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Papers by Walter Maderner

Research paper thumbnail of Cyclic Cohomology for Massive 1+1d-Fermions and Virasoro Algebras

As an important example of characters on 2-summable Fredholm-modules we deal with Schwinger terms... more As an important example of characters on 2-summable Fredholm-modules we deal with Schwinger terms arising in the commutators between charge and axial-charge operators. We discuss 1+1-dimensional massive fermions using the language of quasifree second-quantization and find that for a large class of models the usual δ-Schwinger Term is obtained. Our results confirm calculations within the massive Schwinger-model at finite temperature but disagree with a conjecture concerning anomaly-meltdown. We also investigate the dependence of the Virasoro cocycle in the quasifree second quantization framework on traceclass perturbations (like charging and thermalizing).

Research paper thumbnail of Adiabatic connection and Schwinger term for two-dimensional Majorana-Weyl fermions in static background geometries

Reports on Mathematical Physics, 1994

We investigate the relation between Schwinger term and adiabatic (=Berry) curvature in the case o... more We investigate the relation between Schwinger term and adiabatic (=Berry) curvature in the case of the conformal field theory of Majorana-Weyl fermions in 1+ 1 dimensions. The Schwinger term-known as diffeomorphism anomaly-appears in the commutation relations of the light cone components of the energy-momentum tensor. It will be interpreted as the pullback of the curvature of the Pfaffian (=vacuum) line bundle over the space of complex structures. What is new in our analysis is the appearance of nontrivial adiabatic holonomy in one-particle picture which comes from a left invariant connection on the twofold covering of the diffeomorphism group of the circle. This gives rise to an additional term in the anomalous commutator of the parallel transport ("Gauss law") operators which act on the space of smooth sections in a Fock bundle. As a consequence, many-particle states pick up additional phase besides the phase of the vacuum.

Research paper thumbnail of On the classical origin of the fermionic Schwinger term

Letters in Mathematical Physics, 1994

Research paper thumbnail of On the conformal field theory of the two-dimensional Ising model

Journal of Physics A: Mathematical and General, 1992

An explicit construction of a free and massless Majorana quantum field theory, which exists on a ... more An explicit construction of a free and massless Majorana quantum field theory, which exists on a conformal superworld M is presented. Emphasis is placed on the investigation of the action of an infinite-dimensional group G of spacetime symmetries on M. Starting with a one-particle theory, this action induces a strongly continuous representation of Diffapproximately +(S1) on the single-particle Hilbert space. After second quantization the group of implementers turns out to be a non-trivial central extension of Diffapproximately +(S1) by U(1), and a Schwinger term occurs, which gives rise to the anomalous transformation law of the energy-momentum tensor of the theory. This transformation law is studied, and an interesting connection to geometry is established.

Research paper thumbnail of On spin chains, charges and anomalies

Journal of Physics A: Mathematical and General, 1994

We discuss temperature-effects on anomalies. We deal with an anomaly in a spin-chain model presen... more We discuss temperature-effects on anomalies. We deal with an anomaly in a spin-chain model presented in [1], where in [2] anomaly-meltdown was found. We present a slightly different treatment, that to us seems more plausible from a physical point of view, which shows anomaly persistence. We also discuss a similar problem for U(1)-Schwinger-terms in local current-algebras.

Research paper thumbnail of Can Anomalies Melt?

In our attempts to relate models of quantum field theory and statistical mechanics to aspects of ... more In our attempts to relate models of quantum field theory and statistical mechanics to aspects of non-commutative differential geometry, we study quasifree second quantization of two dimensional external field problems. The implementation of gauge and axial gauge transformations leads to generators fulfilling an algebra with Schwinger term. We define the notion of Fredholm modules within cyclic cohomology and show that in our setting the Schwinger term represents a cyclic cocycle and a 2-summable module is realized. The physically interesting question of the temperature dependence of cocycles is analyzed. We discuss the situation for massive fermions at finite temperature and for a lattice-spin model subject to a magnetic field: our procedure leads to temperature independence of the commutator anomalies, whereas there exists another approach which kills them; we compare the two methods and discuss the appearance of a ’noncommutative ’ cocycle. Contribution the the International Sympo...

Research paper thumbnail of Schwinger terms and cyclic cohomology for massive 1+1-dimensional fermions and Virasoro algebras

An important example of characters on two‐summable Fredholm modules are Schwinger terms arising i... more An important example of characters on two‐summable Fredholm modules are Schwinger terms arising in the commutators between charge and axial‐charge operators. 1+ 1‐dimensional massive fermions are discussed using the language of quasifree second ...

Research paper thumbnail of Cyclic Cohomology for Massive 1+1d-Fermions and Virasoro Algebras

As an important example of characters on 2-summable Fredholm-modules we deal with Schwinger terms... more As an important example of characters on 2-summable Fredholm-modules we deal with Schwinger terms arising in the commutators between charge and axial-charge operators. We discuss 1+1-dimensional massive fermions using the language of quasifree second-quantization and find that for a large class of models the usual δ-Schwinger Term is obtained. Our results confirm calculations within the massive Schwinger-model at finite temperature but disagree with a conjecture concerning anomaly-meltdown. We also investigate the dependence of the Virasoro cocycle in the quasifree second quantization framework on traceclass perturbations (like charging and thermalizing).

Research paper thumbnail of Adiabatic connection and Schwinger term for two-dimensional Majorana-Weyl fermions in static background geometries

Reports on Mathematical Physics, 1994

We investigate the relation between Schwinger term and adiabatic (=Berry) curvature in the case o... more We investigate the relation between Schwinger term and adiabatic (=Berry) curvature in the case of the conformal field theory of Majorana-Weyl fermions in 1+ 1 dimensions. The Schwinger term-known as diffeomorphism anomaly-appears in the commutation relations of the light cone components of the energy-momentum tensor. It will be interpreted as the pullback of the curvature of the Pfaffian (=vacuum) line bundle over the space of complex structures. What is new in our analysis is the appearance of nontrivial adiabatic holonomy in one-particle picture which comes from a left invariant connection on the twofold covering of the diffeomorphism group of the circle. This gives rise to an additional term in the anomalous commutator of the parallel transport ("Gauss law") operators which act on the space of smooth sections in a Fock bundle. As a consequence, many-particle states pick up additional phase besides the phase of the vacuum.

Research paper thumbnail of On the classical origin of the fermionic Schwinger term

Letters in Mathematical Physics, 1994

Research paper thumbnail of On the conformal field theory of the two-dimensional Ising model

Journal of Physics A: Mathematical and General, 1992

An explicit construction of a free and massless Majorana quantum field theory, which exists on a ... more An explicit construction of a free and massless Majorana quantum field theory, which exists on a conformal superworld M is presented. Emphasis is placed on the investigation of the action of an infinite-dimensional group G of spacetime symmetries on M. Starting with a one-particle theory, this action induces a strongly continuous representation of Diffapproximately +(S1) on the single-particle Hilbert space. After second quantization the group of implementers turns out to be a non-trivial central extension of Diffapproximately +(S1) by U(1), and a Schwinger term occurs, which gives rise to the anomalous transformation law of the energy-momentum tensor of the theory. This transformation law is studied, and an interesting connection to geometry is established.

Research paper thumbnail of On spin chains, charges and anomalies

Journal of Physics A: Mathematical and General, 1994

We discuss temperature-effects on anomalies. We deal with an anomaly in a spin-chain model presen... more We discuss temperature-effects on anomalies. We deal with an anomaly in a spin-chain model presented in [1], where in [2] anomaly-meltdown was found. We present a slightly different treatment, that to us seems more plausible from a physical point of view, which shows anomaly persistence. We also discuss a similar problem for U(1)-Schwinger-terms in local current-algebras.

Research paper thumbnail of Can Anomalies Melt?

In our attempts to relate models of quantum field theory and statistical mechanics to aspects of ... more In our attempts to relate models of quantum field theory and statistical mechanics to aspects of non-commutative differential geometry, we study quasifree second quantization of two dimensional external field problems. The implementation of gauge and axial gauge transformations leads to generators fulfilling an algebra with Schwinger term. We define the notion of Fredholm modules within cyclic cohomology and show that in our setting the Schwinger term represents a cyclic cocycle and a 2-summable module is realized. The physically interesting question of the temperature dependence of cocycles is analyzed. We discuss the situation for massive fermions at finite temperature and for a lattice-spin model subject to a magnetic field: our procedure leads to temperature independence of the commutator anomalies, whereas there exists another approach which kills them; we compare the two methods and discuss the appearance of a ’noncommutative ’ cocycle. Contribution the the International Sympo...

Research paper thumbnail of Schwinger terms and cyclic cohomology for massive 1+1-dimensional fermions and Virasoro algebras

An important example of characters on two‐summable Fredholm modules are Schwinger terms arising i... more An important example of characters on two‐summable Fredholm modules are Schwinger terms arising in the commutators between charge and axial‐charge operators. 1+ 1‐dimensional massive fermions are discussed using the language of quasifree second ...