Differential Geometry for Physicists

Hou, Bo‐Yu; Hou, Bingzhe

doi:10.1142/3448

Cited by 27 publications

(31 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is not a new problem, since the authors of many references have considered it from various aspects [12][13][14][15]. By using the definition of Levi-Civita covariant derivative, the second-order covariant derivative of the vierbein ϑ τ r yields a relation…”

Section: The Sky Equation With Source and The Spin-connection Gauge Fmentioning

confidence: 99%

“…The question of unifying the gravitational field with other gauge fields has long been suggested [11][12][13][14] and, in the meantime, such a unification problem was always disturbing physicists. In the literature, the research results have not yet answered unambiguously whether the gravitational and the Yang-Mills fields have the same physical origin.…”

Section: Generalization Of the Gravitational Gauge Theorymentioning

confidence: 99%

“…By means of the gauge approach, the Einstein gravity theory as a field theory of Yang-Mills type will be reformulated. In the literature, though many references have pointed out that the gravitational field is a non-Abelian gauge field [12][13][14][15], yet the field equation in general relativity (GR) is not a Yang-Mills type equation. We find that the reason why the local Lorentz symmetry in the formulation of both metric and Levi-Civita connection does not allow to describe gravity as a Yang-Mills type gauge interaction is because these two gauge interactions are formulated in different languages: specifically, GR is constructed using Levi-Civita connection (in terms of the metric), while the Yang-Mills gauge field theory is described using non-Abelian gauge field connection, which can actually be expressed in terms of the socalled 'Yang-Mills vielbein' (this would be addressed in this paper).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

Shen

2009

Int J Theor Phys

View full text Add to dashboard Cite

A Yang-Mills formulation of Einstein gravity with spin-affine connection as the dynamical variable of gravitational field is suggested based on the Stephenson-KilmisterYang (SKY) equation. A physically interesting property of the present formalism is that the Einstein field equation appears as a first-integral solution to the Yang-Mills type gravitational gauge field equation. The gravitational current density, the law of conservation and the gravitational gauge field strength in vierbein formulation are discussed. The present scheme could provide us with new insight into a possible way to include both Yang-Mills field and gravitational gauge field into one framework of generalized vierbein fields.

show abstract

Section: The Sky Equation With Source and The Spin-connection Gauge Fmentioning

confidence: 99%

Section: Generalization Of the Gravitational Gauge Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

Shen

2009

Int J Theor Phys

View full text Add to dashboard Cite

show abstract

“…A Riemannian metric is first chosen on the manifold of the Lie Group SU(2 n ) (special unitary group) of n-qubit unitary operators with unit determinant [8][9][10][11][12][13][14][15][16][17][18][19][20]. The traceless Hamiltonian serves as a tangent vector to a point on the group manifold of the n-qubit unitary transformation U.…”

Section: Riemannian Geometrymentioning

confidence: 99%

“…In the case of the three-qubit Hamiltonian, there are 4 3 À 1 possible tensor products (corresponding to the dimension of SU(2 3 )), and each term is a 2 3 Â 2 3 matrix. The right-invariant [8][9][10][11]19,20] Riemannian metric for tangent vectors H and J is given by [4] …”

Section: Riemannian Geometrymentioning

confidence: 99%

Quantum computational geodesics

Brandt

2009

Journal of Modern Optics

View full text Add to dashboard Cite

The purpose of this paper is to mathematically investigate characteristics of a geodesic equation describing locally minimum complexity paths in the special unitary group manifold representing the unitary evolution of n qubits associated with a quantum computation. The geodesic equation is a nonlinear differential matrix equation of the Lax type. A simple local initial-value solution is elaborated for the case of three qubits.

show abstract

Coherent state path integral and super‐symmetry for condensates composed of bosonic and fermionic atoms

Mieck

2007

Fortschritte der Physik

View full text Add to dashboard Cite

A super-symmetric coherent state path integral on the Keldysh time contour is considered for bosonic and fermionic atoms which interact among each other with a common short-ranged two-body potential. We investigate the symmetries of Bose-Einstein condensation for the equivalent bosonic and fermionic constituents with the same interaction potential so that a super-symmetry results between the bosonic and fermionic components of super-fields. Apart from the super-unitary invariance U (L|S) of the density terms, we specialize on the examination of super-symmetries for pair condensate terms. Effective equations are derived for anomalous terms which are related to the molecular-and BCS-condensate pairs. A Hubbard-Stratonovich transformation from 'Nambu'-doubled super-fields leads to a generating function with super-matrices for the self-energy whose manifold is given by the orthosympletic supergroup Osp(S, S|2L). A nonlinear sigma model follows from the spontaneous breaking of the orthosymplectic super-group Osp(S, S|2L) to the coset decomposition Osp(S, S|2L)\U (L|S) ⊗ U (L|S). The invariant subgroup U (L|S) for the vacuum or background fields is represented by the density terms in the self-energy whereas the super-matrices on the coset space Osp(S, S|2L)\U (L|S) describe the anomalous molecular and BCS-pair condensate terms. A change of integration measure is performed for the coset decomposition Osp(S, S|2L)\U (L|S) ⊗ U (L|S) , including a separation of density and anomalous parts of the self-energy with a gradient expansion for the Goldstone modes. The independent anomalous fields in the actions can be transformed by the inverse square root Ĝ−1/2 Osp\U of the metric tensor of Osp(S, S|2L)\U (L|S) so that the non-Euclidean integration measure with super-Jacobi-determinant [SDET( ĜOsp\U )] 1/2 can be removed from the coherent state path integral and Gaussian-like integrations remain. The variations of the independent coset fields in the effective actions result in classical field equations for a nonlinear sigma model with the anomalous terms. The dynamics of the eigenvalues of the coset matrices is determined by Sine-Gordon equations which have a similar meaning for the dynamics of the molecular-and BCS-pair condensates as the Gross-Pitaevskii equation for the coherent wave function in BEC phenomena.

show abstract

Differential Geometry for Physicists

Cited by 27 publications

References 0 publications

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation

Quantum computational geodesics

Coherent state path integral and super‐symmetry for condensates composed of bosonic and fermionic atoms

Contact Info

Product

Resources

About