Quantum mechanics in noninertial reference frames: Relativistic accelerations and fictitious forces

One-particle systems in relativistically accelerating reference frames can be associated with a class of unitary representations of the group of arbitrary coordinate transformations, an extension of the Wigner–Bargmann definition of particles as the physical realization of unitary irreducible representations of the Poincaré group. Representations of the group of arbitrary coordinate transformations become necessary to define unitary operators implementing relativistic acceleration transformations in quantum theory because, unlike in the Galilean case, the relativistic acceleration transformations do not themselves form a group. The momentum operators that follow from these representations show how the fictitious forces in noninertial reference frames are generated in quantum theory.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

Independent oscillator model of a heat bath: Exact diagonalization of the Hamiltonian

The problem of a quantum oscillator coupled to an independent-oscillator model of a heat bath is discussed. The transformation to normal coordinates is explicitly constructed using the method of Ullersma. With this transformation an alternative derivation of an exact formula for the oscillator free energy is constructed. The various contributions to the oscillator energy are calculated, with the aim of further understanding this formula. Finally, the limitations of linear coupling models, such as that used by Ullersma, are discussed in the form of some critical remarks.     

Quantum Encoding and Entanglement in Terms of Phase Operators Associated with Harmonic Oscillator

Realization of qudit quantum computation has been presented in terms of number operator and phase operators associated with one-dimensional harmonic oscillator and it has been demonstrated that the representations of generalized Pauli group, viewed in harmonic oscillator operators, allow the qudits to be explicitly encoded in such systems. The non-Hermitian quantum phase operators contained in decomposition of the annihilation and creation operators associated with harmonic oscillator have been analysed in terms of semi unitary transformations (SUT) and it has been shown that the non-vanishing analytic index for harmonic oscillator leads to an alternative class of quantum anomalies. Choosing unitary transformation and the Hermitian phase operator free from quantum anomalies, the truncated annihilation and creation operators have been obtained for harmonic oscillator and it has been demonstrated that any attempt of removal of quantum anomalies leads to absence of minimum uncertainty.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

Caldeira-Leggett model, Landau damping, and the Vlasov-Poisson system

The Caldeira-Leggett Hamiltonian describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas (Landau, 1946 [1]). An invertible linear transformation (Morrison and Pfirsch, 1992 [18]; Morrison, 2000 [19]) is presented that converts solutions of the Caldeira-Leggett model into solutions of the linearized Vlasov-Poisson system.     

Interpretation of the Quantum Measurement: Example of a Linearly Coupled Harmonic Oscillator

We argue that it may be possible to consistently explain the quantum measurement by assuming that the wave function is in one-to-one correspondence with objective physical reality and has no probabilistic interpretation. In the context of such approach we consider the model of a harmonic oscillator linearly coupled to a heat bath and treat the oscillator as the system being measured. Three classes of initial pure states for the bath are considered. Exact expressions for the average values and variances of the oscillator coordinate and momentum as functions of time are considered for each class of pure states. It is shown that these quantities exhibit different asymptotic behavior for different classes of initial states of the bath. In particular, if each mode of the bath is initially in a coherent state, then for an arbitrary initial state of the oscillator the variances of the oscillator coordinate and momentum asymptotically approach the same values as for a coherent state of the free oscillator, while the averages of coordinate and momentum show a Brownian-like behavior. We argue that such behavior shows several features of the quantum measurement and supports our interpretation of the wave function.     

Difficulties with massless particles?

Some difficulties with sharp momentum (one-particle) states for massless particles are indicated, in the framework of unitary irreducible representations of the Poincaré group. It is shown that a Poincaré covariant set of such states requires the introduction, in the spatial direction opposite to the point stabilized, of momentum generalized eigenstates which (when the helicity is nonzero) have a nontrivial orbital transformation. The relevance of these generalized momentum eigenstates for massless theories is then shown.     

Born Reciprocity and the Granularity of Spacetime

The Schrödinger-Robertson inequality for relativistic position and momentum operators X ^μ, P _ν, μ, ν = 0, 1, 2, 3, is interpreted in terms of Born reciprocity and ‘non-commutative’ relativistic position-momentum space geometry. For states which saturate the Schrödinger-Robertson inequality, a typology of semiclassical limits is pointed out, characterised by the orbit structure within its unitary irreducible representations, of the full invariance group of Born reciprocity, the so-called ‘quaplectic’ group U(3, 1) #x2297;_s H(3, 1) (the semi-direct product of the unitary relativistic dynamical symmetry U(3, 1) with the Weyl-Heisenberg group H(3, 1)). The example of the ‘scalar’ case, namely the relativistic oscillator, and associated multimode squeezed states, is treated in detail. In this case, it is suggested that the semiclassical limit corresponds to the separate emergence of spacetime and matter, in the form of the stress-energy tensor, and the quadrupole tensor, which are in general reciprocally equivalent.     

量子阻尼振子的两种解析解法

应用u变换与不变量理论，求解一维量子阻尼振子，两种解法的不同仅在于交换了u变换与相因子的顺序。