Bargmann invariants and geometric phases: A generalized connection
Eqab M. Rabei*
Department of Physics, Mutah University, PostBox 7, Karak, Jordan
Arvind†
Department of Physics, Guru Nanak Dev University, Amritsar 143005, India
N. Mukunda‡
Centre for Theoretical Studies and Department of Physics, Indian Institute of Science, Bangalore 560012, India
R. Simon§
The Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600113, India
We develop the broadest possible generalization of the well known connection between quantummechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in
quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is
shown to be essential for properly understanding geometric phase results in the cases of coherent states and of
Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
I. INTRODUCTION
The geometric phase was originally discovered in the context of cyclic adiabatic quantum-mechanical evolution, governed by the time-dependent Schrödinger equation with a
Hermitian Hamiltonian operator 关1,2兴. Subsequent work has
shown that many of these restrictions can be lifted. Thus the
geometric phase can be defined in nonadiabatic 关3兴, noncyclic and even nonunitary evolution 关4兴. Generalization to the
non-Abelian case has also been achieved 关5兴. Finally the kinematic approach 关6兴 demonstrated that even the Schrödinger equation and a Hamiltonian operator are not needed
for defining the geometric phase. The intimate relationship
between geometric phase and Hamilton’s theory of turns 关7兴
has also been brought out 关8兴.
An important consequence of the kinematic approach has
been to show clearly the close connection between geometric
phases, and a family of quantum-mechanical invariants introduced by Bargmann 关9兴 while giving a new proof of the
Wigner 关10兴 unitary-antiunitary theorem. This connection
depends in an essential way upon the concept of free geodesics in quantum mechanical ray and Hilbert spaces, and the
vanishing of geometric phases for these geodesics.
The purpose of this paper is to generalize this important
link between Bargmann invariants and geometric phases to
the broadest possible extent by going beyond the use of free
geodesics. The key is to characterize in a complete way those
ray space curves with the property that the geometric phase
vanishes for any connected stretch of any one of them. We
*Electronic address: eqab@center.mutah.edu.jo
†
Electronic address: arvind@physics.iisc.ernet.in
Electronic address: nmukunda@cts.iisc.ernet.in
§
Electronic address: simon@imsc.ernet.in
‡
show that this property can be translated into an elementary
and elegant statement concerning the inner product of any
two Hilbert space vectors along any lift of such a ray space
curve. We refer to these as ‘‘null phase curves,’’ and the
generalization of the familiar statement linking Bargmann
invariants and geometric phases is achieved by replacing free
geodesics by such curves. A free geodesic is always a null
phase curve; however, the latter is much more general.
This paper is arranged as follows. Section II recalls the
basic features of the kinematic approach to the geometric
phase; sets up free geodesics in ray and Hilbert spaces;
shows that the geometric phase for any free geodesic vanishes; introduces the Bargmann invariants; and describes
their connection to geometric phases for ray space polygons
bounded by free geodesics. In Sec. III it is argued that it
should be possible to generalize this connection. This motivates the definition and complete characterization of null
phase curves at the Hilbert space level, the previous free
geodesics being a very specific case. It is then shown that
such curves allow us to generalize the previously stated connection to the broadest possible extent. Section IV defines
the concept of constrained geodesics in ray and Hilbert
spaces, the motivation being that in some situations such
curves may in fact be null phase curves. The idea is extremely simple, namely, we limit ourselves to some chosen
submanifolds in ray 共and Hilbert兲 space, and determine
curves of minimum length lying within these submanifolds.
Section V examines several interesting examples to illustrate
these ideas: a submanifold arising out of a linear subspace of
Hilbert space; coherent states for one degree of freedom;
centered Gaussian pure states for one degree of freedom; and
an interesting submanifold in the space of two-mode coherent states. It turns out that in the first case constrained geodesics are just free geodesics, while in the remaining cases
they are very different. This shows that the generalized con-
nection between Bargmann invariants and geometric phases
presented in this paper is just what is needed to be physically
interesting and appropriate. In Sec. VI we present a brief
discussion of these ideas in the differential geometric framework natural to geometric phases, and also develop a direct
ray space description of null phase curves. Section VII contains concluding remarks.
II. CONNECTION BASED ON FREE GEODESICS
Let H be the Hilbert space of states of some quantum
system, R the associated ray space, and ␲ :H→R the corresponding projection. We shall be dealing with 共sufficiently兲
smooth parametrized curves C of unit vectors in H, and their
images C in R. A curve C is described as follows:
C⫽ 兵 ␺ 共 s 兲 苸H兩 储 ␺ 共 s 兲储 ⫽1 s 1 ⭐s⭐s 2 其 傺H.
共2.1兲
Its image C is a curve of pure state density matrices:
␲ 关 C兴 ⫽C傺R,
C⫽ 兵 ␳ 共 s 兲 ⫽ ␺ 共 s 兲 ␺ 共 s 兲 † 兩 s 1 ⭐s⭐s 2 其 .
␺ (h) 共 s 兲 ,
冊
d (h)
␺ 共 s 兲 ⫽0.
ds
共2.3兲
For any curve C傺R, a geometric phase ␸ g 关 C 兴 is defined.
Its calculation is facilitated by going to any lift C, calculating
the total and dynamical phases for C, and taking the difference
␲ 关 C兴 ⫽C:
␸ g 关 C 兴 ⫽ ␸ tot关 C兴 ⫺ ␸ dyn关 C兴 ,
␸ tot关 C兴 ⫽arg„␺ 共 s 1 兲 , ␺ 共 s 2 兲 …,
␸ dyn关 C兴 ⫽Im
冕 冉␺
s2
ds
s1
共 s 兲,
共2.4兲
冊
d
␺共 s 兲 .
ds
冕 再 冏冏
s2
s1
ds
d␺共 s 兲
ds
冏冏 冏 冉
2
⫺ ␺共 s 兲,
d␺共 s 兲
ds
共2.6兲
Thus we have here a plane two-dimensional curve determined by a pair of orthonormal vectors in H, an arc of a
circle. It may be helpful to make the following comment
concerning free geodesics. Given any two ‘‘nonorthogonal’’
points ␳ 1 , ␳ 2 苸R such that Tr( ␳ 1 ␳ 2 )⫽0, we can always
choose unit vectors ␺ 1 , ␺ 2 苸H projecting onto ␳ 1 , ␳ 2 respectively, such that the inner product ( ␺ 1 , ␺ 2 ) is real positive.
Then the free geodesic 共2.6兲 will connect ␺ 1 and ␺ 2 if we
take ␾ 1 ⫽ ␺ 1 and ␾ 2 ⫽ 关 ␺ 2 ⫺ ␺ 1 ( ␺ 1 , ␺ 2 ) 兴 / 兵 1⫺( ␺ 1 , ␺ 2 ) 2 其 1/2.
It is now clear that ␺ (0)⫽ ␺ 1 , and ␺ (s)⫽ ␺ 2 for s
⫽cos⫺1 (␺1 ,␺2)苸(0,␲ /2). It is clear that the curve in H
given by Eq. 共2.6兲 is horizontal and for any two points on it
with 兩 s 1 ⫺s 2 兩 ⬍ ␲ /2, the inner product „␺ (s 1 ), ␺ (s 2 )… is real
positive, so ␺ (s 1 ) and ␺ (s 2 ) are in phase in the Pancharatnam sense 关13兴. From these properties of free geodesics, the
result 关6兴
␸ g 关 free geodesic in R兴 ⫽0
共2.7兲
follows. This can be exploited to connect geometric phases
to Bargmann invariants.
Let ␺ 1 , ␺ 2 , . . . , ␺ n be any n unit vectors in H, no two
consecutive ones being orthogonal, and let ␳ 1 , ␳ 2 , . . . , ␳ n be
their images in R. Then the corresponding n-vertex Bargmann invariant is defined as
⌬ n 共 ␺ 1 , ␺ 2 , . . . , ␺ n 兲 ⫽ 共 ␺ 1 , ␺ 2 兲共 ␺ 2 , ␺ 3 兲 . . . 共 ␺ n , ␺ 1 兲
⫽Tr 共 ␳ 1 ␳ 2 . . . ␳ n 兲 .
共2.8兲
Now we draw n free geodesics in R connecting ␳ 1 to ␳ 2 , ␳ 2
to ␳ 3 , . . . , ␳ n to ␳ 1 . Thus we obtain an n-sided polygon in R
bounded by free geodesics, and we can compute the corresponding geometric phase. Repeatedly exploiting Eq. 共2.7兲
we obtain the basic result 关6兴:
␸g
冋
n⫺vertex polygon in R connecting ␳ 1 to ␳ 2 ,
␳ 2 to ␳ 3 , . . . , ␳ n to ␳ 1 by free geodesics
册
⫽⫺arg⌬ n 共 ␺ 1 , ␺ 2 , . . . , ␺ n 兲 ,
In particular, if C is horizontal ␸ dyn关 C兴 vanishes, and ␸ g 关 C 兴
is just ␸ tot关 C兴 .
Now we define free geodesics in R and H. Given C in R
and any lift C in H, the length of the former can be defined as
the following nondegenerate functional:
L关C兴⫽
共 ␾ 1 , ␾ 1 兲 ⫽ 共 ␾ 2 , ␾ 2 兲 ⫽1, 共 ␾ 1 , ␾ 2 兲 ⫽0.
共2.2兲
Any C in H projecting onto a given C in R is a lift of the
latter. In particular, we have a horizontal lift C (h) if the vectors ␺ (h) (s) along it are such that
冉
␺ 共 s 兲 ⫽ ␾ 1 cos s⫹ ␾ 2 sin s,
冊冏 冎
2 1/2
.
共2.5兲
It is easy to check that the integrand here is independent of
the choice of lift C; it leads to the well known Fubini-Study
metric on R 关11,12兴. Free geodesics in R are those C’s for
which L 关 C 兴 is a minimum for given end points. And by
definition a free geodesic in H is any lift of a free geodesic in
R. It can be shown 关6兴 that any free geodesic in R can be
lifted to H, and the parametrization chosen so that it can be
described as follows:
␳ j ⫽ ␺ j ␺ †j , j⫽1,2, . . . ,n.
共2.9兲
We mention in passing that this result is of considerable
conceptual as well as practical value 关14兴.
In connection with the above result, the following remarks may be made. As is clear from Eq. 共2.8兲, the phases of
the individual vectors ␺ 1 , ␺ 2 ,•••, ␺ n can be freely altered.
We need only assume that successive pairs of unit vectors
are not mutually orthogonal; then the Bargmann invariant is
nonzero and has a well defined phase.
III. GENERALIZED CONNECTION
The definition 共2.8兲 of the Bargmann invariant requires
only the choice of the n vertices ␳ 1 , ␳ 2 , . . . , ␳ n 苸R; consecutive ones need not be connected in any way to form a
closed figure. This suggests that the connection 共2.9兲 between these invariants and geometric phases may apply more
generally, not only in the case where we connect ␳ 1 to ␳ 2 , ␳ 2
to ␳ 3 , . . . , ␳ n to ␳ 1 by free geodesics. We now show that
this is indeed so.
We need to characterize the most general 共smooth兲 curves
C傺R having the property
␸ g 关 any connected portion of C 兴 ⫽0.
共3.1兲
We know that if C is a free geodesic, this property does
follow, but there may be 共indeed there are兲 many other possibilities. We can develop a simple necessary and sufficient
condition on C such that Eq. 共3.1兲 holds.
Given the curve C傺R, let C (h) be a horizontal lift and C
a general lift of C in H. We have
C⫽ 兵 ␳ 共 s 兲 兩 ␳ 共 s 兲 † ⫽ ␳ 共 s 兲 ⭓0, ␳ 共 s 兲 2 ⫽ ␳ 共 s 兲 ,
Tr ␳ 共 s 兲 ⫽1, s 1 ⭐s⭐s 2 其 ,
冉
再
C (h) ⫽ ␺ (h) 共 s 兲 苸H兩 ␲ 共 ␺ (h) 共 s 兲兲 ⫽ ␳ 共 s 兲 ,
␺ (h) 共 s 兲 ,
冊 冎
d (h)
␺ 共 s 兲 ⫽0 ,
ds
C⫽ 兵 ␺ 共 s 兲 苸H兩 ␺ 共 s 兲 ⫽e i ␣ (s) ␺ (h) 共 s 兲 其 .
共3.2兲
Here ␣ (s) is some 共smoothly varying兲 phase angle. For any two points on C with parameter values s and s ⬘ ⬎s we have
␸ g 关 ␳ 共 s 兲 to ␳ 共 s ⬘ 兲 along C 兴 ⫽ ␸ tot关 ␺ (h) 共 s 兲 to ␺ (h) 共 s ⬘ 兲 along C (h) 兴 ⫽arg„␺ (h) 共 s 兲 , ␺ (h) 共 s ⬘ 兲 …
⫽arg„e ⫺i ␣ (s) ␺ 共 s 兲 ,e ⫺i ␣ (s ⬘ ) ␺ 共 s ⬘ 兲 …⫽arg„␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫹ ␣ 共 s 兲 ⫺ ␣ 共 s ⬘ 兲 .
共3.3兲
From this result we see that the necessary and sufficient condition on C to secure the property 共3.1兲 can be expressed in several
equivalent ways, using either an arbitrary lift C of C or a horizontal lift C (h) :
␸ g 关 any connected portion of C 兴 ⫽0 ⇔ arg„␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽ ␣ 共 s ⬘ 兲 ⫺ ␣ 共 s 兲 , any s ⬘ and s
⇔
⳵2
⳵ s ⬘⳵ s
arg„␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽0
⇔ arg „␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽separable in s ⬘ and s
⇔ „␺ (h) 共 s 兲 , ␺ (h) 共 s ⬘ 兲 …⫽real positive, any s ⬘ and s
共3.4兲
⇔ any two points of C (h) are in phase.
Here, separability is to be understood in the additive, and not
in the multiplicative, sense. It is important to recognize that
these characterizations are reparametrization invariant. Any
curve C傺R obeying 共3.4兲 will be called a ‘‘null phase curve
in R,’’ and any lift C of such a C will be called a ‘‘null phase
curve in H.’’ Free geodesics are null phase curves, but the
opposite is not necessarily true.
It may be helpful to make some additional remarks at this
point to clarify the ideas involved. If a curve C傺H is such
that any two points on it 共not too far apart兲 are in phase, then
it is definitely horizontal:
C⫽ 兵 ␺ 共 s 兲 其 : „␺ 共 s 兲 , ␺ 共 s ⬘ 兲 …⫽real positive
⇒
冉
冉
␺共 s 兲,
⇒ ␺共 s 兲,
d␺共 s⬘兲
ds ⬘
冊
C (h) ⫽ 兵 ␺ (h) 共 s 兲 其 ⫽horizontal
冉
⇒ ␺ (h) 共 s 兲 ,
冊
d (h)
␺ 共 s 兲 ⫽0,
ds
⇒ „␺ (h) 共 s 兲 , ␺ (h) 共 s⫹ ␦ s 兲 …⯝1⫹0 共 ␦ s 兲 2 ,
⇒ arg„␺ (h) 共 s 兲 , ␺ (h) 共 s⫹ ␦ s 兲 …⫽0 共 ␦ s 兲 2 .
⫽real
冊
d␺共 s 兲
⫽0
ds
⇒ C horizontal.
The image C⫽ ␲ 关 C兴 is obviously a null phase curve in R,
since Eq. 共3.4兲 is obeyed with ␣ (s)⫽0; therefore, C being a
lift of C is also a null phase curve in H. On the other hand,
for a horizontal curve C (h) 傺H, only ‘‘nearby points’’ are in
phase:
共3.5兲
共3.6兲
However, two general points on C (h) may well not be in
phase, as arg„␺ (s), ␺ (s ⬘ )… could be nonzero. Hence C (h) and
its image ␲ 关 C (h) 兴 may not be null phase curves. For ␲ 关 C (h) 兴
to be a null phase curve, in addition to being horizontal 共a
local property兲, C (h) must possess the global property that for
general s and s ⬘ the inner product „␺ (h) (s), ␺ (h) (s ⬘ )… is real
positive. This is what is captured in conditions 共3.4兲.
We can now generalize the result 共2.9兲 and strengthen it
as follows. Given n unit vectors ␺ 1 , ␺ 2 , . . . , ␺ n 苸H with
␸g
冋
images ␳ 1 , ␳ 2 , . . . , ␳ n 苸R, draw any null phase curves joining consecutive pairs of points ␳ 1 to ␳ 2 , ␳ 2 to ␳ 3 , . . . , ␳ n to
␳ 1 . 共This can certainly be done since in any event free geodesics are available.兲 Then, by exactly the same arguments
that lead to the connection 共2.9兲 we obtain
n⫺sided figure in R with vertices ␳ 1 , ␳ 2 , . . . , ␳ n
and bounded by null phase curves
It must be clear that this is the broadest generalization of the
connection 共2.9兲 that one can obtain. We see that we can
replace each free geodesic belonging to a polygon in R by
any null phase curve, and the geometric phase remains the
same, since the right-hand side of Eq. 共3.7兲 depends on the
vertices alone.
IV. CONSTRAINED GEODESICS AS NULL PHASE
CURVES
We have seen that every free geodesic is a null phase
curve, but the converse is generally not true. Nevertheless,
the former fact inspires the following question: Can we alter
the definition of a free geodesic, based on minimizing the
length functional L 关 C 兴 of Eq. 共2.5兲, in a natural way to obtain other kinds of geodesics, and will they turn out to be null
phase curves as well?
The generalization we explore is the following: instead of
dealing with curves 共of unit vectors兲 in the complete Hilbert
and ray spaces H and R, we restrict ourselves to some
共smooth兲 submanifold M 傺R and consider only curves C
lying in M and connecting pairs of points in M. For such
curves we minimize L 关 C 兴 with respect to variations of C
which stay within M. The resulting curves will naturally be
called ‘‘constrained geodesics,’’ and the question is, do constrained geodesics in some cases turn out to be null phase
curves?
We emphasize that our question is not whether every null
phase curve is a constrained geodesic lying in a suitably
chosen submanifold M 傺R but, rather, whether the latter
curves sometimes have the former property. The physically
important examples presented in the next section show that
our question is indeed interesting. In this section we set up
the general framework to handle constrained geodesics in ray
space.
Given H and R with dim H⫽dim R⫹1 in the real sense,
we consider a submanifold M 傺R of n 共real兲 dimensions
consisting of a 共sufficiently smooth兲 family of unit rays, with
共local兲 real independent and essential coordinates ␰
⫽( ␰ ␮ ), ␮ ⫽1,2, . . . ,n:
M ⫽ 兵 ␳ 共 ␰ 兲 苸R兩 ␰ 苸Rn 其 傺R.
共4.1兲
共We do not indicate explicitly the domain in R over which
␰ may vary.兲 The inverse image of M in H will bring in an
extra phase angle ␣ , and is denoted by M:
n
册
⫽⫺arg ⌬ n 共 ␺ 1 , ␺ 2 , . . . , ␺ n 兲 .
共3.7兲
M⫽ ␲ ⫺1 关 M 兴 ⫽ 兵 ␺ 共 ␰ ; ␣ 兲 苸H兩 ␲ 关 ␺ 共 ␰ ; ␣ 兲兴 ⫽ ␳ 共 ␰ 兲 , ␺ 共 ␰ ; ␣ 兲
⫽e i ␣ ␺ 共 ␰ ;0 兲 其 .
共4.2兲
共Of course each ␺ ( ␰ ; ␣ ) is a unit vector, and ␣ and ␰ ␮ taken
together are local coordinates for M.兲 So in the real sense
dimM⫽n⫹1, and to avoid trivialities we must have 1
⫹n/2⬍ complex dimension of H.
Now we consider a parametrized curve C傺M 傺R, obtained by making the n real variables ␰ ␮ into functions of a
real parameter s:
C⫽ 兵 ␳ „␰ 共 s 兲 …,s 1 ⭐s⭐s 2 其 傺M .
共4.3兲
To lift C to some C傺M傺H, some 共smooth兲 choice of phase
angle ␣ (s) as a function of s must be made, and then we
have
C⫽ 兵 ⌿ 共 s 兲 ⫽ ␺ „␰ 共 s 兲 ; ␣ 共 s 兲 …其 傺M,
␲ 关 C兴 ⫽C.
共4.4兲
Using the definition 共2.5兲 the length L 关 C 兴 can be seen to
involve only the partial derivatives of ␺ ( ␰ ; ␣ ) with respect to
the ␰ ␮ , the dependence on ␣ being trivial and not contributing at all. Therefore, we define
u ␮共 ␰ ; ␣ 兲 ⫽
⳵
⳵␰ ␮
␺ 共 ␰ ; ␣ 兲 , ␮ ⫽1,2, . . . ,n;
u⬜␮ 共 ␰ ; ␣ 兲 ⫽u ␮ 共 ␰ ; ␣ 兲 ⫺ ␺ „␰ ; ␣ ) 共 ␺ 共 ␰ ; ␣ 兲 ,u ␮ 共 ␰ ; ␣ 兲 ….
共4.5兲
Normalization of ␺ ( ␰ ; ␣ ) to unity for all ␰ and ␣ implies
Re„␺ 共 ␰ ; ␣ 兲 ,u ␮ 共 ␰ ; ␣ 兲 …⫽0.
共4.6兲
Now L 关 C 兴 can be expressed as follows:
L关C兴⫽
⫽
冕
冕
s2
s1
s2
s1
ds 冑储 ⌿̇ 共 s 兲储 2 ⫺ 兩 „⌿ 共 s 兲 ,⌿̇ 共 s 兲 …兩 2
ds 冑g ␮ ␯ 共 ␰ 兲 ␰˙ ␮ ␰˙ ␯ ,
g ␮ ␯ 共 ␰ 兲 ⫽Re„u⬜␮ 共 ␰ ; ␣ 兲 ,u⬜␯ 共 ␰ ; ␣ 兲 …,
␰ ␮⫽ ␰ ␮共 s 兲 .
共4.7兲
The parameter dependences of ␰ and ␣ are as in Eq. 共4.4兲.
From the essentiality of ␰ ␮ as coordinates for M, and the
positivity of the metric on H, one easily obtains the following results: the n⫻n matrix 兵 „u⬜␮ ( ␰ ; ␣ ),u⬜␯ ( ␰ ; ␣ )…其 is Hermitian positive definite and independent of ␣ ; and only its real
part 关 g ␮ ␯ ( ␰ ) 兴 , which is symmetric positive definite, enters
into L 关 C 兴 .
To obtain the differential equations for constrained geodesics, we minimize L 关 C 兴 with respect to variations in C
that stay within M. This amounts to minimizing L 关 C 兴 in the
final form given in Eq. 共4.7兲, by making independent variations in the n real functions ␰ ␮ (s); the result is well known
from Riemannian geometry. After making a suitable choice
of the parameter s 共affine parametrization兲, the differential
equations for constrained geodesics become
␰¨ ␮ 共 s 兲 ⫹⌫ ␮ ␯ ␭ 关 ␰ 共 s 兲兴 ␰˙ ␯ 共 s 兲 ␰˙ ␭ 共 s 兲 ⫽0,
⌫ ␮ ␯ ␭ 共 ␰ 兲 ⫽ 21 g ␮ ␳ 共 ␰ 兲关 g ␳␯ ,␭ 共 ␰ 兲 ⫹g ␳ ␭, ␯ 共 ␰ 兲 ⫺g ␯ ␭, ␳ 共 ␰ 兲兴 ,
共4.8兲
关 g ␮ ␯ 共 ␰ 兲兴 ⫽ 关 g ␮ ␯ 共 ␰ 兲兴 ⫺1 ,
g ␳␯ ,␭ 共 ␰ 兲 ⫽
⳵ g ␳␯ 共 ␰ 兲
⳵␰ ␭
.
Here the ⌫’s are the familiar symmetric Christoffel symbols
determined by the ‘‘metric’’ tensor g ␮ ␯ ( ␰ ). Change in scale
and shift of origin are the only remaining freedoms in
choices for parameter s. It is a consequence of the differential equations above that
g ␮ ␯ 关 ␰ 共 s 兲兴 ␰˙ ␮ 共 s 兲 ␰˙ ␯ 共 s 兲 ⫽const.
共4.9兲
A general solution to Eq. 共4.8兲 is uniquely determined by
choices of initial values ␰ ␮ (0), ␰˙ ␮ (0). The resulting ␰ ␮ (s)
determine some constrained geodesic C傺M 傺R, and for
any 共smooth兲 choice of ␣ (s) we get a lift C傺M傺H, which
by definition is a constrained geodesic in H. The meaning of
the ‘‘conservation law’’ 共4.9兲 in terms of Hilbert space vectors is interesting. In terms of the derivative of ⌿(s) with
respect to s, and its component orthogonal to ⌿(s),
⌿̇ 共 s 兲 ⫽
d
␺ „␰ 共 s 兲 ; ␣ 共 s 兲 …⫽ ␰˙ ␮ 共 s 兲 u ␮ 关 ␰ 共 s 兲 ; ␣ 共 s 兲兴
ds
⫹i ␣˙ 共 s 兲 ⌿ 共 s 兲 ,
共4.10兲
⌿̇⬜ 共 s 兲 ⫽⌿̇ 共 s 兲 ⫺⌿ 共 s 兲 „⌿ 共 s 兲 ,⌿̇ 共 s 兲 …⫽ ␰˙ ␮ 共 s 兲 u⬜␮ 关 ␰ 共 s 兲 ; ␣ 共 s 兲兴 ,
V. APPLICATIONS
We look at four examples to illustrate the use of constrained geodesics in the geometric phase context, and to
show the distinction in general between them and null phase
curves.
A. Subspaces of H
Let H0 be a linear subspace of H 共as a complex vector
space兲, and denote by M傺H0 the subset of unit vectors in
H0 . By projection, we obtain the submanifold M
⫽ ␲ 关 M兴 傺R, with the real dimension of M equal to
2⫻ 兵 共complex dimension of H0 )⫺1 其 . In this case, constrained geodesics in M happen to be free geodesics. Given
any two 共nonorthogonal兲 in-phase unit vectors in M, say ␺ 1
and ␺ 2 , the free geodesic connecting them, namely, from Eq.
共2.6兲 the curve C consisting of the vectors
␺ 共 s 兲 ⫽ ␺ 1 cos s⫹
冑1⫺ 共 ␺ 1 , ␺ 2 兲 2
We can then, if we wish, adjust the scale of s so that ⌿̇⬜
becomes a unit vector for all s.
Having set up the basic formalism to determine constrained geodesics, in the next section we look at some
physically motivated examples to see whether they are sometimes null phase curves as well.
共5.1兲
B. Single mode coherent states
We consider the family of coherent states for a single
degree of freedom, described by Hermitian operators q̂, p̂ or
the non-Hermitian combinations â,â † :
â⫽
1
冑2
共 q̂⫹i p̂ 兲 , â † ⫽
1
冑2
共 q̂⫺i p̂ 兲 ,
共5.2兲
关 q̂,p̂ 兴 ⫽i, 关 â,â † 兴 ⫽1.
A general normalized coherent state is labeled with a complex number z and is generated by applying a unitary phase
space displacement operator to the 共Fock兲 vacuum state 兩 0 典 :
冉
冊
1
兩 z 典 ⫽exp共 zâ † ⫺z * â 兲 兩 0 典 ⫽exp ⫺ z * z⫹zâ † 兩 0 典 ,
2
â 兩 z 典 ⫽z 兩 z 典 .
共4.11兲
sin s,
passes entirely through points of M. Hence its image ␲ 关 C兴
⫽C lies entirely within M and, being the free geodesic connecting ␲ ( ␺ 1 ) to ␲ ( ␺ 2 ), it must be the constrained geodesic
as well. In this case, therefore, we do not get anything new.
Conversely, we see that to have a situation where constrained geodesics are different from free ones, the submanifold M 傺R must not arise from a subspace of H in the above
manner. We now look at two such cases, of obvious physical
importance, in which true generalizations of the original
Bargmann invariant-geometric phase connection appear.
we have
g ␮ ␯ 关 ␰ 共 s 兲兴 ␰˙ ␮ 共 s 兲 ␰˙ ␯ 共 s 兲 ⫽const⇒ 储 ⌿̇⬜ 共 s 兲储 ⫽const.
共 ␺ 2⫺共 ␺ 1 , ␺ 2 兲␺ 1 兲
共5.3兲
To conform to the notations of the preceding section, we
introduce real parameters ␰ 1 , ␰ 2 , include a phase angle ␣ ,
and express the above states in terms of q̂ and p̂ as follows
共for ease in writing we use ␰ 1,2 rather than ␰ 1,2):
z⫽
1
冑2
共 ␰ 1 ⫹i ␰ 2 兲 , ␰ 1,2⑀ R:
␺ 共 ␰ ; ␣ 兲 ⫽e i ␣ 兩 z 典 ⫽exp关 i ␣ ⫹i 共 ␰ 2 q̂⫺ ␰ 1 p̂ 兲兴 兩 0 典
冉
冉
冊
冊
i
⫽exp i ␣ ⫺ ␰ 1 ␰ 2 exp共 i ␰ 2 q̂ 兲 exp共 ⫺i ␰ 1 p̂ 兲 兩 0 典
2
i
⫽exp i ␣ ⫹ ␰ 1 ␰ 2 exp共 ⫺i ␰ 1 p̂ 兲 exp共 i ␰ 2 q̂ 兲 兩 0 典 .
2
共5.4兲
关Here we have omitted an s-dependent phase ␣ (s).兴 Each
vector ⌿(s) along this curve is a 共pure兲 coherent state, and
cannot be written as a linear combination of two fixed states
as in Eq. 共2.6兲; so it is immediately clear that this is not a
free geodesic at all.
Now we examine whether this constrained geodesic is a
null phase curve. We find, using the criterion 共3.4兲:
arg共 ⌿ 共 s 兲 ,⌿ 共 s ⬘ 兲兲 ⫽arg具 z 0 ⫹z 1 s 兩 z 0 ⫹z 1 s ⬘ 典
共Note that, as in Eq. 共4.2兲, ␺ ( ␰ ; ␣ ) is a vector in H parametrized by ␰ and ␣ , not a wave function.兲 These various
equivalent forms facilitate further calculations.
The expectation values of q̂ and p̂ in these states are
„␺ 共 ␰ ; ␣ 兲 ,q̂ ␺ 共 ␰ ; ␣ 兲 …⫽ ␰ 1 ,
共 ␺ 共 ␰ ; ␣ 兲 , p̂ ␺ 共 ␰ ; ␣ 兲兲 ⫽ ␰ 2 .
共5.5兲
Now we compute the vectors u ␮ ( ␰ ; ␣ ) and their projections u⬜␮ ( ␰ ; ␣ ) orthogonal to ␺ ( ␰ ; ␣ ), as defined in Eq. 共4.5兲:
冉
冊
1
u 1 ⫽ ⳵ 1 ␺ ⫽⫺i p̂⫺ ␰ 2 ␺ ,
2
冉
冊
1
u 2 ⫽ ⳵ 2 ␺ ⫽i q̂⫺ ␰ 1 ␺ ;
2
共5.6兲
u⬜1 ⫽⫺i 共 p̂⫺ ␰ 2 兲 ␺ , u⬜2 ⫽i 共 q̂⫺ ␰ 1 兲 ␺ .
Here we used Eq. 共5.5兲, and, for simplicity, omitted the arguments ␰ , ␣ in ␺ ,u ␮ ,u⬜␮ . The inner products among the u⬜␮
involve the fluctuations in q̂ and p̂ and the cross term. After
easy calculations we find
共5.7兲
共 u⬜2 ,u⬜2 兲 ⫽„␺ , 共 q̂⫺ ␰ 1 兲 2 ␺ …⫽ 共 ⌬q 兲 2 ⫽1/2.
Therefore, the induced metric tensor in the ␰ 1 ⫺ ␰ 2 plane,
defined in Eq. 共4.7兲, is
1
g ␮␯共 ␰ 兲 ⫽ ␦ ␮␯ ,
2
共5.8兲
namely, it is the ordinary Euclidean metric on R2 . Constrained geodesics in this case are just determined by straight
lines in the ␰ plane, since all ⌫’s vanish:
z 共 s 兲 ⫽z 0 ⫹z 1 s, z 0,1⫽
1
冑2
共 q 0,1⫹i p 0,1兲 :
␰ 1 共 s 兲 ⫽q 0 ⫹q 1 s, ␰ 2 共 s 兲 ⫽p 0 ⫹p 1 s.
共5.9兲
At the Hilbert space level, a constrained geodesic
Cconstr.geo. can be taken to be a curve within the family of
coherent states
Cconstr.geo.⫽ 兵 ⌿ 共 s 兲 ⫽ 兩 z 0 ⫹z 1 s 典 其 .
⫽arg关 exp共 z *
0 z 1 s ⬘ ⫹z 0 z *
1 s 兲兴
⫽ 共 s ⬘ ⫺s 兲 Im z 0* z 1 .
共5.11兲
This is a separable function of s ⬘ and s, so we do have a null
phase curve. We can go from the above Cconstr.geo. to a horizontal curve by adding a phase:
(h)
⫽ 兵 ⌿ ⬘ 共 s 兲 ⫽exp共 ⫺i s Im z 0* z 1 兲 ⌿ 共 s 兲 其 ,
C constr.geo.
共5.10兲
共5.12兲
and then we find that any two points on this curve are in
phase, as expected.
The generalized connection 共3.7兲 in this example now
states: if 兩 z 1 典 , 兩 z 2 典 , . . . , 兩 z n 典 are any n pure coherent states
given by choosing n points in the complex plane, and we join
these points successively by straight lines in the complex
plane so that all along in Hilbert space we deal with individual coherent states and never with superpositions of them,
we have
␸g
冋
n-sided plane polygon with vertices
at the coherent states z 1 ,z 2 , . . . ,z n
册
⫽⫺arg⌬ n 共 兩 z 1 典 , 兩 z 2 典 , . . . , 兩 z n 典 ).
共 u⬜1 ,u⬜1 兲 ⫽„␺ , 共 p̂⫺ ␰ 2 兲 2 ␺ …⫽ 共 ⌬ p 兲 2 ⫽1/2,
共 u⬜1 ,u⬜2 兲 ⫽⫺„␺ , 共 p̂⫺ ␰ 2 兲共 q̂⫺ ␰ 1 兲 ␺ …⫽i/2,
⫽arg关 exp兵 共 z *
0 ⫹z 1* s 兲共 z 0 ⫹z 1 s ⬘ 兲 其 兴
共5.13兲
The case n⫽3 leads to the area formula for the geometric
phase for a triangle in the plane, a very familiar result 关15兴.
From our point of view, the present example is a significant
generalization of the original connection 共2.9兲.
Going further, it is easy to convince oneself that in this
example the most general null phase curve arises in the
above manner; in other words, a given one-parameter family
of coherent states 兵 兩 z(s) 典 其 obeys the separability condition
共3.4兲 if and only if Im z(s) is a linear inhomogeneous expression in Re z(s), so that z(s) describes a straight line in the
complex plane as s varies.
C. Centered Gaussian pure states
This example again deals with one canonical pair q̂,p̂. It
is now more convenient to work with wave functions in the
Schrödinger representation, and not with abstract ket vectors.
The submanifold M傺H consists of normalized Gaussian
wave functions parametrized by two real variables ␰ 1 , ␰ 2 and
a phase angle ␣ defined as follows:
␺ 共 ␰ ; ␣ ;q 兲 ⫽
冉 冊 再
␰2
␲
1/4
冎
i
exp i ␣ ⫹ 共 ␰ 1 ⫹i ␰ 2 兲 q 2 ,
2
␰ 1 苸 共 ⫺⬁,⬁ 兲 , ␰ 2 苸 共 0,⬁ 兲 , ␣ 苸 关 0,2␲ 兲 .
共5.14兲
Normalizability requires that ␰ 2 be strictly positive, so the
combination ␰ 1 ⫹i ␰ 2 is a variable point in the upper half
complex plane. The wave functions u ␮ ( ␰ ; ␣ ;q) are
u 1 共 ␰ ; ␣ ;q 兲 ⫽
⳵
i
␺ 共 ␰ ; ␣ ;q 兲 ⫽ q 2 ␺ 共 ␰ ; ␣ ;q 兲 ,
⳵␰ 1
2
冊
It is clear that to obtain the components u⬜␮ of u ␮ orthogonal
to ␺ , and later to compute the inner products (u⬜␮ ,u⬜␯ ), we
need the expectation values of q 2 and q 4 in the state ␺ .
These are 共omitting for simplicity the arguments of ␺ ):
冉 冊冕
冉 冊冕
␰2
␲
1/2
␰2
共 ␺ ,q 4 ␺ 兲 ⫽
␲
1/2
共 ␺ ,q 2 ␺ 兲 ⫽
⬁
⫺⬁
2
dq q 2 e ⫺ ␰ 2 q ⫽
1
,
2␰2
冉
2
dq q 4 e ⫺ ␰ 2 q ⫽
4 ␰ 22
.
冊
冋 冉 冊 册
冊 册
冋 冉
冊 册
冋 冉
共 u⬜1 ,u⬜1 兲 ⫽
1
1
␺ , q 2⫺
4
2␰2
共 u⬜1 ,u⬜2 兲 ⫽
i
1
␺ , q 2⫺
4
2 ␰2
2
␺ ⫽
1
8 ␰ 22
2
␺ ⫽
1
1
␺ , q 2⫺
4
2 ␰2
i
共5.17兲
;
共5.22a兲
1 2 2
共 ␰˙ ⫺ ␰˙ 兲 ⫽0.
␰2 1 2
共5.22b兲
8␰2
2
g ␮ ␯ 共 ␰ 兲 ⫽Re„u⬜␮ 共 ␰ ; ␣ 兲 ,u⬜␯ 共 ␰ ; ␣ 兲 …⫽
1
8 ␰ 22
1
共5.18兲
This is the well known form of the Lobachevskian metric in
this model of Lobachevsky space 关16兴. Dropping the numerical factor 81 for simplicity, the line element in the upper half
plane is given by
共 d ␰ 21 ⫹d ␰ 22 兲 ,
2
the value of the constant depending on the particular geodesic. After elementary analysis, we find that there are two
families of geodesics:
␰ 1 ⫽const, ␰ 2 ⫽ae bs ,
a⬎0,s苸R;
共5.24a兲
Type II: ␰ 1 ⫽c⫹R cos f 共 s 兲 , ␰ 2 ⫽R sin f 共 s 兲 ,
f 共 s 兲 ⫽2 tan⫺1 共 ae bs 兲 ,
共5.24b兲
Type II: ␰ 1 ⫽c⫹R cos s, ␰ 2 ⫽R sin s,
.
2
8 ␰2
␦ ␮␯ .
共5.23兲
These are both in affinely parametrized form. In Type II it is
simpler to pass to a nonaffine angle type parameter s
苸(0,␲ ), and replace Eq. 共5.24b兲 with
;
2
␺ ⫽
共 ␰˙ 21 ⫹ ␰˙ 22 兲 ⫽const,
c苸R, R⬎0, a⬎0, b⬎0, s苸R.
From these results we obtain the induced metric over M
⫽ ␲ 关 M兴 傺R, described in the upper half complex plane by
the metric tensor
␰2
␰¨ 2 ⫹
Type I:
冉
1
2
˙␰ ␰˙ ⫽0,
␰2 1 2
␰¨ 1 ⫺
␰ 22
i
, 共 ␺ ,u 2 兲 ⫽0;
4␰2
冊
ds 2 ⫽
Using these in Eq. 共4.8兲 we find the following ordinary differential equations to determine geodesics:
1
3
i 2
1
1
1
q ⫺
␺ ,u⬜2 ⫽u 2 ⫽⫺ q 2 ⫺
␺;
2
2␰2
2
2␰2
共 u⬜2 ,u⬜2 兲 ⫽ 共 u 2 ,u 2 兲 ⫽
共5.21兲
We can exploit the fact that these differential equations
lead to the consequence
Now the necessary inner products and projections are easily
found:
u⬜1 ⫽
1
.
␰2
共5.16兲
⬁
⫺⬁
共 ␺ ,u 1 兲 ⫽
共5.20兲
We easily find that the nonvanishing ⌫’s are
⌫ 1 12共 ␰ 兲 ⫽⌫ 2 22共 ␰ 兲 ⫽⫺⌫ 2 11共 ␰ 兲 ⫽⫺
共5.15兲
⳵
1
1
␺ 共 ␰ ; ␣ ;q 兲 ⫽ ⫺q 2 ⫹
␺ 共 ␰ ; ␣ ;q 兲 .
u 2 共 ␰ ; ␣ ;q 兲 ⫽
⳵␰ 2
2
2␰2
冉
g 11共 ␰ 兲 ⫽g 22共 ␰ 兲 ⫽ ␰ 22 , g 12共 ␰ 兲 ⫽0.
c苸R, R⬎0, 0⬍s⬍ ␲ .
共5.25兲
Type I geodesics are straight semi-infinite lines parallel to
the ␰ 2 axis. Type II geodesics are semicircles centered on the
␰ 1 axis and lying above this axis.
In each case we can now ask whether a constrained geodesic in M is a null phase curve. As in the previous example
of coherent states, here too we emphasize that we are concerned with curves within the manifold of centered normalized Gaussian wave functions, and at no stage with linear
combinations of such wave functions. We look at the two
types of constrained geodesics in turn and find these results
共after simple reparametrizations兲:
Type I: ⌿ 共 s 兲 ⫽ ␺ 关 ␰ 1 ⫽a, ␰ 2 ⫽bs; ␣ 共 s 兲兴 ,
共5.19兲
and we must find the corresponding geodesics.
First we compute the nonvanishing ⌫’s. The inverse of
关 g ␮ ␯ ( ␰ ) 兴 has components
arg„⌿ 共 s 兲 ,⌿ 共 s ⬘ 兲 …⫽0;
共5.26a兲
Type II:⌿ 共 s 兲 ⫽ ␺ 关 ␰ 1 ⫽c⫹R cos s, ␰ 2 ⫽R sin s; ␣ 共 s 兲兴 ,
arg„⌿ 共 s 兲 ,⌿ 共 s ⬘ 兲 …⫽ 41 共 s⫺s ⬘ 兲 .
共5.26b兲
关In both cases the choice of phase angle ␣ (s) is irrelevant.兴
So in both cases the criterion 共3.4兲 is obeyed, and both types
of curves in M arising from the two types of geodesics in the
upper half ␰ plane are simultaneously constrained geodesics
and null phase curves.
The statement of the generalized connection 共3.7兲 is clear,
and for illustration we consider the case of just three vertices.
Let A, A ⬘ , and A ⬙ be any three points in the upper half
complex plane, and for any choices of phases ␣ consider the
three normalized centered Gaussian states ␺ (A; ␣ ),
␺ (A ⬘ ; ␣ ⬘ ), and ␺ (A ⬙ ; ␣ ⬙ ). Join A to A ⬘ , A ⬘ to A ⬙ , and A ⬙ to
A by a geodesic of Type I or Type II as appropriate in each
case. This can always be done, and we obtain a hyperbolic
triangle. In M we obtain a triangle with vertices ␳ (A)
⫽ ␲ 关 ␺ (A; ␣ ) 兴 etc., and whose sides are constrained geodesics, and we can state
␸g
冋
triangle in M with vertices
␳共 A 兲,␳共 A⬘兲,␳共 A⬙兲
and sides as constrained geodesics
册
z 1 ⫽cos ␪ , z 2 ⫽e i ␾ sin ␪ , 0⭐ ␪ ⭐ ␲ , 0⭐ ␾ ⭐2 ␲ .
共5.30兲
Therefore, the submanifold M傺H is parametrized by ␪ , ␾
and a phase ␣ and we write
M⫽ 兵 ␺ 共 ␪ , ␾ ; ␣ 兲
⫽e i ␣ 兩 cos ␪ ,e i ␾ sin ␪ 典 兩 0⭐ ␪ ⭐ ␲ ,0⭐ ␾ , ␣ ⭐2 ␲ 其 傺H,
共5.31兲
where the ket on the right is a particular two-mode coherent
state with z† z⫽1:
␺ 共 ␪ , ␾ ; ␣ 兲 ⫽exp共 i ␣ ⫹â †1 cos ␪ ⫹â †2 e i ␾ sin ␪ ⫺1/2兲 兩 0 典 .
共5.32兲
Omitting the arguments ␪ , ␾ , ␣ for simplicity, we easily find
u ␪⫽
⫽⫺arg ⌬ 3 关 ␺ 共 A; ␣ 兲 , ␺ 共 A ⬘ ; ␣ ⬘ 兲 , ␺ 共 A ⬙ ; ␣ ⬙ 兲兴 .
⳵
␺ ⫽ 共 ⫺sin ␪ â †1 ⫹e i ␾ cos ␪ â †2 兲 ␺ ,
⳵␪
共5.27兲
An application of this result has been used elsewhere 关17兴 to
show that the classical Gouy phase 关18兴 in wave optics is
related to a Bargmann invariant and hence is a geometric
phase.
D. Subset of two-mode coherent states
u ␾⫽
⳵
␺ ⫽i e i ␾ sin ␪ â †2 ␺ ;
⳵␾
共5.33a兲
共 ␺ ,u ␪ 兲 ⫽0, 共 ␺ ,u ␾ 兲 ⫽i sin2 ␪ ;
共5.33b兲
u⬜␪ ⫽u ␪ , u⬜␾ ⫽i sin ␪ 共 e i ␾ â †2 ⫺sin ␪ 兲 ␺ .
共5.33c兲
In the preceding two examples, we found that while constrained geodesics differed from free geodesics, they were
nevertheless null phase curves and so led to important instances of Eq. 共3.7兲. This is, however, fortuitous; the really
important objects for our purposes are the null phase curves,
and in a given situation constrained geodesics may well not
be such curves. In our fourth and final example, dealing with
a subset of states for a two-mode system, we will find that
this is exactly what happens. However, we will be able to
completely determine all null phase curves directly, so that
the generalization 共3.7兲 can be meaningfully stated.
For a two-mode system with creation and annihilation operators â ⫹
j ,â j obeying the standard commutation relations
Repeatedly exploiting the eigenvector relation 共5.29兲 and its
adjoint, we compute the inner products among the vectors in
Eq. 共5.33c兲:
关 â j ,â †k 兴 ⫽ ␦ jk , 关 â j ,â k 兴 ⫽ 关 â †j ,a †k 兴 ⫽0, j,k⫽1,2,
The corresponding constrained geodesics are therefore simply great-circle arcs. The question is whether they lead to
null phase curves in M and M.
A general parametrized great-circle arc on S 2 is traced out
by an s-dependent unit vector n̂(s) with polar angles
␪ (s), ␾ (s):
共5.28兲
the general coherent state is labeled with two independent
complex numbers arranged as a column vector z⫽(z 1 ,z 2 ) T :
兩 z典 ⫽exp共 ⫺ 21 z† z⫹z 1 â †1 ⫹z 2 â †2 兲 兩 0 典 ,
共5.29兲
â j 兩 z典 ⫽z j 兩 z典 , j⫽1,2.
Within this family of all normalized coherent states, we now
define a submanifold 共of real dimension three including an
overall phase兲, an ‘‘S 2 worth of states,’’ by taking ␪ , ␾ to be
spherical polar angles on a sphere S 2 and setting z 1 and z 2
equal to the following:
共 u⬜␪ ,u⬜␪ 兲 ⫽1, 共 u⬜␪ ,u⬜␾ 兲 ⫽i cos ␪ sin ␪ , 共 u⬜␾ ,u⬜␾ 兲 ⫽sin2 ␪ .
共5.34兲
Taking the real parts here, we see that the metric induced on
M ⫽ ␲ 关 M兴 ⬃S 2 in R, parametrized by angles ␪ and ␾ , is
just the usual rotationally invariant one:
g ␪␪ 共 ␪ , ␾ 兲 ⫽1, g ␪␾ ⫽0, g ␾␾ 共 ␪ , ␾ 兲 ⫽sin2 ␪ .
共5.35兲
n̂ 共 s 兲 ⫽â cos s⫹b̂ sin s
⫽ 关 sin ␪ 共 s 兲 cos ␾ 共 s 兲 , sin ␪ 共 s 兲 sin ␾ 共 s 兲 , cos ␪ 共 s 兲兴 ,
â,b̂苸S 2 , â•b̂⫽0.
共5.36兲
The corresponding constrained geodesic Cconstr. geo.傺M
共omitting the phase ␣ ) is the curve of coherent states
⌿ 共 s 兲 ⫽ 兩 z 1 共 s 兲 ,z 2 共 s 兲 典 ,
z 1 共 s 兲 ⫽cos ␪ 共 s 兲 ⫽a 3 cos s⫹b 3 sin s,
共5.37兲
z 2 共 s 兲 ⫽e i ␾ (s) sin ␪ 共 s 兲 ⫽ 共 a 1 ⫹i a 2 兲 cos s⫹ 共 b 1 ⫹i b 2 兲 sin s.
To see if this is a null phase curve we compute the phase of
„⌿(s),⌿(s ⬘ )…:
arg„⌿ 共 s 兲 ,⌿ 共 s ⬘ 兲 …⫽arg具 z 1 共 s 兲 ,z 2 共 s 兲 兩 z 1 共 s ⬘ 兲 ,z 2 共 s ⬘ 兲 典
⫽arg兵 exp共 z 1 共 s 兲 z 1 共 s ⬘ 兲 ⫹z 2 共 s 兲 * z 2 共 s ⬘ 兲兲 其
⫽arg兵 exp共关共 a 1 ⫺i a 2 兲 cos s
⫹ 共 b 1 ⫺i b 2 兲 sin s 兴关共 a 1 ⫹i a 2 兲 cos s ⬘
⫹ 共 b 1 ⫹i b 2 兲 sin s ⬘ 兴 兲 其
⫽ 共 â ⵩ b̂ 兲 3 sin共 s ⬘ ⫺s 兲 .
共5.38兲
Unless it vanishes, this is not a separable function of s ⬘ and
s. We conclude that the geodesic 共5.36兲 on S 2 leads to a
constrained geodesic Cconstr.geo.傺M, which is, in general, not
a null phase curve. The only exception is when (â ⵩ b̂) 3 ⫽0,
that is, the geodesic 共5.36兲 on S 2 lies on a meridian of longitude, with â ⵩ b̂ being a vector in the 1-2 plane.
On the other hand, in this example it is quite easy to
explicitly find all null phase curves on M 共and M)! Let ⌫
⫽ 兵 n̂(s) 其 傺S 2 be given, and let us consider the induced curve
C⌫ in M:
C⌫ ⫽ 兵 ⌿ ⌫ 共 s 兲 ⫽ 兩 n 3 共 s 兲 ,n 1 共 s 兲 ⫹i n 2 共 s 兲 典
†
†
⫽exp„⫺ 12 ⫹n 3 共 s 兲 â 1 ⫹ 共 n 1 共 s 兲 ⫹i n 2 共 s 兲兲 â 2 ) 兩 0 典 其 .
able to find all of the latter, and any two points in M can be
connected by some null phase curve, we have succeeded in
providing a nontrivial two-mode example of the generalized
connection 共3.7兲, without using constrained geodesics.
VI. RAY SPACE AND DIFFERENTIAL GEOMETRIC
FORMULATIONS
Very soon after the discovery of the geometric phase, the
differential geometric expressions of its structure and significance were developed 关19,3–5,11兴, by relating it to anholonomy and curvature in a suitable Hermitian line bundle
in quantum mechanical ray space. In this section we provide
a brief discussion of the properties and uses of the new concept of null phase curves at the ray space level and also in
the differential geometric language. Only necessary background material will be recalled, and derivations will be
omitted. Since they may be useful for practical calculations,
where possible local coordinate expressions of important differential geometric objects will be given.
From the preceding sections it is evident that for our purposes it is important to deal with open null phase curves in
general, since it is through them that the connection 共3.7兲 of
the Bargmann invariants to geometric phases is made. Their
definition 共3.4兲 in terms of Hilbert space lifts is quite simple.
Nevertheless, it is of interest to develop a direct ray space
formulation; this can be done essentially via the Bargmann
invariants themselves. From their definition 共2.8兲, it is clear
that any 䉭 2 is real nonnegative, while 䉭 n ’s for n⭓3 are, in
general, complex. On the other hand, it is also known that
any 䉭 n for n⭓4 can be written as the ratio of a suitable
product of 䉭 3 ’s and a suitable product of 䉭 2 ’s:
䉭 n 共 ␺ 1 , ␺ 2 ,•••, ␺ n 兲
共5.39兲
n
⫽
We find that
arg„⌿ ⌫ 共 s 兲 ,⌿ ⌫ 共 s ⬘ 兲 …⫽ 关 n̂ 共 s 兲 ⵩ n̂ 共 s ⬘ 兲兴 3 .
共5.40兲
This will be a separable function of s ⬘ and s if and only if,
for some constants ␤ and ␥ , we have
n 2共 s 兲 ⫽ ␤ n 1共 s 兲 ⫹ ␥ .
共5.41兲
The geometrical interpretation of this is that the projection of
⌫ on the 1-2 plane must be a straight line. In that case, C⌫ is
indeed a null phase curve in M, as we have
arg„⌿ ⌫ 共 s 兲 ,⌿ ⌫ 共 s ⬘ 兲 …⫽ ␥ 关 n 1 共 s 兲 ⫺n 1 共 s ⬘ 兲兴 ,
共5.42兲
which is separable in s ⬘ and s. One can easily see that each
such ⌫ is a latitude circle arc on S 2 corresponding to 共i.e.,
perpendicular to兲 some axis lying in the 1-2 plane, and given
any two points on S 2 , we can always connect them by such a
⌫. In other words, such ⌫ are intersections of S 2 with planes
perpendicular to the 1-2 plane. When such a latitude circle
arc is also a great-circle arc, we recover the result of the
previous paragraph.
The upshot of this example is that here we have a nontrivial illustration of the difference between constrained geodesics and null phase curves. However, since we have been
兿
j⫽3
冒兿
n
䉭 3 共 ␺ 1 , ␺ j⫺1 , ␺ j 兲
j⫽4
䉭 2 共 ␺ 1 , ␺ j⫺1 兲 .
共6.1兲
In this sense the three-vertex Bargmann invariant 䉭 3 is the
basic or primitive one as far as phases are concerned. 关The
basic cyclic invariance of 䉭 n ( ␺ 1 , ␺ 2 ,•••, ␺ n ) is not manifest in Eq. 共6.1兲, but it is not lost either.兴 Guided by these
facts, we give now a direct ray space characterization of null
phase curves.
If C⫽ 兵 ␳ (s) 其 傺R is a null phase curve and C (h)
⫽ 兵 ␺ (h) (s) 其 is a horizontal Hilbert space lift obeying Eq.
共3.4兲, we see immediately that for any choices of parameter
values s,s ⬘ ,s ⬙ ,
䉭 3 关 ␺ (h) 共 s 兲 , ␺ (h) 共 s ⬘ 兲 , ␺ (h) 共 s ⬙ 兲兴 ⫽Tr兵 ␳ 共 s 兲 ␳ 共 s ⬘ 兲 ␳ 共 s ⬙ 兲 其
⫽real and ⭓0;
共6.2兲
and so also for any n parameter values s 1 ,s 2 , . . . ,s n , from
Eq. 共6.1兲,
䉭 n 关 ␺ (h) 共 s 1 兲 , . . . , ␺ (h) 共 s n 兲兴 ⫽Tr兵 ␳ 共 s 1 兲 . . . ␳ 共 s n 兲 其
⫽real and ⭓0.
共6.3兲
As a consequence, by differentiation with respect to
s 2 , . . . ,s n we have
再
Tr ␳ 共 s 1 兲
冎
d␳共 sn兲
d␳共 s2兲
⫽real.
•••
ds n
ds 2
共6.4兲
n
Now, it is known that the geometric phase for any connected
portion of any C can be expressed directly in terms of ␳ (s)
as follows, whether or not C is a null phase curve:
␸ g 关 ␳ 共 s 1 兲 to ␳ 共 s 2 兲 along C 兴
冉 冕
冋再
⫽arg Tr ␳ 共 s 1 兲 P exp
冋
⬁
⫽arg 1⫹
再
兺
n⫽1
⫻Tr ␳ 共 s 1 兲
s1
ds n
d ␳ 共 s n⬘ 兲
ds n⬘
ds
ds
s ⬘n
s1
•••
冊冎册
⬘ •••
ds n⫺1
d ␳ 共 s 1⬘ 兲
ds 1⬘
冎册
冕
s ⬘2
s1
ds 1⬘
,
共6.5兲
⇔Tr兵 ␳ 共 s 兲 ␳ 共 s ⬘ 兲 ␳ 共 s ⬙ 兲 其
⫽real nonnegative, any s,s ⬘ ,s ⬙ .
共6.6兲
Turning now to the specific differential geometric aspects, it
is well known that while the dynamical phase ␸ dyn关 C兴 is an
additive quantity, ␸ g 关 C 兴 does not have this property. On the
manifold of unit vectors in Hilbert space H, there is a oneform A such that
C
A.
共6.7兲
However, referring to the projection ␲ :H→R, A is not the
pull-back via ␲ * of any one-form on the space of unit rays,
and ␸ g 关 C 兴 is not the integral along C of any one-form on R.
In fact, this lack of additivity can be expressed via the Bargmann invariant 䉭 3 . If C 12 connects ␳ 1 to ␳ 2 in R and C 23
connects ␳ 2 to ␳ 3 , then C 12艛C 23 runs from ␳ 1 to ␳ 3 and
␸ g 关 C 12艛C 23兴 ⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 23兴 ⫺B 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 ,
共6.8兲
B 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 ⫽arg䉭 3 共 ␺ 1 , ␺ 2 , ␺ 3 兲 .
More generally, for an 共generally兲 open curve consisting of
(n⫺1) pieces C 12 ,C 23 ,•••C n⫺1,n joining ␳ 1 to ␳ 2 , ␳ 2 to
␳ 3 , •••, ␳ n⫺1 to ␳ n , we generalize Eq. 共6.8兲 to the following:
n⫺1
␸ g 关 C 12艛C 23艛•••艛C n⫺1,n 兴 ⫽
B 3 共 ␺ 1 , ␺ j⫺1 , ␺ j 兲 .
共6.9兲
If we connect ␳ n back to ␳ 1 via C n,1 to get a closed curve of
n pieces, then we have the specific result
⫺B n 共 ␺ 1 , ␺ 2 ,•••, ␺ n 兲 .
C⫽ 兵 ␳ 共 s 兲 其 傺R is a null phase curve
冕
兺
j⫽3
⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 23兴 ⫹•••⫹ ␸ g 关 C n,1兴
where P is the ordering symbol placing later parameter values to the left of earlier ones. If Eq. 共6.2兲 holds on C 关and so
as a consequence Eqs. 共6.3兲 and 共6.4兲 as well兴, we see that at
every stage only real quantities are involved, the geometric
phase in Eq. 共6.5兲 vanishes, and C is a null phase curve. This
leads to the ray space characterization of null phase curves
we are seeking:
␸ dyn关 C兴 ⫽
⫽
␸ g 关 C 12艛C 23艛•••艛C n⫺1,n ,艛C n,1兴
d␳共 s 兲
s1
冕 ⬘冕
s2
s2
B n 共 ␺ 1 , ␺ 2 ,•••, ␺ n 兲 ⫽arg䉭 n 共 ␺ 1 , ␺ 2 , . . . , ␺ n 兲
兺 ␸ g 关 C j, j⫹1 兴
j⫽1
⫺B n 共 ␺ 1 , ␺ 2 ,•••, ␺ n 兲 ,
共6.10兲
Compared to Eq. 共6.9兲, we have one extra ␸ g term on the
right-hand side, but the Bargmann phase term B n is the same.
We see that the lack of additivity shown in all Eqs.
共6.8,6.9,6.10兲 is due to the Bargmann pieces. There is, however, an exception to this general nonadditivity, which occurs in Eq. 共6.8兲 when ␳ 3 ⫽ ␳ 1 and C 12艛C 23 is a closed loop.
Then we find
⳵ 共 C 12艛C 23兲 ⫽0, ␳ 3 ⫽ ␳ 1 :
␸ g 关 C 12艛C 21兴 ⫽ ␸ g 关 C 12兴 ⫹ ␸ g 关 C 21兴 ,
共6.11兲
i.e., ␸ g 关 C 12兴 ⫽ ␸ g 关 C 12艛C 21兴 ⫺ ␸ g 关 C 23兴 .
In the past, this result has been used 关4兴 to relate ␸ g 关 C 兴 for
an open C to ␸ g 关 C艛C ⬘ 兴 for a closed C艛C ⬘ by choosing C ⬘
to be a free geodesic, for then ␸ g 关 C ⬘ 兴 ⫽0. Now we can
generalize this process: if C is an open curve from ␳ 1 to ␳ 2
in R, and C ⬘ is any null phase curve from ␳ 2 back to ␳ 1 , we
have the result
␸ g 关 open curve C 兴 ⫽ ␸ g 关 closed loop C艛C ⬘ 兴 .
共6.12兲
This is the most general way in which an open curve geometric phase can be reduced to a closed loop geometric
phase. More generally, comparing Eqs. 共6.9,6.10兲 valid for
generally open and for a closed curve, we see that if the last
piece C n,1 is a null phase curve, we convert an open curve
geometric phase to a closed loop geometric phase:
␸ g 关 C 12艛C 23艛•••艛C n⫺1,n 兴
⫽ ␸ g 关 C 12艛C 23艛•••艛C n⫺1,n 艛C n,1兴 .
共6.13兲
At this point it is natural to express a closed loop geometric phase as a suitable ‘‘area integral’’ of a two-form, both at
Hilbert and ray space levels. Whereas A is not the pull-back
of any one-form on R, we do have dA⫽ ␲ * ␻ , where ␻ is a
symplectic 共closed, nondegenerate兲 two-form on R. Then, if
C is a closed loop in H, ⳵ C⫽0, so that C⫽ ␲ (C) is a closed
loop in R, we have
␸ g关 C 兴 ⫽
冕
S
dA⫽
冕␻
S
,
共6.14兲
where S and S⫽ ␲ (S) are two-dimensional surfaces in H
and R, respectively, with boundaries C and C :
⳵ S⫽C, ⳵ S⫽C.
共6.15兲
With the help of local coordinates on H and R we get explicit expressions for A,dA and ␻ . Around any point ␳ 0
苸R, and for some chosen ␺ 0 苸 ␲ ⫺1 ( ␳ 0 ), we define an
共open兲 neighborhood N傺R by
N⫽ 兵 ␳ 苸R兩 Tr 共 ␳ 0 ␳ 兲 ⬎0 其 .
共6.16兲
We can introduce real independent coordinates over N as
follows. Let 兵 ␺ 0 ,e 1 ,e 2 , . . . ,e r , . . . 其 be an orthonormal basis for H. Then points in N can be ‘‘labeled’’ in a one-to-one
manner with vectors X苸H orthogonal to ␺ 0 and with norm
less than unity:
冑2
兺r 共 ␤ r ⫺i ␥ r 兲 e r ,
1
2
兺r 共 ␤ r2 ⫹ ␥ r2 兲 ⬍1:
1
␹共 ␤,␥ 兲⫽
储 ␹ 共 ␤ , ␥ 兲储 2 ⫽
␺ 共 ␤ , ␥ 兲 ⫽ ␹ 共 ␤ , ␥ 兲 ⫹ 冑1⫺ 储 ␹ 共 ␤ , ␥ 兲储 2 ␺ 0 ,
共6.17兲
␳ 苸N⇔ ␳ ⫽ ␺ 共 ␤ , ␥ 兲 ␺ 共 ␤ , ␥ 兲 † , for some ␤ , ␥ .
Thus the real independent ␤ ’s and ␥ ’s, subject to the inequality above, are local coordinates for N. They can be extended to get local coordinates for ␲ ⫺1 (N)傺H by including
a phase angle ␣ :
␺苸␲
⫺1
i␣
共 N 兲 ⇔ ␺ ⫽ ␺ 共 ␣ ; ␤ , ␥ 兲 ⫽e ␺ 共 ␤ , ␥ 兲 , 0⭐ ␣ ⬍2 ␲ .
共6.18兲
In these local coordinates over N and ␲ ⫺1 (N) we have the
expressions
A⫽d ␣ ⫹
1
2
兺r 共 ␥ r d ␤ r ⫺ ␤ r d ␥ r 兲 ,
dA⫽
␻⫽
兺r d ␥ r ⵩d ␤ r ,
共6.19兲
兺r d ␥ r ⵩d ␤ r .
The closure and nondegeneracy of ␻ are manifest, so it is a
symplectic two-form on R; and the coordinates ␤ , ␥ realize
the local Darboux or canonical structure for it. On the other
hand, in these ‘‘symplectic’’ coordinates the Fubini-Study
metric is a bit involved. If we combine the ␤ ’s and ␥ ’s into
a single column vector ␩ ⫽( ␤ 1 ␤ 2 . . . ␥ 1 ␥ 2 . . . ) T , then
the length functional L 关 C 兴 of Eq. 共2.5兲 assumes the following local form:
L关C兴⫽
g 共 ␩ 兲 ⫽1⫹
J⫽
冉
1
2
冕
ds 冑␩˙ T g 共 ␩ 兲 ␩˙ ,
␩ ␩T
1
1⫺ ␩ T ␩
2
0
1
⫺1
0
冊
1
⫹ J ␩ ␩ T J,
2
, ␩ T ␩ ⬍2.
共6.20兲
The symplectic matrix J plays a role in this expression for
the metric tensor matrix g( ␩ ). This matrix g( ␩ ) is verified
to be real symmetric positive definite, since one eigenvalue
is (1⫺ 21 ␩ T ␩ ) ⫺1 共eigenvector ␩ ), another eigenvalue is (1
⫺ 12 ␩ T ␩ ) 共eigenvector J ␩ ), and the remaining eigenvalues
are all unity. We appreciate that for considerations of geometric phases and null phase curves this kind of local description is really appropriate, while free geodesics appear
unavoidably complicated.
We also notice that, when H is finite dimensional and the
real dimension of the space R of unit rays is 2n, the symplectic two-form ␻ of Eq. 共6.19兲 is invariant under the linear
matrix group S p(2n,R) acting on the local coordinates ␤ , ␥ .
On the other hand, the integrand of the length functional
L 关 C 兴 in Eq. 共6.20兲 possesses invariance only under
S p(2n,R)艚SO(2n)⯝U(n), which is just the group of
changes in the choice of the vectors 兵 e r 其 which together with
␺ 0 make up an orthonormal basis for H.
Returning now to the discussions in Secs. III and IV, we
can bring in submanifolds M 傺R, M⫽ ␲ ⫺1 (M )傺H, with
local coordinates ␰ ␮ , ␣ as indicated in Eqs. 共4.1兲 and 共4.2兲.
Let i M : M傺H and i M :M 傺R be the corresponding identification maps. Straightforward calculations show that the
pull-backs of A,dA, ␻ in Eq. 共6.15兲 to M and M are locally
given 共with mild abuse of notation兲 by
* A⫽d ␣ ⫹Im„␺ 共 ␰ ; ␣ 兲 ,u ␮ 共 ␰ ; ␣ 兲 … d ␰ ␮ ,
iM
* dA⫽i M* ␻ ⫽Im„u⬜␮ 共 ␰ ; ␣ 兲 ,u⬜␯ 共 ␰ ; ␣ 兲 … d ␰ ␮ ⵩d ␰ ␯
iM
⫽Im关 u ␮ 共 ␰ ; ␣ 兲 ,u ␯ 共 ␰ ; ␣ 兲兴 d ␰ ␮ ⵩d ␰ ␯ .
共6.21兲
We see, as is well known, that while the real symmetric part
of the Hermitian matrix 关 (u⬜␮ ,u⬜␯ ) 兴 determines the metric,
Eq. 共4.7兲, the imaginary antisymmetric part of the same matrix is relevant for symplectic structure and geometric phase,
reinforcing the link between the latter two. 关When M ⫽R
and M⫽H, the ␰ ␮ ’s become the ␤ ’s and ␥ ’s of Eqs. 共6.17兲,
and we immediately recover the expressions 共6.19兲.兴 For our
present purposes, the following comments are pertinent.
While ␻ is closed and nondegenerate, i M* ␻ is closed but
may well be degenerate. An extreme case is when M is an
isotropic submanifold in R, for then i M* ␻ ⫽0. Such a situation can easily arise if, for example, M is described by a
family of real Schrödinger wave functions ␺ ( ␰ ;q). 共A Lagrangian submanifold in R is a particular case of an isotropic
submanifold when the dimension is maximal, namely, half
the real dimension of R.) One may expect that if M is isotropic and C傺M , then C is a null phase curve. However, this
need not always be so, and the situation is as follows. For a
general open curve C 12 from ␳ 1 to ␳ 2 in a general submanifold M, if we can find a null phase curve C 21 from ␳ 2 to ␳ 1
also lying in M, then C 12艛C 21 is a closed loop; if ␲ 1 (M )
⫽0, we can find a two-dimensional surface S苸M having
C 12艛C 21 as boundary. Then from Eq. 共6.12兲 we obtain under these circumstances
␸ g 关 C 12兴 ⫽ ␸ g 关 C 12艛C 21兴 ⫽
冕
S苸M
i*
M ␻.
共6.22兲
Here, as stated above, we had to choose C 21 to be a null
phase curve. 共When ␳ 2 ⫽ ␳ 1 and C 12 is already a closed loop,
there is no need for any C 21 ; it can be chosen to be trivial!兲
If, however, M is an isotropic submanifold, i.e., i *
M ␻ ⫽0
i*
M ␻ ⫽0 ⇔
关and assuming also ␲ 1 (M )⫽0], we can extract some very
interesting consequences for geometric phases, though it
falls short of the vanishing of ␸ g 关 C 兴 for every C傺M . We
have the chain of implications
冕 *␻
S
i M ⫽0, any two dimensional S傺M
⇔ ␸ g 关 C 12艛C 21兴 ⫽0, any ␳ 1 , ␳ 2 ,C 12 ,C 21 in M
⇔ ␸ g 关 C 12兴 unchanged under any continuous deformation of C 12 leaving the end points ␳ 1 , ␳ 2 fixed
Thus, within an isotropic submanifold, the geometric phase
for a general curve depends on the two end points alone.
When the curve chosen is closed, it can be continuously
shrunk to a point 关since, ␲ 1 (M )⫽0] and then its geometric
phase vanishes. One can thus say in summary:
M 傺R,i .M* ␻ ⫽0, ␲ 1 共 M 兲 ⫽0, C傺M :
␸ g 关 C 兴 ⫽0 if ⳵ C⫽0;
共6.24兲
␸ g 关 C 兴 ⫽function of ⳵ C alone, if ⳵ C⫽0.
The main conclusion is that general open curves in an isotropic submanifold need not be null phase curves, but geometric phases are invariant under continuous changes of their
arguments, leaving the end points unchanged. Perhaps this is
not too surprising after all, since the isotropic property is a
two-form condition.
共6.23兲
broadest one possible. The essential concept is that of null
phase curves in Hilbert and ray spaces—the replacement of
free geodesics by such curves leads to our generalization. We
have seen through examples that this wider connection between Bargmann invariants and geometric phases is just
what is needed in several physically relevant situations.
Motivated by the fact that free geodesics are always null
phase curves, we have defined the concept of constrained
geodesics and posed the problem of determining when these
may be null phase curves. We have presented two examples
when this is indeed so and one where they are not the same.
This re-emphasizes the fact that constrained geodesics and
null phase curves are, in principle, different objects, and intensifies the need to find useful characterizations of the
former which may ensure the latter property for them. This is
certain to shed more light on the general questions raised in
this paper, and we plan to return to them in the future.
VII. CONCLUDING REMARKS
ACKNOWLEDGMENTS
We have shown that the familiar connection between the
Bargmann invariants and geometric phases in quantum mechanics, based on the properties of free geodesics in ray and
Hilbert spaces, can be generalized to a very significant extent. In fact we have shown that our generalization is the
One of us 共E.M.R.兲 thanks the Third World Academy of
Sciences, Trieste, Italy for their financial support, the
JNCASR, Bangalore, India for support, and the Center for
Theoretical Studies, IISc., Bangalore for providing facilities
during the completion of this work.
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