A C∗-algebra formulation of gauge transformations of the second kind for the electromagnetic field

Carey, Alan L.; Gaffney, Janice; Hurst, C. A.

doi:10.1016/0034-4877(78)90067-8

Cited by 5 publications

(5 citation statements)

References 7 publications

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“…Finally, let us sketch the entirely analogous development for helicity ±2, that is, linearized Einstein gravity. Here the starting point is the covariant representation U 0,+, (1,1) of P , realized on the space of symmetric tensor fields h µν (x) on Minkowski space, which satisfy the wave equation h µν = 0, with Fourier coefficients in L 2 (R 3 , d ′ p)⊗C 10 . The gauge group G now consists of those non-constant weak solutions ξ µ (x) of the wave equation for which the quantity (δξ…”

Section: ) Andmentioning

confidence: 99%

“…S stands for our P T S). The phase space S is now given by A new feature, indicating that we are now in the setting of quantum field theory, is that, as explained in [10] (following the discussion in [24] for scalar fields), F has a subalgebra with the structure of a local net of C * -algebras in the sense of Haag-Kastler (cf. [22,23]).…”

Section: Rieffel Induction In Electromagnetism 41 Preamblementioning

confidence: 99%

“…Our main problem is finding a quantum analogue of the Marsden-Weinstein reduction procedure. The most convenient setting for doing this is provided by algebraic quantum field theory [22,23], which has already been extensively used in the study of quantum electromagnetism (QEM) [9,10,16,17,19,35]. As explained in the Introduction, this theory can only be used if the gauge is partially fixed.…”

Section: Introductionmentioning

confidence: 99%

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Massless Particles, Electromagnetism, and Rieffel Induction

Landsman

Wiedemann

1995

Rev. Math. Phys.

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The connection between space-time covariant representations (obtained by inducing from the Lorentz group) and irreducible unitary representations (induced from Wigner’s little group) of the Poincaré group is re-examined in the massless case. In the situation relevant to physics, it is found that these are related by Marsden-Weinstein reduction with respect to a gauge group. An analogous phenomenon is observed for classical massless relativistic particles. This symplectic reduction procedure can be (‘second’) quantized using a generalization of the Rieffel induction technique in operator algebra theory, which is carried through in detail for electromagnetism. Starting from the so-called Fermi representation of the field algebra generated by the free abelian gauge field, we construct a new (‘rigged’) sesquilinear form on the representation space, which is positive semi-definite, and given in terms of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a Hilbert Lie group). This eventually constructs the algebra of observables of quantum electromagnetism (directly in its vacuum representation) as a representation of the so-called algebra of weak observables induced by the trivial representation of the gauge group.

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Section: ) Andmentioning

confidence: 99%

Section: Rieffel Induction In Electromagnetism 41 Preamblementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Massless Particles, Electromagnetism, and Rieffel Induction

Landsman

Wiedemann

1995

Rev. Math. Phys.

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“…If for the moment we write (by an abuse of notation) 0« μ (/ μ )+^(0V μ ) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) interpreting oc μ and ξ as the quantised versions (in the Fock representation) of the corresponding classical fields, then it is a simple matter to show that there is an element Γ f eM a @M ξ such that (this calculation appears in [6]). Furthermore as/ranges over the test function space, so Γ f ranges over M a © M ξ .…”

Section: Field Algebrasmentioning

confidence: 99%

“…In a series of previous papers [1][2][3][4][5][6] we have developed a definite metric quantisation procedure for linear boson field theories which contain massless particles. In the case of the free electromagnetic field this rigorises Fermi's original quantisation procedure [7] and among other things provides a natural framework for a discussion of gauge transformations [3], [4]. It has an advantage over the indefinite metric approach in that standard C* algebra and Hubert space methods may be applied.…”

Section: Introductionmentioning

confidence: 99%

A C*-algebra approach to the Schwinger model

Carey

Hurst

1981

Commun.Math. Phys.

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If cutoffs are introduced then existing results in the literature show that the Schwinger model is dynamically equivalent to a boson model with quadratic Hamiltonian. However, the process of quantising the Schwinger model destroys local gauge invariance. Gauge invariance is restored by the addition of a counterterm, which may be seen as a finite renormalisation, whereupon the Schwinger model becomes dynamically equivalent to a linear boson gauge theory. This linear model is exactly soluble. We find that different treatments of the supplementary (i.e. Lorentz) condition lead to boson models with rather different properties. We choose one model and construct, from the gauge invariant subalgebra, a class of inequivalent charge sectors. We construct sectors which coincide with those found by Lowenstein and Swieca for the Schwinger model. A reconstruction of the Hubert space on which the Schwinger model exists is described and fermion operators on this space are defined.

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Quantum Theory of the Free Electromagnetic Field

Hurst

1986

Symmetries in Science II

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A C∗-algebra formulation of gauge transformations of the second kind for the electromagnetic field

Cited by 5 publications

References 7 publications

Massless Particles, Electromagnetism, and Rieffel Induction

Massless Particles, Electromagnetism, and Rieffel Induction

A C*-algebra approach to the Schwinger model

Quantum Theory of the Free Electromagnetic Field

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