Differential geometry and internal symmetryR. Mirman Citation: Journal of Mathematical Physics 16, 2177 (1975); doi: 10.1063/1.522453 View online: http://dx.doi.org/10.1063/1.522453 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/16/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Differential geometry of group lattices J. Math. Phys. 44, 1781 (2003); 10.1063/1.1540713 Lie groups and differential geometry AIP Conf. Proc. 365, 33 (1996); 10.1063/1.50217 On the geometry of spontaneous symmetry breaking J. Math. Phys. 33, 1546 (1992); 10.1063/1.529679 Symmetries of differential equations. II J. Math. Phys. 21, 2046 (1980); 10.1063/1.524714 Dynamical Content of Differential Geometry J. Math. Phys. 8, 2376 (1967); 10.1063/1.1705169

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Differential geometry and internal symmetry R. Mirman

155 East 34 Street, New York. New York 10016 (Received 17 December 1974; final revision 13 March 1975)

It is shown using a generalized Lyra space that the concept of inte~al s~mm~try can. be expressed in the language of differential geometry. By this method invariant and nonmvanant mteractlOns are generated by a gauge formalism.

I. INTRODUCTION

Internal symmetry implies the existence of a set of particles, which corresponds to a representation of some Lie group, all of which behave the same under some interactions, and presumably behave differently under others, We wish here to express the concept of internal symmetry in the terminology of differential geometry and to show that a Lyra space, 1 suitably gen­eralized, can be used for this purpose.

To do this, we require a set of complex tensors, the wavefunctions of the particles, and a set of geometrical quantities, one for each quantum number, which differ­entiates among the tensors. Then we need (at least one) set of transformations forming a Lie group, which mixes the tensors. Finally we want the geometry to lead to interactions, at least one of which is invariant under the transformation and at least one of which is not.

We define a space which has properties leading to the above results. SpeCifically, besides the coordinates x, we introduce variables y and transformations over them.

In addition to these coordinates, we define over the space, functions which behave as tensors in the usu~1 way under x-transformations and also have well-defmed properties under the y-transformations. Under the latter they break up into various classes, and we define further transformations among these classes.

These transformations are all arbitrarily space dependent.

Much of our development is similar to gauge theories (see Camenzind, 2 and references cited therein, for ex­ample). However, for our treatment the assumptions, viewpoint, and language are more geometrical. This may turn out to be more restrictive and suggestive (or differently restrictive and suggestive) than other types of theories.

In particular the geometry has objects (the y's) and transformations over them. As a consequence there are different classes of tensors. and transformations among the classes. Hence internal symmetry flows directly from the geometry,. instead of being added on.

The covariant derivatives then contain terms linking different kinds of tensors (that is interactions). We must introduce one further (phYSical) assumption, Which here is that the tensors are eigenfunctions (in the Dirac formalism sense) of their covariant derivatives. This is a natural generalization, to this space, of the usual quantum mechanical (in the Dirac formalism) require­ment that, for a free particle, the wavefunctions are

2177 Journal of Mathematical Physics, Vol. 16, No. 10, October 1975

eigenfunctions of the ordinary derivative. It seems less ad hoc than the requirement of minimal interactions. In­deed the occurrence of the interaction terms appears to be more natural in this formalism, than in the gauge formalism.

Because the tensors undergo, besides the usual space and internal transformations, another set, those for the y's, there appears an interaction which is symmetry violating.

Hence the geometry implies the presence of internal coordinates and interactions, and it further implies that there are some which give internal invariance and others which break it.

II. THE GEOMETRY

We assume an n-dimensional space and that there are k quantum numbers to be described geometrically. The points are labeled by coordinates xi, ... ,xn. In ad­dition we introduce k auxilary variables Yw' In this space we perform the following types of transformations.

First we can rotate x in the usual manner, leaving the y's fixed, so that

x,i =A~(x)xJ. (1)

We can also leave the x's invariant and transform the y's according to

y' w = sw(x) Yw, (2)

where the s's are mutually independent arbitrary func­tions. These are the gauge transformations. We need the function

(3)

Although it might be useful to consider complex A's (in analogy to a previous analysis3) we restrict our con­siderations here to real ones, and so require thats be positive.

The third type of transformation is conSidered below.

Tensors behave in the usual way under transforma­tions of the x's

T'(j ••• m .. ·=(A-i)f··· A:··· Tp ... q .. •• (4)

We assign every tensor to a class specified by a set of numbers (cil' .. ,c~), where the c's can have any real values. The class is denoted by c, for short, and a ten­sor of class c gauge transforms according to, suppress­ing indices,

(5)

Copyright © 1975 American Institute of Physics 2177

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The final transformation leaves both the x's and y's invariant, operating only on tensors, and changes ten­sors of one class to tensors of another class. Again suppressing space indices, but using an index to indicate the class, we have a set of transformations of the form

(6)

Some, but not necessarily all, of the c's differ between classes Q' and 13. The J's are required to form a Lie algebra, and there are various sets of J's, correspond­ing to different representations. The E'S are arbitrary functions. The capital Latin index denotes the different internal generators, while the Greek one refers to the internal coordinates and its range depends on the representation.

There is a Y for each quantum number operator of the algebra. Thus in SU(3) there are y's for quantum num­bers Y, [z, and [2.

In order to make Eq. (6) gauge invariant, we have to take the set of c's for Jts of the form

for each c of the set, where c",goes with T(a). This means that we consider that the internal symmetry operators carry a charge.

The displacement is defined as

d~=Yl" ·Yk dx.

(7)

(8)

To define the connection, we take a vector, with all c's equal to 1, for which

(9)

in one coordinate system and require its expression in another system. Since

we get

vm + (A -I),!, Ai Vf ,n 'It n

and with

rm = (A-1)m AI fn I f,n

we get

vm n+ rin Vf +0 A(l)n vm = o. • I

So the connection is

fin = rin + 0 5fm A(k)n.

k

We have the usual transformation law for r,

q~ =A:(A -l)~(A-l)~ r~n +A:(A-I)L.

We see that A is a space vector and under the transformation

y" =s'sy

it becomes

A' = a lns's/ax

2178 J. Math. Phys., Vol. 16, No. 10, October 1975

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

so that, in general,

A~ (l) = (A-I)::, Am(l) + iJ Ins" lax'". (18)

The transformation of r is immediate.

The covariant derivative is, for arbitrary c's,

VI;J=vi.J+r~JVa+~c(l)A(l)JVI, (19)

and similarly for tensors.

The metric tensor is defined from the distance squared

ds2 =giJ(YI' "Yk)2dxldxJ=giJd~ide,

and the inverse tensor is defined

(20)

giJgJk = 5~. (21)

We take all the e's for glJ to be - 2 and for gO to be 2.

For Simplicity we further require that the covariant derivative of g, and hence of g7TyZ be zero. Using Eq. (19) with { } the Christoffel symbol, and proceeding in the usual manner we get, assuming that r is symmetric,

(22)

The geometry can be specialized by taking k even and allowing only those transformations for which

(23)

where W is a constant, and so on for all the other pairs. This gives A(l) = - A(2), etc., and the A term disappears from the connection which just becomes the Christoffel symbol.

Above we have written down that part of the connection which gives the rotation of the space indices due to translation. We now look at the part which gives the corresponding rotation of internal indices. The total connection is the sum of the two.

This part is defined from the covariant derivative of a space scalar. It is

(24)

Following Anderson, 4 we require that it transform under the internal transformation in the same way as the scalar does. We find n transforms under these transformations as

(25)

Putting all the parts together, we get the expression for the complete covariant derivative for a vector (and the generalization to higher rank tensors is immediate)

V:;p= V7:. p+{;:'p} V: -zt[A(i)n5;'

+A (i)p5::' - A (i)qg"mgnp ] V'" n

+[0 A(i)pc(i, a)l V:- np~ vsm. i

(26)

The field A is a space vector. Under internal trans­formations A is a scalar, and the representation under which n transforms is determined by the representa­tion of the object for which it is the connection. All these different connections for different representations are related by the rules of decomposition of the product of group representations.

R. Mirman 2178

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The covariant derivative of A is

A(Om;n= Am,n - {,tn}Ap +y [A(j)m~-A{j)nB~ -A{j)Q~gmn]Ap. (27)

These equations are required to be gauge invariant, which places restrictions on the connections. For A all the c's are zero from Eqs. (3) and (15), and thus so are the c's for r from Eq. (18), while for ~m! all the c's are of the form

(28)

from Eq. (21). This leaves Eqs. (23) and (24) invariant under gauge transformations.

To get invariant equations ("relativistic wave equa­tions") for tensor T a , we consider the object If:, which transforms according to the vector representa­tion for the i index (which runs over the dimension of the space) and according to whatever representation a and b belong to (and these run over the dimension of the representation). It is a representative of the direct product of representations going with i, a, and b.

Usually a tensor is written in nonreduced form with a set of indices, but here it is written as an irreducible representation with one index. So its connection is a combination of connections for the fundamental repre­sentation, which we do not write explicitly, but denote by r', which includes the A term.

It is assumed that T is an eigenfunction of the covari­ant derivative with eigenvalue m, which gives the in­variant equation

If: T "~f = If: T "~i +lj~ r~~ T "c + li:[0A {j)jc{j, a)]T '" a j

(29)

Likewise we can convert Eq, (24) to an invariant equation, and get a corresponding equation by forming a curvature for ~.

2179 J. Math. Phys., Vol. 16, No. 10, October 1975

Thus we have an equation describing the tensor, with interactions. The Christoffel symbol can be interpreted as being due to the gravitational field, and the term with with the A as being due to a universal field to which all particles are coupled. The c term is a symmetry break­ing one while the final term gives invariance under in­ternal transformations and so is a symmetry preserving interaction.

The fields to which the particle is coupled themselves obey relativistic wave equations, so that we have a set of coupled equations for the particle and the fields it interacts with.

III. CONCLUSION

We have described internal coordinates, and both symmetry conserving, and symmetry breaking inter­actions in the terminology of differential geometry. Specifically we have introduced a space, defining over it coordinates, auxilary variables, and tensors, and transformations over these quantities. By imposing the requirement that these transformations be arbitrarily space dependent and proceeding as usual in differential geometry, we are led to differential equations, based on the covariant derivative for the geometry, governing the tensors (relativistic wave equations), which link tensors of different classes. Some of their terms are invariant under the internal symmetry transformations, others not.

Internal symmetry and the form of these equations are implied by the geometry. In this space, symmetry, and symmetry breaking, fit naturally.

1D. K. Sen and K. A. Dunn, J. Math. Phys. 12, 578 (1971); D. K. Sen and J. R. Vanstone, J. Math. Phys. 13, 990 (1972).

2M. Camenzind, Intern. J. Theor. Phys. 10, 197 (1974). 3R. Mirman, J. Math. Phys. 15, 374 (1974). 4J • L. Anderson, Principle of RelatiVity Physics (Academic, New York, 1967), p. 44.

R. Mirman 2179

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