Scalar field theory on kappa-Minkowski spacetime and translation and Lorentz invariance
Stjepan Meljanac, Andjelo Samsarov
备注：
22 pages, no figures, revtex4, in new version comments regarding translation invariance and few references are added, accepted for publication in Int. J. Mod. Phys. A
We investigate the properties of kappa-Minkowski spacetime by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg-Weyl algebra. The deformed algebra consists of kappa-Poincare algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an adventages of this approach to consistently construct a star product which has a property that under integration sign it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal, but not also for kappa-Minkowski spacetime. This star product also has generalized trace and cyclic properties and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and by requiring it to be hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachionic modes and basicaly of the very same form. The issue of Lorentz invariance of the theory is also discussed.
We propose a generalized description for the kappa-Poincare-Hopf algebra as a symmetry quantum group of underlying kappa-Minkowski spacetime. We investigate all the possible implementations of (deformed) Lorentz algebras which are compatible with the given choice of kappa-Minkowski algebra realization. For the given realization of kappa-Minkowski spacetime there is a unique kappa-Poincare-Hopf algebra with undeformed Lorentz algebra. We have constructed a three-parameter family of deformed Lorentz generators with kappa-Poincare algebras which are related to kappa-Poincare-Hopf algebra with und…
The algebra of functions on kappa-Minkowski noncommutative spacetime is studied as algebra of operators on Hilbert spaces. The representations of this algebra are constructed and classified. This new approach leads to a natural construction of integration in kappa-Minkowski spacetime in terms of the usual trace of operators.
The differential structure on the kappa-Minkowski spacetime from Jordanian twist of Weyl algebra is constructed, and it is shown to be closed in 4-dimensions in contrast to the conventional formulation. Based on this differential structure, we have formulated a scalar field theory in this kappa-Minkowski spacetime.
We describe the generalized kappa-deformations of D=4 relativistic symmetries with finite masslike deformation parameter kappa and an arbitrary direction in kappa-deformed Minkowski space being noncommutative. The corresponding bicovariant differential calculi on kappa-deformed Minkowski spaces are considered. Two distinguished cases are discussed: 5D noncommutative differential calculus (kappa-deformation in time-like or space-like direction), and 4D noncommutative differential calculus having the classical dimension (noncommutative kappa-deformation in light-like direction). We introduce als…
We introduce the notion of generalized Weyl system, and use it to define *-products which generalize the commutation relations of Lie algebras. In particular we study in a comparative way various *-products which generalize the k-Minkowski commutation relations.
We perform a Noether analysis for a description of translation transformations in 4D kappa-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. Taking properly into account the properties of the differential calculus we arrive at an explicit formula for the conserved charges. We also propose a choice of basis for the 5D calculus which leads to an intuitive description of time derivatives.
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenological consequences. However, the construction of field theories on this space, although operationally well-defined, is plagued with ambiguities. A part of ambiguities can be resolved by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time. To this end we use the twist approach to construct the noncommutative U(1) gauge theory coupled to fermions. However, in this approach we cannot maintain the kappa-Poincaré symmetry; the corresponding symmetry…
Kappa-Minkowski space-time is an example of noncommutative space-time with potentially interesting phenomenology. However, the construction of field theories on this space is plagued with ambiguities. We propose to resolve certain ambiguities by clarifying the geometrical picture of gauge transformations on the kappa-Minkowski space-time in the twist approach. We construct the action for the noncommutative U(1) gauge fields in a geometric way, as an integral of a maximal form. The effective action with the first order corrections in the deformation parameter is obtained using the Seiberg-Witte…
We show that free $κ$-Minkowski space field theory is equivalent to a relativistically invariant, non local, free field theory on Minkowski space-time. The field theory we obtain has in spectrum a relativistic mode of arbitrary mass $m$ and a Planck mass tachyon. We show that while the energy momentum for the relativistic mode is essentially the standard one, it diverges for the tachyon, so that there are no asymptotic tachyonic states in the theory. It also follows that the dispersion relation is not modified, so that, in particular, in this theory the speed of light is energy-independent.
We consider the issue of statistics for identical particles or fields in kappa-deformed spaces, where the system admits a symmetry group G. We obtain the twisted flip operator compatible with the action of the symmetry group, which is relevant for describing particle statistics in presence of the noncommutativity. It is shown that for a special class of realizations, the twisted flip operator is independent of the ordering prescription.
This study of U(1) gauge field theory on the kappa-deformed Minkowski spacetime extends previous work on gauge field theories on this type of noncommutative spacetime. We discuss in detail the properties of the Seiberg-Witten map and the resulting effective action for U(1) gauge theory with fermionic matter expanded in ordinary fields. We construct the conserved gauge current, fix part of the ambiguities in the Seiberg-Witten map and obtain an effective U(1) action invariant under the action of the undeformed Poincare group.
Following the construction of the $κ$-Minkowski space from the bicrossproduct structure of the $κ$-Poincare group, we investigate possible differential calculi on this noncommutative space. We discuss then the action of the Lorentz quantum algebra and prove that there are no 4D bicovariant differential calculi, which are Lorentz covariant. We show, however, that there exist a five-dimensional differential calculus, which satisfies both requirements. We study also a toy example of 2D $κ$-Minkowski space and and we briefly discuss the main properties of its differential calculi.
This study of gauge field theories on kappa-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the Seiberg-Witten formalism. Derivative-valued gauge fields lead to field strength tensors as the sum of curvature- and torsion-like terms. We construct the Lagrangians explicitly to first order in the deformation parameter. This is the first example of a gauge theory that possesses a deformed L…
We investigate a Lie algebra-type $ κ$-deformed Minkowski space-time with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of $ κ$-Minkowski space. The coproduct and the star product corresponding to each of them are found. Utilizing the properties of the {\em{natural}} realization, we construct a scalar field theory on $ κ$-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space.
In this paper we have analyzed the $κ$-deformed Minkowski spacetime through the light of the interference phenomena in QFT where two opposite chiral fields are put together in the same multiplet and its consequences are discussed. The chiral models analyzed here are the chiral Schwinger model, its generalized version and its gauge invariant version, where a Wess-Zumino term were added. We will see that the final actions obtained here are in fact related to the original ones via duality transformations.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincare algebra. These deformations interpolate between Snyder and kappa-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. Deformed Leibniz rule, the coproduct structure and star product are found. Special cases, particularly Snyder and kappa-Minkowski in Maggiore-type realizations are discussed. Our construction leads to a new class of deformed special relativity theories.
The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative spacetimes. In this paper, we formulate Noether's theorem for translations of kappa-Minkowski noncommutative spacetime on the basis of the 5-dimensional kappa-Poincare' covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy-momentum charges. By applying our result to plane waves it follows that the…
We perform a Noether analysis for a description of translation transformations in 4D $κ$-Minkowski noncommutative spacetime which is based on the structure of a 5D differential calculus. The techniques that some of us had previously developed (hep-th/0607221) for a description of translation transformations based on a 4D differential calculus turn out to be applicable without any modification, and they allow us to show that the basis usually adopted for the 5D calculus does not take into account certain aspects of the structure of time translations in $κ$-Minkowski. We propose a change of basi…
It is by now well established that the momentum space dual to the non-commutative $κ$-Minkowski space is a submanifold of de Sitter space. It has been noticed recently that field theories built on such momentum space suffer from a subtle form of Lorentz symmetry breaking. Namely, for any negative energy mode the allowed range of rapidities is bounded above. In this paper we construct a complex scalar field theory with a modified action of Lorentz generators which avoids this problem. For such theory we derive conserved charges corresponding to translational and U(1) symmetries. We also discuss…
A velocity of a point particle in the kappa-Minkowski spacetime is investigated. Characteristic points of the spacetime are that the Poincare group becomes a quantum group with kappa, which is a mass dimension parameter, and is a kind of non-commutative geometry. We consider a particle in a coordinate space instead of it in a momentum space which is discussed in many articles. We see that the particle's velocity has an uncertainty which depends on a length of particle's propagation.
In this paper, we derive Lorentz force and Maxwell's equations on kappa-Minkowski space-time up to the first order in the deformation parameter. This is done by elevating the principle of minimal coupling to non-commutative space-time. We also show the equivalence of minimal coupling prescription and Feynman's approach. It is shown that the motion in kappa space-time can be interpreted as motion in a background gravitational field, which is induced by this non-commutativity. In the static limit, the effect of kappa deformation is to scale the electric charge. We also show that the laws of elec…
We twist the Hopf algebra of igl(n,R) to obtain the kappa-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The kappa-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional kappa-Minkowski spacetime, but its Hopf algebra structure is different from the well known kappa-deformed Poincare algebra in that it has larger symmetry algebra than the kappa-Minkowski case. There are some physical models which consider this symmetry. Incidentally, we obtain the canonical (theta-deformed) non-commutative spacetim…
We construct realizations of the generators of the $κ$-Minkowski space and $κ$-Poincaré algebra as formal power series in the $h$-adic extension of the Weyl algebra. The Hopf algebra structure of the $κ$-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on $κ$-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the $κ$-Minkowski space.
We have constructed noncommutative phi^4 field theory on kappa-Minkowski spacetime. Quantum properties via 2-point functions were analized, and effect of birefringence for the massive scalar field has been found.
General realizations, star products and plane waves for kappa-Minkowski spacetime are considered. Systematic construction of general hermitian realization is presented, with special emphasis on noncommutative plane waves and hermitian star product. Few examples are elaborated and possible physical applications are mentioned.
We present the star-product algebra of the kappa-deformation of Minkowski space and the formulation of Poincare covariant differential calculus. We use these tools to construct a twisted K-cycle over the algebra and a twisted cyclic cocycle.
We examine some alternative possibilities for an action functional for $κ$-Minkowski noncommutative spacetime, with an approach which should be applicable to other spacetimes with coordinate-dependent commutators of the spacetime coordinates ($[x_μ,x_ν]=f_{μ,ν}(x)$). Early works on $κ$-Minkowski focused on $κ$-Poincaré covariance and the dependence of the action functional on the choice of Weyl map, renouncing to invariance under cyclic permutations of the factors composing the argument of the action functional. A recent paper (hep-th/0307149), by Dimitrijevic, Jonke, Moller, Tsouchnika, Wess …
In this article we use the noncommutative (NC) kappa-Minkowski phi^4 model based on the kappa-deformed star product, ({*}_h). The action is modified by expanding up to linear order in the kappa-deformation parameter a, producing an effective model on commutative spacetime. For the computation of the tadpole diagram contributions to the scalar field propagation/self-energy, we anticipate that statistics on the kappa-Minkowski is specifically kappa-deformed. Thus our prescription in fact represents hybrid approach between standard quantum field theory (QFT) and NCQFT on the kappa-deformed Minkow…
We propose two families of differential algebras of classical dimension on kappa-Minkowski space. The algebras are constructed using realizations of the generators as formal power series in a Weyl super-algebra. We also propose a novel realization of the Lorentz algebra so(1,n-1) in terms of Grassmann-type variables. Using this realization we construct an action of so(1,n-1) on the two families of algebras. Restriction of the action to kappa-Minkowski space is covariant. In contrast to the standard approach the action is not Lorentz covariant except on constant one-forms, but it does not requi…
The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, wh…
We unify kappa-Minkowki spacetime and Lorentz algebra in unique Lie algebra. Introducing commutative momenta, a family of kappa-deformed Heisenberg algebras and kappa-deformed Poincare algebras are defined. They are specified by the matrix depending on momenta. We construct all such matrices. Realizations and star product are defined and analyzed in general and specially, their relation to coproduct of momenta is pointed out. Hopf algebra of the Poincare algebra, related to the covariant realization, is presented in unified covariant form. Left-right dual realizations and dual algebra are intr…
In a previous article [arXiv:1011.4258], we established the consistency of isotropic modified Maxwell theory for a finite range of the Lorentz-violating parameter \tilde{kappa}_{tr}, which includes both positive and negative values of \tilde{kappa}_{tr}. As an aside, we mentioned the existence of a physical model which, for low-energy photons, gives rise to isotropic modified Maxwell theory with a positive parameter \tilde{kappa}_{tr} (corresponding to a "slow" photon). Here, we present a related model which gives rise to isotropic modified Maxwell theory with a negative parameter \tilde{kappa…
We investigate the phenomenological consequences of kappa-Minkowski extension of the Standard Model, working in the linear order in inverse $κ$. At this order the *-deformed Lagrangian can be expanded in the series of dimension five operators that have non-trivial transformation properties under the ordinary Lorentz invariance. Such operators cause the Lorentz-violating signatures at low energies, and in particular lead to the anomalous spin precession linked to the external direction. The experimental bounds on this phenomenon then restrict parameter $κ$ to be above 10^{23} GeV, making it dif…
Two one-parameter families of twists providing kappa-Minkowski * -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld's twisting tensor technique. The other one relies on an appropriate extension of "deformed realizations" of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us t…
We construct field theory on noncommutative $κ$-Minkowski space-time. Having the Lorentz action on the noncommutative space-time coordinates we show that the field lagrangian is invariant. We show that noncommutativity requires replacing the Leibnitz rule with the coproduct one.
We present two different quantum deformations for the (anti)de Sitter algebras and groups. The former is a non-standard (triangular) deformation of SO(4,2) realized as the conformal group of the (3+1)D Minkowskian spacetime, while the latter is a standard (quasitriangular) deformation of both SO(2,2) and SO(3,1) expressed as the kinematical groups of the (2+1)D anti-de Sitter and de Sitter spacetimes, respectively. The Hopf structure of the quantum algebra and a study of the dual quantum group are presented for each deformation. These results enable us to propose new non-commutative spacetimes…
The $κ$-Minkoswki space-time provides a quantum noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional $κ$-Minkoswki space-time using the more general framework of Lorentzian noncommutative geometry. We show that this notion allows specific causal relations, but limited by a general constraint which is a quantum generalization of the traditional speed of light limit.
We define a family of star products and involutions associated with $κ$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $κ$-Minkowski space. Furthermore, we give an explicit realization of the action of the $κ$-Poincaré algebra as an involutive Hopf algebra on this repr…
In recent years, different views on the interpretation of Lorentz covariance of non commuting coordinates were discussed. Here, by a general procedure, we construct the minimal canonical central covariantisation of the k-Minkowski spacetime. We then show that, though the usual k-Minkowski spacetime is covariant under deformed (or twisted) Lorentz action, the resulting framework is equivalent to taking a non covariant restriction of the covariantised model. We conclude with some general comments on the approach of deformed covariance.
We present Lie-algebraic deformations of Minkowski space with undeformed Poincaré algebra. These deformations interpolate between Snyder and $κ$-Minkowski space. We find realizations of noncommutative coordinates in terms of commutative coordinates and derivatives. By introducing modules, it is shown that although deformed and undeformed structures are not isomorphic at the level of vector spaces, they are however isomorphic at the level of Hopf algebraic action on corresponding modules. Invariants and tensors with respect to Lorentz algebra are discussed. A general mapping from $κ$-deformed S…
We uncover an infinite number of vacua in two-dimensional quantum field theory, the Klein-Gordon field for simplicity, by conceiving a new mode that is classified by a real positive parameter $κ$. We show each mode has a distinct vacuum, say $κ$-vacuum. This new mode is a generalization of the Unruh-Minkowski mode. Moreover, the Minkowski and Rindler vacua are special cases of the $κ$-vacuum for $κ= 1$ and $κ\rightarrow \infty$, respectively.
We describe $κ$-Minkowski space and its relation to group theory. The group theoretical picture makes it possible to analyze the symmetries of this space. As an application of this analysis we analyze in detail free field theory on $κ$-Minkowski space and the Noether charges associated with deformed spacetime symmetries.
We compare two versions of deformed dispersion relations (energy vs momenta and momenta vs energy) and the corresponding time delay up to the second order accuracy in the quantum gravity scale (deformation parameter). A general framework describing modified dispersion relations and time delay with respect to different noncommutative kappa -Minkowski spacetime realizations is firstly proposed here and it covers all the cases introduced in the literature. It is shown that some of the realizations provide certain bounds on quadratic corrections, i.e. on quantum gravity scale, but it is not exclud…
In this paper, we derive the expression for spectral dimension using a modified diffusion equation in the kappa-deformed space-time. We start with the Beltrami-Laplace operator in the kappa-Minkowski space-time and obtain the deformed diffusion equation. From the solution of this deformed diffusion equation, we calculate the spectral dimension which depends on the deformation parameter `$a$' and also on an integer `$k$', apart from the topological dimension. Using this, we show that, for large diffusion times the spectral dimension approaches the usual topological dimension where as spectral d…
This paper is devoted to detailed investigations of free scalar field theory on $κ$-Minkowski space. After reviewing necessary mathematical tools we discuss in depth the Lagrangian and solutions of field equations. We analyze the spacetime symmetries of the model and construct the conserved charges associated with translational and Lorentz symmetry. We show that the version of the theory usually studied breaks Lorentz invariance in a subtle way: There is an additional trans-Planckian mode present, and an associated conserved charge (the number of such modes) is not a Lorentz scalar.
We consider the Lie group $\mathbb{R}^D_κ$ generated by the Lie algebra of $κ$-Minkowski space. Imposing the invariance of the metric under the pull-back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with $\mathbb{R}^D_κ$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_κ$ relevant for the realization of the non-commutative $κ$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg algebra. Using a suitable set of g…
In this paper we explore the problem of antiparticles in DSR1 and $κ$-Minkowski space-time following three different approaches inspired by the Lorentz invariant case: a) the dispersion relation, b) the Dirac equation in space-time and c) the Dirac equation in momentum space. We find that it is possible to define a map $S_{dsr}$ which gives the antiparticle sector from the negative frequency solutions of the wave equation. In $κ$-Poincaré, the corresponding map $S_{kp}$ is the antipodal mapping, which is different from $S_{dsr}$. The difference is related to the composition law, which is cruci…
We study boost and space-rotation transformations in kappa-Minkowski noncommutative spacetime, using the techniques that some of us had previously developed (hep-th/0607221) for a description of translations in kappa-Minkowski, which in particular led to the introduction of translation transformation parameters that do not commute with the spacetime coordinates. We find a similar description of boosts and space rotations, which allows us to identify some associated conserved charges, but the form of the commutators between transformation parameters and spacetime coordinates is incompatible wit…
In this paper, we derive the Dirac equation in the $κ$-deformed Minkowski space-time. We start with $κ$-deformed Minkowski space-time and investigate the undeformed $κ$-Lorentz transformation valid to all order in the deformation parameter $a$. Using the undeformed $κ$-Lorentz algebra, we obtain the $κ$-deformed Dirac equation, valid to all order in the deformation parameter $a$. In limit $a\rightarrow$0, we get back the correct commutative result.
We consider the general D=4 (10+10)-dimensional kappa-deformed quantum phase space as given by Heisenberg double \mathcal{H} of D=4 kappa-deformed Poincare-Hopf algebra H. The standard (4+4) -dimensional kappa - deformed covariant quantum phase space spanned by kappa - deformed Minkowski coordinates and commuting momenta generators ({x}_{μ},{p}_{μ}) is obtained as the subalgebra of \mathcal{H}. We study further the property that Heisenberg double defines particular quantum spaces with Hopf algebroid structure. We calculate by using purely algebraic methods the explicite Hopf algebroid structur…
The Hilbert space representations of a non-commutative q-deformed Minkowski space, its momenta and its Lorentz boosts are constructed. The spectrum of the diagonalizable space elements shows a lattice-like structure with accumulation points on the light-cone.
I shall recall in historical perspective some results from nineties and show further how $κ$-deformed symmetries and $κ$-Minkowski space inspired DSR (Doubly of Deformed Special Relativity) approach proposed after 2000. As very recent development I shall show how to describe quantum-covariant $κ$-deformed phase spaces by passing from Hopf algebras to Hopf algebroids (arXiv:1507.02612) and I will briefly describe the $κ$-deformations of $AdS_5 \times S^5$ superstring target spaces (arXiv:1510.030.83).
We construct a non-commutative kappa-Minkowski deformation of U(1) gauge theory, following a general approach, recently proposed in JHEP 2008 (2020) 041. We obtain an exact (all orders in the non-commutativity parameter) expression for both the deformed gauge transformations and the deformed field strength, which is covariant under these transformations. The corresponding Yang-Mills Lagrangian is gauge covariant and reproduces the Maxwell Lagrangian in the commutative limit. Gauge invariance of the action functional requires a non-trivial integration measure which, in the commutative limit, do…
I introduce in kappa-Minkowski noncommutative spacetime the basic tools of quantum differential geometry, namely bicovariant differential calculus, Lie and inner derivatives, the integral, the Hodge-star and the metric. I show the relevance of these tools for field theory with an application to complex scalar field, for which I am able to identify a vector-valued four-form which generalizes the energy-momentum tensor. Its closedness is proved, expressing in a covariant form the conservation of energy-momentum.
Boosted and rotated tetrad backgrounds for the Minkowski space are studied in $f(T)$ gravity. We perform Lorentzian perturbations at first order around non-trivial backgrounds and show that some Lorentz modes can exhibit non-trivial dynamics and can propagate in time. This remarkable feature gives evidence of additional mode(s) in the Lorentzian sector which have no precedent in Lorentz-violating modified gravities, however they can cast doubts onto even theoretical viability of these models.
Standard model with intrinsic Lorentz and CPT violations is proposed in a Finsler geometric framework. We present explicitly Lorentz and CPT--breaking Lagrangians of the matter fields and the gauge fields in locally Minkowski spacetime. The Lorentz invariance violation is found to be originated from the spacetime background deviating from the Minkowski one. Similarly, the CPT violation is determined by the behaviors of the locally Minkowski metric under the parity and time reversal operations. To help understanding phenomenologies, we compare the Finslerian model with the standard--model exten…
A sketch of an approach towards Lorentzian spectral geometry (based on joint work with Mario Paschke) is described, together with a general way to define abstractly the quantized Dirac field on such Lorentzian spectral geometries. Moyal-Minkowski spacetime serves as an example. The scattering of the quantized Dirac field by a non-commutative (Moyal-deformed) action of an external scalar potential is investigated. It is shown that differentiating the S-matrix with respect to the strength of the scattering potential gives rise to quantum field operators depending on elements of the non-commutati…
The lattice of integral points of 4-dimensional Minkowski space, together with the inherited indefinite distance function, is considered as a model for discrete space-time. The Lorentz and Poincare groups of this discrete space-time are identified as subgroups of the corresponding Lie groups. The lattice Lorentz group has irreducible projective (including linear) representations which are restrictions of (all) finite-dimensional irreducible projective representations of the Lorentz Lie group and hence can be used to describe all integral and half-odd-integral helicity. The (4-torus) momentum s…
