Gauge Theories under Incorporation of a Generalized Uncertainty Principle

There is considered an extension of gauge theories according to the assumption of a generalized uncertainty principle which implies a minimal length scale. A modification of the usual uncertainty principle implies an extended shape of matter field equations like the Dirac equation. If there is postulated invariance of such a generalized field equation under local gauge transformations, the usual covariant derivative containing the gauge potential has to be replaced by a generalized covariant derivative. This leads to a generalized interaction between the matter field and the gauge field as well as to an additional self interaction of the gauge field. Since the existence of a minimal length scale seems to be a necessary assumption of any consistent quantum theory of gravity, the gauge principle is a constitutive ingredient of the standard model and even gravity can be described as gauge theory of local translations or Lorentz transformations, the presented extension of gauge theories appears as a very important consideration.Comment: 15 page

Nonlocal Quantization Principle in Quantum Field Theory and Quantum Gravity

In this paper a nonlocal generalization of field quantization is suggested. This quantization principle presupposes the assumption that the commutator between a field operator an the operator of the canonical conjugated variable referring to other space-time points does not vanish as it is postulated in the usual setting of quantum field theory. Based on this presupposition the corresponding expressions for the field operators, the eigenstates and the path integral formula are determined. The nonlocal quantization principle also leads to a generalized propagator. If the dependence of the commutator between operators on different space-time points on the distance of these points is assumed to be described by a Gaussian function, one obtains that the propagator is damped by an exponential. This leads to a disappearance of UV divergences. The transfer of the nonlocal quantization principle to canonical quantum gravity is considered as well. In this case the commutator has to be assumed to depend also on the gravitational field, since the distance between two points depends on the metric field.Comment: 10 page

Intersection of Yang-Mills Theory with Gauge Description of General Relativity

An intersection of Yang-Mills theory with the gauge description of general relativity is considered. This intersection has its origin in a generalized algebra, where the generators of the SO(3,1) group as gauge group of general relativity and the generators of a SU(N) group as gauge group of Yang-Mills theory are not separated anymore but are related by fulfilling nontrivial commutation relations with each other. Because of the Coleman Mandula theorem this algebra cannot be postulated as Lie algebra. As consequence, extended gauge transformations as well as an extended expression for the field strength tensor is obtained, which contains a term consisting of products of the Yang Mills connection and the connection of general relativity. Accordingly a new gauge invariant action incorporating the additional term of the generalized field strength tensor is built, which depends of course on the corresponding tensor determining the additional intersection commutation relations. This means that the theory describes a decisively modified interaction structure between the Yang-Mills gauge field and the gravitational field leading to a violation of the equivalence principle.Comment: 12 page

Conformal Gravity on Noncommutative Spacetime

Conformal gravity on noncommutative spacetime is considered in this paper. The presupposed gravity action consists of the Brans-Dicke gravity action with a special prefactor of the term, where the Ricci scalar couples to the scalar field, to maintain local conformal invariance and the Weyl gravity action. The commutation relations between the coordinates defining the noncommutative geometry are assumed to be of canonical shape. Based on the moyal star product, products of fields depending on the noncommutative coordinates are replaced by generalized expressions containing the usual fields and depending on the noncommutativity parameter. To maintain invariance under local conformal transformations with the gauge parameter depending on noncommutative coordinates, the fields have to be mapped to generalized fields by using Seiberg-Witten maps. According to the moyal star product and the thus induced Seiberg-Witten maps the generalized conformal gravity action is formulated and the corresponding field equations are derived.Comment: 10 page

About the Origin of the Division between Internal and External Symmetries in Quantum Field Theory

It is made the attempt to explain why there exists a division between internal symmetries referring to quantum numbers and external symmetries referring to space-time within the description of relativistic quantum field theories. It is hold the attitude that the symmetries of quantum theory are the origin of both sorts of symmetries in nature. Since all quantum states can be represented as a tensor product of two dimensional quantum objects, called ur objects, which can be interpreted as quantum bits of information, described by spinors reflecting already the symmetry properties of space-time, it seems to be possible to justify such an attitude. According to this, space-time symmetries can be considered as a consequence of a representation of quantum states by quantum bits. Internal symmetries are assumed to refer to relations of such fundamental objects, which are contained within the state of one single particle, with respect to each other. In this sense the existence of space-time symmetries, the existence of internal symmetries and their division could obtain a derivation from quantum theory interpreted as a theory of information.Comment: 5 page

Quaternionic Quantization Principle in General Relativity and Supergravity

A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space-time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to $\mathcal{N}=1$ supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.Comment: 26 page

Canonical Quantum Gravity on Noncommutative Spacetime

In this paper canonical quantum gravity on noncommutative space-time is considered. The corresponding generalized classical theory is formulated by using the moyal star product, which enables the representation of the field quantities depending on noncommuting coordinates by generalized quantities depending on usual coordinates. But not only the classical theory has to be generalized in analogy to other field theories. Besides, the necessity arises to replace the commutator between the gravitational field operator and its canonical conjugated quantity by a corresponding generalized expression on noncommutative space-time. Accordingly the transition to the quantum theory has also to be performed in a generalized way and leads to extended representations of the quantum theoretical operators. If the generalized representations of the operators are inserted to the generalized constraints, one obtains the corresponding generalized quantum constraints including the Hamiltonian constraint as dynamical constraint. After considering quantum geometrodynamics under incorporation of a coupling to matter fields, the theory is transferred to the Ashtekar formalism. The holonomy representation of the gravitational field as it is used in loop quantum gravity opens the possibility to calculate the corresponding generalized area operator.Comment: 17 page

Electroweak Theory with a Minimal Length

According to the introduction of a minimal length to quantum field theory which is directly related to a generalized uncertainty principle the implementation of the gauge principle becomes much more intricated. It has been shown in another paper how gauge theories have to be extended in general, if there is assumed the existence of a minimal length. In this paper this generalization of the description of gauge theories is applied to the case of Yang-Mills theories with gauge group SU(N) to consider especially the application to the electroweak theory as it appears in the standard model. The modifications of the lepton-, Higgs- and gauge field sector of the extended Lagrangian of the electroweak theory maintaining local gauge invariance under $SU(2)_L\otimes U(1)_Y$ transformations are investigated. There appear additional interaction terms between the leptons or the Higgs particle respectively with the photon and the W- and Z-bosons as well as additional self-interaction terms of these gauge bosons themselves. It is remarkable that in the quark sector where the full gauge group of the standard model, $SU(3)_c \otimes SU(2)_L\otimes U(1)_Y$, has to be considered there arise coupling terms between the gluons und the W- and Z-bosons which means that the electroweak theory is not separated from quantum chromodynamics anymore.Comment: 20 page

Lowest Landau Level of Relativistic Field Theories in a Strong Background Field

We consider gauge theories in a strong external magnetic like field. This situation can appear either in conventional four-dimensional theories, but also naturally in extra-dimensional theories and especially in brane world models. We show that in the lowest Landau level approximation, some of the coordinates become non-commutative. We find physical reasons to formal problems with non-commutative gauge theories such as the issue with SU(N) gauge symmetries. Our construction is applied to a minimal extension of the standard model. It is shown that the Higgs sector might be non-commutative whereas the remaining sectors of the standard model remain commutative. Signatures of this model at the LHC are discussed. We then discuss an application to a dark matter sector coupled to the Higgs sector of the standard model and show that here again, dark matter could be non-commutative, the standard model fields remaining commutative.Comment: Submitted for the SUSY07 proceedings,4 page

Betrachtungen jenseits des Standardmodells der Teilchenphysik

Gravity with incorporation of additional dimensions and noncommutative geometry.Comment: Diploma Thesis (German), 107 pages, 12 figure