Gennadi Sardanashvily: April 2014

List of Sardanashvily’s blog posts (in English)

Scientific Biography (#)

My conjecture: gravity is not quantized in general (#)

On a notion of the mathematical structure (#)

Is a metric gravitational field non-quantized? (#)

Geometry of the composite bundles (from my Scientific Biography) (#)

What is Nobel Prize in Physics 2013 for? (#)

Who is who among universities in 2013 by THE World University Rankings (#)

Dmitri Ivanenko and Lev Landau – two archival photos (#)

Who is who among universities in 2013 (#)

Sardanashvily Internet addresses (#)

My lectures on supergeometry (#)

Is supersymmetry illusive? (#)

Solvey Conference 1927: They created contemporary physics (#)

Who is who in modern cosmology theory (#)

How we are small in the Universe (#)

Impact Factor 2012 of Journals in mathematical physics (#)

“Albert Einstein” of Vadim Sidur (#)

My review: “Geometric formulation of non-autonomous mechanics” (#)

Quantum field theory: functional integrals as a true measure (from my Scientific Biography) (#)

Against the Impact Factor (#)

Experiments 2013 look promising … (#)

What is Gauge Gravitation Theory about? (#)

Introduction to my book “Advanced Differential Geometry for Theoreticians” (#)

30 Years of “The Gauge Treatment of Gravity”

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What is a classical Higgs field? (#)

Lectures on integrable Hamiltonian systems (#)

My articles in WikipediA (#)

Fibre bundle formulation of time-dependent mechanics (#)

Graded Lagrangian formalism (#)

New Managing Editor of IJGMMP (#)

Any theory is either incomplete or contradictory (#)

Biographic books of WikipediA: Theoretical Physicists, … (#)

Ambarzumyan, Ivanenko and the Sturm – Liouville inverse problem (#)

My favourite painting in the structuralism style (#)

Jet manifold formalism (from my Scientific Biography) (#)

D.Ivanenko’s proton-neutron model of atomic nuclei of 1932 (#)

My review: “Axiomatic quantum field theory” (#)

Different citation indices (#)

My review “Axiomatic classical (prequantum) field theory” (#)

Victor Ambartsumian and Dmitri Ivanenko in history of Quantum Field Theory (#)

My Scientific Biography (#)

Humanity will never leave the Solar system (#)

Introduction to my book “Lectures on Differential Geometry of Modules and Rings”

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Infinite-dimensional differential geometry (#)

My new book on differential geometry of modules and rings (#)

Why to gauge gravity? (#)

Discrete space-time (from my Scientific Biography) (#)

What is a mathematical structure? (#)

The Higgs boson or the Higgs vacuum? (#)

Impact Factor 2011 of Journals in Mathematical Physics (#)

Dmitri Ivanenko’s archive: Nobel Laureates Letters (#)

My book: Generalized Hamiltonian Formalism for Field Theory 2012/06/

Our book in Spanish: D.Ivanenko, G.Sardanashvili, Gravitación 2012/06/

Nobel laureates inscriptions on the walls of Ivanenko's office in Moscow State University 2012/06/

My lectures on mathematical physics 2012/05/

Lagrangian BRST theory (from my Scientific biografy) 2012/05/

Classical mechanics and field theory admit comprehensive geometric formulation 2012/05/

My Library: Completely integrable and superintegrable Hamiltonian systems with noncompact invariant submanifolds 2012/04/

Lagrangian dynamics of higher-dimensional submanifolds 2012/04/

A problem of an inertial reference frame in classical mechanics 2012/04/

My Library: General Noether theorems 2012/04/

On a gauge model of the fifth force 2012/04/

My Library: Jet Manifold Formalism 2012/03/

Freedom is an immanent property of living nature 2012/03/

Review on our book "Geometric and Algebraic Topological Methods in Quantum Mechanics" in Mathematical Reviews 2012/03/

An energy-momentum is not uniquely deffned 2012/03/

My Library: Time-dependent mechanics 2012/02/

Review on our book "Advanced Classical Field Theory" in Mathematical Reviews 2012/02/

My Library: Advanced Classical Field Theory 2012/02/

My Library: Gauge gravitation theory 2012/02/

Is a momentum space of quantum fields Euclidean? 2012/01/

Hierarchy of Noether identities (from my Scientific Biography) 2012/01/

Can contemporary mathematics describe quantum physics? 2012/01/

Why only electromagnetic and gravitational interactions are in classical physics? 2012/01/

“Antropomorphic” mathematics and the crisis of science 2011/12/

What is a reference frame in field theory and mechanics 2011/12/

Covariant (polysymplectic) Hamiltonian field theory (from my Scientific Biography) 2011/12/

Why a classical system admits different non-equivalent quantization 2011/12/

Five fundamental problems of contemporary physics 2011/11/

Integrable Hamiltonian systems: generalization to a case of non-compact invariant submanifolds (from my Scientific Biography) 2011/11/

On a mathematical hypothesis of quantum space-time 2011/11/

Review on our book “Geometric Formulation of Classical and Quantum mechanics” in Mathematical Reviews 2011/11/

II. How we developed gauge gravitation theory (from my Scientific Biography) 2011/11/

I. How we developed gauge gravitation theory (from my Scientific Biography) 2011/11/

On a mathematical hypothesis of the quark confinement 2011/10/

The prespinor model (from my Scientific Biography 2011/10/

Who is who among Universities in 2011 2011/10/

My Scientific Biography: Student period 2011/10/

Illusion of matter 2011/10/

On the strangeness of relativistic mechanics 2011/09/

”Quantum” causality of ancient Greeks 2011/09/

Why a citation list for a theoretician? 2011/09/

What are classical Higgs fields? 2011/09/

The generalized Serre – Swan theorem is a cornerstone of classical field theory 2011/08/

Metric gravity as a non-quantized Higgs field 2011/08/

What are gauge symmetries? 2011/08/

Why connections in classical field theory? 2011/08/

Geometry in quantum theory IV: Modern geometries 2011/07/

Geometry in quantum theory III: Differential geometry of modules and rings 2011/07/

Does Impact Factor show anything? 2011/07/

Geometry in quantum theory II: Infinite-dimensional fiber bundles 2011/07/

Geometry in quantum theory I: Why familiar differential geometry contributes to quantum theory 2011/07/

Problems of gravitation theory: What is a criterion of gravitational singularities? 2011/06/

On geometric formulation of mechanics 2011/06/

Why connections in field theory 2011/06/

What are general covariant transformations? 2011/06/

What is classical field theory really about? 2011/05/

Non-commutative geometry meets a serious problem 2011/05/

What is meant by supergeometry 2011/05/

Quantum field theory: Functional integrals are not true integrals? 2011/05/

What is a discrete space-time? 2011/05/

What is true Equivalence principle? 2011/04/

On relativistic mechanics in a very general setting 2011/04/

Mechanics as particular classical field theory 2011/04/

What is a fundamental science? 2011/04/

Well-known mathematics that theoreticians do not know 2011/04/

What is true Hamiltonian field theory? 2011/04/

Classical field theory is complete: the strict geometric formulation 2011/03/

My teacher Dmitri Ivanenko, a great theoretician of XX century 2011/03/

The Equivalence principle is treated as one of the corner-stones of gravitation theory. However, there exist its different formulations. One separates “weakest”, “weak”, “middle-strong” and “strong” Equivalence principles. All of them are based on the empirical equality of inertial mass, gravitational active and passive charges, and they establish the existence of a certain reference frame, where physical laws would take the known special relativistic form, i. e., a gravitational field effectively disappears.

The “weakest” Equivalence principle is restricted to the motion law of a probe point mass in a uniform gravitational field.

Its sui generis localization is the “weak” Equivalence principle that states the existence of a desired local inertial frame at a given world point. This is the case of equations depending on a gravitational field and its first order derivatives, e. g., the equations of mechanics of probe point masses, and the equations of electromagnetic and Dirac fermion fields.

The “middle-strong” Equivalence principle is concerned with any matter, except a gravitational field, while the “strong” one is applied to all physical laws.

Apparently, only the “weakest” and “weak” Equivalence principles are true. It is the “weak” Equivalence principle that the identification of a gravitational field to a pseudo-Riemannian metric satisfies to. However, the “weak” Equivalence principle provides a necessary, but not sufficient condition of such identification. Moreover, it does not explain the existence of a gravitational field itself.

To overcome these difficulties, we have reformulated the Equivalence principle as follows.

In geometric terms Special Relativity can be characterized as the geometry of Lorentz invariants. Then the Equivalence principle can be formulated to require the existence of Lorentz invariants on a world manifold X. We agree to call it the geometric Equivalence principle. Its requirement holds if and only if the tangent bundle TX of X admits an atlas with Lorentz transition functions, i. e., a structure group of the associated principal bundle LX of frames in TX is reduced to the Lorentz group SO(1,3). By virtue of the well known theorem, this reduction takes place if and only if the quotient bundle LX/SO(1,3)->X admits a global section, which is a pseudo-Riemannian metric on X.

Thus the geometric Equivalence principle provides the necessary and sufficient conditions of the existence of a pseudo-Riemannian metric on a world manifold that we observe as a gravitational field.

Moreover, if a structure group of the frame bundle LX is reduced to the Lorentz group, it always is reduced to the spatial rotation group SO(3). In accordance with the above mentioned theorem, this reduction defines a space-time decomposition of the tangent bundle TX and, thus, makes a world manifold X into a space-time.

The geometric Equivalence principle also provides the necessary condition of the existence of Dirac’s spinor fields, possessing Lorentz symmetries, on a world manifold.

Thus, one can think of an observable Dirac fermion matter as being the underlying physical reason of the geometric Equivalence principle and, consequently, the existence of a pseudo-Riemannian gravitational field.

In gravitation theory, the geometric Equivalence principle characterizes spontaneous symmetry breaking of space-time symmetries and, thus, clarifies the physical nature of a gravitational field as a Higgs field responsible for this symmetry breakdown.

References:

D. Ivanenko, G. Sardanashvily, The gauge treatment of gravity, Physics Reports 94 (1983) 1-45.

G. Sardanashvily, Gauge gravitation theory from the geometric viewpoint, Int. J. Geom. Methods Mod. Phys. 3 (2006) N1 v-xx; arXiv: gr-qc/0512115

WikipediA: Equivalence principle (geometric)

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