The free field

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Gauge Invariance ◽  
Maxwell Equations ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Initial Conditions ◽  
Free Field ◽  
Maxwell Theory ◽  
Gauge Invariant ◽  
Massive Vector ◽  
The One

This chapter studies the structure of Maxwell’s equations in a vacuum and the action from which they are derived, while emphasizing the consequences of their gauge invariance. Gauge invariance, on the one hand, allows one of the components of the magnetic potential to be chosen freely. Here, the chapter shows how the gauge-invariant version of the Maxwell equations in the vacuum can also be derived directly by extremizing. On the other hand, the chapter argues that gauge invariance imposes a constraint on the initial conditions such that in the end the general solution has only two ‘degrees of freedom’. Finally, the chapter develops the Hamiltonian formalisms in the Maxwell theory and compares them to the formalisms using non-gauge-invariant or massive vector fields.

scholarly journals Monodromy defects in free field theories

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Lorenzo Bianchi ◽  
Adam Chalabi ◽  
Vladimír Procházka ◽  
Brandon Robinson ◽  
Jacopo Sisti
Keyword(s):  
Entanglement Entropy ◽  
Free Field ◽  
Operator Product ◽  
Dirac Fermions ◽  
Maxwell Theory ◽  
Conformal Field ◽  
Defect Operator ◽  
Crucial Information ◽  
The One ◽  
Group Flow

Abstract We study co-dimension two monodromy defects in theories of conformally coupled scalars and free Dirac fermions in arbitrary d dimensions. We characterise this family of conformal defects by computing the one-point functions of the stress-tensor and conserved current for Abelian flavour symmetries as well as two-point functions of the displacement operator. In the case of d = 4, the normalisation of these correlation functions are related to defect Weyl anomaly coefficients, and thus provide crucial information about the defect conformal field theory. We provide explicit checks on the values of the defect central charges by calculating the universal part of the defect contribution to entanglement entropy, and further, we use our results to extract the universal part of the vacuum Rényi entropy. Moreover, we leverage the non-supersymmetric free field results to compute a novel defect Weyl anomaly coefficient in a d = 4 theory of free $$ \mathcal{N} $$ N = 2 hypermultiplets. Including singular modes in the defect operator product expansion of fundamental fields, we identify notable relevant deformations in the singular defect theories and show that they trigger a renormalisation group flow towards an IR fixed point with the most regular defect OPE. We also study Gukov-Witten defects in free d = 4 Maxwell theory and show that their central charges vanish.

A quantization of the electromagnetic field

1983 ◽  
Vol 61 (8) ◽  
pp. 1172-1183
Author(s):  
Anton Z. Capri ◽  
Gebhard Grübl ◽  
Randy Kobes
Keyword(s):  
Hilbert Space ◽  
Electromagnetic Field ◽  
Gauge Invariance ◽  
Free Field ◽  
External Source ◽  
Field Level ◽  
Manifest Covariance ◽  
The One ◽  
Physical Spaces

Quantization of the electromagnetic field in a class of covariant gauges is performed on a positive metric Hilbert space. Although losing manifest covariance, we find at the free field level the existence of two physical spaces where Poincaré transformations are implemented unitarily. This gives rise to two different physical interpretations of the theory. Unitarity of the S operator for an interaction with an external source then forces one to postulate that a restricted gauge invariance must hold. This singles out one interpretation, the one where two transverse photons are physical.

scholarly journals DUALITY THROUGH THE SYMPLECTIC EMBEDDING FORMALISM

2007 ◽  
Vol 22 (21) ◽  
pp. 3605-3620 ◽  
Author(s):  
E. M. C. ABREU ◽  
A. C. R. MENDES ◽  
C. NEVES ◽  
W. OLIVEIRA ◽  
F. I. TAKAKURA
Keyword(s):  
Phase Space ◽  
Gauge Invariance ◽  
Zero Mode ◽  
Mass Term ◽  
Gauge Invariant ◽  
Embedding Method ◽  
Symplectic Embedding ◽  
The One ◽  
Topological Term

In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.

scholarly journals C-projection and monopole condensation in QCD

2014 ◽  
Vol 29 ◽  
pp. 1460210
Author(s):  
Y. M. Cho
Keyword(s):  
Effective Action ◽  
Gauge Invariance ◽  
Integral Expression ◽  
Gauge Invariant ◽  
Invariant Integral ◽  
Infra Red ◽  
The Stability ◽  
The One

We show that the monopole condensation is responsible for the confinement. To demonstrate this we present a new gauge invariant integral expression of the one-loop QCD effective action which has no infra-red divergence, and show that the color reflection invariance ("the C-projection") assures the gauge invariance and the stability of the monopole condensation.

scholarly journals ONE-PARTICLE AND COLLECTIVE ELECTRON SPECTRA IN HOT AND DENSE QED AND THEIR GAUGE DEPENDENCE

1998 ◽  
Vol 13 (21) ◽  
pp. 1719-1728 ◽  
Author(s):  
O. K. KALASHNIKOV
Keyword(s):  
Gauge Invariance ◽  
Electron Spectrum ◽  
Particle Spectrum ◽  
Gauge Invariant ◽  
Long Wavelength ◽  
Gauge Dependence ◽  
Electron Spectra ◽  
Zero Momentum ◽  
Collective Spectrum ◽  
The One

The one-particle electron spectrum is found for hot and dense QED and its properties are investigated in comparison with the collective spectrum. It is shown that the one-particle spectrum (in any case its zero momentum limit) is gauge-invariant, but the collective spectrum, being qualitatively different, is always gauge-dependent. The exception is the case m,μ=0 for which the collective spectrum long wavelength limit demonstrates the gauge invariance as well.

scholarly journals The relation between Maxwell, Dirac, and the Seiberg-Witten equations

2003 ◽  
Vol 2003 (43) ◽  
pp. 2707-2734 ◽  
Author(s):  
Waldyr A. Rodrigues
Keyword(s):  
Spinor Field ◽  
Maxwell Equations ◽  
Degrees Of Freedom ◽  
Minkowski Spacetime ◽  
Magnetic Monopoles ◽  
The Other ◽  
Field Equations ◽  
Maxwell Theory ◽  
Hertz Potential ◽  
Form Field

We discuss unsuspected relations between Maxwell, Dirac, and the Seiberg-Witten equations. First, we present the Maxwell-Dirac equivalence (MDE) of the first kind. Crucial to that proposed equivalence is the possibility of solving for ψ (a representative on a given spinorial frame of a Dirac-Hestenes spinor field) the equation F=ψγ21ψ˜, where F is a given electromagnetic field. Such task is presented and it permits to clarify some objections to the MDE which claim that no MDE may exist because F has six (real) degrees of freedom and ψ has eight (real) degrees of freedom. Also, we review the generalized Maxwell equation describing charges and monopoles. The enterprise is worth, even if there is no evidence until now for magnetic monopoles, because there are at least two faithful field equations that have the form of the generalized Maxwell equations. One is the generalized Hertz potential field equation (which we discuss in detail) associated with Maxwell theory and the other is a (nonlinear) equation (of the generalized Maxwell type) satisfied by the 2-form field part of a Dirac-Hestenes spinor field that solves the Dirac-Hestenes equation for a free electron. This is a new result which can also be called MDE of the second kind. Finally, we use the MDE of the first kind together with a reasonable hypothesis to give a derivation of the famous Seiberg-Witten equations on Minkowski spacetime. A physical interpretation for those equations is proposed.

scholarly journals Gauge invariant composite fields out of connections, with examples

2014 ◽  
Vol 11 (03) ◽  
pp. 1450016 ◽  
Author(s):  
C. Fournel ◽  
J. François ◽  
S. Lazzarini ◽  
T. Masson
Keyword(s):  
Field Theory ◽  
Particle Physics ◽  
Vector Fields ◽  
Gauge Field Theory ◽  
Gauge Invariant ◽  
Massive Vector ◽  
Valued Field ◽  
The Standard Model ◽  
Breaking Mechanism ◽  
New Situation

In this paper, we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group-valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.

scholarly journals Killing symmetries as Hamiltonian constraints

2016 ◽  
Vol 13 (04) ◽  
pp. 1650044 ◽  
Author(s):  
Luca lusanna
Keyword(s):  
Vector Field ◽  
Degrees Of Freedom ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Inertial Effects ◽  
Maxwell Theory ◽  
Gauge Invariant ◽  
Minkowski Space Time ◽  
Space Formulation ◽  
Killing Symmetries

The existence of a Killing symmetry in a gauge theory is equivalent to the addition of extra Hamiltonian constraints in its phase space formulation, which imply restrictions both on the Dirac observables (the gauge invariant physical degrees of freedom) and on the gauge freedom. When there is a time-like Killing vector field only pure gauge electromagnetic fields survive in Maxwell theory in Minkowski space-time, while in ADM canonical gravity in asymptotically Minkowskian space-times only inertial effects without gravitational waves survive.

scholarly journals On the one-loop calculations with Reggeized quarks

2017 ◽  
Vol 32 (40) ◽  
pp. 1750207 ◽  
Author(s):  
Maxim Nefedov ◽  
Vladimir Saleev
Keyword(s):  
Field Theory ◽  
Renormalization Group ◽  
Effective Field Theory ◽  
Effective Action ◽  
Gauge Invariance ◽  
Effective Field ◽  
Loop Correction ◽  
Gauge Invariant ◽  
The One ◽  
Regge Limit

The technique of one-loop calculations for the processes involving Reggeized quarks is described in the framework of gauge invariant effective field theory for the Multi-Regge limit of QCD, which has been introduced by Lipatov and Vyazovsky. The rapidity divergences, associated with the terms enhanced by log(s), appear in the loop corrections in this formalism. The covariant procedure of regularization of rapidity divergences, preserving the gauge invariance of effective action is described. As an example application, the one-loop correction to the propagator of Reggeized quark and [Formula: see text]-scattering vertex are computed. Obtained results are used to construct the Regge limit of one-loop [Formula: see text] amplitude. The cancellation of rapidity divergences and consistency of the EFT prediction with the full QCD result is demonstrated. The rapidity renormalization group within the EFT is discussed.

A study of gauge-invariant non-local interactions

1954 ◽  
Vol 223 (1155) ◽  
pp. 468-481 ◽  
Keyword(s):  
Quantum Theory ◽  
Gauge Invariance ◽  
Local Interactions ◽  
Gauge Invariant ◽  
Invariance Properties ◽  
Finite Theory ◽  
Non Local ◽  
The One ◽  
Dimensional Integral

The paper investigates the possibility of introducing ‘non-local’ interactions, i. e. interactions represented by four-dimensional integral operations, in order to eliminate divergences in the quantum theory of interacting fields. In particular, a type of equation is discussed which preserves all the required invariance properties, including gauge invariance and macroscopic causality. It turns out that equations of this type still give divergent results. The origin of these divergences is discussed, and it is shown that if there is any way of formulating a finite theory it would have to be very different from the one investigated here.

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