- Email: [email protected]
On the interaction of spin and torsion
ANNALS
OF PHYSICS
158, 447-475
(1984)
On the Interaction
of Spin and Torsion RAPOPORT
DIEGO
Department
of Mathematics.
Tel Aviv
SHLOMO Department
of Mathematics,
Cambridge,
*
Vniversitwy,
Tel Aviv.
Israel
STERNBERG Harvard University, Massachusetts 02138
Received
March
Oxford
Street,
3. 1984
The geometry of Cartan connections and their relation to the theory of spontaneous symmetry breaking is discussed. In S. Sternberg and T. Ungar [Hadronic J. 1 (1978), 33-361 Cartan connections are used to derive equations of motion for classical particles in the presence of external fields. Examined here is the case of a massive chargeless particle whose spin interacts with the curvature and torsion of a gravitational field. These equations are solved for the case of a constant vector torsion on Minkowski space, and the solution for a vector torsion wave is indicated. 0 1984 Academic Preaa, Inc.
In the standard theory of “minimal coupling” the effect of the electromagnetic field, A, is to modify the Hamiltonian, H, by replacing p by p - eA in H. The choice of H is dictated by other-usually gravitational-forces. Thus, for a charged particle in the presence of a given electromagnetic and gravitational field, the function H is given by H(q, P> = t II P II* = f 1 gtj(q) P’P’ where g, is a given Lorentzian
metric; we replace H by
to obtain the Hamiltonian giving the equations of motion. A similar modification of H describes the equation for a classical particle in the presence of a more general Yang-Mills field (cf. [8] or [ 131). From a conceptual point of view this is somewhat unsatisfactory, in that this approach places the inertial and gravitational forces on a different footing from all the other forces: The gravitational field determines the Hamiltonian, H, while the Yang-Mills field modities H. If we believe in a unified theory of all forces, then we should expect to treat all forces equally. In [ 161 a scheme was presented for obtaining equations of motion for classical particles in the * Current A.P. 55-534.
address: Departamento 09340 Mexico, D.F.
de Matematica,
Universidad
Autonoma
Metropolitana,
Iztapalapa,
447 0003.4916/84
$7.50
Copyright @I 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.
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presence of an external field. This scheme involves the theory of connections and symplectic geometry but does not involve the notion of Hamiltonian or Lagrangian. It treats gravitational and all other forces the same way. We give a rapid outline of the ideas of [ 161. The details will be presented in the first few sections. Let M be a differentiable manifold representing the possible configurations of a classical mechanical system. The basic problem of classical mechanics is to give a description of the possible curves C(f) on M which can arise as actual changes in the configuration. In contrast to Aristotelian physics, classical mechanics says that we need to know more than the actual configuration of a system to determine its future evolution. Thus, in Newton’s laws of motion, one needs to know the momenta and positions in order to predict the future positions-the laws of Newtonian mechanics are given by second order differential equations. For a charged particle in the presence of an electromagnetic field, one must specify an additional “internal variable”-the charge-in order to get the equations of motion. Thus the situation is as follows: one has a second differentiable manifold, E, (the evolution space, to use the terminology of Souriau [12]), together with a smooth map, rr: E +M. One assumes that the map II makes E into a fiber space over M, i.e., that the Jacobian, dx, is everywhere surjective. On E we expect to have a (one-dimensional) foliation-i.e., a system of first order differential equations. The solution curves of this differential system then project onto curves on M, describing the possible evolution of the configuration. Thus the formulation of the laws of mechanics comes down to a description of E and x and the differential system on E. Since the time of Lagrange and Hamilton, it has been realized that the differential equations on E take a special form: one assumes that there is a closed two form, w, on E and that the differential equations are of the form i(r) w = 0, the “null foliation” of cu. This system is always integrable. Of course it may not be one-dimensional. If o is non-degenerate, then < = O-so there are no curves. If w is highly degenerate then the dimension of the null foliation can be large. (This can occur for actual systems. The non-localizability of a classical photon in special relativity is an example, cf. [ 161.) In many interesting cases the dimension of the null foliation is exactly one, and so we get curves. The problem thus becomes: describe E, a, and the closed two form cc).In [ 161 a procedure was devised to give a general description. It involved two mathematical objects-the notion of a connection on a principal bundle with reduction, and symplectic homogeneous spaces. The idea of considering, simultaneously, a principal bundle together with a reduction first appeared in the work of Elie Cat-tan. It is now an essential ingredient in the notion of “spontaneous symmetry breaking” for Yang-Mills fields. Let us give a brief description of Cartan’s point of view. In the classical theory of connections there were two approaches. One, associated primarily with Levi-Civita, stressed
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the role of the tangent bundle as a vector bundle-that is, as a collection of vector spaces ranging smoothly from point to point. A connection was then a means of parallel transport along curves. This notion of parallel transport then gives rise to the concept of covariant derivative, which is so familiar in tensor analysis and general relativity. The principal bundle connected with this approach is the bundle of orthonormal frames, that is, the bundle of all orthonormal bases of all tangent spaces (the set of vierbeine in general relativity). The structure group of this bundle is the orthogonal (or Lorentz) ‘group H, which is a subgroup of Gl(d), where d is the dimension of the base manifold (4 in general relativity). In the classical theory of surfaces, there was another notion intimately related to parallel transport, and that is the development of a surface on a plane along a curve. Intuitively speaking. one “rolled” the surface on the plane along the curve, maintaining first-order contact. This gives an identification of the tangent space to the surface at each point of the curve with the plane. Then parallel translation in the Euclidean geometry of the plane gives the parallel transport in the sense of Levi-Civita along the curve. From this point of view the notion of development is central, and the crucial property of the plane is that it is a homogeneous space in the sense that it admits a transitive group of isometries that includes the full rotation group as the isotropy group of a point. The plane is E(2)/0(2), w h ere E(2) denotes the group of Euclidean motions of the plane. From this point of view, we might also want to study the development of a surface onto a sphere along a curve, where now the sphere is 0(3)/O(2). or the development onto hyperbolic space S/(2, R)/S0(2). This is the starting point for Cartan’s approach. We must not only be given the structure group H, but also a larger group G containing H as a closed subgroup so that G/H is a homogeneous “model” space with which we would like to compare our manifold M. Thus, in general relativity, where H is the Lorentz group, we would take G to be the Poincari group if we were comparing space-time with Minkowski space and G to be the de Sitter group 0( 1,4) if we were comparing space-time with de Sitter space. In the theory of spontaneous symmetry breaking we are also given a pair of principal bundles. For example, in the Weinberg-Salam model we would have the group G = SU(2) X U(1) and H = U(1). In this case, the quotient space G/H = SU(2) - S3 need not be related in any way to the base. In this more general context, it was shown in [ 161 how a connection-a YangMills field-on P, together with a symplectic G-orbit and an H suborbit gives rise to an E, 7~,and w. We shall review this construction in Section 4. If the space G/H does have the same dimension as M, then we can demand more from the connection form on P,. We can demand that its restriction to P, have no kernel. This has the effect of identifying the tangent space to M, at any point, with the tangent space to G/H --giving a precise mathematical formulation to the intuitive idea of “rolling” M along G/H. This was Cartan’s idea of “soldering.” Although Cartan wrote his fundamental papers in the 1920s the precise modern definition of a Cartan connection was first given by Ehresmann [4] in 1950. In sections 1 and 2 we will give an exposition of this notion, following the treatment of Kobayashi [6]. (In Section 3 we describe, as an aside, the relation of the ideas of Section 1 to spontaneous
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symmetry breaking.) If the groups G and H are the affine and orthogonal (or general linear) groups respectively, then the Cartan connection is called an afline connection. The torsion of this afftne connection then enters in a natural way into the equations of motion. We describe this in Section 4. In particular, for general relativity, we obtain an interaction between spin and torsion. We describe this interaction in Section 5. For certain special cases, we solve these equations of motion in Sections 6 and 7.
1, BUNDLES,REDUCTION,
AND SOLDERING
We begin by recalling some basic definitions and facts in order to establish notation. Let G be a Lie group and A4 a differentiable manifold. A principal G bundle over M is a manifold P on which G acts freely (on the right) and such that the quotient of this G action is M. Thus we have a smooth map n: P + M and K’(X) is a G orbit for each x E M. One also assumes that P is locally trivial in that about each x there is a neighbourhood, U, such that a-‘(U) is isomorphic to U x G (with the We shall let R,: P-, P denote right obvious definition of isomorphism). multiplication by aPi R,(P) = pa-‘,
~EP,uEG
so that R, gives a left action of G on P
R,,=R,R,. Let F be some differentiable manifold on which G acts on the left. We can then form the quotient of the product space P X F by the G action; call it F(P).
Let p: P x F -+ F(P) denote passage to the quotient. Then F(P) is also flbered over M by
%@(P,f))=
Z(P)*
F(P) is called the associated bundle (to the G action on F and the principal bundle PI
Let f: P + F be a smooth function satisfying
f(pa)=a-'f(P).
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Then
U(P,f(P)) = (Pa- l, Uf(P>) = tP~-l,ftP~-‘)). Hence p(p, f(p)) is independent of the choice of p E n-‘(x). a section, s, of F(P); i.e., a map b xos=id s: M -+ F(p)
In other words,fdefines
?r(p) =x.
s(x) = P(Pt f(P))
Conversely, given a section s we may define the functionfby and f satisfies f(pu) = a - ‘f(p). Thus
the preceding equations
we may identify the space of sections of F(P) with the space of maps f: P --) F sutisjjyingy(pu) = a-‘f(p).
(1.1)
There are two especialy important cases of this construction: Suppose that F = G/H where H is a closed subgroup of G. So F(P) is a bundle of homogeneous spaces. Letf: P + F satisfy the identity of (1.1) and so be equivalent to a section, s, of F(P). Consider f-‘(H)={pEPIf(p)=HEG/H}.
IfpEf-‘(H)thenf(pu)=a-‘H=HifandonlyifuEH.Thus f -‘(H)=PH
is an H sub-bundle of P, a reduction of the principal G-bundle to an H bundle. Conversely, suppose that PH is an H sub-bundle of P. Then define f: P -+ G/H by f (PH) = H and if P = qa,
4 E PfI
then
f(q) =a-‘H. This is well defined as can easily be checked and defines a function f satisfying the condition of (I. 1). Thus a section of the bundle (G/H)(P) an H bundle, PH.
A second important 595/158/2-12
is the same us a reduction of P to (1.2)
case is where F is a vector space and the action of G on F is
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linear. Then F(P) is a vector bundle. In this case we can consider k-forms, L!, on P in the values in F. We can consider forms which are horizontal in the sense that i(r) J2 = 0 for any vertical tangent vector <
(1.3)
where “vertical” means tangent to the fiber. We can also consider forms which are equivariant in the sense that R,*R=aR
for all
a E G. ’
(1.4)
It is easy to check that (1.1) generalizes to an F-valued k-form on P satisfying (1.3) and (1.4) is equivalent to an F(P) valued k-form on M.
(l-5)
(The case k = 0 of (1.5) is then (l.l).) We can combine the preceding two examples. Suppose that we are given a section s of (G/H)(P), so we get a reduced bundle, Pn. We can consider the vector bundle associated to the adjoint action of H on g/h. This vector bundle can be identified with the bundle of vertical tangent vectors to (G/H)(P) along the section s. Indeed, we can consider (G/H)(P) as the bundle associated to Pn relative to the H action on G/H. On the principal bundle Pn the section s corresponds to the identically constant function, f E H. At the point H we have an identification of T(G/H) with g/h. This identification is consistent with the H action. Thus we may identify [ g/h](P,) with the bundle of vertical tangent vectors to (G/H)(P) along s. Now suppose that w: TP, + g/h is a one form which satisfies (l&3) and (1.4) relative to the group H. Then w can be thought of as a one form on M with values in [ g/h](P,). Thus Let s be a section of (G/H)(P) and PH the corresponding reduced bundle. Let o be a one form on Pn with values in g/h which satisfies (1.3) and (1.4) relative to the group H. Then w can be regarded as a one form on M with values in the bundle of vertical tangent vectors to (G/H)(P) along s.
(l-6)
In the particular case that dim G/H = dim M, we can further demand that the one form w on M give an isomorphism (at all points) between TM and the bundle of vertical tangent vectors. This is the situation considered by Cartan. The form o is called a “soldering form” in this case. For example, suppose that H = O(V) is the orthogonal group of a vector space with non-degenerate scalar product and G = H @ Y is the corresponding group of affine transformations, so G is the semidirect product of H and V. Then G/H = V. A soldering form o then gives an identification of TM with V(Pn). In particular, this puts a (pseudo) Riemann metric on M and also allows us to identify Pn with the bundle of orthogonal frames. Similarly, if we take H = GL(V) then a soldering form o allows us to identify Pt, with the bundle of all frames on M. Conversely, let Pn denote the bundle of frames of a differentiable manifold M,
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where H = GI(R”), n = dim M. Then P, carries a canonical one form w with values in R”, and w satisfies (1.3) and (1.4). Namely o is the so-called “structure form” defined by
t E TPN,
w,(r) = P(d7c,4
p: TIW,~ + R”.
PEPIf,
Of course, we can enlarge the bundle P, to a G-bundle P, = G(P,) and then we are back in the situation described above.
2. CONNECTIONS
AND CARTAN
CONNECTIONS
Let P, be a vector bundle with structure group G. For each r E g, the Lie algebra of G, let [ denote the corresponding vector field on P, given by the right action of G on P,. Recall that a connection on P, can be described as a g-valued one form 8, on P, which satisfies it0 8, = r
5Eg
(2.1 )
R,*O,=Ad,O,
a E G.
(2.2)
and
The horizontal space of a connection 0, at a point p E P, consists of all tangent vectors at p which are annihilated by t9,, i.e., those which satisfy icv)
8, = 0.
(2.3)
Any curve on M lifts to a unique horizontal curve on P, (one whose tangents are everywhere horizontal) once a lift at one point has been specified. This is the notion of parallel transport along a curve. Now suppose that we are in the situation of the preceding section, so we have a reduction of P, to an H bundle, PH, and we are given a connection, 8, on P,. Then the restriction of 8, to P, defines a g-valued one form on P, which satisfies (2.1) with r E h and (2.2) with a E H. As h is an invariant subspace of g under the adjoint action of H on g, we can define the form w = (restriction of 8, to P,)/h as a g/h valued one form on PH satisfying the conditions of the preceding section. If the group H is reductive-r more generally, if h has an H invariant complement, n in g-we can decompose g=h@n
eW’,, =eH+w
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where we have identified n with g/h. Here 0, is an h-valued form and o is an n-valued form. The form 0, satisfies all the conditions for a connection on PH. Thus If the group H is reductive then the restriction of 8, to PH determines an n-valued one form w on PH and a connection 0, on PH. The one form w satisfies (1.3) and (1.4) relative to H.
(2.4)
It is important to note that the horizontal subspaces for the connections 0, and 0, will differ, in general, at points of PH. However, given t9, and w, one can reconstruct BG along PH and hence on all of P,. Thus the data e,, w on PH is equivalent to a connection eG. In particular, if dim G/H = dimM we can consider the condition that w be a soldering form. If this holds, then Bc is known as a Cartan connection. If G is the affine group then a Cartan connection is called an afine connection, cf. [7]. Let F be a vector space on which G acts. We will let Ak(F) denote the space of F-valued R forms, satisfying (1.3) and (1.4). So Ak(F) can be regarded as the space of k forms on M with values in F(P). In particular, we can identify A’(F) with the space of sections of F(P), a space which we shall also denote by r(F). A connection t9 on P defines a covariant derivative d,:Ak(F)*Ak+‘(F) d,Q=dQ-8.0. One also defines the curvature of a connection by curv(f3) = de - j [e, e]. For the case of a reduced bundle PH with a reductive group H, the restriction curt@,) to PH is given by de,+dw-+[e,+w,e,+w]=de,-t[e,,e,]+dw-e,. = curv(8,) + dg,w - 4 [w,
of
W-~[w,w] w].
In particular, for the case of an affine connection [n, n] = 0 so the last term disappears. The term dg,w is known as the torsion of the linear connection 0,. In the Levi-Civita theory of connections the torsion entered as a more or less formal object. It was given, in local coordinates, as the antisymmetric components of the Christoffel symbols. But in the Cartan theory the torsion enters as a component of the curvature. Its geometric meaning is clear: it gives the translational component of parallel transport around an infinitesimal closed curve.
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3. SPONTANEOUS
SYMMETRY
455
BREAKING
In this section we show how the notion of a reduction of a principal bundle arises in the theory of “spontaneous symmetry breaking” (cf. [ 11). Let P, -+ M be a principal bundle with structure group G. There are various geometrical objects that we have associated with P: For a vector space F which is a representation space of G, we have considered the various spaces Ak(F). We shall denote the sum of all these spaces-and for various possible P’s simply A *(PC). We can also consider the space of all connections on P, which we shall denote by Conn(P,). Suppose that M has a (pseudo) Riemann metric and that g is given a G invariant scalar product as is F. Also suppose that we are given some G invariant function V on F. We can then consider various scalar valued functions built out of the geometric objects such as I/42
o E Ak(F)
Ild,42
w E A k(F), 8 E Conn(PG)
IIcud@) II2
0 E Conn(P,)
V(f)
f E A’(F).
If we have a projection of P onto the bundle of orthonormal frames of M, so that T(M) and T*(M) are associated bundles to P,, then we might also have expressions such as
where W is some G invariant polynomial. All of these are G invariant functions on P, and hence define functions on M. Each of the above expressions thus defines a mm L,,L,,
etc., from geometrical
objects to functions on M
L,: Conn(P,)
x A*(P,)-+
Cm(M)
If we then integrate these expressions, or some (linear) combination, L, of them over form we obtain a functional on the space of geometric objects (at least on the set where this integral converges). The study of this functional
M relative to the volume
F=l’
M
Ldvol
is the central object of study in Yang-Mills theories. For example, describing the critical sets of this function is very important. Now F admits a rather large group of symmetries. Let Aut(P,) denote the group of all diffeomorphisms of P, which commute with G and Gau(P,) c Aut(P,) the subgroup consisting of those which project onto the identity on M. Then each of the “Lagrangians” Li and hence the function F is
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invariant under the action of Gau(P,). The idea of “spontaneous symmetry breaking” is to provide a local partial cross-section for the action of Gau(P,) and hence remove some of the redundancy in F. For this we first recall an alternative description of the group Gau(P,): Let G act on itself by conjugation. Then G(P,) is an associated bundle, each fiber of which is a group. Then pointwise multiplication makes P(G(P,)) into a group. This group can be identified with Gau(P,) as follows: An element of Z(G(P,)) can be viewed as a function r: PC + G satisfying
Then the transformation
z(pb) = b -‘z(p) b.
(3.1)
d,(P) = PO>
(3.2)
#=
satisfies
#A@) = @Ob) = M- ‘0) b = VW) b and tir is an element of Gau(P,). Conversely, given #, we can define r by (3.2) and check that it satisfies (3.1). It is immediate from (3.2) that #,,,, = $,, 0 $,, and so we may identify Gau(P,) with T(G(P,)). In what follows it will be convenient to consider several vector spaces U, R, L,... on which G acts, instead of a single F, and single out one of them, U, for special attention which we shall call the “Higgs” space. An element fE Z(U) (that is, a section of the associated bundle U(P,)) is called a “Higgs field.” We assume the following about the action of G on U: That outside of a small G invariant singular set S in U there exists a cross-section, C, to the G action; that is, we assume that U- S = C x (G/K) as G spaces, where K is the (common) isotropy group of all points of C. For example, let G = W(2) and U= C2. Then we can take S = {0} and C to be the set of all vectors of the form ( 9) with r real and positive. Here K = {e}. Let T+(U) c P(v) denote the set of all f which do not intersect S, i.e., f(PG) c U- S. Any such f determines (by projection onto the second factor) a G equivariant map, j P, + G/K. But then
$-l(K) =f-l(C) is a principal K subbundle of P,, i.e., a reduction of P, to a subbundle with structure group K. The group Gau(P,) acts naturally on all geometric objects. In particular, it acts on the space of all sections, T(G/K). We will assume that this space is nonempty and that Gau(P,) acts transitively on it. (For example, in the case that K = e the action is always transitive. Indeed, let& and f2 : P, + G be two functions satisfying
AW) =b-‘f,(p)
f,W) = b -‘h(p)
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457
Then
satisfies t(pb) = b-%(p) and hence defines an element of Gau(P,). f,@(P))
b
Then
=f2(P)f;‘(P)f,oJ)
=fAP)
so 5 carriesf, into f2.) Under our transitivity assumption, all the subbundles f ‘(K) are conjugate under Gau(P,) and thus, in particular, define the same abstract principal K bundle which we shall denote by P,. The choice of particular f then determines an embedding. x P,-f-+P, as a K subbundle. By construction, the G/K component off 0 Tis identically K, so f 0 3= (9, K) where 4: PK -+ C and K acts trivially on C. So 0 can be thought of as a function from M to C. It is clear that we can recover f from 7 and 4. Thus giving f E P+ (U) is the same as giving 7 and 4. NOW let V be any other space on which G acts. By restriction, K also acts on I’ and hence f defines a map, p*, pullback, from V-valued forms on P, to V-valued forms on P,. It is easy to check that (3.3) Notice that for any o E A k(V(P,))
we have
l13*42(d= Il4’(3W) since w is basic. Let us assume that 1 is a K invariant
complement
(3.4)
in g to k so
g=k+l is a K invariant decomposition of g as a vector space. (It is always possible to choose such an 1 if K is compact, for example.) Let t9E Conn(P,). Then we can write
3*e
=
(jl*@k
+
(3*@,.
As we have seen in the preceding section, the first term on the right is an element of Conn(P,) and the second is an element of A ‘(l(P,)). We thus have defined a map Conn(P,)
f’
Conn(P,)
X A ’ (Z(P,)).
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It is clear that at p =f(q)
AND
STERNBERG
we have
curve(P)= cW3*%(d + P~~e,,(3*44q)- f [3*& 3*41(q). Thus we can write down a geometric function L’: Conn(P,) that
(3.5)
x
A ‘(I@‘,)) + F(M) such
a map
(3.6)
L’ 03*0 = )Icurv ell*. In fact, it is clear that we have proved the following: The assignment
r+(U)
X Conn(P,) XA’(l(P,)
X
A *(R)
x/l*@)
f X
+$induces
... z*
P(M,
C)
X
Conn(P,)
x *-a.
For any geometrical function L on the left hand side there exists a geometrical function L’ on the right hand side such that
L’of*=L. In this way, we have reduced the Gau(P,) redundancy inherent in L to a Gau(P,) redundancy in L’. This is known as “spontaneous symmetry breaking” (cf. [ 11). Let us now explain the idea of “acquisition of mass” under spontaneous symmetry breaking. For this purpose, let us examine the function Conn(P,)
X A”(U) + P(M)
given by
Vtf) --)Ild,fll*. Along the sub-bundle f - ‘(C) we have
f= ($9K) so df = d#: TM+
TC
and where 0, denotes the G orbit through c in C. We may identify 1 with the tangent space T(0,). Suppose the cross section has been chosen so that TC, and TO,(,, are always perpendicular. Then
II&f II’W = ll4ll’(g> + PA:(q)
(3.7)
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where /( IIc denotes the metric on I induced by the identification I-
TO,
Let us assume that k and 1 are perpendicular substituting into (3.5) we see that
for the scalar product
on g. Then
Il.curv412= I143*~hl12+ I143*~M2 + terms cubic or higher in (3*0), Let us combine (3.8) with (3.7). Consider
and (3*@,.
a Lagrangian
(3.8)
L of the form
L = /Icurv 811* + a Ild,fl/2. where a is some constant. Then we can write L = L ’ 0 3* where
L~=lI43*~Ml* +a li~,ll:,+iI~~ll~+~l~~3*~~,i~~ + higher order terms. (Here #,, is some constant value of 4 usually determined as the minimum of some function.) In L’, the quadratic terms in 8, look like the Lagrangian for the KleinGordon equation, with a [0,l& giving the “mass terms.” This is the procedure of “mass acquisition” in spontaneous symmetry breaking (cf. [ 11).
4. THE PRESYMPLECTIC STRUCTURE ASSOCIATED TO A CARTAN PAIR In this section we review the construction of [ 161. Let P, be a principal bundle with structure group G. Let Q be a presymplectic manifold, together with a Hamiltonian action of G on Q. By this we mean the following: (i)
There is a two form R on Q satisfying
(ii)
There is a group action of G on Q preserving a*Q=f2
for all
dJ2 = 0. R, so
a E G.
Let < be an element of the Lie algebra, g, of G, and let to denote the corresponding vector field on Q. Condition (ii) implies that D,,R where D denotes the Lie derivative.
= 0
From the general formula
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we see that di(&) 0 = 0. Locally, this is the same as arguing that
@Q)a = d& for some function fs on Q. We strengthen (ii) in a number of ways. First of all we require that fr exist globally, second that it depend linearly on <. Thus we demand that there exists a map @:Q+
g*
with i 0 = d(@, 0
(4.1)
Here g* denotes the dual space of g and ( , ) denotes the pairing of g* with g. The group G acts via the adjoint representation on g, hence by the contragredient to this representation on g*. Our third requirement is that the map @ be equivariant. Thus we require: (iii) There exist an equivariant map @: Q --)g* such that (4.1) holds. Conditions (i), (ii), and (iii) constitute what we mean by a Hamiltonian action of G on the presymplectic manifold Q. The map @J is then called the moment map, cf. [ 121. Now let 8: TP, -+ g be a connection on P,. We can then define the closed form d(@, 0) + R on P,
X
Q. This form satisfies i@(d(@, 6) + fi) =
0
where {denotes the vector field on PG x Q corresponding to r E g. It is also invariant under the action of G. This implies that there exists a (unique) closed two form ua defined on the associated bundle Q(PG) such that P*(Q)
= d(@, 0) + R.
(4.2)
(Recall that p: PG X Q + Q(PG) is the natural projection onto the quotient.) The closed form up thus defines a presymplectic structure, i.e., is a closed two form on Q(PG). However, this closed two form will, in general, have a redundant component to its null foliation. By use of reduction, we can eliminate some of this redundancy. Let H be a closed subgroup of G and P, a reduction of P, to an H subbundle. Let U be a manifold on which H acts and (: U- Q and equivalent H map. Then 4 induces a map i: U(P,) --) Q(PG).
INTERACTION
OF SPIN
We can then pull a, back to U(P,),
AND
TORSION
461
i.e., consider the form
which is closed two form on U(P,). It is the bundle U(P,) together with the form or which was suggested in [ 161 as the general “evolution space” for a mechanical system. Thus the particular mechanical system requires a choice of Q and U. It is a general fact (due to Kostant and Souriau) that every orbit of G acting on q* is a symplectic manifold (and, for many groups, is the most general transitive G orbit), cf. [ 15 ], for example. So a reasonable choice is to take Q to be a coadjoint G orbit and U an H suborbit. This is what we shall do. In case H is reductive, we have the H invariant decomposition g=h+n and dual decomposition g*=lf*+n*. Correspondingly
we have the decompositions (%PH)
= &I + w
(4.3)
and Qc=@h+Qn.
(4.4)
Then we have the projection pu: P.q x u+
U(P,)
and ~c=Pu*((~h,deH)+d~hABH+d~nA~t(~n,d~)+.R).
(4.5)
We want to consider this local expression for (4.5) in the case where w is a soldering form, more particularly, where o is the structure form for an affine connection. Thus we take H = O(V) where dim V = dim M and Y has some nondegenerate scalar product, and G is the corresponding afline group. We will take Q to be a coadjoint orbit of G. The coadjoint orbits for a semi-direct product were first described in [ 151 (see also [8]): Write g* = h* + n* so a typical element of g* is written as (S, p) with S E h* and p E V*. For the case h = o(V), we may identify V with V* using the scalar product /1*V with h and h*. An element of H acts in the standard linear way on II *V and on I’. Translation by u E V sends (S, p) into (S + PW P). Thus ll~ll*, IIS A PII’ (an d more generally (IS A S A S A p/I’, etc.) are all invariants of the affine group action. There is a family of lowest dimensional non-zero orbits-the orbits through the points (0, p). These have dimension 2dim V - 2. They contain a distinguished H
462
RAPOPORT
AND
STERNBERG
suborbit-the dim V- 1 dimensional H orbit through (0, p). This H suborbit is the “sphere” { (0, q) ] (]q I(* = ((p I(‘}. They are the “spinless” orbits. If ]Ip (1’ # 0, so p & p ‘, an important H invariant condition on an H suborbit is the condition
as was shown in [ 161. We shall return to this condition in the next section. We now determine the null foliation of uU. To do so, we can consider the form in parentheses on the right-hand side of (4.5). That is, we need consider its null foliation on PH X U modulo the vector fields coming from the h action. Since H acts diagonally on PH x U and vertically on PH, it is enough to consider this form on pairs of vectors coming from the subspace zEP,,SEh*,pE
w = H* x TU(S,,,
V*,
where HZ denotes the space of horizontal vectors for BH at z E PH. On horizontal vectors, the form 0, = 0, and de, becomes the curvature R, when evaluated on pairs of horizontal vectors, and dw becomes the torsion, T. In (4.5) @,, becomes S and Qv becomes p. We may thus write the form in parentheses in (4.5), when evaluated on A’W, as
(S,R)+(p,T)+d'AwtQ. We can use o to identify HZ with V and hence consider (4.7)
K=(S,R)t(p,T)
as a bilinear form on V. Since H acts transitively on U, we can write the most general element TU,,,p, as t7,,,, where q E h, and v is the corresponding vector field on U and flCs,P) is its value at (S, p). The value of dp on q is just q . p, while the form R evaluated on qCs,P)Art {s,pjis - CC 145 VI>- We can thus regard (4.6) as a two form, b, on V t h given by b(u
t
0) A (v’
t
CT’)) =K(v
A ~‘1
t
(v * P, 2.1’)- (rl’ a P, v> - (S, [rl, r’l>.
(4.8)
The null foliation is then determined by the condition i(u t II) b = 0. These conditions can best be described by setting the V* and h* components of i(v t q) b separately equal to zero. We get the equations i(v)K=--rep
(4.9)
and ?l*S=pAv.
(4.10)
Let us first examine these equations for the case of the simplest H orbits-those with S = 0, p # 0. Then (4.10) says that v is a multiple of p and q . p is determined by (4.9) so the null foliation is one-dimensional. To see what this null foliation is, let
INTERACTION
OF SPIN AND TORSION
463
us consider, in addition to the given connection 19,, the Levi-Civita connection for the underlying metric-the one with torsion zero. Let Ho denote the horizontal subspace for the Levi-Civita connection. Then H and Ho determine two injections of V into (TP,),. This difference thus gives a map J: V -+ h and the torsion, thought of as an element of Horn@ ‘V, V), is given by
Thus with S = 0 in (4.7) K(Ul,
u2)=
(P,J(%) 0, -J(v,)
02).
so (i(P) K)(w) = (PYJ(w) P -J(P)
w>
= (P9 -J(P) w>
since J(w) E h
= V(P) PI w>
since J(p) E h.
So, if we take u = p in (4.9) we see that ?. p=-J(P)P* Let r denote the vector in H corresponding to v. Then the null direction, given by
has a vertical component which exactly compensatesfor the difference between r and the vector to which is horizontal for the Levi-Civita connection. Thus the null trajectories are exactly the geodesicsof the Levi-Civita connections-the geodesicsof the underlying metric. To repeat: in the absenceof spin the torsion has no eflect; the trajectories are the geodesicsof the underlying metric. In the next section we shall see how spin interacts with torsion.
5. THE INTERACTION
OF SPIN WITH TORSION
We now specialize to the case
so H = Lorentz group
464
RAPOPORTAND
STERNBERG
and G = Poincarl
group.
We will study positive mass particles so
MZ = JIpl(* > 0. The G orbit is then determined by the spin s > 0 where 11s A p1l2 = M2S2. We have already considered the case s = 0. So we need only consider the case s > 0. These orits are eight-dimensional. It was suggested in [ 161 that the appropriate H suborbit to choose is the five-dimensional suborbit given by the condition SEA2pl.
We shall make this as our choice. Notice that under the identification A ‘(W lV3) - h this condition becomes
(5.1)
of h* with
S E h,
(5.2)
where h, denotes the isotropy algebra of p (so h, - o(3)). We should observe that since p’ is three-dimensional every element of A2p is decomposable. This has the following consequence: LEMMA. Let S E A2p1 - h, c h so that Im S c R 1*3is two-dimensional. Then for any K E h, the restriction of SK to Im S is a scalar operator, i.e.,
SKSy = pSy
(5.3)
,u==trSK.
(5.4)
where
ProoJ: We may choose a Lorentz frame e,, e, , e2, e3 of R lV3 such that p=meo
(5.5)
and S=se,
Ae,.
(5.6)
Thus Im S is the two-dimensional subspace spanned by e, and e,. We may write the most general KEA2(R113)-h as K=
‘? O
KijeiAej
INTERACTION
so the restriction
OF SPIN
AND
465
TORSION
of K to Im S is
K,,,e,
/“\ e2 + x
+K,,ei@e,+
x jt
i+l,2
fKjzej@e, k.2
of SK to Im S is
and hence the restriction
sK,,,(e,
(5.7)
A e2>’ = 4L2
Q.E.D.
since e, A e, is rotation through 90” and hence (e, A e2)’ = -1 on Im S.
Using this lemma we can now analyse equations (4.9) and (4.10). Take the interior product of (4.10) with p. The right-hand side gives i(p)(p
A v) = m2v - (p, v) p
m’ = (p, p).
(5.8)
For the left-hand side we make use of the fact that under the identification h*,
and under the identification
of A2(lR’,3)
i(p)A
of h with
with h,
.p
AEh,pER’.3.
(5.10)
Then i(p)(rl
. S> = - [rl, Sl . P
=-(yS-Sy)*p since S . p = 0.
= +S(v * P) Substituting
into (4.9) we obtain m2V - (p, u) p = S(q . p) = -S(i(v)
K) = S(Kv)
or
m2v = (p, v) p + S(Kv). From the right-hand side of (5.3) Substituting this into (5.3) gives
we
see that we can write
m2@p + Sy) = m2@ + m2(Sy, p) p + SK(Lp + Sy) or since (Sy, p) = - (y, Sp) = 0,
m2Sy = LSKp + SKSy.
(5.11) v = Lp + Sy.
466
RAPOPORTAND
STERNBERG
Let us assume that the scalar M does not vanish where M=m2-ftrSK.
(5.12)
Then we can solve this last equation uniquely for Sy: Sy=$SKp and so v=I(p+-&SKp) gives the solution of (5.11). Notice that if u = 0, then (4.9) and (4.10) imply that q * p = 0 and q . S = 0. Hence the corresponding tangent vector r + ~o,~, vanishes. We have thus proved THEOREM. Zf we choose the H suborbits for the massive (spinning) orbits according to condition (5.1) then at all points where the scalar (5.12) does not vanish, the null foliation of wv is one dimensional with horizontal component v = w(t) given by (5.13) and vertical component qP,s then determined by (4.9) and (4.10).
We thus get a one-dimensional foliation-a system of ordinary differential equations-n the nine-dimensional manifold U(P,). The projection of these curves onto M then give the trajectories for the motion of a spinning particle in the presence of torsion. In order to get a feel for these trajectories, we should exhibit some cases where we can actually solve the equations of motion. We shall do that in the next sections, where enough symmetry conditions will be imposed to allow a group theoretical solution to the equations of motion: we shall study the effect of torsion on spinning particles in Minkowski space.
6. VECTOR TORSION ON MINKOWSKI
SPACE
It is a well known theorem of Levi-Civita that every (pseudo) Riemann manifold has a unique connection (preserving the metric) with zero torsion. It is equally true, and follows from the proof of the Levi-Civita theorem, cf. [ 14, p. 3341, that given any prescribed torsion tensor T, there exists a unique connection with torsion, T. In the Cartan theory of gravity of [2, 171, the gravitational field is coded into the torsion. In this section we wish to investigate the equation of motion of Section 1 for certain types of connections on Minkowski space, i’t4, (with its flat metric). The bundle of orthonormal frames over M has a global cross section given by a constant frame and flat Levi-Civita connection. Any other connection on M is thus equivalent to giving an h-valued one form, r, on M. (Here h is the Lie algebra of the Lorentz group, but much of what we have to say is valid with h = o(M) and M any
INTERACTION
OF SPIN
AND
467
TORSION
non-degenerate scalar product-space.) We may identify h with n ‘(IV) = n ‘(IR lt3) z n’(M*). Amongst all possible forms t, we shall consider those of the following type. Let p:M-iM* - M be a vector-valued function on M. We can consider the connection r=r,=p
Adx.
(6.1)
Here A denotes exterior multiplication in ,4(M) and dx is the R ‘*3 = M-valued differential form which identifies TM, with M. To avoid confusion about the exterior multiplication of differential forms, we shall temporarily use the notation x for exterior multiplication in /i(M) and so rewrite (6.1) as (6.1)’
t0 = /3 x dx, that is, for any x EM
and any u E TM, - M (r&=~(x)x~E/i~(M)-h.
If e,, e, , e,, e3 is an orthonormal
(6.2)
frame of M then T
Tijei
Zj
X ej
where
5ij = Pi dxj - pj dui. A connection of this form will be called of “vector
(6.3) type.” The curvature
R(r) = dz - [T A z] becomes
d(Pxdx)-(/I,P)dx?dx+@dx)A@xdx). In this expression the symbol 2 means exterior multiplication of differential forms; i.e.,
(6.4 1
multiplication
in A(M)
and exterior
for the commutator
of two rank
(dx 2 dx, u A v) = u A u. The last two terms of (6.4) come from the formula two elements:
[P(x) A u, P(x) A u] = - Ga,8) u A u + C/W), u) P(x) A u - V(x), u) P(x) A u. We should emphasize that in (6.4), p must be regarded as a vector-valued function and not as a linear differential form. For example, the ij component of the first term in (6.4) is
dpi A dXj-d@j
595/158/2
13
A dxi.
468
RAPOPORT
AND
STERNBERG
If b is a constant vector these all vanish and the curvature simplifies to R(z,)=-Ga,p)dxxdx+d0,dx)AGoxdx). If p is light-like,
(6.5)
this simplifies further as @,/I) = 0. One checks that (P,n=PAp=PXP.
Also ((S,dxXdx),uAu)=(S,uAu) and
((S, Go,dx) A (p x dx)), (u A u)) = @ x SP, u x u). Thus (reverting to the A notation) K=P~~+S~~~-IIP~~~S+(S,~(S,~~~~),)
(6.6)
or, for constant p, (6.7)
K=PA~+SP~P-IIPII~S
and for constant, light-like
p, K=pA/I+@AP.
(6.8)
Now for our choice of S E A*p we have trS(pA/3)=0 so tr KS is independent ofp and is an invariant biquadratic has
function of S and /3. One
~~SO(S/~AP)=~(IS/~([~
(6.9)
as can be checked by taking S = se, A e2, for example, and hence tr SK = 2 jlS/3II’ - II/31j2tr S2
(6.10)
and hence, for /3 light-like, tr SK = 2 )IS/311’.
(6.11)
We have (for constant j? and S E /i ‘p’) KP = (B, P)(P + SP) - m*P
(6.12)
INTERACTION
OF SPIN
AND
469
TORSION
and hence (6.13)
SKp = Q?, p) S2p - m2Sp.
The equations (5.13) (when L = 1), (4.9) and (4.10) thus become v = P + ,2 _
1 ,y$,,’
Lu-4 P) S2P -
m’W
(6.14)
v*P=[(PAD)+SPA\lU
(6.15)
q.S=pAv.
(6.16)
These equations are all written relative to the connection r4. We wish to write these equations in terms of the trivialization (and trivial connection) given by a fixed (globally constant) frame of M. This meansthat in (6.15) and (6.16) we must replace rl by q-J(v)=v-v
A/3.
Writing dx/dt for the velocity vector in the flat connection and similarly dp/dt we get
and
dS/dt
dx -&=P+ z
1 m2 - IIW’
[CA P) S’B - m2SPl
= (A A p> v - (v A p) p
(6.17) (6.18)
A=p+V
(6.19)
dS -=AAv+SvAa. dt
(6.20
Applying (6.20) to p and combining with (6.18) shows that dA -z dt
0.
(6.21
In other words A is an invariant of the flow. The invariant vector A should be interpreted as the total momentum of the system. In fact, let G, be the subgroup of the Poincare group which preserves /I so that G, is the semidirect product
470
RAPOPORT
AND
STERNBERG
where HB is the subgroup of the Lorentz group which preserves p: H, = (L 1LB = j?}. The group acts on our bundle by (L, u)(x, (p, S)) = (Lx + u, (LP, LSL -‘))
(6.22)
and this action clearly preserves the presymplectic form. It is easy to check that this action is in fact (pre)-Hamiltonian and the corresponding moment map, @, is given by qx, (P, S)) = (Q
+ Ax), A)
(6.23)
where 71:h* + h,* is the projection dual to the injection of the Lie algebra h, into h. Thus A is the component of the moment map dual to translation and hence represents the (conserved) total linear momentum. (Here “total” means particle plus field.) Now (4 P) = (PP PI since (Sp, /I) = 0 and 64 W = IIWII’
= IIA /I2 -m*
since Sp=O. Thus (p,/?) and I/S/III” are invariants of the motion. But these two functions are also invariant under the action of G, as can be checked directly from (6.22), and these functions are clearly independent. Then the curves of our null foliation all lie in the seven-dimension submanifolds Na,* of the form (P,P)=a>o IlSPll’ =-b*
which are invariant under G,. Now Sfi is a space-like vector since S E A2p’. So llS/3ll= 0 implies that Sp = 0 and hence r.~= p and equations (6.18) and (6.20) become dp z=
o
dS dt=
o
and we get trivial rectilinear motion with no change in p or S. Thus we may restrict attention to the case where I/ ,!$I (1’ < 0. We claim that G, acts freely and transitively on hb- Indeed, the translations act freely and transitively on the x. The group H, acts transitively on the space of p satisfying 11pII 2 = mZ and (p, p) = 0. The subgroup of Ho fixing one such p consists of all rotations in the space-like plane perpendicular to p and /3, But S/3 is a non-zero vector in this plane so the element of the subgroup is determined by its action on S/3. Now our one-dimensional foliation on N0,6 is invariant under G,. If we identify Na,b with G, (by picking a point in Na,J we obtain a left invariant one-dimensional foliation on G,, which must therefore be the orbits of
INTERACTION
OF SPIN
AND
TORSION
471
a one parameter group acting on the right. Translating this back to N,,, we see that the curves of our null foliation must be orbits of one parameter groups of G,. The one parameter group will depend on the initial conditions. To find it, we must simply find the element (B, U) of g, , the Lie algebra of G,, which gives the tangent vector to the trajectory at the initial condition. Without loss of generality, we may assume that x = 0 and then u is determined to be u. We can determine B up to a scalar by the fact that BA = 0 because B must preserve the plane /I = 0 and A = 0. For example, we can, without loss of generality, asstime that we have chosen coordinates so that p=me,
and
P = + (e, - e, ).
Then
S = s1.2elA e2+ s,,,e, A e3+ s2.3e2A e3 and
SO
llwII’ =-f
(d.3 + $,3).
The most general B E h, can be written as B = b,Pe, + b,Pe, + b,,,e, A e, and the condition
gives B = s,T,jle, - s,.,Pe, + me, A e,.
7. MOTION IN THE PRESENCE OF A VECTOR TORSION WAVE
The explicit integration of Section 6 depended on the fact that we had two invariants and a seven dimensional group of symmetries. So, in effect, we had a seven-dimensional group acting transitively and freely on seven-dimensional manifolds. If we had a six-dimensional group and seven-dimensional invariant manifolds we could still say something about the integration procedure. Indeed, let us focus on such a manifold, and assume that coordinates (u, y), u = (u, ,..., us) have been chosen so that the group of symmetries acts transitively on each level surface
472
RAPOPORTAND
STERNBERG
y = const. Then if { =f(~, y) a/ay + g(a/&) is invariant under the group, we can conclude that af/au = 0, so f = f(y) and that, for each fixed y, g(a/au) comes from the Lie algebra. Then y(t) could be found by an integration, and the solution of the equations reduce to solving a system of time dependent differential equations on the group-a phenomenon annular to the nutation effect seen in collective motion [9]. As an example of this situation we consider the case of a vector connection as in the preceding section-so (6.1) holds, but where now p = de” = e* d#;
(7.1)
I.e., jl,=ebf$
j =*o, 1,2,3.
(7.1’)
J
Here we will take 4 = il where 1 is a linear function; for example tp = i(xo - x,).
(7.2)
Then, in (6.4) (j3, dx) = d#. Then, if IId$)l’ = 0 (I is light-like)
(7.3)
then the curvature becomes (from (6.4))
R(r) = e@(l - em) d# A (d# x dx)
(7.4)
and K = (S, R) + (p, T) = e*(l -e”)
S d# A d4 + emp A d$.
(7.5)
We then have tr(SK)=2(1-e@/2)jSd#I(2
(7.6)
M=m2-11-em121JSd~(12.
(7.7)
so, in (5.12),
Then also SKp = e”(1 - e”)(dq4 p) S2 d$ - m’e@S d#
(7.8)
and so, from (5.13), 1 u=P+m2-~l-e~~211Sd~~)2
em[(l - e@)(d& p) S2 d# - m2S d#].
(7.9)
INTERACTION
OF
SPIN
AND
TORSION
473
The equations (4.9) and (4.10) then become rj . p = [e@p d# + e@(l -e@) S d@ A d$] u
(7.10)
?l*S=pAv.
(7.11)
These are written in terms trivialization these become dx z=P+
m2-/l
of the connection
1 -e”/*)1Sd#))*
r = tD. In terms
e@[(l -e@)(d#,p)S*d#-m*Sd#]
of the global
(7.12)
and (7.13)
If we write (7.14)
A=p+(l-em)Sd4 then (7.11) becomes dp -&A
(7.15)
A&v-(vAP)p.
Then (7.11) becomes dS -=AAv++vA/?. dt Applying
(7.16)
/I to (7.16) and combining
with (7.15) we obtain that dA dt=
0
(7.17)
so that A is an invariant vector field of the flow, which we interpret as the total momentum of the system. We consider now G,, the subgroup of the Poincare group P which preserves 4 {a E P/a*qqw)
= $4(w) = I
= $h(w), w EM).
We note that if A E G,, it then preserves d# by the induced action, and in consequence preserves j3. In fact, G, is the semi-direct sum Gm=Hm@R;3
474
RAPOPORT
AND
STERNBERG
where H, and lRiq3 are the subgroups of the Lorentz and translation preserve 4. The group acts on the nine-dimensional bundle U(P,,) by
groups which (7.18)
(L u>(x, (PI S)) = (Lx + u, (LP, LSL - ‘>I.
This action preserves the presymplectic form. It is easy to check that this action is in fact (pre)-Hamiltonian and the corresponding moment map, 4, is given by @(x, (P, s)) = ($S + A A xl, A) where z: h* + hz is the projection dual to the injection of h, into h. Hence, A is the component of the moment map dual to translation and hence represents the conserved total linear momentum. Now
since (Sd#, d4) = 0 and since Sp = 0 we have (1 -e-@)e-@(S/3,A)=
[l -e-m](Sd#,A)=/l
-eem)2JJSd~J~2=~JA~~2-m2~
Thus, (d#,p) and (1 -e@\*(SdQ)(* are invariants of the motion, which are clearly independent functions. They are further invariants by the action of G,. Thus we can consider the six-dimensional group G, acting on the seven-dimensional submanifolds Naqb given by
i(p, d#>= a
(7.19)
11-em12[ISd#112=-b2. We can then integrate by the procedure mentioned in the beginning of this section.
ACKNOWLEDGMENTS The first author would Mathematics Department
like to express his gratitude to Miguel Filli for several discussions of Harvard University for its warm hospitality during his visit.
and to the
Note added in proof. This paper was written in ignorance of some of the important earlier literature on our subject. The authors are indebted to Professor F. Hehl for calling our attention to important relevant references, and to Dr. Baekler and Professor Hehl for interesting discussions relating our work to known results. The idea of using a spinning test particle to measure the torsion of a connection occurs in the paper of Hehl [How does one measure torsion of space-time? Phys. Left. A 36 (1971) 2251. It also occurs in the work of Trautman [Bull. Pal. Sci. Ser. Math. Astr. Phys. 20 (1972), 895j. Our equations (4.9) and (4.10) seem very close to the equations obtained by Yasskin in his Ph.D. thesis (Univ. of Maryland, unpublished). A more recent investigation of this subject can be found in the paper of Yasskin and Stoeger [Phys. Rev. D 21 (1980), 20811. A thorough discussion of the whole question of Poincare gauge fields and Cartan connections can be found in the lectures of Hehl [“Proceedings of the
INTERACTION
OF SPIN
AND
TORSION
415
6th Course of the International School of Cosmology and Gravitation on Spin, Torsion, Rotation and Supergravity, Erice” (Bergmann and desabbata, Eds.), Plenum, New York], in which the reader can find a comprehensive bibliography. We should point out that in much of the literature equations of motion are derived at the field theoretic level from Noether’s theorem, usually with some additional ansatze. Particle motion is then derived as a limiting case in the standard manner. Our equations (4.9) and (4.10) did not involve any special assumptions but came from the general abstract principle enunciated in ( 16 I. They can also be \ 101) to the set of all Cartan derived by applying the method of general covariance (cf. (81 and/or connections taken as the geometrical objects. We shall present the details of this derivation in a forthcoming paper.
REFERENCES I. E. S. ABERS AND B. W. LEE, Phys. Rep. 9 (1973), I. 2. E. CARTAN AND A. EINSTEIN. “Lettres sur le parallelisme absolu, The Einstein-Cartan Correspondence.” (R. Deberer, Ed.). Princeton Univ. Press and The Royal Academy of Sciences of Belgium. Princeton, N.J., 1979. 3. C. DUVAL AND P. HORVATHY, Ann. Phys. (NY) 142 (1982). 10-33. 4. C. EHRESMANN, Les connexions infinitisimals dans un fibre diffirentiable, Colloque de topologie (Brussels. 1950). 5. S. KOBAYASHI, Can. J. M&h. 8 (1956), 145-156. 6. S. KOBAYASHI, Ann. Muf. Pura Appl. 43. No. (4) (1951), 119-194. 7. S. KOBAYASHI AND K. NOMIZU, ‘*Foundations of Differential Geometry.” Vol. I. Wiley, Interscience, New York, 1963. 8. V. GUILLEMIN AND S. STERNBERG, Hadronic J. 1 (1978), 1-323. 9. V. GUILLEMIN AND S. STERNBERG, Ann. Phvs. (NY) 127 (1980). 220-253. 10. V. GU~LLEMIN AND S. STERNBERG, “Symplectic Techniques in Physics,” Cambridge Univ. Press. London/New York, 1984. Il. D. RAPOPORT AND S. STERNBERG, Classical and prequantized mechanics without Lagrangians or Hamiltonians. The Institute for Advanced Studies, Tel Aviv University, preprint No. 808-33, 1983. 12. J. M. SOURIAU. “Structure des Systkmes Dynamiques (Maitrises des Mathtmatiques).” Dunod. Paris, 1970. 13. S. STERNBERG, Proc. Nat. Acad. Sci. U.S.A. 14, No. 12, 5253-5254. 14. S. STERNBERG, “Lectures in Differential Geometry,” Prentice-Hall, Englewood Cliffs, N.J., 1964. 15. S. STERNBERG, Trans. Amer. Math. Sot. 212 (1975). 113-130. 16. S. STERNBERG AND T. UNGAR, Hadronic J. 1 (1978), 33-36. 17. M. TSAMPARLIS, Ph.w. Rev. D 24, No. 6 (1981). 14-5 I.
OF PHYSICS
158, 447-475
(1984)
On the Interaction
of Spin and Torsion RAPOPORT
DIEGO
Department
of Mathematics.
Tel Aviv
SHLOMO Department
of Mathematics,
Cambridge,
*
Vniversitwy,
Tel Aviv.
Israel
STERNBERG Harvard University, Massachusetts 02138
Received
March
Oxford
Street,
3. 1984
The geometry of Cartan connections and their relation to the theory of spontaneous symmetry breaking is discussed. In S. Sternberg and T. Ungar [Hadronic J. 1 (1978), 33-361 Cartan connections are used to derive equations of motion for classical particles in the presence of external fields. Examined here is the case of a massive chargeless particle whose spin interacts with the curvature and torsion of a gravitational field. These equations are solved for the case of a constant vector torsion on Minkowski space, and the solution for a vector torsion wave is indicated. 0 1984 Academic Preaa, Inc.
In the standard theory of “minimal coupling” the effect of the electromagnetic field, A, is to modify the Hamiltonian, H, by replacing p by p - eA in H. The choice of H is dictated by other-usually gravitational-forces. Thus, for a charged particle in the presence of a given electromagnetic and gravitational field, the function H is given by H(q, P> = t II P II* = f 1 gtj(q) P’P’ where g, is a given Lorentzian
metric; we replace H by
to obtain the Hamiltonian giving the equations of motion. A similar modification of H describes the equation for a classical particle in the presence of a more general Yang-Mills field (cf. [8] or [ 131). From a conceptual point of view this is somewhat unsatisfactory, in that this approach places the inertial and gravitational forces on a different footing from all the other forces: The gravitational field determines the Hamiltonian, H, while the Yang-Mills field modities H. If we believe in a unified theory of all forces, then we should expect to treat all forces equally. In [ 161 a scheme was presented for obtaining equations of motion for classical particles in the * Current A.P. 55-534.
address: Departamento 09340 Mexico, D.F.
de Matematica,
Universidad
Autonoma
Metropolitana,
Iztapalapa,
447 0003.4916/84
$7.50
Copyright @I 1984 by Academic Press. Inc. All rights of reproduction in any form reserved.
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presence of an external field. This scheme involves the theory of connections and symplectic geometry but does not involve the notion of Hamiltonian or Lagrangian. It treats gravitational and all other forces the same way. We give a rapid outline of the ideas of [ 161. The details will be presented in the first few sections. Let M be a differentiable manifold representing the possible configurations of a classical mechanical system. The basic problem of classical mechanics is to give a description of the possible curves C(f) on M which can arise as actual changes in the configuration. In contrast to Aristotelian physics, classical mechanics says that we need to know more than the actual configuration of a system to determine its future evolution. Thus, in Newton’s laws of motion, one needs to know the momenta and positions in order to predict the future positions-the laws of Newtonian mechanics are given by second order differential equations. For a charged particle in the presence of an electromagnetic field, one must specify an additional “internal variable”-the charge-in order to get the equations of motion. Thus the situation is as follows: one has a second differentiable manifold, E, (the evolution space, to use the terminology of Souriau [12]), together with a smooth map, rr: E +M. One assumes that the map II makes E into a fiber space over M, i.e., that the Jacobian, dx, is everywhere surjective. On E we expect to have a (one-dimensional) foliation-i.e., a system of first order differential equations. The solution curves of this differential system then project onto curves on M, describing the possible evolution of the configuration. Thus the formulation of the laws of mechanics comes down to a description of E and x and the differential system on E. Since the time of Lagrange and Hamilton, it has been realized that the differential equations on E take a special form: one assumes that there is a closed two form, w, on E and that the differential equations are of the form i(r) w = 0, the “null foliation” of cu. This system is always integrable. Of course it may not be one-dimensional. If o is non-degenerate, then < = O-so there are no curves. If w is highly degenerate then the dimension of the null foliation can be large. (This can occur for actual systems. The non-localizability of a classical photon in special relativity is an example, cf. [ 161.) In many interesting cases the dimension of the null foliation is exactly one, and so we get curves. The problem thus becomes: describe E, a, and the closed two form cc).In [ 161 a procedure was devised to give a general description. It involved two mathematical objects-the notion of a connection on a principal bundle with reduction, and symplectic homogeneous spaces. The idea of considering, simultaneously, a principal bundle together with a reduction first appeared in the work of Elie Cat-tan. It is now an essential ingredient in the notion of “spontaneous symmetry breaking” for Yang-Mills fields. Let us give a brief description of Cartan’s point of view. In the classical theory of connections there were two approaches. One, associated primarily with Levi-Civita, stressed
INTERACTION
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449
the role of the tangent bundle as a vector bundle-that is, as a collection of vector spaces ranging smoothly from point to point. A connection was then a means of parallel transport along curves. This notion of parallel transport then gives rise to the concept of covariant derivative, which is so familiar in tensor analysis and general relativity. The principal bundle connected with this approach is the bundle of orthonormal frames, that is, the bundle of all orthonormal bases of all tangent spaces (the set of vierbeine in general relativity). The structure group of this bundle is the orthogonal (or Lorentz) ‘group H, which is a subgroup of Gl(d), where d is the dimension of the base manifold (4 in general relativity). In the classical theory of surfaces, there was another notion intimately related to parallel transport, and that is the development of a surface on a plane along a curve. Intuitively speaking. one “rolled” the surface on the plane along the curve, maintaining first-order contact. This gives an identification of the tangent space to the surface at each point of the curve with the plane. Then parallel translation in the Euclidean geometry of the plane gives the parallel transport in the sense of Levi-Civita along the curve. From this point of view the notion of development is central, and the crucial property of the plane is that it is a homogeneous space in the sense that it admits a transitive group of isometries that includes the full rotation group as the isotropy group of a point. The plane is E(2)/0(2), w h ere E(2) denotes the group of Euclidean motions of the plane. From this point of view, we might also want to study the development of a surface onto a sphere along a curve, where now the sphere is 0(3)/O(2). or the development onto hyperbolic space S/(2, R)/S0(2). This is the starting point for Cartan’s approach. We must not only be given the structure group H, but also a larger group G containing H as a closed subgroup so that G/H is a homogeneous “model” space with which we would like to compare our manifold M. Thus, in general relativity, where H is the Lorentz group, we would take G to be the Poincari group if we were comparing space-time with Minkowski space and G to be the de Sitter group 0( 1,4) if we were comparing space-time with de Sitter space. In the theory of spontaneous symmetry breaking we are also given a pair of principal bundles. For example, in the Weinberg-Salam model we would have the group G = SU(2) X U(1) and H = U(1). In this case, the quotient space G/H = SU(2) - S3 need not be related in any way to the base. In this more general context, it was shown in [ 161 how a connection-a YangMills field-on P, together with a symplectic G-orbit and an H suborbit gives rise to an E, 7~,and w. We shall review this construction in Section 4. If the space G/H does have the same dimension as M, then we can demand more from the connection form on P,. We can demand that its restriction to P, have no kernel. This has the effect of identifying the tangent space to M, at any point, with the tangent space to G/H --giving a precise mathematical formulation to the intuitive idea of “rolling” M along G/H. This was Cartan’s idea of “soldering.” Although Cartan wrote his fundamental papers in the 1920s the precise modern definition of a Cartan connection was first given by Ehresmann [4] in 1950. In sections 1 and 2 we will give an exposition of this notion, following the treatment of Kobayashi [6]. (In Section 3 we describe, as an aside, the relation of the ideas of Section 1 to spontaneous
450
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symmetry breaking.) If the groups G and H are the affine and orthogonal (or general linear) groups respectively, then the Cartan connection is called an afline connection. The torsion of this afftne connection then enters in a natural way into the equations of motion. We describe this in Section 4. In particular, for general relativity, we obtain an interaction between spin and torsion. We describe this interaction in Section 5. For certain special cases, we solve these equations of motion in Sections 6 and 7.
1, BUNDLES,REDUCTION,
AND SOLDERING
We begin by recalling some basic definitions and facts in order to establish notation. Let G be a Lie group and A4 a differentiable manifold. A principal G bundle over M is a manifold P on which G acts freely (on the right) and such that the quotient of this G action is M. Thus we have a smooth map n: P + M and K’(X) is a G orbit for each x E M. One also assumes that P is locally trivial in that about each x there is a neighbourhood, U, such that a-‘(U) is isomorphic to U x G (with the We shall let R,: P-, P denote right obvious definition of isomorphism). multiplication by aPi R,(P) = pa-‘,
~EP,uEG
so that R, gives a left action of G on P
R,,=R,R,. Let F be some differentiable manifold on which G acts on the left. We can then form the quotient of the product space P X F by the G action; call it F(P).
Let p: P x F -+ F(P) denote passage to the quotient. Then F(P) is also flbered over M by
%@(P,f))=
Z(P)*
F(P) is called the associated bundle (to the G action on F and the principal bundle PI
Let f: P + F be a smooth function satisfying
f(pa)=a-'f(P).
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TORSION
Then
U(P,f(P)) = (Pa- l, Uf(P>) = tP~-l,ftP~-‘)). Hence p(p, f(p)) is independent of the choice of p E n-‘(x). a section, s, of F(P); i.e., a map b xos=id s: M -+ F(p)
In other words,fdefines
?r(p) =x.
s(x) = P(Pt f(P))
Conversely, given a section s we may define the functionfby and f satisfies f(pu) = a - ‘f(p). Thus
the preceding equations
we may identify the space of sections of F(P) with the space of maps f: P --) F sutisjjyingy(pu) = a-‘f(p).
(1.1)
There are two especialy important cases of this construction: Suppose that F = G/H where H is a closed subgroup of G. So F(P) is a bundle of homogeneous spaces. Letf: P + F satisfy the identity of (1.1) and so be equivalent to a section, s, of F(P). Consider f-‘(H)={pEPIf(p)=HEG/H}.
IfpEf-‘(H)thenf(pu)=a-‘H=HifandonlyifuEH.Thus f -‘(H)=PH
is an H sub-bundle of P, a reduction of the principal G-bundle to an H bundle. Conversely, suppose that PH is an H sub-bundle of P. Then define f: P -+ G/H by f (PH) = H and if P = qa,
4 E PfI
then
f(q) =a-‘H. This is well defined as can easily be checked and defines a function f satisfying the condition of (I. 1). Thus a section of the bundle (G/H)(P) an H bundle, PH.
A second important 595/158/2-12
is the same us a reduction of P to (1.2)
case is where F is a vector space and the action of G on F is
452
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linear. Then F(P) is a vector bundle. In this case we can consider k-forms, L!, on P in the values in F. We can consider forms which are horizontal in the sense that i(r) J2 = 0 for any vertical tangent vector <
(1.3)
where “vertical” means tangent to the fiber. We can also consider forms which are equivariant in the sense that R,*R=aR
for all
a E G. ’
(1.4)
It is easy to check that (1.1) generalizes to an F-valued k-form on P satisfying (1.3) and (1.4) is equivalent to an F(P) valued k-form on M.
(l-5)
(The case k = 0 of (1.5) is then (l.l).) We can combine the preceding two examples. Suppose that we are given a section s of (G/H)(P), so we get a reduced bundle, Pn. We can consider the vector bundle associated to the adjoint action of H on g/h. This vector bundle can be identified with the bundle of vertical tangent vectors to (G/H)(P) along the section s. Indeed, we can consider (G/H)(P) as the bundle associated to Pn relative to the H action on G/H. On the principal bundle Pn the section s corresponds to the identically constant function, f E H. At the point H we have an identification of T(G/H) with g/h. This identification is consistent with the H action. Thus we may identify [ g/h](P,) with the bundle of vertical tangent vectors to (G/H)(P) along s. Now suppose that w: TP, + g/h is a one form which satisfies (l&3) and (1.4) relative to the group H. Then w can be thought of as a one form on M with values in [ g/h](P,). Thus Let s be a section of (G/H)(P) and PH the corresponding reduced bundle. Let o be a one form on Pn with values in g/h which satisfies (1.3) and (1.4) relative to the group H. Then w can be regarded as a one form on M with values in the bundle of vertical tangent vectors to (G/H)(P) along s.
(l-6)
In the particular case that dim G/H = dim M, we can further demand that the one form w on M give an isomorphism (at all points) between TM and the bundle of vertical tangent vectors. This is the situation considered by Cartan. The form o is called a “soldering form” in this case. For example, suppose that H = O(V) is the orthogonal group of a vector space with non-degenerate scalar product and G = H @ Y is the corresponding group of affine transformations, so G is the semidirect product of H and V. Then G/H = V. A soldering form o then gives an identification of TM with V(Pn). In particular, this puts a (pseudo) Riemann metric on M and also allows us to identify Pn with the bundle of orthogonal frames. Similarly, if we take H = GL(V) then a soldering form o allows us to identify Pt, with the bundle of all frames on M. Conversely, let Pn denote the bundle of frames of a differentiable manifold M,
INTERACTION
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453
where H = GI(R”), n = dim M. Then P, carries a canonical one form w with values in R”, and w satisfies (1.3) and (1.4). Namely o is the so-called “structure form” defined by
t E TPN,
w,(r) = P(d7c,4
p: TIW,~ + R”.
PEPIf,
Of course, we can enlarge the bundle P, to a G-bundle P, = G(P,) and then we are back in the situation described above.
2. CONNECTIONS
AND CARTAN
CONNECTIONS
Let P, be a vector bundle with structure group G. For each r E g, the Lie algebra of G, let [ denote the corresponding vector field on P, given by the right action of G on P,. Recall that a connection on P, can be described as a g-valued one form 8, on P, which satisfies it0 8, = r
5Eg
(2.1 )
R,*O,=Ad,O,
a E G.
(2.2)
and
The horizontal space of a connection 0, at a point p E P, consists of all tangent vectors at p which are annihilated by t9,, i.e., those which satisfy icv)
8, = 0.
(2.3)
Any curve on M lifts to a unique horizontal curve on P, (one whose tangents are everywhere horizontal) once a lift at one point has been specified. This is the notion of parallel transport along a curve. Now suppose that we are in the situation of the preceding section, so we have a reduction of P, to an H bundle, PH, and we are given a connection, 8, on P,. Then the restriction of 8, to P, defines a g-valued one form on P, which satisfies (2.1) with r E h and (2.2) with a E H. As h is an invariant subspace of g under the adjoint action of H on g, we can define the form w = (restriction of 8, to P,)/h as a g/h valued one form on PH satisfying the conditions of the preceding section. If the group H is reductive-r more generally, if h has an H invariant complement, n in g-we can decompose g=h@n
eW’,, =eH+w
454
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where we have identified n with g/h. Here 0, is an h-valued form and o is an n-valued form. The form 0, satisfies all the conditions for a connection on PH. Thus If the group H is reductive then the restriction of 8, to PH determines an n-valued one form w on PH and a connection 0, on PH. The one form w satisfies (1.3) and (1.4) relative to H.
(2.4)
It is important to note that the horizontal subspaces for the connections 0, and 0, will differ, in general, at points of PH. However, given t9, and w, one can reconstruct BG along PH and hence on all of P,. Thus the data e,, w on PH is equivalent to a connection eG. In particular, if dim G/H = dimM we can consider the condition that w be a soldering form. If this holds, then Bc is known as a Cartan connection. If G is the affine group then a Cartan connection is called an afine connection, cf. [7]. Let F be a vector space on which G acts. We will let Ak(F) denote the space of F-valued R forms, satisfying (1.3) and (1.4). So Ak(F) can be regarded as the space of k forms on M with values in F(P). In particular, we can identify A’(F) with the space of sections of F(P), a space which we shall also denote by r(F). A connection t9 on P defines a covariant derivative d,:Ak(F)*Ak+‘(F) d,Q=dQ-8.0. One also defines the curvature of a connection by curv(f3) = de - j [e, e]. For the case of a reduced bundle PH with a reductive group H, the restriction curt@,) to PH is given by de,+dw-+[e,+w,e,+w]=de,-t[e,,e,]+dw-e,. = curv(8,) + dg,w - 4 [w,
of
W-~[w,w] w].
In particular, for the case of an affine connection [n, n] = 0 so the last term disappears. The term dg,w is known as the torsion of the linear connection 0,. In the Levi-Civita theory of connections the torsion entered as a more or less formal object. It was given, in local coordinates, as the antisymmetric components of the Christoffel symbols. But in the Cartan theory the torsion enters as a component of the curvature. Its geometric meaning is clear: it gives the translational component of parallel transport around an infinitesimal closed curve.
INTERACTION
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3. SPONTANEOUS
SYMMETRY
455
BREAKING
In this section we show how the notion of a reduction of a principal bundle arises in the theory of “spontaneous symmetry breaking” (cf. [ 11). Let P, -+ M be a principal bundle with structure group G. There are various geometrical objects that we have associated with P: For a vector space F which is a representation space of G, we have considered the various spaces Ak(F). We shall denote the sum of all these spaces-and for various possible P’s simply A *(PC). We can also consider the space of all connections on P, which we shall denote by Conn(P,). Suppose that M has a (pseudo) Riemann metric and that g is given a G invariant scalar product as is F. Also suppose that we are given some G invariant function V on F. We can then consider various scalar valued functions built out of the geometric objects such as I/42
o E Ak(F)
Ild,42
w E A k(F), 8 E Conn(PG)
IIcud@) II2
0 E Conn(P,)
V(f)
f E A’(F).
If we have a projection of P onto the bundle of orthonormal frames of M, so that T(M) and T*(M) are associated bundles to P,, then we might also have expressions such as
where W is some G invariant polynomial. All of these are G invariant functions on P, and hence define functions on M. Each of the above expressions thus defines a mm L,,L,,
etc., from geometrical
objects to functions on M
L,: Conn(P,)
x A*(P,)-+
Cm(M)
If we then integrate these expressions, or some (linear) combination, L, of them over form we obtain a functional on the space of geometric objects (at least on the set where this integral converges). The study of this functional
M relative to the volume
F=l’
M
Ldvol
is the central object of study in Yang-Mills theories. For example, describing the critical sets of this function is very important. Now F admits a rather large group of symmetries. Let Aut(P,) denote the group of all diffeomorphisms of P, which commute with G and Gau(P,) c Aut(P,) the subgroup consisting of those which project onto the identity on M. Then each of the “Lagrangians” Li and hence the function F is
456
RAPOPORT
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invariant under the action of Gau(P,). The idea of “spontaneous symmetry breaking” is to provide a local partial cross-section for the action of Gau(P,) and hence remove some of the redundancy in F. For this we first recall an alternative description of the group Gau(P,): Let G act on itself by conjugation. Then G(P,) is an associated bundle, each fiber of which is a group. Then pointwise multiplication makes P(G(P,)) into a group. This group can be identified with Gau(P,) as follows: An element of Z(G(P,)) can be viewed as a function r: PC + G satisfying
Then the transformation
z(pb) = b -‘z(p) b.
(3.1)
d,(P) = PO>
(3.2)
#=
satisfies
#A@) = @Ob) = M- ‘0) b = VW) b and tir is an element of Gau(P,). Conversely, given #, we can define r by (3.2) and check that it satisfies (3.1). It is immediate from (3.2) that #,,,, = $,, 0 $,, and so we may identify Gau(P,) with T(G(P,)). In what follows it will be convenient to consider several vector spaces U, R, L,... on which G acts, instead of a single F, and single out one of them, U, for special attention which we shall call the “Higgs” space. An element fE Z(U) (that is, a section of the associated bundle U(P,)) is called a “Higgs field.” We assume the following about the action of G on U: That outside of a small G invariant singular set S in U there exists a cross-section, C, to the G action; that is, we assume that U- S = C x (G/K) as G spaces, where K is the (common) isotropy group of all points of C. For example, let G = W(2) and U= C2. Then we can take S = {0} and C to be the set of all vectors of the form ( 9) with r real and positive. Here K = {e}. Let T+(U) c P(v) denote the set of all f which do not intersect S, i.e., f(PG) c U- S. Any such f determines (by projection onto the second factor) a G equivariant map, j P, + G/K. But then
$-l(K) =f-l(C) is a principal K subbundle of P,, i.e., a reduction of P, to a subbundle with structure group K. The group Gau(P,) acts naturally on all geometric objects. In particular, it acts on the space of all sections, T(G/K). We will assume that this space is nonempty and that Gau(P,) acts transitively on it. (For example, in the case that K = e the action is always transitive. Indeed, let& and f2 : P, + G be two functions satisfying
AW) =b-‘f,(p)
f,W) = b -‘h(p)
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457
Then
satisfies t(pb) = b-%(p) and hence defines an element of Gau(P,). f,@(P))
b
Then
=f2(P)f;‘(P)f,oJ)
=fAP)
so 5 carriesf, into f2.) Under our transitivity assumption, all the subbundles f ‘(K) are conjugate under Gau(P,) and thus, in particular, define the same abstract principal K bundle which we shall denote by P,. The choice of particular f then determines an embedding. x P,-f-+P, as a K subbundle. By construction, the G/K component off 0 Tis identically K, so f 0 3= (9, K) where 4: PK -+ C and K acts trivially on C. So 0 can be thought of as a function from M to C. It is clear that we can recover f from 7 and 4. Thus giving f E P+ (U) is the same as giving 7 and 4. NOW let V be any other space on which G acts. By restriction, K also acts on I’ and hence f defines a map, p*, pullback, from V-valued forms on P, to V-valued forms on P,. It is easy to check that (3.3) Notice that for any o E A k(V(P,))
we have
l13*42(d= Il4’(3W) since w is basic. Let us assume that 1 is a K invariant
complement
(3.4)
in g to k so
g=k+l is a K invariant decomposition of g as a vector space. (It is always possible to choose such an 1 if K is compact, for example.) Let t9E Conn(P,). Then we can write
3*e
=
(jl*@k
+
(3*@,.
As we have seen in the preceding section, the first term on the right is an element of Conn(P,) and the second is an element of A ‘(l(P,)). We thus have defined a map Conn(P,)
f’
Conn(P,)
X A ’ (Z(P,)).
458
RAPOPORT
It is clear that at p =f(q)
AND
STERNBERG
we have
curve(P)= cW3*%(d + P~~e,,(3*44q)- f [3*& 3*41(q). Thus we can write down a geometric function L’: Conn(P,) that
(3.5)
x
A ‘(I@‘,)) + F(M) such
a map
(3.6)
L’ 03*0 = )Icurv ell*. In fact, it is clear that we have proved the following: The assignment
r+(U)
X Conn(P,) XA’(l(P,)
X
A *(R)
x/l*@)
f X
+$induces
... z*
P(M,
C)
X
Conn(P,)
x *-a.
For any geometrical function L on the left hand side there exists a geometrical function L’ on the right hand side such that
L’of*=L. In this way, we have reduced the Gau(P,) redundancy inherent in L to a Gau(P,) redundancy in L’. This is known as “spontaneous symmetry breaking” (cf. [ 11). Let us now explain the idea of “acquisition of mass” under spontaneous symmetry breaking. For this purpose, let us examine the function Conn(P,)
X A”(U) + P(M)
given by
Vtf) --)Ild,fll*. Along the sub-bundle f - ‘(C) we have
f= ($9K) so df = d#: TM+
TC
and where 0, denotes the G orbit through c in C. We may identify 1 with the tangent space T(0,). Suppose the cross section has been chosen so that TC, and TO,(,, are always perpendicular. Then
II&f II’W = ll4ll’(g> + PA:(q)
(3.7)
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TORSION
where /( IIc denotes the metric on I induced by the identification I-
TO,
Let us assume that k and 1 are perpendicular substituting into (3.5) we see that
for the scalar product
on g. Then
Il.curv412= I143*~hl12+ I143*~M2 + terms cubic or higher in (3*0), Let us combine (3.8) with (3.7). Consider
and (3*@,.
a Lagrangian
(3.8)
L of the form
L = /Icurv 811* + a Ild,fl/2. where a is some constant. Then we can write L = L ’ 0 3* where
L~=lI43*~Ml* +a li~,ll:,+iI~~ll~+~l~~3*~~,i~~ + higher order terms. (Here #,, is some constant value of 4 usually determined as the minimum of some function.) In L’, the quadratic terms in 8, look like the Lagrangian for the KleinGordon equation, with a [0,l& giving the “mass terms.” This is the procedure of “mass acquisition” in spontaneous symmetry breaking (cf. [ 11).
4. THE PRESYMPLECTIC STRUCTURE ASSOCIATED TO A CARTAN PAIR In this section we review the construction of [ 161. Let P, be a principal bundle with structure group G. Let Q be a presymplectic manifold, together with a Hamiltonian action of G on Q. By this we mean the following: (i)
There is a two form R on Q satisfying
(ii)
There is a group action of G on Q preserving a*Q=f2
for all
dJ2 = 0. R, so
a E G.
Let < be an element of the Lie algebra, g, of G, and let to denote the corresponding vector field on Q. Condition (ii) implies that D,,R where D denotes the Lie derivative.
= 0
From the general formula
460
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we see that di(&) 0 = 0. Locally, this is the same as arguing that
@Q)a = d& for some function fs on Q. We strengthen (ii) in a number of ways. First of all we require that fr exist globally, second that it depend linearly on <. Thus we demand that there exists a map @:Q+
g*
with i
(4.1)
Here g* denotes the dual space of g and ( , ) denotes the pairing of g* with g. The group G acts via the adjoint representation on g, hence by the contragredient to this representation on g*. Our third requirement is that the map @ be equivariant. Thus we require: (iii) There exist an equivariant map @: Q --)g* such that (4.1) holds. Conditions (i), (ii), and (iii) constitute what we mean by a Hamiltonian action of G on the presymplectic manifold Q. The map @J is then called the moment map, cf. [ 121. Now let 8: TP, -+ g be a connection on P,. We can then define the closed form d(@, 0) + R on P,
X
Q. This form satisfies i@(d(@, 6) + fi) =
0
where {denotes the vector field on PG x Q corresponding to r E g. It is also invariant under the action of G. This implies that there exists a (unique) closed two form ua defined on the associated bundle Q(PG) such that P*(Q)
= d(@, 0) + R.
(4.2)
(Recall that p: PG X Q + Q(PG) is the natural projection onto the quotient.) The closed form up thus defines a presymplectic structure, i.e., is a closed two form on Q(PG). However, this closed two form will, in general, have a redundant component to its null foliation. By use of reduction, we can eliminate some of this redundancy. Let H be a closed subgroup of G and P, a reduction of P, to an H subbundle. Let U be a manifold on which H acts and (: U- Q and equivalent H map. Then 4 induces a map i: U(P,) --) Q(PG).
INTERACTION
OF SPIN
We can then pull a, back to U(P,),
AND
TORSION
461
i.e., consider the form
which is closed two form on U(P,). It is the bundle U(P,) together with the form or which was suggested in [ 161 as the general “evolution space” for a mechanical system. Thus the particular mechanical system requires a choice of Q and U. It is a general fact (due to Kostant and Souriau) that every orbit of G acting on q* is a symplectic manifold (and, for many groups, is the most general transitive G orbit), cf. [ 15 ], for example. So a reasonable choice is to take Q to be a coadjoint G orbit and U an H suborbit. This is what we shall do. In case H is reductive, we have the H invariant decomposition g=h+n and dual decomposition g*=lf*+n*. Correspondingly
we have the decompositions (%PH)
= &I + w
(4.3)
and Qc=@h+Qn.
(4.4)
Then we have the projection pu: P.q x u+
U(P,)
and ~c=Pu*((~h,deH)+d~hABH+d~nA~t(~n,d~)+.R).
(4.5)
We want to consider this local expression for (4.5) in the case where w is a soldering form, more particularly, where o is the structure form for an affine connection. Thus we take H = O(V) where dim V = dim M and Y has some nondegenerate scalar product, and G is the corresponding afline group. We will take Q to be a coadjoint orbit of G. The coadjoint orbits for a semi-direct product were first described in [ 151 (see also [8]): Write g* = h* + n* so a typical element of g* is written as (S, p) with S E h* and p E V*. For the case h = o(V), we may identify V with V* using the scalar product /1*V with h and h*. An element of H acts in the standard linear way on II *V and on I’. Translation by u E V sends (S, p) into (S + PW P). Thus ll~ll*, IIS A PII’ (an d more generally (IS A S A S A p/I’, etc.) are all invariants of the affine group action. There is a family of lowest dimensional non-zero orbits-the orbits through the points (0, p). These have dimension 2dim V - 2. They contain a distinguished H
462
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suborbit-the dim V- 1 dimensional H orbit through (0, p). This H suborbit is the “sphere” { (0, q) ] (]q I(* = ((p I(‘}. They are the “spinless” orbits. If ]Ip (1’ # 0, so p & p ‘, an important H invariant condition on an H suborbit is the condition
as was shown in [ 161. We shall return to this condition in the next section. We now determine the null foliation of uU. To do so, we can consider the form in parentheses on the right-hand side of (4.5). That is, we need consider its null foliation on PH X U modulo the vector fields coming from the h action. Since H acts diagonally on PH x U and vertically on PH, it is enough to consider this form on pairs of vectors coming from the subspace zEP,,SEh*,pE
w = H* x TU(S,,,
V*,
where HZ denotes the space of horizontal vectors for BH at z E PH. On horizontal vectors, the form 0, = 0, and de, becomes the curvature R, when evaluated on pairs of horizontal vectors, and dw becomes the torsion, T. In (4.5) @,, becomes S and Qv becomes p. We may thus write the form in parentheses in (4.5), when evaluated on A’W, as
(S,R)+(p,T)+d'AwtQ. We can use o to identify HZ with V and hence consider (4.7)
K=(S,R)t(p,T)
as a bilinear form on V. Since H acts transitively on U, we can write the most general element TU,,,p, as t7,,,, where q E h, and v is the corresponding vector field on U and flCs,P) is its value at (S, p). The value of dp on q is just q . p, while the form R evaluated on qCs,P)Art {s,pjis - CC 145 VI>- We can thus regard (4.6) as a two form, b, on V t h given by b(u
t
0) A (v’
t
CT’)) =K(v
A ~‘1
t
(v * P, 2.1’)- (rl’ a P, v> - (S, [rl, r’l>.
(4.8)
The null foliation is then determined by the condition i(u t II) b = 0. These conditions can best be described by setting the V* and h* components of i(v t q) b separately equal to zero. We get the equations i(v)K=--rep
(4.9)
and ?l*S=pAv.
(4.10)
Let us first examine these equations for the case of the simplest H orbits-those with S = 0, p # 0. Then (4.10) says that v is a multiple of p and q . p is determined by (4.9) so the null foliation is one-dimensional. To see what this null foliation is, let
INTERACTION
OF SPIN AND TORSION
463
us consider, in addition to the given connection 19,, the Levi-Civita connection for the underlying metric-the one with torsion zero. Let Ho denote the horizontal subspace for the Levi-Civita connection. Then H and Ho determine two injections of V into (TP,),. This difference thus gives a map J: V -+ h and the torsion, thought of as an element of Horn@ ‘V, V), is given by
Thus with S = 0 in (4.7) K(Ul,
u2)=
(P,J(%) 0, -J(v,)
02).
so (i(P) K)(w) = (PYJ(w) P -J(P)
w>
= (P9 -J(P) w>
since J(w) E h
= V(P) PI w>
since J(p) E h.
So, if we take u = p in (4.9) we see that ?. p=-J(P)P* Let r denote the vector in H corresponding to v. Then the null direction, given by
has a vertical component which exactly compensatesfor the difference between r and the vector to which is horizontal for the Levi-Civita connection. Thus the null trajectories are exactly the geodesicsof the Levi-Civita connections-the geodesicsof the underlying metric. To repeat: in the absenceof spin the torsion has no eflect; the trajectories are the geodesicsof the underlying metric. In the next section we shall see how spin interacts with torsion.
5. THE INTERACTION
OF SPIN WITH TORSION
We now specialize to the case
so H = Lorentz group
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and G = Poincarl
group.
We will study positive mass particles so
MZ = JIpl(* > 0. The G orbit is then determined by the spin s > 0 where 11s A p1l2 = M2S2. We have already considered the case s = 0. So we need only consider the case s > 0. These orits are eight-dimensional. It was suggested in [ 161 that the appropriate H suborbit to choose is the five-dimensional suborbit given by the condition SEA2pl.
We shall make this as our choice. Notice that under the identification A ‘(W lV3) - h this condition becomes
(5.1)
of h* with
S E h,
(5.2)
where h, denotes the isotropy algebra of p (so h, - o(3)). We should observe that since p’ is three-dimensional every element of A2p is decomposable. This has the following consequence: LEMMA. Let S E A2p1 - h, c h so that Im S c R 1*3is two-dimensional. Then for any K E h, the restriction of SK to Im S is a scalar operator, i.e.,
SKSy = pSy
(5.3)
,u==trSK.
(5.4)
where
ProoJ: We may choose a Lorentz frame e,, e, , e2, e3 of R lV3 such that p=meo
(5.5)
and S=se,
Ae,.
(5.6)
Thus Im S is the two-dimensional subspace spanned by e, and e,. We may write the most general KEA2(R113)-h as K=
‘? O
KijeiAej
INTERACTION
so the restriction
OF SPIN
AND
465
TORSION
of K to Im S is
K,,,e,
/“\ e2 + x
+K,,ei@e,+
x jt
i+l,2
fKjzej@e, k.2
of SK to Im S is
and hence the restriction
sK,,,(e,
(5.7)
A e2>’ = 4L2
Q.E.D.
since e, A e, is rotation through 90” and hence (e, A e2)’ = -1 on Im S.
Using this lemma we can now analyse equations (4.9) and (4.10). Take the interior product of (4.10) with p. The right-hand side gives i(p)(p
A v) = m2v - (p, v) p
m’ = (p, p).
(5.8)
For the left-hand side we make use of the fact that under the identification h*,
and under the identification
of A2(lR’,3)
i(p)A
of h with
with h,
.p
AEh,pER’.3.
(5.10)
Then i(p)(rl
. S> = - [rl, Sl . P
=-(yS-Sy)*p since S . p = 0.
= +S(v * P) Substituting
into (4.9) we obtain m2V - (p, u) p = S(q . p) = -S(i(v)
K) = S(Kv)
or
m2v = (p, v) p + S(Kv). From the right-hand side of (5.3) Substituting this into (5.3) gives
we
see that we can write
m2@p + Sy) = m2@ + m2(Sy, p) p + SK(Lp + Sy) or since (Sy, p) = - (y, Sp) = 0,
m2Sy = LSKp + SKSy.
(5.11) v = Lp + Sy.
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Let us assume that the scalar M does not vanish where M=m2-ftrSK.
(5.12)
Then we can solve this last equation uniquely for Sy: Sy=$SKp and so v=I(p+-&SKp) gives the solution of (5.11). Notice that if u = 0, then (4.9) and (4.10) imply that q * p = 0 and q . S = 0. Hence the corresponding tangent vector r + ~o,~, vanishes. We have thus proved THEOREM. Zf we choose the H suborbits for the massive (spinning) orbits according to condition (5.1) then at all points where the scalar (5.12) does not vanish, the null foliation of wv is one dimensional with horizontal component v = w(t) given by (5.13) and vertical component qP,s then determined by (4.9) and (4.10).
We thus get a one-dimensional foliation-a system of ordinary differential equations-n the nine-dimensional manifold U(P,). The projection of these curves onto M then give the trajectories for the motion of a spinning particle in the presence of torsion. In order to get a feel for these trajectories, we should exhibit some cases where we can actually solve the equations of motion. We shall do that in the next sections, where enough symmetry conditions will be imposed to allow a group theoretical solution to the equations of motion: we shall study the effect of torsion on spinning particles in Minkowski space.
6. VECTOR TORSION ON MINKOWSKI
SPACE
It is a well known theorem of Levi-Civita that every (pseudo) Riemann manifold has a unique connection (preserving the metric) with zero torsion. It is equally true, and follows from the proof of the Levi-Civita theorem, cf. [ 14, p. 3341, that given any prescribed torsion tensor T, there exists a unique connection with torsion, T. In the Cartan theory of gravity of [2, 171, the gravitational field is coded into the torsion. In this section we wish to investigate the equation of motion of Section 1 for certain types of connections on Minkowski space, i’t4, (with its flat metric). The bundle of orthonormal frames over M has a global cross section given by a constant frame and flat Levi-Civita connection. Any other connection on M is thus equivalent to giving an h-valued one form, r, on M. (Here h is the Lie algebra of the Lorentz group, but much of what we have to say is valid with h = o(M) and M any
INTERACTION
OF SPIN
AND
467
TORSION
non-degenerate scalar product-space.) We may identify h with n ‘(IV) = n ‘(IR lt3) z n’(M*). Amongst all possible forms t, we shall consider those of the following type. Let p:M-iM* - M be a vector-valued function on M. We can consider the connection r=r,=p
Adx.
(6.1)
Here A denotes exterior multiplication in ,4(M) and dx is the R ‘*3 = M-valued differential form which identifies TM, with M. To avoid confusion about the exterior multiplication of differential forms, we shall temporarily use the notation x for exterior multiplication in /i(M) and so rewrite (6.1) as (6.1)’
t0 = /3 x dx, that is, for any x EM
and any u E TM, - M (r&=~(x)x~E/i~(M)-h.
If e,, e, , e,, e3 is an orthonormal
(6.2)
frame of M then T
Tijei
Zj
X ej
where
5ij = Pi dxj - pj dui. A connection of this form will be called of “vector
(6.3) type.” The curvature
R(r) = dz - [T A z] becomes
d(Pxdx)-(/I,P)dx?dx+@dx)A@xdx). In this expression the symbol 2 means exterior multiplication of differential forms; i.e.,
(6.4 1
multiplication
in A(M)
and exterior
for the commutator
of two rank
(dx 2 dx, u A v) = u A u. The last two terms of (6.4) come from the formula two elements:
[P(x) A u, P(x) A u] = - Ga,8) u A u + C/W), u) P(x) A u - V(x), u) P(x) A u. We should emphasize that in (6.4), p must be regarded as a vector-valued function and not as a linear differential form. For example, the ij component of the first term in (6.4) is
dpi A dXj-d@j
595/158/2
13
A dxi.
468
RAPOPORT
AND
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If b is a constant vector these all vanish and the curvature simplifies to R(z,)=-Ga,p)dxxdx+d0,dx)AGoxdx). If p is light-like,
(6.5)
this simplifies further as @,/I) = 0. One checks that (P,n=PAp=PXP.
Also ((S,dxXdx),uAu)=(S,uAu) and
((S, Go,dx) A (p x dx)), (u A u)) = @ x SP, u x u). Thus (reverting to the A notation) K=P~~+S~~~-IIP~~~S+(S,~(S,~~~~),)
(6.6)
or, for constant p, (6.7)
K=PA~+SP~P-IIPII~S
and for constant, light-like
p, K=pA/I+@AP.
(6.8)
Now for our choice of S E A*p we have trS(pA/3)=0 so tr KS is independent ofp and is an invariant biquadratic has
function of S and /3. One
~~SO(S/~AP)=~(IS/~([~
(6.9)
as can be checked by taking S = se, A e2, for example, and hence tr SK = 2 jlS/3II’ - II/31j2tr S2
(6.10)
and hence, for /3 light-like, tr SK = 2 )IS/311’.
(6.11)
We have (for constant j? and S E /i ‘p’) KP = (B, P)(P + SP) - m*P
(6.12)
INTERACTION
OF SPIN
AND
469
TORSION
and hence (6.13)
SKp = Q?, p) S2p - m2Sp.
The equations (5.13) (when L = 1), (4.9) and (4.10) thus become v = P + ,2 _
1 ,y$,,’
Lu-4 P) S2P -
m’W
(6.14)
v*P=[(PAD)+SPA\lU
(6.15)
q.S=pAv.
(6.16)
These equations are all written relative to the connection r4. We wish to write these equations in terms of the trivialization (and trivial connection) given by a fixed (globally constant) frame of M. This meansthat in (6.15) and (6.16) we must replace rl by q-J(v)=v-v
A/3.
Writing dx/dt for the velocity vector in the flat connection and similarly dp/dt we get
and
dS/dt
dx -&=P+ z
1 m2 - IIW’
[CA P) S’B - m2SPl
= (A A p> v - (v A p) p
(6.17) (6.18)
A=p+V
(6.19)
dS -=AAv+SvAa. dt
(6.20
Applying (6.20) to p and combining with (6.18) shows that dA -z dt
0.
(6.21
In other words A is an invariant of the flow. The invariant vector A should be interpreted as the total momentum of the system. In fact, let G, be the subgroup of the Poincare group which preserves /I so that G, is the semidirect product
470
RAPOPORT
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where HB is the subgroup of the Lorentz group which preserves p: H, = (L 1LB = j?}. The group acts on our bundle by (L, u)(x, (p, S)) = (Lx + u, (LP, LSL -‘))
(6.22)
and this action clearly preserves the presymplectic form. It is easy to check that this action is in fact (pre)-Hamiltonian and the corresponding moment map, @, is given by qx, (P, S)) = (Q
+ Ax), A)
(6.23)
where 71:h* + h,* is the projection dual to the injection of the Lie algebra h, into h. Thus A is the component of the moment map dual to translation and hence represents the (conserved) total linear momentum. (Here “total” means particle plus field.) Now (4 P) = (PP PI since (Sp, /I) = 0 and 64 W = IIWII’
= IIA /I2 -m*
since Sp=O. Thus (p,/?) and I/S/III” are invariants of the motion. But these two functions are also invariant under the action of G, as can be checked directly from (6.22), and these functions are clearly independent. Then the curves of our null foliation all lie in the seven-dimension submanifolds Na,* of the form (P,P)=a>o IlSPll’ =-b*
which are invariant under G,. Now Sfi is a space-like vector since S E A2p’. So llS/3ll= 0 implies that Sp = 0 and hence r.~= p and equations (6.18) and (6.20) become dp z=
o
dS dt=
o
and we get trivial rectilinear motion with no change in p or S. Thus we may restrict attention to the case where I/ ,!$I (1’ < 0. We claim that G, acts freely and transitively on hb- Indeed, the translations act freely and transitively on the x. The group H, acts transitively on the space of p satisfying 11pII 2 = mZ and (p, p) = 0. The subgroup of Ho fixing one such p consists of all rotations in the space-like plane perpendicular to p and /3, But S/3 is a non-zero vector in this plane so the element of the subgroup is determined by its action on S/3. Now our one-dimensional foliation on N0,6 is invariant under G,. If we identify Na,b with G, (by picking a point in Na,J we obtain a left invariant one-dimensional foliation on G,, which must therefore be the orbits of
INTERACTION
OF SPIN
AND
TORSION
471
a one parameter group acting on the right. Translating this back to N,,, we see that the curves of our null foliation must be orbits of one parameter groups of G,. The one parameter group will depend on the initial conditions. To find it, we must simply find the element (B, U) of g, , the Lie algebra of G,, which gives the tangent vector to the trajectory at the initial condition. Without loss of generality, we may assume that x = 0 and then u is determined to be u. We can determine B up to a scalar by the fact that BA = 0 because B must preserve the plane /I = 0 and A = 0. For example, we can, without loss of generality, asstime that we have chosen coordinates so that p=me,
and
P = + (e, - e, ).
Then
S = s1.2elA e2+ s,,,e, A e3+ s2.3e2A e3 and
SO
llwII’ =-f
(d.3 + $,3).
The most general B E h, can be written as B = b,Pe, + b,Pe, + b,,,e, A e, and the condition
gives B = s,T,jle, - s,.,Pe, + me, A e,.
7. MOTION IN THE PRESENCE OF A VECTOR TORSION WAVE
The explicit integration of Section 6 depended on the fact that we had two invariants and a seven dimensional group of symmetries. So, in effect, we had a seven-dimensional group acting transitively and freely on seven-dimensional manifolds. If we had a six-dimensional group and seven-dimensional invariant manifolds we could still say something about the integration procedure. Indeed, let us focus on such a manifold, and assume that coordinates (u, y), u = (u, ,..., us) have been chosen so that the group of symmetries acts transitively on each level surface
472
RAPOPORTAND
STERNBERG
y = const. Then if { =f(~, y) a/ay + g(a/&) is invariant under the group, we can conclude that af/au = 0, so f = f(y) and that, for each fixed y, g(a/au) comes from the Lie algebra. Then y(t) could be found by an integration, and the solution of the equations reduce to solving a system of time dependent differential equations on the group-a phenomenon annular to the nutation effect seen in collective motion [9]. As an example of this situation we consider the case of a vector connection as in the preceding section-so (6.1) holds, but where now p = de” = e* d#;
(7.1)
I.e., jl,=ebf$
j =*o, 1,2,3.
(7.1’)
J
Here we will take 4 = il where 1 is a linear function; for example tp = i(xo - x,).
(7.2)
Then, in (6.4) (j3, dx) = d#. Then, if IId$)l’ = 0 (I is light-like)
(7.3)
then the curvature becomes (from (6.4))
R(r) = e@(l - em) d# A (d# x dx)
(7.4)
and K = (S, R) + (p, T) = e*(l -e”)
S d# A d4 + emp A d$.
(7.5)
We then have tr(SK)=2(1-e@/2)jSd#I(2
(7.6)
M=m2-11-em121JSd~(12.
(7.7)
so, in (5.12),
Then also SKp = e”(1 - e”)(dq4 p) S2 d$ - m’e@S d#
(7.8)
and so, from (5.13), 1 u=P+m2-~l-e~~211Sd~~)2
em[(l - e@)(d& p) S2 d# - m2S d#].
(7.9)
INTERACTION
OF
SPIN
AND
TORSION
473
The equations (4.9) and (4.10) then become rj . p = [e@p d# + e@(l -e@) S d@ A d$] u
(7.10)
?l*S=pAv.
(7.11)
These are written in terms trivialization these become dx z=P+
m2-/l
of the connection
1 -e”/*)1Sd#))*
r = tD. In terms
e@[(l -e@)(d#,p)S*d#-m*Sd#]
of the global
(7.12)
and (7.13)
If we write (7.14)
A=p+(l-em)Sd4 then (7.11) becomes dp -&A
(7.15)
A&v-(vAP)p.
Then (7.11) becomes dS -=AAv++vA/?. dt Applying
(7.16)
/I to (7.16) and combining
with (7.15) we obtain that dA dt=
0
(7.17)
so that A is an invariant vector field of the flow, which we interpret as the total momentum of the system. We consider now G,, the subgroup of the Poincare group P which preserves 4 {a E P/a*qqw)
= $4(w) = I
= $h(w), w EM).
We note that if A E G,, it then preserves d# by the induced action, and in consequence preserves j3. In fact, G, is the semi-direct sum Gm=Hm@R;3
474
RAPOPORT
AND
STERNBERG
where H, and lRiq3 are the subgroups of the Lorentz and translation preserve 4. The group acts on the nine-dimensional bundle U(P,,) by
groups which (7.18)
(L u>(x, (PI S)) = (Lx + u, (LP, LSL - ‘>I.
This action preserves the presymplectic form. It is easy to check that this action is in fact (pre)-Hamiltonian and the corresponding moment map, 4, is given by @(x, (P, s)) = ($S + A A xl, A) where z: h* + hz is the projection dual to the injection of h, into h. Hence, A is the component of the moment map dual to translation and hence represents the conserved total linear momentum. Now
since (Sd#, d4) = 0 and since Sp = 0 we have (1 -e-@)e-@(S/3,A)=
[l -e-m](Sd#,A)=/l
-eem)2JJSd~J~2=~JA~~2-m2~
Thus, (d#,p) and (1 -e@\*(SdQ)(* are invariants of the motion, which are clearly independent functions. They are further invariants by the action of G,. Thus we can consider the six-dimensional group G, acting on the seven-dimensional submanifolds Naqb given by
i(p, d#>= a
(7.19)
11-em12[ISd#112=-b2. We can then integrate by the procedure mentioned in the beginning of this section.
ACKNOWLEDGMENTS The first author would Mathematics Department
like to express his gratitude to Miguel Filli for several discussions of Harvard University for its warm hospitality during his visit.
and to the
Note added in proof. This paper was written in ignorance of some of the important earlier literature on our subject. The authors are indebted to Professor F. Hehl for calling our attention to important relevant references, and to Dr. Baekler and Professor Hehl for interesting discussions relating our work to known results. The idea of using a spinning test particle to measure the torsion of a connection occurs in the paper of Hehl [How does one measure torsion of space-time? Phys. Left. A 36 (1971) 2251. It also occurs in the work of Trautman [Bull. Pal. Sci. Ser. Math. Astr. Phys. 20 (1972), 895j. Our equations (4.9) and (4.10) seem very close to the equations obtained by Yasskin in his Ph.D. thesis (Univ. of Maryland, unpublished). A more recent investigation of this subject can be found in the paper of Yasskin and Stoeger [Phys. Rev. D 21 (1980), 20811. A thorough discussion of the whole question of Poincare gauge fields and Cartan connections can be found in the lectures of Hehl [“Proceedings of the
INTERACTION
OF SPIN
AND
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415
6th Course of the International School of Cosmology and Gravitation on Spin, Torsion, Rotation and Supergravity, Erice” (Bergmann and desabbata, Eds.), Plenum, New York], in which the reader can find a comprehensive bibliography. We should point out that in much of the literature equations of motion are derived at the field theoretic level from Noether’s theorem, usually with some additional ansatze. Particle motion is then derived as a limiting case in the standard manner. Our equations (4.9) and (4.10) did not involve any special assumptions but came from the general abstract principle enunciated in ( 16 I. They can also be \ 101) to the set of all Cartan derived by applying the method of general covariance (cf. (81 and/or connections taken as the geometrical objects. We shall present the details of this derivation in a forthcoming paper.
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