We show that the key problems of quantum measurement theory, namely the reduction of the wave packet of a microsystem and the specification of its quantum state by a macroscopic measuring instrument, may be rigorously resolved within the traditional framework of the quantum mechanics of finite conservative systems. The argument is centred on the generic model of a microsystem, S, coupled to a finite macroscopic measuring instrument, I, which itself is an N-particle quantum system. The pointer positions of I correspond to the macrostates of this instrument, as represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces, or 'phase cells', are the simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables, M, and, crucially, are of astronomically large dimensionalities, which incease exponentially with N. We formulate conditions on the conservative dynamics of the composite (S+I) under which it yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant state of S; and we show that these conditions are fulfilled by the finite version of the Coleman-Hepp model.
Let $V=\mathbb{C}^N$, and $H$ (an observable) a Hermitian linear operator on $V$. Let $v_1,..., v_n$ be an orthonormal basis for $V$. Let $\mathcal{M}$ be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the $v_j$'s. Here we propose a simple model which explains the Born rule and is compatible with entanglement.
We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition (joint work with Domingos Marchetti), the (quantum and classical) Edwards-Anderson (EA) spin glass model and return to equilibrium for a class of spin glass models, which includes the EA model initially in a very large transverse magnetic field.
We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Domingos Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse field.
If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected "quantum Zeno subspace" is called "quantum Zeno dynamics:" for instance, if the meas…
We introduce Quantum Register Algebra (QRA) as an efficient tool for quantum computing. We show the direct link between QRA and Dirac formalism. We present GAALOP (Geometric Algebra Algorithms Optimizer) implementation of our approach. Using the QRA basis vectors definitions given in Section 4 and the framework based on the de Witt basis presented in Section 5, we are able to fully describe and compute with QRA in GAALOP using the geometric product. We illustrate the intuitiveness of this computation by presenting the QRA form for the well known SWAP operation on a two qubit register.
We report on recent developments towards a relativistic quantum mechanical theory of motion for a fixed, finite number of electrons, photons, and their anti-particles, as well as its possible generalizations to other particles and interactions.
We consider the generalized quantum Rabi model with the so-called $A^{2}$-term in the light of the Hepp-Lieb-Preparata quantum phase transition. We investigate the dressed photon in its ground state when the atom-light coupling strength is in the deep-strong coupling regime. We show how the dressed photon appears in the ground state. We dedicate this paper to Pavel Exner and Herbert Spohn on the occasion of their 70th birthdays, and Klaus Hepp on the occasion of his 80th birthday.
This monograph aims to build some new mathematical structures originated from Dyson--Schwinger equations for the description of non-perturbative aspects of gauge field theories whenever bare or running coupling constants are strong enough.
According to mathematical constructivism, a mathematical object can exist only if there is a way to compute (or "construct") it; so, what is non-computable is non-constructive. In the example of the quantum model, whose Fock states are associated with Fibonacci numbers, this paper shows that the mathematical formalism of quantum mechanics is non-constructive since it permits an undecidable (or effectively impossible) subset of Hilbert space. On the other hand, as it is argued in the paper, if one believes that testability of predictions is the most fundamental property of any physical theory, …
From each of Feynman's and von Neumann's postulates to the restricted Feynman path integrals: a mathematical theory of temporally continuous quantum measurements
本文严格证明了在简单近似下，Mensky 的受限 Feynman 路径积分可由 Feynman 量子化公理和 von Neumann 瞬时测量公理导出，并应用于自旋系统及多种量子效应的表述。
This is a sequel to the papers (quant-ph/9910063) and (quant-ph/0004102). The aim of this paper is to give mathematical foundations to Holonomic Quantum Computation (Computer) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093). In 2-qubit case we give an explicit form to non-abelian Berry connection of quantum computational bundle which is associated with Holonomic Quantum Computation, on some parameter space. We also suggest a possibility that not only usual holonomy but also higher-dimensional holonomies must be used to prove a universality of our…
We provide the mathematical structure and a simple, transparent and rigorous derivation of the magnons as elementary quasi-particle excitations at low temperatures and in the infinite spin limit for a large class of Heisenberg ferromagnets. The magnon canonical variables are obtained as fluctuation operators in the infinite spin limit. Their quantum character is governed by the size of the magnetization.
An Euclidean topological space E is homeomorphic to the subset of delta-functions of the space D'(E) of Schwartz distributions on E. Herewith, any smooth function of compact support on E is extended onto D'(E). One can think of these extensions as sue generis quantum deformations. In quantum models, one therefore should replace integration of functions over E with that over D(E).
We give a mathematical definition of quantum field theory on a manifold, and definition of quantization of a classical field theory given by a variational principle.
In the search of a mathematical basis for quantum mechanics, in order to render it self-consistent and rationally understandable, we find that the best approach is to adopt E. Cartan's way for discovering spinors; that is to start from 3-dimensional null vectors and then show how they may be represented by two dimensional spinors. We have now only to go along this path, however in the opposite direction; with these spinors (which are pure) construct bilinearly null vectors: and we find that they naturally generate null vectors of Minkowski momentum space, where Cartan equations defining pure s…
We will examine a particular mathematical derivation in a paper by P. Falkensteiner and H. Grosse (F&G) [1]. In [1] a quantity "delta(A)" is defined. This quantity is generated when the normal ordered generalized charge operator undergoes a unitary transformation. Using standard mathematical techniques F&G convert "delta(A)" from its original form to another form which is suppose to be equivalent. It will be shown here that the two forms are not equivalent and that there is a mathematical inconsistency in their derivation. We will examine the source of this inconsistency and show that it can b…
This thesis describes the mathematical structures of the quantum BRST constraint method. Ultimately, the quantum BRST structures are formulated in a C*-algebraic context, leading to comparison of the quantum BRST and the Dirac constraint method in a mathematically consistent framework. Comparisons of rigorous models leads to the following three consequences: The quantum BRST method and quantum Dirac method of constraints are not equivalent in general; The BRST method does not remove the ghosts in the BRST physical algebra in general. Extra selection criteria are required to select the correct …
This paper collects and extends the lectures I gave at the "XXIV International Fall Workshop on Geometry and Physics" held in Zaragoza (Spain) August 31 - September 4, 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called (C*-) algebrai…
Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.
ETH Zürich（苏黎世联邦理工学院）；Joint Institute for Nuclear Research（联合核研究所）；Swiss Finance Institute（瑞士金融研究所）；University of Geneva（日内瓦大学）
One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory describes entangled decision making, non-commutativity of subse…
The quantum algebra suq(2) is introduced as a deformation of the ordinary Lie algebra su(2). This is achieved in a simple way by making use of $q$-bosons. In connection with the quantum algebra suq(2), we discuss the q-analogues of the harmonic oscillator and the angular momentum. We also introduce q-analogues of the hydrogen atom by means of a q-deformation of the Pauli equations and of the so-called Kustaanheimo-Stiefel transformation.
The mathematical structure of the sheaf of Dedekind real numbers $\RsubD(X)$ for a quantum system is discussed. The algebra of physical qualities is represented by an $O^{*}$ algebra $\mathcal M$ that acts on a Hilbert space that carries an irreducible representation of the symmetry group of the system. $X =\EsubS(\mathcal M)$, the state space for $\mathcal M$, has the weak topology generated by the functions $ a_{Q}(\cdot)$, defined for $\hat A \in \mathcal M_{sa} $ and $\forall \hat ρ\in \EsubS(\mathcal M) $, by $ a_{Q}(\hat ρ) = Tr \hat A \hat ρ$. For any open subset $W$ of $\EsubS(\mathcal…
We construct a mathematical model analogous to quantum field theory, but without the notion of vacuum and without measurable physical quantities. This model is a direct mathematical generalization of scattering theory in quantum mechanics to path integrals with multidimensional trajectories (whose mathematical interpretation has been given in a previous paper). In this model the normal ordering of operators in the Fock space is replaced by the Weyl-Moyal algebra. This model shows to be useful in proof of various results in quantum field theory: one first proves these results in the mathematica…
We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governe…
Determining properties of ground states of spin Hamiltonians remains a topic of central relevance connecting disciplines of mathematical, theoretical and applied physics. In the last few decades, ground state properties of physical systems have been increasingly considered as computational resources. This thesis develops parts of the mathematical apparatus to create (program) ground states relevant for quantum and classical computation. The core findings presented in this thesis (now over a decade old) including that (i) logic operations (gates) can be embedded into the low-energy sector of Is…
A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mil…
We show that the principles of a ''complete physical theory'' and the conclusions of the standard quantum mechanics do not irreconcilably contradict each other as is commonly believed. In the algebraic approach, we formulate axioms that allow constructing a renewed mathematical scheme of quantum mechanics. This scheme involves the standard mathematical formalism of quantum mechanics. Simultaneously, it contains a mathematical object that adequately describes a single experiment. We give an example of the application of the proposed scheme.
The paper contains a differential-geometric foundations for an attempt to formulate Lagrangian (canonical) quantum field theory on fibre bundles. In it the standard Hilbert space of quantum field theory is replace with a Hilbert bundle; the former playing a role of a (typical) fibre of the letter one. Suitable sections of that bundle replace the ordinary state vectors and the operators on the system's Hilbert space are transformed into morphisms of the same bundle. In particular, the field operators are mapped into corresponding field morphisms.
This paper addresses the scattering of a beam of charged particles by an infinitely long magnetic string in the context of the hydrodynamical approach to quantum mechanics. The scattering is qualitatively analyzed by two approaches. In the first approach, the quantum flow is studied via a one-parameter family of complex potentials. In the second approach, the qualitative theory of planar differential equations is used to obtain a one-parameter family of Hamiltonian functions which determine the phase portraits of the systems.
We describe a mathematical link between aspects of information theory, called pairwise comparisons, and discretized gauge theories. The link is made by the notion of holonomy along the edges of a simplex. This correspondance leads to open questions in both field.
The purpose of this paper is to present a mathematical theory of the half-twisted $(0,2)$ gauged linear sigma model and its correlation functions that agrees with and extends results from physics. The theory is associated to a smooth projective toric variety $X$ and a deformation $\sheaf E$ of its tangent bundle $T_X$. It gives a quantum deformation of the cohomology ring of the exterior algebra of $\sheaf E^*$. We prove that in the general case, the correlation functions are independent of `nonlinear' deformations. We derive quantum sheaf cohomology relations that correctly specialize to the …
By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Dirac's abstract Hilbert space formulation, we devote the second part of our paper to the latter approac…
We give an introduction into some aspects of the emerging mathematical theory of many-body localization (MBL) for disordered quantum spin chains. In particular, we discuss manifestations of MBL such as zero-velocity Lieb-Robinson bounds, quasi-locality of the time evolution of local observables, as well as exponential clustering and low entanglement of eigenstates. Explicit models where such properties have recently been verified are the XY and XXZ spin chain, in each case with disorder introduced in the form of a random exterior field. We introduce these models, state many of the available re…
The Higgs mechanism of mass generation is the main ingredient in the contemporary Standard Model and its various generalizations. However, there is no comprehensive theory of spontaneous symmetry breaking. We summarize the relevant mathematical results characterizing spontaneous symmetry breaking phenomena in algebraic quantum theory, axiomatic quantum field theory, group theory, and classical gauge theory.
These are lecture notes for a master-level course given at KTH, Stockholm, in the spring of 2017, with the primary aim of proving the stability of matter from first principles using modern mathematical methods in many-body quantum mechanics. General quantitative formulations of the uncertainty and the exclusion principles of quantum mechanics are introduced, such as the Hardy, Sobolev and Poincaré functional inequalities as well as the powerful Lieb-Thirring inequality that combines these two principles. The notes are aimed to be both fairly self-contained and at the same time complementary to…
In this paper we use the formalism of S.Weinberg in order to construct a mathematical model based on the weak decay of hadrons and nuclei. In particular we consider a model which generalizes the weak decay of the nucleus of the cobalt. We associate with this model a Hamiltonian with cutoffs in a Fock space. The Hamiltonian is self-adjoint and has an unique ground state. By using the commutator theory we get a limiting absorption principle from which we deduce that the spectrum of the Hamiltonian is absolutely continuous above the energy of the ground state and below the first threshold.
We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of symmetries of dynamical systems. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution - only information about invariant combinations of such objects is available. We demonstrate…
The description of bumps in scattering cross-sections by Breit-Wigner amplitudes led in the framework of the mathematical Physics to its formulation as the so-called Resonance-Decay Problem. It consists of a spectraltheoretical component and the connection of this component with the construction of decaying states. First the note quotes a solution for scattering systems, where the absolutely continuous parts of the Hamiltonians are semi-bounded and the scattering matrix is holomorphic in the upper half plane. This result uses the approach developed by Lax and Phillips, where the energy scale i…
One of the predominant challenges when engineering future quantum information processors is that large quantum systems are notoriously hard to maintain and control accurately. It is therefore of immediate practical relevance to investigate quantum information processing with limited physical resources, for example to ask: How well can we perform information processing tasks if we only have access to a small quantum device? Can we beat fundamental limits imposed on information processing with classical resources? This book will introduce the reader to the mathematical framework required to answ…
These are the second 50 issues of This Week's Finds of Mathematical Physics, from April 23, 1995 to March 23, 1997. These issues discuss quantum gravity, topological quantum field theory and other topics in mathematics and physics. They also include two expository series: one on ADE classifications and one on categories and higher categories. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).
The dissertation presents possibilities of applying noncommutative spacetimes description, particularly kappa-deformed Minkowski spacetime and Drinfeld's deformation theory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativistic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa-Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, wh…
These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of $n$-categories to these subjects. However, there are also digressions into Lie algebras, elliptic curves, linear logic and other subjects. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird).
This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE re…
Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms
The Wick rotation provides the standard technique of computing Feynman diagrams by means of Euclidean propagators. Let us suppose that quantum fields in an interaction zone are really Euclidean. In contrast with the well-known Euclidean field theory dealing with the Wightman and Schwinger functions of free quantum fields, we address complete Green's functions of interacting fields, i.e., causal forms on the Borchers algebra of quantum fields. They are the Laplace transform of the Euclidean states obeying a certain condition. If Euclidean states of a quantum field system, e.g., quarks do not sa…
Recent years have seen noteworthy progress in the mathematical formulation of quantum field theory and perturbative string theory. We give a brief survey of these developments. It serves as an introduction to the more detailed collection "Mathematical Foundations of Quantum Field Theory and Perturbative String Theory".
These are the third 50 issues of This Week's Finds of Mathematical Physics, from April 9, 1997 to June 18, 2000. These issues discuss quantum gravity, topological quantum field theory, n-categories and other topics in mathematics and physics. They also contain a series of expository mini-articles on homotopy theory. They were typeset in 2020 by Tim Hosgood. If you see typos or other problems please report them. (I already know the cover page looks weird; you can get a better-looking version at https://math.ucr.edu/home/baez/twf_latex/)
