Negative-energy levels of the Dirac equation in N dimensions

R. Shankar, Harsh Mathur

The Thomas precession in the nonrelativistic limit of the Dirac equation may be attributed to a non-Abelian Berry vector potential. We ask what object produces the non-Abelian potential in parameter space, in the same sense that the Abelian vector potential arising in the adiabatic transport of a nondegenerate level is produced by a monopole (centered at the point where the level becomes degenerate with another) as shown by Berry. We find that it is a meron, an object in four Euclidean dimensions with instanton number \textonehalf{}, centered at the point where the doubly degenerate positive and negative energy levels of the Dirac equation become fourfold degenerate.

Lucas Lamata

The quantum Rabi model has been a useful and pedagogical quantum model in the past decades, sufficiently simple to be solved analytically and intuitively understood, while sufficiently complex as to provide highly non-trivial eigenstates and a practical description of quantum optical platforms for quantum technologies. The Dirac equation, especially when restricted to 1+1 dimensions, is a simple toy model as well, but its easy diagonalization enabled historically to connect the electron spin to the fermionic statistics, among others. Both models share a symmetry at the purely mathematical level, namely, the spectra of each one has a dual equivalent under energy sign change, that I name a mirror dual symmetry. Usually, one quantizes these equations by assuming a ground state energy for the bosonic mode. But there is another option for the interpretation of the Hamiltonian, as I will argue, that is to assume a total symmetry principle, namely, that the total energy is zero at all times, for either the quantum Rabi model or the Dirac equation, and impose the constraint that every positive energy excitation has a mirror excitation of negative energy. This possibility, which was, apparently, ignored in the times when Paul Dirac was studying the implications of his equation, would avoid the worries in the scientific community that the negative energy solutions would decay until minus infinity, thus obviating the necessity to build a highly artificial Dirac sea, and instead impose what has always been successful in Physics, which is the enforcement of symmetry principles. Assuming a total symmetry principle, many of the problems of current Physics, such as renormalization of quantum gravity, dark matter, and dark energy, may possibly be automatically solved. One obvious result would be the automatic cancellation of the zero point energy.

Garnet N. Ord, D. G. C. McKeon

Jens Bolte, Stefan Keppeler

F. Masafumi Toyama, Yukihisa Nogami

Ersan Demiralp

Bound state solutions of the Schrödinger equation have been investigated for n -dimensional ( n ⩾ 2) harmonic oscillator potential decorated with any finite number ( P ) of Dirac delta functions. The potential is radially symmetric and given as , where σ i s are arbitrary real numbers, r 1 < r 2 < ⋅⋅⋅ < r P and r i ∊ (0, + ∞). We have demonstrated that addition of Dirac delta functions lifts the accidental degeneracies of n -dimensional harmonic oscillator energy levels and leaves only the degeneracy due to the radial symmetry. Explicit forms of bound state eigenfunctions and the eigenvalue equation are given for n , l values, where n is the space dimension and l is the degree of n -dimensional spherical harmonics. We have shown that, for given n and l , there are a countably infinite number of bound state energy levels which are continuous functions of ω, σ i s and at most P of them can be negative.