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arXiv:1604.05629

Projective Limits of State Spaces: Quantum Field Theory without a Vacuum

Suzanne Lanéry

Published Apr 19, 2016• 5 citations
arXivHTMLPDF

Abstract

Instead of formulating the states of a Quantum Field Theory (QFT) as density matrices over a single large Hilbert space, it has been proposed by Kijowski [Kijowski, 1977] to construct them as consistent families of partial density matrices, the latter being defined over small 'building block' Hilbert spaces. In this picture, each small Hilbert space can be physically interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce the quantization of a classical field theory to the quantization of finite-dimensional sub-systems, thus sidestepping some of the common ambiguities (specifically, the issues revolving around the choice of a 'vacuum state'), while obtaining robust and well-controlled quantum states spaces. The present letter provides a self-contained introduction to this formalism, detailing its motivations as well as its relations to other approaches to QFT (such as conventional Fock-like Hilbert spaces, path-integral quantization, and the algebraic formulation). At the same time, it can serve as a reading guide to the series of more in-depth articles [arXiv:1411.3589, arXiv:1411.3590, arXiv:1411.3591, arXiv:1510.01926].

DOI
10.48550/arxiv.1604.05629

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Instead of formulating the states of a Quantum Field Theory (QFT) as density matrices over a single large Hilbert space, it has been proposed by Kijowski [Kijowski, 1977] to construct them as consistent families of partial density matrices, the latter being defined over small 'building block' Hilbert spaces. In this picture, each small Hilbert space can be physically interpreted as extracting from the full theory specific degrees of freedom. This allows to reduce the quantization of a classical field theory to the quantization of finite-dimensional sub-systems, thus sidestepping some of the common ambiguities (specifically, the issues revolving around the choice of a 'vacuum state'), while obtaining robust and well-controlled quantum states spaces. The present letter provides a self-contained introduction to this formalism, detailing its motivations as well as its relations to other approaches to QFT (such as conventional Fock-like Hilbert spaces, path-integral quantization, and the algebraic formulation). At the same time, it can serve as a reading guide to the series of more in-depth articles [arXiv:1411.3589, arXiv:1411.3590, arXiv:1411.3591, arXiv:1510.01926].
Abstract The modal interpretation is actually a family of interpretations sharing the feature that a system’s density operator constrains the possibilities for assigning definite values to its observables (Healey 1989, Dieks & Vermaas 1998). We start with a brief review of the basic idea behind these interpretations from an algebraic point of view (Zimba & Clifton 1998, Halvorson & Clifton 1999). Consider a ‘universe’ comprised of a quantum system U, with finitely many degrees of freedom, represented by a Hilbert space Hu = ®i Hi. At any given time, U will occupy a pure vector state x E Hu that determines a reduced density operator Ds on the Hilbert space Hi of any subsystem S U. Let B(Hs) de­
This document is meant as a pedagogical introduction to the modern language used to talk about quantum theory, especially in the field of quantum information. It assumes that the reader has taken a first traditional course on quantum mechanics, and is familiar with the concept of Hilbert space and elementary linear algebra. As in the popular textbook on quantum information by Nielsen and Chuang, we introduce the generalised concept of states (density matrices), observables (POVMs) and transformations (channels), but we also characterise these structures from an algebraic standpoint, which provides many useful technical tools, and clarity as to their generality. This approach also makes it manifest that quantum theory is a direct generalisation of probability theory, and provides a unifying formalism for both fields. The focus on finite-dimensional systems allows for a self-contained presentation which avoids many of the technicalities inherent to the more general C∗C^*C∗-algebraic approach, while being appropriate for the quantum information literature.
A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schrödinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15—to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS , might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms—the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and applications of the geometric approach. The first four chapters contain the standard mathematics required to understand the rest of the material presented: specific areas in colour theory, set theory, probability theory, differential geometry and projective geometry are all covered with an eye to the material that follows. Chapter 5 starts the first real discussion of quantum theory in GQS and serves as an elegant, succinct introduction to the geometry which underlies quantum theory. This may be the most worthwhile chapter for the casual reader who wants to understand the key ideas in this field. Chapter 6 builds on the discussion in Chapter 5, introducing a group theoretic approach to understand coherent states and Chapter 7 describes a geometric tool in the form of an approach to complex projective geometry called 'the stellar representation'. Chapter 8 returns to a more purely quantum mechanical discussion as the authors turn to study the space of density matrices. This chapter completes the discussion which started in Chapter 5. Chapter 9 begins the part of the book concerned with applications of the geometric approach. From this point on the book aims, specifically, to prepare the reader for the material in Chapter 15 beginning with a discussion on the purification of mixed quantum states. In the succeeding chapters a definite choice has been made to present a geometric approach to certain quantum information problems. For example, Chapter 10 contains an extremely well formulated discussion of measurement and positive operator-valued measures with several well illustrated examples and Chapter 11 reopens the discussion of density matrices. Entropy and majorization are again revisited in Chapter 12 in much greater detail than in previous chapters. Chapters 13 and 14 concern themselves with a discussion of various metrics and their relation to the problem of distinguishing between probability distributions and their suitability as probability measures.
This paper uses Schwinger's representation of quantum systems of a finite number of degrees of freedom to formulate discrete path integral representations of the dynamics. The representation starts with an observable with a finite number of distinct eigenvalues. Using that observable Schwinger constructs complementary pairs of finite dimensional unitary operators that are finite dimensional analogs of the irreducible Weyl algebra. These can be decomposed into products of irreducible sub-algebras that reduce to q-bit gates for the case of 2N2^N2N degrees of freedom. In the limit of a large number of discrete degrees of freedom these representations can be used to model quantum systems with continuous degrees of freedom. This limit recovers the continuum Weyl algebra. Path integrals can be formulated in the discrete case. They have the advantage that the number of paths for a finite number of time slices is finite. In this work the path integral is interpreted as the expectation value of a potential functional with respect to a complex probability distribution on the space of paths. An application to scattering from a short-range potential is given. Multi-resolution wavelet bases are used exactly represent a local ϕ4(x)\phi^4(x)ϕ4(x) quantum field theory as a theory with an infinite number of discrete local modes. The discrete path integral is illustrated by computing time evolution when this theory is truncated to two coupled modes.
Abstract The fundamental entity in the algebraic approach to quantum theory is an abstract C*-algebra 𝒜 together with its set of states ℱ, that is, the set of all positive linear functionals on 𝒜. I shall call such a pair an (abstract) quantum system. The theory of C*-algebras and their representations provides the most powerful abstract approach to quantum theory; unlike the propositional lattice approach it also has direct applications in mainstream physics, and in a certain sense is even forced upon us in quantum field theory (OFT) (sec sections (2.11), (2.16)-(2.18)). Mathematically, the subject is a complex of analysis and algebra; it is a rather wild area involving high degrees of infinity. Philosophically it is most strongly associated with opera tionalism; in part this rellects the predilections of its originators, and in part the continuing philosophical appeal of positivism within the mathematics community. (I do not know why this is so.)
This work is devoted to incorporating into QFT the notion that particles and hence the particle states should be localizable in space. It focuses on the case of the Dirac field in 1+1 dimensional flat spacetime, generalizing a recently developed formalism for scalar fields. This is achieved exploiting again the non-uniqueness of quantization process. Instead of elementary excitations carrying definite amounts of energy and momentum, we construct the elementary excitations of the field localized (at some instant) in a definite region of space. This construction not only leads to a natural notion of localized quanta, but also provides a local algebra of operators. Once constructed, the new representation is confronted to the standard global (Fock) construction. In spite of being unitarily inequivalent representations, the localized operators are well defined in the conventional Fock space. By using them we dig up the issues of localization in QFT showing how the globality of the vacuum state is responsible for the lack of a "common sense localization" notion for particles in the standard Fock representation of QFT and propose a method to meet its requirements.