%**	This is a pure LaTeX 2e file [article class]
%**	Filename:	bqft-1-math_CSIVF-2010_report.tex
%**	Last update:    19 October 2010
%**     Final version:  19 October 2010
%**	Title:		A mathematical base for Fibre bundle formulation of
%**			Lagrangian Quantum Field Theory
%**	Purpose:	Report for the Workshop CSIVF 2010 - Full version
%**		:	Geometrical formulation of Quantum Field Theory using
%**			the mathematical theory of fibre bundles
%**
%**	Author (and responsible for all errors):

%***************************************************************************%
%**                                                                       **%
%**	 Article style of BOZHIDAR ZAKHARIEV ILIEV: 		 	  **%
%**                                                                       **%
%**	 Laboratory of Mathematical Modeling in Physics			  **%
%**	 Institute for Nuclear Research and Nuclear Energy                **%
%**	 Bulgarian Academy of Sciences                                    **%
%**	 Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria           **%
%**	 E-mail address: bozho@inrne.bas.bg				  **%
%**      URL: http://theo.inrne.bas.bg/~bozho/	        		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

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\typeout{"Presentation Report for Workshop CSIVF-10 -- 2010"}
\typeout{"A mathematical base for Fibre bundle formulation of}
\typeout{Lagrangian Quantum Field Theory"}
\typeout{by Bozhidar Zakhariev Iliev}
\typeout{Its initial draft version was written during the period}
\typeout{18---30 August 2010}
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% Beginning of the bibliography file bqft-1-math_CSIVF-2010_report.bbl
\begin{filecontents}{bqft-1-math_CSIVF-2010_report.bbl}
\begin{thebibliography}{1}

\small
\pageDuration{9}

     \bibitem{bp-BQM-1}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics. {I}.\
  {Introduction}. {The} evolution transport.
\newblock {\em Journal of Physics A: Mathematical and General},
  34(23):4887--4918, 2001.
\newblock DOI~No.~10.1088/0305-4470/34/23/308\\ http://arXiv.org e-Print
  archive, E-print No.\ quant-ph/9803084, March 1998.

\bibitem{bp-BQM-2}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics. {II}.\
  {Equations} of motion and observables.
\newblock {\em Journal of Physics A: Mathematical and General},
  34(23):4919--4934, 2001.
\newblock DOI~No.~10.1088/0305-4470/34/23/309\\ http://arXiv.org e-Print
  archive, E-print No.\ quant-ph/9804062, April 1998.

\bibitem{bp-BQM-3}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics.
  {III}.\ {Pictures} and integrals of motion.
\newblock {\em Journal of Physics A: Mathematical and General},
  34(23):4935--4950, 2001.
\newblock DOI~No.~10.1088/0305-4470/34/23/310\\ http://arXiv.org e-Print
  archive, E-print No.\ quant-ph/9806046, June 1998.

\bibitem{bp-BQM-4}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics. {IV}.\
  {Mixed} states and evolution transport's curvature.
\newblock {\em International Journal of Modern Physics~A}, 17(2):229--243,
  2002.
\newblock DOI~No.~10.1142/S0217751X02005669\\ http://arXiv.org e-Print archive,
  E-print No.\ quant-ph/9901039, January 1999.

%--------------------------------------
\newslide
%--------------------------------------

\bibitem{bp-BQM-5}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics. {V.\
  Interpretation, summary, and discussion}.
\newblock {\em International Journal of Modern Physics~A}, 17(2):245--258,
  2002.
\newblock DOI~No.~10.1142/S0217751X02005712\\ http://arXiv.org e-Print archive,
  E-print No.\ quant-ph/9902068, February 1999.

\bibitem{bp-BQM-full}
Bozhidar~Z. Iliev.
\newblock Fibre bundle formulation of nonrelativistic quantum mechanics (full
  version).
\newblock \\ http://arXiv.org e-Print archive, E-print No.\ quant-ph/0004041,
  April 2000.

\bibitem{Husemoller}
D.~Husemoller.
\newblock {\em Fibre bundles}.
\newblock McGraw-Hill Book Co., New York-St.\ Louis-San
  Francisco-Toronto-London-Sydney, 1966.
\newblock Other editions: Springer Verlag, 1975, 1994, 1998. Russian
  translation: Mir, Moscow, 1970.

\bibitem{Steenrod}
Norman Steenrod.
\newblock {\em The topology of fibre bundles}.
\newblock Princeton University Press, Princeton, ninth edition, 1974.
\newblock First eddition: Princeton University Press, 1951.

\bibitem{Lang/Manifolds}
Serge Lang.
\newblock {\em Differential manifolds}.
\newblock Springer Verlag, New York, 1985.
\newblock (originally published: Addison-Wesley Pub.\ Co, Reading, Mass. 1972).

\bibitem{bp-TM-general}
Bozhidar~Z. Iliev.
\newblock Transports along maps in fibre bundles.
\newblock JINR Communication E5-97-2, Dubna, 1997.
\newblock 19~pp. \\ http://arXiv.org e-Print archive, E-print No.\
  dg-ga/9709016, September 1997.


\end{thebibliography}

\end{filecontents}
% End of the bibliography file Heisenberg-Relations_CSIVF_report.bbl


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%***************************************************************************%
%**     BEGINNING of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV    **%
%**	Filename: bozhomac.sty                                            **%
%**	Author (and responsible for all errors):                          **%
%***************************************************************************%
%**                                                                       **%
%**	  BOZHIDAR ZAKHARIEV ILIEV:					  **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

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% \theoremstyle{plain}	% plain, definition, or remark
\newtheorem{VarAxm}{Axiom}			% [ AXIOM ]
\newtheorem{VarCor}{Corollary}			% [ COROLLARY ]
\newtheorem{VarCrit}{Criterion} 		% [ Criterion ]
\newtheorem{VarLem}{Lemma}		     	% [ LEMMA ]
\newtheorem{VarPost}{Postulate}			% [ POSTULATE ]
\newtheorem{VarPrin}{Principle}			% [ PRINCIPLE ]
\newtheorem{VarProp}{Proposition}		% [ PROPOSITION ]
\newtheorem{VarThm}{Theorem}			% [THEOREM ]

	\theoremstyle{definition}	%*********************************
%	\theorembodyfont{\rmfamily\slshape} % requires amstex package


\newtheorem{Answ}{Answer}[section]			% [ ANSWER ]

\newtheorem{Assm}{Assumption}[section]			% [ ASSUMPTION ]

\newtheorem{Assr}{Assertion}[section]			% [ ASSERTION ]

\newtheorem{Defn}{Definition}[section]			% [ DEFINITION ]

\newtheorem{Conj}{Conjecture}[section] 			% [ CONJECTURE ]

\newtheorem{Conc}{Conclusion}[section]			% [ CONCLUSION ]

\newtheorem{Conv}{Convention}[section] 			% [ CONVENTION ]

\newtheorem{Exmp}{Example}[section]			% [ EXAMPLE ]

\newtheorem{Exrc}{Exercise}[section]			% [ EXERCISE ]

\newtheorem{Nota}{Notation}[section]			% [ NOTATION ]

\newtheorem{Prob}{Problem}[section] 			% [ PROBLEM ]

\newtheorem{Sol}{Solution}[section] 			% [ SOLUTION ]
%	Versions
\newtheorem{VarAnsw}{Answer}			% [ ANSWER ]
\newtheorem{VarAssr}{Assertion}			% [ ASSERTION ]
\newtheorem{VarAssm}{Assumption}		% [ ASSUMPTION ]
\newtheorem{VarConj}{Conjecture} 		% [ CONJECTURE ]
\newtheorem{VarConc}{Conclusion}		% [ CONCLUSION ]
\newtheorem{VarConv}{Convention}		% [ CONVENTION ]
\newtheorem{VarDefn}{Definition}		% [ DEFINITION ]
\newtheorem{VarExmp}{Example}			% [ EXAMPLE ]
\newtheorem{VarExrc}{Exercise}			% [ EXERCISE ]
\newtheorem{VarNota}{Notation}			% [ NOTATION ]
\newtheorem{VarProb}{Problem} 			% [ PROBLEM ]
\newtheorem{VarSol}{Solution} 			% [ SOLUTION ]

	\theoremstyle{remark}	%*********************************
%	\theorembodyfont{\rmfamily\upshape} % requires amstex package


\newtheorem{Case}{Case}[section] 			% [ CASE ]

\newtheorem{Cond}{Condition}[section] 			% [ CONDITION ]

\newtheorem{Comm}{Comments}[section] 			% [ COMMENTS ]

\newtheorem{Note}{Note}[section] 			% [ NOTE ]

\newtheorem{Obse}{Observation}[section]			% [ OBSEVATION ]

\newtheorem{Ques}{Question}[section]			% [ QUESTION ]

\newtheorem{Rem}{Remark}[section] 			% [ REMARK ]

\newtheorem{Summ}{Summary}[section] 			% [ SUMMARY ]
%	Versions
\newtheorem{VarCase}{Case} 			% [ CASE ]
\newtheorem{VarCond}{Condition}			% [ CONDITION ]
\newtheorem{VarComm}{Comments} 			% [ COMMENT ]
\newtheorem{VarNote}{Note} 			% [ NOTE ]
\newtheorem{VarObse}{Observation}		% [ OBSEVATION ]
\newtheorem{VarQues}{Question}			% [ QUESTION ]
\newtheorem{VarRem}{Remark} 			% [ REMARK ]
\newtheorem{VarSumm}{Summary} 			% [ SUMMARY ]

%	\theoremstyle{remark}	%*********************************
 %	\theorembodyfont{\sffamily\upshape}  requires amstex package

\newtheorem{Ackn}{Acknowledgment}[section]		% [ ACKNOWLEDGMENT ]

\newtheorem{Algr}{Algorithm}[section] 			% [ ALGORITHM ]

\newtheorem{Claim}{Claim}[section] 			% [ CLAIM ]

\newtheorem{Step}{Step}[section] 			% [ STEP ]

%	Versions
\newtheorem{VerAck}{Acknowledgment}		% [ ACKNOWLEDGMENT ]
\newtheorem{VerAlg}{Algorithm} 			% [ ALGORITHM ]
\newtheorem{VerClaim}{Claim} 			% [ CLAIM ]
\newtheorem{VarStep}{Step} 			% [ STEP ]

%++++++++++++++++++++++++++++++++++++++----------------------------------------
%	END of theorem-like environments (BOZHO'S DEFINITIONS):
%++++++++++++++++++++++++++++++++++++++----------------------------------------


%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
%  BEGINNING of New and Renew commands and environments (BOZHO'S DEFINITIONS):

% use to produce exceptional versions of existing environments
%	REDEFINE the corresponding counters and there appearance!!
\newenvironment{Version}[1][Choose a NAME and FONT!]%
		{\vspace{0.0ex}\noindent{#1}}%
		{\vspace{1.2ex}}

% {Proof}: an alternative is the AMS (amsthm package) "proof" environment
%	   but it often does not end correctly.
% Caution: some packages like FoilTeX also (re)define this environtment
\newenvironment{Proof}[1][Proof.]%
		{\vspace{0.0ex}\noindent\textit{#1\hspace*{0.5em}}}%
		{\hspace*{0pt}\hfill\QEDsymbol\vspace{1.2ex}}
% Q.E.D.  symbols. The AMS package `amsthm` provides \qed producing \qedsymbol
\providecommand{\qedsymbol}{$\square$}  % Requires amssymb package
    \newcommand{\QEDsymbol}{$\square$}	% Requires amssymb package
\providecommand{\qed}{\hfill\qedsymbol}
%\newcommand{\QED}[1][6]{\hfill\hbox{\vrule width #1pt height #1pt depth 0pt}}
\newcommand{\QED}{\hspace*{0pt}\hfill Q.E.D.}

% End symbols (besides QED one) for manual usage
\newcommand{\EndAxm}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndPost}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndPrin}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndThm}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndProp}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndLem}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndCor}{\hspace*{0pt}\hfill$\blacksquare$}
\newcommand{\EndCrit}{\hspace*{0pt}\hfill$\blacksquare$}
%
\newcommand{\EndRem}{\hspace*{0pt}\hfill$\blacklozenge$}
\newcommand{\EndComm}{\hspace*{0pt}\hfill$\blacklozenge$}
\newcommand{\EndNote}{\hspace*{0pt}\hfill$\blacklozenge$}
\newcommand{\EndCase}{\hspace*{0pt}\hfill$\blacklozenge$}
%
\newcommand{\EndDefn}{\hspace*{0pt}\hfill$\lozenge$}
\newcommand{\EndConj}{\hspace*{0pt}\hfill$\bigcirc$}
\newcommand{\EndAssr}{\hspace*{0pt}\hfill$\vartriangleright$}
\newcommand{\EndConc}{\hspace*{0pt}\hfill$\between$}
\newcommand{\EndExmp}{\hspace*{0pt}\hfill$\vartriangle$}
\newcommand{\EndExrc}{\hspace*{0pt}\hfill$\vartriangleleft$}
\newcommand{\EndProb}{\hspace*{0pt}\hfill$\heartsuit$}
%
\newcommand{\EndAlgr}{\hspace*{0pt}\hfill$\triangleq$}
\newcommand{\EndClaim}{\hspace*{0pt}\hfill$\triangleq$}
\newcommand{\EndAckn}{\hspace*{0pt}\hfill$\triangleq$}


%	Matrices, vectors, vector spaces:
\newcommand{\Mat}[1]{\boldsymbol{#1}}	% Matrix of a vector or linear map
\newcommand{\Vect}[1]{\boldsymbol{#1}}	% Vector quantity
\providecommand{\openone}{\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}}
				% Identity/Unit Matrix (Taken from RevTeX)

%	Mathematical Abbreviations:
\newcommand{\id}{\mspace{2mu}\mathsf{i}\mspace{-1mu}\mathsf{d}}% identity map
\newcommand{\const}{\mathrm{const}}	%  constant quantity
\newcommand{\iu}   {\mathrm{i}}		%  Imaginary Unit
\newcommand{\e}{\ensuremath{\mathrm{e}}}%  base of natural logarithms
\newcommand{\Lied}{\mathcal{L}}         % Lie Derivative

%	Field (in algebraic sense)
\newcommand{\field}[1][K]{\ensuremath{\mathbb{#1}}}

%	(Lie) groups and algebras
\newcommand{\G}{\mathrm{G}}		% Group G
\newcommand{\g}{\mathfrak{g}}		% Lie algebra of a group G
%
\DeclareMathOperator{\Int}{Int}	% Internal automorphisms x\mapsto axa^{-1}
\DeclareMathOperator{\Ad}{Ad}	% Adjoint representation of a Lie group
\DeclareMathOperator{\ad}{ad}	%   ... the differential of \Ad ...
% \newcommand{\exp}{\mathrm{exp}}	% exponential mapping; LaTeX predefined
\DeclareMathOperator{\Lie}{Lie}	% Lie algebra of ...
%
\DeclareMathOperator{\GL}{\mathrm{GL}}	 % General Linear (Group)
\DeclareMathOperator{\gl}{\mathfrak{gl}} % General Linear (Group) Lie algebra
\DeclareMathOperator{\SL}{\mathrm{SL}}	 % Special Linear (Group)
\DeclareMathOperator{\slg}{\mathfrak{sl}}% Special Linear (Group) Lie algebra
\DeclareMathOperator{\GA}{\mathrm{GA}}	 % General Affine(Group)
\DeclareMathOperator{\ga}{\mathfrak{ga}} % General Affine(Group) Lie algebra
\DeclareMathOperator{\T}{\mathrm{T}}	 % general Translation (Group)
%
\DeclareMathOperator{\Ug}{\mathrm{U}}	% Unitary (Group)
\DeclareMathOperator{\ug}{\mathfrak{u}}	% Unitary (Group) Lie algebra
\DeclareMathOperator{\SU}{\mathrm{SU}}	% Special Unitary (Group)
\DeclareMathOperator{\su}{\mathfrak{su}}% Special Unitary (Group) Lie algebra
%
\DeclareMathOperator{\Og}{\mathrm{O}}	% Orthogonal (Group)
\DeclareMathOperator{\og}{\mathfrak{o}}	% Orthogonal (Group) Lie algebra
\DeclareMathOperator{\SO}{\mathrm{SO}}	% Special Orthogonal (Group)
% \DeclareMathOperator{\so}{\mathfrak{so}}% Special Orthogonal (Group) Lie algebra
% \providecommand{\so}{\,\mathfrak{so}\,}% \so appears in some packages

%	Ordinary and Partial differentials:
\newcommand{\ordinary}{\mathrm{d}}	% ordinary differential
\newcommand{\od}      {\ordinary}       % Ordinary Differential (abbreviation)
% the command \partial is predefined    % Partial Differential
\newcommand{\pd}      {\partial}        % Partial Differential  (abbreviation)
\newcommand{\Id}[1][\,]{#1\mathrm{d}}	% Integral Differential, adjustable front space
%
\DeclareMathOperator{\ed} {\mathsf{d}}	% External Differential (operator)
\DeclareMathOperator{\edo}{\mathsf{d}}	% External Differential Operator
\DeclareMathOperator{\odo}{\ordinary}	% Ordinary Differential Operator
\DeclareMathOperator{\pdo}{\partial}    % Partial Differential Operator

%	left index(es) in \mathnormal:
\newcommand{\lindex}[3][{}]{{}\mspace{3mu}{\vphantom{#1}}%
	_{#2}^{#3}\mspace{-0.8mu}{#1}}
%	left index(es) in \mathrm:
\newcommand{\lindexrm}[3][{}]{{}\mspace{3mu}{\vphantom{#1}}%
	_{\mathrm{#2}}^{\mathrm{#3}}\mspace{-0.8mu}{#1}}
%	right indexes with space bellow/over them
\newcommand{\Sprindex}[3][]{{#1}_{\hphantom{#3}#2}^{#3}}% Sup: space bellow it
\newcommand{\Sbrindex}[3][]{{#1}_{#2}^{\hphantom{#2}#3}}% Sub: space over it

%	Differential geometry
%	Fields (in differential geometric sense; particular sections}
\newcommand{\Fields}[1]{\ensuremath{\mathfrak{#1}}}
%       Algebra (as an algebraic structure)
\newcommand{\algebra}[1]{\ensuremath{\boldsymbol{#1}}}
%	Dimensions with respect to some (algebraic) field
\newcommand{\dimR}[1][R]{\dim_{\field[#1]}}% Real dimension
\newcommand{\dimC}[1][C]{\dim_{\field[#1]}}% Complex dimension
\newcommand{\dimK}[1][K]{\dim_{\field[#1]}}% Dimension with resp. to \field K
%	Christoffel symbols
\newcommand{\Christoffel}[3][]{\genfrac{\{}{\}}{0pt}{}{#2}{#3}_{#1}}


%	Physical Abbreviations:
\newcommand{\ih}{\mathrm{i}\hbar}% imaginary unit times Planck constant/{2\pi}
\newcommand{\iih}{\frac{1}{\ih}} % the inverse to \ih (see previous command)
			      % =1/(imaginary unit times Planck constant/2\pi)

%	Additional description of the title page
   \newcommand{\ShortTitle}[1]{\textbf{\textsl{Short title:} #1}}
   \newcommand{\Subject}[1]   {\textit{\textbf{Subject Classes:\\}#1}}
   \newcommand{\MSC}[2][2000]{\begin{tabular}{@{}c@{}}%
	\textit{\textbf{#1 MSC numbers:}}\\ #2 \end{tabular} }
   \newcommand{\PACS}[2][2003]{\begin{tabular}{@{}c@{}}%
	\textit{\textbf{#1 PACS numbers:}}\\ #2 \end{tabular} }
   \newcommand{\KeyWords}[1]  {\textit{\textbf{Key-Words:\\}#1}}

%	Nice formatting of (subject) index
\newcommand{\ff}[1]{#1ff}% e.g., "64ff" in the Index ("and following pages")
\newcommand{\nn}[1]{#1n} % e.g., "64n"  in the Index (footNote item)
\newcommand{\defined}[2][{}]{\textbf{#2}#1{}}%
				% Index page(s) where a concept is defined
% use "see{\emph{also} ...}" to obtain the version "see also"
% of the "see" command for MakeIndex

%	Sans Serif capital Latin letters & their possible usage
\newcommand{\Dsf}{\mathsf{D}}		% Derivation along paths of sections
					% or of the tensor algebra
\newcommand{\Esf}{\mathsf{E}}
\newcommand{\Isf}{\mathsf{I}}		% Parallel transport assigned to a
					% transport I along paths
\newcommand{\Psf}{\mathsf{P}}		% Parallel transport along path
\newcommand{\Rsf}{\mathsf{R}}		% parallel transport cuRvature

%	Black board capital Latin letters & their possible usage
\newcommand{\Pbb}{\mathbb{P}}		% Parallelism structure

%	Slashed and backslashed symbols (normal and diagonal variant):
\newcommand{\slashed}[1]{\ensuremath{#1\mspace{-9.2mu}/}}%slashed small symbol
\newcommand{\Slashed}[1]{\ensuremath{#1\mspace{-11.3mu}/}}% for BIG symbols
\newcommand{\backslashed}[1]{\ensuremath{#1\mspace{-9.4mu}\backslash}}% small
\newcommand{\Backslashed}[1]{\ensuremath{#1\mspace{-11.0mu}\backslash}}% BIG
\newcommand{\varslashed}[1]{\ensuremath{#1\mspace{-11.8mu}\diagup}} % var
\newcommand{\varSlashed}[1]{\ensuremath{#1\mspace{-14.9mu}\diagup}} % var
\newcommand{\varbackslashed}[1]{\ensuremath{#1\mspace{-12.4mu}\diagdown}}% var
\newcommand{\varBackslashed}[1]{\ensuremath{#1\mspace{-14.2mu}\diagdown}}% var

%	Latin abbreviations:
\newcommand{\ie} {i.e.\ }		%  That is (Latin: It Est)
\newcommand{\cf} {cf.\ }		%  Compare with (Latin: ConFer)
\newcommand{\eg} {e.g.\ }		%  For example (Latin: Exapli gratia)
\newcommand{\etc}{etc.\ }		%  And so on (Latin: et cetera)
\newcommand{\viz}{viz.\ }		%  That is (Latin: vidilizet)
\newcommand{\vv}{v.v.\ }		%  Inversely (Latin: vice versa)
\newcommand{\vive}{\emph{vice versa}}	%  Inversely (Latin: vice versa)

%	No breakable dashes:
\newcommand{\Ndash}{\nobreakdash--}	     %no breakable dash (Number range)
\newcommand{\ndash}{\nobreakdash-\hspace{0pt}}% no breakable normal dash

%	Special brackets
\newcommand{\slb}{[\![}       % Special Left  Bracket (like black board "[")
\newcommand{\srb}{]\!]}	      % Special Right Bracket (like black board "]")
\newcommand{\clangle}{\mspace{0.0mu}\prec\mspace{-5.3mu}}%calligraphic \langle
\newcommand{\crangle}{\mspace{-5.3mu}\succ\mspace{0.0mu}}%calligraphic \rangle

%	Misc
\newcommand{\bs}[1]{\boldsymbol{#1}}	% Boldface Symbols (AMS style)
\newcommand{\xlrarrow}[2][12mu]{% extensible left-right arrow over an object
    \overset{%
	\xleftarrow{\hphantom{#2}\mspace{-2mu}}%
	\mspace{-#1}\mspace{-#1}%
	\xrightarrow{\mspace{2mu}}%
   	    }%
   	    {{#2}}%
   }% optional argument: equals 1/2 times the overlapping of the 2 arrows
    % Ignore the internal overfull warnings!


%		END of new commands and environments
%  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<


%	>=================================>
%	BEGINNING of Mathematical Operators  (requires amsopn.sty package)

% Set theory and topology
 \DeclareMathOperator{\diam}{diam}	% diameter

% Mappings
 \DeclareMathOperator{\Ker}{Ker}	% Kernel of a mapping
% \DeclareMathOperator{\ker}{ker}	% kernel	; LaTeX predefined
\DeclareMathOperator{\Image}{\mathrm{Im}}% Image of a mapping
 \DeclareMathOperator{\rank}{rank}	% rank
 \DeclareMathOperator{\im}{im}		% image
%\DeclareMathOperator{\Im}{Im}		% Image;LaTeX predefined,Complex numbers
 \DeclareMathOperator{\degree}{deg}	% degree ;-\deg which is LaTeX predefined
 \DeclareMathOperator{\pr}{pr}		% projection
 \DeclareMathOperator{\End}{End}	% set of Endomorphisms
 \DeclareMathOperator{\Aut}{Aut}	% set of Automorphisms

% Complex numbers:
 \DeclareMathOperator{\Imp}{Im}		% Imaginary part; LaTeX predefined
 \DeclareMathOperator{\Rep}{Re}		% Real part     ; LaTeX predefined

% Matrices/Linear operators
 \DeclareMathOperator{\tr}{tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Tr}{Tr}		% Trace (spur) of matrices/operators
 \DeclareMathOperator{\Spur}{Sp}	% Spur (trace) of matrices/operators
 \DeclareMathOperator{\diag}{diag}	% Diagonal matrix

% Manifolds
 \DeclareMathOperator{\Tan}{T}		% the Tangent functor

% Fibre bundles
 \DeclareMathOperator{\Sec}{Sec}	% set of Sections of a fibre bundle
 \DeclareMathOperator{\PSec}{PSec}	% set of Sections of ... along Paths
 \DeclareMathOperator{\Morf}{Mor}	% set of Morphisms of a fibre bundle
 \DeclareMathOperator{\PMorf}{PMor}	% set of Morphisms of... along Paths
 \DeclareMathOperator{\MorfSec}{MorSec}	% set of Morphism of the set of
					% Sections of a bundle
 \DeclareMathOperator{\Diff}{Diff}	% set of Diffeomorphisms
 \DeclareMathOperator{\Lift}{Lift}	% set of Liftings
 \DeclareMathOperator{\PLift}{PLift}	% set of Liftings along Paths
 \DeclareMathOperator{\Path}{P}		% set of Paths
 \DeclareMathOperator{\PF}{PF}		% set of Functions along Paths
% \DeclareMathOperator{\pr}{\mathrm{pr}}% projection mapping, defined above
 \DeclareMathOperator{\morf}{mor}	% Morphism fibre bundle of a bundle

% Exponents
%\DeclareMathOperator{\exp}{exp}	% exponent: LaTeX predefined
 \DeclareMathOperator{\Exp}{Exp}	% exponent
 \DeclareMathOperator{\Pexp}{Pexp}	% Path/Time ordered exponent
 \DeclareMathOperator{\Texp}{Texp}	% Time/Path ordered exponent

%	Differential vector operators:
\DeclareMathOperator{\diver}{div}	% divergence
\DeclareMathOperator{\grad}{grad}	% gradient
\DeclareMathOperator{\rot}{rot}		% rotor

%	Ordinary and Partial differentials: 	these are defined above
% \DeclareMathOperator{\odo}{\mathrm{d}}  % Ordinary Differential Operator
% \DeclareMathOperator{\pdo}{\partial}    % Partial Differential Operator

% Misilaneous
 \DeclareMathOperator{\sign}{sign}	% sign of a real number
 \DeclareMathOperator{\sgn} {sgn}	% sign of a real number
 \DeclareMathOperator{\Supp}{Supp}	% Support of a function
 \DeclareMathOperator{\supp}{supp}	% Support of a function

%	<==================================<
%	END of Mathematical Operators  (requires amsopn.sty package)



%	>=================================>
%	BEGINNING of Text representation (lay-out) commands

% Putting a point after section number in the text titles (not in .toc file)
\renewcommand{\@seccntformat}[1]{\csname the#1\endcsname.\hspace{1.0em}}

%	Equation numbering and referring
\numberwithin{equation}{section}
% \renewcommand{\theequation}{\thesection.\arabic{equation}}
% \setcounter{equation} {0}	% put this after every \section command
				% to be used only with the above eqn. No. def

% Equation reference number in (...)
% AMS (amsmath package) provides \eqref{...} in upright/roman font in (...)
%	but it often add unnecessary space at the end of the reference.
\providecommand{\eqref}[1]{{\rmfamily\upshape(\ref{#1})}}
     \newcommand{\eref}[1]{{\rmfamily\upshape(\ref{#1})}}

% Table reference number
\numberwithin{table}{section}
\renewcommand{\thetable}{\arabic{section}.\arabic{table}}

% Figure reference number
\numberwithin{figure}{section}
\renewcommand{\thefigure}{\arabic{section}.\arabic{table}}

%	<==================================<
%	END of Text representation (lay-out) commands


%***************************************************************************%
%**       END of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV        **%
%***************************************************************************%
\end{filecontents}
% End of the file bozhomac.sty


% Beginning of the file bozhlogo.sty
\begin{filecontents}{bozhlogo.sty}
%***************************************************************************%
%**     BEGINNING of LaTeX2e logo package of BOZHIDAR ZAKHARIEV ILIEV     **%
%**	Filename: bozhlogo.sty                                            **%
%**	Author (and responsible for all errors): Bozho                    **%
%***************************************************************************%
%**                                                                       **%
%**       This file defines the commands \bozho and \BOZHO                **%
%**       which are the logos of BOZHIDAR ZAKHARIEV ILIEV:                **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**		    (DON'T USE WITHOUT PERMISSION)			  **%
%**                                                                       **%
%***************************************************************************%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                       %%
%% 		Beginning of \CharacterTable                             %%
%%                                                                       %%
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z  %%
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z  %%
%%   Digits        \0\1\2\3\4\5\6\7\8\9                                  %%
%%   Exclamation   \!     Double quote  \"     Hash (number) \#          %%
%%   Dollar        \$     Percent       \%     Ampersand     \&          %%
%%   Acute accent  \'     Left paren    \(     Right paren   \)          %%
%%   Asterisk      \*     Plus          \+     Comma         \,          %%
%%   Minus         \-     Point         \.     Solidus       \/          %%
%%   Colon         \:     Semicolon     \;     Less than     \<          %%
%%   Equals        \=     Greater than  \>     Question mark \?          %%
%%   Commercial at \@     Left bracket  \[     Backslash     \\          %%
%%   Right bracket \]     Circumflex    \^     Underscore    \_          %%
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|          %%
%%   Right brace   \}     Tilde         \~}                              %%
%%                                                                       %%
%% 		End of \CharacterTable                                   %%
%%                                                                       %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{bozhlogo}[2010/08/26 BOZHIDAR Z. ILIEV's LaTeX2e BOZHO logo]


\newlength{\BO}		\newlength{\HO}
\newlength{\UP}		\newlength{\DOWN}	\newlength{\MIDDLE}


\newcommand{\bozho}{\leavevmode\hbox{\slshape\bfseries%
%
%
\settowidth{\BO}{BO}\settowidth{\HO}{HO}\settoheight{\UP}{BOZHO}%
\settowidth{\MIDDLE}{/}\settodepth{\DOWN}{/}%
%
\addtolength{\UP}{+0.15\UP}%
\addtolength{\BO}{+\MIDDLE}%
\rule[\UP]{\BO}{0.15ex}\hspace{-\BO}\raisebox{-0.12\UP}{BO}%
\hspace{+0.09em}\raisebox{+0.17\UP}{/}%
%
\hspace{-0.24em}\raisebox{+0.71\UP}{$\bullet$}%
%
\hspace{-0.32em}\hspace{-1.14\MIDDLE}\raisebox{-0.4\UP}{$\bullet$}%
%
\hspace{-0.30em}%
\addtolength{\DOWN}{-0.41\DOWN}%
\addtolength{\HO}{+1.5\MIDDLE}%
\rule[-\DOWN]{\HO}{0.15ex}%
\addtolength{\HO}{-\MIDDLE}%
\hspace{-\HO}\hspace{+0.18em}\raisebox{+0.17\UP}{HO}%
%
}}

\newcommand{\BOZHO}% requires "amsmath" package
{\bozho$^{^{\text{\textregistered}\,} \text{\texttrademark} }$}

% testing for different font sizes:
%\Huge\BOZHO\hspace{2em}\huge\BOZHO\hspace{1.9em}\LARGE\BOZHO\\
%\Large\BOZHO\hspace{1.8em}\large\BOZHO\hspace{1.7em}\normalsize\BOZHO
%\small\BOZHO\hspace{1.6em}\footnotesize\BOZHO
%\scriptsize\BOZHO\hspace{1.5em}\tiny\BOZHO\hspace{1.5em}\BOZHO

%***************************************************************************%
%**     	END of LaTeX2e logo package of BOZHIDAR ZAKHARIEV ILIEV   **%
%***************************************************************************%
\end{filecontents}
% End of the file bozhlogo.sty

% <<====	END OF EXTERNAL FILES	<<====	<<====	<<====	<<====





% Start of the preambule
%%
\ProvidesFile{Heisenberg-Relations_CSIVF_report.tex}%
      [2006/08/22 Presentation using TeXPower]



% Info text, if needed
% \def\MyNewText{Text to appear where this command is put}

% Enable all color emphasis and highlighting options; use a light background and slifonts.
\PassOptionsToPackage{coloremph,colormath,colorhighlight,lightbackground%
% ,printout% for draft/printing without dynamics effects
}{texpower}
\RequirePackage{tpslif	onts}

% TeXPower options:
% display (dynamic features), printout (default, No dynamics), verbose (info)
% oldfiltering (in the text on/of: \oldfillteringon/off)
% fixcolorstack (color stack correction, iff there are color problems)
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% pspecialsallowed (true if PS specials may be used)
% display (true if display option is given)
% TPcolor (true if hilighting option is given)

% document class "powersem" options
% display (off oll features of "seminar")
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%%%%%%%%								%%%%%%
%+++++++>>>	Beginning of page layout	+++++++>>>

% New commands
% Numbering of the PARTS
% \renewcommand{\thepart}{\Alph{part}}

% New commands
% \newcommand{\U}{{\ope{U}}}  		% vector field in momentum picture
% \newcommand{\tU}{{\tope{U}}}		% vector field in Heisenberg picture
% \newcommand{\tu}{{\Tilde{u}}}		% Fourier image of a
					% vector field in Heisenberg picture
\newcommand{\ta}{{\Tilde{a}}}		% linearly independent Fourier image of
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%	\psi				% (spin 1/2) particle spinor
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\newcommand{\Hil}{\mathcal{F}}		% usual Hilbert space/standard fibre
\newcommand{\HilB}{(\bHil,\proj,\base,\Hil)}	% Hilbert fibre bundle
\newcommand{\Hilb}{(\bHil,\proj,\base)}	% Hilbert fibre bundle (without fibre)
	\newcommand{\bHil}{\mathit{F}}	% (total) bundle Hilbert space
	\newcommand{\proj}{\pi}		% Hilbert bundle projection
	\newcommand{\base}{\mathit{M}}	% Hilbert bundle base space

% Hamiltonians
 \newcommand{\Ham}{\mathcal{H}}	% usual Hamiltonian
 \newcommand{\bHam}{\mathit{H}}	% bundle Hamiltonian morphism
%
% \newcommand{\HamM}{\boldsymbol{\Ham}} % the matrix of the usual Hamiltonian
% \newcommand{\bHamM}{\boldsymbol{\bHam}} % the matrix of the bundle Hamiltonian
%
% \newcommand{\mHam}{\boldsymbol{\Ham}^\mathbf{m}}   % usual matrix Hamiltonian
% \newcommand{\mbHam}{\boldsymbol{\bHam}\!^\mathbf{m}}%bundle matrix Hamiltonian

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\newcommand{\dyn}[1]{\pmb{\mathbb{#1}}}	% dynamical variable ----->>>

% Operators (in the fibre= standard Hilbert space)
% \newcommand{\ope}[1]{\mathcal{#1}}		 % operator (1 argument var.)
\newcommand{\ope}[2][{}]{\lindex[\mathcal{#2}]{}{#1}} % operator with left
 			% superscript [optional]; possibly adjust space
% versions of \ope[]{}
\newcommand{\tope}[2][{}]{\ope[#1]{\Tilde{#2}}} % operator with Tilde
						% used in Heisenberg picture
%
% \newcommand{\mope}[1]{\boldsymbol{\mathcal{#1}}}    % matrix of operator

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% \newcommand{\mor}[1]{\mathit{#1}}		 % morphism (1 argument var.)
\newcommand{\mor}[2][{}]{\lindex[\mspace{-2.8mu}\mathit{#2}]{}{#1}}
			     % morphism with left superscript [optional];
			     % possibly adjust the space parameter
%\newcommand{\mor}[2][{}]{{}#1\mathit{#2}} % morphism with [optional] left
			     % parameter
% versions of \mor[]{}
\newcommand{\hmor}[2][{}]{\mor[#1]{\Hat{#2}}}	% morphism with Hat
\newcommand{\tmor}[2][{}]{\mor[#1]{\Tilde{#2}}}	% morphism with Tilde
						% used in Heisenberg picture
%
% \newcommand{\mmor}[1]{\boldsymbol{\mathit{#1}}}	 % matrix of morphism

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\begin{slide}
% the \end{slide} is at the end of all slides in the present group of slides
  \title
  {% Main Title
    A mathematical base for fibre bundle formu-\\lation of Lagrangian
quantum field  theory
% \\
%     {% Subtitle
%       \normalfont
%       the subtitle
%       }%
    }


\author{
Bozhidar Z.\ Iliev\\
\hspace*{-0.6em}
Laboratory of Mathematical Modeling in Physics\\
\hspace*{-0.7em}
Institute for Nuclear Research and Nuclear Energy\\
Bulgarian Academy of Sciences\\
% Boul.\ Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria
\url{E-mail: bozho@inrne.bas.bg}\\
\href{http://theo.inrne.bas.bg/~bozho/}
{URL: http://theo.inrne.bas.bg/$\sim$bozho/}
}

\date{%
\begin{center}\highlighttext{10$^\text{th}$
		International Workshop on
		\\ \ \ Complex Structures,
		\\ \ \ Integrability and Vector Fields
		}
\end{center}
\vspace{-0.5ex}\normalfont
\highlightenhanced{
\href{http://theo.inrne.bas.bg/csvf/csivf2010/}{URL:
      http://theo.inrne.bas.bg/csvf/csivf2010/}
}
}

  \maketitle


%--------------------------------------
\newslide
%--------------------------------------
\pageTransitionSplitHO% Split Horizontally Outside

  \minisec{Abstract}% some title on a slide

	This report presents a differential-geometric foundeations for an
attempt to formulate Lagrangian (canonical) quantum field theory on fibre
bundles. In it the standard Hilbert space of quantum field theory is replace
with a Hilbert bundle; the former playing a role of a (typical) fibre of the
letter one. Suitable sections of that bundle replace the ordinary state vectors
and the operators on the system's Hilbert space are transformed into morphisms
of the same bundle. In particular, the field operators are mapped into
corresponding field morphisms.

%--------------------------------------
\newslide
%--------------------------------------


 % Table of contnets slide(s)
%  \tableofcontents

\end{slide}


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%%%%%%%%%%%
\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside

%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->---->-->-->>
\section {Introduction}
\label{Introduction}
%			BEGINNING OF SECTION~\ref{Introduction}

	The purpose of this work is to be presented grounds for a
consistent formulation of quantum field theory in terms of fibre bundles. The
 ideas for that goal are shared
from~[1-5],
where the quantum mechanics is formulated
on the geometrical language of fibre bundle theory.
 %--------------------------------------
\newslide
%--------------------------------------

\underline{The plan:}\\

At first we present some basic definitions. Special
attention will be paid on the Hilbert bundles, which will replace the Hilbert
spaces of the ordinary quantum field theory, and the metric structure in them.

\pause

\emph{Next we considered an isomorphism between the fibres of a Hilbert
bundle, called the bundle transport. It will play a central role in this
investigation.}

\pause

\highlightenhanced{Then a motivation is presented why the (Hilbert) fibre
bundles are a natural scene for a mathematical formulation of quantum
field theory.}

   %	The report closes with some comments.


\end{slide}

\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside

%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection

\section{Fibre Bundles. Hilbert bundles}
	\label{Subsect2.1}

	A \emph{bundle} is a triple $(E,\pi,B)$ of sets $E$ (bundle space) and
$B$ (base) and  surjective mapping
$\pi\colon E\to B$ (projection). If $b\in B$,
 $\pi^{-1}(b)$ is the fibre over $b$ and, if
	 $Q\subseteq B$,
$(E,\pi,B)|_Q := (\pi^{-1}(Q),\pi|_{\pi^{-1}(Q)},Q)$
is the restriction on $Q$ of a bundle $(E,\pi,B)$.
	A \emph{section} of $(E,\pi,B)$ is a mapping
$\sigma\colon B\to E$ such that $\pi\circ\sigma=\id_B$, where  $\id_Z$
is the identity mapping of a set $Z$, and their set is denoted by
$\Sec(E,\pi,B)$.
 The set of morphisms of $(E,\pi,B)$ is\[
\Morf(E,\pi,B)
 := \{
	(\varphi,f)| \varphi\colon E\to E,\ f\colon B\to B,\
	\pi\circ\varphi = f\circ\pi
    \} .
\]
The set
of all $B$-morphisms (strong morphisms) of $(E,\pi,B)$
is
\(
 \Morf_B(E,\pi,B)
 := \{\varphi|\varphi\colon E\to E,\ \pi\circ\varphi=\pi \} .
\)
%--------------------------------------
\newslide
%--------------------------------------

	The \emph{bundle of point\ndash restricted morphisms} is defined by
	\begin{align*}
E_0 : &= \{
 (\varphi_b,f)\,|\,
	\varphi_b = \varphi|_{\pi^{-1}(b)},
	\ b\in B,\ (\varphi,f)\in\Morf(E,\pi,B)
	\}
\\
 & = \{
	(\varphi_b,f) \,|\, \varphi_b\colon\pi^{-1}(b)\to \pi^{-1}(f(b)),
	\ b\in B,\ f\colon B\to B
	\}
	\end{align*}
and  $\pi_0\colon E_0\to B$ with $\pi_0(\varphi_b,f):=b$ for
$(\varphi_b,f)\in E_0$.

\pause

 The \emph{bundle $\morf_B(E,\pi,B)$ of
(point\ndash) restricted morphisms over} $B$ of $(E,\pi,B)$ has a
bundle space
	\begin{align*}
E_{0}^{B} : &=
 \{
   \varphi_b \,|\, \varphi_b=\varphi|_{\pi^{-1}(b)},\
	b\in B,\ \varphi\in\Morf_B(E,\pi,B)
 \}
\\
 & = \{ \varphi_b \,|\, \varphi_b\colon \pi^{-1}(b)\to\pi^{-1}(b),\ b\in B \}
	\end{align*}
and projection $\pi_{0}^{B}\colon E_{0}^{B}\to B$ with
\(
\pi_{0}^{B}(\varphi_b) := b,
\)
\( \varphi_b\in E_{0}^{B} .
\)
\pause
There is a natural bijection
\(
\Morf_B(E,\pi,B)  \xrightarrow{\ \chi\ } \Sec\bigl(\morf_B(E,\pi,B)\bigr) .
\)

%--------------------------------------
\newslide
%--------------------------------------

	A map $\Sec(E,\pi,B)\to\Sec(E,\pi,B)$ is called
\emph{morphism of  $\Sec(E,\pi,B)$}. Their set is
$\MorfSec(E,\pi,B)$.

\pause


       	\begin{Defn}	\label{Defn2.1}
	A \emph{Hilbert (fibre) bundle} is a vector bundle whose fibres over
the base are isomorphic Hilbert spaces.
	\end{Defn}

\pause

	Let $\HilB$ be a Hilbert bundle and $\bHil_x:=\proj^{-1}(x)$. Let
$l_x\colon\bHil_x\to\Hil$, $x\in\base$, be the \emph{point\ndash trivializing
isomorphisms} defined by the decomposition functions via
 $\phi_W|_x(x,\psi)=:l_x^{-1}(\psi)\in\proj^{-1}(x)$ for every $\psi\in\Hil$.
 \\
	Let $\clangle\,\cdot\, | \,\cdot\,\crangle\colon\Hil\times\Hil\to\mathbb{R}$ be
the scalar product in $\Hil$ and
\(
\langle\,\cdot\, | \,\cdot\,\rangle_x
	\colon\bHil_x\times\bHil_x\to\mathbb{R}
\)
be the scalar product in  $\bHil_x$.
 A Hilbert bundle with compatible
vector and metric structure,
\(
\langle\varphi_x | \psi_x\rangle_x
		= \clangle l_x(\varphi_x) | l_x(\psi_x) \crangle
\)
for every $\varphi_x,\psi_x\in\bHil_x$,
 is called \emph{compatible Hilbert bundle}.

%--------------------------------------
\newslide
%--------------------------------------


In such a bundle we have
	\begin{gather}	\label{2.1}
\langle \,\cdot\,|\,\cdot\, \rangle _x=
\clangle l_{x}\,\cdot\,|l_{x}\,\cdot\, \crangle, \qquad x\in \base
%	\end{equation}
\\
%	\begin{equation}
	\tag{\ref{2.1}$^\prime$}	\label{2.1'}
\clangle \,\cdot\,|\,\cdot\, \crangle =
\langle l_{x}^{-1}\,\cdot\,|l_{x}^{-1}\,\cdot\, \rangle _x, \qquad x\in \base.
	\end{gather}
  	Beginning from now on, \emph{only
compatible Hilbert bundles will be employed}.

\pause

	Defining the \emph{Hermitian conjugate} mapping
\( \mor{A}_{x}^{\ddag}\colon \Hil\to \bHil_x \) of
\( \mor{A}_x\colon  \bHil_x \to \Hil \) by
 %	\begin{equation}	\label{2.3}
\(
\langle \mor{A}_{x}^{\ddag}\varphi|\chi_x \rangle_x :=
\clangle \varphi | \mor{A}_x\chi_x\crangle,
\)
\(
 \varphi\in\Hil,
\)
\( \chi_x\in \bHil_x,
%	\end{equation}
\)
we find ($\dag$ denotes conjugation in $\Hil$)
	\begin{equation}	\label{2.4}
\mor{A}_{x}^{\ddag} =
l_{x}^{-1}\circ \left( \mor{A}_x\circ l_{x}^{-1} \right)^\dag .
	\end{equation}
 	We call a mapping $\mor{A}_x\colon\bHil_x\to\Hil$ \emph{unitary} if
it has an inverse $\mor{A}^{-1}$ and
 %	\begin{equation}	\label{2.5}
\(
\mor{A}_{x}^{\ddag} = \mor{A}_{x}^{-1} .
%	\end{equation}
\)
\newslide

The isometric isomorphisms $l_x\colon\bHil_x\to\Hil$ are unitary
in this sense:
	\begin{equation}	\label{2.6}
l_{x}^{\ddag} = l_{x}^{-1}.
	\end{equation}
\pause
   	The \emph{Hermitian conjugate}
mapping to $\mor{A}_{x\to y}\in$
\(
\{\mor{C}_{x\to y}\colon\bHil_x\to\bHil_y,\ x,y\in\base\}
\)
is
\( \mor{A}_{x\to y}^{\ddag}\colon \bHil_x\to \bHil_y \)
such that
%	\begin{equation}	\label{2.7}
\(
\langle \mor{A}_{x\to y}^{\ddag} \mor{\Phi}_x| \mor{\Psi}_y \rangle_y :=
\langle \mor{\Phi}_x| \mor{A}_{y\to x}\mor{\Psi}_y \rangle_x,
\ \mor{\Phi}_x\in \bHil_x,\  \mor{\Psi}_y\in \bHil_y.
 %	\end{equation}
\)
Its explicit form is
	\begin{equation}	\label{2.8}
\mor{A}_{x\to y}^{\ddag} =
l_{y}^{-1}\circ
\left( l_x\circ \mor{A}_{y\to x}\circ l_{y}^{-1} \right)^\dag
\circ l_x.
	\end{equation}
As $(\ope{A}^\dag)^\dag\equiv \ope{A}$ for any
$\ope{A}\colon \Hil\to\Hil$, we have
	\begin{equation}	\label{2.9}
\bigl( \mor{A}_{x\to y}^{\ddag} \bigr)^\ddag = \mor{A}_{x\to y} .
	\end{equation}


	A mapping $\mor{A}_{x\to y}$ is called \emph{Hermitian} if
	\begin{equation}	\label{2.11}
\mor{A}_{x\to y}^\ddag = \mor{A}_{x\to y}.
	\end{equation}
The mappings
\( l_{x\to y}:=l_y^{-1}\circ l_x \)
are Hermitian.
 A mapping
\(
\mor{A}_{x\to y}\colon \bHil_x\to\bHil_y
\)
is called \emph{unitary}
if it has a left inverse mapping
 \(
\mor{A}_{x\to y}^{-1}\colon \bHil_y\to\bHil_x
\)
and
	\begin{equation}	\label{2.12}
\mor{A}_{x\to y}^\ddag = \mor{A}_{y\to x}^{-1} .
	\end{equation}
        The last equation is equivalent with
	\begin{equation}	\tag{\ref{2.12}$^\prime$}	\label{2.12'}
\langle\mor{A}_{y\to x}\,\cdot\, | \mor{A}_{y\to x}\,\cdot\,\rangle _x
   = \langle\,\cdot\, | \,\cdot\,\rangle _y
   \colon\bHil_y\times\bHil_y\to\mathbb{C},
	\end{equation}
\ie the unitary mappings are
\emph{fibre\ndash isometric}.
The following mappings are unitary
	\begin{equation}	\label{2.13}
l_{x\to y}:=l_{y}^{-1}\circ l_x  \colon\proj^{-1}(x)\to\proj^{-1}(y) .
	\end{equation}

\newslide

	Let $\mor{A} \in \Morf\HilB$.
The \emph{Hermitian conjugate}
bundle morphism $\mor{A}^\ddag$ to $\mor{A}$ is defined by
%	\begin{equation}	\label{2.15}
\(
\langle \mor{A}^\ddag\mor{\Phi}_x | \mor{\Psi}_x \rangle_x :=
\langle \mor{\Phi}_x | \mor{A}\mor{\Psi}_x \rangle_x,
\qquad \mor{\Phi}_x,\mor{\Psi}_x\in \bHil_x.
\)
%	\end{equation}
Thus
	\begin{equation}	\label{2.16}
\mor{A}_{x}^{\ddag} := \mor{A}^\ddag\big|_{\bHil_x} =
l_{x}^{-1}\circ\left(l_x\circ \mor{A}_x\circ l_{x}^{-1}\right)^\dag\circ l_x.
	\end{equation}
  %  \pause
    	A bundle morphism $\mor{A}$ is
\emph{Hermitian}
if \(\mor{A}_{x}^{\ddag}=\mor{A}_x\), i.e.\  if
	\begin{equation}	\label{2.17}
\mor{A}^\ddag=\mor{A},
	\end{equation}
and it is called
\emph{unitary}
if \(\mor{A}_{x}^{\ddag}=\mor{A}_x^{-1}\), i.e.\  if
	\begin{equation}	\label{2.18}
\mor{A}^\ddag=\mor{A}^{-1}.
	\end{equation}
which is equivalent to
 %	\begin{equation}	\label{2.19}
\(
\langle\mor{A}\,\cdot\, | \mor{A}\,\cdot\,\rangle_x = \langle\,\cdot\, | \,\cdot\,\rangle_x
   \colon\bHil_x\times\bHil_x\to\mathbb{C}.
%	\end{equation}
\)
Hence the unitary morphisms are fibre-metric compatible, \ie they are
\emph{isometric} .



%           END OF subsection
%<<<<<-----------------------------------------------------------------------
\end{slide}


\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection


\section{The bundle transport}
	\label{Subsect2.2}



	Let $(E,\pi,B,\mathcal{E})$ be a \field-vector  $C^1$ bundle,
$\field=\field[R],\field[C]$.
	\begin{Defn}	\label{Defn2.2}
The \emph{bundle transport} in  $(E,\pi,B,\mathcal{E})$ is a mapping
$l\colon(x,y)\mapsto l_{x\to y}$, $x,y\in B$, where the mapping
	\begin{equation}	\label{2.20}
l_{x\to y}\colon \pi^{-1}(x)\to\pi^{-1}(y),
	\end{equation}
called \emph{(bundle) transport from $x$ to} $y$, is defined by
	\begin{equation}	\label{2.21}
l_{x\to y} := l_y^{-1} \circ l_x
	\end{equation}
with  $l_x\colon\pi^{-1}(x)\to\mathcal{E}$, $x\in B$, being the
point-trivializing isomorphisms of $(E,\pi,B,\mathcal{E})$.
	\end{Defn}

\pause

	Some of the properties of the bundle transport:
%=============================
\newslide
%===========================
	\begin{alignat}{2}				\label{2.22}
l_{x\to y}\circ l_{z\to x} &=
			l_{z\to y},&\  x,y,z&\in B,
\\							\label{2.23}
l_{x\to x} &= \id_{\pi^{-1}(x)}, & x&\in B,
\\							\label{2.24}
l_{x\to y}(\lambda u + \mu v)
  &= \lambda l_{x\to y}u + \mu l_{x\to y}v,
	& \lambda,\mu &\in \field,\  u,v\in{\pi^{-1}(x)},
\\							\label{2.25}
\left(l_{x\to y}\right)^{-1} &= l_{y\to x}.
	& x,y&\in B.
	\end{alignat}


	The bundle transport is Hermitian and unitary in a sense that such are
 $l_{x\to y}$, $x,y\in B$.


%==============
\newslide
%====================

A section $\mor{X}\in\Sec(E,\pi,B,\mathcal{E})$ is
$l$-\emph{transported} if for some (and hence any) $x\in
B$ and every $y\in B$ is fulfilled
       \begin{equation}		\label{2.26}
\mor{X}(y) = l_{x\to y} ( \mor{X}(x) )
\text{ with } \mor{X}(x):=\mor{X}_x ,\
\mor{X}\colon x\mapsto \mor{X}_x .
	\end{equation}
Such a section is uniquely defined by specifying its value at a
single point.  If $\ope{X}_0$ is a fixed  vector in the fibre
$\mathcal{E}$, the section $\mor{X}$ given via
	\begin{equation}	\label{2.27}
\mor{X}(x) = l_x^{-1} (\ope{X}_0)
	\end{equation}
is $l$-transported. Sections of this kind will represent the states of quantum
fields in our approach.

%========================
\newslide
%=========================

	Let $(U,\kappa)$, be a
chart in a neighborhood $U$ of $x\in B$ and $x(\varepsilon,\mu)\in U$,
$\varepsilon \in \field[R]$ and
$\mu=1,\dots,\dim B$, is such that
$\kappa^\nu(x(\varepsilon,\mu))=\kappa^\nu(x)+\varepsilon\delta_\mu^\nu$ with
 $\delta_\mu^\nu=1$ for
$\nu=\mu$ and $\delta_\mu^\nu=0$ for $\nu\not=\mu$.
The bundle transport generates derivations
	\begin{gather}	\label{2.28}  \nonumber
\mor{D}_\mu\colon \Sec^1(E,\pi,B) \to \Sec^0(E,\pi,B)  ,
\ \mu=1,\dots,\dim B
%	\end{equation}
\\           \nonumber
 %	\begin{equation}	\label{2.29}
(\mor{D}_\mu \mor{Y})(x)
:=
\mor{D}_\mu\big|_x (\mor{Y})
:=
\lim_{\varepsilon\to 0}
\frac {l_{x(\varepsilon,\mu)\to x}
\bigl(\mor{Y}(x(\varepsilon,\mu))\bigr) - \mor{Y}(x)}
      {\varepsilon} .
	\end{gather}
	The bundle transport also generates the derivations
 \begin{multline}\nonumber   %	\label{2.30}
\Hat{\mor{D}}_\mu\colon
\bigl\{
\hmor{A}\in\MorfSec^1(E,\pi,B) \text{ is of class } C^1 \text{ and }
%		\\
		\ \hmor{A}(\,\cdot\,)=\mor{A}\circ(\,\cdot\,)
        	\text{ with\nonumber } \mor{A}\in\Morf_B^1(E,\pi,B)
\bigr\}
\\
\to
\bigl\{
\hmor{A}\in\MorfSec^0(E,\pi,B) \text{ is of class } C^0 \text{ and }
%		\\
		\hmor{A}(\,\cdot\,)=\mor{A}\circ(\,\cdot\,)
		\text{ with } \mor{A}\in\Morf_B^0(E,\pi,B)
\bigr\} ,
%	\end{multline}
 \\
   %	\begin{equation} \label{2.31}  \nonumber
\Hat{\mor{D}}_\mu (\hmor{A}) := [\mor{D}_\mu,\hmor{A}]_{\_}
=
\mor{D}_\mu\circ \hmor{A} - \hmor{A}\circ \mor{D}_\mu
	\end{multline}
  $\hmor{A} \in \MorfSec^1(E,\pi,B)$ generated by
$\mor{A}\in\Morf_B^1(E,\pi,B)$.

%==========
\newslide
%=============

	 $\mor{D}_\mu$ and $\Hat{\mor{D}}_\mu$ are \emph{linear} and
 ($f$ is a $C^1$ function on $B$)
	\begin{align}					\label{2.32}
\mor{D}_\mu(f\mor{Y}) &= \frac{\pd f}{\pd x^\mu} \mor{Y}
+ f \mor{D}_\mu(\mor{Y}),
\\							\label{2.33}
\Hat{\mor{D}}_\mu(\hmor{A}\circ \hmor{C})
&=
(\Hat{\mor{D}}_\mu(\hmor{A}))\circ \hmor{C}
	+\hmor{A}\circ(\Hat{\mor{D}}_\mu(\hmor{C})) .
	\end{align}

\pause

  \begin{Prop}	\label{Prop2.1}
If $\mor{Y}$ is a $C^1$ section, $\mor{A}$ a $C^1$ morphism of $(E,\pi,B)$,
and $\hmor{A}(\,\cdot\,)=\mor{A}\circ(\,\cdot\,)$ is the morphisms of $\Sec(E,\pi,B)$
generated by $\mor{A}$, then
	\begin{align}					\label{2.34}
(\mor{D}_\mu \mor{Y})(x)
    & = l_x^{-1}\Bigl( \frac{\pd (l_x(\mor{Y}(x)))}{\pd x^\mu} \Bigr)
      =\bigl( l_x^{-1}\circ\frac{\pd}{\pd x^\mu}\circ l_x \bigr) (\mor{Y}(x)) ,
\\							\label{2.35}
\bigl( (\Hat{\mor{D}}_\mu \hmor{A})(\mor{Y}) \bigr) (x)
	&= \Big( l_x^{-1}\circ
		\frac{\pd (l_x\circ \mor{A}_x \circ l_x^{-1})}{\pd x^\mu}
	   \circ l_x \Bigr) (\mor{Y}(x)).
	\end{align}
	\end{Prop}

%======================
\newslide
%======================

	If $\{x^\mu\}\mapsto\{x^{\prime\,\mu}\}$ is a coordinate changes, then
$\mor{D}_\mu$ and $\Hat{\mor{D}}_\mu$ behave like basic (tangent) vector
fields over $B$:
	\begin{equation}	\label{2.38}
\mor{D}_\mu\mapsto \mor{D}'_\mu
= \frac{\pd x^\nu}{\pd x^{\prime\mu}} \mor{D}_\nu,
\quad
\Hat{\mor{D}}_\mu\mapsto \Hat{\mor{D}}'_\mu
= \frac{\pd x^\nu}{\pd x^{\prime\mu}} \Hat{\mor{D}}_\nu .
	\end{equation}
`Covariant derivatives'  along a vector field
$V=V^\mu\frac{\pd}{\pd x^\mu}$ tangent to $B$: $\mor{D}_V:=V^\mu
\mor{D}_\mu$ and $\Hat{\mor{D}}_V:=V^\mu\Hat{\mor{D}}_\mu$.
\pause\vspace*{-1.0ex}
	The  transport $l$ induces a `\emph{transport along the
identity mapping of} $B$' in the bundle of point\ndash restricted
morphisms $\morf_B(E,\pi,B)=(E_0,\pi_0,B)$. This is a map
$\lcl\colon(x,y)\mapsto \lcl_{x\to y}$, $x,y\in B$, with
\(
\lcl_{x\to y} \colon\pi_0^{-1}(x)\to\pi_0^{-1}(y) ,
\)
(\emph{transport from  $x$ to $y$ in} $\morf_B(E,\pi,B)$)  defined by
	\begin{equation}	\label{2.39} \nonumber
\lcl_{x\to y} (\chi_x)
:=
l_{x\to y}\circ\chi_x \circ l_{y\to x}
=
l_y^{-1}\circ(l_x\circ\chi_x \circ l_x^{-1})\circ l_y
\colon\pi^{-1}(y)\to\pi^{-1}(y)
	\end{equation}
for $\chi_x\colon\pi^{-1}(x)\to\pi^{-1}(x)$.

%======================
\newslide
%====================

 The transports $\lcl_{x\to y}$ are
\field\ndash \emph{linear} and satisfy:
	\begin{alignat}{2}				\label{2.40}
\lcl_{x\to y}\circ \lcl_{z\to x} &=
			\lcl_{z\to y},&\qquad  x,y,z&\in B,
\\							\label{2.41}
\lcl_{x\to x} &= \id_{ \{ \pi^{-1}(x)\to\pi^{-1}(x) \} }, & x&\in B,
\\							\label{2.42}
\bigl( \lcl_{x\to y} \bigr)^{-1} &= \lcl_{y\to x},
	& x,y&\in B.
	\end{alignat}
$\lcl$ is the \emph{transport associated to the bundle transport}
$l$.% (in $\morf_B(E,\pi,B)$).
\\
	A morphism $\mor{A}\in\Morf_B(E,\pi,B)$ is
$\lcl$-transported if the restrictions
$\mor{A}_x:=\mor{A}|_{\pi^{-1}(x)}$ satisfy the equation
	\begin{equation}	\label{2.43}
\mor{A}_y
=\lcl_{x\to y} (\mor{A}_x)
=l_{x\to y} \circ \mor{A}_x\circ l_{y\to x},
\qquad x,y\in B.
	\end{equation}
For fixed $x\in B$ and
$\chi_0\colon\pi^{-1}(x)\to\pi^{-1}(x)$,
% $\lcl_x(\chi_0)\in\Morf_B(E,\pi,B)$ such that
	\begin{equation}	\label{2.44}
\lcl_x(\chi_0)|_{\pi^{-1}(y)}
:= \lcl_{x\to y}(\chi_0)
= l_{x\to y} \circ \chi_0 \circ l_{y\to x}
	\end{equation}
is $\lcl$-transported $B$-morphism.

%========================
\newslide
%======================

 For fixed $x$, any
morphism  $\mor{A}\in\Morf_B(E,\pi,B)$ defines the $\lcl$-transported morphism
	\begin{equation}	\label{2.45} \nonumber
\lcl_x(\mor{A})
:= \lcl_x(\mor{A}_x) \colon y\mapsto (\lcl_x\mor{A})_y
:=\lcl_{x\to y}\mor{A}_x
=l_{x\to y}\circ \mor{A}_x\circ l_{y\to x}.
	\end{equation}
	The transport
$\lcl$ induces derivations $\lcD_\mu$ on the set
$\Sec^1(\morf_B(E,\pi,B))$ via
	\begin{equation}	\label{2.46}
(\lcD_\mu \mor{A})(x)
:=
\lim_{\varepsilon\to 0}
\frac {\lcl_{x(\varepsilon,\mu)\to x}
	\bigl(\mor{A}_{x(\varepsilon,\mu)}\bigr) - \mor{A}_x}
      {\varepsilon}
	\end{equation}
with $x(\varepsilon,\mu)$ defined above and
$\mor{A}\in\Sec^1(\morf_B(E,\pi,B))$.
We have
	\begin{equation}	\label{2.47}
(\lcD_\mu \mor{A})(x)
	= l_x^{-1}\circ
	\frac{\pd (l_x\circ \mor{A}(x)\circ l_x^{-1})}{\pd x^\mu}
	   \circ l_x.
	\end{equation}


%========================
\newslide
%====================

	Now we shall present some expressions in (local) bases which may be a
little more familiar to the physicists.

\pause

	Let $\{e_i(x)\}$ be a basis in $\pi^{-1}(x)$, $\{f_i\}$ a basis  in
$\mathcal{F}$ with $\pd_\mu f_i = 0$, and  $\Mat{l}_x$ and $\Mat{l}(y,x)$ be
the matrices of respectively $l_x$ and $l_{x\to y}$ in them, \ie if
 $l_x(e_i(x)) =: \Sprindex[l_x]{i}{j} f_j$
and
 $l_{x\to y}(e_i(x)) =: \Sprindex[l]{i}{j}(y,x) e_i(y)$.
Then
 $\Mat{l}_x:=[\Sprindex[l_x]{i}{j}]$
and
 $\Mat{l}(y,x):=[\Sprindex[l]{i}{j}(y,x)] = \Mat{l}^{-1}(y) \Mat{l}(x)$
and, for $\mor{Y}=\mor{Y}^i e_i$,
	\begin{gather}	\label{2.b1}
(\mor{D}_\mu\mor{Y})(x)
=
\Bigl(
\frac{\pd\mor{Y}^i(x)}{\pd x^\mu} +
	\Sprindex[\Gamma]{j\mu}{i}(x) \mor{Y}^j(x)
\Bigr) e_i(x)
%	\end{equation}
 \\
%	\begin{equation}	\label{2.b2}
(\mor{D}_\mu e_j)(x) =: \Sprindex[\Gamma]{j\mu}{i}(x) e_i(x)
 %	\end{gather}
  %	\begin{equation}
\\ \label{2.b3}
\Gamma_{\mu}(x)
:=
[ \Sprindex[\Gamma]{j\mu}{i}(x) ]
=
\frac{\pd\Mat{l}(y,x)}{\pd x^\mu}\Big|_{y=x}
=
\Mat{l}_x^{-1}
\frac{\pd\Mat{l}(x)}{\pd x^\mu}
	\end{gather}
$\Sprindex[\Gamma]{j\mu}{i}$ are the \emph{coefficients}
(\emph{components}) of $l$ (resp.\ $\mor{D}_\mu$).

%===================
\newslide
%==================
	Under the changes
 $e_i\mapsto e'_i=C_{i}^{j}e_j$ and $x^\mu\mapsto x^{\prime\,\mu}$,
$C=[C_{i}^{j}]$ being non\ndash degenerate matrix\ndash valued  $C^1$
function, the matrices $\Gamma_\mu$ transform
into
	\begin{equation}	\label{2.b4}
\Gamma'_{\mu}(x)
:=
\Bigl(
	C^{-1}(x) \Gamma_\nu(x) C(x)
      + C^{-1}(x)\frac{\pd C(x)}{\pd x^\nu}
\Bigr)
\frac{\pd x^\nu}{\pd x^{\prime\,\mu}}.
	\end{equation}

	For a $C^1$ section $\mor{A}$, we find the matrix of
$\Hat{\mor{D}}_\mu\hmor{A}$ in $\{e_i\}$ as
	\begin{equation}	\label{2.b5}
[ (\Hat{\mor{D}}_\mu\hmor{A})_{i}^{j}]|_x
=
\Bigl(
\frac{\pd\Mat{\mor{A}}}{\pd x^\mu}
+ [\Gamma_{\mu}(x),\Mat{\mor{A}}]_{\_}
\Bigr)\Big|_{x}
=
\Mat{l}_{x}^{-1} \frac{\pd\Mat{\ope{A}}(x)}{\pd x^\mu} \Mat{l}_x
	\end{equation}
with $\Mat{\mor{A}}$ being the matrix of $\mor{A}$ in $\{e_i\}$.

%			END OF SECTION~\ref{Sect2}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<

\end{slide}



\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection



\end{slide}



\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection
%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>

\section[On bundle formulation of QFT]
{On bundle formulation of quantum field theory}
\label{Sect3}
%			BEGINNING OF SECTION~\ref{Sect3}


	A quantum field $\bs\varphi$ is an operator-valued vector distribution
, $\bs\varphi=(\bs\varphi_1,\dots,\bs\varphi_n)$ with
$n\in\field[N]$ and $\bs\varphi_i$, $i=1,\dots,n$, being operator\ndash valued
distributions called components of $\bs\varphi$, such that
	\begin{multline}	\label{3.3}
\bs\varphi(f)
=
\int_\base \Id^4y\sum_{i}^{} \varphi_i(y) f^i(y)
=
\sum_i \bs\varphi_i(f^i),
\\
\bs\varphi_i(g)
:=
\int_{\base} \Id^4y \varphi_i(y) g(y)
	\end{multline}
for a vector test function $f=(f^1,\dots,f^n)$,
 $\varphi_i$ -- the non-smeared fields acting in $\ope{F}$  and
$f^i,g\colon\base\to\field$ -- test functions.
We adopt the Heisenberg picture
of motion.

%==================
\newslide
%===================

In the  bundle description the system's Hilbert space $\ope{F}$ of states is
replaced with a Hilbert fibre bundle $\HilB$ (of states) with Minkowski
spacetime $\base$ as a base, projection $\proj\colon\bHil\to\base$, fibres
$\proj^{-1}(x)$ with $x\in\base$, and the system's ordinary Hilbert space
$\ope{F}$ as a (standard, typical) fibre. Let $\{l_x : x\in\base\}$ be a set of
(linear) isomorphisms $l_x\colon\bHil_x\to\Hil$. Given an observer $O_x$ at
$x\in\base$, if a system is characterized by a state vector
$\ope{X}\in\ope{F}$, the vector
	\begin{equation}	\label{3.4}
\mor{X}(x) := l_x^{-1}(\ope{X})
	\end{equation}
should be considered as its state vector relative to $O_x$. Hereof the state
vector $\ope{X}$ is replace with a state section $\mor{X}\in\Sec\HilB$ such
that $\mor{X}\colon x\mapsto\mor{X}(x)=l_x^{-1}(\ope{X})$.

%====================
\newslide
%=================

 At a point
$x\in\base$, the bundle analogue $\mor{A}_x$ of an operator
$\ope{A}(x)\colon\ope{F}\to\ope{F}$ can be defined by requiring the transition
$\ope{A}(x)\mapsto\mor{A}_x$ to preserve the scalar products:
	\begin{equation}	\label{3.5}
\clangle \ope{X}|\ope{A}(x)(\ope{Y}) \crangle
=
\langle \mor{X}(x)|\mor{A}_x(\mor{Y}(x)) \rangle _x,
	\end{equation}
where $\ope{X},\ope{Y}\in\ope{F}$, $x\in\base$,
$\mor{X}(x)=l_x^{-1}(\ope{X})$,
$\mor{Y}(x)=l_x^{-1}(\ope{Y})$,
and $\clangle \,\cdot\,|\,\cdot\, \crangle$ and $\langle \,\cdot\,|\,\cdot\, \rangle_x$
are the scalar products in the Hilbert spaces $\ope{F}$ and $\proj^{-1}(x)$,
 which are connected by
\begin{gather}	\label{3.6}
\langle \,\cdot\,|\,\cdot\, \rangle _x=
\clangle l_{x}\,\cdot\,|l_{x}\,\cdot\, \crangle ,
%	\end{equation}
% or
%	\begin{equation}
       %	\tag{\ref{3.6}$^\prime$}  %	\label{3.6'}
\qquad
\clangle \,\cdot\,|\,\cdot\, \crangle =
\langle l_{x}^{-1}\,\cdot\,|l_{x}^{-1}\,\cdot\, \rangle _x .
	\end{gather}
If $\ope{A}(x)$ represents a dynamical
variable $\dyn{A}$, the observed (mean)
value of $\dyn{A}$ is independent of the way we compute (or describe) it
and
	\begin{equation}	\label{3.7}
\mor{A}_x = l_x^{-1}\circ\ope{A}(x)\circ l_x .
	\end{equation}

%========================
\newslide
%======================

	In this way, we see that to an \emph{operator}
$\ope{A}(x)\colon\ope{F}\to\ope{F}$ there corresponds a \emph{morphism}
	\begin{equation}	\label{3.8}
\mor{A}\in\Morf_{\base}\HilB
	\end{equation}
of the system's Hilbert bundle with $\mor{A}|_{\proj^{-1}(x)}=\mor{A}_x$.


%======================
\newslide
%=========================

       In particular, the non-smeared components
$\varphi_i$ of a quantum field $\bs\varphi$ change to
$\mor{\Phi}_i\in\Morf_{\base}\HilB$ with
	\begin{equation}	\label{3.9}
\mor{\Phi}_i|_{\proj^{-1}(x)} := l_x^{-1}\circ\varphi_i(x)\circ l_x.
	\end{equation}
To the smeared fields $\bs\varphi_i$ should correspond
	\begin{equation}	\label{3.10}
\bs{\mor{\Phi}}_i\colon x\mapsto \bs{\mor{\Phi}}_i|_x
:=
l_x^{-1}\circ\bs\varphi_i(\,\cdot\,)\circ l_x.
	\end{equation}
so that the smeared field $\bs{\mor{\Phi}}$ becomes a mapping
	\begin{equation}	\label{3.11}
\bs{\mor{\Phi}} \colon x\mapsto \bs{\mor{\Phi}}|_x
=
l_x^{-1}\circ\bs\varphi(\,\cdot\,)\circ l_x,
	\end{equation}
%so that
	\begin{equation}	\label{3.12}
\bs{\mor{\Phi}}_x(f)
=
\int_\base\Id^4y \sum_{i}^{}
	\lcl_{y\to x} (\mor{\Phi}_i(y))f^i(y)  ,
	\end{equation}
the transport $\lcl$
in $\morf_{\base}\HilB$ is defined via~\eref{2.39}.

%=======================
\newslide
%====================

	The morphisms, like $\mor{A}$, are not the
exact objects we need. Since in
the ordinary theory the operators like, $\ope{A}$, act on state vectors,
like $\ope{X}$, we should expect the bundle analogue $\hmor{A}$ of $\ope{A}$
to act on a state section $\mor{X}$ producing again a section
$\hmor{A}(\mor{X})\in\Sec\HilB$. As in the module
$\Sec\HilB$ there is a natural scalar product
$\langle \,\cdot\, |\,\cdot\, \rangle$
with values in the $\field[C]$\ndash valued functions , the r.h.s.\
of~\eref{3.5} should be identified with the value at $x\in\base$ of the scalar
product $\langle\mor{X}|\hmor{A}(\mor{Y}) \rangle$.  Combining these ideas, we
conclude that
	\begin{equation}	\label{3.13}
\hmor{A}(\mor{X}) = \mor{A}\circ \mor{X} .
	\end{equation}
Hereof, the $\hmor{A}$ is the morphism in $\MorfSec\HilB$ generated by
 $\mor{A}\in\Morf_{\base}\HilB$ whose restriction on $\proj^{-1}(x)$
is %~\eref{3.7}.
\(
\mor{A}_x = l_x^{-1}\circ\ope{A}(x)\circ l_x .
\)

%=============
\newslide
%++++==============

	Assuming $\hmor{A}$ to be the `right' bundle analogue of $\ope{A}$, we
conclude that
	\begin{equation}	\label{3.14}
\hmor{A}(\mor{X})\colon x\mapsto \hmor{A}(\mor{X}) |_x
= (\mor{A}\circ\mor{X})(x)
= \mor{A}_x(\mor{X}(x))
= l_x^{-1}( \ope{A}(x)(\ope{X}) ) ,
	\end{equation}
\ie the image of $\ope{A}(x)(\ope{X})$ according to~\eref{3.3} is exactly the
value at $x$ of $\hmor{A}(\mor{X})$ and
	\begin{equation}	\label{3.15}
\clangle \ope{X}|\ope{A}(x)(\ope{Y}) \crangle
=
\langle \mor{X}|\hmor{A}(\mor{Y}) \rangle|_x
	\end{equation}
which expresses the invariance of the scalar products when we replace state
vectors with state sections. Since the scalar products define the observed
(expectation, mean) values of physically observable quantities, the
last equality expresses the independence of these values of the way we
calculate them.


%			END OF SECTION~\ref{Sect3}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<




\end{slide}





\begin{slide}
\pageTransitionDissolve% Dissolve

%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
\section {Conclusion}
\label{Conclusion}
%			BEGINNING OF SECTION~\ref{Conclusion}


	The present investigation is a continuation of the
fibre bundle formulation of quantum physics begun
in~[1-5].
Here we have applied a slightly different approach to the
quantum field theory. The basic idea is the
Hilbert space of states to be replaced with a Hilbert bundle with it as a
typical fibre, then all quantities of the ordinary quantum field theory are
mapped into their bundle analogues by means of the bundle transport, which
is an internal object of any particular  bundle, or other
mappings build from it.

\pause

	The realization of such a procedure is not unique. The main
reason being that the quantum field theory has not a unique generally
accepted formulation.


%			END OF SECTION~\ref{Conclusion}
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<
\end{slide}


\begin{slide}
% \pageTransitionSplitHI% Split Horizontally Inside
\pageTransitionGlitter{315}

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%-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->-->>
%			BEGINNING OF BIBLIOGRAPHY
% \addcontentsline{toc}{section}{References}
 \bibliography{bozhopub,bozhoref}
 \bibliographystyle{unsrt}
% \addcontentsline{toc}{subsubsection}{This article ends at page}
%\addtocontents{toc}{}
%			END OF BIBLIOGRAPHY
%<<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<--<

\end{slide}

\pagetransitionDissolve

    \begin{slide}
The present report is available on the Internet as:\\
\begin{itemize}
\item
\LaTeX2e file at the following URL:\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.tex}
     {http://theo.inrne.bas.bg/$\sim$bozho/}\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.tex}
     {Publications$\_$lists/Publication$\_$Files/}\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.tex}
     {/bqft-1-math$\_$CSIVF-2010$\_$report.tex}

\item
Acrobat pdf file at the following URL:\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.pdf}
     {http://theo.inrne.bas.bg/$\sim$bozho/}\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.pdf}
     {Publications$\_$lists/Publication$\_$Files/}\\
\href{http://theo.inrne.bas.bg/~bozho/Publications_lists/Publication_Files/bqft-1-math_CSIVF-2010_report.pdf}
     {/bqft-1-math$\_$CSIVF-2010$\_$report.pdf}
\end{itemize}
    \end{slide}





\end{document}


% arXiv HAT
\\
Title:  A mathematical base for Fibre bundle formulation of Lagrangian Quantum Field Theory
Authors: Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear
	Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria)
Comments: 12 LaTeX pages. The packages AMS-LaTeX and amsfonts are required.
	For other papers on the same topic, view
	http://theo.inrne.bas.bg/~bozho/
Report-No:
MSC-class: 53C80, 53C99, 53Z05, 55R99, 91T99, 81Q70
PACS 02.40.Ma, 02.90.+p (Primary), 03.70.+k, 11.10.-z (Secondary)
\\
      The paper contains a differential-geometric foundations for an attempt
to formulate Lagrangian (canonical) quantum field theory on fibre bundles. In
it the standard Hilbert space of quantum field theory is replace with a Hilbert
bundle; the former playing a role of a (typical) fibre of the letter one.
Suitable sections of that bundle replace the ordinary state vectors and the
operators on the system's Hilbert space are transformed into morphisms of the
same bundle. In particular, the field operators are mapped into corresponding
field morphisms.

\\

