If the electromagnetic interactions are invariant under charge conjugation, $C$, then singlet positronium (Ps) can only decay into an even number of photons, while triplet Ps can only decay into an odd number of photons. In this paper we consider a model Hamiltonian which violates $C$ invariance but conserves parity. If $\lambda$ is the coupling constant in the proposed $C$-noninvariant interaction, then the branching ratio of the forbidden decay, $1{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\to 4\gamma$, to the allowed decay, $1{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\to 3\gamma$, has been found to be $F_T^{4\gamma }=1.2\mathrm{}\times {10}^{-5\mathrm{}}{\lambda }^2$.

An experimental upper limit on $F_T^{4\gamma }$ would provide a novel check of $C$ invariance in electromagnetic interactions, and would also yield new information on the $\overline{e}e\gamma \gamma \gamma \gamma$ interaction. At present, indirect limits on $F_T^{4\gamma }$ show it to be less than about ${10}^{-2\mathrm{}}$. We are now working on an experiment which is designed to reduce this upper limit to ${10}^{-5\mathrm{}}$-${10}^{-6\mathrm{}}$. If successful, it will subject the proposed interaction, or any similar interaction leading to $1{}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}\to 4\gamma$, to a stringent test. An outline of the experiment is included in this paper.

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