The Weyl Lagrangian is the massless Dirac Lagrangian. The dynamical variable in the Weyl Lagrangian is a spinor field. We provide a mathematically equivalent representation in terms of a different dynamical variable — the coframe (an orthonormal tetrad of covector fields). We show that when written in terms of this dynamical variable, the Weyl Lagrangian becomes remarkably simple: it is the wedge product of axial torsion of the teleparallel connection with a teleparallel lightlike element of the coframe. We also examine the issues of U(1)-invariance and conformal invariance. Examination of the latter motivates us to introduce a positive scalar field (equivalent to a density) as an additional dynamical variable; this makes conformal invariance self-evident.

1.
D.
Vassiliev
,
Phys. Rev. D
75
,
025006
(
2007
).
2.
E.
Cartan
and
A.
Einstein
,
Letters on Absolute Parallelism
(
Princeton University Press
,
Princeton
,
1979
).
3.
A.
Unzicker
and
T.
Case
(unpublished).
5.
E.
Cosserat
and
F.
Cosserat
,
Théorie des Corps Déformables
, Librairie Scientifique A. Hermann et Fils,
Paris
,
1909
. Reprinted by Cornell University Library.
6.
J. M.
Ball
,
A.
Taheri
, and
M.
Winter
,
Calculus Var. Partial Differ. Equ.
14
,
1
(
2002
).
7.
C.
Liu
and
F.
Lin
,
J. Partial Differ. Equ.
14
,
289
(
2001
).
8.
J. M.
Ball
,
London Analysis and Probability Seminar
, 25 October
2007
(unpublished).
9.
É.
Cartan
,
C. R. Acad. Sci. (Paris)
174
,
593
(
1922
).
10.
H. F. M.
Goenner
,
Living Rev. Relativ.
7
,
2
(
2004
).
11.
F. W.
Hehl
and
Yu. N.
Obukhov
,
Ann. Fond. Louis Broglie
32
,
157
(
2007
).
12.
A. D.
King
and
D.
Vassiliev
,
Class. Quantum Grav.
18
,
2317
(
2001
).
13.
D.
Vassiliev
,
Gen. Relativ. Gravit.
34
,
1239
(
2002
).
14.
D.
Vassiliev
,
Ann. Phys. (Leipzig)
14
,
231
(
2005
).
15.
V.
Pasic
and
D.
Vassiliev
,
Class. Quantum Grav.
22
,
3961
(
2005
).
16.
E. T.
Newman
and
R.
Penrose
,
J. Math. Phys.
3
,
566
(
1962
).
17.
A. L.
Besse
,
Einstein Manifolds
(
Springer-Verlag
,
Berlin
,
1987
).
18.

The group B2 can, in fact, be characterized as the nontrivial Abelian subgroup of the Lorentz group. See Appendix B in Ref. 1, for details.

19.
A.
Dimakis
and
F.
Müller-Hoissen
,
J. Math. Phys.
26
,
1040
(
1985
).
20.
A.
Dimakis
and
F.
Müller-Hoissen
,
Phys. Lett. A
142
,
73
(
1989
).
21.
A.
Dimakis
and
F.
Müller-Hoissen
,
Class. Quantum Grav.
7
,
283
(
1990
).
22.
J. B.
Griffiths
and
R. A.
Newing
,
J. Phys. A
3
,
269
(
1970
).
23.
D.
Vassiliev
, in
Proceedings of the 11th Marcel Grossmann Meeting on General Relativity
, edited by
H.
Kleinert
and
R. T.
Jantzen
(
World Scientific
,
Singapore
,
2008
), Pt. B, pp.
1245
1247
.
24.
J.
Burnett
,
O.
Chervova
, and
D.
Vassiliev
, in
Analysis, Partial Differential Equations and Applications, The Vladimir Maz’ya Anniversary Volume
,
Series Operator Theory: Advances and Applications
Vol.
193
, edited by
A.
Cialdea
,
F.
Lanzara
, and
P. E.
Ricci
(
Birkhäuser
,
Basel, Switzerland
,
2009
), pp.
15
29
.
25.
O.
Chervova
and
D.
Vassiliev
(unpublished).
26.
J.
Frauendiener
,
Class. Quantum Grav.
8
,
1881
(
1991
).
27.
J. M.
Nester
,
J. Math. Phys.
33
,
910
(
1992
).
28.
G. Y.
Chee
,
Phys. Rev. D
68
,
044006
(
2003
).
29.
30.
F. W.
Hehl
,
J. D.
McCrea
,
E. W.
Mielke
, and
Y.
Ne’eman
,
Phys. Rep.
258
,
1
(
1995
).
31.
F. W.
Hehl
,
J.
Nitsch
, and
P.
von der Heyde
, in
General Relativity and Gravitation
, edited by
A.
Held
(
Plenum
,
New York
,
1980
), Vol.
1
, pp.
329
355
.
32.
F.
Gronwald
and
F. W.
Hehl
,
Quantum gravity
(Erice,
1995
);
The Science and Culture Series—Physics
(
World Scientific
,
River Edge, NJ
,
1996
), Vol.
10
, pp.
148
198
.
33.
U.
Muench
,
F.
Gronwald
, and
F. W.
Hehl
,
Gen. Relativ. Gravit.
30
,
933
(
1998
).
34.
V. C.
de Andrade
,
L. C. T.
Guillen
, and
J. G.
Pereira
, in
Proceedings of the Ninth Marcel Grossmann Meeting on General Relativity
, edited by
H.
Kleinert
and
R. T.
Jantzen
(
World Scientific
, Singapore,
2002
).
35.
M.
Blagojević
,
Gravitation and Gauge Symmetries
,
Series in High Energy Physics, Cosmology and Gravitation
(
IOP
,
Bristol
,
2002
).
36.
Yu. N.
Obukhov
and
J. G.
Pereira
,
Phys. Rev. D
67
,
044016
(
2003
).
You do not currently have access to this content.