Michael Mueger - Academia.edu (original) (raw)

Papers by Michael Mueger

Philosophy of Physics, 2007

Algebraic quantum field theory provides a general, mathematically precise description of the stru... more Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by S. Doplicher, R. Haag, and J. E. Roberts (DHR); and we give an alternative proof of Doplicher and Roberts' reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to Roberts and the abstract duality theorem for symmetric tensor * -categories, a self-contained proof of which is given in the appendix. * Forthcoming in Handbook of the Philosophy of Physics, edited by Jeremy Butterfield and John Earman. HH wishes to thank: Michael Müger for teaching him about the Doplicher-Roberts Theorem; the editors for their helpful feedback and patience; and David Baker, Tracy Lupher, and David Malament for corrections. MM wishes to thank Julien Bichon for a critical reading of the appendix and useful comments.

International Journal of Mathematics, 2008

We show that the left regular representation π l of a discrete quantum group (A, ∆) has the absor... more We show that the left regular representation π l of a discrete quantum group (A, ∆) has the absorbing property and forms a monoid (π l ,m,η) in the representation category Rep(A, ∆).

Algebras and Representation Theory, 2000

We give a pedagogical survey of those aspects of the abstract representation theory of quantum gr... more We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor * -category with conjugates is equivalent to the category of finite dimensional non-degenerate *representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka-Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf * -algebras.

Communications in Mathematical Physics, 2001

We describe the structure of the inclusions of factors A(E) ⊂ A(E ′ ) ′ associated with multi-int... more We describe the structure of the inclusions of factors A(E) ⊂ A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂ A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

Philosophy of Physics, 2007

Algebraic quantum field theory provides a general, mathematically precise description of the stru... more Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by S. Doplicher, R. Haag, and J. E. Roberts (DHR); and we give an alternative proof of Doplicher and Roberts' reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to Roberts and the abstract duality theorem for symmetric tensor * -categories, a self-contained proof of which is given in the appendix. * Forthcoming in Handbook of the Philosophy of Physics, edited by Jeremy Butterfield and John Earman. HH wishes to thank: Michael Müger for teaching him about the Doplicher-Roberts Theorem; the editors for their helpful feedback and patience; and David Baker, Tracy Lupher, and David Malament for corrections. MM wishes to thank Julien Bichon for a critical reading of the appendix and useful comments.

International Journal of Mathematics, 2008

We show that the left regular representation π l of a discrete quantum group (A, ∆) has the absor... more We show that the left regular representation π l of a discrete quantum group (A, ∆) has the absorbing property and forms a monoid (π l ,m,η) in the representation category Rep(A, ∆).

Algebras and Representation Theory, 2000

We give a pedagogical survey of those aspects of the abstract representation theory of quantum gr... more We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor * -category with conjugates is equivalent to the category of finite dimensional non-degenerate *representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka-Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well known general result concerning discrete multiplier Hopf * -algebras.

Communications in Mathematical Physics, 2001

We describe the structure of the inclusions of factors A(E) ⊂ A(E ′ ) ′ associated with multi-int... more We describe the structure of the inclusions of factors A(E) ⊂ A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂ A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.