arXiv subjects

G. Rudolph

Publications and source records attributed to G. Rudolph.

At least 19 recordsLinked to original sources

arXiv2021

On a connection used in deformation quantization

Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.

math-ph↗

arXiv2020

Quasicharacters, recoupling calculus and costratifications of lattice quantum gauge theory

We study the algebra of invariant representative functions over the N-fold Cartesian product of copies of a compact Lie group G modulo the action of conjugation by the diagonal subgroup. We construct a basis of invariant representative functions referred to as quasicharacters. The form of the quasicharacters depends on the choice of a reduction scheme. We determine the multiplication law of quasicharacters and express their structure constants in terms of recoupling coefficients. Via this link, the choice of the reduction scheme acquires an interpretation in terms of binary trees. We show explicitly that the structure constants decompose into products over primitive elements of 9j symbol type. For SU(2), everything boils down to the combinatorics of angular momentum theory. In the final part, we show that the above calculus enables us to calculate the matrix elements of bi-invariant operators occuring in quantum lattice gauge theory. In particular, both the quantum Hamiltonian and the orbit type relations may be dealt with in this way, thus, reducing both the construction of the costratification and the study of the spectral problem to numerical problems in linear algebra. We spell out the spectral problem for G=SU(2) and we present sample calculations of matrix elements of orbit type relations for the gauge groups SU(2) and SU(3). The methods developed may be useful in the study of virtually all quantum models with polynomial constraints related to some symmetry.

math-ph↗

arXiv2008

A gauge model for quantum mechanics on a stratified space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is $\mathrm{SU}(2)$, we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.

hep-th↗

arXiv2008

Parton fragmentation in the vacuum and in the medium

We present the mini-proceedings of the workshop on ``Parton fragmentation in the vacuum and in the medium'' held at the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*, Trento) in February 2008. The workshop gathered both theorists and experimentalists to discuss the current status of investigations of quark and gluon fragmentation into hadrons at different accelerator facilities (LEP, B-factories, JLab, HERA, RHIC, and Tevatron) as well as preparations for extension of these studies at the LHC. The main physics topics covered were: (i) light-quark and gluon fragmentation in the vacuum including theoretical (global fits analyses and MLLA) and experimental (data from e+e-, p-p, e-p collisions) aspects, (ii) strange and heavy-quark fragmentation, (iii) parton fragmentation in cold QCD matter (nuclear DIS), and (iv) medium-modified fragmentation in hot and dense QCD matter (high-energy nucleus-nucleus collisions). These mini-proceedings consist of an introduction and short summaries of the talks presented at the meeting.

hep-ph↗

arXiv2008

On the Topology of the Reduced Classical Configuration Space of Lattice QCD

We study the topological structure of the quotient of $SU(3)\times SU(3)$ by diagonal conjugation. This is the simplest nontrivial example for the classical reduced configuration space of chromodynamics on a spatial lattice in the Hamiltonian approach. We construct a cell complex structure of the quotient in such a way that the closures of strata are subcomplexes and we compute the homology and cohomology groups of the strata and their closures.

hep-th↗

arXiv2006

A Lattice Gauge Model of Singular Marsden-Weinstein Reduction. Part I. Kinematics

The simplest nontrivial toy model of a classical SU(3) lattice gauge theory is studied in the Hamiltonian approach. By means of singular symplectic reduction, the reduced phase space is constructed. Two equivalent descriptions of this space in terms of a symplectic covering as well as in terms of invariants are derived.

hep-th↗

arXiv2004

On the Structure of the Observable Algebra of QCD on the Lattice

The structure of the observable algebra ${\mathfrak O}_Λ$ of lattice QCD in the Hamiltonian approach is investigated. As was shown earlier, ${\mathfrak O}_Λ$ is isomorphic to the tensor product of a gluonic $C^{*}$-subalgebra, built from gauge fields and a hadronic subalgebra constructed from gauge invariant combinations of quark fields. The gluonic component is isomorphic to a standard CCR algebra over the group manifold SU(3). The structure of the hadronic part, as presented in terms of a number of generators and relations, is studied in detail. It is shown that its irreducible representations are classified by triality. Using this, it is proved that the hadronic algebra is isomorphic to the commutant of the triality operator in the enveloping algebra of the Lie super algebra ${\rm sl(1/n)}$ (factorized by a certain ideal).

hep-th↗

arXiv2004

On the Stratified Classical Configuration Space of Lattice QCD

The stratified structure of the configuration space $\mb G^N = G \times ... \times G$ reduced with respect to the action of $G$ by inner automorphisms is investigated for $G = SU(3) .$ This is a finite dimensional model coming from lattice QCD. First, the stratification is characterized algebraically, for arbitrary $N$. Next, the full algebra of invariants is discussed for the cases $N = 1$ and $N =2 .$ Finally, for $N = 1$ and $N =2 ,$ the stratified structure is investigated in some detail, both in terms of invariants and relations and in more geometric terms. Moreover, the strata are characterized explicitly using local cross sections.

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arXiv2004

Charge Superselection Sectors for QCD on the Lattice

We study quantum chromodynamics (QCD) on a finite lattice $Λ$ in the Hamiltonian approach. First, we present the field algebra ${\mathfrak A}_Λ$ as comprising a gluonic part, with basic building block being the crossed product $C^*$-algebra $C(G) \otimes_α G$, and a fermionic (CAR-algebra) part generated by the quark fields. By classical arguments, ${\mathfrak A}_Λ$ has a unique (up to unitary equivalence) irreducible representation. Next, the algebra ${\mathfrak O}^i_Λ$ of internal observables is defined as the algebra of gauge invariant fields, satisfying the Gauss law. In order to take into account correlations of field degrees of freedom inside $Λ$ with the ``rest of the world'', we have to extend ${\mathfrak O}^i_Λ$ by tensorizing with the algebra of gauge invariant operators at infinity. This way we construct the full observable algebra ${\mathfrak O}_Λ .$ It is proved that its irreducible representations are labelled by ${\mathbb Z}_3$-valued boundary flux distributions. Then, it is shown that there exist unitary operators (charge carrying fields), which intertwine between irreducible sectors leading to a classification of irreducible representations in terms of the ${\mathbb Z}_3$-valued global boundary flux. By the global Gauss law, these 3 inequivalent charge superselection sectors can be labeled in terms of the global colour charge (triality) carried by quark fields. Finally, ${\mathfrak O}_Λ$ is discussed in terms of generators and relations.

hep-th↗

arXiv2003

Polynomial super-gl(n) algebras

We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in the $gl(n)$ generators. Specifically, we investigate `type I' super-$gl(n)$ algebras, having odd generators transforming in a single irreducible representation of $gl(n)$ together with its contragredient. Admissible structure constants are discussed in terms of available $gl(n)$ couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the $n$-dimensional defining representation, with odd generators $Q_{a}, \bar{Q}{}^{b}$, and even generators ${E^{a}}_{b}$, $a,b = 1,...,n$, a three parameter family of quadratic super-$gl(n)$ algebras (deformations of $sl(n/1)$) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulfilled. For these quadratic super-$gl(n)$ algebras, the construction of Kac modules, and conditions for atypicality, are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and space-time supersymmetry, are discussed.

hep-th↗

arXiv2002

Charge Superselection Sectors for Scalar QED on the Lattice

The lattice model of scalar quantum electrodynamics (Maxwell field coupled to a complex scalar field) in the Hamiltonian framework is discussed. It is shown that the algebra of observables ${\cal O}(Λ)$ of this model is a $C^*$-algebra, generated by a set of gauge-invariant elements satisfying the Gauss law and some additional relations. Next, the faithful, irreducible and non-degenerate representations of ${\cal O}(Λ)$ are found. They are labeled by the value of the total electric charge, leading to a decomposition of the physical Hilbert space into charge superselection sectors. In the Appendices we give a unified description of spinorial and scalar quantum electrodynamics and, as a byproduct, we present an interesting example of weakly commuting operators, which do not commute strongly.

hep-th↗

arXiv2002

On the gauge orbit space stratification (a review)

First, we review the basic mathematical structures and results concerning the gauge orbit space stratification. This includes general properties of the gauge group action, fibre bundle structures induced by this action, basic properties of the stratification and the natural Riemannian structures of the strata. In the second part, we study the stratification for theories with gauge group $\rmSU(n)$ in space time dimension 4. We develop a general method for determining the orbit types and their partial ordering, based on the 1-1 correspondence between orbit types and holonomy-induced Howe subbundles of the underlying principal $\rmSU(n)$-bundle. We show that the orbit types are classified by certain cohomology elements of space time satisfying two relations and that the partial ordering is characterized by a system of algebraic equations. Moreover, operations for generating direct successors and direct predecessors are formulated, which allow one to construct the set of orbit types, starting from the principal one. Finally, we discuss an application to nodal configurations in topological Chern-Simons theory.

hep-th↗

arXiv2002

Spectral triples and differential calculi related to the Kronecker foliation

Following ideas of Connes and Moscovici, we describe two spectral triples related to the Kronecker foliation, whose generalized Dirac operators are related to first and second order signature operators. We also consider the corresponding differential calculi $Ω_D$, which are drastically different in the two cases. As a side-remark, we give a description of a known calculus on the two-dimensional noncommutative torus in terms of generators and relations.

math-ph↗

arXiv2001

On a Certain Stratification of the Gauge Orbit Space

For a principal $\rmSU(n)$-bundle over a compact manifold of dimension $2,3,4$, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of $\rmSU(n)$ for which a certain characteristic equation is solvable. Depending on the base manifold, this equation leads to a linear, bilinear, or quadratic Diophantine equation.

math-ph↗

arXiv2001

On the Gauss Law and Global Charge for QCD

The local Gauss law of quantum chromodynamics on a finite lattice is investigated. It is shown that it implies a gauge invariant, additive law giving rise to a gauge invariant ${\mathbb Z}_3$-valued global charge in QCD. The total charge contained in a region of the lattice is equal to the flux through its boundary of a certain ${\mathbb Z}_3$-valued, additive quantity. Implications for continuous QCD are discussed.

hep-th↗

arXiv1999

Towards an Effective Field Theory of QED

A procedure for reducing the functional integral of QED to an integral over bosonic gauge invariant fields is presented. Next, a certain averaging method for this integral, giving a tractable effective quantum field theory, is proposed. Finally, the current-current propagator and the chiral anomaly are calculated within this new formulation. These results are part of our programme of analyzing gauge theories with fermions in terms of local gauge invariants.

hep-th↗

arXiv1997

Gauge Invariant Formulation and Bosonisation of the Schwinger Model

The functional integral of the massless Schwinger model in $(1+1)$ dimensions is reduced to an integral in terms of local gauge invariant quantities. It turns out that this approach leads to a natural bosonisation scheme, yielding, in particular the famous `bosonisation rule'' and giving some deeper insight into the nature of the bosonisation phenomenon. As an application, the chiral anomaly is calculated within this formulation.

hep-th↗

arXiv1996

Effective Bosonic Degrees of Freedom for One-Flavour Chromodynamics

We apply an earlier formulated programme for quantization of nonabelian gauge theories to one-flavour chromodynamics. This programme consists in a complete reformulation of the functional integral in terms of gauge invariant quantities. For the model under consideration two types of gauge invariants occur -- quantities, which are bilinear in quarks and antiquarks (mesons) and a matrix-valued covector field, which is bilinear in quarks, antiquarks and their covariant derivatives. This covector field is linear in the original gauge potential, and can be, therefore, considered as the gauge potential ``dressed'' in a gauge invariant way with matter. Thus, we get a complete bosonization of the theory. The strong interaction is described by a highly non-linear effective action obtained after integrating out quarks and gluons from the functional integral. All constructions are done consequently on the quantum level, where quarks and antiquarks are anticommuting objects. Our quantization procedure circumvents the Gribov ambiguity.

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