then Vk+Vk + give the usual five operators for the ^-interaction. We have previously taken as interaction hamiltonian a linear combination 1 V~G 2 C fc (7*+n + ),in which the constants Ck must be real in order that V be hermitian. 2 ' 3 Trigg and Feenberg 4 have given the generalization to the (hermitian) linear combinations using complex constants C kUsing the conventional transformations of the wave functions in the Dirac equation, 5 it is easily verified that the interactions according to Eqs. (2) and (3) are invariant for (continuous) lorentz transformations, reflection of space, and reversal of time.We now give two types of symmetries that can be imposed on the interaction./. First condition of symmetry.-The interaction energy must be invariant, if we transform all four particles to the anti-particles. (This is done by taking charge-conjugated solutions. 3 ) In this way a transformed interactionis obtained from Eq. (3). Biedenharn and Rose have obtained this result by employing Wigner's time-reversal operator, a procedure equivalent to the use of charge conjugation here. In order that Eqs. (3) and (4) have the same meaning, the coefficients C k must be real, so that Eq. (3) reduces to Eq. (2). This is especially clear if we consider the lower sign expressions (5) occurring in the formulas, taking the polarization into account (see below). These expressions would change their sign, when taking everywhere the "anti-particles." 6 II. Second condition of symmetry.-We give two alternative formulations a and b of a second condition of symmetry, proposed earlier, (a) The processes of negatron and positron emission must be symmetrical in such a way that if coulomb interaction is neglected, the expressions for the interaction energy HfT and Hp + are equal (possibly with the exception of a phase factor e % &) if the following conditions are satisfied: the wave functions of the emitted (positive and negative) electrons and neutrinos must be physically equivalent, and ^/(p) and \pi(n) for negatron emission must be respectively the same as \pf(n) and \pi(p) for positron emission (i initial; / final; n neutron; p proton). (b) The interaction energy must be invariant (or differ only by a phase factor e 1^) , when we take positions as real particles, negatrons as holes, and perform a corresponding change for the neutrinos, but not for the nucleons (see formulas (65), (70), and (75) of reference 1).The consequences of symmetry principle i7 are as. follows: 1. Using (2), we have either combinations of S, A, and P only, or combinations of V and T only. 2. Using (3), one obtains, 4 with ei=e 4 =e5=-1, and €2=e 3 = l,We have also calculated the transition probabilities for ^-emission taking into account the orientation of the nucleus, the polarization of the emitted electron, and the e-v angular correlation (for allowed transitions). 7 We then get cross terms that differ in sign for 0 + and (3~ emission, with coefficients (5), but for k, l-1, 2, 3, 4 only. The pseudoscalar (k -5) gives no cross terms with the other invariants f...
