. A CONSTRUCTIVE DEFINITION FOR BILINEAR FORMS ON SPINORS Jorg Schray, Charles H-T. Wang School of Physics and Chemistry, Lancaster University Lancaster, LA1 4 YB, UI<
[email protected] .lanes. ac. uk
[email protected]. ac. uk
Abstract. The following article gives a constructive definition of a bilinear form on a space of spinors. The construction is purely algebraic terms without reference to a matrix basis. The new definition is of interest both in principle, indicating how structures of simple algebras are inherited by their minimal ideals, and for the practical implementation of a symbolic yet efficient way of implementing Clifford algebras in a computer algebra system as is shown in [1). An analogous construction for the charge conjugation operation is indicated. Explicit connections to the standard definition and development of bilinear forms on spinors are made throughout.
1. Introduction
This article grew out of the desire to lift the mystery enshrouding an enigmatic Clifford element C that happens to show up in the definition of bilinear forms on spinors in (p. 63 ref. 2] .1 (This is one of the cases where an existence theorem is nice as long as you do not want to do an actual computation.) The beauty of a representation independent, algebraic approach to spinors seemed to be spoiled, since one supposedly needs a matrix basis to get one's hands on C. In the construction of bilinear forms presented here, as well as in a similar construction of the charge conjugation operation which is based on the same principles, this beauty is restored. Apart from this noble endeavor, the constructions are very useful for implementing bilinear forms and charge conjugation on spinors in computer algebra packages in a purely symbolic yet efficient manner. 1 The element in the reference is actually named J, but there are already too many j's in this paper.
Advances in Applied Clifford Algebras 6 No. 2, 151-158 (1996)
152
A Constructive Definition for Bilinear Forms on .....
J. Schray, C. H-T. Wang
Section 2 contains some introductory material and presents the usual definition to bilinear forms on spinors. Section 3 presents our constructive definition and compares it to the previous one. Section 4 gives an example appliC.) The symmetry factor ( describes the symmetry properties of the bilinear form, i.e., whether it is symmetric or symplectic, hermitian or antihermitian:
(¢, 1/J)
c- 1 v q;.J v 1/J = ( c- 1 v q;.J v 1/J v c- 1 .J v c (C- 1 v(C- 1 v1/J.J v¢).1 vC ((1/J,¢)jc.
(8)
3. A Constructive Definition of Bilinear Forms There are many invertible elements C E 2t that satisfy Eq. (4) but the bilinear forms arising from those only differ by a factor in 1). This observation should raise some suspicion about how much information about C actually enters into the definition of the bilinear form in Eq. (3). In the approach presented in this section we suggest that the :D-module structure of lP v2lviP.r provides a both simpler and more fundamental avenue to introduce bilinear forms than the one above. Of course, both approaches are completely equivalent.
154
A Constructive Definition for Bilinear Forms on .....
J. Schray, C. H-T. Wang
We examine the role of c- 1 in (c/J, 7/J) (see Eq. (3)). 7/J E 2LvlP implies 'f/;.7 E JP.7 v 2L, which allows us to replace 'f/;.7 by JP:f v 7/Jj. Since (c/J, 'f/;) E lP v 2L v 1P we may multiply by lP from the left, whence
c- 1 v IP:r v c/J:r v 7/J. (9) 1 So once we know lP v c- v JP.7 E 1P v 2{ v JP:l, the bilinear form is completely (c/J, 7/J) = IP v
determined. However, by Eq. (4) lP v Qt v jp.7
= lP v mv c v lP v c-l = 1) v c- 1
(10)
is a 1-dimensionalleft ::D-module. Therefore, there is a simple, concise, constructive definition of bilinear forms. Namely any non-zero element W E 1P v mvfP.7
w v c/J.7 v 7/J = p lP v c- 1 v jp.7 v c/J.7 v 7/J = p (c/J, 7/J) (11) defines a non-degenerate bilinear form that agrees with ( ·, -) up to a factor in p E 1), which is determined by W = p lP v c- 1 v JP.7, and all bilinear forms derived from .J can be obtained in this fashion. Note that this definition eliminated any redundant information contained in C. While different C's satisfying Eq. (4) may define the same bilinear form, there is a 1-1 correspondence between elements of lP v 2L v JP.7 and bilinear forms. Furthermore the symmetry of the bilinear form defined in Eq. (11) depends on the anti-automorphism iw on ::D induced by W and .J: Jjww = W v J.7. (12)
(c/J, 7/J)w
:=
Since the left ::D-module lP B E ::D such that
v
2{ v JP.7 is 6bviously stable under :1, we have
w:r =ow.
(13)
With this definition we see that iw is an involution if B is in the center of ::D: (ojw)jww W v (ojw).7 = ,(ojw v w:r).7 = (ojw v W v (B- 1 ).7).7
= (w v o-:r v (o- 1 ):r):r = o- 1 J w:r (14) The behavior of the bilinear form ( ·, ·)w under exchange of arguments can be expressed in terms of iw and B:
(c/J, 7/J)w w:r
w
v
c/J:r v 7/J v w:r = w
v
(W v '1/J:r v c/J):r
w v ((7/J, c/J)w ):r ~ (7/J, cfJ){;w (7/J, c/J >W' o- 1 w:r ('f/;,c/J){;B-1
(15)
Advances in Applied Clifford Algebras 6, No. 2 (1996)
155
Therefore, choosing wE lP v !ZL JP:f such that w:r = ±W induces products of the corresponding symmetry. It is actually simple to find an appropriate W. If 1) is a division algebra over the reals, the left i)-module lP v 2t v JP:f ,is in particular a vector space over lR with .J acting on it. Since .1 2 = 1, there are !R-linear projectors !(1 ± .J) splitting lP v 2t v JP:f into a direct sum of two !R-subspaces, one is fixed by .J, the other one is inverted by .J. For 1) = JR, lP v 2t v JP:f is !-dimensional and the symmetry is fixed. The complexification of this case is obtained, if 1) = (]) and .J is (/)-linear, i.e., .J does not involve complex conjugation. Otherwise both symmetries are possible. An algorithm for constructing a bilinear form with a specified symmetry has to generate a non-zero element W ·of lP v 2t· v JP:f. If W ± W :r is non-zero an appropriate element is found otherwise further JR.- linearly independent elements of lP v 2t v JP:f need to be generated. In any case fixing the symmetry of the bilinear form still leaves a real scale factor undetermined. Since for the quaternionic case, 1) = !HI, one of the spaces !(1 ± .J)(IP v 2t v JP:f) has an !R-dimension bigger than 1, there are bilinear forms with the same symmetry factor fJ = ±1, but with different induced involutions iw. These involutions are related by an automorphism of lHI, which can be seen from the relationship of iw and ic: Jic
= p-lJiw p.
(Note that
ic
=
iw'
( 16)
for W' = lP
v
c- 1 v IP:r = p- 1 W.)
4. Example
Whereas the previous section is applicable to any simple algebra we will now turn to Clifford algebras for our example. In particular, we will demonstrate the usefulness of the alternative construction by classifying spin invariant (/)bilinear forms on Dirac spinors in even dimension n = 2m, where the complex Clifford algebra is indeed simple. Since we assume (/)-linearity in both arguments the signature of the metric is irrelevant and may be chosen Euclidean, i.e., we consider the Clifford algebra Cl(n) over the complex vector space V = (J)n with non-degenerate metric g. Let { ei} be an orthonormal basis of V, which implies that fori= j, for i f. j.
(17)
Then
(18)
156
A Constructive Definition for Bilinear Forms on . . . . .
J. Schray, C. H-T. Wang
--~--------~--------~
e,
is a primitive idempotent. For the involutwns .:J = which just interchanges the order of products, and .:J where ., is the automorphism that inverts V we obtain
=e.,,
Therefore we have a non-zero element W E 1P ~ Qt v
W =
IPvel v·
· ·vern vfP.:l =
JP.:l
in both cases
2~ (e1 +iem+I) (e2+iem+2) ···(em +ie2m ).(20)
To obtain the symmetry of the mduced bilinear forms we consider
e
e
So for even m both involutions and TJ are equivalent and they are inequivalent for odd m. The induced bilinear form is symmetric for .:J = and m 0, 1 mod 4 or .:J = TJ and m 0, 3 mod 4, otherwise it is antisymmetric.
=
e
=
e
5. The Principle of Inheritance
The ideas that give rise to the alternative approach to bilinear forms on spinors as described in section 3 can be tra~ed to a deeper relationship between (anti-)automorphisms of a simple algebra and maps on its minimal ideals. We restrict ourselves to the presentation of the logical structure of the theory below, since a complete derivation is in close analogy to the development in section 3 Consider a general (anti-)automorphism* (.:!) of a simple algebra Qt. (Complex conjugation and one of the standard involutions, e.g., may serve as examples in the case of Clifford algebras.) Given such maps on Qt we observe a first inheritance from the simple algebra to its minimal ideals. Namely, on any any minimall eft idea Qt v 1P (or equivalently primitive idempotent 1P ) we have an induced map
e,
,c:
Qt vfP---+ Qt v
1/J
r-+
1/Jc
(22)
fP
:=
1/J* v B,
for any non-zero element B E dual
JP* v
7:fPvQt---+QtvfP
1/J
r-+ {/; :=
w v '1/J.:J,
Qt v 1P, and from a minimalleft ideal to its
(23)
Advances in Applied Clifford Algebras 6, No. 2 (1996)
157
for any non-zero element of W E lP v Q( v JP:f. Since both JP* v Q( v lP and 1P v Q( v JP:f are 1-dimensionaJ one-sided 1}-modules, namely a right and a left one, the maps (22) and (23) are defined up to elements pf 1). The crucial .properties of these maps are:
(24) (Of course, as our notation is suggesting, these are defining properties for spinor charge conjugation and spinor adjoint, if * is identified with complex conjugation and .1 is identified with one of the standard involutions.) In a next step of inheritance we have induced maps on 1) ::::: lP v Q( v lP, namely an automorphism b
(25) and an antiautomorphism j
(26)
*
which we have implicitly defined for t5 E 1). If and .J are involutory then b and j can be made involutory by choosing B* v B = ±P and W .7 = ± W, which is always possible for 1) IR.,
