Bargmann invariants, null phase curves, and a theory ... - CORE

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, Universita` di Bologna, INFM and INFN, Via Irnerio 46, 40126 Bologna, Italy G. Marmo^§

Dipartimento di Scienze Fisiche, Universita` di Napoli Federico II and INFN, Via Cinzia, 80126 Napoli, Italy G. Morandi^储

Dipartimento di Fisica, Universita` 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 mechan- ics, 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 sys- tematic 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 adia- baticity, 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 con- text, it has gradually become clear that the GP is largely kinematical in content关4兴. In the process, important connec- tions 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-

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 Schro¨- dinger 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 gen- eral兲can be defined as a derived object directly as a suitable two-dimensional surface integral of the symplectic two-form

*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

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 defi- nition, are seen to be immediate consequences of the present one.

We shall make frequent reference to the geodesic arc con- necting 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 Hil- bert 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 be- gins 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 ki- nematic definition of the GP is also shown, and the connec- tions 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 struc- ture 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 equa- tions 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 de- scription of geodesics, respectively.

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 ofHmay be finite, say n⫽1,2, . . . , or infinite. The inner product and the norm for vectors

␺^,␾^{, . . . in} Hare written as (␺^,␾^{) and} 储␺储, respectively.

The subset of unit vectors in His defined by

B⫽兵␺苸H兩储␺储⫽1其傺H. 共2.1兲

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 withH

andBis denoted byR. It is the quotient ofBwith 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 Rare represented by pure state density matri- ces, or one-dimensional projections,␳; and there is a projec- tion ␲ ^fromBtoR:

␲^: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 Rin 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s₁⭐s⭐s₂其^傺B;

C⫽␲共C兲⫽兵␳^共s兲⫽␺共s兲␺共s兲^†苸R兩s₁⭐s⭐s₂其^傺R. 共2.4兲 In general, we assume that the parameter s varies over a closed finite interval 关s₁,s₂兴傺R, exceptions will be indi- cated. We permit strictly monotonic reparametrizations of these curves,

s→s

⬘

⫽f共s兲, d f共s兲

ds ⬎0, 共2.5兲

and impose other smoothness conditions as appropriate and described below. We also permit smooth local phase changes along a curve Cto lead to a new curveC

⬘

^,

C

⬘

^⫽兵␺

⬘

共s兲⫽e^i␣(s)␺共s兲兩s₁⭐s⭐s₂其^{傺B 共2.6兲} having the same image inR:

␲共C

⬘

兲⫽␲共C兲⫽C傺R. 共2.7兲 Thus, both CandC

⬘

are lifts, fromRtoB, of C.

We now define three classes of curves, with different smoothness conditions, as follows.

Class I:

„␺共s₁兲,␺共s₂兲…⫽0

␺共s兲,␳共s兲 continuous and piecewise once differentiable.

共2.8兲 Class II:

„␺共s兲,␺共s

⬘

兲…⫽0, any s,s

⬘

苸关s₁,s₂兴

␺共s兲,␳共s兲 continuous once differentiable. 共2.9兲

Class III:

„␺共s₁兲,␺共s₂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 geo- metric phases can be defined; class-II curves subject to fur- ther conditions are null phase curves; and class-III curves obeying suitable differential equations are geodesics. In each case, both C傺Band C⫽␲⁽C)傺Rwill 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 proper- ties 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␻ ^onRrelated to dA via pullback

dA⫽␲*␻^. 共2.11兲 The one-form A is essentially defined by giving its integral along any curveC傺Bof class I:

冕

CA⫽Im

冕

s₁ s₂

ds

冉

^{␺共s兲,d␺ds共s兲}

冊

^⫽⫺i

冕

^s1 s₂

ds

冉

^{␺共s兲,d␺ds共s兲}

冊

^.

共2.12兲 IfChappens to be a closed loop with␺^(s2)⫽␺^(s1) so that its projection␲⁽C)⫽C傺Ris also a closed loop, we have

冖

CA⫽

冕

SdA⫽

冕

S␻^, 共2.13兲 whereSis any smooth two-dimensional surface inBhaving 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 ofSand S are determined by the directions ofCand C, respectively. It is to be emphasized that␻ ^{is not} exact, so A is not the pullback via␲*of any one-form onR. Since the two-form␻^onRwill 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 canoni- cal 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 onHvia its defining representa- tion 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 ge- neric 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 vari- ous lower dimensions. From the general Kostant-Kirillov- Souriau theory of coadjoint orbits of Lie groups G, it is known that each orbit共generic or exceptional兲is a symplec- tic 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 repre- sentation of U(n). The actions of U(n) onBand onRgiven by

g苸U共n兲:␺苸B→g␺苸B 共2.15兲 and

␳苸R→g␳^g⫺1苸R 共2.16兲 are both transitive. Therefore, for any choice of a fiducial vector ␺0苸B, we have an onto map␮0:U(n)→Bgiven 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 direc- tion, evidently, the pullback ␮0*allows us to take A and dA onBto appropriate forms on U(n). From Eq.共2.12兲A onB 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␳0g^†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␳0dg^†∧dg兲⫽i Tr共␳0g^†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 X₁ and X₂ are two left-invariant vector fields on u(n) associated with Hermitian matrices

␶1and␶2 in u(n), i_X

jg^†dg⫽i␶j, j⫽1,2, 共2.21兲

then a short calculation shows

␮0*dA共X₁,X₂兲⫽共dA兲„␮0*^共X1兲,␮0*^共X2兲…

⫽i_X

2i_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 con- nection between A,dA onBand the Maurer-Cartan forms on U(n), mentioned in the previous paragraph, are made ex- plicit关15兴.

We now recall the definition of the GP according to the kinematic approach关4兴. IfCis a class-I curve with image C, then the GP for C is

␸g关C兴⫽arg„␺共s₁兲,␺共s₂兲…⫺

冕

CA. 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 liftCused 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 defini- tion 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␺^(s1),␺^(s2) ofCare mutually orthogo- nal, 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 s₀苸(s₁,s₂),␺^(s0) may be orthogonal to either

␺^(s1) or␺^(s2) or both. Assume␺^(s0) is indeed orthogonal to ␺^(s1), and let s₀ be a point on Cand C at which these curves are differentiable. If for sufficiently small ⑀ ^both

␺^(s0⫺⑀^{) and} ␺^(s0⫹⑀) are not orthogonal to ␺^(s1), then the GP’s are defined for the portions of C running from s₁to s₀⫾⑀, and they obey

␸g关C for s₁⭐s⭐s₀⫹⑀兴⫺␸g关C for s₁⭐s⭐s₀⫺⑀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 „␺^(s1),␺^(s2)… 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 inBandRas 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 dimensionH⭓,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 succes- sive 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 curveC傺Bof 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…

⫽Tr„␳共s兲␳共s

⬘

兲␳共s

⬙

兲…⫽real positive

⇔Tr„␳共s兲关␳共s

⬘

兲,␳共s

⬙

兲兴…⫽0, any s,s

⬘

^,s

⬙

苸关s₁,s₂兴. 共3.3兲 For convenience, when this condition is obeyed, we refer to bothCand C as NPC’s.

It is immediately evident that any connected subset or portion of an NPC, say running from s₃ to s₄, where 关s₃,s₄兴傺关s₁,s₂兴, is also an NPC.

From Eq.共B13兲of Appendix B, it follows that every geo- desic inR共and any lift of it inB) is an NPC. Since any two points␳1,␳2苸Rcan 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 con- dition that no two vectors along it should be mutually or- thogonal. 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s₀兲,␺共s兲,␺共s

⬘

兲…⫽real positive, any fixed s₀苸关s₁,s₂兴, any s,s

⬘

苸关s₁,s₂兴. 共3.4兲 Let now C be an NPC. We derive a fundamental formula for the integral of A along any lift Cof C. For any chosen reference point s₀苸关s₁,s₂兴, choose some ␺0(s₀) 苸␲^⫺1„␳^(s0)…. By definition of class II, for any s苸关s₁,s₂兴 and any choice of ␺^(s)苸␲^⫺1„␳^(s)…, the scalar product

„␺0(s₀),␺^(s)… is nonzero. Adjust the phase of ␺^{(s) to get}

␺0(s)苸␲^⫺1„␳^(s)… such that

„␺0共s₀兲,␺0共s兲…⫽real positive, any s苸关s₁,s₂兴. 共3.5兲 This gives us a particular lift C0⫽兵␺0(s)兩s₁⭐s⭐s₂其 ^{of C} with the property

„␺0共s兲,␺0共s

⬘

兲…⫽real positive, any s,s

⬘

^苸关s₁,s₂兴. 共3.6兲 We see this by setting s

⬙

^⫽s0 in the definition共3.3兲and then using Eq.共3.5兲. This means that any two points onC0 are in phase in the Pancharatnam sense 关6兴, a nonlocal property;

and furthermoreC0 is horizontal, a local property,

C0:

冉

^{␺0共s兲,dsd ␺0共s兲}

冊

^{⫽0, any s苸关s1,s2兴. 共3.7兲}

Therefore, for the end points of C0 and for the integral of A along C0, we have

arg„␺0共s₁兲,␺0共s₂兲…⫽0,

冕

C0

A⫽0. 共3.8兲

Now let C⫽兵␺^{(s)⫽ei␣(s)}␺0(s)其 be a general lift of C ob- tained from C0 by a 共sufficiently smooth兲 local phase trans- formation. ForCwe find in place of Eq.共3.8兲,

arg„␺共s₁兲,␺共s₂兲…⫽␣共s₂兲⫺␣共s₁兲,

冕

CA⫽⫺i

冕

s₁ s₂

ds

冉

^{␺共s兲,d␺ds共s兲}

冊

^⫽

冕

^s1 s₂

dsd␣共s兲 ds , 共3.9兲 that is,

冕

CA⫽arg„␺共s₁兲,␺共s₂兲…. 共3.10兲 This is the basic property of NPC’s that we will use repeat- edly.

Concerning the construction of the particular ‘‘Pancharat- nam lift’’ C0 of the NPC C⫽兵␳^(s)⫽␺^(s)␺^(s)†其^{傺R, we} may add the following remark. The lift C0is completely de- termined once a choice of␺0(s₀) at the reference point s₀ is made. Any alteration of␺0(s₀) 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s₀兲⫽␺共s兲e^{⫺iarg„␺0(s0),␺(s)…}, 共3.11兲 where N(s) is a real positive normalization factor,

N共s兲⫽兩„␺0共s₀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s₀兲,␺0共s兲…⫽N共s兲„␺0共s₀兲,␳共s兲␺0共s₀兲…

⫽兩„␺0共s₀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s₀兲,␳共s

⬘

兲␺0共s₀兲…

⫽N共s兲N共s

⬘

兲Tr„␳共s₀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 liftsC,C

⬘

^{of C,C}

⬘

^from␺1 to␺2and␺2to

␺1, respectively. The unions C艛C

⬘

傺R,C艛C

⬘

傺B are closed loops. Let S傺Rbe any smooth two-dimensional sur- face with boundary ⳵^S⫽C艛C

⬘

. Combining Eq.共2.11兲with the basic property共3.10兲for bothCandC

⬘

^{, we find}

C,NPC ␳1 to␳2, C

⬘

^NPC ␳2 to␳1, ⳵^S⫽C艛C

⬘

^,

冕

S␻⫽

冖

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 liftsC,C

⬘

are each of class II, since they are NPC’s, the closed loops C艛C

⬘

^andC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 differ- entiability.

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 liftsCj, j⫹1running from␺j to␺j⫹1, we find using Eq. 共3.10兲repeatedly

⳵^S⫽C₁₂艛C₂₃艛 . . .艛C_m1, C_{j, j⫹1} NPC’s␳j to ␳j⫹1;

冕

S␻⫽j

兺

⫽1

m

冕

C_{j, j⫹1}A⫽arg⌬m共␺1,␺2, . . . ,␺m兲. 共3.15兲 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. Definition of the GP

With this preparation we are able to define the GP␸g关C兴 for any class-I curve C傺Rfrom ␳1 to ␳2. We choose any NPC C

⬘

^傺Rfrom␳2to␳1, so that C艛C

⬘

is a class-I closed loop in R, and then choose any two-dimensional surface S傺Rwith⳵^S⫽C艛C

⬘

^{. Then}␸g关C兴 is defined as the inte- gral of␻ ^{over S}关17兴:

C class I ␳1to␳2, C

⬘

^NPC ␳2 to ␳1, ⳵^S⫽C艛C

⬘

^;

␸g关C兴⫽⫺

冕

^{S␻. 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 no- tation. For any curves C,C, we denote by˜,CC˜ 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 ⇔C˜ ,C˜ NPC,

␸g关C˜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

place of C

⬘

^{to compute} ␸g关C兴. Choose any surface S

⬘

傺R with boundary ⳵^S

⬘

⫽C˜

⬘

艛C

⬙

^{. Then S}艛S

⬘

has boundary

⳵^(S艛S

⬘

^)⫽C艛C

⬙

. Using Eq.共3.14兲for the pair C˜

⬘

^,C

⬙

^we have

⳵^S⫽C艛C

⬘

^, ⳵^S

⬘

⫽C˜

⬘

艛C

⬙

^, ⳵共S艛S

⬘

兲⫽C艛C

⬙

^;

␸g关C兴⫽⫺

冕

S␻⫽⫺

冕

S␻⫺

冕

S⬘␻⫽⫺

冕

S艛S⬘␻^. 共3.19兲 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Rwhich could be open, i.e.,␳2⫽␳1. In case␳2⫽␳1and 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Rwith⳵^S⫽C, and directly have

⳵^S⫽C, ⳵^C⫽⳵C⫽0;

␸g关C兴⫽⫺

冕

S␻⫽⫺

冖

CA. 共3.20兲 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 class I ␳1 to ␳2, C

⬘

^NPC ␳2 to ␳1, ⳵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, letCbe 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

S␻⫽⫺

冖

C艛C⬘A

⫽⫺

冕

CA⫺

冕

C⬘A⫽arg共␺1,␺2兲⫺

冕

CA, 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Rbe images of␺1,␺2,␺3苸B, no two in either triplet being mutually orthogonal. Join them pairwise by NPC’s: C₁₂ from ␳1 to ␳2, and C₂₃ from ␳2 to ␳3, and C₃₁from␳3to␳1, their liftsC12from␺1 to␺2,C23from␺2

to␺3,andC31 from␺3to␺1. Now both C₁₂艛C₂₃艛C₃₁and

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;

␸g关C₁₂艛C₂₃兴⫽␸g关C₁₂艛C₂₃艛C₃₁兴

⫽⫺

冖

C12艛C23艛C31

A⫽⫺

冕

C12

A⫺

冕

C23

A⫺

冕

C31

A

⫽⫺arg⌬3共␺1,␺2,␺3兲. 共3.23兲 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C₁₂艛C₂₃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Bas before, let C₁₂and C₂₃be any class-I curves, not necessarily NPC’s from ␳1 to ␳2 and ␳2 to ␳3, respectively. Next let C₂₁

⬘

^,C32

⬘

^,C31

⬘

be NPC’s from ␳2 to␳1, ␳3 to␳2, and ␳3 to

␳1, respectively. These are needed to define the GP’s which appear below. Finally, we choose two-dimensional surfaces S₁,S₂,S₃ with boundaries

⳵^S1⫽C₁₂艛C₂₁

⬘

^, ⳵^S2⫽C₂₃艛C₃₂

⬘

^,

⳵^S3⫽C˜

21

⬘

艛C˜

32

⬘

艛C₃₁

⬘

^,

⳵共S₁艛S₂艛S₃兲⫽C₁₂艛C₂₃艛C₃₁

⬘

^. 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

␸g关C₁₂艛C₂₃兴⫺␸g关C₁₂兴⫺␸g关C₂₃兴

⫽⫺

冕

S₁艛S₂艛S₃␻⫹

冕

S₁␻⫹

冕

S₂␻⫽⫺

冕

S₃␻

⫽␸g关C˜

21

⬘

艛C˜

32

⬘

艛C₃₁

⬘

兴

⫽⫺arg⌬3共␺1,␺2,␺3兲,

i.e., ␸g关C₁₂艛C₂₃兴⫽␸g关C₁₂兴⫹␸g关C₂₃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.

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

⳵^S⫽C₁₂艛C₂₁;

␸g关C₁₂艛C₂₁兴⫽␸g关C₁₂兴⫹␸g关C₂₁兴

⫽␸g关C₁₂兴⫺␸g关C˜₂₁兴

⫽⫺

冕

S␻^. 共3.27兲 This will be used in the sequel.

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 vi- sualize 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␳1to␳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 Poincare´ sphere S². If now three points

␳1,␳2,␳3苸R correspond to respective unit vectors nˆ₁,nˆ₂,nˆ₃苸S², then, as is known, arg Tr(␳1␳2␳3) is one-half of the solid angle subtended at the center of S²by the spheri- cal triangle with vertices nˆ₁,nˆ₂,nˆ₃ 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 nˆ苸S² as

␳⫽1

2关1⫹nˆ•␴ជ_兴, 共4.1兲 and then

Tr兵␳共s兲关␳共s

⬘

兲,␳共s

⬙

兲兴其⫽i

2nˆ共s兲•nˆ共s

⬘

兲⫻nˆ共s

⬙

兲. 共4.2兲 Parametrizing the nˆ’s with spherical polar angles ␪^,␾ ^{on S2} in the usual way, and assuming with no loss of generality nˆ(s)⫽(0,0,1), the vanishing of nˆ(s)•nˆ(s

⬘

⁾⫻nˆ(s

⬙

^{) for all} independent s

⬘

^,s

⬙

^{amounts to}

x₂共s

⬘

兲

x₁共s

⬘

兲⫽tan␾共s

⬘

兲⫽const. 共4.3兲 Hence, the NPC condition corresponds, after rotating to the configuration nˆ(s)⫽(0,0,1), to␾⫽const; and we find that NPC’s are great circle arcs or geodesics on S². Given any two nonantipodal points on S², 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 geo- desics 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Rof ␳1 de- fined in the manner of Eq.共A.8兲. We can therefore use a local description ofR(␳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⫽e₁. According to Eq.

共A.10兲we can express␺2 in the form

␺2⫽e₁cos␪0⫹e₂sin␪0,

共e₁,e₂兲⫽0, 储e₂储⫽1, 共4.5兲 so ␺1⫽e₁,e₂ form an orthonormal pair. As in Appendix A we supplement e₁,e₂ by further vectors e₃,e₄, . . .苸B, ter- minating with e_n if dimensionH⫽n is finite, such that 兵␺1

⫽e₁,e₂,e₃, . . .其 is an orthonormal basis for H. Let C

⫽兵␺^{(s)兩s1⭐s⭐s2}其 be a lift of C from␺1 to␺2. For␺^(s), we write

␺共s兲⫽x₁共s兲e₁⫹x₂共s兲e₂⫹x₃共s兲e₃⫹•••, x₁共s兲⫽0, x₁共s兲cos␪0⫹x₂共s兲sin␪0⫽0, 兩x₁共s兲兩²⫹兩x₂共s兲兩²⫹兩x₃共s兲兩²⫹•••⫽1, s₁⭐s⭐s₂.

共4.6兲 The coefficients x₁(s),x₂(s), . . . must be continuous once differentiable. At the end points, they have real values

x共s₁兲⫽共1,0,0,0, . . .兲,

x共s₂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R^m. Assume x(s) to have real components, with x_m⫹1(s)⫽x_m⫹2(s)⫽•••⫽0 for s 苸关s₁,s₂兴. Thus the first m components of x(s) describe a moving point on S^m⫺1, with x₁(s) and x₁(s)cos␪0

⫹x₂(s)sin␪0 nonzero throughout. Let us further limit our- selves when s₁⬍s⬍s₂ to vectors on S^m⫺1 with all compo- nents strictly positive. That is, generalizing the positive quadrant and octant in two and three dimensions, we define S_⫹^m⫺1⫽兵^{x苸Sm⫺1兩x1,x2, . . . ,xm⬎0}其^{傺Sm⫺1, 共4.8兲} and choose

x共s兲苸S_⫹^m⫺1, s₁⬍s⬍s₂. 共4.9兲 Then the vectors

␺共s兲⫽x₁共s兲e₁⫹x₂共s兲e₂⫹x₃共s兲e₃⫹•••⫹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s1,s2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 e₃,e₄, . . . ,e_m forming along with␺1⫽e₁ and e₂an orthonormal set inBmay each be freely chosen; and then x(s) for s₁⬍s⬍s₂ is any once- differentiable curve on S_⫹^m⫺1 obeying the boundary condi- 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 deduc- tive 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Rif it has the prop- erty that every continuous once-differentiable curve C傺M is an NPC ? The latter kind of question is clearly not meaning- ful 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s₁⭐s⭐s₂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 cho- sen. Let C0⫽兵␺^{0(s)兩s苸关s1,s2}兴其 be the particular lift of C, from ␺1 to␺2, obeying the condition共3.6兲so

␺0共s₁兲⫽␺1, ␺0共s₂兲⫽␺2,

„␺0共s

⬘

兲,␺0共s兲…⫽real positive, s

⬘

^,s苸关s₁,s₂兴. 共4.12兲

We expand␺0(s) as

␺0共s兲⫽x₁共s兲e₁⫹x₂共s兲e₂⫹␹共s兲,

„e₁,␹共s兲…⫽„e₂,␹共s兲…⫽0, s苸关s₁,s₂兴. 共4.13兲 At this stage, x₁(s) and x₂(s) are complex continuous once- differentiable functions of s, while ␹(s) is a continuous once-differentiable vector in the subspace H_⬜(␺1,␺2)傺H orthogonal to the pair␺1,␺2i.e., to e₁,e₂. At the end points, we have

x₁共s₁兲⫽1, x₂共s₁兲⫽␹共s₁兲⫽0;

x₁共s₂兲⫽cos␪0, x₂共s₂兲⫽sin␪0, ␹共s₂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Bfor all s. From the real positivity of„␺1,␺0(s)… and„␺2,␺0(s)…, we get

x₁共s兲⫽real positive,

x₁共s兲cos␪0⫹x₂共s兲sin␪0⫽real positive, s苸关s₁,s₂兴. 共4.15兲 These imply

x₂共s兲⫽real,

x₁共s兲cos␪0

2 ⫹x₂共s兲sin␪0

2 ⫽real positive, s苸关s₁,s₂兴. 共4.16兲 Thus, x₁(s) can never vanish, while x₂(s) could vanish, as it does at s⫽s₁, or even sometimes be negative. Next from the normalization of␺0(s), we have

储␺0共s兲储⫽1⇔x₁共s兲²⫹x₂共s兲²⫹储␹共s兲储²⫽1, s苸关s₁,s₂兴. 共4.17兲 We therefore parametrize x₁(s) and x₂(s) by

x₁共s兲⫽␴共s兲cos␪共s兲, x₂共s兲⫽␴共s兲sin␪共s兲, 0⬍␴共s兲⭐1, s苸关s₁,s₂兴;

␴共s₁兲⫽␴共s₂兲⫽1, ␪共s₁兲⫽0, ␪共s₂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 x₁-x₂ plane is thus a segment OAB of the unit disk subtend- ing 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),

hence x₁(s) and x₂(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 con- dition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s₁,s₂兴. 共4.21兲 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^e3,e₄, . . .其 for the subspaceH_⬜(␺1,␺2)傺Hsuch that

␹共s兲⫽r⫽

兺

3,4, . . .

x_r共s兲e_r,

x_r共s兲⫽real,

„␹共s

⬘

兲,␹共s兲…⫽_{r⫽3,4, . . .}

兺

^xr共s

⬘

^{兲xr共s兲⫽real,}

储␹共s兲储²⫽_{r⫽3,4, . . .}

兺

^{xr共s兲2⫽1⫺␴共s兲2苸关0,1兲. 共4.22兲}

The orthonormal set 兵^{e3,e4, . . .}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 in- equality to a bound on the magnitude of the last term on the right-hand side of the condition共4.21兲:

FIG. 1. The allowed region in the x1-x2 plane is the segment OAB of the unit disc, excluding the radii OA and OB.

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 some- times 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 compen- sate 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^1/2兩cos„␪共s

⬘

^兲⫺␪共s兲…兩⭐1 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 de- scription 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 preced- ing section correspond to the special simplifying assumption that the functions x₁(s),x₂(s),x₃(s), . . . are all non- negative. Then 兩␪^(s

⬘

⁾