The God has created a man in order that he creates that the God fails to do

Sunday, 16 June 2013

«Теорминимум-XXI». Современный курс теоретической физики.

Этот курс был задуман как современный «Теорминимум-XXI» в качестве альтернативы известному "Курсу теоретической физики" Ландау и Лифшица, который отражает уровень теоретической физики середины прошлого века. Но уже тогда в 70-е годы зародилась совсем другая теоретическая физика, основанная на математическом аппарате дифференциальной геометрии и алгебраической топологии. Она была стимулирована успехами теории калибровочных полей как универсального механизма описания фундаментальных взаимодействий и ее строгой математической формулировкой в терминах геометрии расслоенных пространств.

Расслоения, связности и многообразия струй, суперсимметрии, супергеометрия и некоммутативная геометрия, гомологии и когомологии, солитоны, инстантоны и топологические заряды, многомерные модели, топологическая теория поля, аномалии, квантовые группы и алгебры Хопфа, геометрическое и деформационное квантования, группоиды, алгеброиды и т. д. составляют стандартный контент современных квантовых и полевых моделей. Ничего этого нет ни у Ландау - Лифшица, ни в подавляющем большинстве отечественных университетских учебников и курсов.

Представляемый курс теоретической физики «Современные методы теории поля» включает 5 томов:

Г.А. Сарданашвили, «Современные методы теории поля. 1. Геометрия и классические поля» (УРСС, 1996) (2-е изд. 2011)

Г.А. Сарданашвили, «Современные методы теории поля. 2. Геометрия и классическая механика» (УРСС, 1998)

Г.А. Сарданашвили, «Современные методы теории поля. 3. Алгебраическая квантовая теория» (УРСС, 1999) (2-е изд. 2011)

Г.А. Сарданашвили, «Современные методы теории поля. 4. Геометрия и квантовые поля» (УРСС, 2000)

Г.А. Сарданашвили, «Современные методы теории поля. 5. Гравитация» (УРСС, 1996) (2-е изд. 2011).

Это своего рода адаптированный «Теорминимум-XXI» для тех, кто собирается начать заниматься современной теоретической и математической физикой. Но для профессиональной работы он недостаточен. Экспертное изложение необходимых математических методов и теоретических моделей дано в монографиях:

L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, 2000),

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (World Scientific, 2005),

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, 2009),

G. Giachetta, L. Mangiarotti, G. Sardanashvily, Geometric Methods in Classical and Quantum Mechanics (World Scientific, 2010),

G.Sardanashvily, Lectures on Differential Geometry of Modules and Rings. Application to Quantum Theory (Lambert Academic Publishing, Saarbrucken, 2012),

G.Sardanashvily, Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory (Lambert Academic Publishing, Saarbrucken, 2013),

которые доступны на странице Monographs моего сайта, его дубликата в Google, а также в MendeleY.

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Monday, 10 June 2013

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

G.Sardanashvily, Geometric formulation of non-autonomous mechanics, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1350061 # 


We address classical and quantum mechanics in a general setting of arbitrary time-dependent transformations. Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over a time axis R. Connections on these bundles describe reference frames. Quantum time-dependent mechanics is phrased in geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. 


The technique of symplectic manifolds is well known to provide the adequate Hamiltonian formulation of autonomous mechanics. Its realistic example is a mechanical system whose configuration space is a manifold M and whose phase space is the cotangent bundle T*M of M provided with the canonical symplectic form Om on T*M. Any autonomous Hamiltonian system locally is of this type.

However, this geometric formulation of autonomous mechanics is not extended to mechanics under time-dependent transformations because the symplectic form Om fails to be invariant under these transformations. As a palliative variant, one has developed time-dependent mechanics on a configuration space Q=RxM where R is the time axis. Its phase space RxT*M is provided with the pull-back presymplectic form. However, this presymplectic form also is broken by time-dependent transformations.

We address non-relativistic mechanics in a case of arbitrary time-dependent transformations. Its configuration space is a fibre bundle Q->R. Its velocity space is the first order jet manifold of sections of Q->R. A phase space is the vertical cotangent bundle V*Q of Q->R.

This formulation of non-relativistic mechanics is similar to that of classical field theory on fibre bundles over a base of dimension >1. A difference between mechanics and field theory however lies in the fact that connections on bundles over R are flat, and they fail to be dynamic variables, but describe reference frames.

Note that relativistic mechanics is adequately formulated as particular classical string theory of one-dimensional submanifolds.

Time-dependent integrable Hamiltonian systems and mechanics with time-dependent parameters also are considered.

Sunday, 2 June 2013

Quantum field theory: generating functionals as a measure (from my Scientific Biography)

Third period (1978 - 1990) ...

The fact that a gravitational field by its physical nature is a Higgs field, drew my attention to a general problem of description of a Higgs vacuum. In united models of fundamental interactions, its occurrence is regarded as a kind of phase transition at a certain energy (temperature). I made various attempts to approach this problem [37,62], in particular, developed an idea that a Higgs vacuum is a source of a Higgs gravitational field [51]. However, any substantive theory of a Higgs vacuum still not constructed.

Working on the problem of Higgs vacuum, I met the fact that, in general, there is no mathematically correct formulation of quantum field theory, either axiomatic or in the form of the perturbation theory. The latter is formulated in terms of the so-called functional integrals, but they are not any true integrals, and their properties are defined by analogy with those on finite-dimensional spaces in order to reproduce Feynman diagrams. Pursuing the well-known GNS construction in quantum theory, I knew its expansion to unnormed involutive algebras and, in particular, to an algebra of free fields represented by probe (rapidly decreasing at infinity) functions. These functions form a nuclear space S. Continuous forms of this algebra are generalized functions that make up the dual space S'. A problem was with a description of a system of interacting fields, appeared and disappeared in some instants of time. Such a system is characterized by chronological forms on a space S, but they are not continuous. However, I have found that, after a Wick rotation to Euclidean fields, their chronological form (Euclidean Green function) are continuous. Moreover, they are derived from a generating functional which, by virtue of the well-known theorem, is a measure on a space of generalized functions S'. This construction provides a good mathematical basis of quantum field theory [54,59]. It was extended to fermion fields [57]. However, a problem lies in the fact that it is impossible to write these measures in an explicit form, with the exception of Gaussian measures for fields without interaction. In addition, properties of these measures differ from those adopted for functional integrals in perturbation quantum field theory. For example, there is no translationally invariant Lebesgue measure on S', and I even tried to use this fact for describing a Higgs vacuum [62].

Later I repeatedly returned to attempts to build a generating functional of quantum field theory as a measure, but so far unsuccessfully. However, in the language of measures I described nonequivalent representations of algebras of canonical commutation relations, modeled on nuclear spaces [105].


G. Sardanashvily, True functional integrals in Algebraic Quantum Field Theory, arXiv: hep-th/9410107