the geometric phase from imaginary time translations

Physics Letters A 160 ( 1991 ) 31-35 North-Holland

PHYSICS LETTERS A

The geometric phase from imaginary time translations

Adr i an Stanley The Physics Laboratory, University of Kent at Canterbury, Canterbury CT2 7NZ, UK

Received 22 July 1991; revised manuscript received 26 August 1991; accepted for publication 6 September ! 991 Communicated by J.P. Vigier

The geometric phase is obtained from projection operators of states which have been translated in imaginary time. This leads to a new geometric interpretation of the phase in terms of deformations of the underlying Hilbert space.

I. Introduction

The geometric phase [ 1 ] is a remarkable phenom- enon arising from anholonomy in the parallel trans- port [ 2 ] o f a state vector in a projective Hilbert space [3]. This is equivalent to stating that the space is intrinsically curved. This non-trivial topology is im- posed on the space by the requirement that the state vector is invariant under multiplication by arbitrary complex numbers (i.e. the state is projective geometric) [3 ]. An arbitrary pure state vector will be represented by n complex numbers: l a ) ~ C n. The above condition creates a surjective mapping: C n - * C P n - I thus inducing the non-trivial topology. In this Letter it will be shown how the geometric phase can be extracted from a modified form of projection operator that is a function of a time parameter which is permitted to take complex values (section 3). This allows the space defined by the real-time projection operator (CP n- ~ ) to be viewed as a section through a higher-dimensional space defined by the complex-time projection operator. The phase is then a measure o f the rate o f deformation of the sec- tions with respect to the imaginary time coordinate (section 4).

2. Real-time projection operators

In the von Neumann formulation o f quantum me- chanics [4] , states are represented by projection operators which are essentially pure state density ma- trices containing all possible information concerning

the system [5 ]. A projection operator can be represented as the outer product of a bra and ket vector: P,~ = Iot) ( ot I such that

Tr P,~ = (ot lot ) = 1 (1)

for a normalised non-dissipative system. I f a set of states l i ) is complete and orthonormal, then

E P , = I . (2) i

If the state I o t ) now undergoes a real infinitesimal time translation (generated by any convenient Ham- iltonian), t+8, to lo t+dot ) then

P.+d,~ = I o t + d a > <ot +dotl (3)

and, to first order in 8,

TrP,~'P~+dc,= I (ot lot+dot)12 (4)

= I ( a l o t ) + ( a l d / d t l o t ) S I 2 (5)

= I ( o t l o t ) 1 2 + 2 ( o t l o t ) R e ( a l d / d t l a ) 8

+ 0 ( 8 2 ) . (6)

Utilising a similar expression for Tr P,~.P,_a,~ yields

Re( otld/dtlot)

1 = lira TrP,~'(P~+a,~--P,~_d~) (7)

~ o 4( o t l a )8

Observing that

Tr P,~+d,~ = < ot + d a l o t + d a >

= < a l a > + 2 Re<otld/dtl a > 8 + O ( 8 z) ,

0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

(8)

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Volume 160, number 1 PHYSICS LETTERS A 4 November 1991

we may also write

• 1 R e ( a l d / d t l a ) = ~imo ~-~ Tr(P~+d~-P~_d~ ) (9)

l d = ~ dt Tr P . , (lO)

whence it follows that if ( a [ a ) = 1, then [ 6 ]

R e ( a l d / d t l a ) = 0 , (1 la)

i.e.

l e x p ( ( a l d / d t [ o t ) ) l = 1 . ( l l b )

3. Imaginary t ime translat ions

The imaginary part of the phase may be obtained from considering

( a + d a l a - d a ) and < a - d a l a + d o t ) . (12)

Although these may be written in terms of real de- rivatives of projection operators (Tr P dP^ dP as studied extensively by Avron et al. [8]), such expressions involve two-forms and it is the object of this paper to demonstrate how the geometric phase may be written as the integral of a one-form. To re- cover the imaginary part of the phase one-form, we allow the state to perform an imaginary infinitesimal time translation, t~ t+ iE, and follow a course anal. ogous to that of section 2. By writing

Pa+ida = [ o t + i d a ) ( a + i dal (13)

(where ( a + i d a [ - [ a + i d a ) t ) , it follows that

Tr P~'P~+ida = [ ( a l a + i da ) [2 (14)

= I ( a l a ) + i ( a l d / d t l a ) E I z (15)

= I ( a l a ) 1 2 - 2 ( a l a ) Im(otld/dt la)E

+O(E2), (16)

whence

Im( a[d /d t la )

-1 =lim Tr P~'(P,,+id~--P~-id~) • (17)

, ,o 4 ( a l a ) e

This also has an equivalent form, deduced from considering

d ( a l E ) ( l a ) . l _ i d l a )E) Tr P,~+ia,~ = ( ( a l - i N

= ( a ] a ) - 2 I m ( a [ d / d t [ a ) E + O ( E 2) , (18)

i.e.

I m ( a [ d /d t la )

- 1 =lim--Tr(P~+ie,~-P~ id,~) - (19)

,40 4E

An interesting point comes to light if we compare expressions (8) and (18). To first order in ~,

Tr P,~+d,~ = 1 , (20)

due to the fact that the state is non-dissipative under real time translations. However, to first order in E,

Tr P~+ida = 1 - 2/'E, (21)

where

F = I m ( a l d / d t l a ) , the phase one-form. (22)

This is due to a trade-off in the choice of definition of the dual of the imaginary time translated state. By choosing

( a + i dal-= l a + i da)* (23)

(and thus preserving the traditional dual relation be- tween bra and ket) we are forced to concede that the state is no longer normalised to unity. An alternative definition given by

< a - i d a l - l a + i d a ) *

would imply that

( a - i d a l a + i d a )

- - ( < a l + i d < a l e ) ( l a >

(24)

) + i N [a)E (25)

d - dt (°t ld/dtl°t)E2 (26)

= ( a l o t ) +i ~t ( a l ot)E--O(E 2) (27)

= 1 to first order, (28)

but would preclude us from being able to define the

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V o l u m e 160, n u m b e r 1 P H Y S I C S L E T T E R S A 4 N o v e m b e r 1991

geometric phase (as a one-form) from the associated projection operator, Iot + i dot) ( ot - i dot I-

A noteworthy point is that while Pa+da may be written as P~(t+8) no equivalent expression exists for P~+id~. To write Pa ( t+ ie ) suggests that one may arrive at Pa+ida by the simple substitution of t+ie for t in Pa(t). This is not the case as will become abundantly clear in subsequent examples. In future, Pa- ida will be written as Pa ( t, e ) when the need arises to explicitly state its dependence on both parameters (this choice arises from the fact that Tr Pa+~aa is of the form a-be whereas TrPa_~aa is of the form a+be).

It is important to demonstrate that P~+~aa remains a projection operator:

(Pa+iea)"= ( Iot+i dot) (o t+ i da l )" (29)

= [ot+i dot) (o t+ i dot[

× ( (o t+ i dot [ a + i dot) )" -~ , (30)

thus implying that

Tr(P~+ida)"= ( ( a + i dot [ a + i dot) )"

= (Tr P~+ida) n . (31)

If2i are the eigenvalues of Pa+ida, then this relation states that

2, = (2,)" Vn, (32)

from which it is obvious that there can be at most one non-zero eigenvalue which is the trace. How- ever, it is also obvious that this will not be equal to unity as, in general, it will be e-dependent.

If we now consider the trace as a function of t and e,

Tr Pa-iaa -f(t , e) , (33)

then to first order in e,

f(t, e) ~f(t, O) +e Of(t , E) (34) U~

which gives

Tr(Pa+,da--Pa-,da)=--2'~f(t, ') I , (35)

o r

l 0 ) , = o Im(ot[O/Otlot) = ~ ~ Tr Pa(t, e (36)

_ 1 0_ TrPa id,~l (37) 2 de - [,=o

_ 1 0 ,.i,r pa+id a . (38) 2 de ~=o

Hence the phase calculated for a complete cycle over time T is T

l T OR f lm(ald /d t lo t ) dt=~ -~-~ ~=o' (39) 0

where T

R = l T r ~ Pa(t,e) dt. (40) o

Addressing now the question of how the imaginary time translation is generated leads us to another expression for the phase. Assuming the dynamics of the state is dependent on a linear function of the generators of a semi-simple Lie group, the action of a real time translation, 8, is given by

Pa(t+8) =exp(ia,giS)P,~(t) exp(--iaigt, 8 ) , (41)

where g~ are generators of the Lie algebra (Einstein summation convention assumed). Now, if the time translation is imaginary (ie), the expression may be written

Pa+ie,(t, e)=A(e)P,~(t, 0)At(e)

=exp( -a,g~e)Pa(t, 0) exp(-a~g~e) , (42)

which leads to the following expression for the phase:

T

f lm(otld/dtlot) dt o

= - ½ T T r [a,(g, +gt)p] (43)

= - TTr(aigiP) , (44)

where the second expression holds ifg~ is Hermitian (i.e. the group is unitary) and

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7" 1

P= ~ ~ P,(t,O) dt, 0

(45)

the average value of P , (t, 0) over the cycle. Let us now consider a concrete example with which to il- lustrate the geometric nature of these expressions. A spin-½ particle possessing a magnetic moment in a uniform magnetic field may be represented by a two- spinor lot, and thus by a projection operator, Iot ) ( ot I, in 2 × 2 matrix representation (if the particle is rotating about the z-axis with angular fre- quency to, the angle ~ may be identified with tot). Respectively, these are

cos(½0) sin ( ½ 0)eic°t ] ( 46a )

and

COS2(½0) COS(½0) sin ( ½0)e-i'°'~ cos(½0) sin(½0)e i°" sin2 (10) ]"

(46b)

Subjecting these to an imaginary time translation modifies them to [o t+idot) and Io t+idot) × (o t+ i dot I, respectively, which are, in the pre- vious representations,

cos(½O) "~ sin( ½8)ei°"- °~'/

and

cos2(½0) cos(½0) sin( ½0)e i°'- °"

Obviously,

(47a)

cos(½8) sin( ½0)e-i°~t-~ sin2(½0)e -2~ ]"

(47b)

Tr P~+ia,~ = cos 2 (½0) + sin2( ½0)e -2"~ , (48)

and over a complete cycle the well-known expression for the phase,

y = n ( 1 - c o s 0) = ½£2, (49)

is obtained. The generators of this imaginary time translation can be found through

A ( , ) = ( I 0 e L ) , (50,

and therefore

a,g;= = - ½ t o ( l - a : ) , (51)

which yields the same expression for the phase after substitution in (44).

4. Geometrical interpretation

Let us now consider the geometrical aspect of this approach. The system is a representation of SU (2) and therefore the real-time projection operator may be viewed as being a member of the Lie group U (2) subject to the condition detP~=0. This, of course, precludes projection operators from forming a group but a space may still be defined according to the above restrictions. A general matrix of this sort will be of the form

/*-iq / ,+ iv 2 - p ] ' P 2 " { - / * 2 "~- V 2 = / [ 2 ' (52)

i.e. the space is a two-sphere of radius 2. But 2=½ TrP~,= ½ for a real-time projection operator thereby displaying that

cos2(½0) cos(½0) sin(10)e-i°~" / cos(½0) sin(½0)e i'°' sin2 (½0) ,/

(53) describes a two-sphere of radius ½. The projection operator therefore defines the Hilbert subspace for the state and includes a "size" parameter (in the form of the trace) into the bargain. Clearly for a real-time projection operator Tr P~ is constant; but for an imaginary time translation it will in general be COS2(½0) +sin2( ½0)e -2~, i.e. a function of 0 and c. Therefore, a section at constant E will not, in general, be a sphere but a limagon of revolution which, for small ~ has a radius which varies as

~. = ½ ( 1 -- to~ ) + ½ to~ cos 0. ( 54 )

The greater the angle 0, the greater the phase and, consequently, the greater the deformation from a sphere. Eq. (40) may now be seen in a completely different light: R is the mean radius of the lima¢on of resolution over the circuit and the phase is a measure of the rate of change of this mean radius with

evaluated on the undeformed sphere. As ~ varies

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the space may be visualised as a smooth progression from a cardioid of revolution (E = ~ ) to a sphere (~ = 0) to another cardioid of revolution (c = - ~ ) with the intermediate steps being filled by lima9ons of resolution of varying sizes.

In general, for spin j the spin states are representations of SU(2j+ 1 ) and, although the group man- ifolds are exceedingly more complicated than S 2 (in general, they will be cp2j), an analogous interpretation is possible. As the generators of SU(2j+ 1 ) are traceless, the only contribution to the trace of the projection operator is from the unit matrix. This will be a constant for a real-time non-dissipatory quantum state and will be proportional to the norm of the state vector. Although in general the trace will not readily be identifiable with a quantity as simple as a radius it will nevertheless still serve as a size parameter for the space and the space will, once again, become deformed as imaginary time translations cause it to become ~-dependent.

An explicit form of the metric on this space can be obtained from the expression [ 7 ]

ds2=(TrP~)-l{Tr(dp~)2-½[Tr(dPa)]2}. (55)

For the real-time state given above the metric is simply that on a sphere of radius ½:

ds2= ½ (d02+sin20 d0 2) , (56)

but for the imaginary-time translated state a trivial, if tedious, calculation delivers

ds2= [ ( 1 + e -2") + ( 1 - e -2") cos 0] - l

X { d0 2 + sin20 dO 2 _ ~ ( 2 - 2e- 2,_ e - 4,) sin20 dO 2

- 4 sin 0 e-6"[ 1 - 2 cos 0( 1 - - e -E") ] d0dr/

+ 8 ( 1 - c o s 0)e -4n d ~ } . (57)

In the above expression, tot has been re-replaced by 0 and a new variable, ~/, has been introduced to take the place of the more cumbersome toE.

5. Conclusion

The concept of the geometric phase has been broadened considerably [2,3,6,7] since Berry's original paper [ 1 ]. Rather than extending the scope of the geometric phase, this paper has shown that, through imaginary time translations, there are other possible interpretations of the geometric phase and additional mathematical methods by which the phase may be evaluated.

Acknowledgement

I would like to thank Dr. Lewis Ryder for many helpful discussions and for introducing me to the geometric phase, Professor J.P. Vigier for bringing the work of Avron et al. [8 ] to my attention and Mark Hogarth for helpful comments on the original version of this paper. This work was supported by the Science and Engineering Research Council under grant no. 90807264.

References

[1 ] M.V. Berry, Proc. R. Soc. 392 (1984) 45. [2] B. Simon, Phys. Rev. Lett. 51 (1983) 2167. [3] Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58 (1987)

1593. [4] J. von Neumann, Mathematische Grundlagen der

Quantenmechanik (Springer, Berlin, 1932 ) [ English transl. (Princeton Univ. Press, Princeton, 1955) ].

15] C. Bouchiat and G.W. Gibbons, J. Phys. (Paris) 49 (1988) 187.

[6] K. Wanelik, A universal definition of the geometric phase, to be published.

[ 7 ] J. Samuel and R. Bhandari, Phys. Rev. Lctt. 60 (1988) 2339. [ 8 ] J.E. Avron, A. Reveh and B. Zur, Rev. Mod. Phys. 60 (1988)

873.

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the geometric phase from imaginary time translations

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