Relativistic generalization and extension to the non-Abelian gauge theory of Feynman's proof of the Maxwell equations

Tanimura, Shinji

doi:10.1016/0003-4916(92)90362-p

Cited by 70 publications

(125 citation statements)

References 10 publications

Supporting

Mentioning

123

Contrasting

Order By: Relevance

“…This proof was generalized to the setting of gauge fields by C. R. Lee [3] and an attempt towards a relativistic theory was made by S. Tanimura [4]. Further considerations appear also in a review by J.F Carinena, L.A. Ibort, G.Marmo and A.Stern [2].…”

Section: Introductionmentioning

confidence: 90%

Generally covariant quantum mechanics on noncommutative configuration spaces

Kopf

Paschke²

2007

Journal of Mathematical Physics

View full text Add to dashboard Cite

We generalize the previously given algebraic version of "Feynman's proof of Maxwell's equations" to noncommutative configuration spaces. By doing so, we also obtain an axiomatic formulation of nonrelativistic quantum mechanics over such spaces, which, in contrast to most examples discussed in the literature, does not rely on a distinguished set of coordinates. We give a detailed account of several examples, e.g., C ∞ (Q) ⊗ Mn(C) which leads to nonabelian Yang-Mills theories, and of noncommutative tori T d θ . Moreover we, examine models over the Moyal-deformed plane R 2 θ . Assuming the conservation of electrical charges, we show that in this case the canonical uncertainty relation [x k ,ẋ l ] = ig kl with metric g kl is only consistent if g kl is constant.

show abstract

Section: Introductionmentioning

confidence: 90%

Generally covariant quantum mechanics on noncommutative configuration spaces

Kopf

Paschke²

2007

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…We claim that a matter field, such as quarks and leptons, is defined by a stable localized geometry, which is a topological object in the defining algebra (noncommutative ⋆-algebra) of quantum gravity [7]. First we review in Section IV.A Feynman's view [46][47][48] of the electrodynamics of a charged particle to understand why an extra internal space is necessary to introduce the weak and the strong forces. The extra dimensions appear with a Poisson structure of Lie algebra type implemented with some localizability condition to stabilize the internal space.…”

Section: Outline Of the Papermentioning

confidence: 99%

“…(232) Consider a particle motion defined on R 3 × F with an internal space F whose coordinates are {x i : i = 1, 2, 3} ∈ R 3 and {Q I : I = 1, · · · , n 2 − 1} ∈ F . The dynamics of the particle carrying an internal charge in F [47,48] is defined by a symplectic structure on T * R 3 × F whose commutation relations are given by…”

Section: A Feynman's View On Electrodynamicsmentioning

confidence: 99%

Quantum gravity from noncommutative spacetime

Lee

Yang

2014

Journal of the Korean Physical Society

View full text Add to dashboard Cite

We review a novel and authentic way to quantize gravity. This novel approach is based on the fact that Einstein gravity can be formulated in terms of a symplectic geometry rather than a Riemannian geometry in the context of emergent gravity. An essential step for emergent gravity is to realize the equivalence principle, the most important property in the theory of gravity (general relativity), from U (1) gauge theory on a symplectic or Poisson manifold. Through the realization of the equivalence principle, which is an intrinsic property in symplectic geometry known as the Darboux theorem or the Moser lemma, one can understand how diffeomorphism symmetry arises from noncommutative U (1) gauge theory; thus, gravity can emerge from the noncommutative electromagnetism, which is also an interacting theory. As a consequence, a background-independent quantum gravity in which the prior existence of any spacetime structure is not a priori assumed but is defined by using the fundamental ingredients in quantum gravity theory can be formulated. This scheme for quantum gravity can be used to resolve many notorious problems in theoretical physics, such as the cosmological constant problem, to understand the nature of dark energy, and to explain why gravity is so weak compared to other forces. In particular, it leads to a remarkable picture of what matter is. A matter field, such as leptons and quarks, simply arises as a stable localized geometry, which is a topological object in the defining algebra (noncommutative ⋆-algebra) of quantum gravity.

show abstract

“…It would be of interest to know whether other types of equations of physical interest, for example to the realm of gravity [14], can be developed along similar lines as described here.…”

Section: Maxwell Equations and Poisson Bracketsmentioning

confidence: 99%