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Absence of 3-cocycles in the Dirac monopole problem
Volume 153B, number 4,5
PHYSICS LETTERS
4 April 1985
ABSENCE OF 3-COCYCLES IN THE DIRAC MONOPOLE PROBLEM David G. B O U L W A R E a, S. D E S E R 2 and B. Z U M I N O 3
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Received 18 January 1985
The relation between 3-cocyles and violation of Jacobi identity is investigated for the quantum mechanical system of a charge in the field of a magnetic monopole. We show that, in the string description, there is no violation even without Dirac charge quantization. The alternative description of the vector potential on coordinate patches, which presupposes quantization, also preserves the Jacobi identity.
Introduction. In the application of cohomology theory to physical systems [ 1 ], there are subtleties involved even in the simplest case of models with a finite number of degrees of freedom. Our interest in these matters was stimulated by some recent discussions of the quantum mechanical example of a particle in an external magnetic field, particularly that of a magnetic monopole [ 2 - 5 ] . There, an apparent violation of the Jacobi identity [6] would seem to imply the presence of a non-vanishing 3-cocycle. Such a violation would be surprising in a quantum mechanical system with linear Hilbert space operators where the Jacobi identity's validity is manifest. We therefore propose to reanalyze this question in a somewhat different way than those of refs. [ 2 - 5 ] , and conclude both that the Jacobi identity is not violated and that the 3-cocycle vanishes for the Dirac monopole case, which is an ordinary quantum mechanical system. We will comment at the end on possible formulations which go beyond the standard quantum mechanical framework by introducing a non-associative algebra o f observables. Cocycles. We do not repeat here details o f the cohomology theory background, since it is well covered in 1 Permanent address: Department of Physics, FM15, University of Washington, Seattle, WA 98195, USA. 2 Permanent address: Department of Physics, Brandeis University, Waltham, MA 02254, USA. 3 Permanent address: Physics Department, University of California, Berkeley, CA 94720, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
refs. [ 1 - 5 ] . However, the following salient points will be important• Consider for concreteness the action of a Hilbert space operator V a representing a translation on a state (r'l labelled by eigenvalues r' of a complete set. It is defined by
(1)
Here U is a unitary matrix, whose elements are labelled in general by the other eigenvalues needed to specify the state. In the particle case, where r is a complete set, U is specified by a single numerical phase, the first cocycle o~1 : •
t
U(r', a) = exp [1~1 (r , a)].
(2)
Consequently, the second cocycle a2, defined by the composition law
Ira V b = exp [ia2(r', a, b)] Va+ b
(3a)
degenerates here, by direct calculation to
a2(r',a, b) = (AOtl)(r', a, b) =--Oll(r' + a , b ) - a l ( r ' , a + b ) + a l ( r ' , a
).
(3b)
Note that, because only the exponentials exp(iai) are ever defined, it is always possible (even in this trivial case) to replace (3b) by a 2 = (AO~l)' -= Aol1 + 2rrN2,
(3c)
where N 2 is a (discontinuous) integer-valued function of its arguments (r', a, b). The third and all higher co307
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cycles vanish here because z ~ O t 1 = 0 , however, it is always possible to assign the trivial value 2rrNi to any of these higher cocycles a/. We mention also that a change of phase (such as would arise from a gauge transformation) of the states, (r'l ~ exp [i%(r')] (r'l, produces the change ~1 ~ al - A ~ in the first cocycle, which is therefore not unique. {Of course, a redefinition of the Hilbert space operator, Va -~ exp [ - i a 1(r, a)] Fa, removes 0t1 (r', a) altogether in (1), but this is now a change of operator. The physical significance of such changes is not completely clear to us; this is a point which should be better understood.}
Monopole preliminaries. The above remarks, including the possibility o f adding a discontinuous integer to any cocycle a la (3c) even if that cocycle is trivial, are rather obvious. However, this integer can enter in a disguised way, as we shall see in the monopole field example. The presence of nontrivial 0~2 and a 3 there seems equally dictated by the appearance of a Schwinger term and apparent failure of the Jacobi identity [6] in elementary considerations of the commutation properties of the gauge invariant operator n = p - eA, namely [Tri, Tr]] =ieFi/,
½ei/k[ni , [rr/,rrk] ] = e V ' S .
(4)
There are two ways of dealing with magnetic monopole fields, and one must consistently follow the one chosen to avoid paradoxes. The first is to allow the vector potential A (and hence rt) to be defined everywhere, at the price of a string singularity and therefore use of distributions rather than functions. Then the operator n = - i V - eA is well-defined, as is the magnetic field B which now includes the string's contribution in addition to the radial part B ~ g r / r 3 . The second approach, in which A is defined only on local sections [7], allows one to deal with normal functions, but at the price of excluding the origin, where V ' B fails to vanish. We shall adopt the string approach, but will establish the equivalence of our results to the section method at the end. In particular, the Jacobi identity is always satisfied in both approaches. Furthermore, we shall see that in the string approach, Dirac quantization is not even required to enforce the Jacobi identity, although it is of course needed physically to make the string unobservable and so to preserve rotation invariance. [The section approach requires quantization in order to be globally well-defined.]
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Monopoles - formalism. Consider three different choices of Va, all of which obey (1): I: II:
o~1 = O,
VI = exp(a'V),
~I=ef
r'+a
dr'A,
Vli=exp[a.(V-ieAii)],
r'
III:
a]II=ef A
dS'B,
Vili=expta.(V-ieAiil) ] . (5)
The first choice corresponds to use of the gauge-variant translation operator; the second (with a straight line path in the ct1 integral) differs in its choice of V from VI by an operator redefinition with factor exp [ - i a 1(r', a)]. The third, unconventional, choice differs from the second by a gauge, i.e. by a zero cocycle phase c h a n g e a~ II = a II + A0t0 and by a corresponding difference in potentials in VII and VII I . It introduces a reference point r 0 (distinct from the monopole's location which defines the origin), and the surface integral in its etIII is over the triangle with vertices (r0, r, r + a). The V operator here is exp(ia .g) if there are no monopoles (V.B = 0); when one is present it still has this form in the gauge in which the monopole's string is directed radially outward with respect to r 0. Neither a~I nor 0~111is of course removable by a Aa 0. There is actually a difference between II and III, but it is of the discontinuous 21rN type defined previously.
2-cocycles. Let us now calculate the higher cocycles. In I, all or/> 1 vanish. In II, a 2 = ActI =~ dr.eA = f dS'eB where B of course includes the string contribution. It can also be written as f dS.e'B, together with an additive 27rN where N is the number o f times the string pierces the triangle (r, r + a, r + a + b), if we assume the Dirac charge quantization. In III, ot2 is again a surface integral f dS'eB, but this time over the "cap" of the tetrahedron with vertices (r 0' r, r + a, r + a + b) whose (excluded) base is the (r, r + a, r + a + b) triangle and apex is r o. It too may be written as an integral over B, with a 2n change if the string pierces the "cap" when quantization is assumed. Note that there was no Schwinger term in I, since [p,p] = 0, while in II and III, [hi, rt]] = ieFi] as operator relations. There is no contradiction here; VI is a different operator. The aJ2I and 0~2II actually differ by • an additive 21rN so it is normal that their infinitesimal
Volume 153B, number 4,5
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versions, responsible for Schwinger terms, are identical: this 2rrN must be discarded in the exponential before making the Taylor expansion in the continuous parts of a2. On the other hand, a 2 is not removable: it is a A a l , but this a 1 cannot be changed by anything but a Aa 0 without physically changing the operator V itself. The above remarks apply equally well, of course, to the case of an ordinary (V .B = 0) magnetic field. 3.cocycles. Now we turn to the third cocycle Ct3, whose presence would signal a breakdown of the Jacobi identity (we can emphasize that it only is a nontrivial exp(ia3) which corresponds to a breakdown of associativity). First, note that all three a 3 in (5) vanish simply because a 3 = Aot2 = A2al = 0, up to possible addition of 2nN factors. Indeed, in II we get a 3 :/: 0 if (and only if) we use the stringless B in a2, which is just such an implicit discontinuous addition of 2rrN. In III, we get a 3 = 0 either way: this is obvious with a,I; I = e fcap d S ' B , but is true even with ~2! = e f c a p d S ' B because that a 2 is also the A of an al, namely of a 1 = fa dS'B. As mentioned earlier, one may also deal with monopoles without strings by use of sections [7], i.e. by defining A (as a function rather than distribution) on overlapping coordinate patches. Every such patch must exclude a conical region extending from the monopole to infinity (in which the string could run); since the origin is part of any such region, it is always excluded in this description. Once the origin is excluded then a3 vanishes anyhow, since V ' B now vanishes everywhere in the new manifold excluding the origin. (In the quantum mechanical calculation of ref. [6], the wave function vanishes at the origin, but the present considerations do not involve this fact directly.) More specifically, the choice II gives a3 = e
fdS.B,
where the surface surrounds the origin. This is equal to 2zrN by the Dirac quantization condition and hence is trivial, but n o t to the volume integral f d3r V-B because the origin is n o t in our space. (Thus it is not even possible to discuss the Jacobi identity at the origin here.) Definition III cannot be formulated at all in this context, because the same reference point r 0 cannot be used for all patches, for if it were, the surfaces in the aJ1II integral would intersect an excluded cone.
4 April 1985
In summary, we have shown that the apparent presence of 3-cocycles in the monopole problem is purely an artifact of the trivial ambiguity of any cocycle definition up to a factor 21rN, which does not affect the associativity of operators as defined by exp(ia3). The corresponding infinitesimal version (Jacobi identity violation) is therefore also absent since it is meaningful only for the continuous part of a 3 for which alone the Taylor expansion of the exponential is defined. That the a 3 they obtained was of the trivial 2rrN type was of course known to the authors of refs. [ 2 - 5 ] ;however, we have emphasized here that a straightforward definition of a 3 leads, as it must, to its vanishing identically. Non-associative algebras. Is it possible to define the monopole system on a more abstract basis, in which the operator relations (4) without underlying Hilbert space, are taken as its definition? That is, where V .B does not vanish (e.g., if we use the stringless B), one must give up a conventional quantum mechanical operator description because Hilbert space operators form an associative algebra and hence satisfy the Jacobi identity. This interesting notion of using a non-associative algebra was suggested in ref. [4], and it is indeed unavoidable when V.B :~ 0. The most commonly studied [8] non-associative algebras are the Lie, Jordan and alternative algebras. Of these, Lie algebras obey the Jacobi identity under the appropriate definition of multiplication and Jordan algebras are commutative, leaving only the last as a possibility. We have not seriously attempted to check whether (4) is of this type, or even if it gives a consistent set of rules. However, even if such an interpretation of (4) is viable, one must still show that it is possible to define an acceptable equivalent to quantum mechanics without Hilbert space, perhaps in terms of density matrices alone. Open questions. We conclude with some remarks about each cocycle level. We have already alluded to the operator redefinitions of V which would allow one to alter or even remove a 1 . To what extent are two such V's physically equivalent? Are gauge theories with nontrivial o~2, in which the constraints (or gauge generators) no longer form an algebra [1 ], viable? Do there exist field theoretical examples in which a nontrivial a 3 appears through presence of U-functions
309
Volume 153B, number 4,5
PHYSICS LETTERS
which are not merely phases but infinite-dimensional matrices in the state labels? Abstractly, higher cocycles seem to arise in discussions [9] of geometric quantization; do they have natural realizations in interesting field theories? This research was supported in part by the National Science Foundation under Grant Nos. PHY77-27084, PHY81-18547, and PHY82-01094 supplemented by funds from the National Aeronautics and Space Administration, and in part by the US Department of Energy Contract Nos. DE-AC03-76SF00098, AT0380ER10617, and DE-AC06-81 ER-40048.
310
4 April 1985
References [ 1] L.D. Faddeev, Phys. Lett. 145B (1984) 81; B. Zumino, Nucl. Phys. B, to be published. [2] B. Grossman, Phys. Lett. 152B (1985) 93. [3] B.-Y. and B.-Y. Hou, preprint Northwestern University, Xian (China). [4] R. Jackiw, Phys. Rev. Lett., to be published. [5] Y.-S. Wu andA. Zee, Phys. Lett. 152B (1985) 98. [6] H.J. Lipkin, W.I. Weisberger and M. Peshkin, Ann. Phys. 53 (1969) 203. [7] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [8] N. Jacobson, Jordan algebras, A.M.S. Colloquium Publications, Vol. XXXIX (Providence, RI, 1968); R.D. Schafer, Nonassociative algebras (Academic Press, New York, 1966). [9] A. Lichnerowicz, in: Quantum theory, groups, fields and particles, ed. A.O. Barut (D. Reidel, Dordrecbt, 1983); C.J. Isham, in: Relativity, groups and topology II, eds. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984).
PHYSICS LETTERS
4 April 1985
ABSENCE OF 3-COCYCLES IN THE DIRAC MONOPOLE PROBLEM David G. B O U L W A R E a, S. D E S E R 2 and B. Z U M I N O 3
Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA Received 18 January 1985
The relation between 3-cocyles and violation of Jacobi identity is investigated for the quantum mechanical system of a charge in the field of a magnetic monopole. We show that, in the string description, there is no violation even without Dirac charge quantization. The alternative description of the vector potential on coordinate patches, which presupposes quantization, also preserves the Jacobi identity.
Introduction. In the application of cohomology theory to physical systems [ 1 ], there are subtleties involved even in the simplest case of models with a finite number of degrees of freedom. Our interest in these matters was stimulated by some recent discussions of the quantum mechanical example of a particle in an external magnetic field, particularly that of a magnetic monopole [ 2 - 5 ] . There, an apparent violation of the Jacobi identity [6] would seem to imply the presence of a non-vanishing 3-cocycle. Such a violation would be surprising in a quantum mechanical system with linear Hilbert space operators where the Jacobi identity's validity is manifest. We therefore propose to reanalyze this question in a somewhat different way than those of refs. [ 2 - 5 ] , and conclude both that the Jacobi identity is not violated and that the 3-cocycle vanishes for the Dirac monopole case, which is an ordinary quantum mechanical system. We will comment at the end on possible formulations which go beyond the standard quantum mechanical framework by introducing a non-associative algebra o f observables. Cocycles. We do not repeat here details o f the cohomology theory background, since it is well covered in 1 Permanent address: Department of Physics, FM15, University of Washington, Seattle, WA 98195, USA. 2 Permanent address: Department of Physics, Brandeis University, Waltham, MA 02254, USA. 3 Permanent address: Physics Department, University of California, Berkeley, CA 94720, USA. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
refs. [ 1 - 5 ] . However, the following salient points will be important• Consider for concreteness the action of a Hilbert space operator V a representing a translation on a state (r'l labelled by eigenvalues r' of a complete set. It is defined by
(1)
Here U is a unitary matrix, whose elements are labelled in general by the other eigenvalues needed to specify the state. In the particle case, where r is a complete set, U is specified by a single numerical phase, the first cocycle o~1 : •
t
U(r', a) = exp [1~1 (r , a)].
(2)
Consequently, the second cocycle a2, defined by the composition law
Ira V b = exp [ia2(r', a, b)] Va+ b
(3a)
degenerates here, by direct calculation to
a2(r',a, b) = (AOtl)(r', a, b) =--Oll(r' + a , b ) - a l ( r ' , a + b ) + a l ( r ' , a
).
(3b)
Note that, because only the exponentials exp(iai) are ever defined, it is always possible (even in this trivial case) to replace (3b) by a 2 = (AO~l)' -= Aol1 + 2rrN2,
(3c)
where N 2 is a (discontinuous) integer-valued function of its arguments (r', a, b). The third and all higher co307
Volume 153B, number 4,5
PHYSICS LETTERS
cycles vanish here because z ~ O t 1 = 0 , however, it is always possible to assign the trivial value 2rrNi to any of these higher cocycles a/. We mention also that a change of phase (such as would arise from a gauge transformation) of the states, (r'l ~ exp [i%(r')] (r'l, produces the change ~1 ~ al - A ~ in the first cocycle, which is therefore not unique. {Of course, a redefinition of the Hilbert space operator, Va -~ exp [ - i a 1(r, a)] Fa, removes 0t1 (r', a) altogether in (1), but this is now a change of operator. The physical significance of such changes is not completely clear to us; this is a point which should be better understood.}
Monopole preliminaries. The above remarks, including the possibility o f adding a discontinuous integer to any cocycle a la (3c) even if that cocycle is trivial, are rather obvious. However, this integer can enter in a disguised way, as we shall see in the monopole field example. The presence of nontrivial 0~2 and a 3 there seems equally dictated by the appearance of a Schwinger term and apparent failure of the Jacobi identity [6] in elementary considerations of the commutation properties of the gauge invariant operator n = p - eA, namely [Tri, Tr]] =ieFi/,
½ei/k[ni , [rr/,rrk] ] = e V ' S .
(4)
There are two ways of dealing with magnetic monopole fields, and one must consistently follow the one chosen to avoid paradoxes. The first is to allow the vector potential A (and hence rt) to be defined everywhere, at the price of a string singularity and therefore use of distributions rather than functions. Then the operator n = - i V - eA is well-defined, as is the magnetic field B which now includes the string's contribution in addition to the radial part B ~ g r / r 3 . The second approach, in which A is defined only on local sections [7], allows one to deal with normal functions, but at the price of excluding the origin, where V ' B fails to vanish. We shall adopt the string approach, but will establish the equivalence of our results to the section method at the end. In particular, the Jacobi identity is always satisfied in both approaches. Furthermore, we shall see that in the string approach, Dirac quantization is not even required to enforce the Jacobi identity, although it is of course needed physically to make the string unobservable and so to preserve rotation invariance. [The section approach requires quantization in order to be globally well-defined.]
308
4 April 1985
Monopoles - formalism. Consider three different choices of Va, all of which obey (1): I: II:
o~1 = O,
VI = exp(a'V),
~I=ef
r'+a
dr'A,
Vli=exp[a.(V-ieAii)],
r'
III:
a]II=ef A
dS'B,
Vili=expta.(V-ieAiil) ] . (5)
The first choice corresponds to use of the gauge-variant translation operator; the second (with a straight line path in the ct1 integral) differs in its choice of V from VI by an operator redefinition with factor exp [ - i a 1(r', a)]. The third, unconventional, choice differs from the second by a gauge, i.e. by a zero cocycle phase c h a n g e a~ II = a II + A0t0 and by a corresponding difference in potentials in VII and VII I . It introduces a reference point r 0 (distinct from the monopole's location which defines the origin), and the surface integral in its etIII is over the triangle with vertices (r0, r, r + a). The V operator here is exp(ia .g) if there are no monopoles (V.B = 0); when one is present it still has this form in the gauge in which the monopole's string is directed radially outward with respect to r 0. Neither a~I nor 0~111is of course removable by a Aa 0. There is actually a difference between II and III, but it is of the discontinuous 21rN type defined previously.
2-cocycles. Let us now calculate the higher cocycles. In I, all or/> 1 vanish. In II, a 2 = ActI =~ dr.eA = f dS'eB where B of course includes the string contribution. It can also be written as f dS.e'B, together with an additive 27rN where N is the number o f times the string pierces the triangle (r, r + a, r + a + b), if we assume the Dirac charge quantization. In III, ot2 is again a surface integral f dS'eB, but this time over the "cap" of the tetrahedron with vertices (r 0' r, r + a, r + a + b) whose (excluded) base is the (r, r + a, r + a + b) triangle and apex is r o. It too may be written as an integral over B, with a 2n change if the string pierces the "cap" when quantization is assumed. Note that there was no Schwinger term in I, since [p,p] = 0, while in II and III, [hi, rt]] = ieFi] as operator relations. There is no contradiction here; VI is a different operator. The aJ2I and 0~2II actually differ by • an additive 21rN so it is normal that their infinitesimal
Volume 153B, number 4,5
PHYSICS LETTERS
versions, responsible for Schwinger terms, are identical: this 2rrN must be discarded in the exponential before making the Taylor expansion in the continuous parts of a2. On the other hand, a 2 is not removable: it is a A a l , but this a 1 cannot be changed by anything but a Aa 0 without physically changing the operator V itself. The above remarks apply equally well, of course, to the case of an ordinary (V .B = 0) magnetic field. 3.cocycles. Now we turn to the third cocycle Ct3, whose presence would signal a breakdown of the Jacobi identity (we can emphasize that it only is a nontrivial exp(ia3) which corresponds to a breakdown of associativity). First, note that all three a 3 in (5) vanish simply because a 3 = Aot2 = A2al = 0, up to possible addition of 2nN factors. Indeed, in II we get a 3 :/: 0 if (and only if) we use the stringless B in a2, which is just such an implicit discontinuous addition of 2rrN. In III, we get a 3 = 0 either way: this is obvious with a,I; I = e fcap d S ' B , but is true even with ~2! = e f c a p d S ' B because that a 2 is also the A of an al, namely of a 1 = fa dS'B. As mentioned earlier, one may also deal with monopoles without strings by use of sections [7], i.e. by defining A (as a function rather than distribution) on overlapping coordinate patches. Every such patch must exclude a conical region extending from the monopole to infinity (in which the string could run); since the origin is part of any such region, it is always excluded in this description. Once the origin is excluded then a3 vanishes anyhow, since V ' B now vanishes everywhere in the new manifold excluding the origin. (In the quantum mechanical calculation of ref. [6], the wave function vanishes at the origin, but the present considerations do not involve this fact directly.) More specifically, the choice II gives a3 = e
fdS.B,
where the surface surrounds the origin. This is equal to 2zrN by the Dirac quantization condition and hence is trivial, but n o t to the volume integral f d3r V-B because the origin is n o t in our space. (Thus it is not even possible to discuss the Jacobi identity at the origin here.) Definition III cannot be formulated at all in this context, because the same reference point r 0 cannot be used for all patches, for if it were, the surfaces in the aJ1II integral would intersect an excluded cone.
4 April 1985
In summary, we have shown that the apparent presence of 3-cocycles in the monopole problem is purely an artifact of the trivial ambiguity of any cocycle definition up to a factor 21rN, which does not affect the associativity of operators as defined by exp(ia3). The corresponding infinitesimal version (Jacobi identity violation) is therefore also absent since it is meaningful only for the continuous part of a 3 for which alone the Taylor expansion of the exponential is defined. That the a 3 they obtained was of the trivial 2rrN type was of course known to the authors of refs. [ 2 - 5 ] ;however, we have emphasized here that a straightforward definition of a 3 leads, as it must, to its vanishing identically. Non-associative algebras. Is it possible to define the monopole system on a more abstract basis, in which the operator relations (4) without underlying Hilbert space, are taken as its definition? That is, where V .B does not vanish (e.g., if we use the stringless B), one must give up a conventional quantum mechanical operator description because Hilbert space operators form an associative algebra and hence satisfy the Jacobi identity. This interesting notion of using a non-associative algebra was suggested in ref. [4], and it is indeed unavoidable when V.B :~ 0. The most commonly studied [8] non-associative algebras are the Lie, Jordan and alternative algebras. Of these, Lie algebras obey the Jacobi identity under the appropriate definition of multiplication and Jordan algebras are commutative, leaving only the last as a possibility. We have not seriously attempted to check whether (4) is of this type, or even if it gives a consistent set of rules. However, even if such an interpretation of (4) is viable, one must still show that it is possible to define an acceptable equivalent to quantum mechanics without Hilbert space, perhaps in terms of density matrices alone. Open questions. We conclude with some remarks about each cocycle level. We have already alluded to the operator redefinitions of V which would allow one to alter or even remove a 1 . To what extent are two such V's physically equivalent? Are gauge theories with nontrivial o~2, in which the constraints (or gauge generators) no longer form an algebra [1 ], viable? Do there exist field theoretical examples in which a nontrivial a 3 appears through presence of U-functions
309
Volume 153B, number 4,5
PHYSICS LETTERS
which are not merely phases but infinite-dimensional matrices in the state labels? Abstractly, higher cocycles seem to arise in discussions [9] of geometric quantization; do they have natural realizations in interesting field theories? This research was supported in part by the National Science Foundation under Grant Nos. PHY77-27084, PHY81-18547, and PHY82-01094 supplemented by funds from the National Aeronautics and Space Administration, and in part by the US Department of Energy Contract Nos. DE-AC03-76SF00098, AT0380ER10617, and DE-AC06-81 ER-40048.
310
4 April 1985
References [ 1] L.D. Faddeev, Phys. Lett. 145B (1984) 81; B. Zumino, Nucl. Phys. B, to be published. [2] B. Grossman, Phys. Lett. 152B (1985) 93. [3] B.-Y. and B.-Y. Hou, preprint Northwestern University, Xian (China). [4] R. Jackiw, Phys. Rev. Lett., to be published. [5] Y.-S. Wu andA. Zee, Phys. Lett. 152B (1985) 98. [6] H.J. Lipkin, W.I. Weisberger and M. Peshkin, Ann. Phys. 53 (1969) 203. [7] T.T. Wu and C.N. Yang, Phys. Rev. D12 (1975) 3845. [8] N. Jacobson, Jordan algebras, A.M.S. Colloquium Publications, Vol. XXXIX (Providence, RI, 1968); R.D. Schafer, Nonassociative algebras (Academic Press, New York, 1966). [9] A. Lichnerowicz, in: Quantum theory, groups, fields and particles, ed. A.O. Barut (D. Reidel, Dordrecbt, 1983); C.J. Isham, in: Relativity, groups and topology II, eds. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984).