By using the fact that a plane electromagnetic wave is described by two Lorentz invariant statements, a complex orthogonal representation of the Lorentz group, including charged fields, is discussed. The representation provides the possibility of a combined study of the P, C, T symmetry operations for spin ½ and spin 1 fields. In particular, the chargelessness of the neutrino as a complex fermion field results from a reducible three charge states representation of the complex group.
REFERENCES
1.
E. P. Wigner, Group Theory (Academic Press, Inc., New York, 1959), contains a detailed discussion of antilinear operations in physics.
In connection with a discussion of vacuum expectation values of time‐ordered products of field operators,
D.
Hall
and A. S.
Wightman
have introduced a complex representation of Lorentz group [Kgl. Danske Videnskab. Selskab, Mat.‐Fys. Medd.
31
, No. 5
(1957
)]. Their discussion is, however, meant for quite a different purpose and is not directly related to the present work.2.
The subgroup of the extended group considered here is a four branch group corresponding to four different values of the determinants of R transformations.
3.
This type of equation was first discussed by
J. R.
Oppenheimer
[Phys. Rev.
38
, 725
(1931
)],and
R. H.
Good
[Phys. Rev.
105
, 1914
(1957
)] have investigated further quantum‐mechanical implications of such an equation.Four‐dimensional representations have been discussed by
H. E.
Moses
[Nuovo Cimento. Suppl.
7
, 1
(1958
)];4.
The situation here is, in some sense, similar to the metrical correspondences of the two‐dimensional spinor space with metric and four‐dimensional spinor space with metric β, to Lorentz metric F.
5.
The two‐dimensional spinor space under consideration here is, in some respects, quite different form the two‐dimensional spinor space where one considers the transformation properties of the vector matrix In the former space the condition is not a necessity and the U transformations are directly determined from R transformation. In the latter spinor space the transformation laws of are given by , where the condition is a necessity and S are defined directly in terms of the coefficients Also well‐known properties fo under improper Lorentz transformations draws a sharp line of demarcation between U and S transformations. The said reasons and others, to be discussed in later sections, will allow us to regard U and R transformations as charge space representations of Lorentz group. In particular, for quantized the operators may be related to the representation of the isotopic spin group, where χ has an additional isotopic spin coordinate referring to three charge states.
6.
Actually if we wished we could set up the said one‐two correspondence of R and U transformations as one‐two correspondence of rotations and velocity transformations separately. For example, the rotation and velocity transformations of the complex space affected by the R matrices (not summed over i) and correspond to rotations and velocity transformations of the spinor space affected by the U matrices and , respectively.
7.
8.
The symmetry operation of weak reflection here is not used in the sense introduced by Pauli [Niels Bohr and the Development of Physics (Pergamon Press, New York, 1955] who defined it as a combination of two symmetry operations of (i) reflection of space time coordinates together with transformation of every particle into its antiparticle (strong reflection) and (ii) particle, antiparticle conjugation.
Similar considerations by
J.
Schwinger
were given in Phys. Rev.
82
, 914
(1951
),and also by
G.
Luders
, Kgl. Danske Videnskab. Selskab, Mat.‐Fys. Medd.
28
, No. 5
(1954
).A different approach to a complex four‐dimensional representation of Lorentz group is given by
Res.
Yost
, Helv. Phys. Acta.
30
, 407
(1957
).9.
For a q‐number theory the field must be quantized according to , where , and is the usual propagator of electromagnetic field.
10.
The corresponding situation for the photon field can be described as a “charge conjugation” where particle and antiparticle properties are to be attributed to two different states of polarization.
11.
12.
13.
14.
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© 1961 American Institute of Physics.
1961
American Institute of Physics
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