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Multi-particle Spaces

Daniel Canarutto

Published Oct 5, 2020• 0 citations
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Abstract

Abstract The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.

DOI
10.1093/oso/9780198861492.003.0013

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Abstract The fundamental algebraic notions needed in many-particle physics are exposed. Spaces of free states containing an arbitrary number of particles of many types are introduced. The operator algebra generated by absorption and emission operators is studied as a natural generalisation of standard exterior algebra. The link between the discrete and the distributional formalisms is provided by the spaces of finite linear combinations of semi-densities of Dirac type.
We consider in detail an approach (proposed by the author earlier) where quantum states are described by elements of a linear space over a Galois field, and operators of physical quantities - by linear operators in this space. The notion of Galois fields (which is extremely simple and elegant) is discussed in detail and we also discuss the conditions when our description gives the same predictions as the conventional one. In quantum theory based on a Galois field, all operators are well defined and divergencies cannot not exist in principle. A particle and its antiparticle are described by the same modular irreducible representation of the symmetry algebra. This automatically explains the existence of antiparticles and shows that a particle and its antiparticle are the different states of the same object. As a consequence, a new symmetry arises, and the structure of the theory is considerably simplified. In particular, one can work with only creation operators or only annihilation ones since they are no more independent. In our approach the problem arises whether the existence of neutral elementary particles (e.g. the photon) is compatible with the usual relation between spin and statistics, or in other words whether neutral particles can be elementary or only composite.
Abstract The basic ideas of particle physics are reviewed, in the detector context, by exploiting several notions introduced in previous chapters. Quantum pictures are regarded as related to parallel transport along the detector with respect to a free-particle connection and a modification obtained by adding a quantum interaction morphism. The quantum interaction is seen to arise by merging the classical interaction with a quantum ‘ingredient’ constituted by a special generalised semi-density on particle momenta.
Abstract Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.
Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly [Formula: see text] generators for N identical particles, bosons or fermions, in d dimensions. The generators are called shapes. Each shape is a possible many-body vacuum. Shapes are natural generalizations of the ground-state Slater determinant to more than one dimension. Physical states, including the ground state, are superpositions of shapes with symmetric-function coefficients, for both bosons and fermions. These symmetric functions may be interpreted as bosonic excitations of the shapes. The algebraic structure of Hilbert space described here provides qualitative insights into long-standing issues of many-body physics, including the fermion sign problem and the microscopic origin of bands in the spectra of finite systems.
Abstract The closing chapter details the application of noncommutative Orlicz spaces to Quantum Physics. We specifically show that the (Köthe dual) pair of Orlicz spaces Llog⁡(L+1) and Lcosh−1 may be better suited as home for regular states and observables respectively than the more standard pair of spaces L∞ and L∞. Specifically the geometry of the space L1∩Llog⁡(L+1) is shown to be well suited to identifying those states which admit a well-defined notion of entropy. In fact the geometry of this space may be used to posit a clear definition of entropy valid for the most general von Neumann algebras. Parallel to this the space Lcosh−1 proves to be home for possibly unbounded observables which nevertheless are very regular in the sense of for example having all their moments finite. It is further pointed out that in the case of B(H) this approach yields exactly the Dirac style elementary Quantum Mechanical formalism. The technology thus developed is also used to gain a clearer understanding of relative entropy of states. The fact that the space Lcosh−1 admits unbounded observables allows for an application to the necessarily unbounded field operators of Quantum Field Theory. We specifically show that under very natural criteria, a large number of quantum fields which satisfy a so-called generalised H-bound condition automatically also satisfy an Lcosh−1 regularity condition namely that one may canonically embed the field operators into Lcosh−1 - the natural superspace for the regular observables.
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.