Calculation of the Scattering Amplitude for Resonance Production using Relativistic Gamow Vectors
This paper has been withdrawn by the author in order to implement the referee's suggestions and comments.
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arXiv subjects
H. Kaldass
Publications and source records attributed to H. Kaldass.
QCD on the Cell Broadband Engine
We evaluate IBM's Enhanced Cell Broadband Engine (BE) as a possible building block of a new generation of lattice QCD machines. The Enhanced Cell BE will provide full support of double-precision floating-point arithmetics, including IEEE-compliant rounding. We have developed a performance model and applied it to relevant lattice QCD kernels. The performance estimates are supported by micro- and application-benchmarks that have been obtained on currently available Cell BE-based computers, such as IBM QS20 blades and PlayStation 3. The results are encouraging and show that this processor is an interesting option for lattice QCD applications. For a massively parallel machine on the basis of the Cell BE, an application-optimized network needs to be developed.
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Time Asymmetry in Quantum Physics - I. Theoretical Conclusion from Resonance and Decay-Phenomenology
It is explained how the unification of resonance and decay phenomena into a consistent mathematical theory leads to quantum mechanical time-asymmetry. This provides the theoretical basis for a subsequent paper II in which the interpretation and experimental demonstration of this time-asymmetry is discussed.
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Time Asymmetry in Quantum Physics - II. Experimental Demonstration Using a Single Ion
Quantum physics involves an ensemble of quantum systems, usually one thinks of a large ensemble of identical quantum systems at one single time. In single ion experiments one has a single quantum system at an ensemble of different times. This provides the means of demonstrating the beginning of time of a semigroup evolution for a decaying state.
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Analysis of resonance production using relativistic Gamow vectors
The calculation of an amplitude involving resonance production is presented. This calculation employs for the resonance state a relativistic Gamow vector. It is used for investigating the question of compatibility of the relativistic Gamow vectors kinematics, defined by real 4-velocities and complex mass, with the stable particle kinematics; or in other words, the integration of the Gamow vectors with the conventional Dirac bra-ket formalism. The calculation demonstrates a consistent framework comprising stable and Gamow vectors.
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apeNEXT: A multi-TFlops Computer for Simulations in Lattice Gauge Theory
We present the APE (Array Processor Experiment) project for the development of dedicated parallel computers for numerical simulations in lattice gauge theories. While APEmille is a production machine in today's physics simulations at various sites in Europe, a new machine, apeNEXT, is currently being developed to provide multi-Tflops computing performance. Like previous APE machines, the new supercomputer is largely custom designed and specifically optimized for simulations of Lattice QCD.
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Status of the apeNEXT project
We present the current status of the apeNEXT project. Aim of this project is the development of the next generation of APE machines which will provide multi-teraflop computing power. Like previous machines, apeNEXT is based on a custom designed processor, which is specifically optimized for simulating QCD. We discuss the machine design, report on benchmarks, and give an overview on the status of the software development.
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The apeNEXT project (Status report)
We present the current status of the apeNEXT project. Aim of this project is the development of the next generation of APE machines which will provide multi-teraflop computing power. Like previous machines, apeNEXT is based on a custom designed processor, which is specifically optimized for simulating QCD. We discuss the machine design, report on benchmarks, and give an overview on the status of the software development.
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Time Asymmetric Quantum Theory - II. Relativistic Resonances from S-Matrix Poles
Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the $S$-matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in- and out-scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several ``relativistic Breit-Wigner'' amplitudes, which represent the resonances per se. These Breit-Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.
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Time Asymmetric Quantum Theory - III. Decaying States and the Causal Poincare Semigroup
A relativistic resonance which was defined by a pole of the $S$-matrix, or by a relativistic Breit-Wigner line shape, is represented by a generalized state vector (ket) which can be obtained by analytic extension of the relativistic Lippmann-Schwinger kets. These Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of the Lippmann-Schwinger plane wave scattering states, form only a semigroup of Poincaré transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed.
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The APENEXT project
APENEXT is a new generation APE processor, optimized for LGT simulations. The project follows the basic ideas of previous APE machines and develops simple and cheap parallel systems with multi T-Flops processing power. This paper describes the main features of this new development.
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Relativistic Gamow Vectors I Derivation from Poles of the S-Matrix
A state vector description for relativistic resonances is derived from the first order pole of the $j$-th partial $S$-matrix at the invariant square mass value $\sm_R=(m-iΓ/2)^2$ in the second sheet of the Riemann energy surface. To associate a ket, called Gamow vector, to the pole, we use the generalized eigenvectors of the four-velocity operators in place of the customary momentum eigenkets of Wigner, and we replace the conventional Hilbert space assumptions for the in- and out-scattering states with the new hypothesis that in- and out-states are described by two different Hardy spaces with complementary analyticity properties. The Gamow vectors have the following properties: - They are simultaneous generalized eigenvectors of the four velocity operators with real eigenvalues and of the self-adjoint invariant mass operator $M=(P_μP^μ)^{1/2}$ with complex eigenvalue $\sqrt{\sm_R}$. - They have a Breit-Wigner distribution in the invariant square mass variable $\sm$ and lead to an exactly exponential law for the decay rates and probabilities.
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Relativistic Gamow Vectors II
Motivated by the debate of possible definitions of mass and width of resonances for $Z$-boson and hadrons, we suggest a definition of unstable particles by ``minimally complex'' semigroup representations of the Poincaré group characterized by $(j,{\mathsf s}=(m-iΓ/2)^{2})$ in which the Lorentz subgroup is unitary. This definition, though decidedly distinct from those based on various renormalization schemes of perturbation theory, is intimately connected with the first order pole definition of the $S$-matrix theory in that the complex square mass $(m-iΓ/2)^{2}$ characterizing the representation of the Poincaré semigroup is exactly the position ${\mathsf s}_R$ at which the $S$-matrix has a simple pole. Wigner's representations $(j,m)$ are the limit case of the complex representations for $Γ=0$. These representations have generalized vectors (Gamow kets) which have, in addition to the $S$-matrix pole at ${\mathsf s}=(m-iΓ/2)^{2}$, all the other properties that heuristically the unstable states need to possess: a Breit-Wigner distribution in invariant square mass and a lifetime $τ=\frac{1}Γ$ defined by the exactly exponential law for the decay probability ${\cal P}(t)$ and rate $\dot{\cal P}(t)$ given by an exact Golden Rule which becomes Dirac's Golden Rule in the Born-approximation. In addition and unintended, they have an asymmetric time evolution.
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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 $\hbar/Γ_{R}$.
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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.
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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.
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Relativistic Partial Wave Analysis Using the Velocity Basis of the Poincare Group
The velocity basis of the Poincare group is used in the direct product space of two irreducible unitary representations of the Poincare group. The velocity basis with total angular momentum j will be used for the definition of relativistic Gamow vectors.
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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.
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