## Twisting all the way: from Classical Mechanics to Quantum Fields (2007)

### Cached

### Download Links

Citations: | 20 - 7 self |

### BibTeX

@MISC{Aschieri07twistingall,

author = {Paolo Aschieri and Fedele Lizzi and Patrizia Vitale},

title = {Twisting all the way: from Classical Mechanics to Quantum Fields},

year = {2007}

}

### OpenURL

### Abstract

We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators. In particular commutation relations

### Citations

2100 | Noncommutative geometry
- Connes
- 1994
(Show Context)
Citation Context ...e usual description of spacetime as a pseudo-Riemannian manifold locally modeled on Minkowski space is not adequate anymore, and it has been proposed that it be described by a Noncommutative Geometry =-=[3, 4, 5]-=-. This line of thought has been pursued since the early days of Quantum Mechanics [6], and more recently in [7]-[18] (see also the recent review [19]). In this context two relevant issues are the form... |

513 | Foundations of quantum group theory - Majid - 1995 |

275 | Quantum Field Theory on Noncommutative Spaces, Phys. Rept
- Szabo
- 2003
(Show Context)
Citation Context ...t widely studied form of noncommutativity is the one for which the quantity Θ µν of (1.1) is a constant. This noncommutativity is obtained trought the Grönewold-Moyal-Weyl ⋆-product (for a review see =-=[29]-=-). The product between functions (fields) is given by (f ⋆ h)(x) = exp ( i ∂ θµν 2 ∂x µ ∂ ∂y ν ) f(x)h(y) ∣ ∣ x=y (1.2) with the θ µν -matrix constant and antisymmetric. In particular the coordinates ... |

239 |
Differential calculus on compact matrix pseudogroups (quantum groups
- WORONOWICZ
- 1989
(Show Context)
Citation Context ...ctorspaces Ξ = Ξ⋆, but Ξ⋆ is a ⋆-Lie algebra. We stress that a ⋆-Lie algebra is not a generic name for a deformation of a Lie algebra. Rather it is a quantum Lie algebra of a quantum (symmetry) group =-=[38]-=-, (see [39] for a short introduction and further references). In this respect the deformed Leibniz rule (2.31), that states that only vectorfields (or the identity) can act on the second argument g in... |

223 | The Quantum structure of space-time at the Planck scale and quantum
- Doplicher, Fredenhagen, et al.
- 1995
(Show Context)
Citation Context ...ment which points at a failure of the classical spacetime picture at high energy scales comes from the attempt of conjugating the principles of Quantum Mechanics with those of General Relativity (see =-=[1]-=-, and for a review [2]). If one tries to locate an event with a spatial accuracy comparable with the Planck length, spacetime uncertainty relations necessarily emerge. In total analogy with Quantum Me... |

181 | An introduction to noncommutative spaces and their geometries
- Landi
- 1997
(Show Context)
Citation Context ...e usual description of spacetime as a pseudo-Riemannian manifold locally modeled on Minkowski space is not adequate anymore, and it has been proposed that it be described by a Noncommutative Geometry =-=[3, 4, 5]-=-. This line of thought has been pursued since the early days of Quantum Mechanics [6], and more recently in [7]-[18] (see also the recent review [19]). In this context two relevant issues are the form... |

119 |
On a LorentzInvariant Interpretation of Noncommutative Space-Time
- Chaichian, Kulish, et al.
(Show Context)
Citation Context ...� or c. Since the commutator x µ ⋆ x ν − x ν ⋆ x µ in (1.3) is not Lorentz invariant, the usual notion of Poincaré symmetry is lost. However there is still a symmetry, due to a twisted Poincaré group =-=[32, 33, 34, 35]-=-, a quantum Poincaré Lie algebra and Lie group invariance that implies that fields on noncommutative space are organized according to the same particle representations as in commutative space. We adop... |

107 |
Foundations of Quantum Group Theory,” Cambridge Univ
- Majid
(Show Context)
Citation Context ...ly normalized. In general an element F of UΞ⊗UΞ is a twist if it is invertible, satisfies a cocycle condition and is properly normalized [37] (see [18, 17] for a short introduction; see also the book =-=[38]-=-). The cocycle and the normalization conditions imply associativity of the ⋆-product and the normalization h⋆1 = 1⋆h = h. 42.1 Vectorfields and Tensorfields We now use the twist to deform the spaceti... |

73 |
A gravity theory on noncommutative spaces
- Aschieri, Blohmann, et al.
- 2005
(Show Context)
Citation Context ...ent canonical commutation relations have been considered in the literature [20]-[28]. We here frame this issue in a geometric context and address it by further developing the twist techniques used in =-=[17, 16, 18]-=- in order to formulate a noncommutative gravity theory. We see how noncommutative spacetime induces a noncommutative phase space geometry, equipped with a deformed Poisson bracket. This leads to canon... |

49 | Noncommutative geometry and gravity
- Aschieri, Dimitrijevic, et al.
- 2006
(Show Context)
Citation Context ...ent canonical commutation relations have been considered in the literature [20]-[28]. We here frame this issue in a geometric context and address it by further developing the twist techniques used in =-=[17, 16, 18]-=- in order to formulate a noncommutative gravity theory. We see how noncommutative spacetime induces a noncommutative phase space geometry, equipped with a deformed Poisson bracket. This leads to canon... |

49 |
Constant quasiclassical solutions of the Yang-Baxter quantum equation
- Drinfel’d
- 1983
(Show Context)
Citation Context ...symplectic transformations, and that of the constants of motion of a given Hamiltonian system. These noncommutative spaces and symmetries are obtained by deforming the usual ones via a Drinfeld twist =-=[36]-=-. For example the Drinfeld twist that implements the Moyal-Weyl i − noncommutativity (1.2) is F = e 2 θµν∂µ⊗∂ν . i − In Section 2 we introduce the twist F = e 2 θµν∂µ⊗∂ν and, starting from the princip... |

40 | Differential Calculus on ISOq(N), Quantum Poincaré Algebra and q-Gravity - Castellani |

29 | Gauge and Einstein Gravity from Non-Abelian Gauge Models on Noncommutative Spaces”, Phys - Vacaru |

29 |
Emergent gravity from noncommutative gauge theory
- Steinacker
(Show Context)
Citation Context ...re, and it has been proposed that it be described by a Noncommutative Geometry [3, 4, 5]. This line of thought has been pursued since the early days of Quantum Mechanics [6], and more recently in [7]-=-=[19]-=- (see also the recent review [20]). In this context two relevant issues are the formulation of General Relativity and the quantization of field theories on noncommutative spacetime. There are differen... |

25 | Quantum and braided group Riemannian geometry - Majid - 1999 |

22 | Deforming Einstein’s gravity,” Phys - Chamseddine - 2001 |

22 | On full twisted Poincaré symmetry and QFT on Moyal–Weyl spaces, Phys
- Fiore, Wess
(Show Context)
Citation Context ...ntization of field theories on noncommutative spacetime. There are different proposals for this second issue, and different canonical commutation relations have been considered in the literature [20]-=-=[28]-=-. We here frame this issue in a geometric context and address it by further developing the twist techniques used in [17, 16, 18] in order to formulate a noncommutative gravity theory. We see how nonco... |

21 |
Twisting noncommutative Rd and the equivalence of quantum field theories
- Oeckl
(Show Context)
Citation Context ...� or c. Since the commutator x µ ⋆ x ν − x ν ⋆ x µ in (1.3) is not Lorentz invariant, the usual notion of Poincaré symmetry is lost. However there is still a symmetry, due to a twisted Poincaré group =-=[32, 33, 34, 35]-=-, a quantum Poincaré Lie algebra and Lie group invariance that implies that fields on noncommutative space are organized according to the same particle representations as in commutative space. We adop... |

20 |
Remarks on twisted noncommutative quantum field theory, Phys
- Zahn
(Show Context)
Citation Context ...finition of the R-matrix it can be easily verified that [ ˆ F, ˆ G]⋆ = ˆ F⋆ ˆ G − ¯ R α ( ˆ G)⋆ ¯ Rα( ˆ F) (5.7) which is indeed the ⋆-commutator in Â⋆. This ⋆-commutator (5.3) has been considered in =-=[25]-=- (and was introduced in [42]). We studied four algebras and brackets: (A, { , }), ( Â, [ , ]), (A⋆, { , }⋆), ( Â⋆, [ , ]⋆) . Canonical quantization on noncommutative space is the map �⋆ in the diagram... |

20 |
New concept of relativistic invariance in NC spacetime: twisted Poincaré symmetry and its
- Chaichian, Preˇsnajder, et al.
(Show Context)
Citation Context ...� or c. Since the commutator x µ ⋆ x ν − x ν ⋆ x µ in (1.3) is not Lorentz invariant, the usual notion of Poincaré symmetry is lost. However there is still a symmetry, due to a twisted Poincaré group =-=[32, 33, 34, 35]-=-, a quantum Poincaré Lie algebra and Lie group invariance that implies that fields on noncommutative space are organized according to the same particle representations as in commutative space. We adop... |

19 |
Deformed Coordinate Spaces; Derivatives, lecture given at Balcan Workshop BW2003 on “Mathematical, Theoretical and Phenomenological Challenges Beyond Standard Model”, 29 August-02 September, 2003 Vrnjacka
- Wess
- 2005
(Show Context)
Citation Context |

18 | Symmetry, gravity and noncommutativity
- Szabo
- 2006
(Show Context)
Citation Context ... it be described by a Noncommutative Geometry [3, 4, 5]. This line of thought has been pursued since the early days of Quantum Mechanics [6], and more recently in [7]-[18] (see also the recent review =-=[19]-=-). In this context two relevant issues are the formulation of General Relativity and the quantization of field theories on noncommutative spacetime. There are different proposals for this second issue... |

18 | and Spin-Statistics Relation in Noncommutative Quantum Field Theory”, Phys - Tureanu, “Twist |

18 | Quantum space-time and classical gravity - Madore, Mourad - 1998 |

17 | Noncommutative deformation of four dimensional Einstein gravity - Cardella, Zanon - 2003 |

16 | Gravity on fuzzy space-time
- Madore
(Show Context)
Citation Context ...nymore, and it has been proposed that it be described by a Noncommutative Geometry [3, 4, 5]. This line of thought has been pursued since the early days of Quantum Mechanics [6], and more recently in =-=[7]-=--[18] (see also the recent review [19]). In this context two relevant issues are the formulation of General Relativity and the quantization of field theories on noncommutative spacetime. There are dif... |

16 | Noncommutative Quantum Gravity, Phys - Moffat - 2000 |

16 |
Spin and statistics on the Groenwald-Moyal plane: Pauli-forbidden levels and transitions
- Balachandran, Mangano, et al.
- 2006
(Show Context)
Citation Context ...r also from (5.13) and the quantum analogue of (4.21)), [â(k), â † (k ′ )]⋆ = (2π) d δ(k − k ′ ) . (5.15) In order to compare this expression with similar ones which have been found in the literature =-=[20, 21, 22, 23, 26, 27, 28]-=- it is useful to recall (5.7) and realize the action of the R-matrix. Since R = F −2 we obtain that (5.15) is equivalent to â(k) ⋆ â † (k ′ ) − e −iθij k ′ i kj â † (k ′ ) ⋆ â(k) = (2π) d δ(k − k ′ ) ... |

13 | Twists of Quantum Groups and Noncommutative Field Theory
- Kulish
(Show Context)
Citation Context ...r also from (5.13) and the quantum analogue of (4.21)), [â(k), â † (k ′ )]⋆ = (2π) d δ(k − k ′ ) . (5.15) In order to compare this expression with similar ones which have been found in the literature =-=[20, 21, 22, 23, 26, 27, 28]-=- it is useful to recall (5.7) and realize the action of the R-matrix. Since R = F −2 we obtain that (5.15) is equivalent to â(k) ⋆ â † (k ′ ) − e −iθij k ′ i kj â † (k ′ ) ⋆ â(k) = (2π) d δ(k − k ′ ) ... |

12 |
Twisted conformal symmetry in noncommutative two-dimensional quantum field theory
- Lizzi, Vaidya, et al.
(Show Context)
Citation Context ...r also from (5.13) and the quantum analogue of (4.21)), [â(k), â † (k ′ )]⋆ = (2π) d δ(k − k ′ ) . (5.15) In order to compare this expression with similar ones which have been found in the literature =-=[20, 21, 22, 23, 26, 27, 28]-=- it is useful to recall (5.7) and realize the action of the R-matrix. Since R = F −2 we obtain that (5.15) is equivalent to â(k) ⋆ â † (k ′ ) − e −iθij k ′ i kj â † (k ′ ) ⋆ â(k) = (2π) d δ(k − k ′ ) ... |

12 |
Noncommutative Field Theory from Twisted Fock Space”, Phys
- Bu, Kim, et al.
(Show Context)
Citation Context |

11 |
Statistics and UV-IR mixing with twisted Poincaré invariance, Phys
- Balachandran, Govindarajan, et al.
(Show Context)
Citation Context |

10 | Noncommutative Symmetries and Gravity
- Aschieri
(Show Context)
Citation Context ...Lie algebra Ξ⋆ we have constructed gives rise to the universal enveloping algebra UΞ⋆ of sums of products of vectorfields, with the identification u⋆v− ¯R α (v)⋆ ¯Rα(u) = [u, v]⋆ and coproduct (2.32) =-=[17, 18]-=-. The Hopf (or symmetry) algebras UΞF and UΞ⋆ are isomorphic. Therefore to some extent it is a matter of taste wich algebra one should use. We prefer UΞ⋆ becasue UΞ⋆ naturally arises from the general ... |

10 |
Comments on noncommutative gravity
- Alvarez-Gaume, Meyer, et al.
(Show Context)
Citation Context ... x ν − x ν ⋆ x µ = iθ µν . (1.3) 1There are two approaches to study the symmetries (e.g. Poincaré symmetry) of this noncommutative space. One can consider θ µν as a covariant tensor (see for example =-=[30, 31]-=-), then the Moyal product is fully covariant under Poincaré (indeed linear affine) transformations. Poincaré symmetry is spontaneously broken by the nonzero values θ µν . The other approach is to cons... |

8 |
Letter from Heisenberg to Peierls, in
- Heisenberg
- 1985
(Show Context)
Citation Context ...ki space is not adequate anymore, and it has been proposed that it be described by a Noncommutative Geometry [3, 4, 5]. This line of thought has been pursued since the early days of Quantum Mechanics =-=[6]-=-, and more recently in [7]-[18] (see also the recent review [19]). In this context two relevant issues are the formulation of General Relativity and the quantization of field theories on noncommutativ... |

8 |
Noncommutative spacetime symmetries: Twist versus covariance, Phys
- Gracia-Bondia, Lizzi, et al.
(Show Context)
Citation Context ... x ν − x ν ⋆ x µ = iθ µν . (1.3) 1There are two approaches to study the symmetries (e.g. Poincaré symmetry) of this noncommutative space. One can consider θ µν as a covariant tensor (see for example =-=[30, 31]-=-), then the Moyal product is fully covariant under Poincaré (indeed linear affine) transformations. Poincaré symmetry is spontaneously broken by the nonzero values θ µν . The other approach is to cons... |

8 | Ultraviolet finiteness of the averaged Hamiltonian on the noncommutative Minkowski space,” arXiv:hep-th/0405224
- Bahns
(Show Context)
Citation Context ...can be easily verified that [ ˆ F, ˆ G]⋆ = ˆ F⋆ ˆ G − ¯ R α ( ˆ G)⋆ ¯ Rα( ˆ F) (5.7) which is indeed the ⋆-commutator in Â⋆. This ⋆-commutator (5.3) has been considered in [25] (and was introduced in =-=[42]-=-). We studied four algebras and brackets: (A, { , }), ( Â, [ , ]), (A⋆, { , }⋆), ( Â⋆, [ , ]⋆) . Canonical quantization on noncommutative space is the map �⋆ in the diagram F A ⏐ ↓ � −−−→ bF Â ⏐ ↓ (5.... |

8 |
On the construction of Möller operators for the nonlinear Schrödinger equation, Phys
- Grosse
- 1979
(Show Context)
Citation Context ... the action of the R-matrix. Since R = F −2 we obtain that (5.15) is equivalent to â(k) ⋆ â † (k ′ ) − e −iθij k ′ i kj â † (k ′ ) ⋆ â(k) = (2π) d δ(k − k ′ ) . (5.16) This relation first appeared in =-=[44]-=-. In the noncommutative QFT context it appears in [28], [27], and implicitly in [26] (it is also contemplated in [29] as a second option). On the other hand [22, 23, 24, 29], starting from a different... |

7 |
Lectures on Hopf algebras, quantum groups and twists
- Aschieri
(Show Context)
Citation Context ...Ξ = Ξ⋆, but Ξ⋆ is a ⋆-Lie algebra. We stress that a ⋆-Lie algebra is not a generic name for a deformation of a Lie algebra. Rather it is a quantum Lie algebra of a quantum (symmetry) group [38], (see =-=[39]-=- for a short introduction and further references). In this respect the deformed Leibniz rule (2.31), that states that only vectorfields (or the identity) can act on the second argument g in h⋆g (no hi... |

6 |
Spacetime and fields, a quantum texture," [arXiv:hep-th/0105251]. [3] A. Connes, Noncommutative Geometry
- Doplicher
- 1994
(Show Context)
Citation Context ... failure of the classical spacetime picture at high energy scales comes from the attempt of conjugating the principles of Quantum Mechanics with those of General Relativity (see [1], and for a review =-=[2]-=-). If one tries to locate an event with a spatial accuracy comparable with the Planck length, spacetime uncertainty relations necessarily emerge. In total analogy with Quantum Mechanics, uncertainty r... |

6 |
Deformed Gauge Theories
- Wess
- 608
(Show Context)
Citation Context ...algebra of functions on Rd and the algebra (4.1) become noncommutative with noncommutativity given by i ∂ ∂ − θij the twist (2.2), F = e 2 ∂xi ⊗ ∂x j . The twist lifts to the algebra A of functionals =-=[41]-=- so that this latter too becomes noncommutative. This is achieved by lifting to A the action of the partial derivatives ∂ ∂xi. Explicitly ∂i is lifted to ∂∗ i acting on A as, ∂ ∗ ∫ i G := − d d x ∂iΦ(... |

4 | Noncommutative Quantum Gravity - Moffat - 2000 |

4 | Deforming Einstein’s Gravity - Chamseddine - 2001 |

3 | 2+1 Einstein Gravity as a Deformed Chern-Simons Theory - Bimonte, Musto, et al. |

2 |
Statistics and quantum group symmetries, [arXiv:hepth/9605133]. Published in Quantum groups and quantum spaces
- Fiore, Schupp
- 1997
(Show Context)
Citation Context ...e quantization of field theories on noncommutative spacetime. There are different proposals for this second issue, and different canonical commutation relations have been considered in the literature =-=[20]-=--[28]. We here frame this issue in a geometric context and address it by further developing the twist techniques used in [17, 16, 18] in order to formulate a noncommutative gravity theory. We see how ... |

2 | particles and quantum symmetries,” Nucl - “Identical - 1996 |

1 |
Dynamical symmetries in q deformed quantum mechanics
- Lorek, Wess
- 1995
(Show Context)
Citation Context ...lex conjugates of (3.63). We find also interesting to study deformations H⋆ of the harmonic oscillator Hamiltonian that admit the undeformed angular momentum L as constant of motion. The aim, like in =-=[40]-=-, is to consider new dynamical systems that may be highly nontrivial if thought in commutative space (the equation of motion (3.49) or (3.59) can just be seen as a partial differential equation on com... |

1 |
Comments on noncommutative gravity,” Nucl. Phys. B 753 (2006) 92 [arXiv:hep-th/0605113]. 31
- Alvarez-Gaume, Meyer, et al.
(Show Context)
Citation Context ... x ν − x ν ⋆ x µ = iθ µν . (1.3) 1There are two approaches to study the symmetries (e.g. Poincaré symmetry) of this noncommutative space. One can consider θ µν as a covariant tensor (see for example =-=[31, 32]-=-), then the Moyal product is fully covariant under Poincaré (indeed linear affine) transformations. Poincaré symmetry is spontaneously broken by the nonzero values θ µν . The other approach is to cons... |