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RSS FeedMon, 17 Feb 2020 09:33:19 GMT2020-02-17T09:33:19ZOperations on the Hochschild Bicomplex of a Diagram of Algebras
https://pure.york.ac.uk/portal/en/publications/operations-on-the-hochschild-bicomplex-of-a-diagram-of-algebras(f5f3ad81-50b4-4278-b717-e9830658c214).html
<div style='font-size: 9px;'><div class="rendering rendering_researchoutput rendering_researchoutput_short rendering_workingpaper rendering_short rendering_workingpaper_short"><h2 class="title"><a rel="WorkingPaper" href="https://pure.york.ac.uk/portal/en/publications/operations-on-the-hochschild-bicomplex-of-a-diagram-of-algebras(f5f3ad81-50b4-4278-b717-e9830658c214).html" class="link"><span>Operations on the Hochschild Bicomplex of a Diagram of Algebras</span></a></h2><a rel="Person" href="https://pure.york.ac.uk/portal/en/researchers/eli-hawkins(c329a10b-79e4-4719-973a-837cd5edfd40).html" class="link person"><span>Hawkins, E.</span></a>, <span class="date">3 Feb 2020</span>, (Unpublished).<p class="type"><span class="type_family">Research output<span class="type_family_sep">: </span></span><span class="type_classification_parent">Working paper</span></p></div><div class="rendering rendering_researchoutput rendering_researchoutput_detailsportal rendering_workingpaper rendering_detailsportal rendering_workingpaper_detailsportal"><div class="workingpaper"><h3 class="subheader">Publication details</h3><table class="properties"><tbody><tr class="status"><th>Date</th><td><span class="prefix">Unpublished - </span><span class="date">3 Feb 2020</span></td></tr><tr class="language"><th>Original language</th><td>English</td></tr></tbody></table><h3 class="subheader">Abstract</h3><div class="textblock"> A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an $L_\infty$-algebra. The same results are true for the asimplicial subcomplex and its cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation. </div></div><h3 class="subheader">Bibliographical note</h3><p>60 pages</p></div></div>Mon, 03 Feb 2020 00:00:00 GMThttps://pure.york.ac.uk/portal/en/publications/operations-on-the-hochschild-bicomplex-of-a-diagram-of-algebras(f5f3ad81-50b4-4278-b717-e9830658c214).html2020-02-03T00:00:00ZQuantum electrostatics and a product picture for quantum electrodynamics: or, the temporal gauge revised
https://pure.york.ac.uk/portal/en/publications/quantum-electrostatics-and-a-product-picture-for-quantum-electrodynamics(3b194fd4-6e8d-4b61-a305-ddbb433b05ca).html
<div style='font-size: 9px;'><div class="rendering rendering_researchoutput rendering_researchoutput_short rendering_workingpaper rendering_short rendering_workingpaper_short"><h2 class="title"><a rel="WorkingPaper" href="https://pure.york.ac.uk/portal/en/publications/quantum-electrostatics-and-a-product-picture-for-quantum-electrodynamics(3b194fd4-6e8d-4b61-a305-ddbb433b05ca).html" class="link"><span>Quantum electrostatics and a product picture for quantum electrodynamics: or, the temporal gauge revised</span></a></h2><a rel="Person" href="https://pure.york.ac.uk/portal/en/researchers/bernard-s-kay(c7548bf4-4f31-4a16-8118-7fe78f480861).html" class="link person"><span>Kay, B. S.</span></a>, <span class="date">16 Mar 2020</span>, <span>arXiv</span>, <span class="numberofpages">24 p.</span><p class="type"><span class="type_family">Research output<span class="type_family_sep">: </span></span><span class="type_classification_parent">Working paper</span></p></div><div class="rendering rendering_researchoutput rendering_researchoutput_detailsportal rendering_workingpaper rendering_detailsportal rendering_workingpaper_detailsportal"><div class="workingpaper"><h3 class="subheader">Publication details</h3><table class="properties"><tbody><tr class="status"><th>Date</th><td><span class="prefix">Published - </span><span class="date">16 Mar 2020</span></td></tr><tr><th>Publisher</th><td><span>arXiv</span></td></tr><tr class="numberofpages"><th>Number of pages</th><td>24</td></tr><tr class="language"><th>Original language</th><td>English</td></tr></tbody></table><h3 class="subheader">Abstract</h3><div class="textblock">We introduce a suitable notion of quantum coherent state to describe the electrostatic field of a static classical charge distribution, thereby underpinning the author's 1998 formulae for the inner product of a pair of such states. (We also correct an incorrect factor of 4π.) Contrary to what one might expect, this is non-zero whenever the two total charges are equal, even if the charge distributions themselves are different. We then address the problem of furnishing QED with a "product structure", i.e. a formulation in which there is a total Hamiltonian, arising as a sum of a free electromagnetic Hamiltonian, a free charged-matter Hamiltonian and an interaction term, acting on a total Hilbert space which is the tensor product of an electromagnetic Hilbert space and a charged-matter Hilbert space. (The traditional Coulomb-gauge formulation of QED doesn't have a product structure in this sense because, in it, the longitudinal part of the electric field is a function of the charged matter operators.) Motivated by all this, and both for a charged Dirac field and for a system of non-relativistic charged balls, we transform Coulomb-gauge QED into an equivalent formulation which we call the "product picture". This involves a full Hilbert space which is the tensor product of a Hilbert space of transverse and longitudinal photons with a Hilbert space for charged matter and in this sits a physical subspace (in all states of which the charged matter is entangled with longitudinal photons) on which Gauss's law holds strongly, together with a total Hamiltonian which has a product structure, albeit the electric field operator while self-adjoint on the physical subspace, isn't self-adjoint on the full Hilbert space. The product-picture Hamiltonian resembles the temporal gauge Hamiltonian, but the product picture is free from the difficulties in pre-existing temporal-gauge quantizations.</div></div></div></div>Mon, 16 Mar 2020 00:00:00 GMThttps://pure.york.ac.uk/portal/en/publications/quantum-electrostatics-and-a-product-picture-for-quantum-electrodynamics(3b194fd4-6e8d-4b61-a305-ddbb433b05ca).html2020-03-16T00:00:00Z