On Gauge Invariance of the Bosonic Measure in Chiral Gauge Theories

Gauge invariance of the measure associated with the gauge field is usually taken for granted, in a general gauge theory. We furnish a proof of this invariance, within Fujikawa’s approach. To stress the importance of this fact, we briefly review gauge anomaly cancellation as a consequence of gauge invariance of the bosonic measure and compare this cancellation to usual results from algebraic renormalization, showing that there are no potential inconsistencies. Then, using a path integral argument, we show that a possible Jacobian for the gauge transformation has to be the identity operator, in the physical Hilbert space. We extend the argument to the complete Hilbert space by a direct calculation.

Around averaged mappings

This paper is intended for a general mathematical audience. The examples show how the study of existence of fixed points of averaged mappings $$T_{\lambda }= (1-\lambda )I+ \lambda T$$ T λ = ( 1 - λ ) I + λ T , where $$0<\lambda <1$$ 0 < λ < 1 and I is the identity operator, can help in the study of existence of fixed points of mappings T.

New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

A new class of operators, larger than ∗ \ast -finite operators, named generalized ∗ \ast -finite operators and noted by Gℱ ∗ ( ℋ ) {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ ∗ ( ℋ ) = { ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) : ∥ T A − B T ∗ − λ I ∥ ≥ ∣ λ ∣ , ∀ λ ∈ C , ∀ T ∈ ℬ ( ℋ ) } . {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.

Blind quantum computation with hybrid model

Blind quantum computation (BQC) allows a client who has a few quantum abilities to interact and delegate her quantum computation to a server that has strong quantum computabilities, while the server learns nothing about client’s quantum inputs, algorithms, and outputs. In this article, a new BQC protocol with hybrid model is proposed, using the combination of rotation operators to construct arbitrary quantum gate. Our BQC protocol is divided into two phases. In the first phase, a new kind of function operation is designed and defined, that is, the client sends the classical message [Formula: see text] to the server and then the sever performs a corresponding function operation [Formula: see text], which has been defined. In the second phase, a rotation operator or identity operator is implemented by quantum gate teleportation where the server can’t know which quantum gate has been teleported. Combining these two phases, the server has no idea about client’s quantum algorithms. When the server performs the corresponding operation honestly, the client only needs to perform [Formula: see text] and [Formula: see text] operators.

Quantum Feller semigroup in terms of quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

<p class="Abstract">In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar identity operator. In our case that scalar is equal to one.</p>

Construction of Frames on the Heisenberg Groups

In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.