Poisson bracket relations for generators of canonical transformations are derived directly from the Galilei and Poincare groups of changes of space-time coordinates. The method is simple but rigorous. The meaning of each step is clear because it corr
esponds to an operation in the group of changes of space-time coordinates. Only products and inverses are used; differences are not used. It is made explicitly clear why constants occur in some bracket relations but not in others, and how some constants can be removed, so that in the end there is a constant in the bracket relations for the Galilei group but not for the Poincare group. Each change of coordinates needs to be only to first order, so matrices are not needed for rotations or Lorentz transformations; simple three-vector descriptions are enough. Conversion to quantum mechanics is immediate. One result is a simpler derivation of the commutation relations for angular momentum directly from rotations. Problems are included.
Dependent symmetries, symmetries that depend on the situation of the subsystem in a larger closed system, are explored by looking at simple examples. This is a new kind of symmetry in the open quantum dynamics of a subsystem Each symmetry implies a
particular form for the results of the open dynamics. The forms exhibit the symmetries very simply. It is shown directly, without assuming anything about the symmetry, that the dynamics produces the form, but knowing the symmetry and the form it implies can reduce what needs to be done to work out the dynamics; pieces can be deduced from the symmetry rather that calculated from the dynamics. Symmetries can be related to constants of the motion in new ways. A quantity might be a dependent constant of the motion, constant only for particular situations of the subsystem in the larger system. In particular, a generator of dependent symmetries could represent a quantity that is a dependent constant of the motion for the same situations as for the symmetries. The examples present a variety of possibilities. Sometimes a generator of dependent symmetries does represent a dependent constant of the motion. Sometimes it does not. Sometimes no quantity is a dependent constant of the motion. Sometimes every quantity is.
Simple examples are used to introduce and examine symmetries of open quantum dynamics that can be described by unitary operators. For the Hamiltonian dynamics of an entire closed system, the symmetry takes the expected form which, when the Hamiltonia
n has a lower bound, says that the unitary symmetry operator commutes with the Hamiltonian operator. There are many more symmetries that are only for the open dynamics of a subsystem. Examples show how these symmetries alone can reveal properties of the dynamics and reduce what needs to be done to work out the dynamics. A symmetry of the open dynamics of a subsystem can even imply properties of the dynamics for the entire system that are not implied by the symmetries of the dynamics of the entire system. The symmetries are generally not related to constants of the motion for the open dynamics of the subsystem. There are many symmetries that cannot be seen in the Schrodinger picture as symmetries of dynamical maps of density matrices for the subsystem. There are symmetries of the open dynamics of a subsystem that depend only on the dynamics. In the simplest examples, these are also symmetries of the dynamics of the entire system. There are many more symmetries, of a new kind, that also depend on correlations, or absence of correlations, between the subsystem and the rest of the entire system, or on the state of the rest of the entire system.
Experiments that look for nonlinear quantum dynamics test the fundamental premise of physics that one of two separate systems can influence the physical behavior of the other only if there is a force between them, an interaction that involves momentu
m and energy. The premise is tested because it is the assumption of a proof that quantum dynamics must be linear. Here variations of a familiar example are used to show how results of nonlinear dynamics in one system can depend on correlations with the other. Effects of one system on the other, influence without interaction between separate systems, not previously considered possible, would be expected with nonlinear quantum dynamics. Whether it is possible or not is subject to experimental tests together with the linearity of quantum dynamics. Concluding comments and questions consider directions our thinking might take in response to this surprising unprecedented situation.
The change of the plane of oscillation of a Foucault pendulum is calculated without using equations of motion, the Gauss-Bonnet theorem, parallel transport, or assumptions that are difficult to explain.
An example shows that weak decoherence is more restrictive than the minimal logical decoherence structure that allows probabilities to be used consistently for quantum histories. The probabilities in the sum rules that define minimal decoherence are
all calculated by using a projection operator to describe each possibility for the state at each time. Weak decoherence requires more sum rules. They bring in additional variables, that require different measurements and a different way to calculate probabilities, and raise questions of operational meaning. The example shows that extending the linearly positive probability formula from weak to minimal decoherence gives probabilities that are different from those calculated in the usual way using the Born and von Neumann rules and a projection operator at each time.
Examples of repeatable procedures and maps are found in the open quantum dynamics of one qubit that interacts with another qubit. They show that a mathematical map that is repeatable can be made by a physical procedure that is not.
Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite different f
or the same unitary dynamics in the same situation in the larger system. An affine form is used for both kinds of maps to find necessary and sufficient conditions for inverse maps. All the different maps with the same homogeneous part in their affine forms have inverses if and only if the homogeneous part does. Some of these maps are completely positive; others are not, but the homogeneous part is always completely positive. The conditions for an inverse are the same for maps that are not completely positive as for maps that are. For maps defined for fixed mean values, the homogeneous part depends only on the unitary operator for the dynamics of the larger system, not on any state or mean values or correlations. Necessary and sufficient conditions for an inverse are stated several different ways: in terms of the maps of matrices, basis matrices, density matrices, or mean values. The inverse maps are generally not tied to the dynamics the way the maps forward are. A trace-preserving completely positive map that is unital can not have an inverse that is obtained from any dynamics described by any unitary operator for any states of a larger system.
A Markov approximation in open quantum dynamics can give unphysical results when a map acts on a state that is not in its domain. This is examined here in a simple example, an open quantum dynamics for one qubit in a system of two interacting qubits,
for which the map domains have been described quite completely. A time interval is split into two parts and the map from the exact dynamics for the entire interval is replaced by the conjunction of that same map for both parts. If there is any correlation between the two qubits, unphysical results can appear as soon as the map conjunction is used, even for infinitesimal times. If the map is repeated an unlimited number of times, every state is at risk of being taken outside the bounds of physical meaning. Treatment by slippage of initial conditions is discussed.
Simple examples are constructed that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. One of the two qubits interacts with a third qubit, a control, that is never entangled or correlated with
either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. When the entanglement increases, the map that describes the change of the state of the two entangled qubits is not completely positive. Combination of two independent interactions that individually give exponential decay of the entanglement can cause the entanglement to not decay exponentially but, instead, go to zero at a finite time.
