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Quantisation in multiply connected spaces
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
QUANTISATION IN MULTIPLYCONNECTED SPACES P.A. HORVATHY Centre de Physique Théorique, CNRS, Marseille, France and Université de Pro vence, Aix-Marseille I, France Received 20 November 1979
Quantization in terms of Feynman’s path integrals is studied in geometric terms. Topics covered include: Bohm— Aharonov experiment, the nonintegrable phase factor of Wu and Yang, Schulman—DeWitt--Laidlaw theorem on propagators, equivalence with Souriau—Kostant prequantisation.
1. The problems to be studied. Effects due to the nontrivialtopology of the configuration space were pointed out by Aharonov and Bohm [1] There is a related contribution of Wu and Yang [2] with their “nonintegrable phase factor”. Also, as first pointed out by Schulman [3] and C. DeWitt and Laidlaw [4] path integration in multiply connected spaces involves characters of the homotopy group. Finally, the same conditions are present in the classification theorem of prequantum bundles of Souriau [6] and Kostant [7]. Using the language of differential forms and fiber bundles we propose here to relate the above questions. .
,
2. The problem ofclassical action. Let Q, the configuration space of a classical system, be a multiply connected manifold with homotopy group G. The dynamics is described in the symplectic approach [6] by a presymplectic form a on the evolution space E = T Q X R. Suppose the system admits a global variational description [9] ; a is then exact, i.e. there exists a ~ form 00 on Esatisfying *
‘~‘
*1
More exactly, a regular lagrangian function; (2) is then restricted to 1-forms arising from lagrangians; the Legendre transformation is then well defmed and can be used to identify curves in Q with curvesIn T*Q X R; this identification is tacitly assumed through the entire discussion. I am indebted to David Simms for pointing out this problem (footnote added in proof).
do0
=
(1)
a.
Now, the general solution of eq. (1) is 0 = 0~+ a,
(2)
where a is closed, da = 0, but, in contrast to the simply connected case, not nec~ssarilyexact. (The homotopy group of E is again G, (E, G, P) forming the universal cover of E; E = T X~,R,with the universal covering manifold of Q, F: E E projection.) The effect of a is an additional term in the classical hamiltonian action: for any path ‘y C Q * ~‘
~
-+
S(’y)
:
Jo
=
S0(y) +fa,
‘V
(3)
‘V
where S0(’y) is of course ~ the term a is generally path dependent.0o P”0 D~flote~ ~ 0~ =F*a £ being sim-~ ply connected, there exists a function f: if -÷ R such that df = If)’ is any1(q, pathp, through (q,p, t), then any r) passes exactly one through continuous @“ ~ t) E F lift ~projecting onto ~‘; f 70 = f~O. ffi3 is a closed curve in E through (q, p, t), then its lift f3 through (i’, p, t) ends at g(~’,p, t), where g is the homotopy class Qf 13: g = [13]. As a consequence f~a = 7(g(~,p, t)) f (i’, p, t). ~‘
—
11
Volume 76A, number 1
PHYSICS LETTERS
3. The propagator in multiply connected spaces. According to Feynman [8] the propagator is expressed as a path integral: K(q’, t’; q, t) = fexp[(i/h)S(7)] D7,
(4)
3 March 1980
Theorem (Schulman C. DeWitt—Laidlaw). K(q ,t;q,t)=c where c
=
~(g)K~(q’,t’;q,
(8)
t),
gaG
exp[(i/h)f~a] ,and the partial propagators
are defined as integrals over the ~g’s. where ~ is the “infinite dimensional manifold” of paths 7 lying between q’, t’, and q, tin Q X R [9]. Two expressions S1 (y) = f701 and S2(’y) = f702 of the classical action will be quantum-mechanically equivalent if the corresponding propagators are related by a gauge transformation; this is the case iff for any closed curve 0i 02)] r ~1 = 1 exp [(i/Il)J ( —
iff (1/11) J (01
—
02) = 2irk,
(5)
We conclude that the symplectic description in itself is not sufficient to characterize the quantum behaviour; the additional term a in (2) may change essentially the situation. The expression (6) mathematically a group character is of fundamental importance. The physical meaning of this “path-dependent phase factor” will be clear from the following example. 4. The Bohm—Aharonov experiment and the Wu— Yang factor. Consider the Bohm—Aharonov experiment [1,2] Dropping the variable along the axis of the cylinder the configuration space becomes R2 a disk D, with homotopy group Z. The evolution space is then E = T Q X R, and the symplectic structure has to be that of a free particle, for in Q there is no magnetic field. The “insensitivity” of the symplectic description to this “exterior” magnetic field makes clear that we need some additional information if we want to take seriously the description restricted to Q; for instance, we have to choose a 1-form a, corresponding to the vector potential —
—
.
—
with k integer. If (5) holds we will call 01 and 02 equivalent, 01 02. Let 13 be a closed curve in Q, [13] = g; we see that for a = 0 00 ‘~
—
r
.
~(g) : exp [(i/h)
f
1 a].
(6)
is well defined and is, in fact, a character of the homotopy group. Furthermore, x 1 = X2 ~ 01 02 iff a1 “~‘a 2. Choose any path p from ~ the paths in ~ can be set then in bijection with cycles through q according to 7 = p o 13, the o denoting composition of paths. y~ and ~2 are homotopic iff [13k] = [I~2]; the homotopy classes are thus labelled by elements of G. Denote by the class corresponding tog E G. Evidently we have ~ = U~G ~Pg. For any a we then have S(-y)=So(’y)+fa+fa.
(7)
Note that f~ai: independent of ~ and f13a is constant on homotopy classes. Thus, by (7), (6), using the additivity of the path integral we get:
12
*
1
(9)
a = eA .dq ‘I
a1
-~
a~ iff
—
~2
=
~h/e)2irk k = 0, 1,
...
,(10)
where ~ = ~ A1 dql = 1DB is the magnetic flux, 1~o being a curve homeomorphic to a circle. The Bohm—Aharonov experiment tells us that if (10) is true, we cannot distinguish between the two situations. The group characters are found as
x(n) =
=
exp[in((e/h) ~A1 dq1)] i3o
(exp [i((ern)J
A1 dqi)] °
}~
where we recognize the nth power of the nonintegrable phase factor of Wu and Yang [2].
Volume 76A, number 1
PHYSICS LETTERS
Theorem. The inequivalent quantisations are in 1—1 correspondence with the Wu—Yang phase factor
~ A. dq~l
exp [i(e/h)
(12)
[
j j j. 3o / Note that to identify physically our a, we had to
3 March 1980
the equivalence classes of 1-forms on E and the inequivalent prequantisations. Let
E X S1, ~: ~ E, ~ = ~70~+ dz/iz (j = 1, be two solutions. first that Y1 Y2 in the2)fiber-bundle sense. Suppose There is then an equivalence =
—
map F: Y 1
morphism F between them intertwining the action of U(l) on the fibers and preserving the connection form (see also ref. [10]). If the symplectic form is exact, this construction is always possible and any solution is of the form
1’2, projecting to a symplectomorphism C E; ~213~admits : P131. Pick up ay1 E ~j 2 =F(y1). the (unique) horizontal lift f3~through v~which is. in turn, mapped to the unique horizontal lift 132 of ~2 throughy 2. Now, the restriction ofFto a fiber is a multiplication with an element of the complex unit circle, and thus ~ ~ ~~ ~ )I~1)z0 1Z~ 15 ~1)z~ )‘~1Jz~ ) ~2of (14), the equivalence of the Thus, taking account 0 1’s is established. On the other hand, if 01 02 in our sense, then denote by f : E R the function satisfying hdf = 0~—02; the equivalence of the O,~’sinduces that F(x) = exp [if (x)] is a well-defmed function on E, and one verifies at once that it establishes the equiva-
Y=EX S1, ir: E X S1 -~E, w = lr*0 +th/~z, (13)
lenceof Y1 and Y2 according toF~x,z)~x,F~x).z).
look inside the cylinder [3]. 5. Equivalence with prequantisation. The classification scheme we have presented in terms of equivalence classes of 1-forms onE may be reformulated in fiber bundle language. Prequantising a system, in the sense of Kostant [7] and Souriau [6] , amounts in fact to constructing a U(l) principal bundle Ywith connectionYform w overthe E, bundle whose curvature form is i~*a (ir: E being projection). Two such constructions are equivalent if there exists a diffeo-~
0 being a 1-form on E. Conversely, every 1-form 0 defines a prequantisation. (In the non-exact case the possibility of the prequantum construction depends on the cohomology class of a [6,7,10] , yielding quantisation of spin, Dirac monopoles, etc.) Starting with an w, we may define a covariant derivation on the sections of our bundle and it is meaningful to speak of horizontal lifts. If ‘y C E, through every point of ~ 1(7(0)) passes exactly one horizontal lift 7 = (7’ Yz). Note that exp [(1/
h
r01
)j j
0
— —
~z
1
( )/~~( )
(14)
‘V
If E is multiply connected, the prequantisation problem admits more than one solution, the inequivalent bundles being classified in terms of characters of the homotopy group [6,7] . We propose now to rederive this theorem using our approach, and establishing thus the equivalence between our and Souriau’s construction. Theorem. There is a 1—1 correspondence between
-+
F: E -÷ E. Take any closed sety curve 1(13(o))
—‘°
‘~
.
—
‘~
-+
Corollary (Souriau [6] , Kostant [7]). There is a 1—1 correspondence between the inequivalent prequantisations and the characters of the homotopy group. Remember only that xi = x2iff Ui 02. Furthermore, one could show that the universal covering of E admits a unique prequantisation, the homotopy group has as many isomorphic lifts to it as there are characters of G, and any prequantisation is obtained by factorizing the prequantisation of the universal cover by such a lift. Thus the Bohm—Aharonov experiment provides us ~-‘
with an example where a symplectic system admits an infmite number of inequivalent prequantisations, each of which being observed in Nature. Two situations are physically equivalent iff the prequantisations are equivalent. I am indebted to Jean-Marie Souriau for hospitality and encouragement in Marseille; discussions with Christian Duval are also gratefully acknowledged. 13
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
Note added: I am indebted to C. DeWitt-Morette for calling my attention to a paper of Schulman [11].
[71 B.
References
[8] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, 1965).
[1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [2] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 12,3845. [3] L.S. Schulman, J. Math. Phys. 12 (1971) 304. [41 M.G.G. Laidlaw and C. Morette-DeWitt, Phys. Rev. D3 (1971) 1375. [5] J.S. Dowker, J. Phys. A5 (1972) 936. [6] J.-M. Souriau, Structure des systèmes dynamiques (Dunod, Paris, 1970); Structure of dynamical systems (North-Holland, Amsterdam), to be published.
14
Kostant, Quantization and unitary representations, in: Lectures in modern analysis and applications III, ed. Taam, Lecture notes in mathematics, Vol. 170 (Springer,
1970). [9] P.A. Horváthy and L. Ury, Acta Phys. Hung. 42(1977) 251. [10] D.J. Simms and N.M.J. Woodhouse, Lectures on geometric quantisation, Lecture notes in physics, Vol. 53 (Springer, 1976). [11] L. Schulman, in: Functional integration and its applications, Proc. Intern. Conf. (London, 1974), ed. A.M. Arthurs (Clarendon, 1975).
PHYSICS LETTERS
3 March 1980
QUANTISATION IN MULTIPLYCONNECTED SPACES P.A. HORVATHY Centre de Physique Théorique, CNRS, Marseille, France and Université de Pro vence, Aix-Marseille I, France Received 20 November 1979
Quantization in terms of Feynman’s path integrals is studied in geometric terms. Topics covered include: Bohm— Aharonov experiment, the nonintegrable phase factor of Wu and Yang, Schulman—DeWitt--Laidlaw theorem on propagators, equivalence with Souriau—Kostant prequantisation.
1. The problems to be studied. Effects due to the nontrivialtopology of the configuration space were pointed out by Aharonov and Bohm [1] There is a related contribution of Wu and Yang [2] with their “nonintegrable phase factor”. Also, as first pointed out by Schulman [3] and C. DeWitt and Laidlaw [4] path integration in multiply connected spaces involves characters of the homotopy group. Finally, the same conditions are present in the classification theorem of prequantum bundles of Souriau [6] and Kostant [7]. Using the language of differential forms and fiber bundles we propose here to relate the above questions. .
,
2. The problem ofclassical action. Let Q, the configuration space of a classical system, be a multiply connected manifold with homotopy group G. The dynamics is described in the symplectic approach [6] by a presymplectic form a on the evolution space E = T Q X R. Suppose the system admits a global variational description [9] ; a is then exact, i.e. there exists a ~ form 00 on Esatisfying *
‘~‘
*1
More exactly, a regular lagrangian function; (2) is then restricted to 1-forms arising from lagrangians; the Legendre transformation is then well defmed and can be used to identify curves in Q with curvesIn T*Q X R; this identification is tacitly assumed through the entire discussion. I am indebted to David Simms for pointing out this problem (footnote added in proof).
do0
=
(1)
a.
Now, the general solution of eq. (1) is 0 = 0~+ a,
(2)
where a is closed, da = 0, but, in contrast to the simply connected case, not nec~ssarilyexact. (The homotopy group of E is again G, (E, G, P) forming the universal cover of E; E = T X~,R,with the universal covering manifold of Q, F: E E projection.) The effect of a is an additional term in the classical hamiltonian action: for any path ‘y C Q * ~‘
~
-+
S(’y)
:
Jo
=
S0(y) +fa,
‘V
(3)
‘V
where S0(’y) is of course ~ the term a is generally path dependent.0o P”0 D~flote~ ~ 0~ =F*a £ being sim-~ ply connected, there exists a function f: if -÷ R such that df = If)’ is any1(q, pathp, through (q,p, t), then any r) passes exactly one through continuous @“ ~ t) E F lift ~projecting onto ~‘; f 70 = f~O. ffi3 is a closed curve in E through (q, p, t), then its lift f3 through (i’, p, t) ends at g(~’,p, t), where g is the homotopy class Qf 13: g = [13]. As a consequence f~a = 7(g(~,p, t)) f (i’, p, t). ~‘
—
11
Volume 76A, number 1
PHYSICS LETTERS
3. The propagator in multiply connected spaces. According to Feynman [8] the propagator is expressed as a path integral: K(q’, t’; q, t) = fexp[(i/h)S(7)] D7,
(4)
3 March 1980
Theorem (Schulman C. DeWitt—Laidlaw). K(q ,t;q,t)=c where c
=
~(g)K~(q’,t’;q,
(8)
t),
gaG
exp[(i/h)f~a] ,and the partial propagators
are defined as integrals over the ~g’s. where ~ is the “infinite dimensional manifold” of paths 7 lying between q’, t’, and q, tin Q X R [9]. Two expressions S1 (y) = f701 and S2(’y) = f702 of the classical action will be quantum-mechanically equivalent if the corresponding propagators are related by a gauge transformation; this is the case iff for any closed curve 0i 02)] r ~1 = 1 exp [(i/Il)J ( —
iff (1/11) J (01
—
02) = 2irk,
(5)
We conclude that the symplectic description in itself is not sufficient to characterize the quantum behaviour; the additional term a in (2) may change essentially the situation. The expression (6) mathematically a group character is of fundamental importance. The physical meaning of this “path-dependent phase factor” will be clear from the following example. 4. The Bohm—Aharonov experiment and the Wu— Yang factor. Consider the Bohm—Aharonov experiment [1,2] Dropping the variable along the axis of the cylinder the configuration space becomes R2 a disk D, with homotopy group Z. The evolution space is then E = T Q X R, and the symplectic structure has to be that of a free particle, for in Q there is no magnetic field. The “insensitivity” of the symplectic description to this “exterior” magnetic field makes clear that we need some additional information if we want to take seriously the description restricted to Q; for instance, we have to choose a 1-form a, corresponding to the vector potential —
—
.
—
with k integer. If (5) holds we will call 01 and 02 equivalent, 01 02. Let 13 be a closed curve in Q, [13] = g; we see that for a = 0 00 ‘~
—
r
.
~(g) : exp [(i/h)
f
1 a].
(6)
is well defined and is, in fact, a character of the homotopy group. Furthermore, x 1 = X2 ~ 01 02 iff a1 “~‘a 2. Choose any path p from ~ the paths in ~ can be set then in bijection with cycles through q according to 7 = p o 13, the o denoting composition of paths. y~ and ~2 are homotopic iff [13k] = [I~2]; the homotopy classes are thus labelled by elements of G. Denote by the class corresponding tog E G. Evidently we have ~ = U~G ~Pg. For any a we then have S(-y)=So(’y)+fa+fa.
(7)
Note that f~ai: independent of ~ and f13a is constant on homotopy classes. Thus, by (7), (6), using the additivity of the path integral we get:
12
*
1
(9)
a = eA .dq ‘I
a1
-~
a~ iff
—
~2
=
~h/e)2irk k = 0, 1,
...
,(10)
where ~ = ~ A1 dql = 1DB is the magnetic flux, 1~o being a curve homeomorphic to a circle. The Bohm—Aharonov experiment tells us that if (10) is true, we cannot distinguish between the two situations. The group characters are found as
x(n) =
=
exp[in((e/h) ~A1 dq1)] i3o
(exp [i((ern)J
A1 dqi)] °
}~
where we recognize the nth power of the nonintegrable phase factor of Wu and Yang [2].
Volume 76A, number 1
PHYSICS LETTERS
Theorem. The inequivalent quantisations are in 1—1 correspondence with the Wu—Yang phase factor
~ A. dq~l
exp [i(e/h)
(12)
[
j j j. 3o / Note that to identify physically our a, we had to
3 March 1980
the equivalence classes of 1-forms on E and the inequivalent prequantisations. Let
E X S1, ~: ~ E, ~ = ~70~+ dz/iz (j = 1, be two solutions. first that Y1 Y2 in the2)fiber-bundle sense. Suppose There is then an equivalence =
—
map F: Y 1
morphism F between them intertwining the action of U(l) on the fibers and preserving the connection form (see also ref. [10]). If the symplectic form is exact, this construction is always possible and any solution is of the form
1’2, projecting to a symplectomorphism C E; ~213~admits : P131. Pick up ay1 E ~j 2 =F(y1). the (unique) horizontal lift f3~through v~which is. in turn, mapped to the unique horizontal lift 132 of ~2 throughy 2. Now, the restriction ofFto a fiber is a multiplication with an element of the complex unit circle, and thus ~ ~ ~~ ~ )I~1)z0 1Z~ 15 ~1)z~ )‘~1Jz~ ) ~2of (14), the equivalence of the Thus, taking account 0 1’s is established. On the other hand, if 01 02 in our sense, then denote by f : E R the function satisfying hdf = 0~—02; the equivalence of the O,~’sinduces that F(x) = exp [if (x)] is a well-defmed function on E, and one verifies at once that it establishes the equiva-
Y=EX S1, ir: E X S1 -~E, w = lr*0 +th/~z, (13)
lenceof Y1 and Y2 according toF~x,z)~x,F~x).z).
look inside the cylinder [3]. 5. Equivalence with prequantisation. The classification scheme we have presented in terms of equivalence classes of 1-forms onE may be reformulated in fiber bundle language. Prequantising a system, in the sense of Kostant [7] and Souriau [6] , amounts in fact to constructing a U(l) principal bundle Ywith connectionYform w overthe E, bundle whose curvature form is i~*a (ir: E being projection). Two such constructions are equivalent if there exists a diffeo-~
0 being a 1-form on E. Conversely, every 1-form 0 defines a prequantisation. (In the non-exact case the possibility of the prequantum construction depends on the cohomology class of a [6,7,10] , yielding quantisation of spin, Dirac monopoles, etc.) Starting with an w, we may define a covariant derivation on the sections of our bundle and it is meaningful to speak of horizontal lifts. If ‘y C E, through every point of ~ 1(7(0)) passes exactly one horizontal lift 7 = (7’ Yz). Note that exp [(1/
h
r01
)j j
0
— —
~z
1
( )/~~( )
(14)
‘V
If E is multiply connected, the prequantisation problem admits more than one solution, the inequivalent bundles being classified in terms of characters of the homotopy group [6,7] . We propose now to rederive this theorem using our approach, and establishing thus the equivalence between our and Souriau’s construction. Theorem. There is a 1—1 correspondence between
-+
F: E -÷ E. Take any closed sety curve 1(13(o))
—‘°
‘~
.
—
‘~
-+
Corollary (Souriau [6] , Kostant [7]). There is a 1—1 correspondence between the inequivalent prequantisations and the characters of the homotopy group. Remember only that xi = x2iff Ui 02. Furthermore, one could show that the universal covering of E admits a unique prequantisation, the homotopy group has as many isomorphic lifts to it as there are characters of G, and any prequantisation is obtained by factorizing the prequantisation of the universal cover by such a lift. Thus the Bohm—Aharonov experiment provides us ~-‘
with an example where a symplectic system admits an infmite number of inequivalent prequantisations, each of which being observed in Nature. Two situations are physically equivalent iff the prequantisations are equivalent. I am indebted to Jean-Marie Souriau for hospitality and encouragement in Marseille; discussions with Christian Duval are also gratefully acknowledged. 13
Volume 76A, number 1
PHYSICS LETTERS
3 March 1980
Note added: I am indebted to C. DeWitt-Morette for calling my attention to a paper of Schulman [11].
[71 B.
References
[8] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, 1965).
[1] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485. [2] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 12,3845. [3] L.S. Schulman, J. Math. Phys. 12 (1971) 304. [41 M.G.G. Laidlaw and C. Morette-DeWitt, Phys. Rev. D3 (1971) 1375. [5] J.S. Dowker, J. Phys. A5 (1972) 936. [6] J.-M. Souriau, Structure des systèmes dynamiques (Dunod, Paris, 1970); Structure of dynamical systems (North-Holland, Amsterdam), to be published.
14
Kostant, Quantization and unitary representations, in: Lectures in modern analysis and applications III, ed. Taam, Lecture notes in mathematics, Vol. 170 (Springer,
1970). [9] P.A. Horváthy and L. Ury, Acta Phys. Hung. 42(1977) 251. [10] D.J. Simms and N.M.J. Woodhouse, Lectures on geometric quantisation, Lecture notes in physics, Vol. 53 (Springer, 1976). [11] L. Schulman, in: Functional integration and its applications, Proc. Intern. Conf. (London, 1974), ed. A.M. Arthurs (Clarendon, 1975).