Holonomy in Quaternionic Quantum Mechanics

The generalization of geometric phase for the quantum systems described by quaternionic quantum mechanics is given. The geometry of the quantum cyclic evolution is studied and the quaternionic Berry phase is shown to be given by the holonomy of the suitably defined fiber bundle.     

A unified theory of quantum holonomies

A periodic change of slow environmental parameters of a quantum system induces quantum holonomy. The phase holonomy is a well-known example. Another is a more exotic kind that exhibits eigenvalue and eigenspace holonomies. We introduce a theoretical formulation that describes the phase and eigenspace holonomies on an equal footing. The key concept of the theory is a gauge connection for an ordered basis, which is conceptually distinct from Mead-Truhlar-Berry’s connection and its Wilczek-Zee extension. A gauge invariant treatment of eigenspace holonomy based on Fujikawa’s formalism is developed. Example of adiabatic quantum holonomy, including the exotic kind with spectral degeneracy, are shown.     

Quantum adiabatic theorem and universal holonomy

We give a simplified proof of the quantum adiabatic theorem for a system of possibly degenerate Hamiltonians by taking Berry's phase into account. We also relate the adiabatic transformation to the parallel transport induced by the holonomy in the universal bundle over a Grassman manifold. The special case of a nondegenerate Hamiltonian is precisely the cyclic quantum evolution studied by Aharanov and AnandanThe author is S. Y. Wu     

Off-diagonal quantum holonomy along density operators

Uhlmann's concept of quantum holonomy for paths of density operators is generalised to the off-diagonal case providing insight into the geometry of state space when the Uhlmann holonomy is undefined. Comparison with previous off-diagonal geometric phase definitions is carried out and an example comprising the transport of a Bell-state mixture is given.     

On geometric interpretation of the berry phase

A geometric interpretation of the Berry phase and its Wilczek–Zee non-Abelian generalization are given in terms of connections on principal fiber bundles. It is demonstrated that a principal fiber bundle can be trivial in all cases, while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.     

Geometric phase as a simple closed-form integral

For a general evolution of a quantal system, the geometric phase measured with reference to a given initial state is derived as an integral of a function of the pure state density operator by invoking the Pancharatnam connection continuously.     

Quantum mechanical version of the classical Liouville theorem

In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from |z>to|sz-rz*> corresponds to the motion from a point z(q,p) to another point sz-rz* with |s|2-|r|2=1.The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.     

Methods of geometrical integration in accelerator physics

In the paper we consider a method of geometric integration for a long evolution of the particle beam in cyclic accelerators, based on the matrix representation of the operator of particles evolution. This method allows us to calculate the corresponding beam evolution in terms of two-dimensional matrices including for nonlinear effects. The ideology of the geometric integration introduces in appropriate computational algorithms amendments which are necessary for preserving the qualitative properties of maps presented in the form of the truncated series generated by the operator of evolution. This formalism extends both on polarized and intense beams. Examples of practical applications are described.     

用不变量方法解具有非线性场和原子与场非线性耦合的Jaynes-Cummings模型

采用Lewis-Riesenfeld不变量方法研究了具有非线性场和任意形式原子与场相互作用的Jaynes-Cummings模型.该模型由于具有超对称代数结构,因此其Hamiltonian量可用超对称算符的线性组合表示.在算符N′的本征值子空间,用生成元算符构造出系统的不变量后,利用不变量方法求得了系统的一般波函数和时间演化算符,同时也计算了原子布居数和光子数的时间演化表达式.     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Operating a geometric quantum gate by external controllable parameters

A scheme to perfectly preserve an initial qubit state in geometric quantum computation is proposed for a single-qubit geometric quantum gate in a nuclear magnetic resonance system. At first, by adjusting some magnetic field parameters, one can let the dynamic phase be proportional to the geometric phase. Then, by controlling the azimuthal angle in the initial state, we may realize a geometric quantum gate whose fidelity is equal to one under cyclic evolution. This means that the quantum information is no distortion in the process of geometric quantum computation.     

Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Carathéodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes’s distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Carathéodory distance dh defined by A. In this paper we make precise this link, showing that the equality of d and d _H strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).     

Unoriented WZW Models and Holonomy of Bundle Gerbes

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models. manche meinen lechts und rinks kann man nicht velwechsern werch ein illtum Ernst Jandl [Jan95] K.W. is supported with scholarships by the German Israeli Foundation (GIF) and by the Rudolf und Erika Koch–Stiftung.     

Degeneracy in loop variables

The small algebra of loop functionals, defined by Rovelli and Smolin, on the Ashtekar phase space of general relativity is studied. Regarded as coordinates on the phase space, the loop functionals become degenerate at certain points. All the degenerate points are found and the corresponding degeneracy is discussed. The intersection of the set of degenerate points with the real slice of the constraint surface is shown to correspond precisely the Goldberg-Kerr solutions. The evolution of the holonomy group of Ashtekar's connection is examined, and the complexification of the holonomy group is shown to be preserved under it. Thus, an observable of the gravitational field is constructed.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

On geometric interpretation of the Aharonov–Bohm effect

A geometric interpretation of the Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is demonstrated that the principal fiber bundle can be trivial while the connection and its holonomy group are nontrivial. Therefore, the main role is played by geometric rather than topological effects.     

Fluxes,bundle gerbes and 2-Hilbert spaces

We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their categories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the \(\mathsf {S}\mathsf {U}(2)\) WZW model, providing an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions.     

Superconnections and index theory

We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate η$\eta$-invariants and prove an APS theorem, and construct a geometric determinant line bundle for families of such operators, computing its curvature and holonomy in terms of familiar index theoretic quantities.