%**	This is a pure LaTeX 2e file [article class]
%**	Filename:	Heisenberg-Relations_CSIVF2008_report.tex
%**	Last update:
%**     Final version:
%**	Title:		Heisenberg relations in the general case
%**	Purpose:	Derivation of the Heisenberg relations in general
%**			setting


%**	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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%% Copyright (C) 2003-2005 Hans Fredrik Nordhaug
%%
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%%
%% --------------------------------------------------------------
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%% Copyright (C) 2003-2005 Hans Fredrik Nordhaug
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%%
%% This program is distributed in the hope that it will be useful,
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%-----------------------------------------------------------------------------------------------------------------
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%
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%-----------------------------------------------------------------------------------------------------------------
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%% This is file `__TPcfg.tex',
%% generated with the docstrip utility.
%%
%% The original source files were:
%%
%% texpower-doc.dtx  (with options: `config')
%%
%% --------------------------------------------------------------
%% TeXPower bundle - dynamic online presentations with LaTeX
%% Copyright (C) 1999-2004 Stephan Lehmke
%% Copyright (C) 2003-2005 Hans Fredrik Nordhaug
%%
%% This program is free software; you can redistribute it and/or
%% modify it under the terms of the GNU General Public License
%% as published by the Free Software Foundation; either version 2
%% of the License, or (at your option) any later version.
%%
%% This program is distributed in the hope that it will be useful,
%% but WITHOUT ANY WARRANTY; without even the implied warranty of
%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
%% GNU General Public License for more details.
%% --------------------------------------------------------------
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%% given in the file `00readme.txt'.
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%-----------------------------------------------------------------------------------------------------------------
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% Beginning of the bibliography file Heisenberg-Relations_CSIVF_report.bbl
\begin{filecontents}{Heisenberg-Relations_CSIVF2008_report.bbl}
\begin{thebibliography}{1}

\small
\pageDuration{9}

\bibitem{Bjorken&Drell-2}
J.~D. Bjorken and S.~D. Drell.
\newblock {\em Relativistic quantum fields}, volume~2.
\newblock McGraw-Hill Book Company, New York, 1965.
\newblock Russian translation: Nauka, Moscow, 1978.

\bibitem{Roman-QFT}
Paul Roman.
\newblock {\em Introduction to quantum field theory}.
\newblock John Wiley \& Sons, Inc., New York-London-Sydney-Toronto, 1969.

\bibitem{Bjorken&Drell}
J.~D. Bjorken and S.~D. Drell.
\newblock {\em Relativistic quantum mechanics}, volume 1 and 2.
\newblock McGraw-Hill Book Company, New York, 1964, 1965.
\newblock Russian translation: Nauka, Moscow, 1978.

\bibitem{Bogolyubov&Shirkov}
N.~N. Bogolyubov and D.~V. Shirkov.
\newblock {\em Introduction to the theory of quantized fields}.
\newblock Nauka, Moscow, third edition, 1976.
\newblock In Russian. English translation: Wiley, New York, 1980.

\newslide

\bibitem{bp-MP-book}
Bozhidar~Z. Iliev.
\newblock {\em Langrangian Quantum Field theory in Momentum Picture. Free
  fields}.
\newblock Nova Science Publishers, New York, 2008.
\newblock 306 pages, Hardcover, ISBN-13: 978-1-60456-170-8, ISBN-10:
1-60456-170-X, EAN: 9781604561708.

\bibitem{Bogolyubov&et_al.-AxQFT}
N.~N. Bogolubov, A.~A. Logunov, and I.~T. Todorov.
\newblock {\em Introduction to axiomatic quantum field theory}.
\newblock W.~A. Benjamin, Inc., London, 1975.
\newblock Translation from Russian: Nauka, Moscow, 1969.

\bibitem{Bogolyubov&et_al.-QFT}
N.~N. Bogolubov, A.~A. Logunov, A.~I. Oksak, and I.~T. Todorov.
\newblock {\em General principles of quantum field theory}.
\newblock Nauka, Moscow, 1987.
\newblock In Russian. English translation: Kluwer Academic Publishers,
  Dordrecht, 1989.

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


% Beginning of the file bozhomac.sty
\begin{filecontents}{bozhomac.sty}
%***************************************************************************%
%**     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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% ======================== identification ===================================
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
\ProvidesPackage{bozhomac}[2008/5/12
			   BOZHIDAR Z. ILIEV's LaTeX2e style package]


% ========================  initial code ====================================

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% ---------> End of Package Loading part


% =================== declaration of options ===============================

% =================== execution of options =================================



% =================== main code ============================================


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

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\newtheorem{Lem}{Lemma}[section]     			% [ LEMMA ]

\newtheorem{Post}{Postulate}[section]			% [ PRINCIPLE ]

\newtheorem{Prin}{Principle}[section]			% [ PRINCIPLE ]

\newtheorem{Prop}{Proposition}[section]			% [ PROPOSITION ]

\newtheorem{Thm}{Theorem}[section]			% [ THEOREM ]
%	Versions
% to produce variants of the theorems-like ... numbering Redefine \theVar... !!!
% \theoremstyle{plain}	% plain, definition, or remark
\newtheorem{VarAxm}{Axiom}			% [ AXIOM ]
\newtheorem{VarCor}{Corollary}			% [ COROLLARY ]
\newtheorem{VarCrit}{Criterion} 		% [ Criterion ]
\newtheorem{VarLem}{Lemma}		     	% [ LEMMA ]
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\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}[2007/05/20 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}%
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\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:
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%\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]



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%
%
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% \usepackage{automata}% DFA & Mealy automata drawing
% \usepackage{tplists}% dynamic lists:
%	stepped, flipped, dimmed (vertions of itemize & enumerate)
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% Loads
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% and the packages
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{% No. We'll make do without.
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\ifthenelse{\boolean{psspecialsallowed}}{%
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\begin{slide}
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  \title
  {% Main Title
    Heisenberg relations in the general case\\
% \\
%     {% Subtitle
%       \normalfont
%       the subtitle
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    }


\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{9$^\text{th}$
		International Workshop on
		\\Complex Structures,
		\\Integrability and Vector Fields
		}
\end{center}
\vspace{-0.5ex}\normalfont
\highlightenhanced{
\href{http://theo.inrne.bas.bg/csvf/csivf2008/}{URL:
      http://theo.inrne.bas.bg/csvf/csivf2008/}
}
}

  \maketitle


%--------------------------------------
\newslide
%--------------------------------------
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  \minisec{Abstract}% some title on a slide

	The Heisenberg relations are derived in a quite general setting when
the field transformations are induced by three representations of a given group.
They are considered also in the fibre bundle approach. The results are
illustrated in a case of transformations induced by the Poincar\'e group.

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


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

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


The (global) origin of the Heisenberg relations is in the equations like
    \begin{equation}    \label{2.8-2}
\varphi'_i(r) = U\circ \varphi_i(r)\circ U^{-1} .
    \end{equation}
which connect the components $\varphi_i$ and $\varphi'_i$ of a quantum field
$\varphi$ with respect to two frames of reference. Here $U$ is an operator
acting on the state vectors of the quantum system considered and it is
expected that the transformed field operators  $\varphi'_i$ can be expressed
explicitly by means of $\varphi_i$ via some equations.

%--------------------------------------
\newslide
%--------------------------------------
If the
elements $U$ (of the representation) of the group are labeled by
$b=(b^1,\dots,b^s)\in\field^s$ for some $s\in\field[N]$, we may write $U(b)$
for $U$, and then the corresponding Heisenberg relations are obtained
from the above equation with $U(b)$ for $U$ by differentiating it with respect
to $b^\omega$, $\omega=1,\dots,s$, and then setting $b=b_0$, where
$b_0\in\field^s$ is such that $U(b_0)$ is the identity element.

\pause
	The Heisenberg relations are from pure
geometric origin and the only physics in them is the
motivation leading to the above equation. There are
evidences that to them can be given physical
sense by identifying in them the generators (of the representation)
of the group by the corresponding operators of conserved physical quantities.



\end{slide}

\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside

%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection
\section{The Poincar\'e group}
\label{Subsect6.1}

	Let us have a quantum field with components $\varphi_i$ relative
to two reference frames connected by a general Poincar\'e
transformation
    \begin{equation}    \label{2.3-2}
u '(x)= \Lambda u(x)+a .
    \end{equation}
{\pause\vspace*{-2ex}}%
The
``global' version of the Heisenberg relations is
	\begin{equation}	\label{6.1}
U(\Lambda,a)\circ\varphi_i(x)\circ U^{-1}(\Lambda,a)
=
{D}_i^j(\Lambda,a)\varphi_j(\Lambda x+a) ,
	\end{equation}
where $U$ ($D$) is a representation of the Poincar\'e group on the space
of state vectors (field operators),
 $U(\Lambda,a)$ ($\Mat{D}(\Lambda,a)=[{D}_i^j(\Lambda,a)]$) is the
mapping (the matrix of the mapping) corresponding via $U$ ($D$)
to~\eref{2.3-2}. We
have
\(%     \begin{equation}    \label{6.2}
U(\openone,\Mat{0}) = \id
\) and \(%\qquad
\Mat{D}(\openone,\Mat{0}) = \openone .
\)%     \end{equation}

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


Define
    \begin{subequations}    \label{6.3}
    \begin{align*}    \label{6.3a}
T_\mu
& := \frac{\pd U(\Lambda,a)}{\pd a^\mu}
\Big|_{(\Lambda,a)=(\openone,\Mat{0})}
\quad% \\		    \label{6.3b}
S_{\mu\nu}
& := \frac{\pd U(\Lambda,a)}{\pd \Lambda^{\mu\nu}}
\Big|_{(\Lambda,a)=(\openone,\Mat{0})}
\\		    %\label{6.3c}
H_{j\mu}^{i}
& := \frac{\pd {D}_j^i(\Lambda,a)}{\pd a^\mu}
\Big|_{(\Lambda,a)=(\openone,\Mat{0})}
\quad% \\		    \label{6.3d}
I_{j\mu\nu}^{i}
& := \frac{\pd {D}_j^i(\Lambda,a)}{\pd \Lambda^{\mu\nu}}
\Big|_{(\Lambda,a)=(\openone,\Mat{0})} .
    \end{align*}
    \end{subequations}

	Differentiating~\eref{6.1} relative to $a^\mu$ and setting after that
$(\Lambda,a)=(\openone,\Mat{0})$, we find
    \begin{equation}    \label{6.4}
[T_\mu,\varphi_i(x)]_{\_} = \pd_\mu\varphi_i(x) + H_{i\mu}^{j}\varphi_j(x) ,
    \end{equation}
where $[A,B]_{\_}:=AB-BA$.

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

	For field theories invariant
relative to spacetime translation of the coordinates, \ie
$\Mat{x}\mapsto\Mat{x}+a$, we can suppose that
    \begin{equation}    \label{6.6}
H_{j\mu}^{i} = 0.
    \end{equation}
In this case equation~\eref{6.4} reduces to
    \begin{subequations}    \label{6.7}
    \begin{equation}    \label{6.7a}
[T_\mu,\varphi_i(x)]_{\_} = \pd_\mu\varphi_i(x).
    \end{equation}
\pause
Similarly, differentiation~\eref{6.1} with respect to
$\Lambda^{\mu\nu}$ and putting after that $(\Lambda,a)=(\openone,\Mat{0})$, we
obtain
    \begin{equation}    \label{6.7b}
[S_{\mu\nu},\varphi_i(x)]_{\_}
= x_\mu \pd_\nu\varphi_i(x) - x_\nu \pd_\mu\varphi_i(x) +
I_{i\mu\nu}^{j}\varphi_j(x)   .
    \end{equation}
    \end{subequations}

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


	As we have mentioned earlier, the Heisenberg
relations are from pure geometrical origin.  Recalling that
the translation (resp.\ rotation) invariance of a (Lagrangian) field theory
results to conservation of system's momentum (resp.\ angular momentum) operator
$P_\mu$ (resp.\ $M_{\mu\nu}$) and the correspondences
    \begin{equation}    \label{6.9}
\ih T_{\mu} \mapsto P_{\mu}
\qquad
\ih S_{\mu\nu} \mapsto M_{\mu\nu} ,
    \end{equation}
with $\hbar$ being the Planck's constant, one may suppose
the validity of the Heisenberg relations
    \begin{subequations}    \label{6.10}
    \begin{align}    \label{6.10a}
[P_\mu,\varphi_i(x)]_{\_}
&= \ih\pd_\mu\varphi_i(x)
\\		    \label{6.10b}
[M_{\mu\nu},\varphi_i(x)]_{\_}
&= \ih \{ x_\mu \pd_\nu\varphi_i(x) - x_\nu \pd_\mu\varphi_i(x) +
I_{i\mu\nu}^{j}\varphi_j(x) \} .
    \end{align}
    \end{subequations}
%--------------------------------------
\newslide
%--------------------------------------

However, one should be careful when applying the last two equations  in the
Lagrangian formalism as they are external to it and need a particular proof in
this approach; \eg they hold in the free field
theory, but a general proof seems to be missing.
In the axiomatic quantum field
theory these
equations are identically valid as in it the generators of the translations
(rotations) are identified up to a constant factor with the components of the
(angular) momentum operator, $P_\mu=\ih T_\mu$ ($M_{\mu\nu}=\ih S_{\mu\nu}$).


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


\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection
\section{Internal transformations}
\label{Subsect6.2}


	An internal transformation is a change of the reference
frame $(u,\{e^i\})$, consisting of a local coordinate system $u$ and a frame
$\{e^i\}$ in some vector space $V$, such that the spacetime coordinates remain
unchanged. We suppose that $e^i \colon x\in M\mapsto e^i(x)\in V$, where $M$ is
the Minkowski spacetime and the quantum field $\varphi$ considered takes values
in $V$, \ie $\varphi \colon x\in M\mapsto \varphi(x)=\varphi_i(x)e^i(x)\in V$

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

	Let $G$ be a group whose elements $g_b$ are labeled by $b\in\field^s$
for some $s\in\field[N]$. Consider two reference frames $(u,\{e^i\})$ and $(u',\{e^{\prime\,i}\})$, with
$u'=u$ and $\{e^i\}$ and $\{e^{\prime\,i}\}$ being connected via a matrix
$I^{-1}(b)$, where $I \colon G\mapsto \GL(\dim V,\field)$ is a matrix
representation of $G$ and $I\colon G\ni g_b\mapsto I(b)\in \GL(\dim V,\field)$.
The components of the fields transform into
    \begin{equation}    \label{6.13}
\varphi'_{u,i}(r) = U(b)\circ\varphi_{u,i}(r)\circ U^{-1}(b)
    \end{equation}
where $U$ is a representation of $G$ on the Hilbert space of state vectors and
$U \colon G\ni g_b\mapsto U(b)$.

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

 So that
    \begin{equation}    \label{6.14}
U(b)\circ\varphi_{u,i}(r)\circ U^{-1}(b) = I_i^j(b) \varphi_{u,j} (r) .
    \end{equation}

	Let $g_{b_0}$ is the identity element of $G$ for some $b_0\in\field^s$
and for $b=(b^1,\dots,b^s)$ and $\omega=1,\dots,s$ define
    \begin{equation}    \label{6.14-1}
Q_\omega : = \frac{\pd U(b)}{\pd b^\omega} \Big|_{b=b_0}
\qquad
I_{i \omega}^j : = \frac{\pd I_i^j(b)}{\pd b^\omega} \Big|_{b=b_0} .
    \end{equation}
Differentiation~\eref{6.14} with respect to $b^\omega$ and putting in the
result $b=b_0$, we get the Heisenberg relation
    \begin{equation}    \label{6.15}
[ Q_\omega, \varphi_{u,i}(r) ]_{\_} = I_{i\omega}^j \varphi_{u,j}(r)
    \end{equation}
or, if we identify $x\in\base$ with $r=u(x)$ and omit the subscript $u,$
    \begin{equation}    \label{6.16}
[ Q_\omega, \varphi_{i}(x) ]_{\_} = I_{i\omega}^j \varphi_{j}(x) .
    \end{equation}


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


       \textbf{Example.} Consider
one\ndash dimensional group $G$, $s=1$, when $\omega=1$; so we
identify $b^1$ with $b=(b^1)$. Besides, let
    \begin{equation}    \label{6.17}
I(b) = \openone \exp(f(b)-f(b_0))
    \end{equation}
for some $C^1$ function $f$. Then~\eref{6.16} reduces to
    \begin{equation}    \label{6.18}
[ Q_1, \varphi_{i}(x) ]_{\_} = f'(b_0) \varphi_i(x) ,
    \end{equation}
where $f'(b):=\frac{\od f(b)}{\od b}$.

{\pause\vspace*{-1ex}}%
In particular, if we are dealing with
phase transformations, \ie
    \begin{equation}    \label{6.19}
U(b) = \e^{\frac{1}{\iu e} b Q_1}
\quad
I(b) = \openone \e^{-\frac{q}{\iu e} b}
\qquad
b\in\field[R]
    \end{equation}
for some constants $q$ and $e$ (having a meaning of charge and unit charge,
respectively) and operator $Q_1$ on system's Hilbert space of states (having a
meaning of a charge operator), then we arrive to the familiar equations
    \begin{align}    \label{6.20}
& \varphi'_i(x)
=
\e^{\frac{1}{\iu e} b Q_1}\circ \varphi_i(x) \circ \e^{-\frac{1}{\iu e} b Q_1}
=
\e^{-\frac{q}{\iu e} b} \varphi(x)
\\		    \label{6.21}
& [Q_1,\varphi_i(x)]_{\_} = -q \varphi_i(x) .
    \end{align}

{\pause\vspace*{-2ex}}%
	The considerations in the framework of Lagrangian formalism invariant
under  phase
transformations implies
conservation of the charge operator $Q$ and suggests the correspondence
    \begin{equation}    \label{6.22}
Q_1 \mapsto Q
    \end{equation}
which in turn suggests the Heisenberg relation
    \begin{equation}    \label{6.23}
[Q,\varphi_i(x)]_{\_} = - q \varphi_i(x) .
    \end{equation}


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



\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection
\section{The general case}
\label{Subsect6.3}


	The corner stone of the (global) Heisenberg relations is the equation
    \begin{equation}    \label{6.24}
 U\circ\varphi_{u,i}(r)\circ U^{-1}
=
\frac{\pd( u '\circ u ^{-1})(r)}{\pd r}
	\bigl( A^{-1}( u ^{-1}(r)) \bigr)_i^j
	\varphi_{u,j}( ( u '\circ u ^{-1})(r) )
    \end{equation}
representing the components $\varphi'_{u',i}$ of a quantum field $\varphi$ in a
reference frame $(u,\{e^{\prime\, i}=A_j^i e^j\})$ via its components
$\varphi_{u,j}$ in a frame $(u,\{e^i\})$ in two different way. Here $A=[A_i^j]$
is a non\ndash degenerate matrix\ndash valued function, $r\in\field[R]^4$ and
$\varphi_{u,i}:=\varphi_i\circ u^{-1}$.

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



	Let $G$ be  $s$-dimensional, $s\in\field[N]$, Lie group. Let its
elements be labeled by $b=(b^1,\dots,b^s)\in\field^s$ and $g_{b_0}$ is the
identity element of $G$ for some fixed $b_0\in\field^s$. Let there are given
three representations $H$, $I$ and $U$ of $G$ and:
    \begin{enumerate}
\item
\(
H\colon G\ni g_b\mapsto H_b
\colon \field[R]^{\dim \base}\to \field[R]^{\dim \base}
\) and any change $(U,u)\mapsto(U',u')$ of the charts of $M$ is such
that $u'\circ u^{-1}=H_b$ for some $b\in\field^s$.
\pause
\item
$I \colon G\ni g_b\mapsto I(b)\in\GL(\dim V,\field)$
and any change $\{e_i\}\mapsto\{e'_i=A_i^je_j\}$ of the frames in $V$ is such
that $A^{-1}(x)=I(b)$ for all $x\in\base$ and some $b\in\field^s$.
\pause
\item
 $U \colon G\ni g_b\mapsto U(b)$, where $U(b)$ is an operator on the space of
state vectors, and the changes $(u,\{e_i\})\mapsto(u',\{e'_i\})$ of the
reference frames entail~\eref{2.8-2} with $U(b)$ for $U$.
    \end{enumerate}

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



Under the above hypotheses we have
    \begin{equation}    \label{6.25}
U(b)\circ\varphi_{u,i}(r)\circ U^{-1}(b)
=
\det\Bigl[ \frac{\pd (H_b(r))^i}{\pd r^j} \Bigr]
I_i^j(b) \varphi_{u,j}(H_b(r))
    \end{equation}
which can be called global Heisenberg relation in the particular situation. The
next step is to differentiate this equation with respect to $b^\omega$,
$\omega=1,\dots,s$, and then to put  $b=b_0$ in the result. In this way we
obtain the following (local) Heisenberg relation
    \begin{equation}    \label{6.26}
[U_\omega,\varphi_{u,i}(r)]_{\_}
=
  \Delta_\omega(r)\varphi_{u,i}(r)
+ I_{i\omega}^j \varphi_{u,j}(r)
+ (h_{\omega(r)})^k \frac{\pd \varphi_{u,i}(r)}{\pd r^k} ,
    \end{equation}
where

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



    \begin{subequations}    \label{6.27}
    \begin{align*}    %\label{6.27a}
U_\omega
& := \frac{\pd U(b)}{\pd b^\omega} \Big|_{b=b_0}
\quad%\\		    %\label{6.27b}
\Delta_\omega(r)
 :=
\left.
\frac{\pd \det\Bigl[ \frac{\pd (H_b(r))^j}{\pd r^j} \Bigr] } {\pd b^\omega}
\right|_{b=b_0}
\in  \field[R]^{\dim \base}
\\		    %\label{6.27c}
I_{i\omega}^j
& :=
\frac{\pd I_i^j(b)}{\pd b^\omega} \Big|_{b=b_0} \in\field
\quad%\\		    %\label{6.27d}
h_\omega
 :=
\frac{\pd H_b}{\pd b^\omega} \Big|_{b=b_0}
 \colon \field[R]^{\dim \base}\to \field[R]^{\dim \base} .
    \end{align*}
    \end{subequations}
{\pause\vspace*{-1ex}}%
In this setting the Heisenberg relations corresponding to
Poincar\'e transformations are described by
$b\mapsto(\Lambda^{\mu\nu},a^\lambda)$, $H(b)\mapsto\Lambda$, $a(b)\mapsto a$
and $I(b)\mapsto I(\Lambda)$, so that
 $U_\omega\mapsto(S_{\mu\nu,T_\lambda})$,
 $\Delta_\omega(r)\equiv 0$,
$I_{i\omega}^j\mapsto(I_{i \mu\nu}^j,0)$ and
\(
(h_\omega(r))^k\frac{\pd }{\pd r^k}
\mapsto
r_\mu\frac{\pd }{\pd r^\nu} - r_\nu\frac{\pd }{\pd r^\mu} .
\)
{\pause\vspace*{-1ex}}%
	The case of internal transformations, considered in the previous
subsection, corresponds to $H_b=\id_{\field[R]^{\dim\base}}$ and, consequently,
in it $\Delta_\omega(r)\equiv 0$ and $h_\omega=0$.
\end{slide}



\begin{slide}
\pageTransitionSplitHI% Split Horizontally Inside


%>>>>>-----------------------------------------------------------------------
%           BEGINNING OF subsection
\section{Fibre bundle approach}
\label{Subsect6.4}

	Let a physical field is described as a section
$\varphi \colon \base\to E$ of a vector bundle $(E,\pi,\base)$. Here $\base$ is
a real differentiable (4\ndash)manifold, serving as a
spacetime model, $E$ is the bundle space and $\pi \colon \base\to E$ is the
projection; the fibres $\pi^{-1}(x)$, $x\in\base$.

	Let $(U,u)$ be a chart of $\base$ and $\{e^i\}$ be a frame in
the bundle with domain $U$, \ie
 $e^i \colon x\mapsto e^i(x)\in\pi^{-1}(x)$ with $x$ in the domain of $\{e^i\}$
and $\{e^i(x)\}$  being a basis in $\pi^{-1}(x)$. Below we assume
$x\in U\subseteq \base$. Thus, we have
    \begin{equation}    \label{3.1}
\varphi \colon \base\ni x\mapsto \varphi(x)
=
\varphi_i(x) e^i(x)
=
\varphi_{u,i}(\Mat{x}) e^i(u^{-1}(\Mat{x})) ,
    \end{equation}
where
\(%    \begin{equation}    \label{3.2}
\Mat{x}:=u(x) \ %\qquad
\varphi_{u,i}:=\varphi_i\circ u^{-1} .
\)%    \end{equation}

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



	The origin of the Heisenberg relations on the background of fibre
bundle setting is in the equations
	\begin{align}	\label{6.29}
U\circ\varphi_i(x)\circ U^{-1}
&= (A^{-1})_i^j(x) \varphi_j(x)
\\			\tag{\ref{6.29}$'$}
			\label{6.29'}
U\circ\varphi_{u,i}(\Mat{x})\circ U^{-1}
&= (A^{-1})_i^j(x) \varphi_{u,j}(\Mat{x}) .
	\end{align}


	Consider a Lie group $G$, its
representations $I$ and $U$ and reference frames with the following properties:
    \begin{enumerate}
\pause
\item
$I \colon G\ni g_b\mapsto I(b)\in\GL(\dim V,\field)$
and the changes $\{e^i\}\mapsto\{e'^i=A^i_je^j\}$ of the frames in $V$ are such
that $A^{-1}(x)=I(b)$ for all $x\in\base$ and some $b\in\field^s$.
 \pause
\item
$U \colon g\ni g_b\mapsto U(b)$, $U(b)$ being operator on the space of
state, and the changes $(u,\{e^i\})\mapsto(u',\{e'^i\})$ of the
reference frames entail~\eref{2.8-2} with $U(b)$ for $U$.
    \end{enumerate}

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



Thus equations~\eref{6.29} and~\eref{6.29'} transform into
	\begin{align}	\label{6.30}
U(b)\circ\varphi_i(x)\circ U^{-1}(b)
&= I_i^j(b) \varphi_j(x)
\\			\tag{\ref{6.30}$'$}
			\label{6.30'}
U(b)\circ\varphi_{u,i}(\Mat{x})\circ U^{-1}(b)
&= I_i^j(b) \varphi_{u,j}(\Mat{x}) .
	\end{align}
Differentiating with respect to $b^\omega$ and putting
$b=b_0$, we get the Heisenberg relations
	\begin{align}	\label{6.31}
[U_\omega,\varphi_i(x)]_{\_}
&= I_{i\omega}^j \varphi_j(x)
\\			\tag{\ref{6.31}$'$}
			\label{6.31'}
[U_\omega,\varphi_{u,i}(\Mat{x})]_{\_}
&= I_{i\omega}^j \varphi_{u,j}(\Mat{x}) ,
	\end{align}
where
%     \begin{subequations}    \label{6.32}
%     \begin{align*}    \label{6.32a}
\(
U_\omega
 := \frac{\pd U(b)}{\pd b^\omega} \Big|_{b=b_0}
\quad%\\		    \label{6.32b}
I_{i\omega}^j
%&
 :=
\frac{\pd I_i^j(b)}{\pd b^\omega} \Big|_{b=b_0} .
\)
%     \end{align*}
%     \end{subequations}


We can rewire the Heisenberg relations obtained as
    \begin{equation}    \label{6.33}
[U_\omega,\varphi]_{\_} = I_{i\omega}^{j} \varphi_j e^i .
    \end{equation}

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



	The Poncar\'e transformations are described by
$b\mapsto(\Lambda^{\mu\nu},a^\lambda)$, $U_\omega\mapsto(S_{\mu\nu},T_\lambda)$
and $I_{i\omega}^{j}\mapsto(I_{i\mu\nu}^j,0)$ and, consequently, the
equations~\eref{6.30} and~\eref{6.30'} now read
	\begin{align}	\label{6.33-1}
U(\Lambda,a)\circ\varphi_i(x)\circ U^{-1}(\Lambda,a)
&= I_i^j(\Lambda,a) \varphi_j(x)
\\			\tag{\ref{6.33-1}$'$}
			\label{6.30-1'}
U(\Lambda,a)\circ\varphi_{u,i}(\Mat{x})\circ U^{-1}(\Lambda,a)
&= I_i^j(\Lambda,a) \varphi_{u,j}(\Mat{x}) .
	\end{align}
For instance, we have the Heisenberg relations
    \begin{subequations}    \label{6.34}
    \begin{align}    \label{6.34a}
[T_{\mu},\varphi_i(x)]_{\_} &= 0
\\		    \label{6.34b}
[S_{\mu\nu},\varphi_i(x)]_{\_} &= I_{i\mu\nu}^{j} \varphi_j(x) .
    \end{align}
    \end{subequations}


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

The "physical" versions of these relations are
\begin{subequations}    \label{6.35}
    \begin{align}    \label{6.35a}
[P_{\mu},\varphi_i(x)]_{\_} &= 0
\\		    \label{6.35b}
[M_{\mu\nu},\varphi_i(x)]_{\_} &= I_{i\mu\nu}^{j} \varphi_j(x) .
    \end{align}
    \end{subequations}


These relations contradict to the particle interpretation of quantum field
theory which may be retained if one accepts their non\ndash bundle versions.
	This is possible if the frames used are
connected by linear homogeneous transformations with spacetime constant
matrices, $A(x)=\const$ or $\pd_\mu A(x)=0$.


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

	Since the general case is coordinates independent, it describes also the
fibre bundle version of internal transformations. This explains why
some equations in the both versions are identical  up to the meaning
$\varphi_{u,i}$ and $I_{i\omega}^{j}$. In
particular, for
    \begin{equation}    \label{6.36}
U(b) = \e^{\frac{1}{\iu e} b Q_1}
\quad
I(b) = \openone \e^{-\frac{q}{\iu e} b}
\qquad
b\in\field[R]
    \end{equation}
the Heisenberg relations~\eref{6.31} reduce to
    \begin{equation}    \label{6.37}
[Q_1,\varphi_i(x)]_{\_} = -q \varphi_i(x),
    \end{equation}
which is identical with~\eref{6.21}, but now $\varphi_i$ are the components of
the section $\varphi$ in $\{e^i\}$. The invariant form of the last relations is
    \begin{equation}    \label{6.38}
[Q_1,\varphi]_{\_} = -q \varphi .
    \end{equation}





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





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

	We have shown how the Heisenberg equations arise in the
general case and in particular situations. They are from pure
geometrical origin and one should be careful when applying them to the
Lagrangian formalism in which they are subsidiary conditions, like the Lorentz
gauge in the electrodynamics. Generally they need not to be consistent
with the Lagrangian formalism and their validity should carefully be checked.
For instance, if one starts with field operators in the Lagrangian formalism of
free fields and adds to it the Heisenberg relations concerning the momentum
operator, then the arising scheme is not consistent as in it start to appear
distributions, like the Dirac delta function.



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


\begin{slide}
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 \bibliographystyle{unsrt}
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\end{slide}

\pagetransitionDissolve

    \begin{slide}
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