Topic
Four-vector
About: Four-vector is a research topic. Over the lifetime, 569 publications have been published within this topic receiving 12731 citations.
Papers published on a yearly basis
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Papers
TL;DR: The superposition principle of the wave function is defined in this article, which is the fundamental principle of quantum mechanics that the system of states forms a linear manifold, in which a unitary scalar product is defined.
Abstract: It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.
3,248 citations
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TL;DR: In this article, the authors give an example of a Lorentz invariant discrete space-time, which is not required by the assumption that space time is a continuous space, and show that it is possible to construct a discrete space time with Lorentzi invariance.
Abstract: It is usually assumed that space-time is a continuum. This assumption is not required by Lorentz invariance. In this paper we give an example of a Lorentz invariant discrete space-time.
2,616 citations
TL;DR: In this article, Utiyama's discussion is extended by considering the 10-parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables.
Abstract: An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama's discussion is extended by considering the 10‐parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierbein components hk μ as well as the ``local affine connection'' Ai jμ . The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to Fμν for the electromagnetic field, and the simplest possible form is just the usual curvature scalar density expressed in terms of hk μ and Ai jμ . This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini's Lagrangian. In the absence of matter, it yields the familiar equationsRμν =0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl) which arises from the fact that Ai jμ appears in the matter field Lagrangian, so that the equation of motion relating Ai jμ to hk μ is changed. In particular, this means that, although the covariant derivative of the metric vanishes, the affine connection Γλ μν is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the ``spin density'' Sk ij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual metric theory.
1,874 citations
TL;DR: In this paper, the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations, and it is shown that dilatation preserves the partial ordering on Minkowski space.
Abstract: Causality is represented by a partial ordering on Minkowski space, and the group of all automorphisms that preserve this partial ordering is shown to be generated by the inhomogeneous Lorentz group and dilatations.
414 citations
TL;DR: In this paper, a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorenttz Algebra, was proposed.
Abstract: We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra.
We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028.
As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value.
412 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 4 |
| 2024 | 1 |
| 2023 | 3 |
| 2022 | 24 |
| 2021 | 1 |
| 2020 | 4 |