Asymptotic locality and the structure of local internal symmetries

Noethers theorem shows that continuous global symmetries lead classically to conservation laws. Such symmetries can be divided into spacetime and internal symmetries. The invariance of Minkowski space-time under global Poincaré transformations leads to the conservation of the four-momentum and the total angular momentum. Examples for conserved charges due to internal symmetries are electric and colour charge. The vacuum expectation value of a Noether current is shown to beconserved in a quantum field theory if the symmetry transformation keeps the path-integral measure invariant.

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Canonical brackets of a toy model for the Hodge theory without its canonical conjugate momenta

International Journal of Modern Physics A ◽

10.1142/s0217751x15501158 ◽

2015 ◽

Vol 30 (20) ◽

pp. 1550115 ◽

Cited By ~ 4

Author(s):

D. Shukla ◽

T. Bhanja ◽

R. P. Malik

Keyword(s):

Present Method ◽

Hodge Theory ◽

Rigid Rotor ◽

Normal Ordering ◽

Creation And Annihilation Operators ◽

Basic Concepts ◽

Definition Of ◽

Continuous Symmetries ◽

Internal Symmetries ◽

Theory Lead

We consider the toy model of a rigid rotor as an example of the Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism and show that the internal symmetries of this theory lead to the derivation of canonical brackets amongst the creation and annihilation operators of the dynamical variables where the definition of the canonical conjugate momenta is not required. We invoke only the spin-statistics theorem, normal ordering and basic concepts of continuous symmetries (and their generators) to derive the canonical brackets for the model of a one [Formula: see text]-dimensional (1D) rigid rotor without using the definition of the canonical conjugate momenta anywhere. Our present method of derivation of the basic brackets is conjectured to be true for a class of theories that provide a set of tractable physical examples for the Hodge theory.

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Hermann Weyl's Analysis of the “Problem of Space” and the Origin of Gauge Structures

Science in Context ◽

10.1017/s0269889704000080 ◽

2004 ◽

Vol 17 (1-2) ◽

pp. 165-197 ◽

Cited By ~ 21

Author(s):

Erhard Scholz

Keyword(s):

General Relativity ◽

Dirac Equation ◽

A Priori ◽

Group Extension ◽

Central Idea ◽

German Word ◽

General Relativistic ◽

Empirical Turn ◽

Relativistic Framework ◽

Internal Symmetries

Hermann Weyl (1885–1955) was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben (in all senses of the German word) in the general relativistic framework for the Dirac equation. He preserved the central idea of gauge as a “purely infinitesimal” aspect of (internal) symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations.

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