Bargmann invariants, null phase curves, and a theory of the geometric phase
N. Mukunda*
Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012, India
Arvind†
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
E. Ercolessi‡
Dipartimento di Fisica, Università di Bologna, INFM and INFN, Via Irnerio 46, 40126 Bologna, Italy
G. Marmo§
Dipartimento di Scienze Fisiche, Università di Napoli Federico II and INFN, Via Cinzia, 80126 Napoli, Italy
G. Morandi储
Dipartimento di Fisica, Università di Bologna, INFM and INFN, Viale Berti-Pichat 6/2, 40127 Bologna, Italy
R. Simon¶
The Institute of Mathematical Sciences, CIT Campus, Tharamani, Chennai 600 113, India
We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves
defined entirely in terms of the 共third order兲 Bargmann invariant, and it is shown that these are the curves
natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic
structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency
of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.
I. INTRODUCTION
The evolution of our understanding of the geometric
phase 共GP兲 关1兴 has brought together many aspects of the
basic structure of quantum mechanics, both in Hilbert space
and ray space levels. They include both linear vector space
features and differential geometric features 关2兴. During this
development, on one hand, the original assumptions of adiabaticity, cyclicity, and unitary evolution were relaxed in
stages in significant generalizations 关3兴. On the other hand,
starting from its discovery in an essentially dynamical context, it has gradually become clear that the GP is largely
kinematical in content 关4兴. In the process, important connections to properties of the Bargmann invariants 共BI兲 关5兴, and
even to earlier ideas of Pancharatnam in classical optics 关6兴,
have also been established 关7兴.
It is well known that the ray space 共complex projective
space兲 associated with the Hilbert space of any quantum sys-
*Email address: nmukunda@cts.iisc.ernet.in
†
Permanent address: Department of Physics, Guru Nanak Dev
University, Amritsar 143 005, India.
Email address: xarvind@andrew.cmu.edu
‡
Email address: ercolessi@bo.infn.it
§
Email address: gimarmo@na.infn.it
储
Email address: morandi@bo.infn.it
¶
Email address: simon@imsc.res.in
tem carries a natural Riemannian metric—the Fubini-Study
metric 关8兴—as well as a natural symplectic structure 关9兴. The
former determines a corresponding family of geodesics
which have played an important role in the GP theory in at
least two ways. Initially, they were exploited to show how to
define the GP for noncyclic evolution governed by the Schrödinger equation, essentially by converting such an evolution
to a cyclic one by adding on a geodesic to connect the end
points 关10兴. Later they were found to be useful in showing
that phases of BI’s are particular instances of the GP 关11兴.
The ray space symplectic structure has been known to be
intimately involved with GP’s, specially for cyclic evolution.
Subsequent work has shown that rather than geodesics,
the really basic geometric objects needed to connect BI’s and
GP’s are a family of ray space and Hilbert space curves
which have been named null phase curves 共NPC兲 关12兴. It has
been shown that while geodesics are NPC’s, the latter form a
vastly larger class of curves having little to do with the
Fubini-Study metric or the notion of geodesics. They also
lead to the most general possible connection between BI’s
and GP’s.
The purpose of this paper is to provide a logical basis for
defining the GP and understanding its properties. We will
show that the primitive building block is the BI 共in particular,
the three-vertex BI兲 in terms of which NPC’s can be defined.
Once this is in hand, the GP 共for the noncyclic case in general兲 can be defined as a derived object directly as a suitable
two-dimensional surface integral of the symplectic two-form
on ray space. Thus in this approach, primary importance is
given to BI’s, NPC’s, and ray space symplectic structure.
Earlier definitions of the GP, in particular, the kinematic definition, are seen to be immediate consequences of the present
one.
We shall make frequent reference to the geodesic arc connecting two given points in state space. By this we shall
always mean the shorter geodesic, which is unique assuming
that the given pair of points correspond to nonorthogonal
states.
The contents of this paper are arranged as follows. In Sec.
II, we assemble some details of notation relating to the Hilbert and ray spaces of a general quantum-mechanical system.
We define in a precise manner various classes of smooth
curves needed for further work; recall the natural one-form
on Hilbert space and the symplectic two-form on ray space;
and then the kinematic definition of the GP. Section III begins with the definition and basic properties of the BI, and in
terms of them defines the family of NPC’s in Hilbert and ray
spaces. Some important formulas connecting the two are also
developed. With this preparation, the definition of the GP for
a general open 共sufficiently smooth兲 ray space curve is given,
and its consistency is demonstrated. The recovery of the kinematic definition of the GP is also shown, and the connections between BI’s and GP’s, mediated by the uses of NPC’s,
are brought out. The emphasis in this section is to display the
logical structure of ideas. Section IV explores the properties
of NPC’s from various points of view, emphasizing always
that it is these curves that are natural and basic to the structure of the GP. The fact that they are far more numerous than
geodesics means that their description is very different from
that of the latter; in particular, they cannot be viewed as
solutions to any local finite-order ordinary differential equations at all. Examples of 共infinitely many兲 NPC’s connecting
any two given ray space points; a description of the most
general NPC; of submanifolds every curve in which is a
NPC; and examples of such submanifolds; are all developed.
Section V contains concluding remarks.
There are two appendixes, devoted to basic differential
geometry of Hilbert and ray spaces and to a complete description of geodesics, respectively.
and B is denoted by R. It is the quotient of B with respect to
the equivalence relation ␺ ⬃e i ␣ ␺ among unit vectors for all
real phases ␣ :
R⫽B/U共 1 兲 .
共2.2兲
Elements of R are represented by pure state density matrices, or one-dimensional projections, ␳ ; and there is a projection ␲ from B to R:
␲ :B→R: ␺ 苸B→ ␲ 共 ␺ 兲 ⫽ ␳ ␺ ⫽ ␺␺ † 苸R.
共2.3兲
As is well known, B is a principal U(1) bundle over R,
which in turn for finite dimensions is the complex projective
space CP n⫺1 . The real dimension of R in that case is 2(n
⫺1), and it is also connected and simply connected.
For considerations of geometric phases, null phase curves,
and geodesics, we need to deal with continuous parametrized
curves C傺B, and their projections C⫽ ␲ (C)傺R, obeying
suitable smoothness conditions. They are always directed
curves. We describe them as follows:
C⫽ 兵 ␺ 共 s 兲 苸B兩 s 1 ⭐s⭐s 2 其 傺B;
C⫽ ␲ 共 C兲 ⫽ 兵 ␳ 共 s 兲 ⫽ ␺ 共 s 兲 ␺ 共 s 兲 † 苸R兩 s 1 ⭐s⭐s 2 其 傺R.
共2.4兲
In general, we assume that the parameter s varies over a
closed finite interval 关 s 1 ,s 2 兴 傺 R, exceptions will be indicated. We permit strictly monotonic reparametrizations of
these curves,
s→s ⬘ ⫽ f 共 s 兲 ,
d f 共s兲
⬎0,
ds
共2.5兲
and impose other smoothness conditions as appropriate and
described below. We also permit smooth local phase changes
along a curve C to lead to a new curve C ⬘ ,
C ⬘ ⫽ 兵 ␺ ⬘ 共 s 兲 ⫽e i ␣ (s) ␺ 共 s 兲 兩 s 1 ⭐s⭐s 2 其 傺B
共2.6兲
having the same image in R:
␲ 共 C ⬘ 兲 ⫽ ␲ 共 C兲 ⫽C傺R.
II. NOTATIONAL PRELIMINARIES AND KINEMATIC
DEFINITION OF GEOMETRIC PHASE
We denote by H the complex 共separable兲 Hilbert space
describing the pure states of some quantum system. The
complex dimension of H may be finite, say n⫽1,2, . . . , or
infinite. The inner product and the norm for vectors
␺ , ␾ , . . . in H are written as ( ␺ , ␾ ) and 储 ␺ 储 , respectively.
The subset of unit vectors in H is defined by
B⫽ 兵 ␺ 苸H兩 储 ␺ 储 ⫽1 其 傺H.
共2.1兲
共2.7兲
Thus, both C and C ⬘ are lifts, from R to B, of C.
We now define three classes of curves, with different
smoothness conditions, as follows.
Class I:
„␺ 共 s 1 兲 , ␺ 共 s 2 兲 …⫽0
␺ 共 s 兲 , ␳ 共 s 兲 continuous and piecewise once differentiable.
共2.8兲
Class II:
For dimension H⫽n finite, B is the real Euclidean sphere
S 2n⫺1 of real dimension (2n⫺1). It is both connected and
simply connected. The space of unit rays associated with H
„␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽0, any s,s ⬘ 苸 关 s 1 ,s 2 兴
␺ 共 s 兲 , ␳ 共 s 兲 continuous once differentiable.
共2.9兲
Class III:
„␺ 共 s 1 兲 , ␺ 共 s 2 兲 …⫽0
␺共 s 兲,
␳ 共 s 兲 continuous twice differentiable. 共2.10兲
It will turn out that class-I curves are those for which geometric phases can be defined; class-II curves subject to further conditions are null phase curves; and class-III curves
obeying suitable differential equations are geodesics. In each
case, both C傺B and C⫽ ␲ (C)傺R will be assumed to obey
the same smoothness conditions and both will belong to the
same class. Similarly, reparametrizations and local phase
changes will be assumed to preserve the smoothness properties of each class.
The basic differential geometric objects needed for our
purposes, with suitable notations, are given in Appendix A
关13兴. These are a one-form A on B; its exterior derivative
two-form dA on B; and a closed nondegenerate symplectic
two-form ␻ on R related to dA via pullback
dA⫽ ␲ * ␻ .
共2.11兲
The one-form A is essentially defined by giving its integral
along any curve C傺B of class I:
冕
C
A⫽Im
冕 冉␺
s2
s1
ds
共 s 兲,
冊 冕 冉
d␺共 s 兲
⫽⫺i
ds
s2
s1
ds ␺ 共 s 兲 ,
冊
d␺共 s 兲
.
ds
共2.12兲
If C happens to be a closed loop with ␺ (s 2 )⫽ ␺ (s 1 ) so that
its projection ␲ (C)⫽C傺R is also a closed loop, we have
冖 冕
C
A⫽
S
dA⫽
冕␻
S
,
共2.13兲
where S is any smooth two-dimensional surface in B having
C as boundary, and S is the image of S in R with C as
boundary:
⳵ S⫽C,
S⫽ ␲ 共 S兲 , ⳵ S⫽C.
共2.14兲
The orientations of S and S are determined by the directions
of C and C, respectively. It is to be emphasized that ␻ is not
exact, so A is not the pullback via ␲ * of any one-form on R.
Since the two-form ␻ on R will play a primary role in our
definition of the GP, we add the following comments to help
better understand its nature. In the finite-dimensional case,
dimension H⫽n, we have mentioned that R⫽CP n⫺1 , the
complex projective space of 共complex兲 dimension (n⫺1);
and it is well known that these spaces are important canonical examples of symplectic manifolds 关9兴. This property can
however be also grasped from another point of view. The
unitary group U(n) acting on H via its defining representation has for its Lie algebra u(n), the set of all Hermitian
operators on H. Upon conjugation by elements of U(n)
共which is the adjoint action兲, u(n) is mapped onto itself and
broken up into disjoint orbits. Each orbit is essentially a
coset space U(n)/H, where H is the stability group of any
chosen representative point on the orbit. Most orbits are generic and of real dimension n(n⫺1), being the coset space
U(n)/U(1)⫻U(1)⫻•••⫻U(1) (n factors兲. Apart from
these, there are several exceptional or singular orbits of various lower dimensions. From the general Kostant-KirillovSouriau theory of coadjoint orbits of Lie groups G, it is
known that each orbit 共generic or exceptional兲 is a symplectic manifold; the symplectic two-form is obtained by descent
from the Maurer-Cartan two-form on the group G itself, i.e.,
by quotienting with respect to its kernel 关14兴. In the case at
hand, the pure state density operators for an n-dimensional
quantum system are elements of u(n). They are clearly acted
upon transitively by conjugation with elements of U(n); and
they form a single nongeneric orbit in the Lie algebra, the
stability group H in this case being easily seen to be U(1)
⫻U(n⫺1). We can then identify R with the coset space
U(n)/U(1)⫻U(n⫺1), which is indeed of real dimension
2(n⫺1); and according to the general theory of coadjoint
orbits of Lie groups, it is a symplectic manifold.
Even without appeal to the Kostant-Kirillov-Souriau
共KKS兲 theory and then limiting ourselves to the nongeneric
orbit of pure state density operators, one can directly display
the connection between the forms A and dA on B and the
Maurer-Cartan one- and two-forms on U(n).
Denote by g,g ⬘ , . . . the matrices of the defining representation of U(n). The actions of U(n) on B and on R given
by
g苸U共 n 兲 : ␺ 苸B→g ␺ 苸B
共2.15兲
␳ 苸R→g ␳ g ⫺1 苸R
共2.16兲
and
are both transitive. Therefore, for any choice of a fiducial
vector ␺ 0 苸B, we have an onto map ␮ 0 :U(n)→B given by
␮ 0 :g苸U共 n 兲 →g ␺ 0 苸B.
共2.17兲
This map can be used to take any 共smooth兲 curve ␥
⫽ 兵 g(s) 其 傺G
to
an
image
C⫽ 兵 ␮ 0 (g(s) 其 ⫽ 兵 ␺ (s)
⫽g(s) ␺ 0 其 傺B, and in turn to its projection C⫽ 兵 ␳ (s)
⫽ ␺ (s) ␺ (s) † ⫽g(s) ␺ 0 ␺ †0 g(s) ⫺1 其 傺R. In the reverse direction, evidently, the pullback ␮ 0* allows us to take A and dA
on B to appropriate forms on U(n). From Eq. 共2.12兲 A on B
has the expression
A⫽⫺i ␺ † d ␺ .
共2.18兲
Combining with Eq. 共2.17兲 we get
† †
†
†
␮ 0* A⫽⫺i ␮ *
0 共 ␺ d ␺ 兲 ⫽⫺i ␺ 0 g dg ␺ 0 ⫽⫺i Tr共 ␳ 0 g dg 兲 .
共2.19兲
The expression g † dg is the matrix of left-invariant
Maurer-Cartan one-forms on U(n), so in Eq. 共2.19兲 we have
the expected connection between the one-form A on B and
the Maurer-Cartan one-forms on U(n). A particular linear
combination of the latter, determined by the choice of ␺ 0 , is
picked out. It also follows by taking the exterior derivative of
Eq. 共2.19兲 that
␮ 0* dA⫽⫺i Tr共 ␳ 0 dg † ∧dg 兲 ⫽i Tr共 ␳ 0 g † dg∧g † dg 兲 ,
共2.20兲
so here the matrix of left-invariant Maurer-Cartan two-forms
on U(n) appears. Hence, if X 1 and X 2 are two left-invariant
vector fields on u(n) associated with Hermitian matrices
␶ 1 and ␶ 2 in u(n),
i X j g † dg⫽i ␶ j , j⫽1,2,
共2.21兲
then a short calculation shows
␮*
0 dA 共 X 1 ,X 2 兲 ⫽ 共 dA 兲 „␮ 0 * 共 X 1 兲 , ␮ 0 * 共 X 2 兲 …
⫽i X 2 i X 1 ␮ 0* dA
⫽i Tr共 ␳ 0 关 ␶ 1 , ␶ 2 兴 兲 .
共2.22兲
In particular, we can choose ␶ 1 and ␶ 2 to be two elements
␳ 1 , ␳ 2 苸R and then we get in Eq. 共2.22兲 the result
i Tr( ␳ 0 关 ␳ 1 , ␳ 2 兴 ) which agrees with Eq. 共A.7兲. Thus, the connection between A,dA on B and the Maurer-Cartan forms on
U(n), mentioned in the previous paragraph, are made explicit 关15兴.
We now recall the definition of the GP according to the
kinematic approach 关4兴. If C is a class-I curve with image C,
then the GP for C is
␸ g 关 C 兴 ⫽arg„␺ 共 s 1 兲 , ␺ 共 s 2 兲 …⫺
冕
C
A.
共2.23兲
Clearly, because of the first term on the right, this phase is
defined modulo 2 ␲ ; and as implied by the notation it is a
functional of C independent of the lift C used to compute the
individual terms on the right. Moreover it is unchanged by
any permitted reparametrizations.
In a previous work, Eq. 共2.23兲 was adopted as the definition of the GP, and thereafter null phase curves were defined
and used in various ways 关12兴. Our approach here will be to
regard null phase curves as primitive objects and to define
GP’s in terms of them. This will be done in the following
section.
If the end points ␺ (s 1 ), ␺ (s 2 ) of C are mutually orthogonal, clearly the GP ␸ g 关 C 兴 becomes undefined. This is the
reason behind the condition of nonorthogonality of end
points in the definition 共2.8兲 of class-I curves. It ensures that
for any class-I curve the, GP is well-defined modulo 2 ␲ .
However, the definition 共2.8兲 does not forbid the possibility
that for some s 0 苸(s 1 ,s 2 ), ␺ (s 0 ) may be orthogonal to either
␺ (s 1 ) or ␺ (s 2 ) or both. Assume ␺ (s 0 ) is indeed orthogonal
to ␺ (s 1 ), and let s 0 be a point on C and C at which these
curves are differentiable. If for sufficiently small ⑀ both
␺ (s 0 ⫺ ⑀ ) and ␺ (s 0 ⫹ ⑀ ) are not orthogonal to ␺ (s 1 ), then
the GP’s are defined for the portions of C running from s 1 to
s 0 ⫾ ⑀ , and they obey
␸ g 关 C for s 1 ⭐s⭐s 0 ⫹ ⑀ 兴 ⫺ ␸ g 关 C for s 1 ⭐s⭐s 0 ⫺ ⑀ 兴 ⫽⫾ ␲ ,
共2.24兲
which are equivalent modulo 2 ␲ . Thus, in such a situation
we see that the GP is defined upto just before a point of
orthogonality to the initial point, as well as to a point just
after; and there is a discontinuity of ⫾ ␲ as we cross that
point.
In passing we may mention that even in the context of a
real Hilbert space, the GP survives though in a rudimentary
form 关16兴. The dynamical phase is of course absent, however
the total phase could be an odd multiple of ␲ . In fact, each
time the inner product „␺ (s 1 ), ␺ (s 2 )… passes through zero,
we pick up a contribution ⫾ ␲ , just as in Eq. 共2.24兲. The BI
⌬ 3 ( ␺ 1 , ␺ 2 , ␺ 3 ) 关see below兴 can also have a nontrivial phase,
namely, ⫾ ␲ when it is negative. All these remarks remain
valid also in the case of a complex Hilbert space, if we
restrict ourselves to the real linear span of a set of vectors
taken from an orthonormal basis. The role of such ‘‘real’’
subspaces will become evident in the sequel.
III. NULL PHASE CURVES AND A DEFINITION
OF THE GEOMETRIC PHASE
Our aim now is to define NPC in B and R as the basic or
primitive objects, then define GP’s in terms of them, and
derive their properties in a logically consistent manner. To
begin with, we recall the definition and properties of the
third-order Bargmann invariant, as the NPC definition will
depend on it. For convenience, we divide this section into
further sections.
A. Bargmann invariants „BI…
Given any three mutually nonorthogonal vectors
␺ 1 , ␺ 2 , ␺ 3 苸B, projecting onto ␳ 1 , ␳ 2 , ␳ 3 苸R, the third-order
BI is defined as
⌬ 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 ⫽ 共 ␺ 1 , ␺ 2 兲共 ␺ 2 , ␺ 3 兲共 ␺ 3 , ␺ 1 兲 ⫽Tr共 ␳ 1 ␳ 2 ␳ 3 兲 .
共3.1兲
Its key properties are well known: 共i兲 for dimension H⭓,2 it
is in general complex; 共ii兲 it is cyclically symmetric; 共iii兲 as
the second form shows, it is invariant under independent
phase changes in each of the vectors ␺ 1 , ␺ 2 , ␺ 3 .
Higher-order BI’s can be defined in a similar manner. For
any m vectors ␺ 1 , ␺ 2 ,..., ␺ m 苸B, such that no two successive ones are mutually orthogonal, we have the generally
complex mth order BI
⌬ m 共 ␺ 1 , ␺ 2 , . . . , ␺ m 兲 ⫽ 共 ␺ 1 , ␺ 2 兲共 ␺ 2 , ␺ 3 兲 ••• 共 ␺ m , ␺ 1 兲
⫽Tr共 ␳ 1 ␳ 2 ••• ␳ m 兲 .
共3.2兲
For m⫽2 of course, ⌬ 2 ( ␺ 1 , ␺ 2 ) is real non-negative.
B. Null phase curves
A curve C傺B of class II, with image C傺R, will be said
to be an NPC if for any three vectors on it, the BI is real
positive:
C,CNPC⇔⌬ 3 „␺ 共 s 兲 , ␺ 共 s ⬘ 兲 , ␺ 共 s ⬙ 兲 …
arg„␺ 0 共 s 1 兲 , ␺ 0 共 s 2 兲 …⫽0,
冕
⫽Tr„␳ 共 s 兲 ␳ 共 s ⬘ 兲 ␳ 共 s ⬙ 兲 …⫽real positive
C0
⇔Tr„␳ 共 s 兲关 ␳ 共 s ⬘ 兲 , ␳ 共 s ⬙ 兲兴 …⫽0,
any s,s ⬘ ,s ⬙ 苸 关 s 1 ,s 2 兴 .
共3.3兲
For convenience, when this condition is obeyed, we refer to
both C and C as NPC’s.
It is immediately evident that any connected subset or
portion of an NPC, say running from s 3 to s 4 , where
关 s 3 ,s 4 兴 傺 关 s 1 ,s 2 兴 , is also an NPC.
From Eq. 共B13兲 of Appendix B, it follows that every geodesic in R 共and any lift of it in B) is an NPC. Since any two
points ␳ 1 , ␳ 2 苸R can definitely be connected by a geodesic
关which is moreover unique if Tr( ␳ 1 ␳ 2 )⬎0], we can say that
they can definitely be connected by an NPC. However, as we
will see in the following section, provided dimension H
⭓3, NPC’s are far more numerous than geodesics: there are
infinitely many of them connecting any ␳ 1 , ␳ 2 苸R.
The definition 共2.9兲 of a class-II curve includes the condition that no two vectors along it should be mutually orthogonal. The motivation for this is now understandable: we
need the BI appearing in Eq. 共3.3兲 to be nonzero for any
triplet of vectors on the curve. It will soon emerge that a
somewhat more economical definition of an NPC, which is
however fully equivalent to Eq. 共3.3兲, is
C,CNPC⇔⌬ 3 „␺ 共 s 0 兲 , ␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽real positive,
any fixed s 0 苸 关 s 1 ,s 2 兴 ,
any s,s ⬘ 苸 关 s 1 ,s 2 兴 .
共3.4兲
Let now C be an NPC. We derive a fundamental formula
for the integral of A along any lift C of C. For any chosen
reference point s 0 苸 关 s 1 ,s 2 兴 , choose some ␺ 0 (s 0 )
苸 ␲ ⫺1 „␳ (s 0 )…. By definition of class II, for any s苸 关 s 1 ,s 2 兴
and any choice of ␺ (s)苸 ␲ ⫺1 „␳ (s)…, the scalar product
„␺ 0 (s 0 ), ␺ (s)… is nonzero. Adjust the phase of ␺ (s) to get
␺ 0 (s)苸 ␲ ⫺1 „␳ (s)… such that
„␺ 0 共 s 0 兲 , ␺ 0 共 s 兲 …⫽real positive,
any s苸 关 s 1 ,s 2 兴 .
共3.5兲
This gives us a particular lift C0 ⫽ 兵 ␺ 0 (s) 兩 s 1 ⭐s⭐s 2 其 of C
with the property
„␺ 0 共 s 兲 , ␺ 0 共 s ⬘ 兲 …⫽real positive,
any s,s ⬘ 苸 关 s 1 ,s 2 兴 .
共3.6兲
We see this by setting s ⬙ ⫽s 0 in the definition 共3.3兲 and then
using Eq. 共3.5兲. This means that any two points on C0 are in
phase in the Pancharatnam sense 关6兴, a nonlocal property;
and furthermore C0 is horizontal, a local property,
冉
C0 : ␺ 0 共 s 兲 ,
冊
d
␺ 共 s 兲 ⫽0,
ds 0
any s苸 关 s 1 ,s 2 兴 .
共3.7兲
Therefore, for the end points of C0 and for the integral of A
along C0 , we have
共3.8兲
A⫽0.
Now let C⫽ 兵 ␺ (s)⫽e i ␣ (s) ␺ 0 (s) 其 be a general lift of C obtained from C0 by a 共sufficiently smooth兲 local phase transformation. For C we find in place of Eq. 共3.8兲,
arg„␺ 共 s 1 兲 , ␺ 共 s 2 兲 …⫽ ␣ 共 s 2 兲 ⫺ ␣ 共 s 1 兲 ,
冕
C
A⫽⫺i
冕 冉␺
s2
ds
s1
共 s 兲,
冊 冕
d␺共 s 兲
⫽
ds
s2
ds
s1
d␣共 s 兲
, 共3.9兲
ds
that is,
冕
C
A⫽arg„␺ 共 s 1 兲 , ␺ 共 s 2 兲 ….
共3.10兲
This is the basic property of NPC’s that we will use repeatedly.
Concerning the construction of the particular ‘‘Pancharatnam lift’’ C0 of the NPC C⫽ 兵 ␳ (s)⫽ ␺ (s) ␺ (s) † 其 傺R, we
may add the following remark. The lift C0 is completely determined once a choice of ␺ 0 (s 0 ) at the reference point s 0 is
made. Any alteration of ␺ 0 (s 0 ) by a phase leads to a rigid or
constant phase change of all points along C0 . One can now
see that the rule 共3.5兲 to determine ␺ 0 (s) for general s has
the following quite explicit solution:
␺ 0 共 s 兲 ⫽N 共 s 兲 ␳ 共 s 兲 ␺ 0 共 s 0 兲 ⫽ ␺ 共 s 兲 e ⫺iarg„␺ 0 (s 0 ), ␺ (s)…,
共3.11兲
where N(s) is a real positive normalization factor,
N 共 s 兲 ⫽ 兩 „␺ 0 共 s 0 兲 , ␺ 共 s 兲 …兩 ⫺1 .
共3.12兲
The vector ␺ 0 (s) in Eq. 共3.11兲 is clearly invariant under
changes in phase of ␺ (s). Then both Eqs. 共3.5兲 and 共3.6兲 are
obeyed by the expression 共3.11兲:
„␺ 0 共 s 0 兲 , ␺ 0 共 s 兲 …⫽N 共 s 兲 „␺ 0 共 s 0 兲 , ␳ 共 s 兲 ␺ 0 共 s 0 兲 …
⫽ 兩 „␺ 0 共 s 0 兲 , ␺ 共 s 兲 ) 兩 ⬎0,
共 ␺ 0 共 s 兲 , ␺ 0 共 s ⬘ 兲 …⫽N 共 s 兲 N 共 s ⬘ 兲 „␳ 共 s 兲 ␺ 0 共 s 0 兲 , ␳ 共 s ⬘ 兲 ␺ 0 共 s 0 兲 …
⫽N 共 s 兲 N 共 s ⬘ 兲 Tr„␳ 共 s 0 兲 ␳ 共 s 兲 ␳ 共 s ⬘ 兲 …⬎0,
共3.13兲
since C is given to be an NPC.
Now let C be an NPC from ␳ 1 to ␳ 2 , and C ⬘ an NPC
from ␳ 2 to ␳ 1 . Choose vectors ␺ 1 苸 ␲ ⫺1 ( ␳ 1 ), ␺ 2
苸 ␲ ⫺1 ( ␳ 2 ), and lifts C,C ⬘ of C,C ⬘ from ␺ 1 to ␺ 2 and ␺ 2 to
␺ 1 , respectively. The unions C艛C ⬘ 傺R,C艛C ⬘ 傺B are
closed loops. Let S傺R be any smooth two-dimensional surface with boundary ⳵ S⫽C艛C ⬘ . Combining Eq. 共2.11兲 with
the basic property 共3.10兲 for both C and C ⬘ , we find
C,NPC ␳ 1 to ␳ 2 , C ⬘ NPC ␳ 2 to ␳ 1 ,
⳵ S⫽C艛C ⬘ ,
冕␻ 冖
⫽
S
C艛C ⬘
A⫽arg ⌬ 2 共 ␺ 1 , ␺ 2 兲 ⫽0.
共3.14兲
It is perhaps worth emphasizing that while the curves C,C ⬘
and their lifts C,C ⬘ are each of class II, since they are NPC’s,
the closed loops C艛C ⬘ and C艛C ⬘ may not be of class II,
and in any case they are not expected to be NPC’s. They are
of course class I which requires only piecewise once differentiability.
The generalization of Eq. 共3.14兲 to a string of three or
more successive NPC’s, altogether forming a closed loop,
involves the phase of a nontrivial BI. Thus for any m⭓3, if
␳ 1 , ␳ 2 , . . . , ␳ m 苸R and C j, j⫹1 are NPC’s from ␳ j to ␳ j⫹1
for j⫽1,2, . . . ,m 共with ␳ m⫹1 ⫽ ␳ 1 ), with lifts C j, j⫹1 running
from ␺ j to ␺ j⫹1 , we find using Eq. 共3.10兲 repeatedly
⳵ S⫽C 12艛C 23艛 . . . 艛C m1 , C j, j⫹1 NPC’s ␳ j to ␳ j⫹1 ;
冕␻
S
m
⫽
兺
j⫽1
冕
C j, j⫹1
A⫽arg ⌬ m 共 ␺ 1 , ␺ 2 , . . . , ␺ m 兲 .
place of C ⬘ to compute ␸ g 关 C 兴 . Choose any surface S ⬘ 傺R
with boundary ⳵ S ⬘ ⫽C̃ ⬘ 艛C ⬙ . Then S艛S ⬘ has boundary
⳵ (S艛S ⬘ )⫽C艛C ⬙ . Using Eq. 共3.14兲 for the pair C̃ ⬘ ,C ⬙ we
have
⳵ S⫽C艛C ⬘ ,
␸ g 关 C 兴 ⫽⫺
With this preparation we are able to define the GP ␸ g 关 C 兴
for any class-I curve C傺R from ␳ 1 to ␳ 2 . We choose any
NPC C ⬘ 傺R from ␳ 2 to ␳ 1 , so that C艛C ⬘ is a class-I closed
loop in R, and then choose any two-dimensional surface
S傺R with ⳵ S⫽C艛C ⬘ . Then ␸ g 关 C 兴 is defined as the integral of ␻ over S 关17兴:
C ⬘ NPC ␳ 2 to ␳ 1 ,
␸ g 关 C 兴 ⫽⫺
冕␻ 冕␻ 冕 ␻ 冕
冕␻
.
S
⳵ S⫽C艛C ⬘ ;
共3.16兲
For consistency, we must show that the integral involved
here is independent of the choice of the NPC C ⬘ . Pending
that, we see immediately upon comparing Eq. 共3.14兲 with the
definition 共3.16兲 that for any NPC, the GP vanishes,
C is NPC⇒ ␸ g 关 C 兴 ⫽0.
共3.17兲
The proof of the consistency of the definition 共3.16兲 also
rests on the result 共3.14兲. First, we introduce an item of notation. For any curves C,C, we denote by C̃,C̃ the reversed
curves obtained by traversing them backwards. We note then
that the NPC property is preserved while the GP changes
sign:
C,C NPC ⇔C̃,C̃ NPC,
␸ g 关 C̃ 兴 ⫽⫺ ␸ g 关 C 兴 .
共3.18兲
Now turning to the consistency of Eq. 共3.16兲, let C ⬙ be any
other NPC from ␳ 2 to ␳ 1 , which could have been used in
S
⫽⫺
⫺
S
S⬘
⫽⫺
S艛S ⬘
␻ . 共3.19兲
⳵ S⫽C, ⳵ C⫽ ⳵ C⫽0;
共3.15兲
C. Definition of the GP
⳵ 共 S艛S ⬘ 兲 ⫽C艛C ⬙ ;
Here we used the additivity property for integrals of ␻ over
共nonoverlapping兲 surfaces S and S ⬘ . Thus, the consistency of
the definition 共3.16兲 is established.
All of the above is applicable for a general class-I curve
C傺R which could be open, i.e., ␳ 2 ⫽ ␳ 1 . In case ␳ 2 ⫽ ␳ 1 and
C is a closed loop, there is no need to append any NPC to it
before computing its GP. We choose any lift C also in the
form of a closed loop and any two-dimensional surface
S傺R with ⳵ S⫽C, and directly have
It is because only ⌬ 2 ( ␺ 1 , ␺ 2 ) is known to be always real
positive that we obtain a vanishing right-hand side in the
result 共3.14兲.
C class I ␳ 1 to ␳ 2 ,
⳵ S ⬘ ⫽C̃ ⬘ 艛C ⬙ ,
␸ g 关 C 兴 ⫽⫺
冕␻ 冖
S
⫽⫺
C
共3.20兲
A.
Going back to the general definition 共3.16兲 of ␸ g 关 C 兴 for an
open C, we now compare with Eq. 共3.20兲 for closed C and
draw the conclusion
C ⬘ NPC ␳ 2 to ␳ 1 ,
C class I ␳ 1 to ␳ 2 ,
⳵ 共 C艛C ⬘ 兲
⫽0;
␸ g 关 C 兴 ⫽ ␸ g 关 C艛C ⬘ 兴 .
共3.21兲
The kinematic definition 共2.23兲 for ␸ g 关 C 兴 is immediately
recovered from the present definition 共3.16兲. With reference
to the latter, let C be any lift of C from any ␺ 1 苸 ␲ ⫺1 ( ␳ 1 ) to
any ␺ 2 苸 ␲ ⫺1 ( ␳ 2 ), and let C ⬘ , an NPC, be any lift of the
NPC C ⬘ from ␺ 2 to ␺ 1 . Then using Eqs. 共2.11兲 and 共3.10兲,
we obtain
␸ g 关 C 兴 ⫽⫺
⫽⫺
冕␻ 冖
冕 冕
S
C
⫽⫺
A⫺
C⬘
C艛C ⬘
A
A⫽arg共 ␺ 1 , ␺ 2 兲 ⫺
冕
C
A, 共3.22兲
which is Eq. 共2.23兲.
D. The BI-GP connections
There are two important formulas connecting BI’s and
GP’s. Both of them can be derived from the definition 共3.16兲
with the property 共3.10兲 for NPC’s.
Let ␳ 1 , ␳ 2 , ␳ 3 苸R be images of ␺ 1 , ␺ 2 , ␺ 3 苸B, no two in
either triplet being mutually orthogonal. Join them pairwise
by NPC’s: C 12 from ␳ 1 to ␳ 2 , and C 23 from ␳ 2 to ␳ 3 , and
C 31 from ␳ 3 to ␳ 1 , their lifts C12 from ␺ 1 to ␺ 2 , C23 from ␺ 2
to ␺ 3 ,and C31 from ␺ 3 to ␺ 1 . Now both C 12艛C 23艛C 31 and
C12艛C23艛C31 are closed loops, so we can use Eq. 共3.20兲 to
get
C’s,C ’s NPC’s;
An exception to the general lack of additivity expressed
by Eq. 共3.26兲 occurs when we take ␳ 3 ⫽ ␳ 2 because BI
⌬ 3 ( ␺ 1 , ␺ 2 , ␺ 1 )⫽⌬ 2 ( ␺ 1 , ␺ 2 ) is real positive. In this case, we
have
␸ g 关 C 12艛C 23兴 ⫽ ␸ g 关 C 12艛C 23艛C 31兴
⫽⫺
冖
C12艛C23艛C31
A⫽⫺
冕
C12
A⫺
冕
C23
A⫺
⫽⫺arg ⌬ 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 .
冕
⳵ S⫽C 12艛C 21 ;
A
␸ g 关 C 12艛C 21兴 ⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 21兴
共3.23兲
⫽ ␸ g 关 C 12兴 ⫺ ␸ g 关 C̃ 21兴
C31
This is in fact the most general connection between GP’s and
共phases of兲 BI’s, as discussed elsewhere 关12兴. It takes the
known connection 共3.15兲 between NPC’s and BI’s one step
further and brings in the GP. Equation 共3.23兲 goes with Eq.
共3.15兲 for m⫽3. For m⭓4, we have, using the notations of
Eq. 共3.15兲,
C’s,C ’sNPC’s;
␸ g 关 C 12艛C 23艛•••艛C m1 兴 ⫽⫺
冕
C12
A⫺
冕
C23
A⫺ . . . ⫺
冕
Cm1
A
⫽⫺arg ⌬ m 共 ␺ 1 , ␺ 2 , . . . , ␺ m 兲 .
共3.24兲
The second formula brings out the role of BI’s in showing
the nonadditivity of GP’s and its derivation exploits Eq.
共3.23兲. For triplets of points ␳ 1 , ␳ 2 , ␳ 3 苸R, ␺ 1 , ␺ 2 , ␺ 3 苸B as
before, let C 12 and C 23 be any class-I curves, not necessarily
NPC’s from ␳ 1 to ␳ 2 and ␳ 2 to ␳ 3 , respectively. Next let
⬘ ,C 32
⬘ ,C 31
⬘ be NPC’s from ␳ 2 to ␳ 1 , ␳ 3 to ␳ 2 , and ␳ 3 to
C 21
␳ 1 , respectively. These are needed to define the GP’s which
appear below. Finally, we choose two-dimensional surfaces
S 1 ,S 2 ,S 3 with boundaries
⬘ ,
⳵ S 1 ⫽C 12艛C 21
⬘ ,
⳵ S 2 ⫽C 23艛C 32
⬘ 艛C̃ 32
⬘ 艛C 31
⬘ ,
⳵ S 3 ⫽C̃ 21
⬘ .
⳵ 共 S 1 艛S 2 艛S 3 兲 ⫽C 12艛C 23艛C 31
共3.25兲
Then repeatedly using the definition 共3.16兲 and at the last
step appealing to the result 共3.23兲, we find
⫽⫺
冕
S 1 艛S 2 艛S 3
␻⫹
S1
S2
⫽⫺
共3.27兲
IV. EXAMPLES AND PROPERTIES OF NULL
PHASE CURVES
We have mentioned that for any two points ␳ 1 , ␳ 2 苸R,
there is a geodesic connecting them, which is unique when
Tr( ␳ 1 ␳ 2 )⬎0, and that geodesics are NPC’s. In this section,
we explore NPC’s from several points of view, so as to visualize them better. We will show by explicit construction
that for dimension H⭓3, given ␳ 1 , ␳ 2 苸R with Tr( ␳ 1 ␳ 2 )
⬎0, there are infinitely many 共in a quite nontrivial sense兲
NPC’s connecting ␳ 1 to ␳ 2 . We follow this up by developing
an explicit analytical description, as far as is possible, of the
most general NPC from ␳ 1 to ␳ 2 . Finally, we explore the
differential geometric properties and characterization of
smooth submanifolds M 傺R with the property that every
continuous once-differentiable curve C傺M is an NPC, and
give examples of such submanifolds.
It is instructive to see how the condition dimH⭓3 for the
existence of nontrivial NPC’s arises. For dimH⫽2, the ray
space R is the Poincaré sphere S 2 . If now three points
␳ 1 , ␳ 2 , ␳ 3 苸R correspond to respective unit vectors
n̂ 1 ,n̂ 2 ,n̂ 3 苸S 2 , then, as is known, arg Tr( ␳ 1 ␳ 2 ␳ 3 ) is one-half
of the solid angle subtended at the center of S 2 by the spherical triangle with vertices n̂ 1 ,n̂ 2 ,n̂ 3 关18兴. Thus, for condition
共3.3兲 to be obeyed for any three points on an NPC, this solid
angle must always vanish, so the NPC must be contained
within some great circle. More explicitly, ␳ is expressible in
terms of its representative point n̂苸S 2 as
1
␳ ⫽ 关 1⫹n̂• ␴ជ 兴 ,
2
冕␻ 冕␻ 冕␻
⫹
.
S
This will be used in the sequel.
␸ g 关 C 12艛C 23兴 ⫺ ␸ g 关 C 12兴 ⫺ ␸ g 关 C 23兴
⫽⫺
冕␻
共4.1兲
S3
and then
⬘ 艛C̃ 32
⬘ 艛C 31
⬘兴
⫽ ␸ g 关 C̃ 21
i
Tr兵 ␳ 共 s 兲关 ␳ 共 s ⬘ 兲 , ␳ 共 s ⬙ 兲兴 其 ⫽ n̂ 共 s 兲 •n̂ 共 s ⬘ 兲 ⫻n̂ 共 s ⬙ 兲 . 共4.2兲
2
⫽⫺arg ⌬ 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 ,
i.e., ␸ g 关 C 12艛C 23兴 ⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 23兴
⫺arg ⌬ 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 .
共3.26兲
This is a known result, the purpose here was to derive it as a
logical consequence of Eq. 共3.16兲.
Parametrizing the n̂’s with spherical polar angles ␪ , ␾ on S 2
in the usual way, and assuming with no loss of generality
n̂(s)⫽(0,0,1), the vanishing of n̂(s)•n̂(s ⬘ )⫻n̂(s ⬙ ) for all
independent s ⬘ ,s ⬙ amounts to
x 2共 s ⬘ 兲
x 1共 s ⬘ 兲
⫽tan ␾ 共 s ⬘ 兲 ⫽const.
共4.3兲
Hence, the NPC condition corresponds, after rotating to the
configuration n̂(s)⫽(0,0,1), to ␾ ⫽const; and we find that
NPC’s are great circle arcs or geodesics on S 2 . Given any
two nonantipodal points on S 2 , then an NPC connecting
them is either the corresponding geodesic, or it may explore
some more extended portion of the corresponding great
circle. The vast generalization involved in going from geodesics to NPC’s really shows up only for dimH⭓3.
For convenience, as in the preceding section, the present
one is also divided into further sections.
A. Examples of null phase curves
Let two distinct points ␳ 1 , ␳ 2 苸R with Tr( ␳ 1 ␳ 2 )⬎0 be
given. We will construct examples of class-II curves C傺R
from ␳ 1 to ␳ 2 which are NPC’s. Since by Eq. 共2.9兲 every
point ␳ (s)苸C must obey Tr关 ␳ 1 ␳ (s) 兴 ⬎0, it follows that C
must lie entirely in the neighborhood R( ␳ 1 )傺R of ␳ 1 defined in the manner of Eq. 共A.8兲. We can therefore use a local
description of R( ␳ 1 ) as set up in Appendix A.
Let ␺ 1 苸 ␲ ⫺1 ( ␳ 1 ) and choose ␺ 2 苸 ␲ ⫺1 ( ␳ 2 ) such that
( ␺ 1 , ␺ 2 ) is real positive. Introduce an angle ␪ 0 by
共 ␺ 1 , ␺ 2 兲 ⫽cos ␪ 0 ,
␪ 0 苸 共 0,␲ /2兲 .
共4.4兲
Let us when convenient write ␺ 1 ⫽e 1 . According to Eq.
共A.10兲 we can express ␺ 2 in the form
␺ 2 ⫽e 1 cos ␪ 0 ⫹e 2 sin ␪ 0 ,
共 e 1 ,e 2 兲 ⫽0,
储 e 2 储 ⫽1,
共4.5兲
so ␺ 1 ⫽e 1 ,e 2 form an orthonormal pair. As in Appendix A
we supplement e 1 ,e 2 by further vectors e 3 ,e 4 , . . . 苸B, terminating with e n if dimension H⫽n is finite, such that 兵 ␺ 1
⫽e 1 ,e 2 ,e 3 , . . . 其 is an orthonormal basis for H. Let C
⫽ 兵 ␺ (s) 兩 s 1 ⭐s⭐s 2 其 be a lift of C from ␺ 1 to ␺ 2 . For ␺ (s),
we write
␺ 共 s 兲 ⫽x 1 共 s 兲 e 1 ⫹x 2 共 s 兲 e 2 ⫹x 3 共 s 兲 e 3 ⫹•••,
x 1 共 s 兲 ⫽0,
x 1 共 s 兲 cos ␪ 0 ⫹x 2 共 s 兲 sin ␪ 0 ⫽0,
兩 x 1 共 s 兲 兩 2 ⫹ 兩 x 2 共 s 兲 兩 2 ⫹ 兩 x 3 共 s 兲 兩 2 ⫹•••⫽1,
s 1 ⭐s⭐s 2 .
共4.6兲
The coefficients x 1 (s),x 2 (s), . . . must be continuous once
differentiable. At the end points, they have real values
x 共 s 1 兲 ⫽ 共 1,0,0,0, . . . 兲 ,
x 共 s 2 兲 ⫽ 共 cos ␪ 0 ,sin ␪ 0 ,0,0, . . . 兲 .
共4.7兲
Now choose any integer m苸(3,4, . . . ,n) and consider
the real unit sphere S m⫺1 傺Rm . Assume x(s) to have real
components, with x m⫹1 (s)⫽x m⫹2 (s)⫽•••⫽0 for s
苸 关 s 1 ,s 2 兴 . Thus the first m components of x(s) describe a
moving point on S m⫺1 , with x 1 (s) and x 1 (s)cos ␪0
⫹x2(s)sin ␪0 nonzero throughout. Let us further limit ourselves when s 1 ⬍s⬍s 2 to vectors on S m⫺1 with all components strictly positive. That is, generalizing the positive
quadrant and octant in two and three dimensions, we define
m⫺1
⫽ 兵 x苸S m⫺1 兩 x 1 ,x 2 , . . . ,x m ⬎0 其 傺S m⫺1 ,
S⫹
共4.8兲
and choose
m⫺1
,
x 共 s 兲 苸S ⫹
s 1 ⬍s⬍s 2 .
共4.9兲
Then the vectors
␺ 共 s 兲 ⫽x 1 共 s 兲 e 1 ⫹x 2 共 s 兲 e 2 ⫹x 3 共 s 兲 e 3 ⫹•••⫹x m 共 s 兲 e m
共4.10兲
obey
„␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽x 共 s 兲 •x 共 s ⬘ 兲 ⫽real positive,
s,s ⬘ 苸 关 s 1 ,s 2 兴 .
共4.11兲
Condition 共3.3兲 for C,C to be an NPC is clearly satisfied,
so we have succeeded in constructing infinitely many
NPC’s from ␳ 1 to ␳ 2 . In this construction, the integer
m苸(3,4, . . . ,n), and the vectors e 3 ,e 4 , . . . ,e m forming
along with ␺ 1 ⫽e 1 and e 2 an orthonormal set in B may each
be freely chosen; and then x(s) for s 1 ⬍s⬍s 2 is any oncem⫺1
obeying the boundary condidifferentiable curve on S ⫹
tions 共4.7兲 at the end points.
This great profusion of NPC’s as compared to geodesics,
available only when dimension H⭓3, is an indication that
the former are not solutions to any system of local ordinary
differential equations with some boundary conditions, in the
way familiar with geodesics. It is also clear from the deductive development in Sec. III that it is NPC’s that are basic to
the theory of the GP, and in a sense it is incidental that
geodesics are NPC’s. These remarks lead to the following
interesting questions: given any two distinct nonorthogonal
points ␳ 1 , ␳ 2 苸R, how can we describe in a constructive
sense the most general NPC from ␳ 1 to ␳ 2 ; and how can we
characterize a smooth submanifold M 傺R if it has the property that every continuous once-differentiable curve C傺M is
an NPC ? The latter kind of question is clearly not meaningful in the case of geodesics. We will find that here again the
third-order BI plays a key role.
B. Description of a general null phase curve
Now we develop a description of the most general
NPC C⫽ 兵 ␳ (s) 兩 s 1 ⭐s⭐s 2 其 connecting two given points
␳ 1 , ␳ 2 苸R with Tr( ␳ 1 ␳ 2 )⬎0. We assume vectors
␺ 1,2苸 ␲ ⫺1 ( ␳ 1,2) obeying Eqs. 共4.4兲 and 共4.5兲 have been chosen. Let C0 ⫽ 兵 ␺ 0 (s) 兩 s苸 关 s 1 ,s 2 兴 其 be the particular lift of C,
from ␺ 1 to ␺ 2 , obeying the condition 共3.6兲 so
␺ 0共 s 1 兲 ⫽ ␺ 1 , ␺ 0共 s 2 兲 ⫽ ␺ 2 ,
„␺ 0 共 s ⬘ 兲 , ␺ 0 共 s 兲 …⫽real positive,
s ⬘ ,s苸 关 s 1 ,s 2 兴 .
共4.12兲
We expand ␺ 0 (s) as
␺ 0 共 s 兲 ⫽x 1 共 s 兲 e 1 ⫹x 2 共 s 兲 e 2 ⫹ ␹ 共 s 兲 ,
„e 1 , ␹ 共 s 兲 …⫽„e 2 , ␹ 共 s 兲 …⫽0,
s苸 关 s 1 ,s 2 兴 .
共4.13兲
At this stage, x 1 (s) and x 2 (s) are complex continuous oncedifferentiable functions of s, while ␹ (s) is a continuous
once-differentiable vector in the subspace H⬜ ( ␺ 1 , ␺ 2 )傺H
orthogonal to the pair ␺ 1 , ␺ 2 i.e., to e 1 ,e 2 . At the end points,
we have
x 1 共 s 1 兲 ⫽1, x 2 共 s 1 兲 ⫽ ␹ 共 s 1 兲 ⫽0;
x 1 共 s 2 兲 ⫽cos ␪ 0 , x 2 共 s 2 兲 ⫽sin ␪ 0 , ␹ 共 s 2 兲 ⫽0. 共4.14兲
Now we draw out step by step the implications of the real
positivity condition 共4.12兲, and of ␺ 0 (s)苸B for all s. From
the real positivity of „␺ 1 , ␺ 0 (s)… and „␺ 2 , ␺ 0 (s)…, we get
x 1 共 s 兲 ⫽real positive,
x 1 共 s 兲 cos ␪ 0 ⫹x 2 共 s 兲 sin ␪ 0 ⫽real positive, s苸 关 s 1 ,s 2 兴 .
共4.15兲
These imply
x 2 共 s 兲 ⫽real,
x 1 共 s 兲 cos
␪0
␪0
⫹x 2 共 s 兲 sin ⫽real positive, s苸 关 s 1 ,s 2 兴 .
2
2
共4.16兲
Thus, x 1 (s) can never vanish, while x 2 (s) could vanish, as it
does at s⫽s 1 , or even sometimes be negative. Next from the
normalization of ␺ 0 (s), we have
储 ␺ 0 共 s 兲储 ⫽1⇔x 1 共 s 兲 2 ⫹x 2 共 s 兲 2 ⫹ 储 ␹ 共 s 兲储 2 ⫽1,
s苸 关 s 1 ,s 2 兴 .
共4.17兲
We therefore parametrize x 1 (s) and x 2 (s) by
x 1 共 s 兲 ⫽ ␴ 共 s 兲 cos ␪ 共 s 兲 ,
0⬍ ␴ 共 s 兲 ⭐1,
␴ 共 s 1 兲 ⫽ ␴ 共 s 2 兲 ⫽1,
x 2 共 s 兲 ⫽ ␴ 共 s 兲 sin ␪ 共 s 兲 ,
FIG. 1. The allowed region in the x 1 -x 2 plane is the segment
OAB of the unit disc, excluding the radii OA and OB.
hence x 1 (s) and x 2 (s), can be chosen freely subject to the
bounds and conditions given in Eqs. 共4.18兲 and 共4.20兲.
Now we turn to the more comprehensive positivity condition 共4.12兲,
„␺ 0 共 s ⬘ 兲 , ␺ 0 共 s 兲 …⫽real positive
⇔ ␴ 共 s ⬘ 兲 ␴ 共 s 兲 cos„␪ 共 s ⬘ 兲 ⫺ ␪ 共 s 兲 …
⫹„␹ 共 s ⬘ 兲 ,
␹ 共 s 兲 …⫽real positive,
s ⬘ , s苸 关 s 1 ,s 2 兴 .
An immediate conclusion is that for all s ⬘ and
s,„␹ (s ⬘ ), ␹ (s)… is real. One can show quite easily that this
means the following: there is some orthonormal basis
兵 e 3 ,e 4 , . . . 其 for the subspace H⬜ ( ␺ 1 , ␺ 2 )傺H such that
s苸 关 s 1 ,s 2 兴 ;
␪ 共 s 1 兲 ⫽0,
␹共 s 兲⫽
␪ 共 s 2 兲 ⫽ ␪ 0 . 共4.18兲
Both ␴ (s) and ␪ (s) are continuous once differentiable, and
for the norm of ␹ (s) we have
储 ␹ 共 s 兲储 ⫽„1⫺ ␴ 共 s 兲 2 …1/2苸 关 0,1兲 .
共4.19兲
The positivity conditions 共4.15兲 lead to the allowed range for
␪ (s),
⫺ ␲ /2⫹ ␪ 0 ⬍ ␪ 共 s 兲 ⬍ ␲ /2,
共4.20兲
which exceeds ␲ /2 in extent. The permitted region in the
x 1 -x 2 plane is thus a segment OAB of the unit disk subtending an angle ( ␲ ⫺ ␪ 0 ) at the center, as shown in Fig. 1. The
open arc A to B is included, while the end points A,B, and
the radii OA, OB are excluded. At this point, ␴ (s) and ␪ (s),
共4.21兲
兺
r⫽3,4, . . .
x r共 s 兲 e r ,
x r 共 s 兲 ⫽real,
„␹ 共 s ⬘ 兲 , ␹ 共 s 兲 …⫽
储 ␹ 共 s 兲储 2 ⫽
兺
兺
r⫽3,4, . . .
r⫽3,4, . . .
x r 共 s ⬘ 兲 x r 共 s 兲 ⫽real,
x r 共 s 兲 2 ⫽1⫺ ␴ 共 s 兲 2 苸 关 0,1兲 . 共4.22兲
The orthonormal set 兵 e 3 ,e 4 , . . . 其 , defined of course upto a
real orthogonal transformation, depends in general on the
particular NPC C from ␳ 1 to ␳ 2 that we are describing; it
need not be the same for all NPC’s from ␳ 1 to ␳ 2 . The
information on 储 ␹ (s) 储 leads via the Cauchy-Schwartz inequality to a bound on the magnitude of the last term on the
right-hand side of the condition 共4.21兲:
兩 „␹ 共 s ⬘ 兲 , ␹ 共 s 兲 …兩 ⭐ 兵 „1⫺ ␴ 共 s ⬘ 兲 2 …„1⫺ ␴ 共 s 兲 2 …其 1/2. 共4.23兲
Since, as already mentioned, the range for ␪ (s) in Eq. 共4.20兲
exceeds ␲ /2 in extent, the difference ␪ (s ⬘ )⫺ ␪ (s) can sometimes exceed ␲ /2 in magnitude; this would make the first
term in the inequality 共4.21兲 negative. In that case,
„␹ (s ⬘ ), ␹ (s)… must be positive and large enough to compensate for this, while still subject to Eq. 共4.23兲. We are thus led
to an interesting nonlocal condition on ␴ (s) and ␪ (s):
␴共 s⬘兲␴共 s 兲
兵 „1⫺ ␴ 共 s ⬘ 兲 2 …„1⫺ ␴ 共 s 兲 2 …其
兩 cos„␪ 共 s ⬘ 兲 ⫺ ␪ 共 s 兲 …兩 ⭐1
1/2
if 兩 ␪ 共 s ⬘ 兲 ⫺ ␪ 共 s 兲 兩 ⬎ ␲ /2.
共4.24兲
This condition, being nonlocal, cannot be translated in any
simple way to a further restriction on the so-far allowed
ranges for ␴ (s) and ␪ (s), but must be carried along as a
nontrivial condition to be obeyed by them. For such allowed
choices of ␴ (s) and ␪ (s), ␹ (s) must then be chosen as in Eq.
共4.22兲 ensuring that the inequality 共4.21兲 is obeyed.
This is the extent to which an explicit constructive description of a general NPC from ␳ 1 to ␳ 2 can be given.
Admittedly, it is much less ‘‘complete’’ than the description
we can give for a geodesic, again an indication that NPC’s
form a much larger family of curves than do geodesics. We
can now see that the examples of NPC’s given in the preceding section correspond to the special simplifying assumption
that the functions x 1 (s),x 2 (s),x 3 (s), . . . are all nonnegative. Then 兩 ␪ (s ⬘ )⫺ ␪ (s) 兩 never exceeds ␲ /2, and the
nonlocal conditions 共4.21兲 and 共4.24兲 are automatically
obeyed as even „␹ (s ⬘ ), ␹ (s)… is throughout real nonnegative.
C. Submanifolds of null phase curves
Now we enlarge the scope of our analysis of NPC’s and
ask the question: how can we characterize a smooth submanifold M 傺R if every once-differentiable curve C傺M is
to be an NPC? For brevity let us call such a submanifold an
NPM. To answer this question we first assemble the basic
formula which generalizes Eq. 共3.26兲 and connects GP’s,
BI’s, and two-dimensional surface integrals of the two-form
␻ . Let C傺R be any closed class-I curve, and S傺R any
two-dimensional surface with ⳵ S⫽C. Let ␳ 1 , ␳ 2 , ␳ 3 be any
three pairwise nonorthogonal points chosen in sequence
along C. Denote the successive portions of C from ␳ 1 to ␳ 2 ,
␳ 2 to ␳ 3 and ␳ 3 to ␳ 1 by C 12 ,C 23 , and C 31 , respectively.
Then we have the relation
␸ g 关 C⫽C 12艛C 23艛C 31兴 ⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 23兴 ⫹ ␸ g 关 C 31兴
⫺arg ⌬ 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲
⫽⫺
冕␻
S
共4.25兲
for any ␺ j 苸 ␲ ⫺1 ( ␳ j ), j⫽1,2,3. If as a special case we set
␳ 3 ⫽ ␳ 2 here, recall the definition 共3.16兲 and the fact that
⌬ 2 ( ␺ 1 , ␺ 2 ) is real positive, we recover Eq. 共3.27兲. Thus, the
latter is a particular special case of Eq. 共4.25兲. However, in
the reverse direction, of course, it is not possible to derive
Eq. 共4.25兲 from Eq. 共3.27兲. Now Eq. 共4.25兲 can be generalized to the case where we choose any m⭓4 points
␳ 1 , ␳ 2 , . . . , ␳ m 苸C, located one after the other in a sequence, with no two consecutive ones being orthogonal.
Then in an obvious notation we have
␸ g 关 C⫽C 12艛C 23艛•••艛C m1 兴
⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 23兴 ⫹•••⫹ ␸ g 关 C m1 兴
⫺arg ⌬ m 共 ␺ 1 , ␺ 2 , . . . , ␺ m 兲
⫽⫺
冕␻
,
共4.26兲
m⭓4.
S
This is also derivable from Eq. 共4.25兲 which is the primitive
relation of this kind. The proofs of Eqs. 共4.25兲 and 共4.26兲
follow the pattern of the arguments in Sec. III. In these relations, if all the segments C 12 ,C 23 , . . . ,C m1 are NPC’s, we
recover Eqs. 共3.23兲 and 共3.24兲.
Now let M 傺R be a connected, simply connected smooth
submanifold with dimension M ⭓2 in the real sense 关19兴.
The corresponding identification map is i M :M R. The
pullback to M of the symplectic two-form ␻ on R is
␻ M ⫽i *
M␻.
共4.27兲
This is of course closed but may well be degenerate. In the
extreme case of an isotropic submanifold it vanishes,
M isotropic⇔ ␻ M ⫽0.
共4.28兲
If in addition dim M ⫽(n⫺1), it is a Lagrangian submanifold.
Assume now that M is an NPM. Every once-differentiable
curve C傺M must then be of class II and we get the following two consequences:
M is an NPM⇒M is isotropic,
␻ M ⫽0;
M is an NPM⇒for any ␳ j ⫽ ␲ 共 ␺ j 兲 苸M ,
⌬ 3共 ␺ 1 , ␺ 2 , ␺ 3 兲
共4.29a兲
j⫽1,2,3,
is real positive.
共4.29b兲
The first follows from Eq. 共3.14兲 after specializing to
C,C ⬘ ,S傺M . The second then follows from the first upon
use of Eq. 共4.25兲, again specializing to C 12 ,C 23 ,C 31 ,S傺M .
From Eq. 共4.29b兲 we also see by taking ␳ 3 ⫽ ␳ 2 that any two
points in M are nonorthogonal. This means that every oncedifferentiable curve C傺M is of class II.
Conversely, if we start by assuming Eq. 共4.29b兲 for M, we
deduce
⌬ 3 ( ␺ 1 , ␺ 2 , ␺ 3 ) real positive,
any ␳ j ⫽ ␲ 共 ␺ j 兲 苸M ,
j⫽1,2,3
⇒Tr共 ␳ 1 ␳ 2 兲 ⬎0,
any ␳ 1 , ␳ 2 苸M ;
共4.30a兲
⇒M is an NPM;
共4.30b兲
⇒M is isotropic, ␻ M ⫽0.
共4.30c兲
These three statements are not independent since, by Eqs.
共4.29兲, 共4.30b兲 implies Eq. 共4.30c兲. In any case, we see from
Eqs. 共4.29b兲 and 共4.30b兲 together that the necessary and sufficient condition for M to be an NPM is that ⌬ 3 ( ␺ 1 , ␺ 2 , ␺ 3 )
be real positive if ␳ j ⫽ ␲ ( ␺ j )苸M , j⫽1,2,3. It then follows
that the GP for an open or closed class-I curve lying entirely
in an NPM is zero. Isotropy of M is thus only a necessary but
not a sufficient condition to ensure this property for M. It is
therefore instructive to see how far one can go on the basis of
isotropy alone. We now examine this point.
Let M be a connected, simply connected submanifold in
R, such that Tr( ␳ 1 ␳ 2 )⬎0 for any ␳ 1 , ␳ 2 苸M . As in the preceding section for an NPC, we construct a lift of M to a
submanifold M0 傺 B, in the spirit of the Pancharatnam lift.
Choose any point ␳ 0 苸M and then any ␺ 0 苸 ␲ ⫺1 ( ␳ 0 )傺B.
The lift M0 is then completely and uniquely defined by the
rule 关generalization of Eq. 共3.11兲兴
␳ 苸M → ␺ ⫽
␳␺0
冑Tr共 ␳ 0 ␳ 兲
苸M0 ,
␲共 ␺ 兲⫽␳,
共4.32兲
Of course, ␳ 0 苸M is lifted to ␺ 0 苸M0 . This lift M
→M0 is characterized by the fact that each vector ␺ 苸M0
is in phase with ␺ 0 in the Pancharatnam sense, from Eq.
共4.31兲,
共 ␺0 ,␺ 兲⫽
共␺0 ,␳␺0兲
冑Tr共 ␳ 0 ␳ 兲
⫽ 冑Tr共 ␳ 0 ␳ 兲 ⫽real⬎0.
共 ␺ ⬘ , ␺ 兲 ⫽ 共 ␳ ⬘ ␺ 0 , ␳ ␺ 0 兲 / 冑Tr共 ␳ 0 ␳ ⬘ 兲 Tr共 ␳ 0 ␳ 兲
⫽Tr共 ␳ 0 ␳ ⬘ ␳ 兲 / 冑Tr共 ␳ 0 ␳ ⬘ 兲 Tr共 ␳ 0 ␳ 兲 ,
arg共 ␺ ⬘ , ␺ 兲 ⫽arg ⌬ 3 共 ␺ 0 , ␺ ⬘ , ␺ 兲 .
共4.34兲
Thus, whether or not general pairs of points in M0 are in
phase depends entirely on whether the BI’s ⌬ 3 ( ␺ 0 , ␺ ⬘ , ␺ ) for
triplets of points in M are real positive or complex. This is
the sense in which the lift M →M0 is as nearly a Pancharatnam lift as possible in a general case.
Now let us impose the condition that M be isotropic,
␻ M ⫽.i *
M ␻ ⫽0.
arg共 ␺ 1 , ␺ 2 兲 ⫺
冕
C12
i.e.,
冕
共4.35兲
⬘ are any two class-I
Then Eq. 共3.27兲 shows that if C 12 ,C 12
curves in M from any ␳ 1 苸M to any ␳ 2 苸M , the GP’s are the
same;
A⫽arg共 ␺ 1 , ␺ 2 兲 ⫺
C12
A⫽
冕
C 12
⬘
冕
C 12
⬘
A,
A.
共4.37兲
This means that 共as we have assumed simple connectedness兲 the pullback of A from B to M0 is exact. Denoting the
relevant identification map as i M0 :M0 B, we have
* A⫽d f
␻ M ⫽0⇔i M
0
共4.38兲
for some f 苸F(M0 ). For emphasis we repeat that isotropy
* A is exact, and allows
of M allows us to conclude that i M
0
( ␺ , ␺ ⬘ ) for ␺ , ␺ ⬘ 苸M0 to have a nontrivial phase.
If at this point we assume in addition that M is an NPM,
we immediately see that we have much stronger conclusions;
共 ␺ , ␺ ⬘ 兲 ⫽real positive, any ␺ , ␺ ⬘ 苸M0 ,
* A⫽0.
iM
共4.33兲
However, if we take two general vectors ␺ , ␺ ⬘ 苸M0 , we
find that their inner product is, in general, complex:
共4.36兲
Furthermore, it is clear that this statement exhausts the content of isotropy. Since the lift M →M0 is unique 共given ␳ 0
and ␺ 0 ), we can faithfully transcribe this statement to M0 ,
which is more convenient since we then deal with vectors.
Denote by ␺ 1 and ␺ 2 the lifts of ␳ 1 and ␳ 2 to M0 , and by
⬘ the 共unique兲 lifts of C 12 ,C 12
⬘ to M0 . Then the full
C12 ,C 12
content of Eq. 共4.36兲 is expressed as follows: for any two
⬘ in
points ␺ 1 , ␺ 2 苸M0 , and for any class-I curves C12 ,C 12
M0 connecting them, we have
共4.31兲
so M0 as a subset of B is displayed as
M0 ⫽ 兵 ␳ ␺ 0 / 冑Tr共 ␳ 0 ␳ 兲 , ␳ 苸M 其 傺B.
⬘ 兴 ⫽ ␸ g 关 C 12兴 .
␸ g 关 C 12
0
共4.39兲
These results imply that now M0 is truly a Pancharatnam lift
of M, and they show very effectively the extent to which the
NPM property goes beyond isotropy.
To round out this discussion we give some examples of
submanifolds M 傺R, connected and simply connected, possessing the NPM property. To begin with, we use a construction similar to that used in Sec. IV A to construct families of
NPC’s connecting any two given nonorthogonal points
␳ 1 , ␳ 2 苸R. For any m⭓3 共upto n in case dim H⫽n is finite兲, let e r ,r⫽1,2, . . . ,m, be an orthonormal set of vectors
in H. We first define a submanifold M傺B as consisting of
all real normalized ‘‘positive’’ linear combinations of the e r :
再
m
M⫽ ␺ 共 x 兲 ⫽
m
兺
r⫽1
兺
r⫽1
x r e r 苸B兩 x r real positive,
冎
x r2 ⫽1 傺B,
and then take the projection to get M,
共4.40兲
M ⫽ ␲ 共 M兲 傺R.
共4.41兲
Both M and M are of real dimension (m⫺1), in fact M is
m⫺1
of Eq. 共4.8兲. By construction we see that
essentially S ⫹
m
„␺ 共 x ⬘ 兲 , ␺ 共 x 兲 …⫽x ⬘ •x⫽
兺
r⫽1
x r⬘ x r ⫽real positive,
共4.42兲
and therefore for any three points in M,
⌬ 3 „␺ 共 x 兲 , ␺ 共 x ⬘ 兲 , ␺ 共 x ⬙ 兲 …⫽x•x ⬘ x ⬘ •x ⬙ x ⬙ •x⫽real positive.
共4.43兲
This ensures that every once-differentiable curve C傺M , obtained by setting x⫽x(s) for suitable functions x r (s), is of
class II and also an NPC.
After this abstract example of an NPM, we give two others involving explicit families of Schrödinger wave functions, which are simple but quite relevant. The context is the
family of coherent states of a system of N identical simple
harmonic oscillators. Starting with the normalized vacuum
state 兩 0 典 with the wave function
兩 0 典 → ␺ 0 共 X 兲 ⫽ ␲ ⫺N/4 exp共 ⫺ 21 X•X 兲 ,
N
X•X⫽
兺
j⫽1
x 2j ,
共4.44兲
the spatial translates of 兩 0 典 , 兩 Y 典 say, have wave functions
兩 Y 典 → ␺ Y 共 X 兲 ⫽ ␲ ⫺N/4 exp关 ⫺ 21 共 X⫺Y 兲 • 共 X⫺Y 兲兴 .
共4.45兲
Taken for all Y 苸R n , these states constitute a submanifold
R n in the manifold R 2n of all coherent states; and the
former clearly form an NPM. Clearly, the image of this
manifold of states under any unitary transformation will also
be an NPM. In particular, the submanifold of all momentum
translates of the vacuum state form an NPM.
The second example is in the context of the manifold
S p(2N,R)/U(N) of all squeezed vacuum states obtained as
the orbit of the state 共4.44兲 under the unitary action of the
group Sp(2N,R) of all real linear canonical transformations
关20兴. These states can be characterized by a pair of real symmetric N⫻N matrices u, v with u⬎0, and so they form an
N(N⫹1)-dimensional submanifold in Hilbert space. The
corresponding normalized wave functions are
兩 u, v 典 → ␺ (u, v ) 共 X 兲 ⫽ ␲ ⫺N/4共 det u 兲 1/4 exp兵 ⫺ 21 X T 共 u⫹i v 兲 X 其 ,
共4.46兲
and for general u, v these are complex. This set of wave
functions constitute a generalization of the Poincaré upperhalf plane, with Sp(2N,R) acting on u, v through 共matrix兲
fractional linear transformations. In this set if we now limit
ourselves to those with v ⫽0, all the wave functions
␺ (u,0) (X) are real and all scalar products among them are real
positive. Thus we have another example of an NPM. Again
the image of this submanifold under any unitary transformation retains this property.
V. CONCLUDING REMARKS
We have developed an approach to the theory of the GP in
quantum mechanics, in which the basic ingredient is the
three-vertex BI of quantum mechanics. This invariant leads
to the definition of NPC’s in the Hilbert and ray spaces of
quantum systems. In turn, this lets us define the GP associated with any suitable 共open or closed兲 ray space curve as an
area integral of the ray space symplectic form. The emphasis
has been on the logical basis and consistency of the entire
development, and the crucial role of NPC’s. We have shown
that it is the BI which is the truly fundamental concept underlying all the others. In the course of the development, we
have taken care to define with precision the classes of ray
and Hilbert space curves that one must work with for each
purpose.
NPC’s are a vast and important generalization of the more
familiar family of geodesics in Hilbert and ray spaces, and it
turns out that they truly belong to the theory of the GP. It
happens to be true that geodesics are instances of NPC’s;
however the latter are far more numerous by any measure,
and are intrinsically of a quite different nature. Thus, length
of a curve and its minimization are not at all the relevant
concepts in arriving at NPC’s. This makes their description
considerably more difficult than of geodesics, for which a
differential equation treatment is available. They have deep
properties of a nonlocal nature. The examples of NPC’s and
NPM’s, and their general properties brought out in our discussion, should help in aiding our understanding this important class of quantum mechanical objects.
The Pancharatnam lift M0 of an NPM M is characterized
by the two properties 共4.39兲. The first actually implies the
second. Its structure suggests the following nonlocal operation or construction: pass from the collection of vectors
␺ , ␺ ⬘ , . . . 苸M0 to its real linear hull, i.e., form all real linear combinations of any numbers of vectors in M0 共and then
normalize them to get results in B). This much enlarged
collection of vectors in B is clearly associated with a real
subspace of H all inner products among whose vectors are
real 共but of course not anymore always positive兲. The consideration of NPM’s leads in a natural way to associate real
linear subspaces in H within which the Hermitian scalar
product of H reduces to a real symmetric scalar product.
Such a subspace is clearly ␲ * ␻ isotropic, and we are led to
consider trying to characterize NPM’s via such associated
subspaces.
It is useful to view all this also from another perspective.
The Fubini-Study metric and the symplectic form on the
quantum-mechanical ray space both originate from the
Kahler form, as its real and imaginary parts, respectively
关21兴. While the geodesics stem from the metric, the BI,
NPC’s, and GP are all more naturally related to the symplectic structure. Hence, it is that NPC’s and not geodesics, form
the principal notion in the GP context.
We have seen in Sec. IV that it is only when dim H⭓3
that the true differences between NPC’s and geodesics
emerge. The situations in which the GP has been traditionally
studied in detail have involved two-dimensional symplectic
manifolds: CP1 or S 2 corresponding to a two-level system
such as polarized light or spin-1/2 particles; the twodimensional plane R 2 as in the coherent states of a harmonic
oscillator; and the timelike two-dimensional upper half unit
hyperboloid in (2⫹1)-dimensional space 共equivalently the
unit disc or the upper half complex plane兲, as in the squeezed
vacuum states of an oscillator. In all these cases, there is
simply no extra room in these minimal situations for the
differences between NPC’s and geodesics to show up. Thus,
it is understandable that the geodesics were thought to be
intrinsically relevant to GP discussions, but this status rightfully belongs to the NPC’s.
We have examined elsewhere the conditions under which
constrained geodesics, i.e., geodesics among curves restricted to lie within a given ray space submanifold could
turn out to be NPC’s 关12兴. Important examples when this
happens, and the corresponding BI-GP connections which
are of physical interest, have been given. The more comprehensive account of NPC’s and NPM’s presented in this paper
should enable us to study the connections to constrained geodesics in a more definite manner. We intend to return to this
and related problems elsewhere.
APPENDIX A: BASIC DIFFERENTIAL GEOMETRY
OF RAY SPACE, LOCAL DESCRIPTIONS
For the convenience of the reader we collect here some
basic definitions relating to the unit sphere B傺H, the ray
space R⫽ ␲ (B), defined in Eqs. 共2.1兲 and 共2.3兲, and geometric objects associated with them. Convenient local coordinate
descriptions of some of them are also given. For definiteness,
we may assume that dimension H⫽n is finite.
The space B⫽S 2n⫺1 is a differentiable manifold. At any
point ␺ 0 苸B, the tangent space is
T ␺ 0 B⫽ 兵 ␾ 苸H兩 Re共 ␺ 0 , ␾ 兲 ⫽0 其 .
共A1兲
This is clearly a real linear vector space of dimension (2n
⫺1). The connection one-form A defined on B is specified at
each ␺ 0 苸B as a linear functional on T ␺ 0 B is
␾ 苸T ␺ 0 B:A ␺ 0 共 ␾ 兲 ⫽Im共 ␺ 0 , ␾ 兲 ⫽⫺i 共 ␺ 0 , ␾ 兲 .
共A2兲
Next we turn to the ray space R. At a point ␳ 0 ⫽ ␺ 0 ␺ †0
苸R, the tangent space can be defined and then described
explicitly in terms of H ␺ 0 B:
T ␳ 0 R⫽兵B⫽linear operator on H兩 B † ⫽B, TrB⫽0,
兵 B, ␳ 0 其 ⫽B其⫽ 兵 B⫽ ␾␺ †0 ⫹ ␺ 0 ␾ † 兩 ␾ 苸H ␺ 0 B其 .
This is a real linear vector space of dimension 2(n⫺1). In
the latter form, if we change the representative vector ␺ 0
苸 ␲ ⫺1 ( ␳ 0 ) by a phase, we must keep track of the change in
B if ␾ is unchanged, or alternatively change ␾ by a compensating phase to keep B the same.
The two-form dA on B is the pullback of a two-form ␻ on
R,
dA⫽ ␲ * ␻ .
␻ ␳ 0 共 B,B ⬘ 兲 ⫽⫺iTr共 ␳ 0 关 B,B ⬘ 兴 兲 ⫽2Im共 ␾ , ␾ ⬘ 兲 ,
共A7兲
where ␾ , ␾ ⬘ 苸H ␺ 0 B correspond to B,B ⬘ , respectively, as in
the second line of Eq. 共A5兲.
All the above definitions and expressions are coordinate
independent and intrinsic. Now we give local coordinate descriptions for some of them, which are sometimes useful. For
given ␳ 0 苸R, and some chosen ␺ 0 苸 ␲ ⫺1 ( ␳ 0 ), we define an
open neighborhood R( ␳ 0 ) by
R共 ␳ 0 兲 ⫽ 兵 ␳ 苸R兩 Tr共 ␳ 0 ␳ 兲 ⬎0 其 傺R.
共A8兲
Thus, what are excluded from R( ␳ 0 ) are projections ␳ onto
vectors in B which are orthogonal to ␺ 0 , that is, onto vectors
in H ␺ 0 . The corresponding open subset of B is denoted by
B( ␺ 0 ), though in fact it is determined by ␳ 0 ;
B共 ␺ 0 兲 ⫽ ␲ ⫺1 关 R共 ␳ 0 兲兴 ⫽ 兵 ␺ 苸B兩 共 ␺ 0 , ␺ 兲 ⫽0 其 傺B. 共A9兲
We can give an explicit formula for any ␳ 苸R( ␳ 0 ) as follows:
␳ 苸R共 ␳ 0 兲 ⇔ ␳ ⫽ ␳ 共 ␾ 兲
⫽ 关 ␾ ⫹ 共 1⫺ 储 ␾ 储 2 兲 1/2␺ 0 兴
H ␺ 0 B⫽ 兵 ␾ 苸T ␺ 0 B兩 A ␺ 0 共 ␾ 兲 ⫽0 其
⫻ 关 ␾ ⫹ 共 1⫺ 储 ␾ 储 2 兲 1/2␺ 0 兴 † ,
共A3兲
This is of course a real linear vector space of dimension
2(n⫺1), but it is in a natural sense a complex linear vector
space of dimension (n⫺1), namely, the subspace of H orthogonal to ␺ 0 .
The two-form dA is, at each ␺ 0 苸B, an antisymmetric
bilinear functional on T ␺ 0 B,
␾ , ␾ ⬘ 苸T ␺ 0 B: 共 dA 兲 ␺ 0 共 ␾ , ␾ ⬘ 兲 ⫽2Im共 ␾ , ␾ ⬘ 兲 .
共A6兲
As an antisymmetric bilinear functional on T ␳ 0 R, ␻ is specified by
Therefore, the horizontal subspace of T ␺ 0 B is defined as
⫽ 兵 ␾ 苸H兩 共 ␺ 0 , ␾ 兲 ⫽0 其 傺T ␺ 0 B.
共A5兲
共A4兲
␾ 苸H ␺ 0 B,
储 ␾ 储 ⬍1;
共A10兲
and then
Tr„␳ 0 ␳ 共 ␾ 兲 …⫽„␺ 0 , ␳ 共 ␾ 兲 ␺ 0 …⫽1⫺ 储 ␾ 储 2 ⬎0.
共A11兲
Thus, points in R( ␳ 0 ) are in one-to-one correspondence
with, and are coordinatized by points inside the unit sphere
in the subspace H ␾ 0 B of H. By adding a phase factor e i ␣ ,
we get a local description for B( ␺ 0 ) as
␺ 苸B共 ␺ 0 兲 ⇔ ␺ ⫽ ␺ 共 ␾ , ␣ 兲 ⫽e i ␣ „␾ ⫹ 共 1⫺ 储 ␾ 储 2 兲 1/2␺ 0 …,
␾ 苸H ␺ 0 B,
储 ␾ 储 ⬍1,
0⭐ ␣ ⬍2 ␲ .
共A12兲
Let 兵 ␺ 0 ⫽e 1 ,e 2 , . . . 其 be any orthonormal basis for H.
We expand ␾ 苸H ␺ 0 B as
␾⫽
1
冑2
兺
r⫽2,3, . . .
共 ␤ r ⫺i ␥ r 兲 e r ,
兺
共 ␤ r2 ⫹ ␥ r2 兲 ⬍2.
冉
冊
d
u⬜ 共 s 兲
⫺ 共 ␺ 共 s 兲 ,u 共 s 兲兲
⫽ f 共 s 兲␺共 s 兲,
储 u⬜ 共 s 兲储
ds
共A13兲
共A14兲
d u共 s 兲
⫽ f 共 s 兲␺共 s 兲,
ds 储 u 共 s 兲储
Then 兵 ␤ 2 , ␤ 3 , . . . , ␥ 2 , ␥ 3 , . . . 其 subject to 共A14兲 are real local coordinates, 2(n⫺1) in number over R( ␳ 0 )傺R; while
兵 ␣ , ␤ 2 , ␤ 3 , . . . , ␥ 2 , ␥ 3 , . . . 其 are real local coordinates over
B( ␺ 0 )傺B. In these charts we have the explicit expressions
1
A⫽d ␣ ⫹
2
兺
r⫽2,3, . . .
dA⫽ ␲ * ␻ ⫽
兺
d ␥ r⵩ d ␤ r .
共 ␺ 共 s 兲 ,u 共 s 兲兲 ⫽0,
共A15兲
d 2␺共 s 兲
ds 2
APPENDIX B: PROPERTIES OF GEODESICS
IN RAY SPACE
共B1兲
running from ␳ (s 1 )⫽ ␳ 1 to ␳ (s 2 )⫽ ␳ 2 be a curve of class III,
and assume Tr( ␳ 1 ␳ 2 )⬎0. Let
C⫽ 兵 ␺ 共 s 兲 苸B兩 ␳ 共 s 兲 ⫽ ␺ 共 s 兲 ␺ 共 s 兲 † ,s 1 ⭐s⭐s 2 其 傺B 共B2兲
be any 共class III兲 lift of C. The length of C is the functional
L关 C 兴 ⫽
冕
s2
s1
ds 储 u⬜ 共 s 兲储 ,
u⬜ 共 s 兲 ⫽u 共 s 兲 ⫺„␺ 共 s 兲 ,u 共 s 兲 …␺ 共 s 兲 苸H ␺ (s) B;
u共 s 兲⫽
d
␺ 共 s 兲 苸T ␺ (s) B.
ds
⫽ f 共 s 兲 ␺ 共 s 兲 , f 共 s 兲 real,
冉
储 ␺ 共 s 兲储 ⫽1, ␺ 共 s 兲 ,
We provide here a brief account of the definition, differential equations, and main properties of geodesics in ray
space R. Since they will be found to obey second-order
ordinary differential equations, it is appropriate to work with
curves of class III. Let then
C⫽ 兵 ␳ 共 s 兲 苸R兩 s 1 ⭐s⭐s 2 其 傺R
共B5兲
f 共 s 兲 real.
Next we can use the reparametrization freedom to switch
from the originally given parameter to an affine parameter.
This makes 储 u(s) 储 constant along C, and for the affinely
parametrized horizontal lift C we have in place of Eq. 共B5兲:
共 ␥ rd ␤ r⫺ ␤ rd ␥ r 兲,
r⫽2,3, . . .
共B4兲
f 共 s 兲 arbitrary real.
This equation is naturally covariant with respect to both reparametrizations and local phase transformations. These properties can now be exploited to successively specialise the
choice of the lift C and its parametrization, and thereby simplify the differential equation 共B4兲 关22兴. By a suitable local
phase transformation we can assume that C is a horizontal lift
of C, so we can replace Eq. 共B4兲 by the simpler system
so that the condition 储 ␾ 储 ⬍1 becomes
r⫽2,3, . . .
␦ L关 C 兴 ⫽0⇔
共B3兲
This is both reparametrization and local phase transformation
invariant. We make infinitesimal changes ␦ ␺ (s) in ␺ (s)
along C, inducing changes ␦ ␳ (s) along C, vanishing at the
end points s 1 and s 2 . Requiring ␦ L关 C 兴 ⫽0 we arrive at a
differential equation which must be obeyed if L关 C 兴 is to be
stationary and C is to be a geodesic:
冊
d␺共 s 兲
⫽0,
ds
冐 冐
d␺共 s 兲
⫽const.
ds
共B6兲
A brief analysis shows that this problem is fully equivalent to
a second-order ordinary differential equation with suitable
initial conditions at s⫽s 1 ,
d 2␺共 s 兲
ds 2
储 ␺ 共 s 1 兲储 ⫽1,
⫽⫺
冉
冐 冐
冉 冊
d␺共 s 兲 2
␺共 s 兲,
ds
␺共 s1兲,
d␺共 s 兲
ds
s1
冊
⫽0.
共B7兲
Finally, we make a scale change and shift of origin in the
parameter to make s 1 ⫽0 and 储 d ␺ (s)/ds 储 ⫽1. This completely exhausts the freedom of reparametrizations, and at
this point we may denote the parameter by a special symbol
ᐉ. For the so-defined and parametrized lift C of a geodesic C
in R, we have the solution to Eq. 共B7兲 in the form
␺ 共 ᐉ 兲 ⫽ ␺ 共 0 兲 cos ᐉ⫹ ␺˙ 共 0 兲 sin ᐉ,
储 ␺ 共 0 兲储 ⫽ 储 ␺˙ 共 0 兲储 ⫽1, „␺ 共 0 兲 , ␺˙ 共 0 兲 …⫽0.
共B8兲
This solution is determined by one orthonormal pair of vectors, and every geodesic in R has such a lift in B. The value
of the length functional L关 C 兴 in Eq. 共B3兲, from the starting
point ᐉ⫽0 to a general point ᐉ, is the parameter ᐉ itself:
L关 geodesic C from ᐉ⫽0 to ᐉ 兴 ⫽
冕
ᐉ
0
共B10兲
共iii兲 A horizontal lift C of this geodesic is given by Eq.
共B8兲 by choosing
␺共 0 兲⫽␺1 ,
␺˙ 共 0 兲 ⫽
␺ 2⫺共 ␺ 1 , ␺ 2 兲␺ 1
„1⫺ 共 ␺ 1 , ␺ 2 兲 2 …1/2
.
共B11兲
共iv兲 The end points of C correspond to the parameter values ᐉ⫽0 for ␺ 1 ,ᐉ⫽cos⫺1(␺1 ,␺2)苸关0,␲ /2) for ␺ 2 .
共v兲 The length of this geodesic is cos⫺1(␺1 ,␺2) and is
strictly less than ␲ /2.
共vi兲 For any two points ᐉ,ᐉ ⬘ 苸 关 0,␲ /2) we have
关1兴 M.V. Berry, Proc. R. Soc. London, Ser. A 392, 45 共1984兲; A
very useful reprint collection covering the period upto about
1989 is Geometric Phases in Physics, edited by J. Shapere and
F. Wilczek, 共World Scientific, Singapore, 1989兲.
关2兴 B. Simon, Phys. Rev. Lett. 51, 2167 共1983兲.
关3兴 Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 共1987兲;
J. Samuel and R. Bhandari, ibid. 60, 2339 共1988兲.
关4兴 N. Mukunda and R. Simon, Ann. Phys. 共N.Y.兲 228, 205 共1993兲;
228, 269 共1993兲.
关5兴 V. Bargmann, J. Math. Phys. 5, 862 共1964兲.
关6兴 S. Pancharatnam, Proc.-Indian Acad. Sci., Sect. A 44A, 247
共1956兲.
关7兴 S. Ramaseshan and R. Nityananda, Curr. Sci. 55, 1225 共1986兲;
N. Mukunda and R. Simon, Ref. 关4兴 above.
关8兴 D. Page, Phys. Rev. A 36, 3479 共1987兲; S. Kobayashi and K.
Nomizu, Foundations of Differential Geometry 共Interscience,
New York, 1969兲, Vol. II, Chap. 9.
关9兴 A concise account may be found, for example, V.I. Arnold,
Mathematical Methods of Classical Mechanics 共SpringerVerlag, Berlin, 1978兲, Appendix III.
关10兴 J. Samuel and R. Bhandari, Ref. 关3兴 above.
关11兴 N. Mukunda and R. Simon in Ref. 关4兴 above; R. Simon and N.
Mukunda, Phys. Rev. Lett. 70, 880 共1993兲. For the use of BI’s
in the theory of off-diagonal geometric phases introduced by F.
Pistolesi and N. Manini, ibid. 85, 1585 共2000兲; 85, 3067
共B12兲
and so for any three points ᐉ,ᐉ ⬘ ,ᐉ ⬙ 苸 关 0,␲ /2), we have
⌬ 3 „␺ 共 ᐉ 兲 , ␺ 共 ᐉ ⬘ 兲 , ␺ 共 ᐉ ⬙ 兲 …⫽Tr„␳ 共 ᐉ 兲 ␳ 共 ᐉ ⬘ 兲 ␳ 共 ᐉ ⬙ 兲 …
dᐉ⫽ᐉ. 共B9兲
Going back to the original curve C in Eq. 共B1兲, we identify
␺ (0)⫽ ␺ 1 苸 ␲ ⫺1 ( ␳ 1 ), while ␺ 2 苸 ␲ ⫺1 ( ␳ 2 ) has to be suitably
chosen. A careful analysis of the explicit solution 共B8兲 shows
us that the following results hold.
共i兲 For given ␳ 1 , ␳ 2 苸R obeying Tr( ␳ 1 ␳ 2 )⬎0, there is a
unique geodesic connecting them.
共ii兲 If we make any choice of ␺ 1 苸 ␲ ⫺1 ( ␳ 1 ), we can determine uniquely ␺ 2 苸 ␲ ⫺1 ( ␳ 2 ) by the condition
共 ␺ 1 , ␺ 2 兲 ⫽real positive.
„␺ 共 ᐉ 兲 , ␺ 共 ᐉ ⬘ 兲 …⫽cos共 ᐉ⫺ᐉ ⬘ 兲 ⫽real positive,
⫽real positive.
共B13兲
共vii兲 If we take the limiting case Tr( ␳ 1 ␳ 2 )⫽0, corresponding to ( ␺ 1 , ␺ 2 )⫽0, the geodesic distance from ␳ 1 to
␳ 2 becomes exactly ␲ /2. However there is now no unique
geodesic connecting ␳ 1 to ␳ 2 since, for any ␣ 苸 关 0,2␲ ), the
curve
␺ 共 ᐉ 兲 ⫽ ␺ 1 cos ᐉ⫹e i ␣ ␺ 2 sin ᐉ
共B14兲
is a solution of the variational problem ␦ L关 C 兴 ⫽0, i.e., of
Eq. 共B7兲, and its projection in R runs from ␳ 1 to ␳ 2 as ᐉ
varies over 关 0,␲ /2兴 .
These results are very similar to what are known about
geodesics or great circle arcs on S 2 , which is the ray space R
when dimension H⫽2. Their lengths, in usual units, never
exceed ␲ . For nonantipodal points, there is a unique geodesic with length strictly less than ␲ . For antipodal points we
have a ‘‘2 ␲ worth’’ of geodesics, each of length ␲ .
In a qualitative sense it is easy to see that in a ray space R
of any dimension, two points ␳ 1 and ␳ 2 can never get very
far from one another, since
Tr共 ␳ 1 ⫺ ␳ 2 兲 2 ⫽2 关 1⫺Tr共 ␳ 1 ␳ 2 兲兴 ⭐2.
共B15兲
The properties of geodesics described above are consistent
with this fact.
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
共2000兲, see N. Mukunda, Arvind, S. Chaturvedi, and R. Simon,
Phys. Rev. A 65, 012102 共2001兲.
E.M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A
60, 3397 共1999兲.
These are described also in, for example, Refs. 关2– 4兴 above.
For discussions from the viewpoint of physical applications
see, for instance, A.P. Balachandran, G. Marmo, B.S. Skagerstam, and A. Stern, Applications to Particle Dynamics
共Springer-Verlag, Berlin, 1983兲; Classical Topology and Quantum States 共World Scientific, Singapore, 1991兲.
E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, Int. J.
Mod. Phys. A 16, 5007 共2001兲.
For a recent work on GP’s in real Hilbert spaces see, for instance, J. Samuel and A. Dhar, Phys. Rev. Lett. 87, 260401
共2001兲.
This is to be compared with J. Samuel and R. Bhandari in Ref.
关3兴, where ␳ 2 is connected back to ␳ 1 using the unique geodesic running between them.
This may be regarded as the essential content of the discovery
of S. Pancharatnam in Ref. 关6兴 above.
For discussions of symplectic manifolds and their submanifolds see, for instance, R. Abraham and J.E. Marsden, Foundations of Mechanics 共Benjamin-Cummings, Reading, MA,
1978兲; P. Liberman and C.M. Marle, Symplectic Geometry and
Analytical Mechanics 共Reidel, Dordrecht, 1987兲; G. Marmo,
E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems—A
Differential Geometric Approach to Symmetry and Reduction
共Wiley, New York, 1985兲.
关20兴 R. Simon, E.C.G. Sudarshan, and N. Mukunda, Phys. Rev. A
37, 3028 共1988兲; Arvind, B. Dutta, N. Mukunda, and R. Simon, Pramana, J. Phys. 45, 471 共1995兲.
关21兴 For an exposition of Kaehler structures see, for instance, A.
Weil, Introduction a l’Etude des Varietes Kaehleriennes 共Hermann, Paris, 1971兲; S.S. Chern, Complex Manifolds Without
Potential Theory 共Springer-Verlag, New York, 1979兲.
关22兴 For a detailed discussion see, for instance, N. Mukunda and R.
Simon, Ref. 关4兴 above.