Pierre Grange - Academia.edu (original) (raw)
Papers by Pierre Grange
arXiv (Cornell University), Dec 10, 2018
International audienceThe photon propagator is built in the "usual" LC-gauges within th... more International audienceThe photon propagator is built in the "usual" LC-gauges within the distributional approach with test functions for the gauge fields. It is shown that test functions provide a sound definition, as distributions, of singular terms in (1/n.k)p where n2=0. The relevance of these extensions is exemplified in a test case where an infinite re-summation in powers of 1/n.k leads to known exact results. The effect of residual gauge degrees of freedom on the presence of singular terms in the most general LC-propagator is discussed in the Dirac-Bergmann algorithm
The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on ... more The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on the importance of causality following the work of Epstein and Glaser. Gauge invariant theories demand the extension of the usual translation operation on OPVD, thereby leading to a generalized QED formulation. At D=2 the solvability of the Schwinger model is totally preserved. At D=4 the paracompactness property of the Euclidean manifold permits using test functions which are decomposition of unity and thereby provides a natural justification and extension of the non perturbative heat kernel method (Fujikawa) for abelian anomalies. On the Minkowski manifold the specific role of causality in the restauration of gauge invariance (and mass generation for QED 2) is examplified in a simple way.
We discuss the implementation of Light Cone Field Theory (LCFT) in the causal Bogoliubov-Epstein-... more We discuss the implementation of Light Cone Field Theory (LCFT) in the causal Bogoliubov-Epstein-Glaser finite S-matrix formulation. The benefits result from the simple vacuum structure of LCFT which make possible calculations of the S-matrix in a perturbative-like manner even in the presence of a nontrivial, nonperturbative vacuum.The interdependence of particle and vacuum properties leads to an iterative scheme for the determination of the vacuum part of the Hamilto-nian.
Proceedings of Light Cone 2010: Relativistic Hadronic and Particle Physics — PoS(LC2010), 2010
It is shown that no IR/UV divergences occur in the calculations of the electron and gauge-boson s... more It is shown that no IR/UV divergences occur in the calculations of the electron and gauge-boson self-energies when performed in the Taylor-Lagrange renormalisation scheme. The Lorentz structure of the gauge boson self-energy to one loop is shown to emerge naturally in this scheme, directly at the physical dimension D = 4. Possible consequences on the fate of quadratic divergences in the Standard Model are pointed out.
Proceedings of LIGHT CONE 2008 Relativistic Nuclear and Particle Physics — PoS(LC2008), 2009
Proceedings of LIGHT CONE 2008 Relativistic Nuclear and Particle Physics — PoS(LC2008), 2009
Proceedings of Light Cone 2010: Relativistic Hadronic and Particle Physics — PoS(LC2010), 2010
Modern Physics Letters A, 2018
Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing a... more Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables [Formula: see text], [Formula: see text] pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the ...
Few-Body Systems, 2016
Quantum field theory formulated in terms of light front (LF) variables has a few attractive as we... more Quantum field theory formulated in terms of light front (LF) variables has a few attractive as well as some puzzling features. The latter hindered a wider acceptance of LF methods. In two space-time dimensions, it has been a long-standing puzzle how to correctly quantize massless fields, in particular fermions. Here we show that two-dimensional massless LF fields (scalar and fermion) can be recovered in a simple way as limits of the corresponding massive fields and thereby quantized without any loss of physical information. Bosonization of the fermion field then follows in a straightforward manner and the solvable models can be studied directly in the LF theory. We sketch the LF operator solution of the Thirring-Wess model and also point out the closeness of the massless LF fields to those of conformal field theory.
Nuclear Physics B - Proceedings Supplements, 2002
Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of d... more Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of dynamical zero modes is introduced and shown to lead to a non trivial covariant dispersion relation only for Continuous LC Quantization (CLCQ). The critical exponent η is found to be governed by the behaviour of the infinite volume limit under conformal transformations properties preserving the local LC structure. The β-function is calculated exactly and found non-analytic, with a critical exponent ω = 2, in agreement with the conformal field theory prediction of Calabrese et al.
Eprint Arxiv Math Ph 0310052, Oct 24, 2003
In Continuum Light Cone Quantization (CLCQ) the treatment of scalar fields as operator valued dis... more In Continuum Light Cone Quantization (CLCQ) the treatment of scalar fields as operator valued distributions and properties of the accompanying test functions are recalled. Due to the paracompactness property of the Euclidean manifold these test functions appear as decomposition of unity. The approach is extended to QED Dirac fields in a gauge invariant way. With such test functions the usual triangle anomalies are calculated in a simple and transparent way.
A genuine continuum treatment of the massive # 4 1+1 -theory in light-cone quantization is propos... more A genuine continuum treatment of the massive # 4 1+1 -theory in light-cone quantization is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well defined handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1.5. Conventional perturbation theory at the corresponding order yields r1 = 1, whereas the RG improved fourth order result is r4 = 1.8 ± 0.05. PM 97/18, June 1997 PACS : 11.10.Ef, 11.10.St, 11.30.Rd 1 Introduction The discretized light front quantization (DLCQ) [1] has played an important role in clarifying infrared aspects of the theory which are decisive for the appearance of the vacuum sector field, the LC-counterpart of the nontrivial ground state of ET-qu...
Physics Letters B, 1992
In stochastic quantization of O (N) scalar field theories, the variational scheme of Amundsen and... more In stochastic quantization of O (N) scalar field theories, the variational scheme of Amundsen and Damgaard is analysed in the framework of a 1/N expansion. In the large-N limit the usual saddle point result is retrieved. Successive corrections in 1/N to the trial field are obtained analytically in a simple iterative process, with improved minimization at each step. A solvable toy model is treated explicitly showing very rapid convergence even for N= 3. For the O (N) ~ 4 theory we argue that a minimization order by order in 1/N is the only feasible one in practice.
Few-Body Systems, 2015
Two-dimensional models with massless fermions (Thirring model, Thirring–Wess and Schwinger model,... more Two-dimensional models with massless fermions (Thirring model, Thirring–Wess and Schwinger model, among others) have been solved exactly a long time ago in the conventional (space-like) form of field theory and in some cases also in the conformal field theoretical approach. However, solutions in the light-front form of the theory have not been obtained so far. The primary obstacle is the apparent difficulty with light-front quantization of free massless fermions, where one half of the fermionic degrees of freedom seems to “disappear” due to the structure of a non-dynamical constraint equation. We shall show a simple way how the missing degree of freedom can be recovered as the massless limit of the massive solution of the constraint. This opens the door to the genuine light front solution of the above models since their solvability is related to free Heisenberg fields, which are the true dynamical variables in these models. In the present contribution, we give an operator solution of the light front Thirring model, including the correct form of the interacting quantum currents and of the Hamiltonian. A few remarks on the light-front Thirring–Wess models are also added. Simplifications and clarity of the light-front formalism turn out to be quite remarkable.
Quantum Field Theory with fields as Operator Valued Distributions with adequate test functions, -... more Quantum Field Theory with fields as Operator Valued Distributions with adequate test functions, -the basis of Epstein-Glaser approach known now as Causal Perturbation Theory-, is recalled. Its recent revival is due to new developments in understanding its renormalization structure, which was a major and somehow fatal disease to its widespread use in the seventies. In keeping with the usual way of definition of integrals of differential forms, fields are defined through integrals over the whole manifold, which are given an atlas-independent meaning with the help of the partition of unity. Using such partition of unity test functions turns out to be the key to the fulfilment of the Poincar\'e commutator algebra as well as to provide a direct Lorentz invariant scheme to the Epstein-Glaser extension procedure of singular distributions. These test functions also simplify the analysis of QFT behaviour both in the UV and IR domains, leaving only a finite renormalization at a point rela...
Physics Letters B, 2013
We clarify a few conceptual problems of quantum field theory on the level of exactly solvable mod... more We clarify a few conceptual problems of quantum field theory on the level of exactly solvable models with fermions. The ultimate goal of our study is to gain a deeper understanding of differences between the usual ("spacelike") and light-front forms of relativistic dynamics. We show that by incorporating operator solutions of the field equations to the canonical formalism the spacelike and light front Hamiltonians of the derivative-coupling model acquire an equivalent structure. The same is true for the massive solvable theory, the Federbush model. In the conventional approach, physical predictions in the two schemes disagree. Moreover, the derivative-coupling model is found to be almost identical to a free theory, in contrast to the conventional canonical treatment. Physical vacuum state of the Thirring model is then obtained by a Bogoliubov transformation as a coherent state quadratic in composite boson operators. To perform the same task in the Federbush model, we derive a massive version of Klaiber's bosonization and show that its light-front form is much simpler.
Physical Review D, 2013
We re-analyse the perturbative radiative corrections to the Higgs mass within the Standard Model ... more We re-analyse the perturbative radiative corrections to the Higgs mass within the Standard Model in the light of the Taylor-Lagrange renormalization scheme. This scheme naturally leads to completely finite corrections, depending on an arbitrary scale. The formulation avoids very large individual corrections to the Higgs mass. This illustrates the fact that the so-called fine-tuning problem in the Standard Model is just an artefact of the regularization scheme. It should therefore not lead to any physical interpretation in terms of the energy scale at which new physics should show up, nor in terms of a new symmetry. We analyse the intrinsic physical scales relevant for the description of these radiative corrections.
arXiv (Cornell University), Dec 10, 2018
International audienceThe photon propagator is built in the "usual" LC-gauges within th... more International audienceThe photon propagator is built in the "usual" LC-gauges within the distributional approach with test functions for the gauge fields. It is shown that test functions provide a sound definition, as distributions, of singular terms in (1/n.k)p where n2=0. The relevance of these extensions is exemplified in a test case where an infinite re-summation in powers of 1/n.k leads to known exact results. The effect of residual gauge degrees of freedom on the presence of singular terms in the most general LC-propagator is discussed in the Dirac-Bergmann algorithm
The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on ... more The treatment of fields as operator valued distributions (OPVD) is recalled with the emphasis on the importance of causality following the work of Epstein and Glaser. Gauge invariant theories demand the extension of the usual translation operation on OPVD, thereby leading to a generalized QED formulation. At D=2 the solvability of the Schwinger model is totally preserved. At D=4 the paracompactness property of the Euclidean manifold permits using test functions which are decomposition of unity and thereby provides a natural justification and extension of the non perturbative heat kernel method (Fujikawa) for abelian anomalies. On the Minkowski manifold the specific role of causality in the restauration of gauge invariance (and mass generation for QED 2) is examplified in a simple way.
We discuss the implementation of Light Cone Field Theory (LCFT) in the causal Bogoliubov-Epstein-... more We discuss the implementation of Light Cone Field Theory (LCFT) in the causal Bogoliubov-Epstein-Glaser finite S-matrix formulation. The benefits result from the simple vacuum structure of LCFT which make possible calculations of the S-matrix in a perturbative-like manner even in the presence of a nontrivial, nonperturbative vacuum.The interdependence of particle and vacuum properties leads to an iterative scheme for the determination of the vacuum part of the Hamilto-nian.
Proceedings of Light Cone 2010: Relativistic Hadronic and Particle Physics — PoS(LC2010), 2010
It is shown that no IR/UV divergences occur in the calculations of the electron and gauge-boson s... more It is shown that no IR/UV divergences occur in the calculations of the electron and gauge-boson self-energies when performed in the Taylor-Lagrange renormalisation scheme. The Lorentz structure of the gauge boson self-energy to one loop is shown to emerge naturally in this scheme, directly at the physical dimension D = 4. Possible consequences on the fate of quadratic divergences in the Standard Model are pointed out.
Proceedings of LIGHT CONE 2008 Relativistic Nuclear and Particle Physics — PoS(LC2008), 2009
Proceedings of LIGHT CONE 2008 Relativistic Nuclear and Particle Physics — PoS(LC2008), 2009
Proceedings of Light Cone 2010: Relativistic Hadronic and Particle Physics — PoS(LC2010), 2010
Modern Physics Letters A, 2018
Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing a... more Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables [Formula: see text], [Formula: see text] pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the ...
Few-Body Systems, 2016
Quantum field theory formulated in terms of light front (LF) variables has a few attractive as we... more Quantum field theory formulated in terms of light front (LF) variables has a few attractive as well as some puzzling features. The latter hindered a wider acceptance of LF methods. In two space-time dimensions, it has been a long-standing puzzle how to correctly quantize massless fields, in particular fermions. Here we show that two-dimensional massless LF fields (scalar and fermion) can be recovered in a simple way as limits of the corresponding massive fields and thereby quantized without any loss of physical information. Bosonization of the fermion field then follows in a straightforward manner and the solvable models can be studied directly in the LF theory. We sketch the LF operator solution of the Thirring-Wess model and also point out the closeness of the massless LF fields to those of conformal field theory.
Nuclear Physics B - Proceedings Supplements, 2002
Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of d... more Critical behaviour of the 2D scalar field theory in the LC framework is reviewed. The notion of dynamical zero modes is introduced and shown to lead to a non trivial covariant dispersion relation only for Continuous LC Quantization (CLCQ). The critical exponent η is found to be governed by the behaviour of the infinite volume limit under conformal transformations properties preserving the local LC structure. The β-function is calculated exactly and found non-analytic, with a critical exponent ω = 2, in agreement with the conformal field theory prediction of Calabrese et al.
Eprint Arxiv Math Ph 0310052, Oct 24, 2003
In Continuum Light Cone Quantization (CLCQ) the treatment of scalar fields as operator valued dis... more In Continuum Light Cone Quantization (CLCQ) the treatment of scalar fields as operator valued distributions and properties of the accompanying test functions are recalled. Due to the paracompactness property of the Euclidean manifold these test functions appear as decomposition of unity. The approach is extended to QED Dirac fields in a gauge invariant way. With such test functions the usual triangle anomalies are calculated in a simple and transparent way.
A genuine continuum treatment of the massive # 4 1+1 -theory in light-cone quantization is propos... more A genuine continuum treatment of the massive # 4 1+1 -theory in light-cone quantization is proposed. Fields are treated as operator valued distributions thereby leading to a mathematically well defined handling of ultraviolet and light cone induced infrared divergences and of their renormalization. Although non-perturbative the continuum light cone approach is no more complex than usual perturbation theory in lowest order. Relative to discretized light cone quantization, the critical coupling increases by 30% to a value r = 1.5. Conventional perturbation theory at the corresponding order yields r1 = 1, whereas the RG improved fourth order result is r4 = 1.8 ± 0.05. PM 97/18, June 1997 PACS : 11.10.Ef, 11.10.St, 11.30.Rd 1 Introduction The discretized light front quantization (DLCQ) [1] has played an important role in clarifying infrared aspects of the theory which are decisive for the appearance of the vacuum sector field, the LC-counterpart of the nontrivial ground state of ET-qu...
Physics Letters B, 1992
In stochastic quantization of O (N) scalar field theories, the variational scheme of Amundsen and... more In stochastic quantization of O (N) scalar field theories, the variational scheme of Amundsen and Damgaard is analysed in the framework of a 1/N expansion. In the large-N limit the usual saddle point result is retrieved. Successive corrections in 1/N to the trial field are obtained analytically in a simple iterative process, with improved minimization at each step. A solvable toy model is treated explicitly showing very rapid convergence even for N= 3. For the O (N) ~ 4 theory we argue that a minimization order by order in 1/N is the only feasible one in practice.
Few-Body Systems, 2015
Two-dimensional models with massless fermions (Thirring model, Thirring–Wess and Schwinger model,... more Two-dimensional models with massless fermions (Thirring model, Thirring–Wess and Schwinger model, among others) have been solved exactly a long time ago in the conventional (space-like) form of field theory and in some cases also in the conformal field theoretical approach. However, solutions in the light-front form of the theory have not been obtained so far. The primary obstacle is the apparent difficulty with light-front quantization of free massless fermions, where one half of the fermionic degrees of freedom seems to “disappear” due to the structure of a non-dynamical constraint equation. We shall show a simple way how the missing degree of freedom can be recovered as the massless limit of the massive solution of the constraint. This opens the door to the genuine light front solution of the above models since their solvability is related to free Heisenberg fields, which are the true dynamical variables in these models. In the present contribution, we give an operator solution of the light front Thirring model, including the correct form of the interacting quantum currents and of the Hamiltonian. A few remarks on the light-front Thirring–Wess models are also added. Simplifications and clarity of the light-front formalism turn out to be quite remarkable.
Quantum Field Theory with fields as Operator Valued Distributions with adequate test functions, -... more Quantum Field Theory with fields as Operator Valued Distributions with adequate test functions, -the basis of Epstein-Glaser approach known now as Causal Perturbation Theory-, is recalled. Its recent revival is due to new developments in understanding its renormalization structure, which was a major and somehow fatal disease to its widespread use in the seventies. In keeping with the usual way of definition of integrals of differential forms, fields are defined through integrals over the whole manifold, which are given an atlas-independent meaning with the help of the partition of unity. Using such partition of unity test functions turns out to be the key to the fulfilment of the Poincar\'e commutator algebra as well as to provide a direct Lorentz invariant scheme to the Epstein-Glaser extension procedure of singular distributions. These test functions also simplify the analysis of QFT behaviour both in the UV and IR domains, leaving only a finite renormalization at a point rela...
Physics Letters B, 2013
We clarify a few conceptual problems of quantum field theory on the level of exactly solvable mod... more We clarify a few conceptual problems of quantum field theory on the level of exactly solvable models with fermions. The ultimate goal of our study is to gain a deeper understanding of differences between the usual ("spacelike") and light-front forms of relativistic dynamics. We show that by incorporating operator solutions of the field equations to the canonical formalism the spacelike and light front Hamiltonians of the derivative-coupling model acquire an equivalent structure. The same is true for the massive solvable theory, the Federbush model. In the conventional approach, physical predictions in the two schemes disagree. Moreover, the derivative-coupling model is found to be almost identical to a free theory, in contrast to the conventional canonical treatment. Physical vacuum state of the Thirring model is then obtained by a Bogoliubov transformation as a coherent state quadratic in composite boson operators. To perform the same task in the Federbush model, we derive a massive version of Klaiber's bosonization and show that its light-front form is much simpler.
Physical Review D, 2013
We re-analyse the perturbative radiative corrections to the Higgs mass within the Standard Model ... more We re-analyse the perturbative radiative corrections to the Higgs mass within the Standard Model in the light of the Taylor-Lagrange renormalization scheme. This scheme naturally leads to completely finite corrections, depending on an arbitrary scale. The formulation avoids very large individual corrections to the Higgs mass. This illustrates the fact that the so-called fine-tuning problem in the Standard Model is just an artefact of the regularization scheme. It should therefore not lead to any physical interpretation in terms of the energy scale at which new physics should show up, nor in terms of a new symmetry. We analyse the intrinsic physical scales relevant for the description of these radiative corrections.