Thomas F. Jordan

Dynamics of initially entangled open quantum systems

Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.

A NO INTERACTION THEOREM IN CLASSICAL RELATIVISTIC HAMILTONIAN PARTICLE DYNAMICS

It is shown that a relativistically invariant classical mechanical Hamiltonian description of a system of three (spinless) particles admits no interaction between the particles. If a set of ten functions of the canonical variables of the three‐particle system satisfies the Poisson bracket relations characteristic of the ten generators of the inhomogeneous Lorentz group, and‐with the canonical position variables of the particles‐satisfies the Poisson bracket equations which express the familiar transformation properties of the (time‐dependent) particle positions under space translation, space rotation, and Lorentz transformation, then this set of ten functions can only describe a system of three free particles. A significant part of the proof is valid for a system containing any fixed number of particles. In this general case, a simplified form is established for the Hamiltonian and generators of Lorentz transformations, and it is shown that the generators of space translations and space rotations can be p...

Lorentz-group Berry phases in squeezed light

Abstract Berry phases for the Lorentz group are obtained from transformations that produce squeezed states of the electromagnetic field. They involve phases that could be observed in interference experiments with degenerate parametric amplifiers of light or microwaves.

Quantum correlations do not transmit signals

Abstract That signals cannot be transmitted with the correlations observed in the Einstein-Podolsky-Rosen experiments that violate Bells inequalities is shown to be a simple general property of correlations between separate subsystems of any quantum-mechanical system. It does not depend, for example, on the particular properties of the state of two spin 1 2 particles with total spin zero.

Irreducible Representations of Generalized Oscillator Operators

All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para‐Bose or para‐Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three‐dimensional rotation group in the para‐Fermi case, and with the three generators of the three‐dimensional Lorentz group in the para‐Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.

DYNAMICAL MAPPINGS OF DENSITY OPERATORS IN QUANTUM MECHANICS

The most general dynamical law for a quantum mechanical system is studied with particular reference to the necessary and sufficient conditions for such a law to represent Hamiltonian dynamics. The main results are stated in the form of three theorems.

Berry phases and unitary transformations

Berry phases for spin are defined for any closed loop made by a vector changing direction in three‐dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are m...

Lorentz transformations that entangle spins and entangle momenta

Simple examples are presented of Lorentz transformations that entangle the spins and momenta of two particles with positive mass and spin

Cosmology calculations almost without general relativity

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SIMPLE DERIVATION OF THE NEWTON-WIGNER POSITION OPERATOR

. They apply to indistinguishable particles, produce maximal entanglement from finite Lorentz transformations of states for finite momenta, and describe entanglement of spins produced together with entanglement of momenta. From the entanglements considered, no sum of entanglements is found to be unchanged.