translational transformation of the field. If we shift xi to xi + ǫi, a
field φ(x) becomes
φ(x) → φ(x+ ǫ)
dgφ̃(g)ei(x+ǫ)
iPi(g)
dg(1 + iǫiPi(g))φ̃(g)e
ixiPi(g). (73)
Thus in the momentum representation, the translational transformation corresponds to an
action
P i ⊲ φ̃(g) = P i(g)φ̃(g), P 0 ⊲ φ̃(g) = P 0(g)φ̃(g). (74)
From the requirement that the star product (58) conserve a kind of momentum, the action
on a product of fields should be
P i ⊲ (φ̃(g1)φ̃(g2)) = P
i(g1g2)φ̃(g1)φ̃(g2)
= (P 01P
2 + P
1 + κǫ
2 )φ̃(g1)φ̃(g2), (75)
P 0 ⊲ (φ̃(g1)φ̃(g2)) = (P
2 − κ2P i1P2i)φ̃(g1)φ̃(g2). (76)
This determines the coproduct of P i, P 0 as
∆′(P i) = P 0 ⊗ P i + P i ⊗ P 0 + κǫijkP j ⊗ P k, (77)
∆′(P 0) = P 0 ⊗ P 0 − κ2P i ⊗ Pi. (78)
This coproduct satisfies the coassociativity, which essentially comes from the associativity
of the group multiplication.
From the axiom (44), the counit of P i, P 0 is given by
ǫ′(P i) = ǫ′(P 0) = 0. (79)
Since the conservation of momentum under the coevaluation map (68) requires that the
action of P i on
dg(φ̃(g)⊗ δ
δφ̃(g)
) vanish from (36), the action of P i on δ
δφ̃(g)
must be
P i ⊲
δφ̃(g)
= P i(g−1)
δφ̃(g)
. (80)
In the following, we see that the momentum algebra satisfies the four conditions (46),
(47), (48), (49).
Condition 1 is satisfied since
P i ⊲ Sint
dg1dg2dg3δ(g1g2g3)P
i ⊲ (φ̃(g1)φ̃(g2)φ̃(g3))
dg1dg2dg3δ(g1g2g3)P
i(g1g2g3)(φ̃(g1)φ̃(g2)φ̃(g3))
= 0. (81)
Condition 2 is satisfied since
ψ(P i ⊲ (φ̃(g1)φ̃(g2))) = P
i(g1g2)(φ̃(g2)φ̃(g
2 g1g2)),
P i ⊲ ψ(φ̃(g1)φ̃(g2)) = P
i(g2g
2 g1g2)(φ̃(g2)φ̃(g
2 g1g2)).
Condition 3 is satisfied since
P i ⊲ γ−1(φ̃(g)) =
P 2(g−1)−m2
P i(g)
δφ̃(g−1)
γ−1(P i ⊲ φ̃(g)) =
P 2(g−1)−m2
P i(g)
δφ̃(g−1)
Condition 4 is satisfied since
P i ⊲
δφ̃(g1)
⊗ φ̃(g2)
= P i(g−11 g2) ev
δφ̃(g1)
⊗ φ̃(g2)
= 0. (82)
Thus we find that the effective braided noncommutative field theory of three-dimensional
quantum gravity coupled with spinless particles has the translational symmetry.
Next we consider a rotational symmetry. The rotational symmetry corresponds to an
action
Λ ⊲ φ̃(g) = φ̃(h−1gh), (83)
which is the usual Lie-group one. The action on the tensor product is
Λ ⊲ (φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g1h)⊗ φ̃(h−1g2h). (84)
Thus the coproduct of the rotational symmetry is given by
∆′(Λ) = Λ⊗ Λ. (85)
From the axiom (44), the counit of Λ is given by
ǫ′(Λ) = 1. (86)
Condition 1 is satisfied since
Λ ⊲ Sint
dg1dg2dg3δ(g1g2g3)Λ ⊲ (φ̃(g1)φ̃(g2)φ̃(g3))
dg1dg2dg3δ(g1g2g3)(φ̃(h
−1g1h)φ̃(h
−1g2h)φ̃(h
−1g3h))
=ǫ′(Λ)Sint. (87)
Condition 2 is satisfied since
ψ(Λ ⊲ (φ̃(g1)⊗ φ̃(g2))) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h)
Λ ⊲ ψ(φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h). (88)
Condition 3 is satisfied since
Λ ⊲ γ−1(φ̃(g)) =
P 2(g−1)−m2
δφ̃(h−1g−1h)
γ−1(Λ ⊲ φ̃(g)) =
P 2(h−1g−1h)−m2
δφ̃(h−1g−1h)
P 2(g−1)−m2
δφ̃(h−1g−1h)
. (89)
Condition 4 is satisfied since
δφ̃(g1)
⊗ φ̃(g2)
δφ̃(h−1g1h)
⊗ φ̃(h−1g2h)
= δ(g−11 g2)
= ǫ′(Λ)ev
δφ̃(g1)
⊗ φ̃(g2)
. (90)
Thus we find that this braided noncommutative field theory has also the rotational sym-
metry.
3.4 Twisted Poincaré symmetry of noncommutative field theory
on Moyal plane
In this subsection, we discuss the twisted Poincaré symmetry of noncommutative field theory
on Moyal plane [xµ, xν ] = iθµν .
For example, the action of a φ3 theory is given by
(∂µφ ∗ ∂µφ)(x)−
m2(φ ∗ φ)(x) + λ
(φ ∗ φ ∗ φ)(x)
, (91)
where the star product is given by
φ(x) ∗ φ(x) = e
θµν∂xµ∂
νφ(x)φ(y)
. (92)
In the momentum representation, the action is
(p2 −m2)φ̃(p)φ̃(−p)
dDp1d
µνp2νδ(p1 + p2 + p3)φ̃(p1)φ̃(p2)φ̃(p3)
. (93)
We take X as the space of φ̃(p) and X∗ as that of δ
δφ̃(p)
. Then we take the braided Hopf
algebra as follows:
∆ : φ̃(p) → φ̃(p)⊗̂1+ 1⊗̂φ̃(p), (94)
ǫ : φ̃(p) → 0, (95)
S : φ̃(p) → −φ̃(p). (96)
From γ(∂) = ∂S0 = (p
2 −m2)φ̃(−p),
γ−1(φ̃(p)) =
p2 −m2
δφ̃(−p)
. (97)
Let us consider the twisted Poincaré symmetry [12, 13, 14]. The coproduct and the counit
of the twisted Poincaré algebra is given by
∆′(P µ) = P µ ⊗ 1+ 1⊗ P µ,
ǫ′(P µ) = 0,
∆′(Mµν) =Mµν ⊗ 1+ 1⊗Mµν
θαβ [(δµαP
ν − δναP µ)⊗ Pβ + Pα ⊗ (δ
ν − δνβP µ)],
ǫ′(Mµν) = 0. (98)
Thus the action of the twisted Lorentz algebra on the tensor product is
Mµν ⊲ (φ̃(p1)⊗ φ̃(p2)) =Mµν ⊲ φ̃(p1)⊗ φ̃(p2) + φ̃(p1)⊗Mµν ⊲ φ̃(p2)
θαβ [(δµαP
ν − δναP µ) ⊲ φ̃(p1)⊗ Pβ ⊲ φ̃(p2)
+ Pα ⊲ φ̃(p1)⊗ (δµβP
ν − δνβP µ) ⊲ φ̃(p2)], (99)
where Mµν ⊲ φ̃(p) = i(pµ∂/∂pν − pν∂/∂pµ)φ̃(p) and P µ ⊲ φ̃(p) = pµφ̃(p). The actions of Mµν
and P µ on δ
δφ̃(p)
Mµν ⊲
δφ̃(p)
= i(pµ∂/∂pν − pν∂/∂pµ)
δφ̃(p)
, (100)
P µ ⊲
δφ̃(p)
= −pµ δ
δφ̃(p)
. (101)
One easily finds that three conditions (46), (48), (49) are satisfied, but (47) is not if the
braiding is trivial. In order to keep the invariance, the braiding must be taken as
ψ(φ̃(p1)⊗ φ̃(p2)) = eiθ
αβp2α⊗p1β(φ̃(p2)⊗ φ̃(p1)). (102)
This agrees with the previous proposal [21, 35].
We can easily check that the translational symmetry holds since the coproduct ∆′(P µ)
follows the usual Leibniz rule.
3.5 Relations among correlation functions : Examples
Now we have checked, in all orders of perturbation, that the two theories in the preceding
sections have symmetry relations among correlation functions implied by the Hopf algebra
symmetries. In Section 3.3 we gave how the translational generator acts on a product of
fields in (75), (76) in the momentum representation. Since the physical meaning of this Hopf
algebra transformation is not so clear, it would be interesting to see explicitly the symmetry
relations among correlation functions. The same thing is also true in the case of the twisted
Lorentz symmetry in Section 3.4. In this subsection, we work out explicitly some relations
among correlation functions in the two theories.
In the effective quantum field theory of quantum gravity, the action of the translational
generators on a correlation function is given by
〈φ̃(g1) · · · φ̃(gn)〉 → iǫiPi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 (103)
in the momentum representation, where ǫi is an infinitesimal parameter. Thus we obtain a
relation,
Pi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 = 0. (104)
This is a (modified) momentum conservation law; the correlation function has support only
on the vanishing momentum subspace, Pi(g1 · · · gn) = 0. This all-order relation in the
quantum field theory would be a simple but an important implication of the Hopf algebraic
translational symmetry. This provides a good example of the physical importance of a Hopf
algebraic symmetry: a Hopf algebra symmetry leads to a (modified) conservation law.
It would also be interesting to see the relations in the coordinate representations, where
the fields are defined by φ(x) =
eip·xφ̃(p). As explicitly noted in the preceding subsections,
we stress that the basis of the spaces X of the field variables in the path integrals are
parameterized in terms of momenta, and that φ(x) are defined by some c-number linear
combinations of them. Therefore, an action a ∈ A of a symmetry transformation acts as
a ⊲ φ(x) =
eip·x(a ⊲ φ̃(p)), (105)
and the symmetry relations of correlation functions can be obtained by some inverse Fourier
transformations (with possible non-trivial measures) of those in momentum representations.
For example, in the case of the two point function, after the inverse Fourier transforma-
tion, the relation among correlation functions is given by
〈∂iφ(x1)φ(x2) + φ(x1)∂iφ(x2)〉 = 0, (106)
where we have used the relation (104). Interestingly, this is the usual relation in a transla-
tionally invariant quantum field theory. In the case of the three point function, however, the
relation is given by
〈∂iφ(x1)
1 + κ2∂2φ(x2)
1 + κ2∂2φ(x3) +
1 + κ2∂2φ(x1)∂
iφ(x2)
1 + κ2∂2φ(x3)
1 + κ2∂2φ(x1)
1 + κ2∂2φ(x2)∂
iφ(x3) + iκǫ
1 + κ2∂2φ(x1)∂jφ(x2)∂kφ(x3)
+ iκǫijk∂jφ(x1)
1 + κ2∂2φ(x2)∂kφ(x3)− iκǫijk∂jφ(x1)∂kφ(x2)
1 + κ2∂2φ(x3)
+ κ2∂jφ(x1)∂
jφ(x2)∂
iφ(x3)− κ2∂iφ(x1)∂kφ(x2)∂kφ(x3)
+ κ2∂kφ(x1)∂
iφ(x2)∂
kφ(x3)〉 = 0. (107)
This is quite a non-trivial relation among correlation functions, and would be hard to find,
if the Hopf algebra symmetry in the quantum field theory was not noticed. This would be
another interesting example implying the physical importance of a Hopf algebra symmetry.
In general, the relation has the form,
∂xli − i
κǫijk∂xlj∂xmk +O(κ
2))〈φ(x1) · · ·φ(xn)〉 = 0. (108)
In the κ → 0 limit, the relation approaches the usual relation. Thus the Hopf algebra sym-
metry is a kind of translational symmetry modified by adding κ dependent higher derivative
multi-field contributions.
We can proceed in a similar manner for the twisted Lorentz symmetry. We have a general
form of such a symmetry relation as
Mµν ⊲ 〈φ̃(p1) · · · φ̃(pn)〉 = 0. (109)
In the case of the two point function, the relation is given by
〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2) + φ(x1)(x
ν − xν2∂µ)φ(x2)〉 = 0, (110)
where we have used the momentum conservation. This is the same relation as that in a
Lorentz invariant quantum field theory. In the case of the three point function, the relation
is given by
〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2)φ(x3)
+ φ(x1)(x
ν − xν2∂µ)φ(x2)φ(x3) + φ(x1)φ(x2)(x
ν − xν3∂µ)φ(x3)
iθαµ(∂αφ(x1)∂
νφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂
νφ(x3) + φ(x1)∂αφ(x2)∂
νφ(x3)
− ∂νφ(x1)∂αφ(x2)φ(x3)− ∂νφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂νφ(x2)∂αφ(x3))
iθαν(∂αφ(x1)∂
µφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂
µφ(x3) + φ(x1)∂αφ(x2)∂
µφ(x3)
− ∂µφ(x1)∂αφ(x2)φ(x3)− ∂µφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂µφ(x2)∂αφ(x3))〉 = 0. (111)
In general, the relation among correlation functions has the from,
((x1µ∂x1ν − x1ν∂x1µ) + · · ·+ (xnµ∂xnν − xnν∂xmν) +O(θ))〈φ(x1) · · ·φ(xn)〉 = 0 (112)
in the coordinate representation. The leading terms corresponds to the usual Lorentz trans-
formation xµ → xµ + ǫµνxν .
The above symmetry relations on Moyal plane can be represented in similar manners
as the usual commutative cases, if we use star products. In the papers [24, 25, 26, 27,
28, 29, 30], they have pointed out that in coordinate representation, correlation functions on
Moyal plane should be defined with star products extended to non-coincident points (see also
[43]) instead of usual products since the usual commutative commutation relation [x
i , x
j ] =
0 (i, j = 1, · · · , n) is not invariant under the twisted Poincaré transformation. Carrying
out Fourier transformation of the symmetry relation (109) in momentum representation to
such a noncommutative coordinate representation, we obtain the symmetry relations in star
tensor products. Namely (110) becomes
〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) + φ(x1) ∗ ((x
ν − xν2∂µ)φ(x2))〉 = 0, (113)
and (111) becomes
〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) ∗ φ(x3)
+ φ(x1) ∗ ((xµ2∂ν − xν2∂µ)φ(x2)) ∗ φ(x3) + φ(x1) ∗ φ(x2) ∗ ((x
ν − xν3∂µ)φ(x3))
iθαµ(∂αφ(x1) ∗ ∂νφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2) ∗ ∂νφ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂νφ(x3)
− ∂νφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂νφ(x1) ∗ φ(x2)∂α ∗ φ(x3)− φ(x1) ∗ ∂νφ(x2) ∗ ∂αφ(x3))
iθαν(∂αφ(x1) ∗ ∂µφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2)∂µ ∗ φ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂µφ(x3)
− ∂µφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂µφ(x1) ∗ φ(x2) ∗ ∂αφ(x3)− φ(x1) ∗ ∂µφ(x2) ∗ ∂αφ(x3))〉 = 0.
(114)
More generally we can derive the symmetry relations of correlation functions for tensor
fields φα1···αn(x) ≡ ∂α1 · · ·∂αnφ(x). For example in the case of the three point function of the
tensor fields, the symmetry relation becomes
〈((M1µν)α1···αl
δ1···δlφδ1···δl(x1)) ∗ φβ1···βm(x2) ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ ((M
2µν)β1···βm
δ1···δmφδ1···δm(x2)) ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ φβ1···βm(x2) ∗ ((M
3µν)γ1···γn
δ1···δnφδ1···δn(x3))
θαµ[∂αφα1···αl(x1) ∗ ∂
νφβ1···βm(x2) ∗ φγ1···γn(x3)
+ ∂αφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂
νφγ1···γn(x3)
+ φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂
νφγ1···γn(x3)
− ∂νφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3)
− ∂νφα1···αl(x1) ∗ φβ1···βm(x2)∂α ∗ φγ1···γn(x3)
− φα1···αl(x1) ∗ ∂
νφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)]
θαν [∂αφα1···αl(x1) ∗ ∂
µφβ1···βm(x2) ∗ φγ1···γn(x3)
+ ∂αφα1···αl(x1) ∗ φβ1···βm(x2)∂
µ ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂
µφγ1···γn(x3)
− ∂µφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3)
− ∂µφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂αφγ1···γn(x3)
− φα1···αl(x1) ∗ ∂
µφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)]〉 = 0, (115)
where
(Mµν)α1···αn
β1···βn = (Lµν)α1···αn
β1···βn + (Sµν)α1···αn
β1···βn
(Lµν)α1···αn
β1···βn = i(xµ∂ν − xν∂µ)δα1β1 · · · δαnβn
(Sµν)α1···αn
β1···βn = i(ηνβ1δ{α1
β2 · · · δαn}βn − ηµβ1δ{α1νδα2β2 · · · δαn}βn) (116)
If we bring the operators (M iµν)α1···αn
β1···βn (i = 1, 2, 3) out of the star products, θµν depen-
dent terms are canceled. The final expressions are just the usual Lorentz rotations on the
coordinates and the tensorial indices in the correlation functions. This is fully consistent
with the discussions in [29].
3.6 Origin of Hopf algebra symmetries
To study more the meaning of these additional terms, let us see closer the transformation
properties of the star products. In the latter case, it is known that the θµν dependence of
the twisted Lorentz transformation (99) comes from the Lorentz transformation of θµν itself
[41]. To see this, let us consider an infinitesimal Lorentz transformation, Λµν = δ
ν + ǫ
The transformation of θµν is given by
θµν → θµν + ǫµρθρν + ǫνρθµρ
:= θµν + δθµν . (117)
If one considers not only the transformation of the coordinates, x
′µ = xµ + ǫµνxν , but also
(117), and assumes that φ(x) ∗θ φ(x) and φ′(x′) ∗θ+δθ φ′(x′) be equal, one obtains, after the
Fourier transformation,
φ̃′(p1)⊗ φ̃′(p2)
(ǫµνMµν ⊗ 1+ 1⊗ ǫµνMµν + δθµνPµ ⊗ Pν)
φ̃(p1)⊗ φ̃(p2)
ǫµν∆′Mµν
φ̃(p1)⊗ φ̃(p2), (118)
which agrees with (99). This shows that the additional part of the coproduct of Mµν takes
into account the transformation of the non-dynamical background parameter θµν .
The former case can be discussed in a similar manner. The definition of the star product
is given by
iPi(g1) ⋆x e
ixiPi(g2) = eix
iPi(g1g2), (119)
where we have explicitly indicated the coordinate where the star product is taken. Then we
recognize that ei(x+ǫ)
iPi(g1) ⋆x+ǫ e
i(x+ǫ)iPi(g2) and ei(x+ǫ)
iPi(g1) ⋆x e
i(x+ǫ)iPi(g2) give distinct values.
Namely, if the coordinate of the star product is also shifted,
ei(x+ǫ)
iPi(g1) ⋆x+ǫ e
i(x+ǫ)iPi(g2) = ei(x+ǫ)
iPi(g1g2), (120)
but, if not,
ei(x+ǫ)
iPi(g1) ⋆x e
i(x+ǫ)iPi(g2) = eiǫ
iPi(g1)eiǫ
iPi(g2)eix
iPi(g1g2). (121)
Therefore, if we take the translational transformation as (120), and carry out the same
procedure in deriving (59), we always obtain a translational invariant commutation relation5,
[(x+ ǫ)i, (x+ ǫ)j ]⋆x+ǫ = 2iκǫ
ijk(x+ ǫ)k. (122)
5There is a similar discussion in [42].
Now, assuming that φ(x) ⋆x φ(x) and φ
′(x′) ⋆x′ φ
′(x′) be equal under the translation xi →
′i = xi + ǫi, we obtain, after the Fourier transformation,
φ̃′(g1)φ̃
′(g2) = (1− iǫiPi(g1g2))φ̃(g1)φ̃(g2), (123)
which is the same as (75).
From these two examples, we anticipate that the multi-field contributions in (41) comes
from the transformation properties of the star products.
4 Summary and comments
We have discussed symmetries in noncommutative field theories in the framework of braided
quantum field theory. We have obtained the algebraic conditions for a Hopf algebra to be a
symmetry of a braided quantum field theory, by discussing the conditions for the relations
among correlation functions generated from the transformation algebra to hold. Then we
have applied our discussions to the Poincaré symmetries in the effective noncommutative
field theory of three-dimensional quantum gravity coupled with spinless particles and in
the noncommutative field theory on Moyal plane. In the former case we can understand
the braiding between fields, which was derived from the three-dimensional quantum gravity
computation, from the viewpoint of the translational symmetry of the noncommutative field
theory on a Lie-algebraic noncommutative spacetime. In the latter case we have found that
the twisted Lorentz symmetry on Moyal plane is a symmetry of the quantum field theory only
after the inclusion of the nontrivial braiding factor, which is in agreement with the previous
proposal [28, 35]. Then we have discussed the meaning of the Hopf algebra symmetries from
the viewpoint of coordinate representation.
In the recent research a noncommutative field theory on κ-Minkowski spacetime is dis-
cussed [36]. Since this noncommutativity of the coordinates is given by [x0, xj] = i
xj , this
noncommutative field theory will not have the naive translational symmetry. We may intro-
duce a non-trivial braiding between fields as in the effective field theory discussed in Section
3.3 to keep the momentum conservation. However, while the effective field theory has the
braided category structure because of the invariance of the Haar measure d(g−1g′g) = dg′,
the measure of the momentum space of the field theory on κ-Minkowski spacetime is only
left-invariant [36]. Therefore it is not clear to us whether we can embed this field theory on
κ-Minkowski spacetime into the framework of braided quantum field theory.
Acknowledgments
We would like to thank S. Terashima and S. Sasaki for useful discussions and comments, and
would also like to thank L. Freidel for stimulating discussions and explaining their recent
results during his stay in Yukawa Institute for Theoretical Physics after the 21st Nishinomiya-
Yukawa Memorial Symposium. Y.S. was supported in part by JSPS Research Fellowships
for Young Scientists. N.S. was supported in part by the Grant-in-Aid for Scientific Research
No.13135213, No.16540244 and No.18340061 from the Ministry of Education, Science, Sports
and Culture of Japan.
A The proofs of the formula (20), (21)
We give the proofs of the formula (20), (21) using diagrams. At first we use the formula
êv(∂ ⊗ αβ) = êv(∂ ⊗ α)ǫ(β) + êv(∂ ⊗ β)ǫ(α), (124)
where α, β ∈ X̂. This is clear from the definition of êv.
Figure 14 gives the proof of (20). In the first line, we use the axiom (12), and in the
second line we use the lemma (124). We find the last line from the property of counit.
Next we prove (21). By using the braided Leibniz rule (20) as α ∈ X ⊗ X̂, the left-hand
side of (21) becomes Figure 15. The first term of Figure 15 becomes (ev ⊗ idn−1)(∂ ⊗ idnα)
by using the definition of coproduct (9).
In the second term of Figure 15, we divide X̂ into X ⊗ X̂ and iterate the same as we did
above. For example, if the degree of X̂ is 3, the second term of Figure 15 can be reduced as
in Figure 16. We have used ∆X = X⊗̂1+ 1⊗̂X in the second line of Figure 16. The result
agrees with (21).
In the same way, we can obtain the formula (21) in general.
B The proofs of (27), (28), (29)
From the definition of γ (25), we find that
αaw = −αdiff(γ−1(a)⊗ w), (125)
for a ∈ X and α ∈ X̂ . On the other hand, adding γ−1 and ψ to the braided Leibniz rule
(20) as in Figure 17, we find that
αdiff(γ−1(a)⊗ w) = diff(ψ(α⊗ γ−1(a))w)− (diff ◦ ψ(α⊗ γ−1(a)))w. (126)
Combining (125), (126), we obtain
αaw = −diff(ψ(α⊗ γ−1(a))w) + (diff ◦ ψ(α⊗ γ−1(a)))w. (127)
Integrating the both hand sides of (127) and using (24), we find that
Z(0)(αa) = Z(0)(diff ◦ ψ(α⊗ γ−1(a))). (128)
If α is b ∈ X ,
Z(0)(ba) = Z(0)(diff ◦ ψ(b⊗ γ−1(a)))
= ev ◦ ψ(b⊗ γ−1(a))
= ev ◦ (γ−1 ⊗ id) ◦ ψ(b⊗ a). (129)
Figure 14: The proof of (20).
Figure 15: The left-hand side of (21).
Figure 16: The second term of Figure 15.
Figure 17: The diagram obtained from adding γ−1 and ψ over the braided Leibniz rule.
Thus we obtain (27).
By putting α = 1, it is clear that
1 (a) = 0. (130)
Next we rewrite (128) for α ∈ Xn−1 using the formula (21). Diagrammatically it is
written as in Figure 18. The second equality is due to (21). Thus we obtain that
Z(0)n = (Z
n−2 ⊗ Z
2 ) ◦ ([n− 1]′ψ ⊗ id) (131)
Iterating this, we find (28) for even n and (29) for odd n.
Figure 18: Diagrammatic proof of (131)
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	Introduction
	Review of braided quantum field theory
	Braided categories and braided Hopf algebras
	Braided quantum field theory
	Braided Feynman rules
	Symmetries in braided quantum field theory
	General description of an action
	Symmetry relations among correlation functions and their algebraic descriptions
	Symmetries of the effective noncommutative field theory of three-dimensional quantum gravity coupled with scalar particles
	Twisted Poincaré symmetry of noncommutative field theory on Moyal plane
	Relations among correlation functions : Examples
	Origin of Hopf algebra symmetries
	Summary and comments
	The proofs of the formula (??), (??)
	The proofs of (??), (??), (??)

  Braided quantum field theories proposed by Oeckl can provide a framework for
defining quantum field theories having Hopf algebra symmetries. In quantum
field theories, symmetries lead to non-perturbative relations among correlation
functions. We discuss Hopf algebra symmetries and such relations in braided
quantum field theories. We give the four algebraic conditions between Hopf
algebra symmetries and braided quantum field theories, which are required for
the relations to hold. As concrete examples, we apply our discussions to the
Poincare symmetries of two examples of noncommutative field theories. One is
the effective quantum field theory of three-dimensional quantum gravity coupled
with spinless particles given by Freidel and Livine, and the other is
noncommutative field theory on Moyal plane. We also comment on quantum field
theory on kappa-Minkowski spacetime.

Introduction
Symmetry is one of the most important notions in quantum field theory. In many examples, it
is useful in investigating properties of quantum field theories non-perturbatively, is a guiding
principle in constructing field theories for various purposes such as grand unification, or
gives powerful methods in finding exact solutions. It also plays important roles in actual
renormalization procedures. Therefore it should be interesting to study symmetries also in
noncommutative field theories [1, 2, 3, 4, 5], which may result from some quantum gravity
effects [6].
A difficulty in the study in this direction is the apparent violation of basic symme-
tries such as Poincaré symmetry in the noncommutativity of spacetime. For example, the
Moyal plane [xµ, xν ] = iθµν is translational invariant, but is not Lorentz or rotational
invariant. Another example is the three-dimensional spacetime with noncommutativity
[xi, xj] = iκǫijkxk (i, j, k = 1, 2, 3) [7, 8, 9, 10] with a noncommutativity parameter κ. This
noncommutative spacetime is Lorentz-invariant, but is not invariant under the translational
transformation xi → xi + ai with c-number ai. In fact, a naive construction of noncom-
mutative quantum field theory on this spacetime leads to rather disastrous violations of
energy-momentum conservation [10]: the violations coming from the non-planar diagrams
do not vanish in the commutative limit κ→ 0 as in the UV/IR mixing phenomena [11].
In recent years, however, there has been interesting conceptual progress in understanding
symmetries in noncommutative field theories: the symmetry transformations in noncommu-
tative spacetime are not the usual Lie-algebraic type, but should be generalized to have
Hopf algebraic structures. The Moyal plane was pointed out to be invariant under the
twisted Poincaré transformation in [12, 13, 14] and under the twisted diffeomorphism in
[15, 16, 17, 18]. There have been various proposals to implement the twisted Poincaré in-
variance in quantum field theories [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. As for the
noncommutative spacetime with [xi, xj ] = iκǫijkxk, a noncommutative quantum field theory
was derived as the effective field theory of three-dimensional quantum gravity with matters
[31]. Its essential difference from the naive construction mentioned above is the nontrivial
braiding for each crossing in non-planar Feynman diagrams. With this braiding, there ex-
ists a kind of conserved energy-momentum in the amplitudes, and the energy-momentum
operators have Hopf algebraic structures.
Our aim of this paper is to systematically understand these Hopf algebraic symmetries
and their consequences in noncommutative field theories in the framework of braided quan-
tum field theories proposed by Oeckl [34]. In the usual quantum field theories, symmetries
give non-perturbative relations among correlation functions. We will see that such relations
have natural extensions to the Hopf algebraic symmetries in braided quantum field theories,
and will obtain the four conditions for the relations to hold. These conditions should be
interpreted as the criteria of the symmetries in braided quantum field theories.
This paper is organized as follows. In the following section, we review braided quantum
field theory. This review part follows faithfully the original paper [34], but figures are more
extensively used in the proofs and the explanations to make this paper self-contained and
intuitively understandable. We start with braided category and braided Hopf algebra. Then
correlation functions of braided quantum field theory are represented in terms of them.
Finally braided Feynman rules are given.
In Section 3, we first review the axioms of action1 of an algebra on vector spaces. Then
we consider the relations among correlation functions in braided quantum field theory. We
find that four algebraic conditions are required for the relations to hold. Then, as concrete
examples, we discuss whether the noncommutative field theories mentioned above have the
Poincaré symmetry by checking the four conditions. In the former case, we find that the
twisted Poincaré symmetry is implemented only after the introduction of a non-trivial braid-
ing factor, which agrees with the previous proposal in [21, 35]. In the latter case, we find
that the theory has a kind of translational symmetry, which is different from the usual one
by multi-field contributions. We also give some examples of such relations among correlation
functions and the implications.
The final section is devoted to summary and comments. We comment on quantum
field theory on κ-Minkowski spacetime whose noncommutativity of coordinates is [x0, xj] =
xj (j = 1, 2, 3) [36].
2 Review of braided quantum field theory
2.1 Braided categories and braided Hopf algebras
First of all, we review braided categories and braided Hopf algebras [34, 37]. Braided cate-
gories are composed of an object X , which is a vector space, a dual object X∗, which is a
dual vector space, and morphisms
ev : X∗ ⊗X → k (evaluation), (1)
coev : k → X ⊗X∗ (coevaluation), (2)
where k is a c-number. The composition of the two morphisms in an obvious way makes the
identity. Then the braided categories have also an invertible morphism
ψV,W : V ⊗W → W ⊗ V (braiding), (3)
where V,W are any pair of vector spaces. Generally the inverse of braiding is not equal to
the braiding itself.
The braiding is required to be compatible with the tensor product such that
ψU,V⊗W = (id⊗ ψU,W ) ◦ (ψU,V ⊗ id),
ψU⊗V,W = (ψU,W ⊗ id) ◦ (id⊗ ψV,W ). (4)
1We use the italic symbol to distinguish it from the action S.
Figure 1: The evaluation, coevaluation, braiding and its inverse.
Then the braiding is also required to be intersectional under any morphisms in a Hopf
algebra. For example,
ψZ,W (Q⊗ id) = (id⊗Q)ψV,W for any Q : V → Z,
ψV,Z(id⊗Q) = (Q⊗ id)ψV,W for any Q : W → Z, (5)
where Z is a vector space.
We can represent these axioms in pictorial ways [38]. We write the morphisms, ev, coev,
ψ, downwards as in Figure 1. Thus the axioms (4) are represented as in Figure 2, and the
axioms (5) are represented as in Figure 3.
Next we consider the polynomials of X ,
X̂ :=
Xn, with X0 := 1 and Xn := X ⊗ · · · ⊗X︸ ︷︷ ︸
n times
, (6)
where 1 is the trivial one-dimensional space. X̂ naturally has the structure of a braided
Hopf algebra via
· (product) : X̂⊗̂X̂ → X̂, (7)
η (unit) : k → X̂ ; η(1) = 1, (8)
∆ (coproduct) : X̂ → X̂⊗̂X̂ ; ∆φ = φ⊗̂1+ 1⊗̂φ, and ∆(1) = 1⊗̂1, (9)
ǫ (counit) : X̂ → k ; ǫ(φ) = 0, and ǫ(1) = 1, (10)
S (antipode) : X̂ → X̂ ; Sφ = −φ, and S(1) = 1, (11)
where φ ∈ X . The tensor product ⊗ is the same as the usual product of Xs, while the new
tensor product ⊗̂ is the tensor product of X̂s. The coproduct ∆, counit ǫ, antipode S of the
Figure 2: The axioms of braiding (4).
Figure 3: The axioms of braiding (5).
Figure 4: The axioms of coproduct, counit, antipode for products.
products of Xs are defined inductively by
∆ ◦ · = (·⊗̂·) ◦ (id⊗̂ψ⊗̂id) ◦ (∆⊗̂∆), (12)
ǫ ◦ · = · ◦ (ǫ⊗̂ǫ), (13)
S ◦ · = · ◦ ψ ◦ (S⊗̂S). (14)
These axioms are diagrammatically represented in Figure 4.
2.2 Braided quantum field theory
Next we represent braided quantum field theory [34] in terms of the braided category and
the braided Hopf algebra. We take the vector space X as the space of a field φ(x), where
x denotes a general index for independent modes of the field. Thus X̂ is the space of
polynomials of the fields such as φ(x1)φ(x2) · · ·φ(xn), and 1 correspond to the constant field
of unit. We also take the dual vector space X∗ as the space of differentials δ/δφ(x). We take
the evaluation and the coevaluation as follows,
δφ(x)
⊗ φ(x′) → δ(x− x′), (15)
coev : 1 →
φ(x)⊗ δ
δφ(x)
, (16)
Figure 5: The differentials on X̂.
where the distribution and the integration should symbolically be understood, and their
detailed forms, which may contain non-trivial measures, depend on each case.
The differential on X̂ is defined by
diff := (êv ⊗ id) ◦ (id⊗∆); X∗ ⊗ X̂ → X̂, (17)
where
êv|X∗⊗Xn =
ev for n = 1,
0 for n 6= 1.
Diagrammatically this is given by Figure 5.
To see whether the map diff gives really the differential of products, let us compute the
differential of φ(x)φ(y) as a simple example using the definition (17). This becomes
δφ(x′)
⊗ φ(x)φ(y)
= (êv ⊗ id) ◦ (id⊗∆)
δφ(x′)
⊗ φ(x)φ(y)
= (êv ⊗ id) ◦
δφ(x′)
⊗∆(φ(x)φ(y))
= (êv ⊗ id) ◦
δφ(x′)
⊗ (φ(x)φ(y)⊗̂1
+ φ(x)⊗̂φ(y) + ψ(φ(x)⊗̂φ(y)) + 1⊗̂φ(x)φ(y))
= δ(x′ − x)⊗ φ(y) + (êv ⊗ id) ◦
δφ(x′)
⊗ ψ(φ(x)⊗̂φ(y))
where we have used the axiom (12) in deriving the third line. If the braiding is trivial, we
find that the differential (17) satisfies the usual Leibniz rule.
Figure 6: Diagram of ψn,m.
Generally we find a braided Leibniz rule
∂(αβ) = ∂(α)β + ψ−1(∂ ⊗ α)(β) (20)
∂(α) = (ev ⊗ idn−1)(∂ ⊗ [n]ψα), (21)
where ∂ ∈ X∗, α, β ∈ X̂ , and we have used a simplified notation
∂(α) := diff(∂ ⊗ α). (22)
Here n is the degree of α, and [n]ψ is called a braided integer defined by
[n]ψ := id
n + ψ ⊗ idn−2 + · · ·+ ψn−2,1 ⊗ id + ψn−1,1, (23)
where ψn,m is a braiding morphism given in Figure 6.
The proofs of the formula (20), (21) are in Appendix A.
Now we define a Gaussian integration, which defines the path integral. The definition is
given by ∫
∂(αw) := 0 for ∂ ∈ X∗, α ∈ X̂, (24)
where w ∈ X̂ is a Gaussian weight. In field theory, w is the exponential of the free part of
the action, e−S0 .
In order to obtain a formula for correlation functions, we define a morphism γ : X∗ → X
such that
∂(w) := −γ(∂)w. (25)
This morphism is assumed to be commutative with the braiding as in (5). If w = e−S0 ,
γ(∂) = ∂(S0). In field theory, this is the kinetic part of the action, or the inverse of the
propagator.
Starting from (24), we can represent correlation functions of a free field theory in terms of
the braided category and the braided Hopf algebra. This is the analog of the Wick theorem
in braided quantum field theory. The definition of the free n point correlation function is
given by
Z(0)n (α) :=
, (26)
where the degree of α is n. Algebraically, this is given by
2 = ev ◦ (γ−1 ⊗ id) ◦ ψ, (27)
2n = (Z
n ◦ [2n− 1]′ψ!!, (28)
2n−1 = 0, (29)
where
[2n− 1]′ψ!! := ([1]′ψ ⊗ id2n−1) ◦ ([3]′ψ ⊗ id2n−3) ◦ · · · ◦ ([2n− 1]′ψ ⊗ id), (30)
ψ := id
n + idn−2 ⊗ ψ−1 + · · ·+ ψ−11,n−1
= ψ−11,n−1 ◦ [n]ψ. (31)
The proofs of (27), (28), (29) are in Appendix B.
Next we consider correlation functions with the existence of an interaction. For S =
S0 + λSint, a correlation function is perturbatively given by
Zn(α) =
αe−S∫
α(1− λSint + · · · )e−S0∫
(1− λSint + · · · )e−S0
, (32)
where α ∈ Xn. Introducing a morphism Sint : k → Xk, where k is the degree of Sint, the
correlation function is algebraically given by
n − λZ(0)n+k ◦ (id
n ⊗ Sint) + 12λ
n+2k ◦ (id
n ⊗ Sint ⊗ Sint) + · · ·
1− λZ(0)
◦ Sint + 12λ2Z
2k ◦ (Sint ⊗ Sint) + · · ·
. (33)
Acting Zn on α ∈ Xn, we obtain the correlation function (32). One can obviously extend
Sint to include various interaction terms.
2.3 Braided Feynman rules
From the results in the preceding subsection, a correlation function can be represented by
summation of diagrams obeying the following rules below.
Figure 7: Propagator (left) and vertex (right).
Figure 8: The braiding ψ (left) and its inverse ψ−1 (right).
• An n-point function Zn is a morphism Xn → k. Thus a Feynman diagram starts with
n strands at the top and must be closed at the bottom.
• The propagator Z(0)2 : X ⊗X → k is represented by the left of Figure 7, which is the
abbreviation of Figure 9.
• The interaction vertex Sint : k → Xk is represented by the right of Figure 7. Generally
the order of the strands is noncommutative.
• The two kinds of crossings, which are represented in Figure 8, correspond to the braid-
ing and its inverse.
• Any Feynman diagram is built out of propagators, vertices, and crossings, and is closed
at the bottom.
3 Symmetries in braided quantum field theory
In this section, we discuss symmetries in braided quantum field theory. In order to represent
symmetry transformations on fields, we review general description of an action in Section
Figure 9: The propagator, which is abbreviated in the left figure of Figure 7.
3.1. In Section 3.2, we study relations among correlation functions. We find four conditions
for such relations to follow from a symmetry algebra. In Section 3.3 and 3.4, we treat two
examples of (braided) noncommutative field theories and discuss their Poincaré symmetries.
3.1 General description of an action
We review an action of a general Hopf algebra on vector spaces in a mathematical language
[37, 39].
An action αV is a map αV : A⊗ V → V , where A is an arbitrary Hopf algebra and V is
a vector space (in our case, A is a symmetry algebra, and V = X or X∗). We will denote
the coproduct and the counit of the Hopf algebra2 by ∆′ and ǫ′ to distinguish them from
those of the braided Hopf algebra of fields in Section 2. We do not write all the axioms of
an action, but our important axioms are the following.
• αV satisfies the following condition.
αV ◦ (· ⊗ id) = αV ◦ (id⊗ αV ), (34)
where the equality acts on A ⊗ A ⊗ V . This means that αV ((a · b) ⊗ V ) = αV (a ⊗
(αV (b⊗ V ))), where a, b ∈ A. In short we can write this as
(a · b) ⊲ V = a ⊲ (b ⊲ V ). (35)
• An action on 1, which is in a vector space, is defined by
αV (a⊗ 1) = ǫ′(a)1, (36)
where ǫ′(a) is the counit of an algebra a ∈ A.
• An action on a tensor product of vector spaces V,W is defined by
αV⊗W (a) := ((αV ⊗ αW ) ◦∆′)(a) =
αV (a
(1))⊗ αW (ai(2)), a ∈ A, (37)
where ∆′(a) =
ai(1) ⊗ ai(2) is the coproduct of the Hopf algebra A. In the case of a
usual Lie-algebraic transformation, its coproduct is given by ∆′(a) = a ⊗ 1 + 1 ⊗ a,
where 1 is in A. This gives the usual Leibnitz rule.
• Since a Hopf algebra has the coassociativity that
((∆′ ⊗ id) ◦∆′)(a) = ((id⊗∆′) ◦∆′)(a), (38)
2We omit the antipode.
the action on a tensor product of vector spaces, which is obtained by the multiple
operations of ∆′ on a, is actually unique. An important consequence is that one can
divide the action on a tensor product of vector spaces as
a⊲(V1 ⊗ · · · ⊗ Vk−1 ⊗ Vk ⊗ · · · ⊗ Vn) =∑
ai(1) ⊲ (V1 ⊗ · · · ⊗ Vk−1)⊗ ai(2) ⊲ (Vk ⊗ · · · ⊗ Vn) (39)
for any k.
3.2 Symmetry relations among correlation functions and their al-
gebraic descriptions
The expression of the correlation functions (33) is perturbative in interactions, but is a full
order algebraic description. Therefore we can discuss the symmetry of the theory and the
implied relations among correlation functions by using this expression. We may even expect
that the relations will hold non-perturbatively.
In usual quantum field theory, if a field theory has a certain symmetry, there is a relation
among the correlation functions in the form,
〈φ(x1) · · · δaφ(xi) · · ·φ(xn)〉 = 0, (40)
where δaφ(x) is a variation of a field under a transformation a, on the assumption that the
path integral measure and the action are invariant under the transformation.
If the coproduct of a symmetry algebra is not the usual Lie-algebraic type and thus the
Leibniz rule is deformed, the relation will generally have the form,
c(bi)a 〈φ(x1) · · · δbφ(xi) · · ·φ(xn)〉
+c(bi)(cj)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · ·φ(xn)〉
+c(bi)(cj)(dk)a 〈φ(x1) · · · δbφ(xi) · · · δcφ(xj) · · · δdφ(xk) · · ·φ(xn)〉
+ · · · = 0, (41)
where c···a are some coefficients. Its essential difference from (40) is the multi-field contribu-
tions. In our algebraic language, the relation can be written as
Zn(a ⊲ χ) = ǫ
′(a)Zn(χ), for a ∈ A, χ ∈ Xn. (42)
This is equivalent to Figure 10 in our diagrammatic representation. Then we consider what
an algebraic structure is required for (42) to hold for any a and χ, i.e. the theory is invariant
under the Hopf algebra transformation A.
Figure 10: A relation among correlation functions in the diagrammatic representation. n
is the number of external legs, k is the order of the interaction, and p is the order of the
perturbation. n+ kp is even.
Let us write the coproduct of an element a ∈ A as
∆′(a) =
f s ⊗ gs, (43)
where f s, gs ∈ A. Since the coproduct must satisfy the Hopf algebra axiom [37],
(ǫ′ ⊗ id)∆′(a) = (id⊗ ǫ′)∆′(a) = a, (44)
f s, gs must satisfy
ǫ′(f s)⊗ gs =
f s ⊗ ǫ′(gs) = a. (45)
For all the relations among correlation functions to hold, we find the following four
conditions for any action a ∈ A.
• (Condition 1) Sint must satisfy
a ⊲ Sint = ǫ
′(a)Sint. (46)
• (Condition 2) The braiding ψ is an intertwining operator. That is
ψ(a ⊲ (V ⊗W )) = a ⊲ ψ(V ⊗W ). (47)
• (Condition 3) γ−1 and a are commutative,
a ⊲ (γ−1(V )) = γ−1(a ⊲ V ). (48)
• (Condition 4) Under an action a, the evaluation map follows
ev(a ⊲ (X∗ ⊗X)) = ǫ′(a)ev(X∗ ⊗X). (49)
Condition 1 to 4 are diagrammatically represented in Figure 11. It is clear that, when the
algebra A is generated from a finite number of its independent elements, it is enough for
these generators to satisfy these conditions.
Condition 1 is the requirement of the symmetry at the classical level for the interaction.
We can extend this condition to
(a ⊲ Xn)⊗ Spint = a ⊲ (Xn ⊗ S
int). (50)
The proof is the following. From a coproduct (43) and its coassociativity (39), the right
hand side of (50) is equal to
(f s ⊲ (Xn ⊗ Sp−1int ))⊗ gs ⊲ Sint (51)
Since Condition 1 implies
gs ⊲ Sint = ǫ
′(gs)Sint, (52)
(51) becomes
(f s ⊲ (Xn ⊗ Sp−1int ))⊗ ǫ′(gs)Sint
= a ⊲ (Xn ⊗ Sp−1int )⊗ Sint, (53)
where we have used (45). Iterating this procedure, we obtain the left-hand side of (50).
Condition 2,3,4 can also be extended to
[n + kp− 1]ψ!! ◦ (a ⊲ Xn+kp) = a ⊲ [n+ kp− 1]ψ!! Xn+kp, (54)
(γ−1 ⊗ id)
2 ◦ (a ⊲ Xn+kp) = a ⊲ (γ−1 ⊗ id)
2 Xn+kp, (55)
2 (a ⊲ (X∗ ⊗X)
2 ) = ǫ′(a) ev
2 (X∗ ⊗X)
2 . (56)
We can find that these extended conditions (50), (54), (55), (56) can be represented as
in Figure 12. In the diagrammatic language, the relation among correlation functions holds
if an action can pass downwards through a Feynman diagram and satisfies (36).
3.3 Symmetries of the effective noncommutative field theory of
three-dimensional quantum gravity coupled with scalar parti-
In this subsection, we discuss the Poincaré symmetry of the effective noncommutative field
theory of three-dimensional quantum gravity coupled with scalar particles, which was ob-
tained in [31] by studying the Ponzano-Regge model [40] coupled with spinless particles. The
Figure 11: Conditions 1,2,3, and 4.
Figure 12: A relation among correlation functions is satisfied if the four conditions (46),
(47), (48), (49) are satisfied.
symmetries of this theory is also known as DSU(2), which was discussed in [32, 33]. We first
review the field theory [10, 31].
Let φ(x) be a scalar field on a three-dimensional space x = (x1, x2, x3). Its Fourier
transformation is given by
φ(x) =
dgφ̃(g)e
tr(Xg), (57)
where κ is a constant, X = ixiσi, and g = P 0−iκP iσi ∈ SO(3)3 with Pauli matrices σi. Here∫
dg is the Haar measure of SO(3) and P 0 = ±
1− κ2P iPi by definition. In the following
discussions, we will only deal with the Euclidean case, but the Lorentzian case can also be
treated in a similar manner by replacing SO(3) with SL(2, R).
The definition of the star product is given by
tr(Xg1) ⋆ e
tr(Xg2) := e
tr(Xg1g2). (58)
Differentiating both hands sides of (58) with respect to P i1 := P
i(g1) and P
2 := P
j(g2) and
then taking the limit P i1, P
2 → 0, one finds the SO(3) Lie-algebraic space-time noncommu-
tativity [7, 8, 9],
[xi, xj]⋆ = 2iκǫ
ijkxk. (59)
For example, the action4 of a φ3 theory is
(∂iφ ⋆ ∂iφ)(x)−
M2(φ ⋆ φ)(x) +
(φ ⋆ φ ⋆ φ)(x)
, (60)
where M2 = sin
. Its momentum representation is
P 2(g)−M2
φ̃(g)φ̃(g−1)
dg1dg2dg3δ(g1g2g3)φ̃(g1)φ̃(g2)φ̃(g3), (61)
from which it is straightforward to read the Feynman rules.
Some quantum properties of this scalar field theory were analyzed in [10]. As can be
seen from (59), the naive translational symmetry is violated. In fact, the violation is rather
disastrous. There exists a kind of conserved energy-momentum in the amplitudes of the
tree and the planar loop diagrams, but this energy-momentum is not conserved in the non-
planar loop diagrams. Moreover, the violation of the energy-momentum conservation does
not vanish in the commutative limit κ → 0 due to a mechanism similar to the UV/IR
phenomena [11].
3The identification g ∼ −g is implicitly assumed.
4Since in the Ponzano-Regge model the definition of the weight of partition function is eiS despite of
Euclidean theory, the sign of the mass term is not the usual one.
In the effective field theory of quantum gravity coupled with spinless particles, however,
the Feynman rules contain also a non-trivial braiding rule for each crossing, which comes
from a flatness condition in a graph of intersecting particles [31]. This can be incorporated
as a braiding between the scalar fields,
ψ(φ̃(g1)φ̃(g2)) = φ̃(g2)φ̃(g
2 g1g2), (62)
in the braided quantum field theory.
From the direct analysis of the Feynman graphs with this braiding rule, one can easily find
that the energy-momentum mentioned above is conserved also in the non-planar diagrams.
This suggests the existence of a translational symmetry in the quantum field theory. In
the sequel, we will discuss the embedding of this field theory into the framework of braided
quantum field theory, and will check the four conditions for its translational and rotational
symmetries.
We use the momentum representation, and take X as the space of φ̃(g) and X∗ as that
δφ̃(g)
. We take the braided Hopf algebra of the fields as follows,
∆ : φ̃(g) → φ̃(g)⊗̂1 + 1⊗̂φ̃(g), (63)
ǫ : φ̃(g) → 0, (64)
S : φ̃(g) → −φ̃(g), (65)
ψ : φ̃(g1)⊗ φ̃(g2) → φ̃(g2)⊗ φ̃(g−12 g1g2). (66)
The evaluation and coevaluation maps are given by
δφ̃(g)
⊗ φ̃(g′) → δ(g−1g′), (67)
coev : 1 →
dgφ̃(g)⊗ δ
δφ̃(g)
. (68)
From γ(∂) = ∂S0 = (P
2(g)−m2)φ̃(g−1),
γ−1(φ̃(g)) =
P 2(g−1)−m2
δφ̃(g−1)
. (69)
From the algebraic consistencies in Figure 13, the braidings between X and X∗ and the
braiding between X∗s are determined to be
δφ̃(g1)
⊗ φ̃(g2)
= φ̃(g2)⊗
δφ̃(g−12 g1g2)
, (70)
φ̃(g1)⊗
δφ̃(g2)
δφ̃(g2)
⊗ φ̃(g2g1g−12 ), (71)
δφ̃(g1)
δφ̃(g2)
δφ̃(g2)
δφ̃(g2g1g
. (72)
Figure 13: The algebraic consistency conditions of coevaluation map and X , X∗.
In this derivation, we have used the invariance of the Haar measure d(g−1g′g) = dg′.
Now we consider a translational transformation of the field. If we shift xi to xi + ǫi, a
field φ(x) becomes
φ(x) → φ(x+ ǫ)
dgφ̃(g)ei(x+ǫ)
iPi(g)
dg(1 + iǫiPi(g))φ̃(g)e
ixiPi(g). (73)
Thus in the momentum representation, the translational transformation corresponds to an
action
P i ⊲ φ̃(g) = P i(g)φ̃(g), P 0 ⊲ φ̃(g) = P 0(g)φ̃(g). (74)
From the requirement that the star product (58) conserve a kind of momentum, the action
on a product of fields should be
P i ⊲ (φ̃(g1)φ̃(g2)) = P
i(g1g2)φ̃(g1)φ̃(g2)
= (P 01P
2 + P
1 + κǫ
2 )φ̃(g1)φ̃(g2), (75)
P 0 ⊲ (φ̃(g1)φ̃(g2)) = (P
2 − κ2P i1P2i)φ̃(g1)φ̃(g2). (76)
This determines the coproduct of P i, P 0 as
∆′(P i) = P 0 ⊗ P i + P i ⊗ P 0 + κǫijkP j ⊗ P k, (77)
∆′(P 0) = P 0 ⊗ P 0 − κ2P i ⊗ Pi. (78)
This coproduct satisfies the coassociativity, which essentially comes from the associativity
of the group multiplication.
From the axiom (44), the counit of P i, P 0 is given by
ǫ′(P i) = ǫ′(P 0) = 0. (79)
Since the conservation of momentum under the coevaluation map (68) requires that the
action of P i on
dg(φ̃(g)⊗ δ
δφ̃(g)
) vanish from (36), the action of P i on δ
δφ̃(g)
must be
P i ⊲
δφ̃(g)
= P i(g−1)
δφ̃(g)
. (80)
In the following, we see that the momentum algebra satisfies the four conditions (46),
(47), (48), (49).
Condition 1 is satisfied since
P i ⊲ Sint
dg1dg2dg3δ(g1g2g3)P
i ⊲ (φ̃(g1)φ̃(g2)φ̃(g3))
dg1dg2dg3δ(g1g2g3)P
i(g1g2g3)(φ̃(g1)φ̃(g2)φ̃(g3))
= 0. (81)
Condition 2 is satisfied since
ψ(P i ⊲ (φ̃(g1)φ̃(g2))) = P
i(g1g2)(φ̃(g2)φ̃(g
2 g1g2)),
P i ⊲ ψ(φ̃(g1)φ̃(g2)) = P
i(g2g
2 g1g2)(φ̃(g2)φ̃(g
2 g1g2)).
Condition 3 is satisfied since
P i ⊲ γ−1(φ̃(g)) =
P 2(g−1)−m2
P i(g)
δφ̃(g−1)
γ−1(P i ⊲ φ̃(g)) =
P 2(g−1)−m2
P i(g)
δφ̃(g−1)
Condition 4 is satisfied since
P i ⊲
δφ̃(g1)
⊗ φ̃(g2)
= P i(g−11 g2) ev
δφ̃(g1)
⊗ φ̃(g2)
= 0. (82)
Thus we find that the effective braided noncommutative field theory of three-dimensional
quantum gravity coupled with spinless particles has the translational symmetry.
Next we consider a rotational symmetry. The rotational symmetry corresponds to an
action
Λ ⊲ φ̃(g) = φ̃(h−1gh), (83)
which is the usual Lie-group one. The action on the tensor product is
Λ ⊲ (φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g1h)⊗ φ̃(h−1g2h). (84)
Thus the coproduct of the rotational symmetry is given by
∆′(Λ) = Λ⊗ Λ. (85)
From the axiom (44), the counit of Λ is given by
ǫ′(Λ) = 1. (86)
Condition 1 is satisfied since
Λ ⊲ Sint
dg1dg2dg3δ(g1g2g3)Λ ⊲ (φ̃(g1)φ̃(g2)φ̃(g3))
dg1dg2dg3δ(g1g2g3)(φ̃(h
−1g1h)φ̃(h
−1g2h)φ̃(h
−1g3h))
=ǫ′(Λ)Sint. (87)
Condition 2 is satisfied since
ψ(Λ ⊲ (φ̃(g1)⊗ φ̃(g2))) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h)
Λ ⊲ ψ(φ̃(g1)⊗ φ̃(g2)) = φ̃(h−1g2h)⊗ φ̃(h−1g−12 g1g2h). (88)
Condition 3 is satisfied since
Λ ⊲ γ−1(φ̃(g)) =
P 2(g−1)−m2
δφ̃(h−1g−1h)
γ−1(Λ ⊲ φ̃(g)) =
P 2(h−1g−1h)−m2
δφ̃(h−1g−1h)
P 2(g−1)−m2
δφ̃(h−1g−1h)
. (89)
Condition 4 is satisfied since
δφ̃(g1)
⊗ φ̃(g2)
δφ̃(h−1g1h)
⊗ φ̃(h−1g2h)
= δ(g−11 g2)
= ǫ′(Λ)ev
δφ̃(g1)
⊗ φ̃(g2)
. (90)
Thus we find that this braided noncommutative field theory has also the rotational sym-
metry.
3.4 Twisted Poincaré symmetry of noncommutative field theory
on Moyal plane
In this subsection, we discuss the twisted Poincaré symmetry of noncommutative field theory
on Moyal plane [xµ, xν ] = iθµν .
For example, the action of a φ3 theory is given by
(∂µφ ∗ ∂µφ)(x)−
m2(φ ∗ φ)(x) + λ
(φ ∗ φ ∗ φ)(x)
, (91)
where the star product is given by
φ(x) ∗ φ(x) = e
θµν∂xµ∂
νφ(x)φ(y)
. (92)
In the momentum representation, the action is
(p2 −m2)φ̃(p)φ̃(−p)
dDp1d
µνp2νδ(p1 + p2 + p3)φ̃(p1)φ̃(p2)φ̃(p3)
. (93)
We take X as the space of φ̃(p) and X∗ as that of δ
δφ̃(p)
. Then we take the braided Hopf
algebra as follows:
∆ : φ̃(p) → φ̃(p)⊗̂1+ 1⊗̂φ̃(p), (94)
ǫ : φ̃(p) → 0, (95)
S : φ̃(p) → −φ̃(p). (96)
From γ(∂) = ∂S0 = (p
2 −m2)φ̃(−p),
γ−1(φ̃(p)) =
p2 −m2
δφ̃(−p)
. (97)
Let us consider the twisted Poincaré symmetry [12, 13, 14]. The coproduct and the counit
of the twisted Poincaré algebra is given by
∆′(P µ) = P µ ⊗ 1+ 1⊗ P µ,
ǫ′(P µ) = 0,
∆′(Mµν) =Mµν ⊗ 1+ 1⊗Mµν
θαβ [(δµαP
ν − δναP µ)⊗ Pβ + Pα ⊗ (δ
ν − δνβP µ)],
ǫ′(Mµν) = 0. (98)
Thus the action of the twisted Lorentz algebra on the tensor product is
Mµν ⊲ (φ̃(p1)⊗ φ̃(p2)) =Mµν ⊲ φ̃(p1)⊗ φ̃(p2) + φ̃(p1)⊗Mµν ⊲ φ̃(p2)
θαβ [(δµαP
ν − δναP µ) ⊲ φ̃(p1)⊗ Pβ ⊲ φ̃(p2)
+ Pα ⊲ φ̃(p1)⊗ (δµβP
ν − δνβP µ) ⊲ φ̃(p2)], (99)
where Mµν ⊲ φ̃(p) = i(pµ∂/∂pν − pν∂/∂pµ)φ̃(p) and P µ ⊲ φ̃(p) = pµφ̃(p). The actions of Mµν
and P µ on δ
δφ̃(p)
Mµν ⊲
δφ̃(p)
= i(pµ∂/∂pν − pν∂/∂pµ)
δφ̃(p)
, (100)
P µ ⊲
δφ̃(p)
= −pµ δ
δφ̃(p)
. (101)
One easily finds that three conditions (46), (48), (49) are satisfied, but (47) is not if the
braiding is trivial. In order to keep the invariance, the braiding must be taken as
ψ(φ̃(p1)⊗ φ̃(p2)) = eiθ
αβp2α⊗p1β(φ̃(p2)⊗ φ̃(p1)). (102)
This agrees with the previous proposal [21, 35].
We can easily check that the translational symmetry holds since the coproduct ∆′(P µ)
follows the usual Leibniz rule.
3.5 Relations among correlation functions : Examples
Now we have checked, in all orders of perturbation, that the two theories in the preceding
sections have symmetry relations among correlation functions implied by the Hopf algebra
symmetries. In Section 3.3 we gave how the translational generator acts on a product of
fields in (75), (76) in the momentum representation. Since the physical meaning of this Hopf
algebra transformation is not so clear, it would be interesting to see explicitly the symmetry
relations among correlation functions. The same thing is also true in the case of the twisted
Lorentz symmetry in Section 3.4. In this subsection, we work out explicitly some relations
among correlation functions in the two theories.
In the effective quantum field theory of quantum gravity, the action of the translational
generators on a correlation function is given by
〈φ̃(g1) · · · φ̃(gn)〉 → iǫiPi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 (103)
in the momentum representation, where ǫi is an infinitesimal parameter. Thus we obtain a
relation,
Pi(g1 · · · gn)〈φ̃(g1) · · · φ̃(gn)〉 = 0. (104)
This is a (modified) momentum conservation law; the correlation function has support only
on the vanishing momentum subspace, Pi(g1 · · · gn) = 0. This all-order relation in the
quantum field theory would be a simple but an important implication of the Hopf algebraic
translational symmetry. This provides a good example of the physical importance of a Hopf
algebraic symmetry: a Hopf algebra symmetry leads to a (modified) conservation law.
It would also be interesting to see the relations in the coordinate representations, where
the fields are defined by φ(x) =
eip·xφ̃(p). As explicitly noted in the preceding subsections,
we stress that the basis of the spaces X of the field variables in the path integrals are
parameterized in terms of momenta, and that φ(x) are defined by some c-number linear
combinations of them. Therefore, an action a ∈ A of a symmetry transformation acts as
a ⊲ φ(x) =
eip·x(a ⊲ φ̃(p)), (105)
and the symmetry relations of correlation functions can be obtained by some inverse Fourier
transformations (with possible non-trivial measures) of those in momentum representations.
For example, in the case of the two point function, after the inverse Fourier transforma-
tion, the relation among correlation functions is given by
〈∂iφ(x1)φ(x2) + φ(x1)∂iφ(x2)〉 = 0, (106)
where we have used the relation (104). Interestingly, this is the usual relation in a transla-
tionally invariant quantum field theory. In the case of the three point function, however, the
relation is given by
〈∂iφ(x1)
1 + κ2∂2φ(x2)
1 + κ2∂2φ(x3) +
1 + κ2∂2φ(x1)∂
iφ(x2)
1 + κ2∂2φ(x3)
1 + κ2∂2φ(x1)
1 + κ2∂2φ(x2)∂
iφ(x3) + iκǫ
1 + κ2∂2φ(x1)∂jφ(x2)∂kφ(x3)
+ iκǫijk∂jφ(x1)
1 + κ2∂2φ(x2)∂kφ(x3)− iκǫijk∂jφ(x1)∂kφ(x2)
1 + κ2∂2φ(x3)
+ κ2∂jφ(x1)∂
jφ(x2)∂
iφ(x3)− κ2∂iφ(x1)∂kφ(x2)∂kφ(x3)
+ κ2∂kφ(x1)∂
iφ(x2)∂
kφ(x3)〉 = 0. (107)
This is quite a non-trivial relation among correlation functions, and would be hard to find,
if the Hopf algebra symmetry in the quantum field theory was not noticed. This would be
another interesting example implying the physical importance of a Hopf algebra symmetry.
In general, the relation has the form,
∂xli − i
κǫijk∂xlj∂xmk +O(κ
2))〈φ(x1) · · ·φ(xn)〉 = 0. (108)
In the κ → 0 limit, the relation approaches the usual relation. Thus the Hopf algebra sym-
metry is a kind of translational symmetry modified by adding κ dependent higher derivative
multi-field contributions.
We can proceed in a similar manner for the twisted Lorentz symmetry. We have a general
form of such a symmetry relation as
Mµν ⊲ 〈φ̃(p1) · · · φ̃(pn)〉 = 0. (109)
In the case of the two point function, the relation is given by
〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2) + φ(x1)(x
ν − xν2∂µ)φ(x2)〉 = 0, (110)
where we have used the momentum conservation. This is the same relation as that in a
Lorentz invariant quantum field theory. In the case of the three point function, the relation
is given by
〈(xµ1∂ν − xν1∂µ)φ(x1)φ(x2)φ(x3)
+ φ(x1)(x
ν − xν2∂µ)φ(x2)φ(x3) + φ(x1)φ(x2)(x
ν − xν3∂µ)φ(x3)
iθαµ(∂αφ(x1)∂
νφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂
νφ(x3) + φ(x1)∂αφ(x2)∂
νφ(x3)
− ∂νφ(x1)∂αφ(x2)φ(x3)− ∂νφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂νφ(x2)∂αφ(x3))
iθαν(∂αφ(x1)∂
µφ(x2)φ(x3) + ∂αφ(x1)φ(x2)∂
µφ(x3) + φ(x1)∂αφ(x2)∂
µφ(x3)
− ∂µφ(x1)∂αφ(x2)φ(x3)− ∂µφ(x1)φ(x2)∂αφ(x3)− φ(x1)∂µφ(x2)∂αφ(x3))〉 = 0. (111)
In general, the relation among correlation functions has the from,
((x1µ∂x1ν − x1ν∂x1µ) + · · ·+ (xnµ∂xnν − xnν∂xmν) +O(θ))〈φ(x1) · · ·φ(xn)〉 = 0 (112)
in the coordinate representation. The leading terms corresponds to the usual Lorentz trans-
formation xµ → xµ + ǫµνxν .
The above symmetry relations on Moyal plane can be represented in similar manners
as the usual commutative cases, if we use star products. In the papers [24, 25, 26, 27,
28, 29, 30], they have pointed out that in coordinate representation, correlation functions on
Moyal plane should be defined with star products extended to non-coincident points (see also
[43]) instead of usual products since the usual commutative commutation relation [x
i , x
j ] =
0 (i, j = 1, · · · , n) is not invariant under the twisted Poincaré transformation. Carrying
out Fourier transformation of the symmetry relation (109) in momentum representation to
such a noncommutative coordinate representation, we obtain the symmetry relations in star
tensor products. Namely (110) becomes
〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) + φ(x1) ∗ ((x
ν − xν2∂µ)φ(x2))〉 = 0, (113)
and (111) becomes
〈((xµ1∂ν − xν1∂µ)φ(x1)) ∗ φ(x2) ∗ φ(x3)
+ φ(x1) ∗ ((xµ2∂ν − xν2∂µ)φ(x2)) ∗ φ(x3) + φ(x1) ∗ φ(x2) ∗ ((x
ν − xν3∂µ)φ(x3))
iθαµ(∂αφ(x1) ∗ ∂νφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2) ∗ ∂νφ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂νφ(x3)
− ∂νφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂νφ(x1) ∗ φ(x2)∂α ∗ φ(x3)− φ(x1) ∗ ∂νφ(x2) ∗ ∂αφ(x3))
iθαν(∂αφ(x1) ∗ ∂µφ(x2) ∗ φ(x3) + ∂αφ(x1) ∗ φ(x2)∂µ ∗ φ(x3) + φ(x1) ∗ ∂αφ(x2) ∗ ∂µφ(x3)
− ∂µφ(x1) ∗ ∂αφ(x2) ∗ φ(x3)− ∂µφ(x1) ∗ φ(x2) ∗ ∂αφ(x3)− φ(x1) ∗ ∂µφ(x2) ∗ ∂αφ(x3))〉 = 0.
(114)
More generally we can derive the symmetry relations of correlation functions for tensor
fields φα1···αn(x) ≡ ∂α1 · · ·∂αnφ(x). For example in the case of the three point function of the
tensor fields, the symmetry relation becomes
〈((M1µν)α1···αl
δ1···δlφδ1···δl(x1)) ∗ φβ1···βm(x2) ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ ((M
2µν)β1···βm
δ1···δmφδ1···δm(x2)) ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ φβ1···βm(x2) ∗ ((M
3µν)γ1···γn
δ1···δnφδ1···δn(x3))
θαµ[∂αφα1···αl(x1) ∗ ∂
νφβ1···βm(x2) ∗ φγ1···γn(x3)
+ ∂αφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂
νφγ1···γn(x3)
+ φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂
νφγ1···γn(x3)
− ∂νφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3)
− ∂νφα1···αl(x1) ∗ φβ1···βm(x2)∂α ∗ φγ1···γn(x3)
− φα1···αl(x1) ∗ ∂
νφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)]
θαν [∂αφα1···αl(x1) ∗ ∂
µφβ1···βm(x2) ∗ φγ1···γn(x3)
+ ∂αφα1···αl(x1) ∗ φβ1···βm(x2)∂
µ ∗ φγ1···γn(x3)
+ φα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ ∂
µφγ1···γn(x3)
− ∂µφα1···αl(x1) ∗ ∂αφβ1···βm(x2) ∗ φγ1···γn(x3)
− ∂µφα1···αl(x1) ∗ φβ1···βm(x2) ∗ ∂αφγ1···γn(x3)
− φα1···αl(x1) ∗ ∂
µφβ1···βm(x2) ∗ ∂αφγ1···γn(x3)]〉 = 0, (115)
where
(Mµν)α1···αn
β1···βn = (Lµν)α1···αn
β1···βn + (Sµν)α1···αn
β1···βn
(Lµν)α1···αn
β1···βn = i(xµ∂ν − xν∂µ)δα1β1 · · · δαnβn
(Sµν)α1···αn
β1···βn = i(ηνβ1δ{α1
β2 · · · δαn}βn − ηµβ1δ{α1νδα2β2 · · · δαn}βn) (116)
If we bring the operators (M iµν)α1···αn
β1···βn (i = 1, 2, 3) out of the star products, θµν depen-
dent terms are canceled. The final expressions are just the usual Lorentz rotations on the
coordinates and the tensorial indices in the correlation functions. This is fully consistent
with the discussions in [29].
3.6 Origin of Hopf algebra symmetries
To study more the meaning of these additional terms, let us see closer the transformation
properties of the star products. In the latter case, it is known that the θµν dependence of
the twisted Lorentz transformation (99) comes from the Lorentz transformation of θµν itself
[41]. To see this, let us consider an infinitesimal Lorentz transformation, Λµν = δ
ν + ǫ
The transformation of θµν is given by
θµν → θµν + ǫµρθρν + ǫνρθµρ
:= θµν + δθµν . (117)
If one considers not only the transformation of the coordinates, x
′µ = xµ + ǫµνxν , but also
(117), and assumes that φ(x) ∗θ φ(x) and φ′(x′) ∗θ+δθ φ′(x′) be equal, one obtains, after the
Fourier transformation,
φ̃′(p1)⊗ φ̃′(p2)
(ǫµνMµν ⊗ 1+ 1⊗ ǫµνMµν + δθµνPµ ⊗ Pν)
φ̃(p1)⊗ φ̃(p2)
ǫµν∆′Mµν
φ̃(p1)⊗ φ̃(p2), (118)
which agrees with (99). This shows that the additional part of the coproduct of Mµν takes
into account the transformation of the non-dynamical background parameter θµν .
The former case can be discussed in a similar manner. The definition of the star product
is given by
iPi(g1) ⋆x e
ixiPi(g2) = eix
iPi(g1g2), (119)
where we have explicitly indicated the coordinate where the star product is taken. Then we
recognize that ei(x+ǫ)
iPi(g1) ⋆x+ǫ e
i(x+ǫ)iPi(g2) and ei(x+ǫ)
iPi(g1) ⋆x e
i(x+ǫ)iPi(g2) give distinct values.
Namely, if the coordinate of the star product is also shifted,
ei(x+ǫ)
iPi(g1) ⋆x+ǫ e
i(x+ǫ)iPi(g2) = ei(x+ǫ)
iPi(g1g2), (120)
but, if not,
ei(x+ǫ)
iPi(g1) ⋆x e
i(x+ǫ)iPi(g2) = eiǫ
iPi(g1)eiǫ
iPi(g2)eix
iPi(g1g2). (121)
Therefore, if we take the translational transformation as (120), and carry out the same
procedure in deriving (59), we always obtain a translational invariant commutation relation5,
[(x+ ǫ)i, (x+ ǫ)j ]⋆x+ǫ = 2iκǫ
ijk(x+ ǫ)k. (122)
5There is a similar discussion in [42].
Now, assuming that φ(x) ⋆x φ(x) and φ
′(x′) ⋆x′ φ
′(x′) be equal under the translation xi →
′i = xi + ǫi, we obtain, after the Fourier transformation,
φ̃′(g1)φ̃
′(g2) = (1− iǫiPi(g1g2))φ̃(g1)φ̃(g2), (123)
which is the same as (75).
From these two examples, we anticipate that the multi-field contributions in (41) comes
from the transformation properties of the star products.
4 Summary and comments
We have discussed symmetries in noncommutative field theories in the framework of braided
quantum field theory. We have obtained the algebraic conditions for a Hopf algebra to be a
symmetry of a braided quantum field theory, by discussing the conditions for the relations
among correlation functions generated from the transformation algebra to hold. Then we
have applied our discussions to the Poincaré symmetries in the effective noncommutative
field theory of three-dimensional quantum gravity coupled with spinless particles and in
the noncommutative field theory on Moyal plane. In the former case we can understand
the braiding between fields, which was derived from the three-dimensional quantum gravity
computation, from the viewpoint of the translational symmetry of the noncommutative field
theory on a Lie-algebraic noncommutative spacetime. In the latter case we have found that
the twisted Lorentz symmetry on Moyal plane is a symmetry of the quantum field theory only
after the inclusion of the nontrivial braiding factor, which is in agreement with the previous
proposal [28, 35]. Then we have discussed the meaning of the Hopf algebra symmetries from
the viewpoint of coordinate representation.
In the recent research a noncommutative field theory on κ-Minkowski spacetime is dis-
cussed [36]. Since this noncommutativity of the coordinates is given by [x0, xj] = i
xj , this
noncommutative field theory will not have the naive translational symmetry. We may intro-
duce a non-trivial braiding between fields as in the effective field theory discussed in Section
3.3 to keep the momentum conservation. However, while the effective field theory has the
braided category structure because of the invariance of the Haar measure d(g−1g′g) = dg′,
the measure of the momentum space of the field theory on κ-Minkowski spacetime is only
left-invariant [36]. Therefore it is not clear to us whether we can embed this field theory on
κ-Minkowski spacetime into the framework of braided quantum field theory.
Acknowledgments
We would like to thank S. Terashima and S. Sasaki for useful discussions and comments, and
would also like to thank L. Freidel for stimulating discussions and explaining their recent
results during his stay in Yukawa Institute for Theoretical Physics after the 21st Nishinomiya-
Yukawa Memorial Symposium. Y.S. was supported in part by JSPS Research Fellowships
for Young Scientists. N.S. was supported in part by the Grant-in-Aid for Scientific Research
No.13135213, No.16540244 and No.18340061 from the Ministry of Education, Science, Sports
and Culture of Japan.
A The proofs of the formula (20), (21)
We give the proofs of the formula (20), (21) using diagrams. At first we use the formula
êv(∂ ⊗ αβ) = êv(∂ ⊗ α)ǫ(β) + êv(∂ ⊗ β)ǫ(α), (124)
where α, β ∈ X̂. This is clear from the definition of êv.
Figure 14 gives the proof of (20). In the first line, we use the axiom (12), and in the
second line we use the lemma (124). We find the last line from the property of counit.
Next we prove (21). By using the braided Leibniz rule (20) as α ∈ X ⊗ X̂, the left-hand
side of (21) becomes Figure 15. The first term of Figure 15 becomes (ev ⊗ idn−1)(∂ ⊗ idnα)
by using the definition of coproduct (9).
In the second term of Figure 15, we divide X̂ into X ⊗ X̂ and iterate the same as we did
above. For example, if the degree of X̂ is 3, the second term of Figure 15 can be reduced as
in Figure 16. We have used ∆X = X⊗̂1+ 1⊗̂X in the second line of Figure 16. The result
agrees with (21).
In the same way, we can obtain the formula (21) in general.
B The proofs of (27), (28), (29)
From the definition of γ (25), we find that
αaw = −αdiff(γ−1(a)⊗ w), (125)
for a ∈ X and α ∈ X̂ . On the other hand, adding γ−1 and ψ to the braided Leibniz rule
(20) as in Figure 17, we find that
αdiff(γ−1(a)⊗ w) = diff(ψ(α⊗ γ−1(a))w)− (diff ◦ ψ(α⊗ γ−1(a)))w. (126)
Combining (125), (126), we obtain
αaw = −diff(ψ(α⊗ γ−1(a))w) + (diff ◦ ψ(α⊗ γ−1(a)))w. (127)
Integrating the both hand sides of (127) and using (24), we find that
Z(0)(αa) = Z(0)(diff ◦ ψ(α⊗ γ−1(a))). (128)
If α is b ∈ X ,
Z(0)(ba) = Z(0)(diff ◦ ψ(b⊗ γ−1(a)))
= ev ◦ ψ(b⊗ γ−1(a))
= ev ◦ (γ−1 ⊗ id) ◦ ψ(b⊗ a). (129)
Figure 14: The proof of (20).
Figure 15: The left-hand side of (21).
Figure 16: The second term of Figure 15.
Figure 17: The diagram obtained from adding γ−1 and ψ over the braided Leibniz rule.
Thus we obtain (27).
By putting α = 1, it is clear that
1 (a) = 0. (130)
Next we rewrite (128) for α ∈ Xn−1 using the formula (21). Diagrammatically it is
written as in Figure 18. The second equality is due to (21). Thus we obtain that
Z(0)n = (Z
n−2 ⊗ Z
2 ) ◦ ([n− 1]′ψ ⊗ id) (131)
Iterating this, we find (28) for even n and (29) for odd n.
Figure 18: Diagrammatic proof of (131)
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http://arxiv.org/abs/hep-th/9212151
http://arxiv.org/abs/hep-th/0605113
http://arxiv.org/abs/hep-th/0607221
http://arxiv.org/abs/hep-th/0109162
	Introduction
	Review of braided quantum field theory
	Braided categories and braided Hopf algebras
	Braided quantum field theory
	Braided Feynman rules
	Symmetries in braided quantum field theory
	General description of an action
	Symmetry relations among correlation functions and their algebraic descriptions
	Symmetries of the effective noncommutative field theory of three-dimensional quantum gravity coupled with scalar particles
	Twisted Poincaré symmetry of noncommutative field theory on Moyal plane
	Relations among correlation functions : Examples
	Origin of Hopf algebra symmetries
	Summary and comments
	The proofs of the formula (??), (??)
	The proofs of (??), (??), (??)
