Reduction of dynamical systems is closely related with symmetry. The purpose of this article is to show that Fourier analysis both on compact Lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing. The reduction procedure is applied to systems of many identical particles lying in R3 which admit the action of a rotation group SO(3) and of a symmetric or permutation group.
Reduction of dynamical systems is closely related with symmetry. The purpose of this article is to show that Fourier analysis both on compact Lie groups and on finite groups serves as a reduction procedure for quantum systems with symmetry on an equal footing. The reduction procedure is applied to systems of many identical particles lying in R3 which admit the action of a rotation group SO(3) and of a symmetric or permutation group.
Quantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a three-dimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group S3 as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the S3 action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.
Lie symmetry group methods provide a useful tool for the analysis of differential equations in a variety of areas in physics and applied mathematics. The nature of symmetry is that it provides information on properties which remain invariant under transformation. In differential equations this invariance provides a route toward complete integrations, reductions, linearisations and analytical solutions which can evade standard techniques of analysis. In this thesis we study two problems in quantum mechanics from a symmetry perspective: We consider for pedagogical purposes the linear time dependent Schrodinger equation in a potential and provide a symmetry analysis of the resulting equations. Thereafter, as an original contribution, we study the group theoretic properties of the density matrix equation for the quantum Brownian motion of a free particle interacting with a bath of harmonic oscillators. We provide a number of canonical reductions of the system to equations of reduced dimensionality as well as several complete integrations.
We study deterministic and quantum dynamics from a constructive “finite” point of view, since the introduction of the continuum or other actual infinities in physics poses severe conceptual and technical difficulties, and while all of these concepts are not really needed in physics, which is in fact an empirical science. Particular attention is paid to the symmetry properties of discrete systems. For a consistent description of the symmetries of dynamical systems at different time instants and the symmetries of various parts of such systems, we introduce discrete analogs of gauge connections. These gauge structures are particularly important to describe the quantum behavior. The symmetries govern the fundamental properties of the behavior of dynamical systems. In particular, we can show that the moving soliton-like structures are inevitable in a deterministic (classical) dynamical system, whose symmetry group breaks the set of states into a finite number of orbits of the group. We demonstrate that the quantum behavior is a natural consequence of symmetries of dynamical systems. This behavior is a result of the fundamental inability to trace the identity of indistinguish-able objects during their evolution. Information is only available on invariant statements and values related with such objects. Using general mathematical arguments, any quantum dynamics can be shown to reduce to a sequence of permutations. The quantum interferences occur in the invariant subspaces of permutation representations of the symmetry groups of dynamical systems. The observables can be expressed in terms of permutation invariants. We also show that in order to describe quantum phenomena it is sufficient to use cyclotomic fields—the minimal extensions of natural numbers suitable for quantum mechanics, instead of a non-constructive number system—the field of complex numbers. The finite groups of symmetries play the central role in this review. In physics there is an additional reason for such groups to be of interest. Numerous experiments and observations in particle physics point to the importance of finite groups of relatively low orders in a number of fundamental processes. The origin of these groups has no explanation within presently recognized theories, such as the Standard Model.
I describe the role of symmetry in two quantum algorithms, with a focus on how that symmetry is made manifest by the Fourier transform. The Fourier transform can be considered in a wider context than the familiar one of functions on Rn or Z/nZ; instead it can be defined for an arbitrary group where it is known as representation theory. The first quantum algorithm solves an instance of the hidden subgroup problem--distinguishing conjugates of the Borel subgroup from each other in groups related to PSL(2; q). I use the symmetry of the subgroups under consideration to reduce the problem to a mild extension of a previously solved problem. This generalizes a result of Moore, Rockmore, Russel and Schulman[33] by switching to a more natural measurement that also applies to prime powers. In contrast to the first algorithm, the second quantum algorithm is an attempt to use naturally continuous spaces. Quantum walks have proved to be a useful tool for designing quantum algorithms. The natural equivalent to continuous time quantum walks is evolution with the Schrodinger equation, under the kinetic energy Hamiltonian for a massive particle. I take advantage of quantum interference to find the center of spherical shells in high dimensions. Any implementation would be likely to take place on a discrete grid, using the ability of a digital quantum computer to simulate the evolution of a quantum system. In addition, I use ideas from the second algorithm on a different set of starting states, and find that quantum evolution can be used to sample from the evolute of a plane curve. The method of stationary phase is used to determine scaling exponents characterizing the precision and probability of success for this procedure.
Symmetry plays a central role throughout physics, from Fourier analysis on discrete lattices to the classification of elementary particles in the standard model. In finite quantum systems, symmetries are often discrete, such as reflection and rotational symmetries. Here, we present the SIM-GRAPH method (Symmetric Ising Models - Graph Reduction And Projected Hamiltonians), which uses such symmetries to efficiently calculate ground-state observables of interacting quantum systems. By projecting all interactions onto a symmetric subset of the system, the method effectively reduces the number of sites, thus providing an exponential speedup over standard exact diagonalization. Furthermore, we introduce two extensions that broaden the applicability of the method and allow for a controlled trade-off between computational cost and accuracy.
Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations