See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/238920600

Geometric phase in the causal quantum theories

Article in Physics Letters A-December 1997

DOI: 10.1016/S0375-9601(97)0072+-X

CITATIONS

6 1 author:

Gonzalo G de Polavieja Champalimaud Neuroscience Program 156 PUBLICATIONS 2,815 CITATIONS

ELSEVIER

Physics Letters A 236 (1997) 296-300

Geometric phase in the causal quantum theories

Gonzalo Garcia de Polavieja

Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford 0X1 3QZ, dry, UK

Received 3 June 1997; revised manuscript received 9 September 1997; accepted for publication 10 September 1997

Communicated by P.R. Holland

Abstract

The projective-geometric formulation of geometric phase for any ensemble in the causal quantum theories is given. This formulation generalizes the standard formulation of geometric phase to any causal ensemble including the cases of a single causal trajectory, the experimental geometric phase and the classical geometric phase. © 1997 Elsevier Science B.V.

The geometric phase is mathematically a property of any complex inner product space. This kinematic formulation has been developed by Aitchison and Wanclik [1] and Mukunda and Simon [2]. Hilbert space being central to standard quantum theory, the quantum geometric phase has great relevance. From the kinematic approach, the important cases for quantum theory of the Samuel-Bhandari [3] noncyclic geometric phase, the Aharonov-Anandan [4] geometric phase for cyclic motion and the Berry [5] geometric phase for adiabatic cyclic motion have been shown to be particular cases [1,2], The quantum geometric phase has also been semiclassically related by Berry [6] to the classical Hannay [7] angle defined only for integrable systems. A difficulty arises in the standard formulation of the quantum geometric phase by the inability to define the geometric phase for arbitrary subensembles. This could be overcome in the causal quantum theories [8-12], However, the Hilbert space structure is secondary in these theories and they might appear at first sight to be unable to accommodate general definitions of geometric phase.

In this Letter we show that the causal theories and therefore classical mechanics in their classical limit allow a projective-geometric formulation and therefore the causal geometric phase can be obtained from the general kinematic approach to geometric phases. A general formulation is presented that generalizes the standard quantum geometric phase to a geometric phase valid for any causal quantum ensemble. Recently, Parmenter and Valentine [13] have proposed, by analogy with the cyclic geometric phase, a phase for a special class of Bohmian trajectories but it remains open in their approach if this phase is projective-geometric in the same sense as the standard geometric phase and unique with respect to the standard geometric phase. Sjoqvist and Carlsen [14] have obtained the geometric phase for the decoherent wavefunction. In this Letter we show that these two cases and their generalizations as well as other cases can be obtained from a single general formulation that is geometric from the outset and unique with respect to the standard geometric phase.

We have organised this Letter as follows. We briefly discuss the kinematic approach to geometric phases. Secondly we obtain the general expression

0375-9601/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PU S0375-9601(97)00724-X

G. Garda de Polauieja /Physics Letters A 236 (1997) 296-300

297

for the geometric phase for arbitrary causal ensembles. Then we show how previous results in the literature can be obtained from our general expression along with their generalizations and new results. As particular cases, we obtain the standard quantum geometric phase, the geometric phase for a single

trni#*rtnrv    nhac^ fnr pYnprimpti.

—- — —j--,

tal situations, the geometric phase for initially structureless ensembles and the classical geometric phase.

The geometric phase is a property of the path tp{s) of the complex inner product space /. The two properties that define the geometric phase are (a) reparametrization invariance, invariance under the transformation s -» £(s) with £ a smooth monotonic function of s, which means that it is independent of the rate of change with s at which the path is traversed and (b) global phase invariance, invariance under the transformation <p(s)    with

z(s) an arbitrary nonzero smooth C ‘-valued function of s. We denote the corresponding projection map by n-.l-rp defined by n(z(s)tp(s)) = J7(<p(s)). Thus the complex inner product space curve X — {<p(s)|s e [sj,s₂]} projects on to the smooth projective complex inner product space p curve C— IJ(X). The real geometric phase obeying the above two properties is a functional of the curve C of the form [1,2]

y_g[C] = arg(<<p(s_t)|«p{s₂)>)

Im j ²ds{

The geometric phase for a particular theory is obtained with (1) and the evolution of the vector <p. In the following we consider the causal quantum theories. 'The state of a system of N particles is given in the causal quantum theories by the trajectories d_t of the N particles and by the guiding field d₂,.,., d_N, t) — h#^r|exp(iS//i) solution of the Schrodinger equation. We consider the geometric phase for a causal ensemble P that obeys the continuity equation

9F    .

— + V >(dP) =0.

The geometric phase for the vector <p

P^l/2exp(iS/h) with the parameter s being the time t in (1) is of the form

y_g[C] - arg((P^!/2(0)P^1/2(t)exp(iAS/^)>)

⁽³⁾

where { ) denotes integration over the space of definition of the causal distribution P. Expression (3) is the geometric phase for any arbitrary causal ensemble.

To obtain the geometric phase associated with a particular causal ensemble P, the form of the ensemble has to be substituted in (3). For the causal ensemble being the full quantum ensemble, P = hFj², (3) reduces to the standard quantum geometric phase that numerically coincides with the definition of Samuel and Bhandari [3] for noncyclic evolution, the Aharonov-Anandan [4] geometric phase for cyclic motion and to the Berry [5] phase for adiabatic cyclic motion.

In the following we derive the geometric phase associated with a cyclic Bohmian trajectory q(t) which is a solution of q = VS/m\_q^_q(t) with period T. For the ensemble P in (3) of the form <5(# -q{T)) — 8(q — q(0)) and using dS/dt = dS/dt + q ■ VS expression (3) reduces to m ,    m _fT

■d^cl-y^-^d«-T/„»^2d'-    (⁴>

Parmenter and Valentine [13] proposed (4) by analogy with the standard cyclic geometric phase for the particular class of cyclic Bohmian trajectories (a) with period T = nr with n an integer, (b) associated with a cyclic wavefunction with period t, 'P(t) = ^(Oje¹² with z a real number and (c) the wavefunction is made up of discrete eigenstates with commensurate frequencies. Moreover, in their approach (d) it is not demonstrated that the phase in (4) is projective-geometric and unique with respect to the standard quantum geometric phase and (e) they speculate that it is related in general with adiabaticity. It is clear in our derivation that (4) is more general as restrictions (a)-(c) are unnecessary. Our derivation has also the advantages of being geometric from the outset and unique with respect to the standard geometric phase as it is also obtained from the same formulation. Concerning the role of adiabaticity, it is true that the geometric phase was first discovered by

298

G Garcia de Polavieja/Physics Letters A 236 tl997) 296-300

Berry [5] in the context of adiabatic evolution but subsequent generalizations [1-4] showed that adia-baticity need not play any role. Our derivation of the causal geometric phase has those kinematic approaches as a starting point and therefore adiabaticity is not a necessary condition for its existence.

Another interesting case is that of experimental situations which are treated in the causal theories as any other physical interactions. A detailed treatment of the causal description of measurements can be found in Refs. [8-10]. After the interaction has taken place, the apparatus particle enters a particular packet and as far as the particle is concerned the other packets can be ignored. The wavefunction after the interaction can be writen as

x,y,t) = ilf{x,t)4>(y,t) + 'P²~{x_ty,t),    (5)

with x the coordinate of the system and y that of the environment. <P and ¹ have disjoint y support and the actual environmental configuration y(r) = Y(t) belongs to the support of <f>. The effective wavefunction for this case has thus the form

Vy = 4><P(y = Y(t)).    (6)

The causal theories permit a full analysis of the whole experimental situation beyond (5) and (6) for the genera] case

*_r(t) = V(x,y = Y(t),t).    (7)

This is the wavefunction for the system x conditioned to the environment y = F(r). The geometric phase for the relevant ensemble in the general experimental situation is then obtained substituting the effective distribution P = iVy]² for the wavefunction (7) in the general expresion (3) and expression (8) of Sjoqvist and Carlsen [14] is obtained.

We now discuss the case of initially structureless causal ensembles obeying

P(jc,0) = (SV)”¹ f dxP(x,0) = P(x,0)

hv

and |¹f^r(x,0)|²= |^(jc,0)|². In a way analogous to classical statistical mechanics, Valentini [15] has obtained an //-theorem in Bohmian mechanics to show that S < 0 with S— — /dx/^,ln(P/|lP'|²) reaching its maximum S = 0 for P — I'P]². This suggests that P-* 1^1² and therefore the geometric phase of the initial structureless distribution would tend to that of the full quantum distribution in the coarse-grained sense, y_g[P] —* ■y_g[(^,f^f I²]. Two differences between the derivation of Valentini and that of classical statistical mechanics are that Bohmian mechanics is defined in position space and not in phase space and that the derivation breaks down for stationary states. These two differences can be removed in a causal quantum theory obtained from the coherent state representation of the Schrodinger equation we have recently proposed [12] and that we discuss in the following to study the classical limit of the quantum geometric phase.

Bohmian mechanics has an ontology close to that of classical mechanics and it might be expected that the classical geometric angle and even a classical geometric phase can be obtained in the classical limit. It is possible to obtain a position space classical geometric phase by using (3) for ensembles P for which the classical limit condition

h² v²h/>i

2m |^|

holds. However, no direct relation with the classical results is obtained as Bohmian mechanics is a position space theory and its classical limit is therefore a position space classical mechanics. Direct contact with the classical geometric phase and angle is obtained from a causal quantum theory in phase space we have recently obtained [12] from the coherent state representation of the Schrodinger equation [1618] ‘ ‘

0

\h~{q,p\V) = (q,p\H(Q,P)\V)    (8)

for a Hamiltonian of the general form H = P²/2m + V(Q) and

(q,p\QW) = (q + ihV_p)(q,p\'P),    (9)

{q,p\PYP) = -\hV_q{q,p\^).    (10)

We substitute the polar form

^(q,p,t) = R(q,pj)exp[(i/h)S(q,p,t)]

for the wavefunction in (8). We consider the expan-

299

300

References

[1]    l.J.R, Aitchison, K. Wanelik, Proc. R. Soc. A 439 (1992) 25.

[2]    M. Mukunda, R. Simon, Ann. Phys. 228 (1993) 205.

[3J J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339.

[4]    Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.

[5]    M.V. Berry, Proc. R. Soc. A 392 (1984) 45.

16) M.V. Beny, J. Phys. A 18 (1985) 15.

[7]    J.H. Hannay, J. Phys. A 18 (1985) 221.

[8]    D. Bohm, Phys. Rev. 85 (166) (1952) ISO.

f9] D. Bohm, B.I. Hiley, The Undivided Universe, Routledge, London, (993.

[10] P R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press, Cambridge, 1993

[11]    S.T. Epstein, Phys. Rev. 89 (1952) 319; 91 (1953) 985.

[12]    G. Garcia de Polavieja, Phys. Lett. A 220 (1996) 303; Found. Phys. Lett. 9 (1996) 411.

[13]    R.H. Parmenter, R.W. Valentine, Phys. Lett. A 219 (1996) 7.

[14]    E. Sjoqvist, H. Carlsen, Phys. Rev. A 56 (1997) 1638.

[15]    A Valentini, Phys. Lett. A 156 (1991) 5.

[16]    R.J. Glauber, Phys. Rev. 131 (1963) 2766.

[17]    J.R. Klauder, B.S. Skagerstaro, (Eds.) Coherent States, World Scientific, Singapore, 1985.

[18]    Go. Torres-Vega. J.H. Frederik, J. Chem. Phys. 98 (1992) 3103.

[19]    G. Garcia de Polavieja, Classical geometric phase for mte-grable and nonintegrable systems, submitted.

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/238920600

Geometric phase in the causal quantum theories

Article in Physics Letters A-December 1997

DOI: 10.1016/S0375-9601(97)0072+-X

CITATIONS

6 1 author:

Gonzalo G de Polavieja Champalimaud Neuroscience Program 156 PUBLICATIONS 2,815 CITATIONS

ELSEVIER

Physics Letters A 236 (1997) 296-300

Geometric phase in the causal quantum theories

Gonzalo Garcia de Polavieja

Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford 0X1 3QZ, dry, UK

Received 3 June 1997; revised manuscript received 9 September 1997; accepted for publication 10 September 1997

Communicated by P.R. Holland

Abstract

The projective-geometric formulation of geometric phase for any ensemble in the causal quantum theories is given. This formulation generalizes the standard formulation of geometric phase to any causal ensemble including the cases of a single causal trajectory, the experimental geometric phase and the classical geometric phase. © 1997 Elsevier Science B.V.

The geometric phase is mathematically a property of any complex inner product space. This kinematic formulation has been developed by Aitchison and Wanclik [1] and Mukunda and Simon [2]. Hilbert space being central to standard quantum theory, the quantum geometric phase has great relevance. From the kinematic approach, the important cases for quantum theory of the Samuel-Bhandari [3] noncyclic geometric phase, the Aharonov-Anandan [4] geometric phase for cyclic motion and the Berry [5] geometric phase for adiabatic cyclic motion have been shown to be particular cases [1,2], The quantum geometric phase has also been semiclassically related by Berry [6] to the classical Hannay [7] angle defined only for integrable systems. A difficulty arises in the standard formulation of the quantum geometric phase by the inability to define the geometric phase for arbitrary subensembles. This could be overcome in the causal quantum theories [8-12], However, the Hilbert space structure is secondary in these theories and they might appear at first sight to be unable to accommodate general definitions of geometric phase.

In this Letter we show that the causal theories and therefore classical mechanics in their classical limit allow a projective-geometric formulation and therefore the causal geometric phase can be obtained from the general kinematic approach to geometric phases. A general formulation is presented that generalizes the standard quantum geometric phase to a geometric phase valid for any causal quantum ensemble. Recently, Parmenter and Valentine [13] have proposed, by analogy with the cyclic geometric phase, a phase for a special class of Bohmian trajectories but it remains open in their approach if this phase is projective-geometric in the same sense as the standard geometric phase and unique with respect to the standard geometric phase. Sjoqvist and Carlsen [14] have obtained the geometric phase for the decoherent wavefunction. In this Letter we show that these two cases and their generalizations as well as other cases can be obtained from a single general formulation that is geometric from the outset and unique with respect to the standard geometric phase.

We have organised this Letter as follows. We briefly discuss the kinematic approach to geometric phases. Secondly we obtain the general expression

0375-9601/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PU S0375-9601(97)00724-X

G. Garda de Polauieja /Physics Letters A 236 (1997) 296-300

297

for the geometric phase for arbitrary causal ensembles. Then we show how previous results in the literature can be obtained from our general expression along with their generalizations and new results. As particular cases, we obtain the standard quantum geometric phase, the geometric phase for a single

trni#*rtnrv    nhac^ fnr pYnprimpti.

—- — —j--,

tal situations, the geometric phase for initially structureless ensembles and the classical geometric phase.

The geometric phase is a property of the path tp{s) of the complex inner product space /. The two properties that define the geometric phase are (a) reparametrization invariance, invariance under the transformation s -» £(s) with £ a smooth monotonic function of s, which means that it is independent of the rate of change with s at which the path is traversed and (b) global phase invariance, invariance under the transformation <p(s)    with

z(s) an arbitrary nonzero smooth C ‘-valued function of s. We denote the corresponding projection map by n-.l-rp defined by n(z(s)tp(s)) = J7(<p(s)). Thus the complex inner product space curve X — {<p(s)|s e [sj,s₂]} projects on to the smooth projective complex inner product space p curve C— IJ(X). The real geometric phase obeying the above two properties is a functional of the curve C of the form [1,2]

y_g[C] = arg(<<p(s_t)|«p{s₂)>)

Im j ²ds{

The geometric phase for a particular theory is obtained with (1) and the evolution of the vector <p. In the following we consider the causal quantum theories. 'The state of a system of N particles is given in the causal quantum theories by the trajectories d_t of the N particles and by the guiding field d₂,.,., d_N, t) — h#^r|exp(iS//i) solution of the Schrodinger equation. We consider the geometric phase for a causal ensemble P that obeys the continuity equation

9F    .

— + V >(dP) =0.

The geometric phase for the vector <p

P^l/2exp(iS/h) with the parameter s being the time t in (1) is of the form

y_g[C] - arg((P^!/2(0)P^1/2(t)exp(iAS/^)>)

⁽³⁾

where { ) denotes integration over the space of definition of the causal distribution P. Expression (3) is the geometric phase for any arbitrary causal ensemble.

To obtain the geometric phase associated with a particular causal ensemble P, the form of the ensemble has to be substituted in (3). For the causal ensemble being the full quantum ensemble, P = hFj², (3) reduces to the standard quantum geometric phase that numerically coincides with the definition of Samuel and Bhandari [3] for noncyclic evolution, the Aharonov-Anandan [4] geometric phase for cyclic motion and to the Berry [5] phase for adiabatic cyclic motion.

In the following we derive the geometric phase associated with a cyclic Bohmian trajectory q(t) which is a solution of q = VS/m\_q^_q(t) with period T. For the ensemble P in (3) of the form <5(# -q{T)) — 8(q — q(0)) and using dS/dt = dS/dt + q ■ VS expression (3) reduces to m ,    m _fT

■d^cl-y^-^d«-T/„»^2d'-    (⁴>

Parmenter and Valentine [13] proposed (4) by analogy with the standard cyclic geometric phase for the particular class of cyclic Bohmian trajectories (a) with period T = nr with n an integer, (b) associated with a cyclic wavefunction with period t, 'P(t) = ^(Oje¹² with z a real number and (c) the wavefunction is made up of discrete eigenstates with commensurate frequencies. Moreover, in their approach (d) it is not demonstrated that the phase in (4) is projective-geometric and unique with respect to the standard quantum geometric phase and (e) they speculate that it is related in general with adiabaticity. It is clear in our derivation that (4) is more general as restrictions (a)-(c) are unnecessary. Our derivation has also the advantages of being geometric from the outset and unique with respect to the standard geometric phase as it is also obtained from the same formulation. Concerning the role of adiabaticity, it is true that the geometric phase was first discovered by

298

G Garcia de Polavieja/Physics Letters A 236 tl997) 296-300

Berry [5] in the context of adiabatic evolution but subsequent generalizations [1-4] showed that adia-baticity need not play any role. Our derivation of the causal geometric phase has those kinematic approaches as a starting point and therefore adiabaticity is not a necessary condition for its existence.

Another interesting case is that of experimental situations which are treated in the causal theories as any other physical interactions. A detailed treatment of the causal description of measurements can be found in Refs. [8-10]. After the interaction has taken place, the apparatus particle enters a particular packet and as far as the particle is concerned the other packets can be ignored. The wavefunction after the interaction can be writen as

x,y,t) = ilf{x,t)4>(y,t) + 'P²~{x_ty,t),    (5)

with x the coordinate of the system and y that of the environment. <P and ¹ have disjoint y support and the actual environmental configuration y(r) = Y(t) belongs to the support of <f>. The effective wavefunction for this case has thus the form

Vy = 4><P(y = Y(t)).    (6)

The causal theories permit a full analysis of the whole experimental situation beyond (5) and (6) for the genera] case

*_r(t) = V(x,y = Y(t),t).    (7)

This is the wavefunction for the system x conditioned to the environment y = F(r). The geometric phase for the relevant ensemble in the general experimental situation is then obtained substituting the effective distribution P = iVy]² for the wavefunction (7) in the general expresion (3) and expression (8) of Sjoqvist and Carlsen [14] is obtained.

We now discuss the case of initially structureless causal ensembles obeying

P(jc,0) = (SV)”¹ f dxP(x,0) = P(x,0)

hv

and |¹f^r(x,0)|²= |^(jc,0)|². In a way analogous to classical statistical mechanics, Valentini [15] has obtained an //-theorem in Bohmian mechanics to show that S < 0 with S— — /dx/^,ln(P/|lP'|²) reaching its maximum S = 0 for P — I'P]². This suggests that P-* 1^1² and therefore the geometric phase of the initial structureless distribution would tend to that of the full quantum distribution in the coarse-grained sense, y_g[P] —* ■y_g[(^,f^f I²]. Two differences between the derivation of Valentini and that of classical statistical mechanics are that Bohmian mechanics is defined in position space and not in phase space and that the derivation breaks down for stationary states. These two differences can be removed in a causal quantum theory obtained from the coherent state representation of the Schrodinger equation we have recently proposed [12] and that we discuss in the following to study the classical limit of the quantum geometric phase.

Bohmian mechanics has an ontology close to that of classical mechanics and it might be expected that the classical geometric angle and even a classical geometric phase can be obtained in the classical limit. It is possible to obtain a position space classical geometric phase by using (3) for ensembles P for which the classical limit condition

h² v²h/>i

2m |^|

holds. However, no direct relation with the classical results is obtained as Bohmian mechanics is a position space theory and its classical limit is therefore a position space classical mechanics. Direct contact with the classical geometric phase and angle is obtained from a causal quantum theory in phase space we have recently obtained [12] from the coherent state representation of the Schrodinger equation [1618] ‘ ‘

0

\h~{q,p\V) = (q,p\H(Q,P)\V)    (8)

for a Hamiltonian of the general form H = P²/2m + V(Q) and

(q,p\QW) = (q + ihV_p)(q,p\'P),    (9)

{q,p\PYP) = -\hV_q{q,p\^).    (10)

We substitute the polar form

^(q,p,t) = R(q,pj)exp[(i/h)S(q,p,t)]

for the wavefunction in (8). We consider the expan-

299

300

References

[1]    l.J.R, Aitchison, K. Wanelik, Proc. R. Soc. A 439 (1992) 25.

[2]    M. Mukunda, R. Simon, Ann. Phys. 228 (1993) 205.

[3J J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339.

[4]    Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593.

[5]    M.V. Berry, Proc. R. Soc. A 392 (1984) 45.

16) M.V. Beny, J. Phys. A 18 (1985) 15.

[7]    J.H. Hannay, J. Phys. A 18 (1985) 221.

[8]    D. Bohm, Phys. Rev. 85 (166) (1952) ISO.

f9] D. Bohm, B.I. Hiley, The Undivided Universe, Routledge, London, (993.

[10] P R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press, Cambridge, 1993

[11]    S.T. Epstein, Phys. Rev. 89 (1952) 319; 91 (1953) 985.

[12]    G. Garcia de Polavieja, Phys. Lett. A 220 (1996) 303; Found. Phys. Lett. 9 (1996) 411.

[13]    R.H. Parmenter, R.W. Valentine, Phys. Lett. A 219 (1996) 7.

[14]    E. Sjoqvist, H. Carlsen, Phys. Rev. A 56 (1997) 1638.

[15]    A Valentini, Phys. Lett. A 156 (1991) 5.

[16]    R.J. Glauber, Phys. Rev. 131 (1963) 2766.

[17]    J.R. Klauder, B.S. Skagerstaro, (Eds.) Coherent States, World Scientific, Singapore, 1985.

[18]    Go. Torres-Vega. J.H. Frederik, J. Chem. Phys. 98 (1992) 3103.

[19]    G. Garcia de Polavieja, Classical geometric phase for mte-grable and nonintegrable systems, submitted.