Andreas Doering - Academia.edu (original) (raw)

Papers by Andreas Doering

We discuss some conceptual issues that any approach to quantum gravity has to confront. In partic... more We discuss some conceptual issues that any approach to quantum gravity has to confront. In particular, it is argued that one has to find a theory that can be interpreted in a realist manner, because theories with an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitably forces an instrumentalist interpretation on us, we suggest that a theory of quantum gravity should not be based on the Hilbert space formalism. We briefly sketch the topos approach, which makes use of the internal logic of a topos associated with a quantum system and comes with a natural (neo-)realist interpretation. Finally, we make some remarks on the relation between system logic and metalogic.

We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the pos... more We show that an atomic orthomodular lattice L can be reconstructed up to isomorphism from the poset B(L) of Boolean subalgebras of L. A motivation comes from quantum theory and the so-called topos approach, where one considers the poset of Boolean sublattices of L=P(H), the projection lattice of the algebra B(H) of bounded operators on Hilbert space.

In arXiv:1212.2613, we associated a presheaf \Sigma^A with each unital C*-algebra A. The spectral... more In arXiv:1212.2613, we associated a presheaf \Sigma^A with each unital C*-algebra A. The spectral presheaf \Sigma^A generalises the Gelfand spectrum of an abelian unital C*-algebra. In the present article, we consider one-parameter groups of automorphisms of the spectral presheaf, in particular those arising from one-parameter groups of inner automorphisms of the algebra. We interpret the spectral presheaf as a (generalised) state space for a quantum system and show how one can use flows on the spectral presheaf and on associated structures to describe the time evolution of non-relativistic quantum systems, both in the Schr\"odinger picture and the Heisenberg picture.

To each unital C*-algebra A we associate a presheaf \Sigma^A, called the spectral presheaf of A, ... more To each unital C*-algebra A we associate a presheaf \Sigma^A, called the spectral presheaf of A, which can be regarded as a generalised Gelfand spectrum. We develop a categorical notion of local duality and show that there is a contravariant functor from the category of unital C*-algebras to a suitable category of presheaves containing the spectral presheaves. We clarify how much algebraic information about a C*-algebra is contained in its spectral presheaf. A nonabelian unital C*-algebra A that is neither isomorphic to C^2 nor to B(C^2) is determined by its spectral presheaf up to quasi-Jordan isomorphisms. For a particular class of unital C*-algebras, including all von Neumann algebras with no type I_2 summand, the spectral presheaf determines the Jordan structure up to isomorphisms.

In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" ... more In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(\Omega), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space \Omega. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf \Sigma\ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of \Sigma. We show that using the spectral presheaf \Sigma\ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.

Accepted for publication in Communications in Mathematical Physics

Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or ... more Observables of a quantum system, described by self-adjoint operators in a von Neumann algebra or affiliated with it in the unbounded case, form a conditionally complete lattice when equipped with the spectral order. Using this order-theoretic structure, we develop a new perspective on quantum observables. In this first paper (of two), we show that self-adjoint operators affiliated with a von Neumann algebra can equivalently be described as certain real-valued functions on the projection lattice of the algebra, which we call q-observable functions. Bounded self-adjoint operators correspond to q-observable functions with compact image on non-zero projections. These functions, originally defined in a similar form by de Groote, are most naturally seen as adjoints (in the categorical sense) of spectral families. We show how they relate to the daseinisation mapping from the topos approach to quantum theory. Moreover, the q-observable functions form a conditionally complete lattice which is shown to be order-isomorphic to the lattice of self-adjoint operators with respect to the spectral order. In a subsequent paper, we will give an interpretation of q-observable functions in terms of quantum probability theory, and using results from the topos approach to quantum theory, we will provide a joint sample space for all quantum observables.

We introduce a new notion of entropy for quantum states, called contextual entropy, and show ... more We introduce a new notion of entropy for quantum states, called contextual entropy, and show how it unifies Shannon and von Neumann entropy. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension 3 or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.

Accepted for publication in "Logic & Algebra in Quantum Computing", Lecture Notes in Logic, published by the Association for Symbolic Logic in conjunction with Cambridge University Press

To each quantum system, described by a von Neumann algebra of physical quantities, we associate a... more To each quantum system, described by a von Neumann algebra of physical quantities, we associate a complete bi-Heyting algebra. The elements of this algebra represent contextualised propositions about the values of the physical quantities of the quantum system.

In "Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework", eds. F. Finster et al., Birkhäuser, Basel, 65--96 (2011)

The so-called topos approach provides a radical reformulation of quantum theory. Structurally, qu... more The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a classical system, and a quantity-value object, generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -- hence the `values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as `unsharp values'. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.

Journal of Mathematical Physics 53, 032101 (2012)

We show how probabilities can be treated as truth values in suitable sheaf topoi. The scheme deve... more We show how probabilities can be treated as truth values in suitable sheaf topoi. The scheme developed in this paper is very general and applies to both classical and quantum physics. On the quantum side, the results are a natural extension of our existing work on a topos approach to quantum theory. Earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective.

Accepted for publication in Houston Journal of Mathematics

For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that f... more For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that for an order-isomorphism f:AbSub(M)->AbSub(N) between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan *-isomorphism g:M->N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of a von Neumann algebra not isomorphic to C^2 and without type I_2 summands is determined by the poset of its abelian subalgebras, and has implications in recent approaches to foundational issues in quantum mechanics.

In "Deep Beauty", ed. Hans Halvorson, Cambridge University Press, Cambridge, 207--238 (2011)

We provide a conceptual discussion and physical interpretation of some of the quite abstract cons... more We provide a conceptual discussion and physical interpretation of some of the quite abstract constructions in the topos approach to physics. In particular, the daseinisation process for projection operators and for self-adjoint operators is motivated and explained from a physical point of view. Daseinisation provides the bridge between the standard Hilbert space formalism of quantum theory and the new topos-based approach to quantum theory. As an illustration, we will show all constructions explicitly for a three-dimensional Hilbert space and the spin-z operator of a spin-1 particle. This article is a companion to the article by Isham in the same volume.

In "Proceedings of Workshop on Quantum Physics and Logic (QPL'09)", eds. B. Coecke, P. Panangaden, P. Selinger, Electronic Notes in Theoretical Computer Science 270, Issue 2, 59--77 (2011)

The topos approach to the formulation of physical theories includes a new form of quantum logic. ... more The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.

Adv. Sci. Lett. 2, Number 2, Special Issue on Quantum Gravity, Cosmology and Black Holes, ed. Martin Bojowald, 291--301 (2009)

After a brief introduction to the spectral presheaf, which serves as an analogue of state space i... more After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure induces a unique state of the von Neumann algebra. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.

In "New Structures for Physics", ed. Bob Coecke, Springer Lecture Notes in Physics 813, 753--940, Springer, Heidelberg (2011)

The goal of this paper is to summarise the first steps in developing a fundamentally new way of c... more The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''.
Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity AAA with an arrow breveAphi:SiphimapRphi\breve{A}_\phi:\Si_\phi\map\R_\phibreveAphi:SiphimapRphi where Siphi\Si_\phiSiphi and Rphi\R_\phiRphi are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, tauphi\tau_\phitauphi. We discuss two different types of language that can be attached to a system,$S$. The first, PLS\PL{S}PLS, is a propositional language; the second, LS\L{S}LS, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of PLS\PL{S}PLS we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf Sig\SigSig--the topos quantum analogue of a classical state space. The topos concerned is SetH\SetH{}SetH: the category of contravariant set-valued functors on the category (partially ordered set) V\V{}V of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space Hi\HiHi.

In "Quantum Field Theory, Competitive Models", eds. B. Fauser, J. Tolksdorf, E. Zeidler, 25--47, Birkhäuser (2009)

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formul... more Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a `mathematical universe' with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.

Journal of Mathematical Physics 49, 053518 (2008)

This paper is the fourth in a series whose goal is to develop a fundamentally new way of building... more This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. The previous papers in this series are concerned with implementing this programme for a single system. In the present paper, we turn to considering a collection of systems: in particular, we are interested in the relation between the topos representation for a composite system, and the representations for its constituents. We also study this problem for the disjoint sum of two systems. Our approach to these matters is to construct a category of systems and to find a topos representation of the entire category.

Journal of Mathematical Physics 49, 053517 (2008)

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing t... more This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In paper II, we studied the topos representations of the propositional language PL(S) for the case of quantum theory, and in the present paper we do the same thing for the, more extensive, local language L(S). One of the main achievements is to find a topos representation for self-adjoint operators. This involves showing that, for any physical quantity A, there is an arrow brevedeltao(A):SigmapSR\breve{\delta}^o(A):\Sig\map\SRbrevedeltao(A):SigmapSR, where SR\SRSR is the quantity-value object for this theory. The construction of brevedeltao(A)\breve{\delta}^o(A)brevedeltao(A) is an extension of the daseinisation of projection operators that was discussed in paper II. The object SR\SRSR is a monoid-object only in the topos, tauphi\tau_\phitauphi, of the theory, and to enhance the applicability of the formalism, we apply to SR\SRSR a topos analogue of the Grothendieck extension of a monoid to a group. The resulting object, kSR\kSRkSR, is an abelian group-object in tauphi\tau_\phitauphi. We also discuss another candidate, PRmathR\PR{\mathR}PRmathR, for the quantity-value object. In this presheaf, both inner and outer daseinisation are used in a symmetric way. Finally, there is a brief discussion of the role of unitary operators in the quantum topos scheme.

Journal of Mathematical Physics 49, 053516 (2008)

This paper is the second in a series whose goal is to develop a fundamentally new way of construc... more This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper, we study in depth the topos representation of the propositional language, PL(S), for the case of quantum theory. In doing so, we make a direct link with, and clarify, the earlier work on applying topos theory to quantum physics. The key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf--the topos quantum analogue of a classical state space. In the second part of the paper we change gear with the introduction of the more sophisticated local language L(S). From this point forward, throughout the rest of the series of papers, our attention will be devoted almost entirely to this language. In the present paper, we use L(S) to study `truth objects' in the topos. These are objects in the topos that play the role of states: a necessary development as the spectral presheaf has no global elements, and hence there are no microstates in the sense of classical physics. Truth objects therefore play a crucial role in our formalism.

Journal of Mathematical Physics 49, 053515 (2008)

This paper is the first in a series whose goal is to develop a fundamentally new way of construct... more This paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics arises when the topos is the category of sets. Other types of theory employ a different topos. In this paper we discuss two different types of language that can be attached to a system, S. The first is a propositional language, PL(S); the second is a higher-order, typed language L(S). Both languages provide deductive systems with an intuitionistic logic. The reason for introducing PL(S) is that, as shown in paper II of the series, it is the easiest way of understanding, and expanding on, the earlier work on topos theory and quantum physics. However, the main thrust of our programme utilises the more powerful language L(S) and its representation in an appropriate topos.