Related papers: Quantum Algebras and Quantum Physics
A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation…
An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the…
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical…
In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully…
Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a…
We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.
Let us consider a Lie (super)algebra $G$ spanned by $T_{\alpha}$ where $T_{\alpha}$ are quantum observables in BV-formalism. It is proved that for every tensor $c^{\alpha_1...\alpha_k}$ that determines a homology class of the Lie algebra…
The states of the physical algebra, namely the algebra generated by the operators involved in encoding and processing qubits, are considered instead of those of the whole system-algebra. If the physical algebra commutes with the interaction…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert…
In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of…
We define a "quantum relation" on a von Neumann algebra M \subset B(H) to be a weak* closed operator bimodule over its commutant M'. Although this definition is framed in terms of a particular representation of M, it is effectively…
The approximations of classical mechanics resulting from quantum mechanics are richer than a correspondence of classical dynamical variables with self-adjoint Hilbert space operators. Assertion that classical dynamic variables correspond to…
We show that the approximate quasiorthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of…
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace both in the case where there are not central charges…
Quantum reference frames are needed in quantum theory for much the same reasons that reference frames are in classical theories: to manifest invariance in line with fundamental relativity principles and to provide a basis for the definition…
We study quantum equivalents of non-commutative operators in quantum mechanics. Any matrix "$B$" satisfying the non-commuting relation $[A,B]\neq 0$ with "$A$", can be used via $B^{-1} AB$ to reproduce eigenvalues of "$A$". This…
We consider the quantum mechanics of a particle on a noncommutative two-sphere with the coordinates obeying an SU(2)-algebra. The momentum operator can be constructed in terms of an $SU(2)\times SU(2)$-extension and the Heisenberg algebra…
The universal enveloping algebra U(g) of a Lie algebra g acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or "quantum group") is a deformation of a universal…
In this paper, we demonstrate the equivalence between the complex Hilbert space and real Kahler space formulations of quantum mechanics. Complex numbers play an important role in the traditional formulation of quantum mechanics in complex…
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…