Quantum mechanics from self-interaction

Mark Kenneth Mitchell

Sergio Giardino

The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the possibility of self-interacting physical systems. The self-interaction precludes confining states, and also imposes non-stationary quantum states, both of them representing novel situations that cannot be observed in terms of quantum wave functions. These results remark the differences between quaternionic quantum mechanics ($\mathbbm H$QM) and complex quantum mechanics ($\mathbbm C$QM), and also establish a method of solving the wave equation that may be applied to a variety of different cases.

Shan Gao

It has been widely thought that the ontology of quantum mechanics is real, physical fields. In this paper, I will present a new argument against the field ontology of quantum mechanics by analyzing one-body systems such as an electron. First, I argue that if the physical entity described by the wave function of an electron is a field, then this field is massive and charged. Next, I argue that if a field is massive and charged, then any two parts of the field in space will have gravitational and electromagnetic interactions, while the existence of such self-interactions for an electron contradicts quantum mechanics and experimental observations. This poses a serious difficulty for the field ontology of quantum mechanics. Third, I argue that a particle ontological interpretation of the wave function may avoid the difficulty by providing a plausible explanation of the non-existence of self-interactions. According to this explanation, the wave function of an electron is a description of the state of the random motion of the electron as a particle, and in particular, the modulus squared of the wave function gives the probability density that the electron appears in every possible position in space. Finally, I answer a major objection to the explanation of the non-existence of self-interactions in terms of particle ontology.

Hilbrand Johannes Groenewold

Francisco Antônio Bezerra Coutinho, Yukihisa Nogami, F. Masafumi Toyama

We explore generalized point interactions in one-dimensional quantum mechanics with two coupled channels. They represent possible self-adjoint extensions of the nonrelativistic kinetic-energy operator. Assuming time-reversal invariance we find a family of self-adjoint extensions with seven parameters. This can be compared with the one-channel case in which the corresponding number of parameters is three.

H. J. Groenewold