We show two independent theoretical methods which lead to generate eigen values of a composite two-particles system associated with the non-locality of EPR effect which was experimentally proved by Aspect and others.One method is to use the quantum groups (Woronowicz and Podles)and the other is to use the two-dimensional kappa-Poincare groups.The representation of Hilbert spaces of these groups has eigen functions and eigen values with real numbers on one dimensional spaces.The range of eigen values is real number,so that the composite two particles system in quantum groups can macroscopicly exists and it has a nonlocal effect.
We show two independent theoretical methods which lead to generate eigen values of a composite two-particles system associated with the non-locality of EPR effect which was experimentally proved by Aspect and others.One method is to use the quantum groups (Woronowicz and Podles)and the other is to use the two-dimensional kappa-Poincare groups.The representation of Hilbert spaces of these groups has eigen functions and eigen values with real numbers on one dimensional spaces.The range of eigen values is real number,so that the composite two particles system in quantum groups can macroscopicly exists and it has a nonlocal effect.
"The problem of providing a physical interpretation of a relativistic quantum theory is discussed. It is shown that a single operator corresponding to a localized measurement on the states of the-theory is sufficient for a significant part (perhaps all) of this interpretation. The application of this method of interpretation to the relativistic quantum theory defined on the representation space of an irreducible representation of a higher symmetry group containing the Poincare group is discussed. It is concluded that the higher symmetry is, in fact, of little use for this interpretation. This .leads in a natural way to the problem of finding some of these ""quasilocal"" operators on the space of states of a given relativistic quantum theory. 1'/ This problem is specialized to the problem of constructing a model relativistic quantum theory of a certain type. The procedure used is to attempt to construct a representation of the generators of the Poincare group on the Fock space X of real scalarbosons in such a way that quasilocal observables are easily expressible in terms of the canonical creation-destruction operators. The structure of the model permits the construction of the generators to be approached by successively satisfying their structure relations in each of the spaces ~(n) specified by the eigenvalues of the free field particle number operator. These structure relations are satisfied in~(l) and~(2). The mathematical details of the problem increase rapidly with n, and no explicit solution is obtained for n > 2. Arguments are given which make it plausible that the structure relations in~(n), with n> 2, may be satisfied by a large class of generators. Several trial solutions in the space ~(3) are described, and the difficulties associated with them are discussed."
A noncanonical quantum system, consisting of two nonrelativistic particles, interacting via a harmonic potential, is considered. The center-of-mass position and momentum operators obey the canonical commutation relations, whereas the internal variables are assumed to be the odd generators of the Lie superalgebra sl(1,2). This assumption implies a set of constraints in the phase space, which are explicitly written in the paper. All finite-dimensional irreducible representations of sl(1,2) are considered. Particular attention is paid to the physical representations, i.e., the representations corresponding to Hermitian position and momentum operators. The properties of the physical observables are investigated. In particular, the operators of the internal Hamiltonian, the relative distance, the internal momentum, and the orbital momentum commute with each other. The spectrum of these operators is finite. The distance between the constituents is preserved in time. It can take no more than three different values. For any non-negative integer or half-integer l there exists a representation, where the orbital momentum is l (in units of 2ℏ). The position of any one of the particles cannot be localized, since the operators of the coordinates do not commute with each other. The constituents are smeared with a certain probability within a finite surface, which moves with a constant velocity together with the center of mass.
The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell's inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations.
The eigenstate existing in the problem of EPR paradox is introduced.The commutation of related operators is analyzed and the wave-function in coordinate representation of the eigenstate is verified and its physical signification is discussed.The eigenstate is expended in the direct product space of the moment eigenstates of a two-particle system,subsequently the wave-function in momentum representation is obtained.The relation of the form of the wave-function in momentum representation and the selection of the origin of coordinates,including its physical meaning was analyzed.Results showed that the eigenstate was a entangled state of the coordinate and momentum.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.