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Between classical and quantum
, 2008
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Mean field dynamics of fermions and the timedependent HartreeFock equation
, 2002
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The classical limit of quantum theory
"... Abstract. For a quantum observable A¯h depending on a parameter ¯h we define the notion “A¯h converges in the classical limit”. The limit is a function on phase space. Convergence is in norm in the sense that A¯h → 0 is equivalent with ‖A¯h ‖ → 0. The ¯hwise product of convergent observables conve ..."
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Cited by 7 (1 self)
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Abstract. For a quantum observable A¯h depending on a parameter ¯h we define the notion “A¯h converges in the classical limit”. The limit is a function on phase space. Convergence is in norm in the sense that A¯h → 0 is equivalent with ‖A¯h ‖ → 0. The ¯hwise product of convergent observables converges to the product of the limiting phase space functions. ¯h −1 times the commutator of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the ¯hwise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed.
Quantum spin chains with quantum group symmetry, preprint  KULTF94/8
"... We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasilocal) tails of the action, we find that there are no actions of a pro ..."
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We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasilocal) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The nonlocality arises from the ordering of factors in the quantum group C*algebra, and can be made onesided, thus allowing semilocal actions on a half chain. Under such actions, localized quantum group invariant elements remain localized. Hence the notion of interactions invariant under the quantum group and also under translations, recently studied by many authors, makes sense even though there is no global action of the quantum group. We consider a class of such quantum group invariant interactions with the property that there is a unique translation invariant ground state. Under weak locality assumptions, its GNS representation carries no unitary representation of the quantum group.
Simulating Arbitrary PairInteractions by a Given Hamiltonian: GraphTheoretical Bounds on the Time Complexity
, 2001
"... We use an nspin system with permutation symmetric zzinteraction for simulating arbitrary pairinteraction Hamiltonians. The calculation of the required time overhead is mathematically equivalent to a separability problem of nqubit density matrices. We derive lower and upper bounds in terms of chr ..."
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We use an nspin system with permutation symmetric zzinteraction for simulating arbitrary pairinteraction Hamiltonians. The calculation of the required time overhead is mathematically equivalent to a separability problem of nqubit density matrices. We derive lower and upper bounds in terms of chromatic index and the spectrum of the interaction graph. The complexity measure defined by such a computational model is related to gate complexity and a continuous complexity measure introduced in a former paper. We use majorization of graph spectra for classifying Hamiltonians with respect to their computational power. 1
Spontaneous Symmetry Breaking in Quantum Systems: Emergence or Reduction?∗
, 2013
"... Dedicated to the memory of Gérard Emch (1936–2013) Beginning with Anderson (1972), spontaneous symmetry breaking (ssb) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics, since in theory no finite system should display it. Even the correspondence bet ..."
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Cited by 2 (0 self)
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Dedicated to the memory of Gérard Emch (1936–2013) Beginning with Anderson (1972), spontaneous symmetry breaking (ssb) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics, since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called ‘Earman’s Principle’, a genuine physical effect (viz. ssb) seems theoretically recovered only in some idealization (namely the thermodynamic limit), disappearing as soon as the idealization is removed. We review the wellknown arguments that (at first sight) no finite system can exhibit ssb, using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite and infinitevolume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which
Macroscopic observables and the Born rule I. Long
, 2008
"... Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutat ..."
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Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C∗algebra of observables is empirically accessible only through associated commutative C∗algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of singlecase probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) in replacing infinite tensor products of Hilbert spaces by continuous fields of C∗algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables. 1
Macroscopic observables and the Born rule
, 2008
"... Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutat ..."
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Dedicated to the memory of Bernd Kuckert (1968–2008) We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C ∗algebra of observables is empirically accessible only through associated commutative C ∗algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of singlecase probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) in replacing infinite tensor products of Hilbert spaces by continuous fields of C ∗algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.