Probability Interpretation for Klein-Gordon Fields

We give an explicit construction of a positive-definite invariant inner-product for the Klein-Gordon fields, thus solving the old problem of the probability interpretation of Klein-Gordon fields without having to restrict to the subspaces of the positive-frequency solutions. Our method has a much wider domain of application and enjoys a remarkable uniqueness property. We explore its consequences for the solutions of the Wheeler-DeWitt equation associated with the FRW-massive-real-scalar-field models. The problem of the lack of a consistent probability interpretation for Klein-Gordon fields is almost as old as quantum mechanics. Indeed this problem has never been completely solved, but rather put aside by Dirac’s discovery of the method of second quantization. It also provided one of the main motivations for the introduction of the Dirac equation. It took several decades for this problem to reemerge as a fundamental obstacle in trying to devise a consistent interpretation for the solutions of the Wheeler-DeWitt equation in quantum cosmology. Unlike the case of Klein-Gordon fields, here the method of second quantization and Dirac’s trick of considering an associated first order field equation do not lead to a satisfactory solution [1, 2, 3, 4]. There has been various partial solutions or rather attempts to avoid this problem. These are either based on the invariant but indefinite Klein-Gordon inner-product which was originally considered in Bryce DeWitt’s pioneering article [5] on quantum cosmology and further developed by ∗E-mail address: [email protected]