Description of Unstable Systems in Relativistic Quantum Mechanics in the Lax-Phillips Theory

We discuss some of the experimental motivation for the need for semigroup decay laws and the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and, therefore, all physical properties of the resonant states can be computed. We show that the parametrized relativistic quantum theory is a natural setting for the realization of the Lax-Phillips theory.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Time Asymmetry and Quantum Theory of Resonances and Decay

These notes review a consistent and exact theory of quantum resonances and decay. Such a theory does not exist in the framework of traditional quantum mechanics and Dirac's formulation. But most of its ingredients have been familiar entities, like the Gamow vectors, the Lippmann-Schwinger (in- and out-plane wave) kets, the Breit-Wigner (Lorentzian) resonance amplitude, the analytically continued S-matrix, and its resonance poles. However, there are inconsistencies and problems with these ingredients: exponential catastrophe, deviations from the exponential law, causality, and recently the ambiguity of the mass and width definition for relativistic resonances. To overcome these problems the above entities will be appropriately defined (as mathematical idealizations). For this purpose we change just one axiom (Hilbert space and/or asymptotic completeness) to a new axiom which distinguishes between (in-)states and (out)observables using Hardy spaces. Then we obtain a consistent quantum theory of scattering and decay which has the Weisskopf-Wigner methods of standard textbooks as an approximation. But it also leads to time-asymmetric semigroup evolution in place of the usual, reversible, unitary group evolution. This, however, can be interpreted as causality for the Born probabilities. Thus we obtain a theoretical framework for the resonance and decay phenomena which is a natural extension of traditional quantum mechanics and possesses the same arrow-of-time as classical electrodynamics. When extended to the relativistic domain, it provides an unambiguous definition for the mass and width of the Z-boson and other relativistic resonances.     

相对论粒子的自旋算符

发展了关于相对论态自旋算符的系统理论.考虑了具有非零静质量的粒子情况.对带自旋的相对论粒子,通常的自旋算符需换为相对论的自旋算符.在Poincar啨群不可约表示的框架里,构造了适用于粒子任意正则态的自旋算符,称为运动自旋.本文的讨论限于量子力学.随后将在量子场论中对此作进一步深入研究.     

Representations of the Poincaré Semigroup and Relativistic Causality

This paper provides the mathematical tools for addressing issues of two kinds of causality in relativistic scattering theory: general causality, i.e., an effect can only be measured after its cause, and Einstein causality, i.e., no propagation of probability outside of the forward light cone. Starting from Wigner's unitary irreducible representations of the Poincaré group for noninteracting, one particle states, we describe the mathematical tools necessary to describe scattering states, the Lippmann-Schwinger Dirace kets, and to describe resonances and decaying states, the relativistic Gamow ket. An important step for their derivations is the Hardy space hypothesis. Investigating the transformation properties of scattering and resonance states under the dynamical Poincaré semigroup reveals that both kinds of causality result from this hypothesis about nature of the spaces of states and observables.     

Euclidean Fermi fields with a hermitean Feynman-Kac-Nelson formula. I

We construct free, Euclidean, spin one-half, quantum fields with the following properties: (i) CAR; (ii) Symanzik positivity; (iii) Osterwalder-Schrader positivity; (iv) no doubling of particle or spin states. They admit the recovery of the relativistic Dirac field by the Osterwalder-Schrader technique. We then formally parametrize interacting theories by a natural class of Hermitean, Euclidean actions, and obtain a simple, Hermitean, Feynman-Kac-Nelson formula. The interacting theory formally obeys all the properties (i)–(iv), and admits the reconstruction of a physical Hilbert space, including a Hermitean, contraction semigroup for the Wick rotated time evolution. We propose a system of axioms for the interacting theory.     

Relativistic quantum protocols: Bit commitment and coin tossing

New relativistic quantum protocols realizing the bit commitment and coin tossing schemes are proposed. The protocols are based on the idea that spatially extended nonstationary orthogonal quantum states inaccessible for measurement cannot be unambiguously distinguished. As the states are transmitted from the region controlled by one party to the region accessible for measurement by the other party, the states become reliably distinguishable when accessed as a whole. Essential points of the protocol are both the quantum character of states and the existence of an ultimate signal propagation speed dictated by special relativity.     

Relativistic Spin Operators

A systematic theory on the appropriate spin operators for the relativistic states is developed.For a massive relativistic particle with arbitrary nonzero spin,the spin operator should be replaced with the relativistic one,which is called in this paper as moving spin.Further the concept of moving spin is discussed in the quantum field theory.A new operator,field quanta spin is defined and in terms of the generators of Poincare group the moving spin of field system is constructed.It is shown that,in virtue of the two operators,problems in quantum field concerned spin can be neatly settled.     

Qubit portrait of qudit states and Bell inequalities

A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. In view of the qubit-portrait method, the Bell inequalities for two qubits and two qutrits are discussed within the framework of the probability-representation of quantum mechanics. A semigroup of stochastic matrices is associated with tomographic-probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as an ansatz to provide a necessary condition for the separability of quantum states.     

Irreversible evolution in quantum logics

The notion of dynamical semigroup is introduced in the quantum logic scheme on the set of the states. Under suitable nonempty mathematical assumptions it is shown that a Heisenberg picture exists equivalent to the Schrödinger one and having many aspects similar to those of the Hilbert case.     

Relativistic quantum bit commitment in real-time

A relativistic quantum exchange protocol making it possible to implement a bit commitment scheme is realized. The protocol is based on the idea that in the relativistic case the propagation of a field into a region of space accessible for measurement requires, in contrast to the nonrelativistic case, a finite time that depends on the structure of the states. The protocol requires one classical and several quantum communication channels. It turns out that it is possible in principle to preserve the secret bit for as long a period of time desired and with probability as close to 1 as desired.     

Complex energies and beginnings of time suggest a theory of scattering and decay

Many useful concepts for a quantum theory of scattering and decay (like Lippmann-Schwinger kets, purely outgoing boundary conditions, exponentially decaying Gamow vectors, causality) are not well defined in the mathematical frame set by the conventional (Hilbert space) axioms of quantum mechanics. Using the Lippmann-Schwinger equations as the takeoff point and aiming for a theory that unites resonances and decay, we conjecture a new axiom for quantum mechanics that distinguishes mathematically between prepared states and detected observables. Suggested by the two signs ±i? of the Lippmann-Schwinger equations, this axiom replaces the one Hilbert space of conventional quantum mechanics by two Hardy spaces. The new Hardy space theory automatically provides Gamow kets with exponential time evolution derived from the complex poles of the S-matrix. It solves the causality problem since it results in a semigroup evolution. But this semigroup brings into quantum physics a new concept of the semigroup time t = 0, a beginning of time. Its interpretation and observations are discussed in the last section.     

On the resistance of relativistic quantum cryptography in open space at finite resources

The security of keys for the basic nonrelativistic BB84 protocol has been examined for more than 15 years. A simple proof of security for the case of a single-photon source of quantum states and finite sequences has been only recently obtained using entropy uncertainty relations. However, the existing sources of states are not strictly single-photon. Since sources are not single-photon and losses in a quantum channel??open space??are not a priori known and vary, nonrelativistic quantum cryptographic systems in open space cannot guarantee the unconditional security of keys. Recently proposed relativistic quantum cryptography removes fundamental constraints associated with non-single-photon sources and losses in open space. The resistance of a fundamentally new family of protocols for relativistic quantum key distribution through open space has been analyzed for the real situation with finite lengths of transmitted sequences of quantum states. This system is stable with real sources of non-single-photon states (weakened laser radiation) and arbitrary losses in open space.     

Relativity from quantum theory

Starting from the fact that the geodesic structure of the projective space of quantum pure states gives a natural explanation for the fundamentality of spin 1/2 systems in relativistic quantum theory, and making use of the inducing construction for infinite-dimensional representations of groups in vector bundles over projective space, a proposal for unifying physical theory in terms of a possible derivation of relativistic physics from pure quantum theory is presented.     

Free-space quantum cryptography using multiphoton states: Secure key distribution to satellites

Whereas quantum cryptography ensures security by virtue of complete indistinguishability of nonorthogonal quantum states, attenuation in quantum communication channels and the unavailability of single-photon sources present major problems. In view of these difficulties, the security of quantum cryptography can change from unconditional to conditional. Since the restrictions imposed by nonrelativistic quantum mechanics and used to formulate key distribution protocols have been largely exhausted, new principles are required. The fundamental relativistic causality principle in quantum cryptography can be used to propose a new approach to ensuring unconditional security of quantum cryptosystems that eliminates the aforementioned difficulties. Quantum cryptosystems of this kind should obviously be called relativistic. It is shown that relativistic quantum cryptosystems remain unconditionally secure: first, attenuation in a quantum communication channel can only reduce the key generation rate, but not the security of the key; second, the source may not generate pure single-photon states, and a nonzero single-photon probability will suffice. The scheme remains secure even if the contribution of a single-photon component is arbitrarily small. This formally implies that a state may be characterized by an arbitrarily large mean photon number. The single-photon probability affects only the key generation rate, but not security.     

Relativistic semigroup,the lorentz group,and tachyons. IV

The algebraic and topological properties of the relativistic semigroup are discussed. Its probability-theoretical features establish that the relativistic semigroup belongs to the type of complex Markov structures. From the functional point of view, the relativistic semigroup is a compact Lie semigroup which is contracting in partial spaces. Principles of measurability, observability, and stochasticity are formulated, and these lead to a space-time structure of complex Markov kind. Thus, a certain probability-theoretical gnosiology is also possible in the theory of relativity.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55–58, August, 1977.I thank D. D. Ivanenko for valuable advice and numerous discussions.     

A relativistic formulation of the Einstein-Podolsky-Rosen paradox

The Einstein-Podolsky-Rosen (EPR) paradox and the correlated states it introduced comprise one of the central interpretive problems of quantum mechanics. Because of the apparent nonlocal character of this paradox, it should be given a relativistic treatment. The purpose of this paper is to provide such a treatment.     

Wave functions in geometric quantization

A geometrical way is described to associate quantum states in the sense of geometric quantization to wave functions in the quantum mechanical sense for each relativistic elementary particle. Explicit computations are made in a number of cases: Klein-Gordon and Dirac equations, neutrino and antineutrino Weyl equations, and very general cases of massive and massless particles of arbitrary spin. In this later case one is led in a canonical way to Penrose wave equations.