
1. 





A Closed Form Solution for Quantum Oscillator
Perturbations Using Lie Algebras
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By
Clark Alexander © 2011. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 3

Table of Contents


• 2 Algebraic preliminaries.....2 


• 3 Using Lie algebras to determine perturbed eigenvalues.....3 


• 4 Lie algebras up to order one in λ.....5 


• 5 Explicit computations.....5 


• 6 Extending the method.....8 






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Abstract


We give a new solution to a wellknown problem, that of computing perturbed eigenvalues for quantum
oscillators. This article is nearly self contained and begins with all the necessary algebraic tools to make the
subsequent calculations. We define a new family of Lie algebras relevant to making computations for perturbed
(anharmonic) oscillators, and show that the only two formally closed solutions are indeed harmonic oscillators
themselves.Through elementaryc ombinatorics and noncanonical forms of wellknown Lie algebras, we are able to
obtain a fully closed form solution for perturbed eigenvalues to first order.
MSC 2010: 37K30, 70G65, 81Q05. 



2. 





Unitary Braid Matrices: Bridge between Topological
and Quantum Entanglements
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By
B. Abdesselam and A. Chakrabarti © 2010. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 2

Table of Contents

• 1 Introduction (two faces of unitary braid matrices).....1 


• 2 Unitary braid matrices and their actions.....3 


• 3 Computation of quantum entanglements.....6 


• 4 Odd dimensions.....10 


• 5 Entanglement via a special coupling of 3 spins.....11 






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Abstract


Braiding operators corresponding to the third Reidemeister move in the theory of
knots and links are realized in terms of parametrized unitary matrices for all dimensions.
Two distinct classes are considered. Their (nonlocal) unitary actions on separable pure
product states of three identical subsystems (i.e., the spin projections of three particles)
are explicitly evaluated for all dimensions. This, for our classes, is shown to generate
entangled superposition of four terms in the base space. The 3body and 2body entanglements
(in three 2body subsystems), the 3 tangles, and 2 tangles are explicitly
evaluated for each class. For our matrices, these are parametrized. Varying parameters
they can be made to sweep over the domain (0,1). Thus, braiding operators corresponding
to over and undercrossings of three braids and, on closing ends, to topologically
entangled Borromean rings are shown, in another context, to generate quantum entanglements.
For higher dimensions, starting with different initial triplets one can entangle
by turns, each state with all the rest. A specific coupling of three angular momenta is
briefly discussed to throw more light on three body entanglements. 



3. 





On Deformed Quantum Mechanical Schemes and Value
Equations Based on the Spacespace Noncommutative
HeisenbergWeyl Group
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By
L. Roman Juarez and Marcos Rosenbaum © 2010. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 2

Table of Contents


• 2 The WWGM phasespace quantum mechanics based on the spacespace noncommutative HeisenbergWeyl Lie algebra.....3 


• 3 The StratonovichWeyl correspondence for the spacespace noncommutative HeisenbergWeyl Lie group.....9 


• 4 The Berezin quantization procedure by means of involution operators and its application to the spacespace noncommutative HeisenbergWeyl algebra.....14 


• 5 Starvalue equations for phasespace quantum mechanics based on the spacespace noncommutative HeisenbergWeyl group.....18 





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Abstract


We investigate the WeylWignerGroenewoldMoyal, the Stratonovich, and the
Berezin group quantization schemes for the spacespace noncommutative Heisenberg Weyl
group. We show that the product for the deformed algebra of Weyl functions for
the first scheme is different than that for the other two, even though their respective
quantum mechanics' are equivalent as far as expectation values are concerned, provided
that some additional criteria are imposed on the implementation of this process. We
also show that it is the product associated with the Stratonovich and the Berezin
formalisms that correctly gives the Weyl symbol of a product of operators in terms of
the deformed product of their corresponding Weyl symbols. To conclude, we derive the
stronger valued equations for the 3 quantization schemes considered and discuss the
criteria that are also needed for them to exist. 



4. 





Hilberts Idea of a Physical Axiomatics: The
Analytical Apparatus of Quantum Mechanics
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By
Yvon Gauthier © 2010. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 2

Table of Contents

• 1 Introduction (the consistency of physical theories).....1 


• 2 Quantum mechanics.....2 






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We discuss the Hilbert program for the axiomatization of physics in the context of
what Hilbert and von Neumann came to call the analytical apparatus and its conditions of
reality. We suggest that the idea of aphysical logic is the basis for a physical mathematics
and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert.
Finite probability theory requires finite derivations in the measurement theory of QM
and we give a polynomial formulation of local complementation for the metric induced
on the topology of the Hilbert space. The conclusion hints at a constructivist physics. 



5. 





Lie Symmetries and Exact Solutions of a Class of
thin Film Equations
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By
Roman Cherniha and Liliia Myroniuk © 2010. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 2

Table of Contents


• 2 Lie symmetry of equation (1.1).....3 


• 3 Symmetry reduction and exact solutions.....12 






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Abstract


A symmetry group classification for fourthorder reactiondiffusion equations, allowing
for both secondorder and fourthorder diffusion terms, is carried out. The fourthorder
equations are treated, firstly, as systems of secondorder equations that bear some
resemblance to systems of coupled reactiondiffusion equations with cross diffusion, secondly,
as systems of a secondorder equation and two firstorder equations. The paper
generalizes the results of Lie symmetry analysis derived earlier for particular cases of
these equations. Various exact solutions are constructed using Lie symmetry reductions
of the reactiondiffusion systems to ordinary differential equations. The solutions include
some unusual structures as well as the familiar types that regularly occur in symmetry
reductions, namely, selfsimilar solutions, decelerating and decaying traveling waves, and
steady states. 



6. 





Some Thoughts on Geometries and on the Nature of
the Gravitational Feld
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By
Eduardo a. NotteCuelloa, Roldao Da Rochab, and Waldyr a. Rodrigues Jr © 2010. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 2

Table of Contents


• 2 Torsion as a description of gravity.....4 


• 3 A comment on Einstein most happy though.....10 


• 4 A model for the gravitational eld represented by the nonmetricity of a connection.....11 






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Abstract


This paper shows how a gravitational field generated by a given energymomentum
distribution (for all realistic cases) can be represented by distinct geometrical structures
(Lorentzian, teleparallel, and nonnull nonmetricity spacetimes) or that we even can dispense
all those geometrical structures and simply represent the gravitational field as a
field in Faraday's sense living in Minkowski spacetime. The explicit Lagrangian density
for this theory is given, and the field equations (which are Maxwell's like equations) are
shown to be equivalent to Einstein's equations. Some examples are worked in detail in
order to convince the reader that the geometrical structure of a manifold (modulus some
topological constraints) is conventional as already emphasized by Poincar'e long ago, and
thus the realization that there are distinct geometrical representations (and a physical
model related to a deformation of the continuum supporting Minkowski spacetime) for
any realistic gravitational field strongly suggests that we must investigate the origin of
its physical nature. We hope that this paper will convince readers that this is indeed the
case. 



7. 










Abstract


The use of the data assimilation technique to identify optimal topography is discussed
in frames of timedependent motion governed by nonlinear barotropic ocean model. Assimilation
of artificially generated data allows to measure the influence of various error
sources and to classify the impact of noise that is present in observational data and model
parameters. The choice of length of the assimilation window in 4DVar is discussed. It is
shown that using longer window lengths would provide more accurate ocean topography.
The topography defined using this technique can be further used in other model runs
that start from other initial conditions and are situated in other parts of the model's attractor. 



8. 





On Nonergodic Property of Bose gas with Weak Pair
Interaction
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By
D. V. Prokhorenko © 2009. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 1

Table of Contents


• 2 The algebra of canonical commutative relations.....3 


• 3 The von Neumann dynamics.....6 


• 4 Dynamics of correlations.....6 


• 5 The tree of correlations.....8 


• 6 The general theory of renormalization of U(t,∞)).....11 


• 7 The Friedrichs diagrams.....13 


• 8 The BogoliubovParasiuk renormalization prescriptions.....15 


• 9 Proof of the theoremconstruction.....16 


• 10 Derivation of nonergodic property from main result.....19 


• 11 Examples, chain diagrams.....23 


• 12 Notes on Bogoliubov derivation of Boltzmann equations.....28 






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In this paper we prove that Bosegas with weak pair interaction is nonergodic system.
In order to prove this fact we consider the divergences in some nonequilibrium diagram
technique. These divergences are analogous to the divergences in the kinetic equations
discovered by Cohen and Dorfman. We develop the general theory of renormalization of
such divergences and illustrate it with some simple examples. The fact that the system
is nonergodic leads to the following consequence: to prove that the system tends to the
thermal equilibrium we should take into account its behavior on its boundary. In this
paper we illustrate this thesis with the Bogoliubov derivation of the kinetic equations.




9. 





Newtons Laws for a Biquaternionic Model of the
ElectroGravimagnetic
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By
L. A. Alexeyeva © 2009. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 1

Table of Contents


• 2 Hamiltonian form of Maxwell equations.....2 



• 3 Differential algebra of biquaternions: Bigradients.....3 


• 4 Biwave equations: Cauchy problem.....4 


• 5 Biquaternions of Afield.....5 


• 6 Lorentz transformations.....6 


• 7 Lorentz transformation of biwave equations.....7 


• 8 The third Newton law: The power and density of actingforces.....8 


• 9 The second Newton law: Transformations equation.....9 


• 11 Modified Maxwell equations: Scalar resistance field.....10 


• 12 Stress pseudotensor: Equations of EGMambiences.....11 


• 13 First thermodynamics law.....12 


• 14 The total field equations and interaction energy.....13 






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By using the Hamiltonian form of the Maxwell equations, a biquaternionic model
of electrogravimagnetic (EGM) field is proposed. The EGMfield equations, generating
different charges and currents, are constructed. The field analogs of three Newton's laws
are formulated for free and interacting charges and currents, as well as the total field
of interaction. The Lorentz invariance of the EGMfield equations is investigated (in
particular, the chargecurrent conservation law). It is shown that at the presence of field
interaction, this law differs from the wellknown one. A new modification of the Maxwell
equations is proposed with the scalar resistance field in the biquaternion EGMfield
tension. Relativistic transformations of mass and chargecurrent densities, forces, and
their powers are constructed. The solution of the Cauchy problem is given for equation
of chargecurrent transformations. 



10. 





Models of Damped Oscillators in Quantum
Mechanics
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By
Ricardo CorderoSoto , Erwin Suazo , and Sergei K. Suslov © 2009. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 1

Table of Contents


• 2 The first two models.....3 


• 3 The gauge transformations.....4 


• 4 Separation of variables for a shifted harmonic oscillator.....6 


• 5 The factorization method for shifted harmonic oscillator.....8 


• 6 Dynamics of energyrelated expectation values.....9 


• 7 A relation with the classical damped oscillations.....13 


• 8 The third model.....14 


• 9 Momentum representation.....14 





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We consider several models of the damped oscillators in nonrelativistic quantum mechanics
in a framework of a general approach to the dynamics of the timedependent
Schrodinger equation with variable quadratic Hamiltonians. The Green functions are
explicitly found in terms of elementary functions and the corresponding gauge transformations
are discussed. The factorization technique is applied to the case of a shifted
harmonic oscillator. The time evolution of the expectation values of the energyrelated
operators is determined for two models of the quantum damped oscillators under consideration.
The classical equations of motion for the damped oscillations are derived for
the corresponding expectation values of the position operator.




11. 





Infinite Lie Algebras and Dual Pairs in 4D CFT
Models
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By
Ivan Todorov © 2009. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 1

Table of Contents


• 2 How do 2D CFT methods work in higher dimensions?.....3 


• 3 Fourpoint functions and conformal partial wave expansions.....5 


• 4 Infinitedimensional Lie algebras and real division rings.....6 


• 5 Fock space representation of the dual pair L(F) × U(N, F).....9 





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It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal
fields, however, which naturally arise in conformal operator product expansions, do generate
infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit
(highly reducible) unitary positive energy representations in a Focks pace. The multiplicity of
their irreducible components is governed by a compact gauge group.The mutually commuting
observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of
local scalar fields of conformal dimension two in four spacetime dimensions the associated
dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M.
Nikolov, K.H. Rehren, and the author. 



12. 





Nonassociative Quantum Theory on Octooctonion
Algebra
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By
Jens Koplinger © 2009. Ashdin Publishing
Journal of Physical Mathematics (JPM), Volume 1

Table of Contents



• 3 Nonassociative decomposition of the spin operator.....3 


• 4 Dirac equation on nonassociative algebra.....5 


• 5 Dimensional reduction program (DRP).....7 



• 7 Summary and outlook.....13 





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Abstract


Using octooctonions (i.e.,octonions with octonion coefficients O×O),this paper expresses
select findings from nonassociative quantum theory in harmonized notation: Nonrelativistic
and relativistic spin operators, Pauliand Diracmatrices, Dirac equation with electromagnetic
and gravitational field, and dimensional reduction from quaternionic spin. A generalization
of the dimensional reduction program is proposed to argue that octooctonion algebra is wide
enough to model a speculated quantum theory that contains all symmetries of the Standard
Model, together with fourdimensional Euclidean quantum gravity. The most narrow candidate for such a formulation consists of four generalized Dirac matrices and a fourdimensional
operator space with associated fields and charges. Algebraic properties of this relation will
be discussed, together with a landscape choice between all possible octooctonionic relations
of similar kind. 

