Quantum anti-zeno paradox

We establish an exact differential equation for the operator describing time-dependent measurements continuous in time and obtain a series solution. Suppose the projection operator E(t) = U(t)EUdagger(t) is measured continuously from t = 0 to T, where E is a projector leaving the initial state unchanged and U(t) a unitary operator obeying U(0) = 1. We prove that the probability of always finding E(t) = 1 from t = 0 to T is unity. If U(t) not equal1, the watched kettle is sure to "boil."     

Quantum Field Formalism for the Electromagnetic Interaction of Composite Particles in a Nonrelativistic Gauge Model

A classical nonrelativistic U(1) × U(1) gauge field model for the electromagnetic interaction of composite particles is proposed and the quantum formalism is constructed. This gauge model containing a Chern–Simons U(1) field and the electromagnetic U(1) field can be coupled to both a bosonic or a fermionic matter field. We explicitly consider the second case, a composite fermion system in the presence of an electromagnetic field, and we carry out the canonical quantization by the Dirac method. The path integral approach is developed and the Feynman rules are established. A simplified model is considered. As an alternative path integral method, the BRST formalism for this gauge model is also treated.     

A Categorification of Quantum \widehat{sl_2}

In this paper, we categorify the algebraU_q(\widehat{sl_2}) with the same approach as in [A. Lauda, Adv. Math. (2010), arXiv:math.QA/0803.3652;
M. Khovanov, Comm. Algebra11 (2001) 5033]. The algebra \dot{U}=\dot{U}_q(\widehat{sl_2}) is obtained fromU_q(\widehat{sl_2}) by adjoining a collection of orthogonal idempotents 1_λ,λ \in P, in which P is the weight lattice ofU_q(\widehat{sl_2}). Under such construction the algebraU is decomposed into a direct sum \bigoplus_{\lambda\in P} 1_{\lambda'}{U} 1_λ. We set the collection of λ\in P as the objects of the categoryU, 1-morphisms fromλ toλ' are given by 1_λ' }k U1_λ, and 2-morphisms are constructed by some semilinear form defined onU. Hence we get a 2-category {\cal U} from the algebraU_q{(\widehat{sl_2})}.     

Multi-types of Skyrmions in SU（N） Quantum Hall System

The skyrmions in SU（N） quantum Hall （QH） system are discussed. By analyzing the gauge field structure and the topological properties of this QH system it is pointed out that in the SU（N） QH system there can exist （N-1） types of skyrmion structures, instead of only one type of skyrmions. In this paper, by means of the Abelian projections according to the （N - 1） Cartan subalgebra local bases, we obtain the （N - 1） U（1） electromagnetic field tensors in the SU（N） gauge field of the QH system, and then derive （N - 1） types of skyrmion structures from these U（1） sub-field tensors. Furthermore, in light of the C-mapping topological current method, the topological charges and the motion of these skyrmions are also discussed.     

Bicross-Product Structure of Affine Quantum Groups

We show that the affine quantum group is isomorphic to a bicross-product central extension of the quantum loop group by a quantum cocycle in R-matrix form.     

Quantum critical transport near the Mott transition

We perform a systematic study of incoherent transport in the high temperature crossover region of the half filled one-band Hubbard model. We demonstrate that the family of resistivity curves displays characteristic quantum critical scaling of the form ρ(T, δU) = ρ(c)(T)f(T/T?(δU)), with T?(δU) ~ |δU|(zν), and ρ(c)(T) ~ T. The corresponding β function displays a "strong coupling" form β ~ ln(ρ(c)/ρ), reflecting the peculiar mirror symmetry of the scaling curves. This behavior, which is surprisingly similar to some experimental findings, indicates that Mott quantum criticality may be acting as the fundamental mechanism behind the unusual transport phenomena in many systems near the metal-insulator transition.     

Elementary Quantum Gates Based on Intrinsic Interaction Hamiltonian

A kind of new operators, the generalized pseudo-spin operators are introduced and a universad intrinsic Hamiltonian of two-qubit interaction is studied in terms of the generalized pseudo-spin operators. A fundamental quantum gate U（θ） is constructed based on the universal Hamiltonian and shown that the roles of the new quantum gate U （θ） is equivalent, functionally, to the joint operation of Hadamard and C-Not gates.     

Quantum cohomology of partial flag manifoldsF_{n_1 } \ldots _{n_k }

We compute the quantum cohomology rings of the partial flag manifolds . The inductive computation uses the idea of Givental and Kim [1]. Also we define a notion of the vertical quantum cohomology ring of the algebraic bundle. For the flag bundle (E) associated with the vector bundleE this ring is found.     

Two Types of Infinite-Dimensional Representations of the Quantum Algebra OSPq(l, 4)

In this letter, we discuss the quantum superalgebra OSP_q(l,4). With the help of the SU_q(2)×U(1) basis, we construct two types of representations for the quantum superalgebra, called the-1/2-representation and the 1/2-representation. Both of them are infinitedimensional.     

Quantum mechanics as a stochastic process with aU(1) degree of freedom

A stochastic theory equivalent to the nonrelativistic quantum mechanics is constructed. A geometric manifestation of U(1) local gauge invariance is proposed. The stochastic theory is not of the type of Nelson's stochastic mechanics.Work supported in part by the US-Israel Binational Science Foundation (BSF).     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

The Embedding of Quantum Algebra U(gl(n+1)) in Irreducible Representations of U(gl(n+1))

Using a formula derived by Jimbo, we express the generators of U(gl(n+1)), the q-analogue of U(gl(n+1)) in an irreducible representation with Gel'fand basis, as functions of the generators of U(gl(n+1)).     

Quantum criticality of D-wave quasiparticles and superconducting phase fluctuations

We present finite temperature (T) extension of the (2+1)-dimensional QED (QED3) theory of under-doped cuprates. The theory describes nodal quasiparticles whose interactions with quantum proliferated hc/2e vortex-antivortex pairs are represented by an emergent U(1) gauge field. Finite T introduces a scale beyond which the spatial fluctuations of vorticity are suppressed. As a result, the spin susceptibility of the pseudogap state is bounded by T2 at low T and crosses over to approximately T at higher T, while the low-T specific heat scales as T2, reflecting the thermodynamics of QED3. The Wilson ratio vanishes as T-->0; the pseudogap state is a "thermal (semi)metal" but a "spin-charge dielectric." This non-Fermi liquid behavior originates from two general principles: spin correlations induced by "gauge" interactions of quasiparticles and fluctuating vortices and the "relativistic" scaling of the T=0 fixed point.     

A New Quantum Analog of the Brauer Algebra

We introduce a new algebra depending on two nonzero complex parameters z and q such that its specialization at z=q ⁿ and q=1 coincides the Brauer algebra. We show that the action of the new algebra commutes with the representation of the twisted deformation of the enveloping algebra U(o _n) in the tensor power of the vector representation.     

Quantum Marginal Inequalities and the Conjectured Entropic Inequalities

A conjecture – the modified super-additivity inequality of relative entropy – was proposed in Zhang et al. (Phys. Lett. A 377:1794–1796, 2013): There exist three unitary operators \(U_{A}\in \mathrm {U}(\mathcal {H}_{A}), U_{B}\in \mathrm {U}(\mathcal {H}_{B})\) , and \(U_{AB}\in \mathrm {U}(\mathcal {H}_{A}\otimes \mathcal {H}_{B})\) such that $$\mathrm{S}\left(U_{AB}\rho_{AB}U^{\dagger}_{AB}||\sigma_{AB}\right)\geqslant \mathrm{S}\left(U_{A}\rho_{A}U^{\dagger}_{A}||\sigma_{A}\right) + \mathrm{S}\left(U_{B}\rho_{B}U^{\dagger}_{B}||\sigma_{B}\right), $$ where the reference state σ is required to be full-ranked. A numerical study on the conjectured inequality is conducted in this note. The results obtained indicate that the modified super-additivity inequality of relative entropy seems to hold for all qubit pairs.     

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.     

Quantum Currents in the coset Space SU(2)/U(1)

We propose a rational quantum deformed nonlocal currentsin the homogeneous space SU(2)k/U(1),and in terms of it and a free boson field a representation for the Drinfeld currents of Yangian double at a general level k=c is obtained.In the classical limit h→0,the quantum nonlocal currents become SU(2)k parafermion,and the realization of Yangian double becomes the parafermion realization of SU(2)k current algebra.     

A Relativistic Gauge Theory of Nonlinear Quantum Mechanics and Newtonian Gravity

A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schrödinger picture of a given field theory. While, for simplicity, we study the example of a \(\mathcal{U}(1)\) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schrödinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. A probabilistic interpretation (Born’s rule) holds, provided the underlying model is scale free.     

Irreducible Representations of Quantum sl(3) Enveloping Algebra

By referring to the construction method of finding representations of classical U(n) group, we developed a parallel method with which one can construct irreducible representations of quantum sl(3) enveloping algebra explicitly.     

Quantum Entanglement in Ground State of Extended Hubbard Model

Metal-insulator and CDW-SDW transitions are studied in the one-dimensional Extended Hubbard Model at half-filling by analysing the behaviour of local entanglement in fermionic systems. 1D traditional Hubbard model exhibits metal-insulator transition at critical point U_c = 0, where local entanglement reaches its maximum value. Moreover, a transition between charge- and spin-density- wave (CDW-SDW) occurs in 1D Extended Hubbard Model tUV with long-range interaction at straight line U = 2 V. The analysis of our obtained results shows that CDW-SDW transition has curious properties whose can be used in quantum information processing.