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Journal of Mathematical Physics

Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics

Arno Böhm

Published Dec 1, 1981• 175 citations
Publisher

Abstract

After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.

DOI
10.1063/1.524871

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American Journal of Physics•Dec 1, 1989•108 citations
After a summary of the Rigged Hilbert space formulation of quantum mechanics and a brief statement of its advantages over von Neumann’s formulation, a mathematically correct definition of Gamow’s exponentially decaying vectors as generalized energy eigenvectors is suggested. It is shown that exponentially decaying vectors are obtained from the S-matrix poles in the lower half of the second sheet and exponentially growing vectors from the S-matrix poles in the upper half of the second sheet. Decaying ’’state’’ vectors are defined as functionals over half of the space of physical states and growing ’’state’’ vectors are defined as functionals over the other half. On functionals over these subspaces, the dynamical group of time development splits into two semigroups, one for t ≳ 0 and the other for t < 0. The generalized basis system connected with the spectrum of the Hamiltonian is transformed into a new basis system in which the exponentially decaying component of the density matrix is separated.
Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
In the first part, Gamow vectors from S-matrix poles of higher multiplicity are derived in analogy to the Gamow vectors describing resonances from first-order poles. With these vectors a density operator is constructed that describes resonances associated with higher-order poles that obey an exponential decay law. It turns out that this operator formed by higher-order Gamow vectors has a unique structure. In the second part, the Gamow vectors are used in order to describe decaying and resonance states at a potential barrier of finite width. The exact numerical solutions of the S-matrix poles are presented and the out-going boundary condition for the Gamow vector description and consequences from that are discussed.
Gamow vectors in non-relativistic quantum mechanics are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues. Like the Dirac kets, they are mathematically well defined in the Rigged Hilbert Space. Gamow kets are derived from the resonance poles of the S-matrix. They have a Breit-Wigner energy distribution, an exponential decay law, and are members of a basis vector expansion whose truncation gives the finite dimensional effective theories with a complex Hamiltonian matrix. They also have an asymmetric time evolution described by a semigroup generated by the Hamiltonian, which expresses a fundamental quantum mechanical arrow of time. These Gamow kets are generalized to relativistic Gamow vectors by extrapolating from the Galilei group to the Poincare group. This leads to semigroup representations of the Poincare group which are characterized by spin j and complex invariant mass square. In these non-unitary representations the Lorentz subgroup is unitarily represented and the four-momenta are "minimally complex" in the sense that the four-velocity is real. The relativistic Gamow vectors have all the properties listed above for the non-relativistic Gamow vectors and are therefore ideally suited to describe relativistic resonances and quasistable particles.
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states ϕ+ϕ^+ϕ+ and the out-states (observables) ψ−ψ^-ψ− of the S-matrix elements (ψ−,ϕ+)=(ψout,Sϕin)(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$, {ψ−}={ϕ+}(=)˝\{ψ^-\}=\{ϕ^+\}(=\H){ψ−}={ϕ+}(=)˝​, the RHS theory chooses the boundary conditions~: ϕ+∈Φ−⊂⊂˝Φ−×ϕ^+\inΦ_-\subset\H\subsetΦ_-^\timesϕ+∈Φ−​⊂⊂˝​Φ−×​, ψ−∈Φ+⊂⊂˝Φ+×ψ^-\inΦ_+\subset \H\subset Φ_+^\timesψ−∈Φ+​⊂⊂˝​Φ+×​, where Φ−Φ_-Φ−​ (Φ+Φ_+Φ+​) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states ϕ+ϕ^+ϕ+ and observables ψ−ψ^-ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets ∣E±∈Φ∓×|E^{\pm}\inΦ_{\mp}^{\times}∣E±∈Φ∓×​ (solutions of the Lippmann-Schwinger equation with ±iε\pm iε±iε respectively) and the Gamow vectors ∣ER−iΓ/2±∈Φ∓×|E_R-iΓ/2^\pm\inΦ_{\mp}^\times∣ER​−iΓ/2±∈Φ∓×​. The Gamow vectors ∣ER−iΓ/2−|E_R-iΓ/2^-∣ER​−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.
Gamow vectors are generalized eigenvectors of the Hamiltonian with complex eigenvalues that describe exponentially decaying (or growing) states. The energy wavefunctions corresponding to Gamow vectors have a pole immediately below (or above) the real axis in the complex energy plane. Although complex energy values were introduced more than half a century ago for the theory of alpha decay, they have become disreputable and have been banished from quantum mechanics textbooks because of mathematical problems and incorrect physical interpretation. Developments in modern mathematics have now provided a mathematical foundation that has led to the correct physical interpretation. In this article, energy eigenvectors with complex eigenvalues are first introduced by explicitly considering a specific simple decaying system. Then, an elementary account of the properties of Gamow vectors is given. The results from the decaying system provide a motivation for the introduction of Gamow vectors and some of the background information needed to understand their properties.