Gamow vectors for barrier wells

We study four examples of Gamow vectors in one-dimensional potential barriers, namely square barriers and delta barriers. We show that resonances appear when the potential has at least two relative maxima and investigate the emergence of double resonances given rise to Gamow–Jordan vectors as well.     

Examples of Gamow vectors

We present three selected physical examples of Gamow vectors to illustrate their physical significance, namely, central potential scattering, one-dimensional lattice with impurities in electric field (Stark) and a simple model for unstable elementary particles (Friedrichs).     

A new formulation of relativistic quantum field theory

Quantum field theory is reformulated in sucha manner that a complete set of ocillators for modes with both positive and negative energies are introduced. The theory leads to the proper connection between spin and statistics as in the standard formulation, but it implements the time reversal transformation and the TCP transformation as linear unitary transformations. Negative energy particles in the initial states are identified with antiparticles in the final state with reversed motion (andvice versa) as far as scattering amplitudes are concerned. A covariant perturbation theory is developed which yields scattering amplitudes which are essentially the same as in the usual theory.     

A model for an interacting quantum field

We present here a close nonlinear analog to the free quantum field of Bose statistics, in which the linear one-particle space is replaced by a nonlinear infinite-dimensional Hermitian symmetric space D, and the quantum field is constructed as a Hilbert space of holomorphic functions on D.     

The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

ABC formula and R-operation for quantum processes with unstable states

For a wide class of Hamiltonians used in quantum field theory and statistical physics, we obtain an explicit formula describing the behavior of the vacuum expectation of the evolution operator on large, but finite, time intervals. This formula also holds for processes with unstable states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 183–201, November, 2006.     

Explicit construction of a time superoperator for quantum unstable systems

A time superoperator T conjugate to the Liouville superoperator L_H=[H,] is constructed for a quantum system with one excited state or unstable particle. While there is no time operator conjugate to the Hamiltonian in the wave function space due to the positivity of energy, T may exist in the density matrix space as the spectrum of L_H covers all the real axis. This is the first example of an observable that can only be formulated in the Liouville–von Neumann space of density matrices. In our example the expectation value of T gives the lifetime of the unstable particle. Once the time superoperator is obtained it is easy to define an entropy superoperator.     

Bogoliubov group variables in relativistic field theory

We define the Bogoliubov variables for strongly coupled systems that are invariant under the Poincaré group in (1+1)-dimensional space-time. This allows us to achieve a compatibility between taking the conservation laws into account exactly and developing a regular perturbation theory. We perform the secondary quantization in terms of the Bogoliubov variables and discuss the problem of reducing the number of states of the field. We also discuss the conditions for validity of the perturbation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 242–251, May, 1997.