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Jean Thierry-Mieg

Publications and source records attributed to Jean Thierry-Mieg.

13 recordsLinked to original sources
arXiv2025

Conformal invariance of antisymmetric tensor field theories in any even dimension

Using a theorem of Jackiw and Pi expressing the delicate balance of the spin and the orbital momentum, we systematically classify the flat-space massless Lagrangian quantum field theories that are invariant under the global conformal group SO(D,2). We recover in a uniform way the facts that scalars and spinors are invariant in any dimension, and that gauge p-tensors are invariant only in 2 p + 2 dimensions. This case includes the Maxwell theory in 4 dimensions and the Kalb-Ramond 2-forms theory in 6 dimensions. We then construct two new classes of Lagrangians extending the Avdeev-Chizhov self-dual tensor model to higher dimensions, one class using a symmetric metric and the other a skew metric in internal space. Finally, we prove in the same uniform way that both classes are conformal invariant in any even dimension. In 4 dimensions, these self-dual tensors naturally couple to the chiral Fermions of the standard model.

hep-th↗
arXiv2024

A new SU(2/1) supergroup with determinant 1 explains many mysteries of the weak interactions

Taken as a classification paradigm completing the standard model, a new compact form of the SU(2/1) supergroup explains many mysterious properties of the weak interactions: the maximal breaking of parity, the fractional charges of the quarks, the cancellation of the quantum field theory anomalies, and ties together the existence of the right neutrinos and of the heavier Fermions. This compact supergroup is constructed by exponentiating the matrices representing the leptons and the quarks which form a semi-direct sum of Kac modules of the real superalgebra su(2/1,R) such that the overall trace of the $U(1)$ weak-hypercharge $Y$ vanishes. Remarkably, all the elements of this supergroup have Berezinian 1 and determinant 1. In practice, $Tr(Y)=0$ simply means that the electric charge of the hydrogen atom is zero.

hep-th↗
arXiv2024

Antisymmetric tensor fields: actions, symmetries and first order Duffin-Kemmer-Petiau formulations

Analyzing the representations of the Lorentz group, we give a systematic count and construction of all the possible Lagrangians describing an antisymmetric rank two tensor field. The count yields two scalars: the gauge invariant Kalb-Ramond model, equivalent to the sigma model and familiar from super gravity and string theory, and the conformally invariant Avdeev-Chizhov model, which describes self-dual tensors. The count also includes a third invariant, a pseudoscalar, which is an antisymmetrized form of the Avdeev-Chizhov Lagrangian, first noticed in the $SU(2/1)$ superalgebraic model of the weak interactions. This model is also conformally invariant, and naturally implements the Landau $CP$ symmetry. Then, by extending the Duffin-Kemmer-Petiau 10 component formalism, we recover the model Lagrangians as first order systems. To complete the analysis we classify all local Lorentz invariant potentials (mass terms and quartic couplings) for charged antisymmetric tensor fields coupled to a Yang-Mills field.

hep-th↗
arXiv2023

Construction of matryoshka nested indecomposable N-replications of Kac-modules of quasi-reductive Lie superalgebras, including the sl(m/n) and osp(2/2n) series

We construct a new class of finite dimensional indecomposable representations of simple superalgebras which may explain, in a natural way, the existence of the heavier elementary particles. In type I Lie superalgebras sl(m/n) and osp(2/2n), one of the Dynkin weights labeling the finite dimensional irreducible representations is continuous. Taking the derivative, we show how to construct indecomposable representations recursively embedding N copies of the original irreducible representation, coupled by generalized Cabibbo angles, as observed among the three generations of leptons and quarks of the standard model. The construction is then generalized in the appendix to quasi-reductive Lie superalgebras.

math.RT↗
arXiv2022

Explicit construction of the finite dimensional indecomposable representations of the simple Lie-Kac $SU(2/1)$ superalgebra and their low level non diagonal super Casimir operators

All finite dimensional irreducible representations of the simple Lie-Kac super algebra SU(2/1) are explicitly constructed in the Chevalley basis as complex matrices. For typical representations, the distinguished Dynkin label is not quantized. We then construct the generic atypical indecomposable quivers classified by Marcu, Su and Germoni and typical indecomposable N-generations block triangular extensions for any irreducible module and any integer N. In addition to the quadratic and cubic super-Casimir operators $C_2$ and $C_3$, the supercenter of the enveloping algebra contains a chiral ghost super-Casimir operator T of mixed order (2,4)in the odd generators, proportional to the superidentity grading operator $χ$, and satisfying $T = χ\;C_2$ and we define a new factorizable chiral-Casimir $T^-=C_2(1-χ)/2=(UV+WX)(VU+XW)$ where (U,V,W,X) are the odd generators. In most indecomposable cases, the super-Casimirs are non diagonal. We compute their pseudo-eigenvalues.

hep-th↗
arXiv2021

Scalar anomaly cancellation reveals the hidden superalgebraic structure of the quantum chiral SU(2/1) model of leptons and quarks

At the classical level, the SU(2/1) superalgebra offers a natural description of the elementary particles: leptons and quarks massless states, graded by their chirality, fit the smallest irreducible representations of SU(2/1). Our new proposition is to pair the left/right space-time chirality with the superalgebra chirality and to study the model at the one-loop quantum level. If, despite the fact that they are non-Hermitian, we use the odd matrices of SU(2/1) to minimally couple an oriented complex Higgs scalar field to the chiral Fermions, novel anomalies occur. They affect the scalar propagators and vertices. However, these undesired new terms cancel out, together with the Adler-Bell-Jackiw vector anomalies, because the quarks compensate the leptons. The unexpected and striking consequence is that the scalar propagator must be normalized using the antisymmetric super-Killing metric and the scalar-vector vertex must use the symmetric d_aij structure constants of the superalgebra. Despite this extraordinary structure, the resulting Lagrangian is actually Hermitian.

hep-th↗
arXiv2021

Chirality, a new key for the definition of the connection and curvature of a Lie-Kac super-algebra

A natural generalization of a Lie algebra connection, or Yang-Mills field, to the case of a Lie-Kac superalgebra, for example SU(m/n), just in terms of ordinary complex functions and differentials, is proposed. Using the chirality $χ$ which defines the supertrace of the superalgebra: $STr(...) = Tr (χ...)$, we construct a covariant differential: $D = χ(d + A) + Φ$, where A is the standard even Lie-subalgebra connection 1-form and $Φ$ a scalar field valued in the odd module. Despite the fact that $Φ$ is a scalar, $Φ$ anticommutes with $(χA)$ because $χ$ anticommutes with the odd generators hidden in $Φ$. Hence the curvature $F = DD$ is a superalgebra-valued linear map which respects the Bianchi identity and correctly defines a chiral parallel transport compatible with a generic Lie superalgebra structure.

hep-th↗
arXiv2021

SU(2/1) superchiral self-duality: a new quantum, algebraic and geometric paradigm to describe the electroweak interactions

We propose an extension of the Yang-Mills paradigm from Lie algebras to internal chiral superalgebras. We replace the Lie algebra-valued connection one-form $A$, by a superalgebra-valued polyform $\widetilde{A}$ mixing exterior-forms of all degrees and satisfying the chiral self-duality condition $\widetilde{A} = {}^*\widetilde{A} \,χ$, where $χ$ denotes the superalgebra grading operator. This superconnection contains Yang-Mills vectors valued in the even Lie subalgebra, together with scalars and self-dual tensors valued in the odd module, all coupling only to the charge parity CP-positive Fermions. The Fermion quantum loops then induce the usual Yang-Mills-scalar Lagrangian, the self-dual Avdeev-Chizhov propagator of the tensors, plus a new vector-scalar-tensor vertex and several quartic terms which match the geometric definition of the supercurvature. Applied to the $SU(2/1)$ Lie-Kac simple superalgebra, which naturally classifies all the elementary particles, the resulting quantum field theory is anomaly-free and the interactions are governed by the super-Killing metric and by the structure constants of the superalgebra.

hep-th↗
arXiv2019

Connections between physics, mathematics and deep learning

Starting from the Fermat's principle of least action, which governs classical and quantum mechanics and from the theory of exterior differential forms, which governs the geometry of curved manifolds, we show how to derive the equations governing neural networks in an intrinsic, coordinate invariant way, where the loss function plays the role of the Hamiltonian. To be covariant, these equations imply a layer metric which is instrumental in pretraining and explains the role of conjugation when using complex numbers. The differential formalism also clarifies the relation of the gradient descent optimizer with Aristotelian and Newtonian mechanics and why large learning steps break the logic of the linearization procedure. We hope that this formal presentation of the differential geometry of neural networks will encourage some physicists to dive into deep learning, and reciprocally, that the specialists of deep learning will better appreciate the close interconnection of their subject with the foundations of classical and quantum field theory.

cs.LG↗
arXiv2018

XOR_p A maximally intertwined p-classes problem used as a benchmark with built-in truth for neural networks gradient descent optimization

A natural p-classes generalization of the eXclusive OR problem, the subtraction modulo p, where p is prime, is presented and solved using a single fully connected hidden layer with p-neurons. Although the problem is very simple, the landscape is intricate and challenging and represents an interesting benchmark for gradient descent optimization algorithms. Testing 9 optimizers and 9 activation functions up to p = 191, the method converging most often and the fastest to a perfect classification is the Adam optimizer combined with the ELU activation function.

cs.LG↗
arXiv2008

Quantum construction of a unitary SU(2/1) model of the electro-weak interactions with 2 Higgs doublets

The interactions and even the number of the Higgs scalar fields are not fixed in the SU(2)U(1) standard model of the electro-weak interactions and the intrinsically chiral nature of the weak interactions is not explained. Embedding SU(2)U(1) into the Lie super-algebra SU(2/1) fills these gaps. The 2 smallest representations of SU(2/1) adequately describe the electron, neutrino, up and down quarks and correlate their chiralities with their U(1) charges, and the Higgs fields have the quantum numbers of the odd generators. But so far, there was an apparent conflict with unitarity, because the quark representation is not Hermitian and the super-Killing metric is not positive definite. We solve this paradox by assuming the existence of 2 complex Higgs doublets minimally coupled to the Fermions via the chiral projections of the odd generators of SU(2/1). We find that Lagrangian induced by the Fermion loops is unitary, thanks to the balance between the leptons and the quarks needed to cancel the triangle anomaly and that the super-Jacobi identity guarantees that the photon remains massless after symmetry breaking. In addition, the Lagrangian has a classical geometric interpretation in terms of the curvature of the corresponding Hermitian algebra. Assuming that the relative strength of the scalar and vector couplings does not depend on the number of families constrains the mass of the Higgs to $M^2_{H^0_1} + M^2_{H^0_2} = 32/9 M^2_W = 2 (107.2 GeV)^2$. Contrary to grand-unified or Wess-Zumino super-symmetric models, the SU(2/1) internal super-unification does not predict any unobserved particle besides the 2 Higgs doublets.

hep-ph↗
arXiv2006

Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra

In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space.

hep-th↗