More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space.
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The government is organized unitarily, thus the central government delegates powers to the departments without a constitutional need for consent. The current structure of Haiti's political system was set forth in the Constitution of March 29, 1987.
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In 1958 Kuroda showed that the Weyl–von Neumann theorem is also true if the Hilbert–Schmidt class is replaced by any Schatten class Sp with p ≠ 1. For S1, the trace-class operators, the situation is quite different. The Kato–Rosenblum theorem, proved in 1957 using scattering theory, states that if two bounded self-adjoint operators differ by a trace-class operator, then their absolutely continuous parts are unitarily equivalent. In particular if a self-adjoint operator has absolutely continuous spectrum, no perturbation of it by a trace-class operator can be unitarily equivalent to a diagonal operator.
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He developed the idea of a characteristic root of a quaternion matrix (an eigenvalue) and shows that they must exist. He also shows that a quaternion matrix is unitarily-equivalent to a triangular matrix. In 1956 he became a Senior Mathematician at Stanford Research Institute.
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Within the formalism of QFT such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the practitioner of QFT is confronted with the so-called choice problem, namely the problem of choosing the 'right' representation among a non-denumerable set of inequivalent representations.
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In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.
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Proof: is a semisimple Lie algebra and thus every element in it is a linear combination of commutators of some pairs of elements, otherwise the derived algebra would be a proper ideal. Moreover, any square matrix with zero trace is unitarily equivalent to a square matrix with diagonal consisting of all zeros.
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The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self- adjoint operators, (P,Q), satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on L^2(\R). The Hille–Yosida theorem generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.
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The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.
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In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.
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One of the interesting property of this notion lies in the fact that two isotypical representations are either quasi-equivalent or disjoint (in analogy with the fact that irreducible representations are either unitarily equivalent or disjoint). This can be understood through the correspondence between factor representations and minimal central projection (in a von Neumann algebra),.Dixmier Two minimal central projections are then either equal or orthogonal.
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Many important linear operators which occur in analysis, such as differential operators, are unbounded. There is also a spectral theorem for self-adjoint operators that applies in these cases. To give an example, every constant-coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed, the unitary operator that implements this equivalence is the Fourier transform; the multiplication operator is a type of Fourier multiplier.
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The government is organized unitarily, thus the central government delegates powers to the departments without a constitutional need for consent. The current structure of Haiti's political system was set forth in the Constitution of Haiti on 29 March 1987. The current president is Jovenel Moïse, elected in 2016 and sworn in on February 7, 2017. In 2010, there were 7,000 people in the Haitian National Police.
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Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA∗ = A∗A then there exists a unitary matrix U such that UAU∗ is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV∗ is diagonal with positive entries.
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Concerning the consequences of Haag's theorem, this observation implies the following: Since QFT does not attempt to predict fundamental parameters such as particle masses or coupling constants, potentially harmful effects arising from unitarily inequivalent representations remain absorbed inside the empirical values that stem from measurements of these parameters (at a given length scale) and that are readily imported into QFT. They thus remain invisible to the practitioner of QFT.
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The Millennium problem requires the proposed Yang-Mills theory to satisfy the Wightman axioms or similarly stringent axioms. There are four axioms: ;W0 (assumptions of relativistic quantum mechanics) Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space. The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space.
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We assume this normalization in what follows. The idea of the Stone—von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples. Example 14.5 To obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations.
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Sklar, Lawrence (2000), Theory and Truth: Philosophical Critique within Foundational Science. Oxford University Press. David Wallace has compared the merits of conventional QFT with those of algebraic QFT (AQFT) and observed that ... AQFT has unitarily inequivalent representations even on spatially finite regions, but this unitary inequivalence only manifests itself with respect to expectation values on arbitrary small spacetime regions, and these are exactly those expectation values which don't convey real information about the world. Wallace, David (2011).
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Hence, the observers will see different quantum states and thus different vacua. In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation relations. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices.
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The government is organized unitarily, thus the central government delegates powers to the departments without a constitutional need for consent. The current structure of Haiti's political system was set forth in the Constitution of Haiti on 29 March 1987. Haitian politics have been contentious: since independence, Haiti has suffered 32 coups. Haiti is the only country in the Western Hemisphere to undergo a successful slave revolution; however, a long history of oppression by dictators such as François Duvalier and his son Jean-Claude Duvalier has markedly affected the nation.
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In addition to the opinion of the prosecutor, which was based on a law providing for the definitive release of prisoners with good behavior and imprisonment for more than twenty years, documents with praise from officials and from P2's own management regarding his conduct in prison, unitarily strengthened the decision. He plasticized the original release license and always carried a copy with him, along with a list of ten dreams that he would like to accomplish before he died. "In my escape, they [the police] would always stop me," he complained, jokingly.
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The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerations of quantization, the deformation extension from classical to quantum mechanics.
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In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means that if G acts unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. The formal definition, introduced by David Kazhdan (1967), gives this a precise, quantitative meaning. Although originally defined in terms of irreducible representations, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual.
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If the protocol is unconditionally concealing, then Alice can unitarily transform these states into each other using the properties of the Schmidt decomposition, effectively defeating the binding property. One subtle assumption of the proof is that the commit phase must be finished at some point in time. This leaves room for protocols that require a continuing information flow until the bit is unveiled or the protocol is cancelled, in which case it is not binding anymore.A. Kent: Secure classical Bit Commitment using Fixed Capacity Communication Channels More generally, Mayers' proof applies only to protocols that exploit quantum physics but not special relativity.
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As was already noticed by Haag in his original work, it is the vacuum polarization that lies at the core of Haag's theorem. Any interacting quantum field (including non-interacting fields of different masses) is polarizing the vacuum, and as a consequence its vacuum state lies inside a renormalized Hilbert space H_R that differs from the Hilbert space H_F of the free field. Although an isomorphism could always be found that maps one Hilbert space into the other, Haag's theorem implies that no such mapping would deliver unitarily equivalent representations of the corresponding CCR, i.e. unambiguous physical results.
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Yes. Given a Borel function h, one can define an operator h(T) by specifying its behavior on the basis: : h(T) e_k = h(\lambda_k) e_k. In general, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator : [T \psi](x) = f(x) \psi(x) acting on L2 of some measure space. The domain of T consists of those functions for which the above expression is in L2. In this case, one can define analogously : [h(T) \psi](x) = [h \circ f](x) \psi(x).
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His argument assumes the unitarity of the AdS/CFT correspondence which implies that an AdS black hole that is dual to a thermal conformal field theory. When announcing his result, Hawking also conceded the 1997 bet, paying Preskill with a baseball encyclopedia "from which information can be retrieved at will." According to Roger Penrose, loss of unitarity in quantum systems is not a problem: quantum measurements are by themselves already non-unitary. Penrose claims that quantum systems will in fact no longer evolve unitarily as soon as gravitation comes into play, precisely as in black holes.
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South Florida Water Management District v. Miccosukee Tribe, 541 U.S. 95 (2004), was a U.S. Supreme Court case involving the application of the National Pollutant Discharge Elimination System (NPDES) of the Clean Water Act. The Supreme Court remanded the case for further determination to resolve the question over the validity of the distinction between the two bodies of water at issue (a canal and an undeveloped wetland) and the Government's broader "unitary waters" argument that all water bodies that are "navigable waters" under the Clean Water Act should be considered "unitarily" for purposes of NPDES permitting.South Florida Water Management Dist. v.
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For the purpose of the administrative and tax jurisdiction, the country is territorially divided in 16 circles. To each of the circles correspond a first instance administrative court of circle and a tributary court, being that both of which may be aggregated, in that case being unitarily designated "administrative and tax courts". At the present, only the Lisbon circle has a separate administrative and a tributary court, with the rest of the circles each having a single administrative and tax court. The judges of the Supreme Administrative Court have the title of conselheiros, those of the central administrative courts have the title of desembargadores and those of the first instance courts have the title of juízes de direito.
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It is expected that a necessary condition for f(T) to make sense is f be defined on the spectrum of T. For example, the spectral theorem for normal matrices states every normal matrix is unitarily diagonalizable. This leads to a definition of f(T) when T is normal. One encounters difficulties if f(λ) is not defined for some eigenvalue λ of T. Other indications also reinforce the idea that f(T) can be defined only if f is defined on the spectrum of T. If T is not invertible, then (recalling that T is an n x n matrix) 0 is an eigenvalue. Since the natural logarithm is undefined at 0, one would expect that ln(T) can not be defined naturally.
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In quantum field theory (QFT), traditional proposals for "the most basic known structures" divide into "particle interpretations" such as ascribing reality to the Fock space of particles, and "field interpretations" such as considering the quantum wavefunction to be identical to the underlying reality. Varying interpretations of quantum mechanics provide one complication; another, perhaps minor, complication is that neither fields nor particles are completely localized in standard QFT. A third, less obvious, complication is that "unitarily inequivalent representations" are endemic in QFT; for example, the same patch of spacetime can be represented by a vacuum by an inertial observer, but as a thermal heat bath by an accelerating observer that perceives Unruh radiation, raising the difficult question of whether the vacuum structure or heat bath structure is the real structure, or whether both of these inequivalent structures are separately real. Another example, which does not require the complications of curved spacetime, is that in ferromagnetism, symmetry-breaking analysis results in inequivalent Hilbert spaces.
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For the purposes of determining whether there has been "any addition of any pollutant to navigable waters from any point source," the Government argued that all water bodies that fall within the Act's definition of "navigable water" should be viewed unitarily for purposes of NPDES permitting requirements. Since the Act requires NPDES permits only when a pollutant is added to navigable waters, the Government contended that such permits are not required when water from one navigable body is discharged, unaltered, into another navigable water. Nonetheless, because neither the District nor the Government raised the "unitary waters" approach before the Eleventh Circuit or in their briefs respecting certiori, the U.S. Supreme Court declined to resolve the argument here and the unitary waters argument was remanded to the lower court. In addition, the U.S. Supreme Court concluded that summary judgment is appropriate only where there is no genuine issue of material fact but some factual issues remain unresolved in this case.
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