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Dynamical locality of the free Maxwell field
Mathematical Physics · Submitted 27 Mar 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The reduced theories are both locally covariant and dynamically local.
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The abstract, sentence by sentence
We consider the non-interacting source-free Maxwell field, described both in terms of the vector potential and the field strength. Starting from the classical field theory on contractible globally hyperbolic spacetimes, we extend the classical field theory to general globally hyperbolic spacetimes in two ways to obtain a "universal" theory and a "reduced" theory. The quantum field theory in terms of the unital $*$-algebra of the smeared quantum field is then obtained by an application of a suitable quantisation functor. We show that the universal theories fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical and non-trivial centres in the quantum case. The reduced theories are both locally covariant and dynamically local. These models provide new examples relevant to the discussion of how theories should be formulated so as to describe the same physics in all spacetimes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the non-interacting source-free Maxwell field, described both in terms of the vector potential and the field strength. | background 0.95 | 0.36 | 0.02 | asserted 0.96 |
| s1 | Starting from the classical field theory on contractible globally hyperbolic spacetimes, we extend the classical field theory to general globally hyperbolic spacetimes in two ways to obtain a "universal" theory and a "reduced" theory. | method 0.81 | 0.93 | 0.05 | asserted 0.99 |
| s2 | The quantum field theory in terms of the unital $*$-algebra of the smeared quantum field is then obtained by an application of a suitable quantisation functor. | method 0.98 | 0.79 | 0.03 | asserted 1.00 |
| s3 | We show that the universal theories fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical and non-trivial centres in the quantum case. | result 1.00 | 0.94 | 0.31 | asserted 0.99 |
| s4 | The reduced theories are both locally covariant and dynamically local. | result 1.00 | 0.91 | 0.53 | asserted 1.00 |
| s5 | These models provide new examples relevant to the discussion of how theories should be formulated so as to describe the same physics in all spacetimes. | interpretation 0.99 | 0.51 | 0.09 | asserted 0.94 |
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The paper
Original abstract
We consider the non-interacting source-free Maxwell field, described both in terms of the vector potential and the field strength. Starting from the classical field theory on contractible globally hyperbolic spacetimes, we extend the classical field theory to general globally hyperbolic spacetimes in two ways to obtain a "universal" theory and a "reduced" theory. The quantum field theory in terms of the unital $*$-algebra of the smeared quantum field is then obtained by an application of a suitable quantisation functor. We show that the universal theories fail local covariance and dynamical locality owing to the possibility of having non-trivial radicals in the classical and non-trivial centres in the quantum case. The reduced theories are both locally covariant and dynamically local. These models provide new examples relevant to the discussion of how theories should be formulated so as to describe the same physics in all spacetimes.
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How the literature uses it
10 arXiv papers cite this in 28 sentences
Papers citing it · 14 sentences without a confident label
Builds on it 1 paper
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We obtain explicit formulas involving Green operators, see in particular Proposition [ref] and Remark [ref], which generalize the earlier non-homotopical results in [ref] .
Relative Cauchy evolution for linear homotopy AQFTs · Introduction and summary · 0.91
Compares with it 2 papers
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Furthermore, when passing to homology and considering linear gauge field observables of the form $φ= δ_Mω∈(M)_0$ with $ω∈Ω^2_(M_+)$, our expression in [ref] agrees with the relative Cauchy evolution for the field strength tensor of Maxwell's theory computed in [ref] .
Relative Cauchy evolution for linear homotopy AQFTs · Example: Linear quantum Yang-Mills theory · 1.00
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In contrast to the situation for Klein-Gordon theory explained in Remark [ref], our model in Theorem [ref] b) for linear Yang-Mills theory as a homotopy AQFT on $$ is not naturally weakly equivalent to existing models in the literature that consider only gauge-invariant on-shell observables, see e.g. [ref] .
Linear Yang-Mills theory as a homotopy AQFT · Functoriality and homotopy AQFT axioms · 1.00
Cites it as support 2 papers
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In degree $n=0$, the linear map $H_0(f_∗): H_0(^(M)) →H_0(^(N))$ is the usual push-forward along $f$ of linear gauge-invariant on-shell observables for linear Yang-Mills theory, which is known to be an isomorphism, see e.g. [ref] .
Linear Yang-Mills theory as a homotopy AQFT · Functoriality and homotopy AQFT axioms · 0.99
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This explains why `topological observables' are absent from quantizations of Maxwell theory obeying these properties [ref] .
Algebraic quantum field theory in curved spacetimes · Conditions on states · 0.99
Cites it as background 5 papers
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Let us start by recalling a well-known result for the Cauchy problem of the Faraday tensor, see e.g. [ref] and also [ref] for details on how to treat initial data of not necessarily compact support.
Abelian duality on globally hyperbolic spacetimes · Cauchy problem · 0.94
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Recently, a number of works on the structure of classical and quantum field theory on curved spacetimes [ref] have made use of de Rham cohomology with spacelike compact supports.
Cohomology with causally restricted supports · Introduction · 0.92
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Most of these studies focused on the case of Yang-Mills theory with structure group $$ or $U(1)$, see e.g. [ref] , but there also exist similar developments for e.g. linearized gravity [ref] and linearized supergravity [ref] .
Linear Yang-Mills theory as a homotopy AQFT · Introduction and summary · 0.87
What it cites · 11 sentences without a confident label
Contests 2 papers
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The latter functor actually yields degenerate presymplectic spaces; however, as pointed out in [ref] , the the treatment of affine theories used in [ref] should be corrected; once this is done their approach would coincide with our reduced theory.
Quantized Abelian principal connections on Lorentzian manifolds · Extensions to non-contractible spacetimes · contest 0.84
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The latter functor actually yields degenerate presymplectic spaces; however, as pointed out in [ref] , the the treatment of affine theories used in [ref] should be corrected; once this is done their approach would coincide with our reduced theory.
Locally covariant quantum field theory with external sources · Extensions to non-contractible spacetimes · contest 0.84
Compares with 3 papers
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This differs from other, more global, approaches like [ref] insofar as we are led to our global theory (on non-contractible spacetimes) by local reasoning.
Quantum field theory on affine bundles · Introduction · 1.00
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This differs from other, more global, approaches like [ref] insofar as we are led to our global theory (on non-contractible spacetimes) by local reasoning.
Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law · Introduction · 1.00
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This differs from other, more global, approaches like [ref] insofar as we are led to our global theory (on non-contractible spacetimes) by local reasoning.
Quantized Abelian principal connections on Lorentzian manifolds · Introduction · 1.00
Cites as background 6 papers
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Dynamical locality has been established for a number of theories, including the massive free scalar field [ref] , the non-minimally coupled scalar field and the enlarged algebra of Wick polynomials [ref] , the Dirac field [ref] and also the inhomogeneous Klein–Gordon theory [ref] ; for a more detailed summary, see Section [ref].
Locally covariant quantum field theory with external sources · Introduction · 0.83
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Approaches which do not make such assumptions are [ref] , which treats field strengths, [ref] , which treats the vector potential, and [ref] , which discusses the Gupta-Bleuler formalism in curved spacetimes with the intention to couple the Dirac field with the electromagnetic field.
Gupta-Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times · Introduction · 0.92
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Approaches which do not make such assumptions are [ref] , which treats field strengths, [ref] , which treats the vector potential, and [ref] , which discusses the Gupta-Bleuler formalism in curved spacetimes with the intention to couple the Dirac field with the electromagnetic field.
Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law · Introduction · 0.92
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