Derivation of an effective Lagrangian from a generalized scalar curvature

ANNALS

OF PHYSICS

82, 264-297 (1974)

Derivation of an Effective Lagrangian from a Generalized Scalar Curvature*+ T. A. BARNEBEY* Department of Physics, Uniuevsity of Caiifornik, Los Angeles, Carifrnib

90024

Received December 27, 1972

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral W(n) x W(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections. The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.

I. INTRODUCTION The utility of phenomenological or effective Lagrangians in duplicating the low energy mass and coupling constant relations of current algebra has been amply demonstrated [l-5]. These Lagrangians are also of interest as potential starting points of detailed particle theories [2]. The construction of effective Lagrangians is usually somewhat arbitrary, however. For example, the number of fields included and the particular Lagrangian terms used to define their properties are usually determined by the physical process being studied, rather than by any underlying general concepts. It may be argued that there is nothing wrong with this in principle, that this is the very nature of phenom* Supported in part by the National Science Foundation. t Adapted from a dissertation submitted in partial satisfaction of the requirements Ph.D. degree, Department of Physics, University of California, Los Angeles. * Present address: 1711 Purdue Ave. #9, Los Angeles, California 90024.

264 coppright AU rights

0 1974 by Academic Press. Inc. of reproduction in any form reserved.

for the

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enology. But still, it is possible to ask whether the interaction terms generally employed are to be found in some sort of primary effective Lagrangian built on a broader foundation. The purpose of the present paper is to answer this question affirmatively. The guiding idea of this work is that the boson fields in the theory are the irreducible parts of affine connections similar to, but more general than, the connection in Einstein’s gravitational field theory. A Lagrangian for the bosons is found by computing the scalar curvature associated with these connections. This view is borrowed from Finkelstein and Ramsay [6], although the method of implementing it is different. Two features of the Finkelstein and Ramsay work invite an extension of their formulations. (a) Their gauge group is not chiral SU(n) x W(n). Since this is the group usually considered by phenomenologists, a direct comparison of their curvature with effective Lagrangian theories is not feasible. (b) The FinkelsteinRamsay theory is not as minimal as could be desired, because tensor fields are included in the connections even though they are unnecessary for the correct connection transformation laws. One result of the present work is that the tensor terms may be omitted without losing any fields normally appearing in effective Lagrangians if the gauge group is taken to be chiral SU(n) x W(n). Hence features (a) and (b) are simultaneously eliminated. Generally speaking, a spirit or principle of minima&y is observed throughout this paper. Terms are discarded whenever it is possible to do so without spoiling the properties of the quantities which contain them or if the theory is not thereby reduced to a triviality. The theory developed here is invariant with respect to: (1) arbitrary relabelings of the global coordinates xU which denote the spacetime points, (2) the choice of local Lorentz frames that may be defined at each spacetime point, and (3) x@-dependent chiral SU(n) x SU(n) gauge transformations. In order that the spinor part of the Lagrangian may satisfy the invariance conditions, vector fields are minimally coupled to the spinors in the usual way. It is then necessary to relate the new fields, the spinor connections, to the affine connections used to parallel transfer global vectors from point to point in xU-space. The succeeding sections are devoted to a systematic exposition of the spinor-vector connection problem and the consequences of its solution. In Section II, the behavior of the spinor connections under the various types of transformations is investigated. Connections associated with local Lorentz vectors and global vectors are defined in Section III, and their properties are delineated. To this end, it is necessary to find a set of vectors with well defined gauge transformation laws, and the vectors chosen are the gauge currents of the spinor Lagrangian. Relations between the spinor connections and vector connections are postulated

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T. A. BARNEBEY

in Section 1II.D by comparing their respective transformation rules. It turns out that comparisons between spinor connections and the connections associated with local Lorentz vectors are particularly easy to make. Section IV contains the construction of a generalized scalar curvature which is postulated to be the boson Lagrangian. A specific world model is presented in Section V. It is the consequence of certain simple assumptions regarding the forms of the generalized aKine connections. In accordance with the minimality principle, no fields are included in the connections which do not contribute to their inhomogeneous transformation laws. In Section VI it is shown that the world model of Section V contains a successful phenomenological Lagrangian.

II.

SPINOR

CONNECTIONS

A. Covariant Derivatives of Spinors The program of constructing a theory invariant under local transformations begun here by assembling an invariant spinor Lagrangian. # denotes the spinor field and x is its Dirac adjoint, i.e.

is

It is supposed that these fields are affected by three groups of transformations follows.

as

transformation

effect

global coordinate

1G’= w>,

(2.la)

local Lorentz

SW

X’W) = xc-4 x’ zzzx&y-l = S(x) $Kx),

(2.lb)

local gauge

f(x)

= R-‘(x) gr),

(2.lc)

x’ = XL-1.

The matrix S in Eqs. (2.lb) is the usual spinor representation of a Lorentz transformation, except that it may depend upon the global coordinates xs. Appendix A contains a detailed account of local Lorentz transformations. R and L are, as yet, unspecified local gauge transformation matrices. By:definition, the generalized derivatives of the spinor fields, notated y31rand xlll , respectively, are (2.2a) h = 44 + Qd and (2.2b) Xlu = ad + xKu -

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CURVATURE-LAGRANGIAN

It is demanded that $1, and xi, be covariant under all of the above groups; that is, these rules must obtain: transformation

effect

global coordinate

= #,“(X) dx”/axu’

Xlu’(X’) = XIV(X) axyaxu

hA’(4= w h,(x)9 XL = XIP $4,‘(x)= R-‘(x)h(x), XII&’ = xIuL-l.

local Lorentz local gauge The transformation ments (2.3).

#,,‘(x’)

global coordinate

local gauge

(2.3b) (2.3~)

properties of the connections Q, and K, follow from require-

transformation

local Lorentz

(2.3a)

effect &‘(x’) K,‘(x’)

= =

Q”(X)

aX”/aP’

K”(X)

aX”/aX”’

(2.4a)

Q,,’ = SQ,S-l

+ S a,$-l

K,,’ = SK,F

- S a,SF

(2.4b)

Q,’ = R-‘Q,R K,’ = L-‘K;,L

+ R-l a,R - L a&‘.

(2.4~)

B. Invariance of the Spinor Lagrangian The spinor Lagrangian density 5?s is taken to be =% = W(-W2[xV3h

- ~~91.

(2.5)

The gamma matrices in this expression are given by yJJ 3 Xi”(X) yi,

(2.6)

where the &y(x)‘s, i = 0, 1, 2, 3, form a complete, orthogonal set of world vectors (see Appendix A), and the ya’s are the standard constant Dirac matrices. g is the determinant of the symmetric world metric; metrics and their properties will be discussed in Section 1V.B. Since yu is a global vector, it follows immediately from transformation rules (2.la) and (2.3a) that -Eosis a world scalar density, i.e., a scalar with weight one. It is also easy to demonstrate with rules (2.lb) and (2.3b) and formulas from Appendix A that it is invariant with respect to local coordinate changes.

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T. A. BARNEBEY

From Eqs. (2.1~) and (2.3~) it may be inferred that a gauge change transforms 9s into a new Lagrangian -Eps’given by 9s‘ = (i/2)( -&y~2[XL-ly’R-1q4U

- &L-ly’R-l$].

Since the theory is to be invariant under gauge transformations, related by LYR = Y,

R and L must be (2.7)

where y stands for any one of the four Dirac matrices. C. The Gauge Group The gauge group may now be specified. It is assumed that R and L have these forms: R = a,U(r) + azV(r) (2.8a) L = a,V(l) + alU(l),

(2.8b)

where al = (1 - y5)/2

a, = (1 + yW,

(2.9)

and y5 is the usual fifth Dirac matrix, whose properties are reviewed in Appendix A. Note that the projection matrices a, and al satisfy these equations a,,s,,t

ar.s-kT = 0,

(2.1Oa)

a7 - al = y6.

(2. lob)

= 4.1 ,

a, + 4 = 1, Since YSY+ condition

YY5

= 0,

(2.7) when applied to expansions (2.8) yields U-ll(r) = U(l)

(2.11a)

V-l(r) = V(l).

(2.1 lb)

and

There are now only two independent matrices, so it is possible to simplify the notation. Put U = U(r) (2.12a) and v s V(2); (2.12b)

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269

then, because of Eqs. (2.1 l), U(I) = u-1

(2.13a)

V(r) = v-1,

(2.13b)

R = a,U + azV-l

(2.14a)

L = a,V -I- aJJ-I.

(2.14b)

and and R and L, Eqs. (2.8), become

and It is further assumed that U and V are &Y(n) transformation matrices. If the generators of W(n) are assigned the symbols F, a = 1,2,..., n2 - 1, then the T’S obey these commutation relations:

(Ta,7”) = 2iC&+,

(2.15)

in which the C’s are of course the SU(n) structure constants. The generators of the complete group under consideration may be chosen as

and Their commutation

rules follow from Eq. (2.15) above and Eqs. (2.10a); they are (T,a,c3 Tyb,d

=

(7,a.c3 T;,,)

= 0.

2iCabd.1

and

So evidently the gauge group is chiral W(n)

x SU(n) [7, 81.

D. Connection Decompositions 1. The Connection Components Transformation Eqs. (2.4b, c) imply that those parts of connections Q, and K,, which transform inhomogeneously are linear in the generators of spinor Lorentz rotations and in the gauge group generators. The homogeneously transforming parts of the connections shall be omitted from future considerations, since they are unnecessary for local invariance. This step is in accordance with the principle of minimality stated in Section I.

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T.

A.

BABNEBEY

The assumed linear form may be made partially explicit by these expressions: = (&/4)

Q,

(2.16a) (2.16b)

ei,,i

K, = Kj,“(3&/4).

In analogy with R and L. (see Eqs. (2.8)), the components Q*,j and Kjui may be further expanded as follows: eiuj =

(2.17a) (2.17b)

+ a~eiud0

avQidr)

Kjuu” = a,Kj,i(r)

+ qK,,“(Z),

where Q,JJ and Kipi(‘J are all W(n) matrices. From Eqs. (2.4a) it is evident that the correct global coordinate transformation rules for the components QUi and Kjuui are Qiu;(x’) Kjf’(X’)

= (ax”/ax@‘) pvj(x)

(2.18a) (2.18b)

= K,.Yi(X)(aX"/aX"'),

with similar relations holding for the r, I subcomponents. 2. Local Lorentz Transformations

The behavior of the coefficients defined above under local Lorentz transformations is implied by the transformation laws for Q, and K, . Q&a and Kiui(3’ are found by reading off the coefficients of (r&/4) a,.,z and (~4J4) a,,l on the right-hand sides of Eqs. (2.4b). The procedure is facilitated by recalling Eq. (2. lob) and noting sap = a”ka,(u-l)kj(yiyj/4), where agj(x) is the spacetime dependent matrix which Lorentz transforms local vectors. The latter formula is most easily proved by building up an arbitrary rotation from infinitesimal ones. To complete the calculation it is only necessary to use some properties of S which are reviewed in Appendix A. The results are given below. Q”uz

(1) (a-‘)“j

Qkuz

Ki,i ('I)' = aik

(f)

(cl-l)*

if

det a = +l,

if

det a = -1;

if

deta=

if

det a = -1.

(2.19a)

+ dk a,(a-i)kj

+l, (2.19b)

- ~,k au+2-yki

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In order to conserve space, it will henceforth be understood, unless otherwise stated, that only proper Lorentz transformations, with det a = +l, are to be considered, so only the first member of each pair of transformations above will be required. Utilizing expansions (2.17), these may be compactly summarized in the forms e(,j’ = u$pi,&z-l)~j + ai~a,(u-l)f ) (2.20a) Kj,i’

= c.~~‘“K,,~(u-~): - ~~“&(a-~),~.

(2.2Ob)

Of course, all future conclusions involving local Lorentz symmetry would still be valid even if more general transformations were allowed. 3. Local Gauge Transformations To learn how the components of the spinor connections are affected by a gauge change, the coefficients of yiy’/4 and y*yi/4 must be compared on the two sides of transformation Eqs. (2.4~). But this cannot be done immediately, because the inhomogeneous terms found there are not linear expansions in the Lorentz group generators. This situation is easily remedied however if the terms in question are first multiplied by the unit spinor matrix written in the form 1 = (yn5/4)

6j” = (y5yJ4)

85.

Then the desired properties of Qiui and KjUi are seen to be Qi,,j’ = R-lQiUjR

+ aijR-la,R

(2.21a)

Kj,ir z LKjuiL-l

- La,L-1 8.3 -

(2.21b)

and

The subcomponents Q,,(9 and Kju”(3 obey similar gauge transformation rules which may be derived from Eqs. (2.21) and expansions (2.14) and (2.17) with the help of Eqs. (2.10a). They shall not be displayed, since they are not needed in the ensuing development. 4. Dependent Components It is possible to postulate a consistent relation between Q,j and Kjri and thus reduce the number of independent fields in the theory. Applying the principle of minimality, this possibility shall be adopted. Consider the quantity yQiui+y-l

272

T.

A. BARNEBEY

where y stands for any one of the four Dirac matrices and + indicates hermitian adjoint in gauge space. Now recall that a Lorentz transformation matrix such as uij satisfies the relation aij = (u-‘)ji ) from which it follows that

Similarly, R and L are unitary as a consequence of the unitarity of U and V (see Eqs. (2.14)); from these facts, along with Eq. (2.7) and the ei,lj transformation rules, Eqs. (2.18a), (2.20a), and (2.21a), it may be inferred that the object above transforms according to transformation

effect

global coordinate

(yQiuj(x’)+ y-l)’ = (yQiv,(x)+ y-l) ax”/W’,

local Lorentz

(y@uj+y-l)’

= ~~“(yQ”,~+y-‘)(u-~)~

(2.22a)

- u&,(u-~)~~, (2.22b)

local gauge

(Ypfij+Y-l)’

= ~(~p,~+p)

~-1 - Lapq..

(2.22c)

A comparison of Eqs. (2.22) with the analogous transproperties of &,i, i.e., with Eqs. (2.18b), (2.2Ob), and (2.21b), shows that the quantity Kjpp - Y@~~+Y-~ transforms homogeneously under all groups. Hence the relation Kj,i

=

Y@uj+Y-l

(2.23)

is a completely invariant one, and is hereby postulated. In this way the number of boson degrees of freedom becomes halved. E. Conjugation The finding in the previous paragraph motivates the definition of a symmetry operation which is presented here. The significance of this operation is discussed in Section V.C., where it is shown to be related to local Lorentz reflections. Let A be a matrix with gauge space indices. By definition, DEFINITION. conjugate of A, denoted (A), is given by (A) = yA+y-l,

the

(2.24)

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273

y being any one of Dirac’s matrices and + denoting the gauge space hermitian adjoint. (More exactly, there are four possible conjugations, one for each Dirac matrix. All statements regarding conjugation are true for any member of this class of operations.) It follows from the definition that the conjugation operation has these algebraic properties: <(A + m

= Go + @)

CAB) = GOGO GO = A.

(2.25a) (2.25b) (2.25~)

The conjugation notation allows Eqs. (2.7) and (2.23) to be written in the slightly more sophisticated forms (R) = L

(2.26)

and (Qiei)

= Kiui.

(2.27)

The necessary information about spinor connections has been obtained in this section. Vector connections are taken up in the next section.

III.

VECTOR CONNECTIONS

A. Vector Currents 1. Dejinitions In order to proceed with the plan outlined in Section I, the transformation properties of the affine connections associated with vectors must be found. Such transformation rules are generally obtained by constructing generalized derivatives of vector fields and then demanding that they be covariant, but this method cannot be implemented here until some specific vectors obeying known gauge transformation laws have been chosen. The natural models for gauge transforming vectors are the gauge currents derivable from the spinor Lagrangian. Therefore, their properties are explored below. The change in 9,) denoted S9”, , induced by an x”-independent, infinitesimal gauge transformation R =I

+i6R,

L=l+i6L

274

T.

A. BARNEBEY

may be written in the form

Clearly, the quantity 6.P introduced here is a conserved current, since .PS is gauge invariant, i.e., SPS = 0. The formula for the spinor Lagrangian, Eq. (2.5), and spinor transformation rules (2.1~) together with the Lagrange equations lead to the relations SJ” = (-g>lj2 xy” 6Ry% = -(-g)1/2

x r%Ly’+,

the second equality being a consequence of condition (2.7). The basic current J” is now chosen to be the matrix coefficient of 6R; therefore J” 5 (-&l/2

I,&“.

(3.1)

Various gauge components Jau of the current may be projected out of this expression with the following operation: J,.p = tr r,J” , inwhichr, ,A = 1, 2 ,..., 2(n2 - l), stands for any one of the chiral W(n) x W(n) generators and tr means normalized trace (tr 1 = 1) on all gauge matrix indices. The outer product in the formula for J”, Eq. (3.1), is a matrix with spinor as well as W(n) indices, so it is not a local Lorentz scalar. But global currents which are also local scalars may be formed as follows: J“(r) = sp a,.Ju

(3.2a)

J“(1) = sp utJ“,

(3.2b)

where a7 and al are defined by Eqs. (2.9) and sp indicates trace on spinor indices only. Although this particular set of vectors turns out to be convenient, it is not considered essentially significant; the transformation laws it provides are to be thought of as typical of some abstract collection of vectors. In the ensuing discussions it shall prove simplest to consider the currents JQ in the combinations ‘a

J”(R) = a~‘@) + aJ”(I)

(3.3a)

: J“(L) = u,.J”(Z) + qJ“(r).

(3.3b)

and

AN

2. Transformation

EFpECm

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CURVATURE-LAGRANGIAN

Properties

Spinor transformation rules (2.1) combine with the forms of R and L given in Eqs. (2.14) yield transformation laws for the currents defined above. effect

transformation global coordinate

J~(~;x’)‘=det(~)-$$J.(~;x);

local Lorentz

J”

local gauge

R’ L

0

(3.4a) if

det a = +I,

if

det a = -1;

(3.4b)

=

J”(R)’ = R-lJ”(R) Jp(L)’ = LJ“(L)

R,

(3.4c)

L-l.

The factor (-g)‘12 in definition (3.1) means that the currents are not strictly global vectors, but rather global vector densities, i.e., vectors with weight one, as evidenced by the det(ax/W) term in Eq. (3.4a). As was done in the spinor connection case, attention shall usually be limited to proper Lorentz rotations in dealing with vector connections, so that only the first of Eqs. (3.4b) shall be considered unless otherwise specified. Once again, this limitation is not essential; it merely economizes the presentation. B. Covariant Derivatives of Vectors 1. Global Vectors It happens that covariant derivatives of J”($ are easily constructed in spite of the fact that these current densities have very different transformation properties under the three groups in question. The appropriate derivatives, denoted J”(~)I, , are defined by J”(R),, = auJa(R) + p,$(R) JB(R) - J”(R) I’;@(R)

(3Sa)

J”(L)rp = a,J”(L) + J’(L) l-‘&(L)‘-

(3.5b)

and r,B,(L) J”(L),

and a straightforward application of Eqs. (3.4) shows that these derivatives are covariant as long as the vector connections r;“(f) obey the rules given below.

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T. A. BARNBBEY

transformation global coordinate

effect l-Z” (f ; xj’

= rfB (; ; x) g

g,

$

axa ----*axp av' axu' ax0 ay ax"' 3 local Lorentz local gauge

(3.6a) (3.6b)

r;“(R)’

= R-II’;,,(R)

r;v(Ly= qv(q

R + R-l a,R tip, ~-1 -

6,“~

(3.6~)

a,L-1.

The third term on the right in each of Eqs. (3.5) has the same form as the extra term known from riemannian geometry to be present in the covariant derivatives of vector densities. It is interesting to note that, written in the order indicated, these terms guarantee gauge covariance in addition to the usual covariance under global coordinate transformations. In this sense, the currents associated with the spinor Lagrangian are seen to be a fortunate choice of representative vectors. 2. Local Vectors

In the interest of treating vectors on the same footing as the locally transforming spinors, the properties of local Lorentz vectors and their associated connections are delineated here. Let Ji(R) and P(L) be the components of the current densities P(f) with respect to an arbitrary local Lorentz frame specified by hui (see Appendix A); i.e., Ja (9 Then the local vector transformation the global vectors, Eqs. (3.4).

5s XPJ’ (f). laws follow immediately

from the rules for

effect

transformation global coordinate

.P (,” ; x’)’ = det (g)

local Lorentz

Ji (9

local gauge

(3.7)

= aijJj

(9;

Ji(R)’ = R-lJi(R) R, Js(L)’ = LJ”(L) L-l.

P (f ; x); (3.8b)

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CURVATURE-LAGRANGIAN

Similarly, generalized derivatives of the local vectors may be defined as those tensors which comprise the local components of the global vector derivatives. More exactly, J” (f),,

= haiJi (;)

. IlL

Together with the expressions for the global vector derivatives, Eqs. (34, definitions (3.7) and (3.9) above imply that the local vector derivatives Ji(i)l, are given by (3.10a)

Ji(R),,

= a,Ji(R)

+ Ciu,Ji(R)

- Ji(R) r;,,(R)

Ji(L),,

= &J”(L)

+ Ji(L) Djui - l-‘,B,(L) Ji(L),

and

where the local vector connections CiUi and Djui are related to r;“(R) according to

(3.10b) and r;“(L) (3.11a)

and Djpi = AimauAai + hsjr&(L)

(3.11b)

A’= .

By utilizing the transformation properties of the F’s, Eqs. (3.6), it may be shown that CiUi and Diui transform as follows. effect

transformation

(3.12a) Ciuj

local Lorentz local gauge

=

a~~ck”,(a-l)zj

+

Djwi’ = ajkDkwz(a-l)~

dkau(a-l)kj

- aikt3u(a-l)ki;

Ciuj’ = R-WuiR

+ R-l a,R Pi ,

Djui’ = LDjuiL-l

_ ,j;L &L-l;

)

(3.12b) (3.12~)

and it is a simple matter to check that derivatives (3.10) are in fact covariant under all groups, given transformation rules (3.8), (3.6), and (3.12). C. Conjugation

The effect of a conjugation operation acting on spinor connections was found in Section 1I.E.; in the present subsection the effects of this operation upon the vector connections are defined.

278

T. A.

BARNEBEY

To do this, it is first noted that when conjugation current densities (3.3), the results are

(J” (f))

definition

(2.24) is applied to

= J” (;)+,

or, since the currents are observables and therefore must be hermitian, (J.(f))=

J”(f;).

(3.13)

(Incidently, recall from Eqs. (3.4b) that an improper local Lorentz transformation also interchanges the R and L currents. This is the first indication that the conjugation operation corresponds to a reflection of the local Lore& coordinates.) Next it is demanded that conjugation and parallel transfer in x*-space commute. In other words, the current densities are to preserve their conjugation parity when they are transferred from one place to another. The changes in the global currents, 6 ,,Ja@, produced by a shift in position 6x11are, according to Eqs. (3.5) 6 ,,J”(R) = --sx“ FzB(R) JB(R) + J=(R) l-‘;,,(R) 6x”

(3.14a)

6 aJ”(L) = - JB(L) r&(L)

(3.14b)

and 6x” + 6x” l-$(L) J”(L),

and the conditions to be imposed have this form:

S,,= @,,J). By utilizing Eqs. (3.13), (3.14), and the algebraic properties of the conjugation operation, Eqs. (2.25), it is concluded that (3.15) The relations between local and global vectors, Eqs. (3.7), and between their respective connections, Eqs. (3.1 l), lead to these additional rules:

(3.16) The attributes of the vector connections found above will be compared with those of the spinor connections in the next subsection.

AN EFFECTIVE CURVATURE-LAGRANGIAN

D. Spinor Connection-Vector

279

Connection Relations

So far, the spinor and vector connections have been studied as separate entities. As a consequence of these investigations, it is now possible to postulate consistent relations between them, a consistent relation being one which is covariant under all of the transformation groups. As might be expected, it is easiest to match the spinor connections (which transform locally) with local vector connections. An inspection of the spinor connection transformation laws, Eqs. (2.18), (2.20), and (2.21), and the analogous rules obeyed by the local vector connections, Eqs. (3.12), shows that CiUi and DjUi transform in the same ways as @uj and Kj,i, respectively. Hence the identifications Ciuj = Qiuj

(3.17a)

Djpi = KjlLi

(3.17b)

and are completely covariant, and are hereby postulated. Note that these equations are also consistent with the conjugation properties of the two sets of connections, Eqs. (2.27) and (3.16), as of course they must be. Relations (3.17) form the essential bridges between the spinor and vector connections. Through them it will be possible to discover how the boson fields in the scalar curvature Lagrangian interact with the spinors. IV. THE GENERAL BOSON LAGRANGIAN A. Introduction The standard Lagrangian density of general relativity is (-#/2 GUYguv [9], g” being the inverse of the world metric and GUYbeing the contracted curvature. In order to construct a generalization of this Lagrangian, the analogs of gu” and G,,” must first be found. That is the purpose of the next two subsections. B. Metrics Let p(R) and p(L) be a conjugate pair of global vectors which gauge transform like current densities P(R) and J”(L) respectively. Then (P(R))

= s”(L),

(4.1)

and, under a gauge transformation, f“(R)’ = R-l&R)

R

(4.2a)

and P(L),

= Lcp(L) L-l.

(4.2b)

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T. A. BARNEBEY

Generalized scalar “lenths” of p(R) and p(L) may be defined by

tr 511(R) guv@) 5y(R)

(4.3a)

tr P(L) g,,(L) 4YL>,

(4.3b)

and

where tr denotes a trace on all gauge space indices and the newly introduced properties quantities g& are generalized world metrics. The transformation of the new metrics are found by requiring the above lengths to be invariant under all groups and applying the p(z) transformation rules, Eqs. (4.2). effect

transformation

g,, (,” ; x) g

;

(4.4a)

global coordinate

g,, (,” ; xr)’ = $

local Lorentz

g&w (;)’ = &” (f);

(4.4b)

local gauge

g,,(R)’ = R-kv@) 4 g,,W' = km L-l*

(4-k)

It is further demanded that the two scalar products, Eqs. (4.3), be conjugates of each other; more precisely,

When properties (2.25) of the conjugation operation are taken into account along with e(R) conjugation rule (4. I), this condition leads to the relation (g,,(R))

= g&9.

(4.5)

Quite generally, the metrics may be written in the form (4.6) in which

and

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CURVATURE-LAGRANGIAN

281

This is clearly a decomposition into irreducible parts in the sense that the two terms on the right in Eqs. (4.4) cannot be transformed into each other under any of the three groups. It is now assumed that &(@

= &W

= l&v 3

(4.7)

where 1 is the unit matrix in gauge space and g,,“(x) is a real, numerical function. This assumption is invariant, as a simple application of transformation laws (4.4) shows. Finally, it is assumed that g,, is the symmetric world metric of general relativity. Accordingly, g,, and its contravariant counterpart gpV,defined by g,, P” = &by, shall be used to raise and lower global tensor indices. For example,

C. Curvatures

Suppose parallel transfer laws (3.14) are used to move J”(R) and J”(L) around an infinitesimal loop in x”-space with area element &P’. The resulting changes in the current densities, W($-), turn out to be W(R)

= -ss”“[r~“~(R)

P(R) - J”(R) q&“(R)]

and

r;*(R) and r;,,,(L) are clearly generalizations of the usual gravitational curvature [IO] while &,(i?) and Q,,,(L) arise solely from the gauge group, as evidenced by the fact that they would vanish in its absence.

282

T.

A. BARNEBEY

In analogy with general relativity, P,,,(L) are postulated to be [lo].

generalized contracted curvatures P,,(R) and

r,“(R)

= cd0

(4.9a)

r,“(L)

= clu).

(4.9b)

and

Their transformation laws follow from curvature definitions rules for global vector connections, Eqs. (3.6). transformation

(4.8) above and the

effect

global coordinate

(4.10a)

local Lorentz r,,,(R)’ = R-ll-‘,,(R)

local gauge

r,,(L)’ = u,,(L)

R, L-I.

(4.1Oc)

Note that the generalized curvatures are tensors under all groups, a property which is appropriate for extensions of the general relativistic curvature tensor. For completeness, the behavior of r,,(R) under conjugation is recorded below. It is a consequence of curvature definitions (4.8), (4.9), and connection conjugation rules (3.15). (4.11) 029) = rdL). D. The Curvature Lugrangian The boson Lagrangian

5?” is postulated to be

=%= i (-We tr V’,,(R) gW) + P’(L) r,,(R)1 + 4 C-P2 tr kv@) 8W

+ g’“-‘(L)g&)1,

(4.12)

where a and b are arbitrary numbers. The last two terms, unlike the first two, have no parallel in standard general relativity, but they are necessary in order that the Lagrange equations for g,,(R), g,,(L) and the antisymmetric parts of the rUV’s be nontrivial. The boson Lagrangian defined above is invariant under all groups, as may be seen by applying transformation rules (4.4) and (4.10). It has also been constructed

AN EFFECTIVE CURVATURE-LAGRANGIAN

283

to be invariant under the conjugation operation; this is readily checked by utilizing Eqs. (4.5) and (4.11). Conjugation invariance may be regarded as an extension of the hermitian adjoint-transposition invariance introduced by Finkelstein [6], which in turn generalized Einstein and Kaufman’s transposition invariance [l 11. The similarity between conjugation and Finkelstein’s operation becomes apparent when connection conjugation rule (3.15) is compared with Eq. (2.7) of [6]. V. A WORLD MODEL A. The Connections The very broad theory in the previous sections is specialized here to a theory of fermions, vector mesons, and gravitons, and collected into a single Lagrangian in the next subsection. Fields carrying spin higher than two are omitted. The procedure is to postulate particular forms for the global vector connections and then apply relations developed earlier. Irreducible parts of the connections are introduced by these equations:

where the number g is a coupling constant. The new fields GE,(f), P,, , R, , A, , and L, appearing above satisfy the conditions

G:v(;)+ = G%(3, G;v(“,, = GT;(“,), p,+ = P,) R,+ = R, ,

Apt = A, ) L,+ = L, ,

(5.2)

so the three terms contributing to each r,& are, from left to right, hermitiansymmetric, antihermitian-symmetric, and antihermitian-antisymmetric. Since the inhomogeneous parts of global connection transformation rules (3.6a, c) are symmetric and anti-hermitian respectively, any hermitian-antisymmetric parts of I’;“(R) or r;“(L) would transform homogeneously if they were included. Therefore, the assumption that they vanish, which was tacitly made in Eqs. (5.1) is a covariant one, and in accordance with the minimality principle of Section I.

284

T.

A.

BARNEBEY

Each term in the connection expressions transforms into itself under all groups according to laws derived from the properties of I’;,(R) and F;,(L), Eqs. (3.6); hence the label “irreducible parts.” effect

transformation global coordinate

(5.3a) local Lorentz

(5.3b)

local gauge

(5.3c)

global coordinate

(5.4a)

local Lorentz

(5.4b)

local gauge

(5.k)

Also, since the global connections conjugate according to Eq. (3.15), it is found that

= -A,, (RJ

(5.5a)

= -L,.

The connections are now further simplified

by imposing two final conditions:

G:,(R) = G%(L) = lG:,,

(5.6)

AN

EFFECTIVE

where Gg, is the gravitational

CURVATURE-LAGRANGIAN

285

connection field, and tr P, = tr R, = 0,

(5.7a)

tr (1, = tr L, = 0.

(5.7b)

It is readily shown by utilizing transformation rules (5.3) that Eq. (5.6) is completely invariant. Since the inhomogeneous parts of transformations (5.4~) are linear in the gauge group generators, and therefore traceless, any nontraceless parts of the connection fields would be gauge tensors. Hence conditions (5.7) eliminate the last vestiges of tensor fields from the vector connections, and they represent the last application of the minimality principle. The result of the assumptions made in this subsection is a theory which is more minimal than the previous theories [6] it is modeled upon in the sense that no fields are present that do not perform an essential function. At this point the complement of boson fields consists of gravitation and four vector gauge fields, two in the R connection and two in the L connection. B. The Complete Lagrangian It is finally possible to utilize the various relations developed previously and assemble a coherent theory. Basically, the boson Lagrangian provides the equations of motion for the irreducible parts of the vector connections, while their interactions with the spinors are found through the spinor connection-vector connection relations. When Q,, K, expansions (2.16), spinor-vector connection relations (3.17), local-global connection relations (3.11) and global connection expressions (5.1) are all substituted into the spinor Lagrangian, Eq. (2.5), it becomes

=%= (i/2)(-W” [XV% + To+ (k/2) PJ # - XE + L + t&/2) 4 ~“$1, (5.8) where r, = (rd/4)

hi,a,Pj

(5.9a)

I, = -A”jauXi,(yjyi/4).

(5.9b)

and Note that G”,,(f), d, , and L, do not enter into this expression for -Es . G%(R) and G;,(L) cancel each other as a consequence of condition (5.6), so the spinors interact with the gravitational field only through the factor ( -$j)1/2 and the X terms given by Eq. (5.9). The disappearance of (1, and L, is due to the asymmetry of the connection terms in which they appear (see Eqs. (5.1)). The contracted curvatures associated with the world model of interest are found

286

T.

A. BARNEBEY

by using the assumed forms of the connections, Eqs. (5. I), in curvature definitions (4.8) and (4.9). When the results, together with metric forms (4.6) and (4.7), are substituted into boson Lagrangian (4.12), it becomes

- @/W-kY’*

Wf’,y - WV - W’, - 4. , p, - WI g’W

+ x

WE~Y2

tr{g,,(R)gY’-‘(R) + g”-‘(L)g&l

W>(-W2

Wp, - RuIIPv- &I g”’ + gWL - Ll[~y - Llh (5.10)

where the constants denoted b, c, d are simply related to the numbers a and b occurring in Eq. (4.12), GUYis the usual contracted Riemann tensor and

puv= a,p,- u, + (WW, ,pJ,

(5.11a)

R,, = %& - a,& + WW,

, %I,

(5.11b)

4,

,-A),

(5.1 Ic)

= 44,

- ad6 - WW,

Luy = auk - up

- WW,

, a.

(5.1 Id)

By employing the transformation rules for the P,, , R, , A, , and L, fields, Eqs. (5.4), it is easily shown that the new quantities defined above are tensors under all groups. Explicitly: transformation

effect p,Yw>’ RJx’)

ay

P,(X),

-Iax”’

R,(x),

ax0 aY

A,(x),

ax0

= w

global coordinate fL”W

(5.12a)

Lpy(x’)’ I = -33 -Iax”’ L,(x);

local Lorentz

local gauge

Pw’ = P,Y 3

4’

= 4,

,

R,w’ = R,, ,

L’

= LY ;

(5.12b)

(5.12~)

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CURVATURE-LAGRANGIAN

287

The complete Lagrangian JZ stemming from assumptions (5.1) is simply 9 = 3s + =% ,

(5.13)

Z’s being given by Eq. (5.8) and 6pB by (5.10). 2 is simultaneously a global scalar density, a local Lorentz scalar and an invariant under local guage transformations. C. Conjugation and Local Lorentz Reflections In analogy with spinor connection expansions (2.17), P,, and A, may be expanded in terms of y6 projection matrices.

p, = al.P,tr) + alp,(l)

(5.14a)

A, = a,&>

(5.14b)

+ aJ,tO.

Of course, the components P,(I) and A,(i) are not all independent; conjugation relation (55a) together with definition (2.24) of conjugation and the hermiticity of P, and A, implies

pu0; = -A, (,‘).

(5.15)

The behavior of the components introduced above with respect to an improper local Lorentz transformation may be inferred from the vector connection expressions, Eqs. (5.1), the global-local connection relations, Eqs. (3.1 l), the postulated vector-spinor connection relations, Eqs. (3.17) and the Lorentz transformation properties of the spinor connections, Eqs. (2.19). These verious equations combine to yield the following behavior under a local Lorentz reflection:

PLa (3’ = pw(I, and (1, (1)’ = (1, (f)

(det a = -l),

which, with the help of component be rewritten in the form

relations (5.15) and expansions (5.14), may

P,’ = -A,

(5.17a)

and A,’ = -P, 595/82/x-19

(det a = -1)

(5.17b)

288

T. A. BABNEBEY

When the Lore& reflection rules above are compared with the effects of conjugation as given by Eq. (5.5a), it is seen that each of the two operations duplicates the action of the other. In this sense, conjugation and local Lorentz reflections correspond. Similarly, it may be shown that the analogous statement regarding the fields R, and L, is also valid. And in fact, by continuing the development in this direction, the conjugation invariance of the theory is found to be equivalent to local Lorentz reflection invariance. D. Local Vectors and Axial Vectors Consider yet another pair of fields pu and au , which are introduced relations:

by these

P, = pu + y5au

(5.18a)

4

(5.18b)

= -pu + Pa, .

With the help of expansions (5.14) and the definitions is found that pp = U/W,(r)

+

of a, and al , Eq. (2.9), it

P,(Ol

and

a, = U/NP,(r) - p,W and from the appropriate properties of components P,,(a, Eqs. (5.16a), the improper Lorentz transformation rules of p,, and a,, are easily derived. ‘-

Plb - Pu

a,’ = -a,

(det a = -1).

Clearly, pu is a local scalar and a, is a local pseudoscalar. Expressed differently, the components pi and ai, which appear in the local Lorentz frame expansions PP = hu”ipi

and au = Auia i form a local Lorentz vector and axial vector respectively. Of course, a similar vector and axial vector could also be associated with R, and L, , but they shall not be explicitly displayed because R, and L, may be used instead to represent pseudoscalars, as is shown in Section V1.C. This subsection completes the discussion of the world model and its general properties. Its relationship to effective Lagrangian theories is contained in the following section.

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289

CURVATURE-LAGRANGIAN

VI. THE PHENOMENOLOGICAL

LAGRANGIAN

A. Introduction It shall be shown below that a successful phenomenological theory is contained within the scalar curvature Lagrangian given by Eq. (5.13). In order to obtain the theory of interest, two steps are taken: (1) Attention is restricted to the flat spacetime limit of the Lagrangian, and (2) the terms in the boson Lagrangian, Eq. (5. IO), which involve the commutators (6.la) and (-44 - L, 9 A” - -L)

(6.lb)

are dropped. Step (1) is taken simply because there are as yet no phenomenological theories known to the author which include the effects of gravitation, and since it is with such theories that the present results are to be compared, the curvature of spacetime shall be ignored. Step (2) also leads to a final Lagrangian which is more directly comparable with phenomenological theories. It cannot, however, be justified as a limiting process. Rather, the attitude adopted in taking this step may be stated as follows. An invariant boson Lagrangian is to be constructed and interaction terms are needed; they are to be selected from among the several separately invariant terms found in the generalized scalar curvature, Eq. (5.10); all terms except those containing the quantities (6.1) are chosen. Of course this is a tentative procedure which is being followed for the sake of obtaining a familiar effective Lagrangian. There is always the possibility that the omitted terms provide a direction in which the usual theories could be usefully expanded, but this direction has yet to be explored. B. The Basic Lagrangian In accordance with (1) and (2) above, the limits Gv + 0,

h,i -+ 8 I4

and g IA”+ %v (T,,” is the Minkowski metric) are taken in Eq. (5.13) and the appropriate commutator terms are dropped. Also, specific values are chosen for the constants b, c, d. The Lagrangian then becomes 3 = -Ep, + % ,

(6.2a)

290

T.

A.

BARNEBEY

with 3s = m)[xr”(a;T and gB = -(l/8)

tr(P,,

+ &/2) PJ * - XK + W2) 4) e/4 - +RJ(P

+ (mz/4) tr(P, - R,)(P

- $W) - (l/8) tr(ll,,

(6.2b)

- &,y)(~~y

- RU) + (m2/4) tr(A, - L&P

- +P’)

- P).

(6.2~)

Note that g,,(R) and g,,(L) have been eliminated via their equations of motion; this simplify&g step is p>rmissible in the sense that it leaves the Lagrange equations of the vector fields unaffected. Obviously, the newly introduced symbol m stands for the bare mass of the vector mesons. In the limit above the x”-dependent structure of spacetime disappears and global and local Lorentz transformations become synonymous. All tensors become Lorentz tensors and the spacetime transformations under which 8 is invariant are the usual constant Lorentz rotations. Also, the fields pU and czrrwhich compose P, and (1, (see Eq. (5.18)) now transform as a vector and axial vector, respectively. The effect of a local gauge change upon the various fields in the Lagrangian is already known. The spinors transform according to Eqs. (2.lc), the vectors according to Eqs. (5.4~) and the tensor transformation laws are given by Eqs. (5.12~). Of course, the theory is still invariant under local gauge transformations that satisfy condition (2.7), and this is true even though the vector fields have finite masses. C. Special Gauges and Pseudoscalar Fields

It is evident upon inspection of Lagrangian (6.2) that R, and L, do not directly interact with the spinors, and it is tempting to try to eliminate them from the theory. One way of accomplishing this would be to make the covariant assumptions P, = R, and A, = L, , but then the vector fields in the theory would be massless. Since vector mesons have measurable nonzero masses, this idea could not lead to a phenomenological Lagrangian. An alternative procedure is to assume that R, and L, vanish. However, because of the inhomogeneity of transformation laws (54c), such an assumption is not gauge invariant; R, and L, can only vanish within one special class of gauges at a time (the members of such a class differing from each other by x”-independent rotations in gauge space). It is hereby supposed that such a class does exist. A member of this class shall be called a vector gauge, and fields in a vector gauge shall be distinguished by the symbol ‘. The assumption being made may then be written & = L, = 0, (6.3a) Is,, = I?+, = 0. (6.3b)

AN

EFFECTIVE

291

CURVATURE-LAGRANGIAN

Together with rules (5.4~) and (5.12c), it implies (6.4a)

R, = (2/ig) R-V,R, L, = -(2/ig)

Lap

(6.4b)

and R,, = L,, = 0,

(6.44

where R and L are now to be interpreted as representing a transformation between a vector gauge and any other gauge. Now consider a gauge which differs from a vector gauge by a purely chiral transformation, that is, a transformation satisfying R = L EE Q-l.

(6.5)

The matrix Q introduced here may of course be written as a function of spacetime dependent group parameters 4”(x), k = 1,2,..., n2 - 1, that is,

and it is shown in Appendix B that as a consequence of the chirality of the transformation represented by Sz, the 4’s are pseudoscalar fields. From this fact and expressions (6.4a, b) it is seen that pseudoscalars appear in Lagrangian (6.2~) whenever gauges obtainable from a vector gauge via chiral transformations are contemplated, and it is natural to designate these special gauges as the physical gauges. Then the pseudoscalar group parameters 4”(x) may be regarded as measuring the “angle” between a vector gauge and a physical gauge. With all of the above assumptions and ideas taken into account, i.e., Eqs. (6.3-6.5), the boson Lagrangian, Eq. (6.2~) is found in a general gauge to be -9” = -(l/8)

tr[P,,P’

+ n,,+4““] + (m2/4) tr[P, - RJ2 + (m2/4) tr[cl, - LJ2,

whereas in a vector gauge it is (6.6)

and in a physical gauge it becomes -% = -(l/8)

tr[Pw,PuY + A,Jlry]

+ (m2/4) tr[-4, + (2/ig)

+ (m2/4) tr[P,, - (2/ig)

sz-la,w.

Qa,Wy (6.7)

Since the fields R, and L, do not enter into spinor Lagrangian (6.2b), it clearly has the same appearance in all gauges.

292

T.

A.

BARNEBEY

D. Gauge Covariance It is interesting to note that the special gauge forms of the boson Lagrangian given by Eqs. (6.6) and (6.7) are actually equivalent to expressions derived in the more usual approaches [3-51 by simply adding a mass term to the Yang-Mills theory [12]. However, the present development does not agree with the earlier theories in so far as gauge transformation properties are concerned. The massive Yang-Mills Lagrangian and equations of motion are not gauge invariant, whereas the generalized scalar curvature is invariant and yields covariant equations of motion. But the curvature contains more fields than the Yang-Mills Lagrangian, and it is by placing constraints upon some of these fields that agreement between the two approaches is reached in any particular gauge. The discovery that certain gauges can have special physical significance even though the Lagrangian is invariant has a precedent in the massless Yang-Mills case. The author has found classical solutions to the massless Yang-Mills equation which exhibit localized energy densities only in one particular class of gauges [13]. E. Conclusion The physical gauge Lagrangian developed in this paper and contained in Eqs. (6.2a), (6.2b), and (6.7) may be easily cast into the following simple form: 2 = i [xWV;#

- xVzy’“*]

+ -$ tr(V;Q)(V*W)+

- t tr[P”,Puv + A,&“] + -$ (sZV;)(sZV-W)+,

(6.8)

where the covariant derivative operators V; and VL are given by

and v,e 3 (a, + + q, respectively. $P represents a theory which describes fermions, vector and axial vector mesons and pseudoscalar mesons, and which possesses vector dominance in the sense that the pseudoscalars do not directly interact with the spinors. Lagrangion (6.8) is essentially identical to the primary Lagrangian from which Finkelstein et al. have derived a well-determined phenomenological theory that reproduces all the usual low energy W(2) and W(3) results [4, 5, 141. The present

293

AN ETFECnVE CURVATURE-LAGRANGIAN

formulation differs only in that the right-left (i.e., P, - ll,) symmetry is more explicit; compare, for example, Eq. (6.8) above with Eq. (1.2) of [4]. To summarize, the local invariance group has been expanded to include chiral SU(n> x W(n) transformations, and it has been shown that the generalized scalar curvature contains a useful effective Lagrangian.

APPENDIX

A: LOCAL LORENTZ FRAMES

1. Local Lorentz Tensors The formalism in the text is required to be independent of the labeling of spacetime points, i.e., it is to possess global or world covariance, but the arbitrary coordinates xU thus allowed are not the directly measured ones. Rather, measurements are made with respect to an arbitrary Lorentz frame constructed in the neighborhood of a particular spacetime point. The theory is also required to be independent of the choice of these local frames, that is, locally covariant. Local Lorentz frames and their associated local tensors and spinors are discussed in this appendix, the general approach being adopted from a paper by Robertson [15]. Choose four contravariant world vector fields XUi(x), i = 0, 1, 2, 3, and define four covariant vector fields X,,(X) by A~&

= qij

(A.la)

and huj&j7p = au,,

(A.Ib)

where 7;1~,and qii are the Minkowski metrics. Let Au and B,, denote a pair of arbitrary world vectors. They may of course be expanded in terms of the x’s; explicitly: A” = )cuJ’, B, = &Bi, 64.2) and with the help of Eq. (A.la) these expressions may be inverted to yield the following formulas for the world scalar components A* and P:

By utilizing expansions (A.2) and orthonormality condition product of global vectors A” and B, is found to have the form AuB 11= A”B$,

(A.la),

the scalar

= A”Bo - A - B ,

which is seen to be the Lorentz scalar product with A* and B’ playing the roles of Lorentz vectors. This identification is now made more complete.

294

T.

A.

BARNEBEY

Note that Eqs. (A.l) do not completely determine Pi and hut ; linear combinations of them can also be orthonormal and form complete sets. But if a new collection of basis vectors h~‘i and Xui’ is defined by (A.4a) and (A.4b) then the coefficients a$ are restricted by the requirement that Eqs. (A.l) must remain valid among the X’s. In particular, this condition implies

Clearly, since qij is the Minkowski metric, a,j is a Lorentz rotation matrix. Furthermore, as a consequence of Eqs. (A.3) and the invariance of world vectors with respect to changes in the local Lorentz frames, sets of components such as Ai and Bi are really Lorentz vectors as hinted above; for instance, Eqs. (A.3) and (A.4b) combine to give the following Lorentz transformation rule for Ai: Ai’ = aijAj ,

(A.3

where dj

f

qakakzq$j.

The basis vectors h and the matrices a( represent local Lorentz frames and Zocal Lore& transformations, respectively, in the sense that, in general, both sets of quantities are functions of the spacetime coordinates xU. Many of the formulas above become neater if the metrics qij and $1 are used to raise and lower local Lorentz indices in the usual way; e.g., Ai

= u--r)

ijh

Ai = qijAj,

Y3 9

f&j

G

fZik7)kj,

etc.

Then, for example, Eqs. (A.l) become

It is now a simple matter to develop a local tensor calculus. Local vectors transform according to Eq. (A.5), higher rank tensors transform as AiAj,

AfAjAk,...,

etc *,

and all such quantities are global scalars if they comprise the local components world tensors.

of

AN

EFFECTIVE

295

CURVATURE-LAGRANGIAN

2. Local Lorentz Spinors

The local spinor calculus may be taken over directly from the usual spinor theory, which already contains the mechanics of Lorentz transformations. The only innovation is that the transformations are now allowed to be spacetime dependent. Reviewing briefly, if 16 is a spinor, then under a local Lorentz transformation represented by the spinor matrix S(X), # changes according to the rule

S is related to the vector transformation plyis

matrix

aij by

= aij(x) yj

with solution qx)

=

e(l14wqiLd,

where the o’s are spacetime dependent Lorentz group parameters satisfying

and the y’s are the usual constant Dirac matrices which obey the anticommutation rules yiyj + yjyi = 2.p. 04.7) The spinor fields are assumed to be global scalars, and of course the constant Dirac matrices are scalars under both global and local transformations. The hermitian spinor matrix y5 is defined by y5 = (i/4!) ~~~~~~~~~~~~~ = iy”y1y2y3,

and its properties include (y512 = 1, y5yi + ygy5 = 0, and S-ly5S = det ay5. In the text, some additional Dirac matrices are employed which transform as world vectors. They are defined by yu(x) = &i(X) yi

and

Yu(X)

z hifi(X)

Yi

*

T. A. BARNEBEY

296

These expressions, along with Eqs. (A.6) and (A.7), imply Y”YY+ y”y’ = 26P”) so the y@‘s appear to be the natural generalizations of the usual Dirac matrices. Note that these objects are neither constants nor local scalars, however.

APPENDIX

B: LORENTZ REFLECTIONS OF THE

CHIRAL TRANSFORMATION PARAMETERS

According to rules (5.4c) a chiral gauge transformation, i.e., one which satisfies condition (6.5), produces the following changes in the fields P, and /1, : P,’ = J2PJ2-l

+ (2/ig) GVJF,

A;

- (2/ig)

= Q-lA,Q

i-2-1au52.

(B.la) (B.lb)

In this appendix the local Lorentz reflection operation shall be denoted by * in order to distinguish it from the gauge transformation. With the new notation, reflection Eqs. (5.17) become Fu = -A,

(B.2a)

and (det a = -1).

cr, = -P,

(B.2b)

By reflecting Eqs. (B.l) and applying rules (B.2) above, A,’ = ztkl,~-l

- (2/ig) s”ia,a-1

P,’ = &lPuG

+ (2/ig) C%,fi

and

are found. But the latter relations must be consistent with Eqs. (B.l); therefore, si = 52-l.

(B-3)

The fields c$“(x) which parameterize the chiral transformation are coordinates in the group space, and it is always possible to choose the origin of coordinates such that a<-$) = In-l(4).

297

AN EFFECTIVE CURVATURE-LAGRANGIAN When this is done, Eq. (B.3) is seen to imply QC&

= Q(-$1

or C$ = -$ so the group parameters stated in the text.

(det a =

of the chiral

-1),

transformation

are pseudoscalar

fields

as

ACKNOWLEDGMENT The author is grateful to Robert J. Finkelstein for suggesting this work and for the abundant, valuable advice and guidance he provided while it was in progress.

REFERENCES 1. 2. 3. 4. 5.

J. SCHWINGER, Phys. Rev. 152 (1966), 1219. S. WEINBERG, Phys. Rev. Left. 18 (1967), 188. J. WESS AND B. ZUMINO, Phys. Rev. 163 (1967), 1727. R. FINKEL~TEIN AND L. STAUNTON, Physica 47 (1970), 182. R. FINKELSTEIN, L. STAUNTON, AND J. HILGEVOORD, Phys. Rev. D 1 (1970), references to other effective Lagrangian

6. R. FINKEL~TEIN

AND W. RAMSAY,

Ann.

2832 (Extensive

theories may be found in this paper.) of Physics

(NY.)

21 (1963),

408.

(Relevant

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