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17 Quantum Effects, Line Bundles, and Holonomy Groups
77 Quantum Effects, Line Bundles, and Holonomy Groups
QUANTUM EFFECTS The quantum behavior of particles occurs in experiments in which particles exhibit a wavelike nature manifested by certain interference effects, a simple case of which we now describe. A beam of identical particles is split into two parts at a point S. Or we can consider S as a source point for beams of particles, all of which are reflected back from a barrier except for the beams which reach the points A and B, where there are slits in the barrier. The points A and B then act as secondary sources allowing particles to go off in all directions (see Fig. 17.1). We now place a detection screen at a distance d from the barrier (see Fig. 17.2). Let us coordinatize the screen by x. We are interested in the following question.
FIGURE 17.1
354
355
PROBABILITY AMPLITUDE PHASE FACTORS
PROBABI LlTY AMPLITUDES Of the total number of particles which reach the screen in a certain time interval, what fraction arrive at x? This fraction can be interpreted as the probability that a particular particle will arrive at x. A model which correctly describes this is based on three principles written down by Feynman [ 113: (a) There is a complex function @(x)so that lCD(x)lzis the probability that a particle will arrive at x. The complex number @(x)is called the probability amplitude for the recording of the arrival of the particle at x. (b) For each path p there is a probability amplitude CD, so that lCD,12 represents the probability that the particle traveled along p. If pi is the path li followed by ti,then @(x) = CD,, Thus, if an event can result from several different processes, you add the probability amplitudes to obtain the probability amplitude for the event. This is in contrast to classical probability theory in which you would add the probabilities of the possible processes which produce the event. (c) If the path consists of two parts such as p = A u t,then CD, = On@(.
+
PROBABILITY AMPLITUDE PHASE FACTORS Thus we obtain interference effects. If CD,,(x) A , ( ~ ) d @then ~(~),
(apl+
@,2)(@,,
+ 6,J
=
A:
+
=
Al(x)ei@l(x) and CD,(x)
=
+ 4 + 2A1’42 cos (41 - 4 2 )
so that \O(x)[’ varies between ( A , A,)’ and ( A , - A,), depending upon - $z. If 6 = 2nn, we get a maximum probability the phase diflerence 6 =
FIGURE 17.2
356
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
point and if S = (2n + 1)7c, we get a minimum probability point. Of course, A , and A , must be such that f @ ( x ) m d x= 1, but A , and A , have no effect on the locations along the screen of the max and min probability points. The location of these points relates to the properties of atomic particles and fields interacting with them. For example, suppose we know the locations of the max-min points for certain particles in an inertial frame subject to no force. When a field is turned on which interacts with the particles, the locations of the max-min points will shift. This is what we want to calculate. Thus we will not worry about A in @ = Ae'4 but will see how to determine the 4. We will refer to ei@as the probability amplitude phase factor.
DeBROGLIE AND FEYNMAN The idea of associating a wave disturbance with a particle motion was developed by Louis deBroglie in 1924. Later Richard Feynman gave us the principle that the probability amplitude phase factor of a path q(t) in the configuration space of a classical mechanical system with kinetic energy function T(q, q) and potential energy function U(q)is exp(i 1(T(q,4) - U(q))dt). The deBroglie wave for a particle of mass m in its rest frame is @(t,x,y, z ) = @oe-2"i", where v = mc2/h, c being the speed of light and h Planck's constant. We include the minus sign in the exponent to be in agreement with Feynman's prescription for particles moving with a velocity much smaller than c as we shall see. Using the Lorentz transformation (11.14) we see that the deBroglie wave of a particle of mass m in an inertial frame in which the particle is moving with velocity u parallel to the x-axis is @(t,x,y , z) = exp [-2niv{yt - y(u/c2)x}],
where y = l / , / l _ O . Now let p be any classical path of such a particle, so that p is given by t = t, x = xo + vt, t, < t < t g . Then @(B)= @ ( A ) . exp [ -2niv(t, - t , ) , / m ] , so that the phase change of @ from point A = (t,, xo + ut,) to B = (t,, xo + ut,) is S, = - 2 n v ~ , ~ ,where T, is the elapsed proper time along p from A to B. This is the phase change of the deBroglie wave along the classical path of the particle. Following Feynman, we shall take the probability amplitude phase factor of any future pointing timelike path p to be exp( - 2niv AT).
PHASE FACTORS AND 1 -FORMS Now let 6 be the curve in phase space obtained from using the relativistic energy momentum relations. Recall that
357
PHASE FACTORS AND 1 -FORMS
where v that
= dp/dt.
Since the metric is ds2 = c2 dt2 - dx2 - d y 2 - dz2, we get
If 8 is the canonical 1-form on phase space, then
Thus, j,- 8 = r n c 2 , / m ( A t ) ,so that 6,, = -(27c/h) culation is valid on any Lorentzian space-time.
Jp
8. In fact, this cal-
LEMMA 17.1 Let A4 be any four-dimensional manifold with a Lorentz metric g . If p(t) is a timelike curve in M and p(t) in T*M is defined by g(p(t))(md p / d q ) = fi(t), then, for a particle of rest mass m, the probability amplitude phase factor of p is exp( (- i/h) jo 0). If p is future pointing, 0 = mc2(A.rh where
so
is the elapsed proper time along p (h = (1/27c)h). REMARK 17.2 The lemma gives the phase factor if there are no external fields influencing the particles, otherwise it must be modified. We will see how to do this in the case of the electromagnetic field. Note that the gravitational field is included in the lemma since it is represented by replacing the space time of special relativity with a general Lorentzian manifold.
We now return to the situation of Fig. 17.2. With pi = ,Ii u Li, the elapsed time of p 1 equals the elapsed time p2 in our reference frame. Since we are assuming that the initial speeds are the same, we get that the proper time of 2, = the proper time of 1,. Now letting L, and L2 also denote the length of the paths, and writing v1 = t ,j A t , u2 = [ , / A t , where At is the elapsed time of path el and e,, the phase difference is
If L, = e,, then let u = u1 = u2 and we get 6 = 0 so 0 is a maximum probability point. Let us find the closest minimum probability point above 0
358
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
assuming that u << c and that d is much larger than the distance between A and B so that d, - d , is small. Using the binomial expansion, we get that 6 z (2.nmv/h)(t1- t,).Setting this equal to -n, we get that the minimum point closest to 0 from above occurs when t, - el = h/2rnu. Because of the small value of h, this value will be too small to be detected for particles that we are used to seeing in daily existence. It is only for atomic particles that h/2rnu is of detectable magnitude. REMARK 17.3 While the heuristic derivation of the formula was, via deBroglie, associated with the motion of a particle moving along a classical trajectory, we cannot consistently think in this way in dealing with quantum behavior. We have seen that the statement “the particle must go along path t,or path t,” would lead us to add the probabilities instead of the probability amplitudes. We cannot apply the principles of classical particle mechanics to curves and points internal to the experiment unless we include the measurement of the values of these quantities as part of the experiment. This inclusion may then change the outcome of the experiment. In addition, there is the Heisenberg uncertainty principle which states that (ApNAx) 2 h, where Ap is the uncertainty in the measurement of the momentum component in the xdirection, Ax is the uncertainty in the xposition, and h is Planck’s constant. For more details on these matters, see Feynman [ll].
COW AND BOHM-AHARANOV EXPERIMENTS We now consider a second interference effect (Fig. 17.3), where the elapsed times of all four paths are the same. We wish to calculate the phase difference between the paths SAD and SBD. Of course, if no fields interact with the particles, the phase difference is 0, since the proper times of the paths are equal. However, we assume there is a uniform gravitational field of strength
FIGURE 17.3
359
COW AND BOHM-AHARANOV EXPERIMENTS
tl in the negative z direction. The form of the proper time metric which represents such a field is
dz2 =f(z) dt2 - (~(z)/c’) dz2 - (1/c2)dXz,
wheref(z) = 1 - (2az/c2) and g(z) = l/f(z). The derivation of this is given in [13]. Such an experiment has been carried out with neutrons by Colella, Overhauser, and Werner and is referred to as the COW experiment. For a discussion see [ 131. Let z1 be the height of BD and z2 the height of SA. Since the metric is invariant under translations in the x direction, the proper time of AD equals the proper time of BD and hence cancel in the phase difference. Now t(SA)
= Jf(z2)(At)’
{
=At I----
- ( 1/c2)(Ax)’
c2
cz
(where v = Ax/At)
so that
But (zz - z,)(Ax) 17.1,
= a,
the area enclosed by SADB, so we get, using Lemma
6 z (maa/hv)2n, which is the COW result. We consider one more example; that of Fig. 17.2, when the particles have a charge q and there is an electromagnetic field present. The formula for the probability amplitude phase factor must be modified as follows. In the presence of an electromagnetic field with a 4-potential A = A , dx”, the probability amplitude phase factor associated with p is defined as di’*)fl, where B=j6(-D-qA).
(17.1)
Here the coordinates are xo = ct, x1 = x, xz = y, x3 = z. Also A , = ($/c, -A), where - V 4 is the electric field if A = 0, and curl A is the magnetic field if 4 = 0 (see Eq. (12.8)).Of course, the 4-potential could be changed by adding df, where f is a scalar function, which would then change the phase
360
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
FIGURE 17.4
factor. But observable effects have to do with the phase difference, which is an integral around a closed curve and hence, is unaffected by df. Also, this prescription only works if there is a global 4-potential in the region of interest. This would not be the case, for example, in the field of a magnetic monopole (see Example 12.11). We will formulate a geometrical interpretation of the phase difference calculation which will not require a global 4-POten tial. The “correctness” of this definition in describing the phyiical world has been established by experiments such as those measuring the Bohm-Aharanov effect, in which a long solenoid carrying an electric current is placed as shown in Fig. 17.4. Here the result is very striking because the electromagnetic field (but not the 4-potential) is negligible along the particle paths. For a discussion see [lo, 351.
COMPLEX LINE BUNDLES AND HOLONOMY We will describe the phase difference calculations in terms of a principal S’-bundle associated with a certain GMS. At this point we ask the reader to work exercises (16.5) and (16.6). Suppose that n: P -+ M is a principal S’-bundle over M , and a is a connection on P. Now s’ acts naturally by complex multiplication on the complex plane C so we can form the associated bundle F = PxSlC.pr: PxSl@-+ M is called the complex line bundle associated with n:P -+ M . Defining
361
COMPLEX LINE BUNDLES AND HOLONOMY
z [ u , w ] = [u, z w ] gives multiplication of elements of
9 by complex numbers. Note that z[ue-ie, eiew] = eiezw] = [u, zw]. In fact, P,is a onedimensional complex vector space V m E M . If m(t), a I t I b, is a curve in M , then we have the parallel transport operators T,:Fm(o) + Frn(,) (see Definition 16.14). Suppose that o: V P is a section of P over the open set I/ c M , and that m(t) is in V for a I t I b. We have 0: V x S' -+ PI V by @(m,z ) =o(m)z, o? = @*a, & = o*a as discussed in Chapter 16. We have -+
&(m,z ) = d(m)
+ i d0
(where z = e"), by Eq. (16.5) and Exercise 16.5. As in Exercise 16.6 we write a = ip, where fl is an ordinary (real-valued) I-form on P, = (D*p = -%, a^ = o*fl = -id. Now a horizontal lift of m(t) is given by u(t) = (m(t),z(t)), aI tI b, where 0 = o?(m,z)(&, i)= d(m)& id and z(a) = zo is the initial condition. Therefore, i0 = - a*(m)&= - ifi(m)k, which gives = - fl(m)ni. Thus, z(t) = zo exp( - i I),where c, is m(s), a I sI t. Now we also get $:I/ x C -+ 91 I/ by $(m, w) = [o(m),w], so I,-' Tt $ can be written as
sCc
I+-' T, $(m(a), w) = 0
0
+
( (
0
0
m(t), exp ( - i P j j W ) -
(17.2)
Thus we have the basic formula for parallel transport in terms of a section o: V + P and I =o*(fl). For closed curves in M , we have the concept of holonomy operator, which we now describe for any principal bundle. DEFINITION 17.4 Let 7c:P-+ M be a principal G-bundle with connection a, m E M , and C , the set of all piecewise smooth curves y : [a, b] -+ M such that in = y(a) = y(b). For u E 7c- ' ( m )we define H i : C , + G as follows: Let yeC, and let y" be the horizontal lift of y with initial condition y"(a) = u. Now y"(b)E 7c- '(in),so there is a unique g E G satisfying f ( b ) = y"(a)g. Then we define H i ( y ) = y.
LEMMA 17.5 (a) The image Hz(C,) is a subgroup of G. (b) If u1 = ug, then H",'(y) = g-'(H:(y))g.
We leave the proof as an exercise. Hi(C,) is called the holonomy group of a at u. If 7c: P -+ M is a principal S'-bundle, with connection a = ip, we see, by Lemma 17.5, that Hi is independent of the choice of u E 7c-'(m), so we get H,: C , -+ S1. If F = P xs,C and y E C,, y ( t ) defined for a I tI b, then T,,:9, -+ F ,is given by multiplication, Tb(<)= H,(y)<, V< E 9,.
362
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
INTEGRAL CONDITION FOR CURVATURE FORM In Exercise 16.6 we saw that there is a unique closed 2-form x on M so that n * ( ~=) dfl. Let m E M and c:V+ P a section of P with m E V Suppose that y E C,, y(t), a I t I b, lies entirely within Vand that y is the boundary of an oriented compact 2-submanifold R c V . Now we saw that the horizontal lift u(t)= (y(t), z ( t ) ) with u(a) = (m, 1) is given by
Thus, u(b) = (m,exp(- i j, b), so that H,(y) = exp( - i 1, p). But by Stokes's theorem H,(y) = exp( - i SR 1)since x = d g by Exercise 16.6. PROPOSITION 17.6 Suppose that y E C , is the boundary of an oriented compact two-dimensional submanifold R c M . Then H,(y) = eie, where 8 = - j R x . PROOF: We have given the proof in the case where R is contained in the domain of a section of P. If not, break R up into finitely many pieces, each of which is obtained in the domain of a section. I COROLLARY 17.7 Suppose that K is an oriented two-dimensional sphere embedded as a submanifold of M . Then x = 2nn: for some integer n.
SK
PROOF: Fix m E K . For E small, let y,:[O, 11 -+ K be a parametrized circle of radius E with y,(O) = m = ~ ~ ( 1Then ) . y, divides K into two parts K , and D,, as shown in Fig. 17.5. Now assume that y, is parametrized so that its orientation is that induced from D,.Then the orientation is the negative of that induced from K , . By Proposition 17.7,
FIGURE 17.5
363
INTEGRAL CONDITION FOR CURVATURE FORM
x + 0 so we conclude that H,(y,) + 1. Thus, But, as E --t 0, for some integer n, which gives x = 2nn = 2nn, as asserted.
IK
SK, x + 2nn
Note that if the 2-sphere K is the boundary of a three-dimensional compact region B in M , then j K x = J B dx by Stokes’s theorem, so that jK x = 0 since d x = 0. Thus, if every 2-sphere in M is the boundary of a three-dimensional compact manifold in M , then Corollary 17.7 places no restriction on x. If Q: S 2 + A4 is any smooth mapping, then we get the pullback bundle 5 a*(P)+ S 2 , 6:Q*(P)+ P , and ic?*(p) gives a connection on this bundle (see Exercise 16.9). Applying Corollary 17.7 to o*(P) shows that j S 2 a*(x)= 2nn for some integer n. Suppose that M is a simply connected manifold and 6 is a closed 2-form on M such that js2~ * is6an integer for every smooth mapping Q: S2 + M . Then, there are collections THEOREM 17.8
where each r/; c M is open and di is a 1-form on where hiis a function on & n V,, such that (a) (b) (c) (d) (e)
v, and {Ljlr/I n 6 # a},
M = u{&li~Z}, dd, = 6 on di - h j = dfij on 5 n 5,
v,
f-. = -f..
11 JC ’ if V# n 5 n V, = $3then f i j V;n Q nV,.
+ fjk + f k ,
is an integral constant on
PROOF: We will indicate the ideas involved; the details require considerable knowledge of algebraic topology [16, 29, 311. Choose an open cover { V l i E l } for M such that each V#,qn5, n F n V, (if nonempty) is contractible to a point. For each i, there is a 1-form 6, on K, such that d6, = G.If n 5 # 0, then d(6, - S j ) = 0 on n 5, so there is a function gij on r/; n satisfying 6, - d j = dg,,. O n r/I n 5 n V,,d(gij g j k g k i ) = 0 so that g i j g j k gki = a i j k , a constant. The collection & = {aijkl& n Vj n V, # a} defines an element of f i 2 ( M ,R), the secondCech cohomology group with real coefficients. Now nn,(M)= 0 implies that n,(M) N H”,M, Z), the second singular homology group with integral coefficients. Define, for Q: S2 + M in n,(M), &(a) = j S z a*6. Then, since M is simply connected, C;, defines an element of the singular cohomology group with integral coefficients H ; ( M , Z ) . Finally, i: Z -+ R gives i,: H ; ( M , Z ) +
v
+
v
+
+
+
364
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
H?(M, R ) and there is an isomorphism 4: H?(M, R ) + kZ(M,R), which gives $(i,(G)) = 15.It follows that there is a collection of numbers {A,l V;- n 5# so that if we takeAj = gij + Aij, then (e) is satisfied. I
a}
REMARK: Suppose that 15 is a closed 2-form on a manifold M . Then [h] E H 2 ( M ) ,the second deRham cohomology group of M . If h satisfies the conclusion of Theorem 17.8, then we will write [I51 E H 2 ( M , 2)and say that [h] is an integral cohomology class.
We want to apply these ideas to the interference effects discussed at the beginning of this chapter. They all involve interactions which, for classical particles, are described by the GMS (Sm2,w + (z*)*6) of Example 15.18. We will see that the evolution of the probability amplitude phase factor will be given by parallel transport in a complex line bundle over a subset of S,, and that x cc (w + (r*)*6). There is an additional problem, however. The curves in Sm2 are lifts (using the relativistic energy momentum relations) of curves in spacetime and these lifts are not continuous, due to jumps in the momentum at certain points. Thus we must define a unique parallel transport along certain discontinuous curves. This is achieved by noting that we have e*: S,, + Q, where Q is a region of spacetime and i:x = 0, where iq:(e*)-'(q) + Smz is inclusion, for all q E Q. We will need the following lemma. LEMMA 17.9 Suppose that 7c:P + M is a principal S'-bundle, that M is connected and simply connected, and that a is a connection on P such that da = 0. Then, for each m E M , H,: C, + S' is given by H,(y) = 1, for each y E C,. Thus, if n is any other point in M , and we define T,,: F, + 8, by parallel transport along any piecewise, smooth curve from m to n, then T,, is independent of the choice of curve. Of course, 8 denotes the associated complex line bundle. PROOF: Fix m E M and uo E P,. Recall that, if y: [a, b] + M is an element of C, and u: [a, b] + P is the horizontal lift of y satisfying ~ ( a=) uo, then u(b) = u,H,(y). Now P , Z S' by uog -+ g so that H,(C,) E { u E P,lu is accessible from uo along the horizontal subbundle %}. Since M is simply connected, H,(C,) is a connected subgroup of S'. Thus H,(C,) = S' or H,(C,) = ( 1 ) . If H,(C,) = S', then every u E P , is %-accessible from uo. But da = 0 implies that % is integrable, so, since X is transverse to P,, we have a contradiction to the inaccessibility theorem. This establishes the first part of the conclusion. The fact that T,, is independent of the curve now follows directly. I
365
PARALLEL TRANSPORT DETERMINED BY A FOLIATION
BUNDLE DESCRIPTION OF PHASE FACTOR CALCU LATl ON We now present the bundle description of the phase factor calculation. We assume we have a GMS ( M , G) and T*: M -,Q such that dc3 = 0 and (PF1) [(1/2n)c3] E H 2 ( M ,Z ) , (PF2) M is simply connected, (PF3) if M , = (z*)-'(q) V q E Q, then $5 = 0, where i,: M , inclusion, (PF3) each M , is connected and simply connected.
-+
M is
We have seen that (PF1) is necessary for 6 to be obtained from the curvature form of a principal S'-bundle over M . In fact it is also sufficient. This follows by Theorem 17.8 and Exercise 17.7. Let n:P -,M be the principal S'-bundle and CI the connection on P constructed in Exercise 17.7, and let 9 be the associated complex line bundle.
Parallel Transport Determined By A Foliation DEFINITION 17.10 Suppose that y = (yl, y 2 , . . . , y k ) is a finite sequence of smooth curves in M so that if m i= initial point of y i is in M a i , ni = end point of y i is in M b i ,then bi = ai+ for i = 1,2, . . . ,k - 1. For any points rn E M a , and n E M,, we define Tmn(y):Fm + cFn by
Tmn = Tnkn
0
7'7,'
Tnk - j m k
0
Tyk
~
I o
. . '
0
TnZrn,
0
Ty, Tn,mz Tyj 0
0
0
Tmmt.
Here T Y Jdenotes parallel transport along y j and the T n J _ l m areJ given by Lemma 17.9 using property (PF2). Now assume that Q is a simply connected region in spacetime and that the structural group of the orthonormal frame bundle of Q has been reduced to the proper Lorentz group 9;(see Chapter 11). This means that spacetime is an oriented 4-manifold with a preferred time direction. We now define M
=
{ p E T*Qlpppp,= rn2 and p is future pointing},
so that M is an open connected subset of Sm2. Also, take 6 = -(l/h)(w + (z*)*6) where 6 = -qF as in Example 15.18. We assume that 6 is such that 6 satisfies (PFI) (see Remark 17.12). The interference effects we discussed related to a situation in spacetime similar to that described by Fig. 17.3. Let
366
17. QUANTLIM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
y, v be the lifts of SAD and SBD to M as described in Lemma 17.1. Then we where m is a fixed element of M a , get T,,(y): @, + @, and T,,,(v):F,+ F,, and n a fixed element of M,. Choose 5 E F,. Write = Tm,,(y)([) and Q2 = T,,,(v)(~). These depend upon t and do not determine unique complex numbers. However, there is a unique eigE S1 such that Q2 = eid@,,and this 6 is the phase difference which we calculated in the interference effects discussed at the beginning of this chapter. The validity of this claim follows from Lemma 17.1, Eq. (17.2),and Eq. (17.3).Note that 6 = - d B where B = -8 qA, if there is a 4-potential A defined on all of Q (the region of interest in spacetime). Now eigt= T,,,(y)-' T,,,(v)(t) so we get 0
THEOREM 17.1 1 The phase difference is 6, where g,(v u (-7)) = eid,g,,, being the extended holonomy operator, which is defined using the extended parallel transport operators of Definition 17.10, and is determined by the connection tl and the decomposition B = { M , I q E Q } .
R EM A R KS-G EO M ETR IC QUANTI ZATl 0 N REMARK 17.12 (1) The replacement tl -+ CI + in*(df) for f:M + R does not change the eidof Theorem 17.11. However, if n l ( M ) # 0, then different connections could produce different phase differences. (2) [(1/27r)6] E H 2 ( M ,2) is no restriction if H 2 ( M , R ) = 0. However, for an electron in the field of a magnetic monopole (see Example 12.11), H 2 ( M , R ) # 0 and we get the Dirac condition 2kq/h = integer. (3) We want to emphasize that our procedure depended not only upon the GMS ( M , G)but also upon (a) the method of lifting paths from Q to M , (b) the decomposition B = { M , I q E Q}. Actually, Q = M / B , so it is the decomposition of M by submanifolds, on each of which 153 restricts to be zero, that is essential. (4) The development we have given is related to the geometric quantization procedure of Konstant and Souriau. For each fixed time t , let M , be the time slice and o,= GIM,. One considers wave functions $, which are the complex line bundle determined by w,, and are parallel sections of F,, 1K = 0 VK E 9,. Such on each of the leaves of some foliation B,. Again 6 a 9, is called a polarization of M,. One is interested in more than the simple interference effects we have discussed: for example, in obtaining the quantized energy levels of bound systems. The standard approach to quantum mechanics requires an inner product ($;, $,") between wave functions. This will be of the form $:$,". Thus, the product of two wave functions should be a differential form. This leads to a redefinition of wave functions as sections
sp
367
HOLONOMY AND CURVATURE FOR GENERAL LIFE GROUPS
of Ft tensor product with certain -$-forms (the square roots of differential forms). It is necessary (because of topological considerations) to generalize 9, to complex polarizations which are certain subbundles of the complexified tangent bundle T M ORC of M . This work is very intricate and the topic of recent and continuing research. For a nice discussion see [14, Chapter 51. ( 5 ) To carry through our development for a general GMS the construction of a polarization 9'is the difficult part. One can usually obtain a lift of curves in Q = M / 9 to M by considering classical trajectories. For example, which decomposition should one use for the GMS (FE, gE)constructed in Chapter 16 to describe a particle interacting with a general gauge field (suppose the group is G = S0(3))?
HOLONOMY A N D CURVATURE FOR GENERAL LIE GROUPS We now present generalizations of the preceding results about holonomy for principal S'-bundles to the case of principal G-bundles, for any Lie group G. Recall that if n:P + M is a principal G-bundle, M is a connection on P, we get H i : C, + G and H;(C,) is called the holonomy group of ct at u. Here u E n-'(rn). Let C: = {y E C,ly can be deformed continuously to m}. Hk(C2) c H;(C,) is called the restricted holonomy group of M at u. Let G, = H;(C,) and G: = H;(C;). GI] is a normal subgroup of G, and G,/GZ is countable.
LEMMA 17.13
PROOF: If yl, y 2 E C, we define y 2 y1 to be the curve obtained by first going around y1 and then around y 2 . Also, 7'; will be the curve obtained by traversing y1 in the opposite direction. Let g E G, and h E GI] be obtained from curves y and v. The y v y can be contracted to y y which can be contracted to a point. Thus g - l h g E GZ. Let n , ( M , rn) be the fundamental group of M with base point m. If y E n,(M), choose a representative y1 E C,. Then Hi(?,) E G,. If y 2 E C , is homotopic to yl, then y;' y 2 represents y-l 0 y = 1 in n , ( M , m).Thus H;(yl)-'H;(y2) E GZ, and we get H : nl(M, rn) G,/G:, a surjection. Since M is a second countable manifold, nl(M, m) is countable. I 0
'
(1
0
-'
0
0
--f
Now let Q, from u to
."I.
=
{."
E
PI there is a horizontal piecewise smooth curve y" in P
THEOREM 17.14 Suppose that G, is a closed subgroup of G, and that M is connected. Then Q. is a reduction of P to a principal subbundle with group G,.
368
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
PROOF: The assumption that G , is a closed subgroup of G (so that it is a submanifold and a Lie group by the proof of Theorem 14.11) makes this theorem a straightforward exercise. [
For a more general theorem which does not assume that G, is closed in G see [17]. From this reference we also obtain THEOREM 17.15 G , is a Lie group and G t is the connected component of the identity. Let 3 be the Lie algebra of G and 3, the Lie algebra of GI]. Then 9, is spanned by all elements of the form Da(ii)((, [) where u" E Q. and (, ( are horizontal tangent vectors at fi (recall that Da is the curvature form of the connection a). COROLLARY 17.16 (a) Suppose that M is connected, U E P, and G , = (e).Then there exists a global section 6:M -+ P , and thus that P is equivalent to M x G . (b) Suppose that M is connected and simply connected and that Da = 0. Then there exists a global section 6:M -+ P , so that P i s equivalent to M x G. PROOF: (a) Let m = n(u).For n E M choose a smooth curve y in M from rn to n. Let 7 be the horizontal lift of y with initial condition u. Define o(n)= endpoint of 7. Since G, = (e), this is independent of the choice of y. The smoothness of 6 can be seen using local trivializations. (b) Since M is simply connected, G, = GE by the proof of Lemma 17.13. But Dcc = 0 means that G t = (e) by Theorem 17.15. [ REMARK 17.17 In Theorem 17.11 a quantum phase difference was expressed in terms of a holonomy operator on a principal S'-bundle. The generalization to the nonabelian case is as follows: Let n:P -+ M be a principal G-bundle with a connection a, and F a set of states, carrying an action of G and a G-invariant inner product ( F can be a complex vector space with a complex-valued inner product). Form the bundle of states B = P x G F as described in Chapter 16. Let m E M , y E C,, and 4 E F,, It,is.a simple exercise to show that, if we choose u E P,, then the quantity (v, H:(y)v), where 4 = [u, u ] , is independent of the choice of u. It is this quantity which is eis in the previous discussion.
EXERCISES 17.1 Show that condition (PF2) implies that any two connections a, and a, on P such that da, = da, are related by a, = a, + in*(df)for some function f :M -+ R.
369
EXERCl S ES
17.2 Show that 2kqlh = integer for a electron in the field of a magnetic monopole. See Remark 17.12(2). 17.3 Answer the question raised in Remark 17.12(5) with G
= SO(3).
17.4 Supply the proof of Theorem 17.14. 17.5 Supply the proof of Lemma 17.5. 17.6 Let n:P --+ M be a principal S'-bundle and CI, = ibl, u2 = ip2 two connections on P. We have seen that there are closed 2-forms xI, x 2 on M so that "*(xi) = It/?,. Suppose that K is an oriented twodimensional sphere embedded as a submanifold of M . Show that JK x1 = S K x 2 .
17.7 Suppose that 6 is a closed 2-form on M such that [(1/2n)G] E H 2 ( M , Z ) is an integral cohomology class. Let {(v,, S,)}, {hj}be the collections described in the conclusion of Theorem 17.8. Define
C = ((i, m, z)li E I, m E and (i, m, z)
N
( j , m, ezffifl,f(m)z) if m E
(1) Show that
-
6 , and z E Sl)
6 n 5.
is an equivalence relation.
Writing [i, rn, z] = { ( j ,m, zl)l(i,m, z) {[i, m, z]li E I , m E 6 , z E Sl}, 71:
P -+ M
by
-
( j , m, z,)), we define P =
n([i,m, z ] ) = m,
and [i, m, z]z, = [i, m, zz,].
For each i E I , we have 4,: x S' Then q5J:1 4,: n x S' -+ 6 n (m,e2ffifij(m)z). Show that 0
-+
P by +,(m, z) = [ i , rn, z]. x S' by
4J:1
0
q5i(m, z) =
( 2 ) There is a unique differential structure on P such that each a diffeomorphism onto an open set of P.
4; is
(3) n:P -+M is a principal S'-bundle with the S'-action defined above.
Define a T,S' valued 1-form Eion 1/; x S1 by
E,(m, z)(ril,i) = 2ni4(m)riz + Ti. Show that (4) There is a unique T,S'-valued connection 1-form CI on P satisfying
370
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
(a) &+a = oZi Vi E I , (b) da = n*(i6). See also Exercise 17.8. 17.8 In the case that M = S2, the 2-sphere, and [(1/2n)6] E H 2 ( M , Z ) a direct construction of the desired principal S1-bundle can be given which is simpler than that of Exercise 17.7, because a system similar to {(K, Si)} can be explicitly specified. We can write S2 = D , u D,, where D , is the upper hemisphere, D, is the lower hemisphere and D, u D , N S'. In fact, assume we have a parametrization D, n D , = {eie10 5 6' 5 2n}. On each Di there is a l-form di such that d6, = (1/2n)6. {(D,, h1), (D,, a,)) plays the role of Si)} in Exercise 17.7 (we do not need that D , n D, is contractable in this construction). Define g : D , n D, + S' by
{(v,
ss,
Then y(0) = 1 and g(2n) = exp(i 6)= e2nin= 1 = g(0) so that g is well defined. Now let P = {[i, rn, z ] 1 i = 1,2, rn E Di, z E S'}, where [l, rn, z] = [2, rn, g(m)z] if rn E D , n D,; define an S1-action on P by [i, rn, z]zl = [i, rn, zz,], and define n:P + S2 by n([i,m, z ] ) = m. There is a unique differential structure on P such that c $ ~ :Di x S' + P by 4i(rn,z ) = [i, m, z ] gives diffeomorphisms between manifolds with boundary, and we obtain a principal S'-bundle. There is a unique connection a on P such that +;(a) = 2niSi + 6, where 9 is the canonical 1-form on S'.
QUANTUM EFFECTS The quantum behavior of particles occurs in experiments in which particles exhibit a wavelike nature manifested by certain interference effects, a simple case of which we now describe. A beam of identical particles is split into two parts at a point S. Or we can consider S as a source point for beams of particles, all of which are reflected back from a barrier except for the beams which reach the points A and B, where there are slits in the barrier. The points A and B then act as secondary sources allowing particles to go off in all directions (see Fig. 17.1). We now place a detection screen at a distance d from the barrier (see Fig. 17.2). Let us coordinatize the screen by x. We are interested in the following question.
FIGURE 17.1
354
355
PROBABILITY AMPLITUDE PHASE FACTORS
PROBABI LlTY AMPLITUDES Of the total number of particles which reach the screen in a certain time interval, what fraction arrive at x? This fraction can be interpreted as the probability that a particular particle will arrive at x. A model which correctly describes this is based on three principles written down by Feynman [ 113: (a) There is a complex function @(x)so that lCD(x)lzis the probability that a particle will arrive at x. The complex number @(x)is called the probability amplitude for the recording of the arrival of the particle at x. (b) For each path p there is a probability amplitude CD, so that lCD,12 represents the probability that the particle traveled along p. If pi is the path li followed by ti,then @(x) = CD,, Thus, if an event can result from several different processes, you add the probability amplitudes to obtain the probability amplitude for the event. This is in contrast to classical probability theory in which you would add the probabilities of the possible processes which produce the event. (c) If the path consists of two parts such as p = A u t,then CD, = On@(.
+
PROBABILITY AMPLITUDE PHASE FACTORS Thus we obtain interference effects. If CD,,(x) A , ( ~ ) d @then ~(~),
(apl+
@,2)(@,,
+ 6,J
=
A:
+
=
Al(x)ei@l(x) and CD,(x)
=
+ 4 + 2A1’42 cos (41 - 4 2 )
so that \O(x)[’ varies between ( A , A,)’ and ( A , - A,), depending upon - $z. If 6 = 2nn, we get a maximum probability the phase diflerence 6 =
FIGURE 17.2
356
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
point and if S = (2n + 1)7c, we get a minimum probability point. Of course, A , and A , must be such that f @ ( x ) m d x= 1, but A , and A , have no effect on the locations along the screen of the max and min probability points. The location of these points relates to the properties of atomic particles and fields interacting with them. For example, suppose we know the locations of the max-min points for certain particles in an inertial frame subject to no force. When a field is turned on which interacts with the particles, the locations of the max-min points will shift. This is what we want to calculate. Thus we will not worry about A in @ = Ae'4 but will see how to determine the 4. We will refer to ei@as the probability amplitude phase factor.
DeBROGLIE AND FEYNMAN The idea of associating a wave disturbance with a particle motion was developed by Louis deBroglie in 1924. Later Richard Feynman gave us the principle that the probability amplitude phase factor of a path q(t) in the configuration space of a classical mechanical system with kinetic energy function T(q, q) and potential energy function U(q)is exp(i 1(T(q,4) - U(q))dt). The deBroglie wave for a particle of mass m in its rest frame is @(t,x,y, z ) = @oe-2"i", where v = mc2/h, c being the speed of light and h Planck's constant. We include the minus sign in the exponent to be in agreement with Feynman's prescription for particles moving with a velocity much smaller than c as we shall see. Using the Lorentz transformation (11.14) we see that the deBroglie wave of a particle of mass m in an inertial frame in which the particle is moving with velocity u parallel to the x-axis is @(t,x,y , z) = exp [-2niv{yt - y(u/c2)x}],
where y = l / , / l _ O . Now let p be any classical path of such a particle, so that p is given by t = t, x = xo + vt, t, < t < t g . Then @(B)= @ ( A ) . exp [ -2niv(t, - t , ) , / m ] , so that the phase change of @ from point A = (t,, xo + ut,) to B = (t,, xo + ut,) is S, = - 2 n v ~ , ~ ,where T, is the elapsed proper time along p from A to B. This is the phase change of the deBroglie wave along the classical path of the particle. Following Feynman, we shall take the probability amplitude phase factor of any future pointing timelike path p to be exp( - 2niv AT).
PHASE FACTORS AND 1 -FORMS Now let 6 be the curve in phase space obtained from using the relativistic energy momentum relations. Recall that
357
PHASE FACTORS AND 1 -FORMS
where v that
= dp/dt.
Since the metric is ds2 = c2 dt2 - dx2 - d y 2 - dz2, we get
If 8 is the canonical 1-form on phase space, then
Thus, j,- 8 = r n c 2 , / m ( A t ) ,so that 6,, = -(27c/h) culation is valid on any Lorentzian space-time.
Jp
8. In fact, this cal-
LEMMA 17.1 Let A4 be any four-dimensional manifold with a Lorentz metric g . If p(t) is a timelike curve in M and p(t) in T*M is defined by g(p(t))(md p / d q ) = fi(t), then, for a particle of rest mass m, the probability amplitude phase factor of p is exp( (- i/h) jo 0). If p is future pointing, 0 = mc2(A.rh where
so
is the elapsed proper time along p (h = (1/27c)h). REMARK 17.2 The lemma gives the phase factor if there are no external fields influencing the particles, otherwise it must be modified. We will see how to do this in the case of the electromagnetic field. Note that the gravitational field is included in the lemma since it is represented by replacing the space time of special relativity with a general Lorentzian manifold.
We now return to the situation of Fig. 17.2. With pi = ,Ii u Li, the elapsed time of p 1 equals the elapsed time p2 in our reference frame. Since we are assuming that the initial speeds are the same, we get that the proper time of 2, = the proper time of 1,. Now letting L, and L2 also denote the length of the paths, and writing v1 = t ,j A t , u2 = [ , / A t , where At is the elapsed time of path el and e,, the phase difference is
If L, = e,, then let u = u1 = u2 and we get 6 = 0 so 0 is a maximum probability point. Let us find the closest minimum probability point above 0
358
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
assuming that u << c and that d is much larger than the distance between A and B so that d, - d , is small. Using the binomial expansion, we get that 6 z (2.nmv/h)(t1- t,).Setting this equal to -n, we get that the minimum point closest to 0 from above occurs when t, - el = h/2rnu. Because of the small value of h, this value will be too small to be detected for particles that we are used to seeing in daily existence. It is only for atomic particles that h/2rnu is of detectable magnitude. REMARK 17.3 While the heuristic derivation of the formula was, via deBroglie, associated with the motion of a particle moving along a classical trajectory, we cannot consistently think in this way in dealing with quantum behavior. We have seen that the statement “the particle must go along path t,or path t,” would lead us to add the probabilities instead of the probability amplitudes. We cannot apply the principles of classical particle mechanics to curves and points internal to the experiment unless we include the measurement of the values of these quantities as part of the experiment. This inclusion may then change the outcome of the experiment. In addition, there is the Heisenberg uncertainty principle which states that (ApNAx) 2 h, where Ap is the uncertainty in the measurement of the momentum component in the xdirection, Ax is the uncertainty in the xposition, and h is Planck’s constant. For more details on these matters, see Feynman [ll].
COW AND BOHM-AHARANOV EXPERIMENTS We now consider a second interference effect (Fig. 17.3), where the elapsed times of all four paths are the same. We wish to calculate the phase difference between the paths SAD and SBD. Of course, if no fields interact with the particles, the phase difference is 0, since the proper times of the paths are equal. However, we assume there is a uniform gravitational field of strength
FIGURE 17.3
359
COW AND BOHM-AHARANOV EXPERIMENTS
tl in the negative z direction. The form of the proper time metric which represents such a field is
dz2 =f(z) dt2 - (~(z)/c’) dz2 - (1/c2)dXz,
wheref(z) = 1 - (2az/c2) and g(z) = l/f(z). The derivation of this is given in [13]. Such an experiment has been carried out with neutrons by Colella, Overhauser, and Werner and is referred to as the COW experiment. For a discussion see [ 131. Let z1 be the height of BD and z2 the height of SA. Since the metric is invariant under translations in the x direction, the proper time of AD equals the proper time of BD and hence cancel in the phase difference. Now t(SA)
= Jf(z2)(At)’
{
=At I----
- ( 1/c2)(Ax)’
c2
cz
(where v = Ax/At)
so that
But (zz - z,)(Ax) 17.1,
= a,
the area enclosed by SADB, so we get, using Lemma
6 z (maa/hv)2n, which is the COW result. We consider one more example; that of Fig. 17.2, when the particles have a charge q and there is an electromagnetic field present. The formula for the probability amplitude phase factor must be modified as follows. In the presence of an electromagnetic field with a 4-potential A = A , dx”, the probability amplitude phase factor associated with p is defined as di’*)fl, where B=j6(-D-qA).
(17.1)
Here the coordinates are xo = ct, x1 = x, xz = y, x3 = z. Also A , = ($/c, -A), where - V 4 is the electric field if A = 0, and curl A is the magnetic field if 4 = 0 (see Eq. (12.8)).Of course, the 4-potential could be changed by adding df, where f is a scalar function, which would then change the phase
360
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
FIGURE 17.4
factor. But observable effects have to do with the phase difference, which is an integral around a closed curve and hence, is unaffected by df. Also, this prescription only works if there is a global 4-potential in the region of interest. This would not be the case, for example, in the field of a magnetic monopole (see Example 12.11). We will formulate a geometrical interpretation of the phase difference calculation which will not require a global 4-POten tial. The “correctness” of this definition in describing the phyiical world has been established by experiments such as those measuring the Bohm-Aharanov effect, in which a long solenoid carrying an electric current is placed as shown in Fig. 17.4. Here the result is very striking because the electromagnetic field (but not the 4-potential) is negligible along the particle paths. For a discussion see [lo, 351.
COMPLEX LINE BUNDLES AND HOLONOMY We will describe the phase difference calculations in terms of a principal S’-bundle associated with a certain GMS. At this point we ask the reader to work exercises (16.5) and (16.6). Suppose that n: P -+ M is a principal S’-bundle over M , and a is a connection on P. Now s’ acts naturally by complex multiplication on the complex plane C so we can form the associated bundle F = PxSlC.pr: PxSl@-+ M is called the complex line bundle associated with n:P -+ M . Defining
361
COMPLEX LINE BUNDLES AND HOLONOMY
z [ u , w ] = [u, z w ] gives multiplication of elements of
9 by complex numbers. Note that z[ue-ie, eiew] = eiezw] = [u, zw]. In fact, P,is a onedimensional complex vector space V m E M . If m(t), a I t I b, is a curve in M , then we have the parallel transport operators T,:Fm(o) + Frn(,) (see Definition 16.14). Suppose that o: V P is a section of P over the open set I/ c M , and that m(t) is in V for a I t I b. We have 0: V x S' -+ PI V by @(m,z ) =o(m)z, o? = @*a, & = o*a as discussed in Chapter 16. We have -+
&(m,z ) = d(m)
+ i d0
(where z = e"), by Eq. (16.5) and Exercise 16.5. As in Exercise 16.6 we write a = ip, where fl is an ordinary (real-valued) I-form on P, = (D*p = -%, a^ = o*fl = -id. Now a horizontal lift of m(t) is given by u(t) = (m(t),z(t)), aI tI b, where 0 = o?(m,z)(&, i)= d(m)& id and z(a) = zo is the initial condition. Therefore, i0 = - a*(m)&= - ifi(m)k, which gives = - fl(m)ni. Thus, z(t) = zo exp( - i I),where c, is m(s), a I sI t. Now we also get $:I/ x C -+ 91 I/ by $(m, w) = [o(m),w], so I,-' Tt $ can be written as
sCc
I+-' T, $(m(a), w) = 0
0
+
( (
0
0
m(t), exp ( - i P j j W ) -
(17.2)
Thus we have the basic formula for parallel transport in terms of a section o: V + P and I =o*(fl). For closed curves in M , we have the concept of holonomy operator, which we now describe for any principal bundle. DEFINITION 17.4 Let 7c:P-+ M be a principal G-bundle with connection a, m E M , and C , the set of all piecewise smooth curves y : [a, b] -+ M such that in = y(a) = y(b). For u E 7c- ' ( m )we define H i : C , + G as follows: Let yeC, and let y" be the horizontal lift of y with initial condition y"(a) = u. Now y"(b)E 7c- '(in),so there is a unique g E G satisfying f ( b ) = y"(a)g. Then we define H i ( y ) = y.
LEMMA 17.5 (a) The image Hz(C,) is a subgroup of G. (b) If u1 = ug, then H",'(y) = g-'(H:(y))g.
We leave the proof as an exercise. Hi(C,) is called the holonomy group of a at u. If 7c: P -+ M is a principal S'-bundle, with connection a = ip, we see, by Lemma 17.5, that Hi is independent of the choice of u E 7c-'(m), so we get H,: C , -+ S1. If F = P xs,C and y E C,, y ( t ) defined for a I tI b, then T,,:9, -+ F ,is given by multiplication, Tb(<)= H,(y)<, V< E 9,.
362
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
INTEGRAL CONDITION FOR CURVATURE FORM In Exercise 16.6 we saw that there is a unique closed 2-form x on M so that n * ( ~=) dfl. Let m E M and c:V+ P a section of P with m E V Suppose that y E C,, y(t), a I t I b, lies entirely within Vand that y is the boundary of an oriented compact 2-submanifold R c V . Now we saw that the horizontal lift u(t)= (y(t), z ( t ) ) with u(a) = (m, 1) is given by
Thus, u(b) = (m,exp(- i j, b), so that H,(y) = exp( - i 1, p). But by Stokes's theorem H,(y) = exp( - i SR 1)since x = d g by Exercise 16.6. PROPOSITION 17.6 Suppose that y E C , is the boundary of an oriented compact two-dimensional submanifold R c M . Then H,(y) = eie, where 8 = - j R x . PROOF: We have given the proof in the case where R is contained in the domain of a section of P. If not, break R up into finitely many pieces, each of which is obtained in the domain of a section. I COROLLARY 17.7 Suppose that K is an oriented two-dimensional sphere embedded as a submanifold of M . Then x = 2nn: for some integer n.
SK
PROOF: Fix m E K . For E small, let y,:[O, 11 -+ K be a parametrized circle of radius E with y,(O) = m = ~ ~ ( 1Then ) . y, divides K into two parts K , and D,, as shown in Fig. 17.5. Now assume that y, is parametrized so that its orientation is that induced from D,.Then the orientation is the negative of that induced from K , . By Proposition 17.7,
FIGURE 17.5
363
INTEGRAL CONDITION FOR CURVATURE FORM
x + 0 so we conclude that H,(y,) + 1. Thus, But, as E --t 0, for some integer n, which gives x = 2nn = 2nn, as asserted.
IK
SK, x + 2nn
Note that if the 2-sphere K is the boundary of a three-dimensional compact region B in M , then j K x = J B dx by Stokes’s theorem, so that jK x = 0 since d x = 0. Thus, if every 2-sphere in M is the boundary of a three-dimensional compact manifold in M , then Corollary 17.7 places no restriction on x. If Q: S 2 + A4 is any smooth mapping, then we get the pullback bundle 5 a*(P)+ S 2 , 6:Q*(P)+ P , and ic?*(p) gives a connection on this bundle (see Exercise 16.9). Applying Corollary 17.7 to o*(P) shows that j S 2 a*(x)= 2nn for some integer n. Suppose that M is a simply connected manifold and 6 is a closed 2-form on M such that js2~ * is6an integer for every smooth mapping Q: S2 + M . Then, there are collections THEOREM 17.8
where each r/; c M is open and di is a 1-form on where hiis a function on & n V,, such that (a) (b) (c) (d) (e)
v, and {Ljlr/I n 6 # a},
M = u{&li~Z}, dd, = 6 on di - h j = dfij on 5 n 5,
v,
f-. = -f..
11 JC ’ if V# n 5 n V, = $3then f i j V;n Q nV,.
+ fjk + f k ,
is an integral constant on
PROOF: We will indicate the ideas involved; the details require considerable knowledge of algebraic topology [16, 29, 311. Choose an open cover { V l i E l } for M such that each V#,qn5, n F n V, (if nonempty) is contractible to a point. For each i, there is a 1-form 6, on K, such that d6, = G.If n 5 # 0, then d(6, - S j ) = 0 on n 5, so there is a function gij on r/; n satisfying 6, - d j = dg,,. O n r/I n 5 n V,,d(gij g j k g k i ) = 0 so that g i j g j k gki = a i j k , a constant. The collection & = {aijkl& n Vj n V, # a} defines an element of f i 2 ( M ,R), the secondCech cohomology group with real coefficients. Now nn,(M)= 0 implies that n,(M) N H”,M, Z), the second singular homology group with integral coefficients. Define, for Q: S2 + M in n,(M), &(a) = j S z a*6. Then, since M is simply connected, C;, defines an element of the singular cohomology group with integral coefficients H ; ( M , Z ) . Finally, i: Z -+ R gives i,: H ; ( M , Z ) +
v
+
v
+
+
+
364
17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
H?(M, R ) and there is an isomorphism 4: H?(M, R ) + kZ(M,R), which gives $(i,(G)) = 15.It follows that there is a collection of numbers {A,l V;- n 5# so that if we takeAj = gij + Aij, then (e) is satisfied. I
a}
REMARK: Suppose that 15 is a closed 2-form on a manifold M . Then [h] E H 2 ( M ) ,the second deRham cohomology group of M . If h satisfies the conclusion of Theorem 17.8, then we will write [I51 E H 2 ( M , 2)and say that [h] is an integral cohomology class.
We want to apply these ideas to the interference effects discussed at the beginning of this chapter. They all involve interactions which, for classical particles, are described by the GMS (Sm2,w + (z*)*6) of Example 15.18. We will see that the evolution of the probability amplitude phase factor will be given by parallel transport in a complex line bundle over a subset of S,, and that x cc (w + (r*)*6). There is an additional problem, however. The curves in Sm2 are lifts (using the relativistic energy momentum relations) of curves in spacetime and these lifts are not continuous, due to jumps in the momentum at certain points. Thus we must define a unique parallel transport along certain discontinuous curves. This is achieved by noting that we have e*: S,, + Q, where Q is a region of spacetime and i:x = 0, where iq:(e*)-'(q) + Smz is inclusion, for all q E Q. We will need the following lemma. LEMMA 17.9 Suppose that 7c:P + M is a principal S'-bundle, that M is connected and simply connected, and that a is a connection on P such that da = 0. Then, for each m E M , H,: C, + S' is given by H,(y) = 1, for each y E C,. Thus, if n is any other point in M , and we define T,,: F, + 8, by parallel transport along any piecewise, smooth curve from m to n, then T,, is independent of the choice of curve. Of course, 8 denotes the associated complex line bundle. PROOF: Fix m E M and uo E P,. Recall that, if y: [a, b] + M is an element of C, and u: [a, b] + P is the horizontal lift of y satisfying ~ ( a=) uo, then u(b) = u,H,(y). Now P , Z S' by uog -+ g so that H,(C,) E { u E P,lu is accessible from uo along the horizontal subbundle %}. Since M is simply connected, H,(C,) is a connected subgroup of S'. Thus H,(C,) = S' or H,(C,) = ( 1 ) . If H,(C,) = S', then every u E P , is %-accessible from uo. But da = 0 implies that % is integrable, so, since X is transverse to P,, we have a contradiction to the inaccessibility theorem. This establishes the first part of the conclusion. The fact that T,, is independent of the curve now follows directly. I
365
PARALLEL TRANSPORT DETERMINED BY A FOLIATION
BUNDLE DESCRIPTION OF PHASE FACTOR CALCU LATl ON We now present the bundle description of the phase factor calculation. We assume we have a GMS ( M , G) and T*: M -,Q such that dc3 = 0 and (PF1) [(1/2n)c3] E H 2 ( M ,Z ) , (PF2) M is simply connected, (PF3) if M , = (z*)-'(q) V q E Q, then $5 = 0, where i,: M , inclusion, (PF3) each M , is connected and simply connected.
-+
M is
We have seen that (PF1) is necessary for 6 to be obtained from the curvature form of a principal S'-bundle over M . In fact it is also sufficient. This follows by Theorem 17.8 and Exercise 17.7. Let n:P -,M be the principal S'-bundle and CI the connection on P constructed in Exercise 17.7, and let 9 be the associated complex line bundle.
Parallel Transport Determined By A Foliation DEFINITION 17.10 Suppose that y = (yl, y 2 , . . . , y k ) is a finite sequence of smooth curves in M so that if m i= initial point of y i is in M a i , ni = end point of y i is in M b i ,then bi = ai+ for i = 1,2, . . . ,k - 1. For any points rn E M a , and n E M,, we define Tmn(y):Fm + cFn by
Tmn = Tnkn
0
7'7,'
Tnk - j m k
0
Tyk
~
I o
. . '
0
TnZrn,
0
Ty, Tn,mz Tyj 0
0
0
Tmmt.
Here T Y Jdenotes parallel transport along y j and the T n J _ l m areJ given by Lemma 17.9 using property (PF2). Now assume that Q is a simply connected region in spacetime and that the structural group of the orthonormal frame bundle of Q has been reduced to the proper Lorentz group 9;(see Chapter 11). This means that spacetime is an oriented 4-manifold with a preferred time direction. We now define M
=
{ p E T*Qlpppp,= rn2 and p is future pointing},
so that M is an open connected subset of Sm2. Also, take 6 = -(l/h)(w + (z*)*6) where 6 = -qF as in Example 15.18. We assume that 6 is such that 6 satisfies (PFI) (see Remark 17.12). The interference effects we discussed related to a situation in spacetime similar to that described by Fig. 17.3. Let
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17. QUANTLIM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
y, v be the lifts of SAD and SBD to M as described in Lemma 17.1. Then we where m is a fixed element of M a , get T,,(y): @, + @, and T,,,(v):F,+ F,, and n a fixed element of M,. Choose 5 E F,. Write = Tm,,(y)([) and Q2 = T,,,(v)(~). These depend upon t and do not determine unique complex numbers. However, there is a unique eigE S1 such that Q2 = eid@,,and this 6 is the phase difference which we calculated in the interference effects discussed at the beginning of this chapter. The validity of this claim follows from Lemma 17.1, Eq. (17.2),and Eq. (17.3).Note that 6 = - d B where B = -8 qA, if there is a 4-potential A defined on all of Q (the region of interest in spacetime). Now eigt= T,,,(y)-' T,,,(v)(t) so we get 0
THEOREM 17.1 1 The phase difference is 6, where g,(v u (-7)) = eid,g,,, being the extended holonomy operator, which is defined using the extended parallel transport operators of Definition 17.10, and is determined by the connection tl and the decomposition B = { M , I q E Q } .
R EM A R KS-G EO M ETR IC QUANTI ZATl 0 N REMARK 17.12 (1) The replacement tl -+ CI + in*(df) for f:M + R does not change the eidof Theorem 17.11. However, if n l ( M ) # 0, then different connections could produce different phase differences. (2) [(1/27r)6] E H 2 ( M ,2) is no restriction if H 2 ( M , R ) = 0. However, for an electron in the field of a magnetic monopole (see Example 12.11), H 2 ( M , R ) # 0 and we get the Dirac condition 2kq/h = integer. (3) We want to emphasize that our procedure depended not only upon the GMS ( M , G)but also upon (a) the method of lifting paths from Q to M , (b) the decomposition B = { M , I q E Q}. Actually, Q = M / B , so it is the decomposition of M by submanifolds, on each of which 153 restricts to be zero, that is essential. (4) The development we have given is related to the geometric quantization procedure of Konstant and Souriau. For each fixed time t , let M , be the time slice and o,= GIM,. One considers wave functions $, which are the complex line bundle determined by w,, and are parallel sections of F,, 1K = 0 VK E 9,. Such on each of the leaves of some foliation B,. Again 6 a 9, is called a polarization of M,. One is interested in more than the simple interference effects we have discussed: for example, in obtaining the quantized energy levels of bound systems. The standard approach to quantum mechanics requires an inner product ($;, $,") between wave functions. This will be of the form $:$,". Thus, the product of two wave functions should be a differential form. This leads to a redefinition of wave functions as sections
sp
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HOLONOMY AND CURVATURE FOR GENERAL LIFE GROUPS
of Ft tensor product with certain -$-forms (the square roots of differential forms). It is necessary (because of topological considerations) to generalize 9, to complex polarizations which are certain subbundles of the complexified tangent bundle T M ORC of M . This work is very intricate and the topic of recent and continuing research. For a nice discussion see [14, Chapter 51. ( 5 ) To carry through our development for a general GMS the construction of a polarization 9'is the difficult part. One can usually obtain a lift of curves in Q = M / 9 to M by considering classical trajectories. For example, which decomposition should one use for the GMS (FE, gE)constructed in Chapter 16 to describe a particle interacting with a general gauge field (suppose the group is G = S0(3))?
HOLONOMY A N D CURVATURE FOR GENERAL LIE GROUPS We now present generalizations of the preceding results about holonomy for principal S'-bundles to the case of principal G-bundles, for any Lie group G. Recall that if n:P + M is a principal G-bundle, M is a connection on P, we get H i : C, + G and H;(C,) is called the holonomy group of ct at u. Here u E n-'(rn). Let C: = {y E C,ly can be deformed continuously to m}. Hk(C2) c H;(C,) is called the restricted holonomy group of M at u. Let G, = H;(C,) and G: = H;(C;). GI] is a normal subgroup of G, and G,/GZ is countable.
LEMMA 17.13
PROOF: If yl, y 2 E C, we define y 2 y1 to be the curve obtained by first going around y1 and then around y 2 . Also, 7'; will be the curve obtained by traversing y1 in the opposite direction. Let g E G, and h E GI] be obtained from curves y and v. The y v y can be contracted to y y which can be contracted to a point. Thus g - l h g E GZ. Let n , ( M , rn) be the fundamental group of M with base point m. If y E n,(M), choose a representative y1 E C,. Then Hi(?,) E G,. If y 2 E C , is homotopic to yl, then y;' y 2 represents y-l 0 y = 1 in n , ( M , m).Thus H;(yl)-'H;(y2) E GZ, and we get H : nl(M, rn) G,/G:, a surjection. Since M is a second countable manifold, nl(M, m) is countable. I 0
'
(1
0
-'
0
0
--f
Now let Q, from u to
."I.
=
{."
E
PI there is a horizontal piecewise smooth curve y" in P
THEOREM 17.14 Suppose that G, is a closed subgroup of G, and that M is connected. Then Q. is a reduction of P to a principal subbundle with group G,.
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17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
PROOF: The assumption that G , is a closed subgroup of G (so that it is a submanifold and a Lie group by the proof of Theorem 14.11) makes this theorem a straightforward exercise. [
For a more general theorem which does not assume that G, is closed in G see [17]. From this reference we also obtain THEOREM 17.15 G , is a Lie group and G t is the connected component of the identity. Let 3 be the Lie algebra of G and 3, the Lie algebra of GI]. Then 9, is spanned by all elements of the form Da(ii)((, [) where u" E Q. and (, ( are horizontal tangent vectors at fi (recall that Da is the curvature form of the connection a). COROLLARY 17.16 (a) Suppose that M is connected, U E P, and G , = (e).Then there exists a global section 6:M -+ P , and thus that P is equivalent to M x G . (b) Suppose that M is connected and simply connected and that Da = 0. Then there exists a global section 6:M -+ P , so that P i s equivalent to M x G. PROOF: (a) Let m = n(u).For n E M choose a smooth curve y in M from rn to n. Let 7 be the horizontal lift of y with initial condition u. Define o(n)= endpoint of 7. Since G, = (e), this is independent of the choice of y. The smoothness of 6 can be seen using local trivializations. (b) Since M is simply connected, G, = GE by the proof of Lemma 17.13. But Dcc = 0 means that G t = (e) by Theorem 17.15. [ REMARK 17.17 In Theorem 17.11 a quantum phase difference was expressed in terms of a holonomy operator on a principal S'-bundle. The generalization to the nonabelian case is as follows: Let n:P -+ M be a principal G-bundle with a connection a, and F a set of states, carrying an action of G and a G-invariant inner product ( F can be a complex vector space with a complex-valued inner product). Form the bundle of states B = P x G F as described in Chapter 16. Let m E M , y E C,, and 4 E F,, It,is.a simple exercise to show that, if we choose u E P,, then the quantity (v, H:(y)v), where 4 = [u, u ] , is independent of the choice of u. It is this quantity which is eis in the previous discussion.
EXERCISES 17.1 Show that condition (PF2) implies that any two connections a, and a, on P such that da, = da, are related by a, = a, + in*(df)for some function f :M -+ R.
369
EXERCl S ES
17.2 Show that 2kqlh = integer for a electron in the field of a magnetic monopole. See Remark 17.12(2). 17.3 Answer the question raised in Remark 17.12(5) with G
= SO(3).
17.4 Supply the proof of Theorem 17.14. 17.5 Supply the proof of Lemma 17.5. 17.6 Let n:P --+ M be a principal S'-bundle and CI, = ibl, u2 = ip2 two connections on P. We have seen that there are closed 2-forms xI, x 2 on M so that "*(xi) = It/?,. Suppose that K is an oriented twodimensional sphere embedded as a submanifold of M . Show that JK x1 = S K x 2 .
17.7 Suppose that 6 is a closed 2-form on M such that [(1/2n)G] E H 2 ( M , Z ) is an integral cohomology class. Let {(v,, S,)}, {hj}be the collections described in the conclusion of Theorem 17.8. Define
C = ((i, m, z)li E I, m E and (i, m, z)
N
( j , m, ezffifl,f(m)z) if m E
(1) Show that
-
6 , and z E Sl)
6 n 5.
is an equivalence relation.
Writing [i, rn, z] = { ( j ,m, zl)l(i,m, z) {[i, m, z]li E I , m E 6 , z E Sl}, 71:
P -+ M
by
-
( j , m, z,)), we define P =
n([i,m, z ] ) = m,
and [i, m, z]z, = [i, m, zz,].
For each i E I , we have 4,: x S' Then q5J:1 4,: n x S' -+ 6 n (m,e2ffifij(m)z). Show that 0
-+
P by +,(m, z) = [ i , rn, z]. x S' by
4J:1
0
q5i(m, z) =
( 2 ) There is a unique differential structure on P such that each a diffeomorphism onto an open set of P.
4; is
(3) n:P -+M is a principal S'-bundle with the S'-action defined above.
Define a T,S' valued 1-form Eion 1/; x S1 by
E,(m, z)(ril,i) = 2ni4(m)riz + Ti. Show that (4) There is a unique T,S'-valued connection 1-form CI on P satisfying
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17. QUANTUM EFFECTS, LINE BUNDLES, AND HOLONOMY GROUPS
(a) &+a = oZi Vi E I , (b) da = n*(i6). See also Exercise 17.8. 17.8 In the case that M = S2, the 2-sphere, and [(1/2n)6] E H 2 ( M , Z ) a direct construction of the desired principal S1-bundle can be given which is simpler than that of Exercise 17.7, because a system similar to {(K, Si)} can be explicitly specified. We can write S2 = D , u D,, where D , is the upper hemisphere, D, is the lower hemisphere and D, u D , N S'. In fact, assume we have a parametrization D, n D , = {eie10 5 6' 5 2n}. On each Di there is a l-form di such that d6, = (1/2n)6. {(D,, h1), (D,, a,)) plays the role of Si)} in Exercise 17.7 (we do not need that D , n D, is contractable in this construction). Define g : D , n D, + S' by
{(v,
ss,
Then y(0) = 1 and g(2n) = exp(i 6)= e2nin= 1 = g(0) so that g is well defined. Now let P = {[i, rn, z ] 1 i = 1,2, rn E Di, z E S'}, where [l, rn, z] = [2, rn, g(m)z] if rn E D , n D,; define an S1-action on P by [i, rn, z]zl = [i, rn, zz,], and define n:P + S2 by n([i,m, z ] ) = m. There is a unique differential structure on P such that c $ ~ :Di x S' + P by 4i(rn,z ) = [i, m, z ] gives diffeomorphisms between manifolds with boundary, and we obtain a principal S'-bundle. There is a unique connection a on P such that +;(a) = 2niSi + 6, where 9 is the canonical 1-form on S'.