Quantum Probability Theory and the Foundations of Quantum Mechanics

Fröhlich, Jürg; Schubnel, Baptiste

doi:10.1007/978-3-662-46422-9_7

Cited by 27 publications

(43 citation statements)

References 65 publications

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“…After all, the measurement process is not part of the path integral description, and thus not described by it. It would be quite interesting to understand how these effects would play out in the context of quantum probability theory [47].…”

Section: Detecting Symmetry Breaking Without a Sourcementioning

confidence: 99%

Brout–Englert–Higgs physics: From foundations to phenomenology

Maas

2019

Progress in Particle and Nuclear Physics

107

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Brout-Englert Higgs physics is one of the most central and successful parts of the standard model. It is also part of a multitude of beyond-the-standard-model scenarios.The aim of this review is to describe the field-theoretical foundations of Brout-Englert-Higgs physics, and to show how the usual phenomenology arises from it. This requires to give a precise and gauge-invariant meaning to the underlying physics. This is complicated by the fact that concepts like the Higgs vacuum expectation value or the separation between confinement and the Brout-Englert-Higgs effect loose their meaning beyond perturbation theory. This is addressed by carefully constructing the corresponding theory space and the quantum phase diagram of theories with elementary Higgs fields and gauge interactions.The physical spectrum needs then to be also given in terms of gauge-invariant, i. e. composite, states. Using gauge-invariant perturbation theory, as developed by Fröhlich, Morchio, and Strocchi, it is possible to rederive conventional perturbation theory in this framework. This derivation explicitly shows why the description of the standard model in terms of the unphysical, gaugedependent, elementary states of the Higgs and W -bosons and Z-boson, but also of the elementary fermions, is adequate and successful.These are unavoidable consequences of the field theory underlying the standard model, from which the usual picture emerges. The validity of this emergence can only be tested non-perturbatively. Such tests, in particular using lattice gauge theory, will be reviewed as well. They fully confirm the underlying mechanisms.In this course it will be seen that the structure of the standard model is very special, and qualitative changes occur beyond it. The extension beyond the standard model will therefore also be reviewed. Particular attention will be given to structural differences arising for phenomenology. Again, non-perturbative tests of these results will be reviewed.Finally, to make this review self-contained a brief discussion of issues like the triviality and hierarchy problem, and how they fit into a fundamental field-theoretical formulation, is included. If there is no gauge-invariant way of defining the vacuum expectation value of the Higgs, the whole standard procedure [14] of doing BEH physics seems to collapse on itself at first. This seems to question the whole basis of the current theoretical description of experiments, despite its huge success [10]. But it does not. Rather, it can be shown that a treatment taking the gauge-dependence into account will lead to, essentially, the same results for the standard model.How this works out is exactly the core of this review: The construction of a fully gauge-invariant description of this physics, and how the standard phenomenology follows from it. This starts in section 3. There, a gauge-invariant description of the theory (3) will be set up. This is the first step of developing a comprehensive, field-theoretical consistent view on BEH physics. This view is continued by establis...

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Section: Detecting Symmetry Breaking Without a Sourcementioning

confidence: 99%

Brout–Englert–Higgs physics: From foundations to phenomenology

Maas

2019

Progress in Particle and Nuclear Physics

107

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“…We then sketch a technique based on the use of time-dependent Hamiltonians controlled from the outside; see, e.g., [17]. Finally, we describe the frequently used method to produce many essentially identical copies of the system S and performing state selection with the help of projective measurements of some physical quantity, A = A * , of S and subsequently keeping only those copies of S that correspond to a specific eigenstate of A we want S to prepare in; see [15] and references given there.…”

Section: Aim Of the Paper Models And Summary Of Resultsmentioning

confidence: 99%

“…For the theorist, this method obviously poses the problem of first understanding what "projective measurements" are; (see, e.g, [15] and refs. given there).…”

Section: Preparation Of States Via Adiabatic Evolutionmentioning

confidence: 99%

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The preparation of states in quantum mechanics

Fröhlich

Schubnel

2016

Journal of Mathematical Physics

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Abstract. The important problem of how to prepare a quantum mechanical system, S, in a specific initial state of interest -e.g., for the purposes of some experiment -is addressed. Three distinct methods of state preparation are described. One of these methods has the attractive feature that it enables one to prepare S in a preassigned initial state with certainty; i.e., the probability of success in preparing S in a given state is unity. This method relies on coupling S to an open quantum-mechanical environment, E, in such a way that the dynamics of S ∨ E pulls the state of S towards an "attractor", which is the desired initial state of S. This method is analyzed in detail. Aim of the Paper, Models and Summary of ResultsThe main problem addressed in this paper is how one may go about preparing a given, spatially localized quantum-mechanical system, S, in a specific initial state of interest in performing some observation or experiment on S. It is our impression that it is difficult to find serious discussion and analysis of this important foundational problem in the literature. In particular, it appears to be ignored in most text books on introductory quantum mechanics. The purpose of our paper is to make a modest contribution towards elucidating some solutions of this problem.After sketching several alternative techniques that can be used to prepare S in a desired initial state, we will turn our attention to the method studied primarily in this paper: By turning on suitable external fields, etc., we attempt to tune the dynamics of S so as to have the property that the state we want S to prepare in, denoted Ω S , is the ground state of the given dynamics; we then weakly couple S to a dispersive environment, E, (e.g., the quantized lattice vibrations of a crystal, or the electromagnetic field) chosen in such a way that, in the vicinity of S, the composed system, S ∨ E, relaxes to the ground state of S ∨ E. By letting the strength of the interaction between S and E tend to zero sufficiently slowly in time, we can manage to asymptotically decouple S from E and have S approach its own ground state, which is the desired state Ω S , as time t tends to ∞. The method for preparing a quantummechanical system in a specific state sketched here has the advantage that it is very robust: It has the attractive property that S approaches the desired state Ω S with probability 1, as time tends to ∞. Moreover, the speed of approach of the state of S to Ω S can be estimated quite explicitly. (In the following, ω denotes the expectation with respect to a state Ω, i.e., ω(·) = Ω, (·)Ω , and we will also use the expression "state" for ω.)Instead of engaging in a general abstract discussion of the problem of preparation of states in quantum mechanics, we explain our ideas and insights on the rather concrete example of a system S with a finite-dimensional Hilbert space of pure state vectors, a simple caricature of a very heavy "atom", coupled to a free massless scalar quantum field (e.g., a quantized field of phonons or "photons"). Mat...

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“…Quantum-mechanical models of physical systems interacting with long sequences of probes that are subsequently subject to direct (i.e., projective) measurements are of fundamental interest in studies of quantum filtering and control (see, e.g., [1,2]) and of the foundations of quantum mechanics (see [3,4,5,6,7,8,9,10,11,12], among others). There is extensive literature on such models, and one may wonder whether something new about these matters can still be added.…”

Section: Introductionmentioning

confidence: 99%

Perturbation Theory for Weak Measurements in Quantum Mechanics, Systems with Finite-Dimensional State Space

Ballesteros

Crawford

Fraas

et al. 2018

Ann. Henri Poincaré

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The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator N with finite spectrum is analysed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with N and a small perturbation not commuting with N . The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with N . If the Hamiltonian commutes with the observable N (i.e., if the perturbation vanishes) the state of the system approaches an eigenstate of N , as the number of direct measurements tends to ∞. If the perturbation term in the Hamiltonian does not commute with N the system exhibits "jumps" between different eigenstates of N . We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is re-scaled appropriately a maximum likelihood estimate of N approaches a Markovian jump process on the spectrum of N , as the strength of the perturbation tends to 0.

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Quantum Probability Theory and the Foundations of Quantum Mechanics

Cited by 27 publications

References 65 publications

Brout–Englert–Higgs physics: From foundations to phenomenology

Brout–Englert–Higgs physics: From foundations to phenomenology

The preparation of states in quantum mechanics

Perturbation Theory for Weak Measurements in Quantum Mechanics, Systems with Finite-Dimensional State Space

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