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A. Bohm

Publications and source records attributed to A. Bohm.

29 records · Page 2Linked to original sources
arXiv1999

Semigroup Representations of the Poincare Group and Relativistic Gamow Vectors

Gamow vectors are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues $(E_{R}\mp iΓ/2)$ describing quasistable states. In the relativistic domain this leads to Poincaré semigroup representations which are characterized by spin $j$ and by complex invariant mass square ${\mathsf{s}}={\mathsf{s}}_{R}=(M_{R}-\frac{i}{2}Γ_{R})^{2}$. Relativistic Gamow kets have all the properties required to describe relativistic resonances and quasistable particles with resonance mass $M_{R}$ and lifetime $\hbar/Γ_{R}$.

hep-th↗
arXiv1999

Rigged Hilbert Space Resonances and Time Asymmetric Quantum Mechanics

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 $ϕ^+$ and the out-states (observables) $ψ^-$ of the S-matrix elements $(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})$ any elements of the HS $\H$, $\{ψ^-\}=\{ϕ^+\}(=\H)$, the RHS theory chooses the boundary conditions~: $ϕ^+\inΦ_-\subset\H\subsetΦ_-^\times$, $ψ^-\inΦ_+\subset \H\subset Φ_+^\times$, where $Φ_-$ ($Φ_+$) 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 $ϕ^+$ and observables $ψ^-$ provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets $|E^{\pm}\inΦ_{\mp}^{\times}$ (solutions of the Lippmann-Schwinger equation with $\pm iε$ respectively) and the Gamow vectors $|E_R-iΓ/2^\pm\inΦ_{\mp}^\times$. The Gamow vectors $|E_R-iΓ/2^-$ 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.

quant-ph↗
arXiv1999

Relativistic Gamow Vectors

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.

hep-th↗
arXiv1999

Time Asymmetric Quantum Physics

Mathematical and phenomenological arguments in favor of asymmetric time evolution of micro-physical states are presented.

quant-ph↗
arXiv1998

Quantum Theory in the Rigged Hilbert Space-Irreversibility from Causality

After a review of the arrows of time, we describe the possibilities of a time-asymmetry in quantum theory. Whereas Hilbert space quantum mechanics is time-symmetric, the rigged Hilbert space formulation, which arose from Dirac's bra-ket formalism, allows the choice of asymmetric boundary conditions analogous to the retarded solutions of the Maxwell equations for the radiation arrow of time. This led to irreversibility on the microphysical level as exemplified by decaying states or resonances. Resonances are mathematically represented by Gamow kets, functionals over a space of very well-behaved (Hardy class) vectors, which have been chosen by a boundary condition (outgoing for decaying states). Gamow states have all the properties that one heuristically needs for quasistable states. For them a Golden Rule can be derived from the fundamental probabilities that fulfills the time-asymmetry condition which could not be realized in the Hilbert space.

quant-ph↗
arXiv1997

Hilbert Space or Gelfand Triplet - Time Symmetric or Time Asymmetric Quantum Mechanics

Intrinsic microphysical irreversibility is the time asymmetry observed in exponentially decaying states. It is described by the semigroup generated by the Hamiltonian $\QTR{it}{H}$ of the quantum physical system, not by the semigroup generated by a Liouvillian $\QTR{it}{L}$ which describes the irreversibility due to the influence of an external reservoir or measurement apparatus. The semigroup time evolution generated by $\QTR{it}{H}$ is impossible in the Hilbert Space (HS) theory, which allows only time symmetric boundary conditions and an unitary group time evolution. This leads to problems with decay probabilities in the HS theory. To overcome these and other problems (non-existence of Dirac kets) caused by the Lebesgue integrals of the HS, one extends the HS to a Gel'fand triplet, which contains not only Dirac kets, but also generalized eigenvectors of the self-adjoint $\QTR{it}{H}$ with complex eigenvalues ($E_R-iΓ/2$) and a Breit-Wigner energy distribution. These Gamow states $ψ^G$ have a time asymmetric exponential evolution. One can derive the decay probability of the Gamow state into the decay products described by $Λ$ from the basic formula of quantum mechanics $\QTR{cal}{P}(t)=Tr(|ψ^G> < ψ^G|Λ)$, which in HS quantum mechanics is identically zero. From this result one derives the decay rate $\QTR{group}{\dot c}(t)$ and all the standard relations between $\QTR{group}{\dot c}(0)$, $Γ$ and the lifetime $τ_R$ used in the phenomenology of resonance scattering and decay. In the Born approximation one obtains Dirac's Golden Rule.

quant-ph↗
arXiv1997

Jordan Blocks and Exponentially Decaying Higher Order Gamow States

In the framework of the rigged Hilbert space, unstable quantum systems associated with first order poles of the analytically continued S-matrix can be described by Gamow vectors which are generalized vectors with exponential decay and a Breit-Wigner energy distribution. This mathematical formalism can be generalized to quasistationary systems associated with higher order poles of the S-matrix, which leads to a set of Gamow vectors of higher order with a non-exponential time evolution. One can define a state operator from the set of higher order Gamow vectors which obeys the exponential decay law. We shall discuss to what extend the requirement of an exponential time evolution determines the form of the state operator for a quasistationary microphysical system associated with a higher order pole of the S-matrix.

quant-ph↗
arXiv1997

Gamow-Jordan Vectors and Non-Reducible Density Operators from Higher Order S-Matrix Poles

In analogy to Gamow vectors that are obtained from first order resonance poles of the S-matrix, one can also define higher order Gamow vectors which are derived from higher order poles of the S-matrix. An S-matrix pole of r-th order at z_R=E_R-iΓ/2 leads to r generalized eigenvectors of order k= 0, 1, ... , r-1, which are also Jordan vectors of degree (k+1) with generalized eigenvalue (E_R-iΓ/2). The Gamow-Jordan vectors are elements of a generalized complex eigenvector expansion, whose form suggests the definition of a state operator (density matrix) for the microphysical decaying state of this higher order pole. This microphysical state is a mixture of non-reducible components. In spite of the fact that the k-th order Gamow-Jordan vectors has the polynomial time-dependence which one always associates with higher order poles, the microphysical state obeys a purely exponential decay law.

quant-ph↗
arXiv1995

Time-Reversal and Irreversibility

The time reversal and irreversibility in conventional quantum mechanics are compared with those of the rigged Hilbert space quantum mechanics. We discuss the time evolution of Gamow and Gamow-Jordan vectors and show that the rigged Hilbert space case admits a new kind of irreversibility which does not appear in the conventional case. The origin of this irreversibility can be traced back to different initial-boundary conditions for the states and observables. It is shown that this irreversibility does not contradict the experimentally tested consequences of the time-reversal invariance of the conventional case but instead we have to introduce a new time reversal operator.

quant-ph↗
arXiv1995

The Rigged Hilbert Space Formulation of Quantum Mechanics and its Implications for Irreversibility

Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.

quant-ph↗