Methods in Approximation N. Statistical Analysis of Random Fields A. Optimal Filtering V. Table of contents Foreword.
Basic Notation. Markov chains. Stochastic differential equations. Semi-groups and the Trotter product formula. The Feynman integral. Feynman integral and stochastic differential equations. Random perturbations of the classical mechanics. Complex dynamics and coherent states. Quantum non-linear oscillations. Feynman integral on analytic submanifolds. Interaction with the environment.
Lindblad equation and stochastic Schroedinger equation. Hamiltonian time evolution of the density matrix. Stochastic representation of the Lindblad time evolution. Decoherence and estimates on dissipative dynamics. Diffusive behaviour of the Wigner function and decoherence. Scattering and tunnelling in an environment. III, Nos 2—7. This massive work on rings of operators was very influential and continues to have an impact in pure mathematics, mathematical physics, and the foundations of physics.
It is usually assumed that the von Neumann algebra contains the identity operator. A von Neumann algebra is a factor, if its center which is the set of elements that commute with all elements of the algebra is trivial, meaning that it only contains scalar multiples of the identity element. Moreover, von Neumann showed in his reduction-theory paper that all von Neumann algebras that are not factors can be decomposed as a direct sum or integral of factors.
Subtypes were not distinguished for these factors until the s and s — see Chapter 3 of Sunder or Chapter 5 of Connes for details. A brief explanation for this shift is provided below. It is ironic that the predominant view now seems to be that the type-III factors are the most relevant class for physics particularly for QFT and quantum statistical mechanics.
This point is elaborated further in Section 4. One has to do with a property of the trace operation, which is the operation appearing in the definition of the probabilities for measurement results the Born rule , and the other with domain problems that arise for unbounded observable operators.
The trace of the identity is infinite when the separable Hilbert space is infinite-dimensional, which means that it is not possible to define a correctly normalized a priori probability for the outcome of an experiment i. By definition, the a priori probability for an experiment is that in which any two distinct outcomes are equally likely.
Thus, the probability must be zero for each distinct outcome when there is an infinite number of such outcomes, which can occur if and only if the space is infinite dimensional. Later, von Neumann changed the basis for his expressed reason for dissatisfaction with infinite dimensional Hilbert spaces from probabilistic to algebraic considerations Birkhoff and von Neumann , p. The problem with unbounded operators arises from their only being defined on a merely dense subset of the set elements of the space.
That is to say, these factor types have a finite trace operation and are not plagued with the domain problems of unbounded operators. But, as was already mentioned, it now seems that this was not the best move either. He wanted to use this structure to bring together the three key elements that were mentioned above: the algebraic approach to quantum mechanics, quantum logics, and rings of operators. He sought to forge a strong conceptual link between these elements and thereby provide a proper foundation for generalizing quantum mechanics that does not make essential use of Hilbert space unlike rings of operators.
Unfortunately, it turns out that the class of continuous geometries is too broad for the purposes of axiomatizing quantum mechanics. The class must be suitably restricted to those having a transition probability. It turns out that there is then no substantial generalization beyond the separable Hilbert space framework. An unpublished manuscript that was finished by von Neumann in was prepared and edited by Israel Halperin, and then published as von Neumann A review of the manuscript by Halperin was published in von Neumann —, Vol. In that review, Halperin notes the following:.
On the contrary, his work on rings of operators does provide significant light to the way forward.
See any care plans, options and policies that may be associated with this product. Ballentine, L. The projection-valued measure associated with A , E A , is then. Sign In or Create an Account. On the other hand, the variance of p i at any time can be evaluated as:. See "Does many-worlds violate conservation of energy?
The upshot of subsequent developments is that von Neumann settled on the wrong factor type for the foundations of physics. It was used extensively by physicists and it inspired some powerful mathematical developments in functional analysis. This came about as follows. Since then, it has been extended to a variety of different contexts in the quantum domain including decay phenomena and the arrow of time. The mathematical developments of Schwartz, Gelfand, and others had a substantial effect on QFT as well. Distribution theory was taken forward by Wightman in developing the axiomatic approach to QFT from the mids to the mids.
Although these developments were only indirectly influenced by Dirac, by way of the mathematical developments that are associated with his formal approach to quantum mechanics, there are other elements of his work that had a more direct and very substantial impact on the development of QFT. In the s, Dirac developed a Lagrangian formulation of quantum mechanics and applied it to quantum fields , and the latter inspired Feynman to develop the path-integral approach to QFT.
The mathematical foundation for path-integral functionals is still lacking Rivers , pp, — , though substantial progress has been made DeWitt-Morette et al. Despite such shortcomings, it remains the most useful and influential approach to QFT to date.
In the s, Dirac developed a form of quantum electrodynamics that involved an indefinite metric — see also Pauli in that connection. This had a substantial influence on later developments, first in quantum electrodynamics in the early s with the Gupta-Bluer formalism, and in a variety of QFT models such as vector meson fields and quantum gravity fields by the late s — see Chapter 2 of Nagy for examples and references.
It then became widely known by way of his textbook Dirac , which was based on a series of lectures on quantum mechanics given by Dirac at Cambridge University. This textbook saw three later editions: the second in , the third in , and the fourth in The fourth edition has been reprinted many times.
Its staying power is due, in part, to another innovation that was introduced by Dirac in the third edition, his bra-ket formalism. He first published this formalism in Dirac , but the formalism did not become widely used until after the publication of the third edition of his book. Another key element, the notion of a nuclear space, was developed by Grothendieck This notion made possible the generalized-eigenvector decomposition theorem for self-adjoint operators in rigged Hilbert space — for the theorem see Gelfand and Vilenken , pp.
It is unfortunate that their chosen name for this mathematical structure is doubly misleading. First, there is a natural inclination to regard it as denoting a type of Hilbert space, one that is rigged in some sense, but this inclination must be resisted. Second, the term rigged has an unfortunate connotation of illegitimacy, as in the terms rigged election or rigged roulette table , and this connotation must be dismissed as prejudicial.
There is nothing illegitimate about a rigged Hilbert space from the standpoint of mathematical rigor or any other relevant standpoint. A more appropriate analogy may be drawn using the notion of a rigged ship: the term rigged in this context means fully equipped. But this analogy has its limitations since a rigged ship is a fully equipped ship, but as the first point indicates a rigged Hilbert space is not a Hilbert space, though it is generated from a Hilbert space in the manner now to be described. It is also the inductive limit of a sequence of Hilbert spaces in which the topologies get rapidly coarser with increasing n.
Two other points are worth noting. First, dual pairs of this sort can also be generated from a pre-Hilbert space, which is a space that has all the features of a Hilbert space except that it is not complete, and doing so has the distinct advantage of avoiding the partitioning of functions into equivalence classes in the case of functions spaces.
The term rigged Hilbert space is typically used broadly to include dual pairs generated from either a Hilbert space or a pre-Hilbert space. As already noted, these operators have no eigenvalues or eigenvectors in a separable Hilbert space; moreover, they are only defined on a dense subset of the elements of the space and this leads to domain problems.
These undesirable features also motivated von Neumann to seek an alternative to the separable Hilbert space framework for quantum mechanics, as noted above. The key result is known as the nuclear spectral theorem and it is also known as the Gelfand-Maurin theorem. The rigging based on the choice of a nuclear operator that determines the test function space can result in different sets of generalized eigenvalues being associated with an operator. But there are situations in which it is desirable for an operator to have complex eigenvalues.
Of course, it is impossible for a self-adjoint operator to have complex eigenvalues in a Hilbert space. Soon after the development of the theory of rigged Hilbert spaces by Gelfand and his associates, the theory was used to develop a new formulation of quantum mechanics. It was later demonstrated that the rigged Hilbert space formulation of quantum mechanics can handle a broader range of phenomena than the separable Hilbert space formulation.
The Prigogine school developed an alternative characterization of the arrow of time using the rigged Hilbert space formulation of quantum mechanics Antoniou and Prigogine Kronz , used this formulation to characterize quantum chaos in open quantum systems. Castagnino and Gadella used it to characterize decoherence in closed quantum systems.
The algebraic formulations of quantum mechanics that were developed by von Neumann and Segal did not change the way that quantum mechanics was done. Nevertheless, they did have a substantial impact in two related contexts: QFT and quantum statistical mechanics. The key difference leading to the impact has to do with the domain of applicability. The domain of quantum mechanics consists of finite quantum systems, meaning quantum systems that have a finite number of degrees of freedom.
Whereas in QFT and quantum statistical mechanics, the systems of special interest — i. Dirac was the first to recognize the importance of infinite quantum systems for QFT, which is reprinted in Schwinger Segal , p. The key advantage of the algebraic approach, according to Segal , pp.
Segal notes , p. Segal notes this advantage in response to a result obtained by Haag , that field theory representations of free fields are unitarily inequivalent to representations of interacting fields. The key mathematical difference, according to Segal, is that von Neumann was working with a weakly closed ring of operators meaning that the ring of operators is closed with respect to the weak operator topology , whereas Segal is working with a uniformly closed ring of operators closed with respect to the uniform topology.
It is crucial because it has the following interpretive significance, which rests on operational considerations:. However, the use of physical equivalence to show that unitarily inequivalent representations are not physically significant has been challenged; see Kronz and Lupher , Lupher , and Ruetsche The algebraic approach has proven most effective in quantum statistical mechanics. It is extremely useful for characterizing many important macroscopic quantum effects including crystallization, ferromagnetism, superfluidity, structural phase transition, Bose-Einstein condensation, and superconductivity.
A good introductory presentation is Sewell , and for a more advanced discussion see Bratteli and Robinson , In algebraic quantum statistical mechanics, an infinite quantum system is defined by specifying an abstract algebra of observables. A particular state may then be used to specify a concrete representation of the algebra as a set of bounded operators in a Hilbert space.
Given an automorphism group, each KMS state corresponds to a representation of the algebra of observables that defines the system, and each of these representations is unitarily inequivalent to any other.
Thus, type-III factors play a predominant role in algebraic quantum statistical mechanics. In algebraic QFT, an algebra of observables is associated with bounded regions of Minkowski spacetime and unbounded regions including all of spacetime by way of certain limiting operations that are required to satisfy standard axioms of local structure: isotony, locality, covariance, additivity, positive spectrum, and a unique invariant vacuum state.
The resulting set of algebras on Minkowski spacetime that satisfy these axioms is referred to as the net of local algebras. It has been shown that special subsets of the net of local algebras — those corresponding to various types of unbounded spacetime regions such as tubes, monotones a tube that extends infinitely in one direction only , and wedges — are type-III factors. Of particular interest for the foundations of physics are the algebras that are associated with bounded spacetime regions, such as a double cone the finite region of intersection of a forward and a backward light cone.
As a result of work done over the last thirty years, local algebras of relativistic QFT appear to be type III von Neuman algebras see Halvorson , pp. One important area for interpretive investigation is the existence of a continuum of unitarily inequivalent representations of an algebra of observables. Attitudes towards unitarily inequivalent representations differ drastically in the philosophical literature. In Wallace unitarily inequivalent representations are not considered a foundational problem for QFT, while in Ruetsche , Lupher and Kronz and Lupher unitarily inequivalent representations are considered physically significant.
In the early s, theoretical physicists were inspired to axiomatize QFT. One motivation for axiomatizing a theory, not the one for the case now under discussion, is to express the theory in a completely rigorous form in order to standardize the expression of the theory as a mature conceptual edifice. Another motivation, more akin to the case in point, is to embrace a strategic withdrawal to the foundations to determine how renovation should proceed on a structure that is threatening to collapse due to internal inconsistencies. One then looks for existing piles fundamental postulates that penetrate through the quagmire to solid rock, and attempts to drive home others at advantageous locations.
Properly supported elements of the superstructure such as the characterization of free fields, dispersion relations, etc. The latter need not be razed immediately, and may ultimately glean supportive rigging from components not yet constructed. In short, the theoretician hopes that the axiomatization will effectively separate sense from nonsense, and that this will serve to make possible substantial progress towards the development of a mature theory. Grounding in a rigorous mathematical framework can be an important part of the exercise, and that was a key aspect of the axiomatization of QFT by Wightman.
It was further refined in the late s by Bogoliubov, who explicitly placed axiomatic QFT in the rigged Hilbert space framework Bogoliubov et al. Rigged Hilbert space entered the axiomatic framework by way of the domain axiom, so this axiom will be discussed in more detail below. In QFT, a field is characterized by means of an operator rather than a function. A field operator may be obtained from a classical field function by quantizing the function in the canonical manner — see Mandl , pp.
Field operators that are relevant for QFT are too singular to be regarded as realistic, so they are smoothed out over their respective domains using elements of a space of well-behaved functions known as test functions.
There are many different test-functions spaces Gelfand and Shilov , Chapter 4. It was later determined that some realistic models require the use of other test-function spaces. Streater and Wightman As noted earlier, the appropriateness of the rigged Hilbert space framework enters by way of the domain axiom. Concerning that axiom, Wightman says the following in the notation introduced above, which differs slightly from that used by Wightman.
In Bogoliubov et al. This serves to justify a claim they make earlier in their treatise:. In both approaches, a field is an abstract system having an infinite number of degrees of freedom.
Sub-atomic quantum particles are field effects that appear in special circumstances. In algebraic QFT, there is a further abstraction: the most fundamental entities are the elements of the algebra of local and quasi-local observables, and the field is a derived notion. The term local means bounded within a finite spacetime region, and an observable is not regarded as a property belonging to an entity other than the spacetime region itself. The term quasi-local is used to indicate that we take the union of all bounded spacetime regions.
In short, the algebraic approach focuses on local or quasi-local observables and treats the notion of a field as a derivative notion; whereas the axiomatic approach as characterized just above regards the field concept as the fundamental notion. The two approaches are mutually complementary — they have developed in parallel and have influenced each other by analogy Wightman For a discussion of the close connections between these two approaches, see Haag , p. Those criticisms motivated mathematically inclined physicists to search for a mathematically rigorous formulation of QFT.
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