%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% example compuscript proc3.tex from editor proceedings macro package:
% http://www.worldscientific.com/style/ws-procs975x65_2e_master.zip
% svailable at
% http://www.worldscientific.com/style/proceedings_style.shtml
% ** modified for MG11 contributors **
% ** by bob jantzen 19-sep-2006 **
% since the current proceedings macros assume all contributions are to be
% joined into a single master latex document, requiring a TableOfContents
% entry and chaptebib.sty for treating the contributions as chapters in
% the document. They also forgot the standard cite.sty file to deal with
% their citations intelligently. This example contains no figure.
% One must consult the example file: ws-procs975x65.tex for this.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% For technical support please email: ykoh@wspc.com.sg (or) rajesh@wspc.com.sg
%% The content, structure, format and layout of this style file is the
%% property of World Scientific Publishing Co. Pte. Ltd.
%% Copyright 2005 by World Scientific Publishing Co.
%% All rights are reserved.
%%
%% Proceedings Trim Size: 9.75in x 6.5in
%% Text Area: 8in (include runningheads) x 5in
%% Main Text is 10/13pt
%% Last Modified: 7-7-05
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\documentclass[draft]{ws-procs975x65}
\documentclass{ws-procs975x65}
\begin{document}
%to switch ON running title
%\markboth{L. Hatcher}{Quantum States from Tangent Vectors}
%\wstoc{Quantum States from Tangent Vectors}{L. Hatcher}
\title{QUANTUM STATES FROM TANGENT
VECTORS\footnote{This research has been partially supported by
DGICYT grant PB/6/FS/97.}}
\author{LEE HATCHER\footnote{He is the author of
several articles and textbook chapters on sustainability in business
and technology.}}
\address{Instituto de F\'\i{s}ica Corpuscular (CSIC--UVEG),\\
Apartado de Correos 22085, %\\
Valencia 46071, Spain %\\
and\\
Max-Planck-Institut f\"ur Gravitationsphysik,
Albert-Einstein-Institut,\\
D-14476 Golm, Germany\\
\email{jmisidro@ific.uv.es}}
\author{ROBERT JANTZEN\footnote{\TeX\ amateur and occasional MG series
proceedings editor, who edited this sample for the contributors of MG11 2006.}}
\address{Dept of Mathematical Sciences, Villanova University, Villanova, PA 19087--4681 USA\\
\email{robert.jantzen@villanova.edu}}
% WARNING. in standard latex cls file formatting, at this point
% \maketitle would typeset the above titlepage information
% but WS has chosen to be nonstandard and have each line typeset
% as it is digested.
% no abstract is necessary.
% \bodymatter below resets the footnote counter and symbols after
% possible use in the title matter.
\begin{abstract}
We argue that tangent vectors to classical phase space give rise to
quantum states of the corresponding quantum mechanics. This is
established for the case of complex, finite-dimensional, compact,
classical phase spaces $\mathcal{C}$, by explicitly constructing
Hilbert-space vector bundles over $\mathcal{C}$. We find that these
vector bundles split as the direct sum of two holomorphic vector
bundles: the holomorphic tangent bundle $T(\mathcal{C})$, plus a
complex line bundle $N(\mathcal{C})$. Quantum states (except the
vacuum) appear as tangent vectors to $\mathcal{C}$. The vacuum state
appears as the fibrewise generator of $N(\mathcal{C})$. Holomorphic
line bundles $N(\mathcal{C})$ are classified by the elements of ${\rm
Pic}({\cal C})$, the Picard group of $\mathcal{C}$. In this way ${\rm
Pic}({\cal C})$ appears as the parameter space for nonequivalent
vacua. Our analysis is modelled on, but not limited to, the case when
$\mathcal{C}$ is complex projective space ${\bf CP}^n$.
\end{abstract}
\bodymatter
\section{Introduction}\label{intro}
Fibre bundles are powerful tools to formulate the gauge theories of
fundamental interactions and gravity.\cite{LIBAZCA} The question
arises whether or not quantum mechanics may also be formulated
fibre bundles.\footnote{The powerful tools of the gauge theories.}
Important physical motivations call for such a formulation.
In quantum mechanics one aims at constructing a Hilbert-space vector
bundle over classical phase space. In geometric quantisation this
goal is achieved in a two-step process that can be very succinctly
summarised as follows. One first constructs a certain holomorphic
line bundle (the {\it quantum line bundle}\/) over classical phase
space. Next one identifies certain sections of this line bundle as
defining the Hilbert space of quantum states. Alternatively one may
skip the quantum line bundle and consider the one-step process of
directly constructing a Hilbert-space vector bundle over classical
phase space. Associated with this vector bundle there is a principal
bundle whose fibre is the unitary group of Hilbert space.
Standard presentations of quantum mechanics usually deal with the case
when this Hilbert-space vector bundle is trivial. Such is the case,
e.g., when classical phase space is contractible to a point. However,
it seems natural to consider the case of a nontrivial bundle as well.
Beyond a purely mathematical interest, important physical issues that
go by the generic name of {\it dualities}\cite{VAFA} motivate the
study of nontrivial bundles.
Given a certain base manifold and a certain fibre, the trivial bundle
over the given base with the given fibre is unique. This may mislead
one to conclude that quantisation is also unique, or independent of
the observer on classical phase space. In fact the notion of duality
points precisely to the opposite conclusion, i.e.~to the
nonuniqueness of the quantisation procedure and to its dependence on
the observer.\cite{VAFA}
Clearly a framework is required in order to accommodate dualities
within quantum mechanics.\cite{VAFA} Nontrivial Hilbert-space vector
bundles over classical phase space provide one such framework. They
allow for the possibility of having different, nonequivalent
quantisations, as opposed to the uniqueness of the trivial
bundle.\footnote{The framework of the vector bundles.}
\enlargethispage*{6pt}
However, although nontriviality is a necessary condition, it is by no
means sufficient. A flat connection on a nontrivial bundle would still
allow, by parallel transport, to canonically identify the
Hilbert-space fibres above different points on classical phase
space. This identification would depend only on the homotopy class of
the curve joining the basepoints, but not on the curve itself. Now
flat connections are characterised by {\it constant}\/ transition
functions,\cite{KN} this constant being always the identity in the
case of the trivial bundle. Hence, in order to accommodate dualities,
we will be looking for {\it nonflat}\/ connections.
First, we want to obtain the
wave functions of a generalized pendulum under time-dependent
gravitation by making use of a unitary transformation and the LR
invariant method. As an example, we consider a generalized pendulum
with\break
exponentially increasing mass and constant gravitation. Second,
we want to present a canonical approach for the generalized
time-dependent pendulum which is based on the use of a time-dependent
canonical transformation and an auxiliary\break
transformation.
\section{Properties of ${\bf CP}^n$ as a Classical Phase
Space \label{cipienne} }
We will consider a classical mechanics whose phase space $\mathcal{C}$
is complex, projective $n$-dimensional space ${\bf CP}^n$. The
following properties are well known.\cite{KN}
Let $Z^1,\ldots,Z^{n+1}$ denote homogeneous coordinates on ${\bf
CP}^n$. The chart defined by $Z^k\neq 0$ covers one copy of the open
set $\mathcal{U}_k={\bf C}^n$. On the latter we have the holomorphic
coordinates $z^j_{(k)}=Z^j/Z^k$, $j\neq k$; there are $n+1$ such
coordinate charts. ${\bf CP}^n$ is a K\"ahler manifold with respect
to the Fubini-Study metric. On the~chart $(\mathcal{U}_k, z_{(k)})$
the K\"ahler potential reads
\begin{equation}
K(z^j_{(k)},\bar z^j_{(k)})
=\log\left(1+\sum_{j=1}^n z^j_{(k)}\bar z^j_{(k)}\right)\,.
\label{fubst}
\end{equation}
The singular homology ring $H_*({\bf CP}^n,\mathbb{Z})$ contains the
nonzero subgroups
\begin{equation}
H_{2k}({\bf CP}^n,\mathbb{Z})=\mathbb{Z}\,,\qquad k=0,1,\ldots,n\,,
\label{oncero}
\end{equation}
while
\begin{equation}
H_{2k+1}({\bf CP}^n,\mathbb{Z})=0\,,\qquad k=0,1,\ldots, n-1\,.
\label{oncerox}
\end{equation}
We have ${\bf CP}^{n}={\bf C}^n\cup {\bf CP}^{n-1}$, with ${\bf
CP}^{n-1}$ a hyperplane at infinity. Topologically, ${\bf CP}^{n}$ is
obtained by attaching a (real) $2n$-dimensional cell to ${\bf
CP}^{n-1}$. ${\bf CP}^n$ is simply connected,
\begin{equation}
\pi_1({\bf CP}^n)=0\,,
\label{grfund}
\end{equation}
it is compact, and inherits its complex structure from that on ${\bf
C}^{n+1}$. It can be regarded as the Grassmannian manifold
\begin{equation}
{\bf CP}^n={\rm U}(n+1)/({\rm U}(n)\times {\rm U}(1))
=S^{2n+1}/{\rm U}(1)\,.
\label{facx}
\end{equation}
Let $\tau^{-1}$ denote the {\it tautological bundle}\/ on ${\bf
CP}^n$. We recall that $\tau^{-1}$ is defined as the subbundle of the
trivial bundle ${\bf CP}^n\times {\bf C}^{n+1}$ whose fibre at $p\in
{\bf CP}^n$ is the line in ${\bf C}^{n+1}$ represented by $p$. Then
$\tau^{-1}$ is a holomorphic line bundle over ${\bf CP}^n$. Its dual,
denoted $\tau$, is called the {\it hyperplane bundle}. For any $l\in
\mathbb{Z}$, the $l$th power $\tau^l$ is also a holomorphic line
bundle over ${\bf CP}^n$. In fact every holomorphic line bundle $L$
over ${\bf CP}^n$ is isomorphic to $\tau^l$ for some $l\in\mathbb{Z}$;
this integer is the first Chern class of $L$.
\subsection{Computation of ${\rm dim}\,H^0({\bf CP}^n,
\mathcal{O}(1))$}
\label{xcompt}
\noindent
Next we present a quantum-mechanical computation of ${\rm
dim}\,H^0({\bf CP}^n,\mathcal{O}(1))$ without resorting to sheaf
cohomology. That is, we compute ${\rm dim}\,\mathcal{H}$ when $l=1$
and prove that it coincides with the right-hand side.
Starting with $\mathcal{C}={\bf CP}^{0}$, i.e.~a point $p$ as
classical phase space, the space of quantum rays must also reduce to a
point. Then the corresponding Hilbert space is $\mathcal{H}_1={\bf
C}$. The only state in $\mathcal{H}_1$ is the vacuum
$|0\rangle_{l=1}$. Henceforth, for brevity, we drop the Picard class
index from the vacuum.
\subsection{Representations}\label{wrepp}
The $(n+1)$-dimensional Hilbert space may be
regarded as a kind of {\it defining representation}, in the sense of
the representation theory of ${\rm SU}(n+1)$ when $n>1$. To make this
statement more precise we observe that one can replace unitary groups
with special unitary groups in Eq.~(\ref{facx}). Comparing our results
with those of Sec.~\ref{cipienne} we conclude that the quantum line
bundle $\mathcal{L}$ now equals $\tau$,
\begin{equation}
\mathcal{L}=\tau\,,
\label{ddttx}
\end{equation}
because $l=1$. This is the smallest value of $l$ that produces a
nontrivial $\mathcal{H}$, gives a one-dimensional
Hilbert space when $l=0$. So our $\mathcal{H}$ spans an
$(n+1)$-dimensional representation of ${\rm SU}(n+1)$, that we can
identify with the defining representation. There is some ambiguity
here since the dual of the defining representation of ${\rm SU}(n+1)$
is also $(n+1)$-dimensional. This ambiguity is resolved by convening
that the latter is generated by the holomorphic sections of the {\it
dual}\/ quantum line bundle
\begin{equation}
\mathcal{L}^*=\tau^{-1}\,.
\label{ddttxx}
\end{equation}
On the chart $\mathcal{U}_j$, $j=1,\ldots,n+1$, the dual of the
defining representation is the linear span of the covectors
\begin{equation}
\langle (j)0|\,,\qquad \langle (j)0|A_i(j)\,,\qquad i=1,2,\ldots,n\,.
\label{pannx}
\end{equation}
Taking higher representations is equivalent to considering the
principal \hbox{${\rm SU}(n+1)$}-bundle (associated with the vector
${\bf C}^{n+1}$-bundle) in a representation higher than the defining
one. We will see next that this corresponds to having $l>1$ in our
choice of the line bundle $\tau^l$.
\section{Tangent Vectors as Quantum States}\label{tvqs}
The converse is not true, as exemplified by the vacuum. Let us
generalise and replace ${\bf CP}^n$ with an arbitrary classical phase
space $\mathcal{C}$. We would like to write,
\begin{equation}
\mathcal{QH}(\mathcal{C})=T(\mathcal{C})\oplus N(\mathcal{C})\,,
\label{adccxj}
\end{equation}
where $N(\mathcal{C})$ is a holomorphic line bundle on $\mathcal{C}$,
whose fibre is generated by the vacuum state, and $T(\mathcal{C})$ is
the holomorphic tangent bundle. Does Eq.~(\ref{adccxj}) hold in
general?
\begin{table} %Table~1
\tbl{This table gives the QES condition and the number of moving
poles of $\chi$ for each combination of $b_{1}$ and $b^{\prime}_1$
for the Khare--Mandal model.}
{\begin{tabular}{@{}cccccc@{}}
\hline\\
&&&&&\\[-15pt]
Set & $b1$ & $b1'$ & $n=\lambda_{1}-b_{1}-b_{1}'$
& Condition o $M$ & QES Condition \\
& (Rad/s) & (Rad/s) \\[2pt]
\hline\\
&&&&&\\[-15pt]
1 & 1/4 & 1/4 & $\dfrac{M}{2}-\dfrac{1}{2}$
& $M={\rm odd}$, $M\geq 1$ & $M=2n+1$ \\[8pt]
3 & 3/4 & 1/4 & $\dfrac{M}{2}-1$
& $M={\rm even}$, $M\geq 2$ & $M=2n+1$ \\[8pt]
4 & 1/4 & 3/4 & $\dfrac{M}{2}-1$
& $M={\rm even}$, $M\geq 2$ & $M=2n+1$ \\[8pt]
\hline
\end{tabular} \label{tp1}}
\end{table}
The answer is also affirmative provided that
$\mathcal{C}$ is a complex $n$-dimensional, compact, symplectic
manifold, whose complex and symplectic structures are
compatible. Notice that $\mathcal{C}$ is not required to be K\"ahler;
examples of Hermitian but non-K\"ahler spaces are Hopf
manifolds.\cite{KN} Let $\omega$ denote the symplectic form. Then
$\int_\mathcal{C}\omega^n < \infty$ thanks to compactness,
\begin{equation}
\int_\mathcal{C}\omega^n=n+1\,.
\label{chepelotudo}
\end{equation}
Let us cover $\mathcal{C}$ with a {\it finite}\/ set of holomorphic
coordinate charts $(\mathcal{W}_k,w_{(k)})$, \hbox{$k=1,\ldots,r$};
the existence of such an atlas follows from the compactness of
$\mathcal{C}$. We can pick an atlas such that $r$ is minimal;
compactness implies that $r\geq 2$.
From Table~\ref{tp1}, we see that sets~1 and 2 are valid only when $M$ is odd
and sets~3 and 4 are valid only when $M$ is even.
\section{Discussion}\label{dicu}
Quantum mechanics is defined on a Hilbert space of states whose
construction usually assumes a global character on classical phase
space. Under {\it globality}\/ we understand, as explained in
Sec.~\ref{intro}, the property that all coordinate charts on classical
phase space are quantised in the same way.
A novelty of our approach is the local character of the Hilbert space:
there is one on top of each Darboux coordinate chart on classical
phase space. The patching together of these Hilbert-space fibres on
top of each chart may be global (trivial bundle) or local (nontrivial
bundle). In order to implement duality transformations we need a
nonflat bundle (hence nontrivial). Flatness would allow for a
canonical identification, by means of parallel transport, of the
quantum states belonging to different fibres.
A duality thus arises as the possibility of having two or more,
apparently different\-, quantum-mechanical descriptions of the same
physics. Mathematically, a duality arises as a nonflat, quantum
Hilbert-space bundle over classical phase space. This notion implies
that the concept of a quantum is not absolute, but relative to the
quantum theory used to measure it.\cite{VAFA} That is, duality
expresses the relativity of the concept of a quantum. In particular,
{\it classical}\/ and {\it quantum}\/, for long known to be deeply
related\cite{PERELOMOV} are not necessarily always the same for all
observers on phase space.
\section*{Acknowledgments}
It is a great pleasure to thank J. de Azc\'arraga for encouragement
and support. Technical discussions with U. Bruzzo and
M. Schlichenmaier are gratefully acknowledged. The author thanks
S. Theisen and Max-Planck-Institut f\"ur Gravitationsphysik,\-
Albert-Einstein-Institut, for hospitality. This work has been
partially supported\- by research grant BFM2002-03681 from Ministerio
de Ciencia y Tecnolog\'\i{a}, by EU FEDER funds, by Fundaci\'on Marina
Bueno and by Deutsche Forschungsgemeinschaft.
\eject
\appendix{Appendix}
We can insert an Appendix here and includes equations which are
numbered as Eq.~(\ref{app1}),
\begin{equation}
\frac{4\pi}{3}r_{ij}^3\cdot\frac{4\pi}{3}p_{ij}^3
=\frac{h^3}{4}\,.
\label{app1}
\end{equation}
\subappendix{Subsection of Appendix}
\begin{equation}
\frac{5\pi}{10}r_{ij}^2\cdot\frac{5\pi}{10}p_{ij}^7
=\frac{h^3}{4}\,.
\label{app2}
\end{equation}
The answer is trivially affirmative when $\mathcal{C}$ is an analytic
submanifold of ${\bf CP}^n$. Such is the case, e.g., of the embedding
of ${\bf CP}^n$ within ${\bf CP}^{n+l}$,
Grassmann manifolds provide another example.
\begin{thebibliography}{00}
%1.
\bibitem{LIBAZCA} J. de Azc\'arraga and J. Izquierdo, {\it Lie Groups,
Lie Algebras, Cohomology and some Applications in Physics} (Cambridge
Univ. Press, 1995).
%2.
\bibitem{VAFA} C. Vafa, hep-th/9702201.
%3.
\bibitem{KN} S. Kobayashi and K. Nomizu, {\it Foundations of
Differential Geometry} (Wiley, 1996).
%4.
\bibitem{MATONE} G. Bertoldi, A. Faraggi and M. Matone, {\it
Class. Quantum Grav.} {\bf 17}, 3965 (2000).
%5.
\bibitem{MINIC} D. Minic and C. Tze, {\it Phys. Rev.} {\bf D68},
061501 (2003); hep-th/0309239; hep-th/0401028; V. Jejjala, D. Minic
anc C. Tze, gr-qc/0406037.
%6.
\bibitem{CARROLL} R. Carroll, quant-ph/0406203.
%7.
\bibitem{MEX} M. Montesinos and G. T. del Castillo, quant-ph/0407051.
%8.
\bibitem{MPLA} J.M. Isidro, {\it Phys. Lett.} {\bf A301}, 210 (2002);
{\it J. Phys. A\/}: {\it Math. Gen.} {\bf 35}, 3305 (2002); {\it
Mod. Phys. Lett.} {\bf A18}, 1975 (2003); {\it ibid.} {\bf A19}, 349
(2004); hep-th/0407161.
%9.
\bibitem{SCHLICHENMAIER} M. Schlichenmaier, Berezin-Toeplitz
quantization and Berezin's symbols for arbitrary compact K\"ahler
manifolds, in {\it Coherent States}, {\it Quantization and Gravity},
eds.~M.~Schlichenmaier {\it et~al.} (Polish Scientific Publishers PWN,
2001).
%10.
\bibitem{LIBSCHL} M. Schlichenmaier, {\it An Introduction to Riemann
Surfaces, Algebraic Curves and Moduli\- Spaces} (Springer, 1989).
%11.
\bibitem{PERELOMOV} A. Perelomov, {\it Generalized Coherent States and
their Applications} (Springer, 1986).
\end{thebibliography}
\vfill
%\pagebreak
\end{document}