arXiv subjects

N. P. Landsman

Publications and source records attributed to N. P. Landsman.

36 records · Page 2Linked to original sources

arXiv1998

Hall's coherent states, the Cameron-Martin theorem, and the quantization of Yang-Mills theory on a circle

We discuss the classical and quantum reduction to the space of physical degrees of freedom of Yang--Mills theory on a circle (so that space-time is a cylinder). Although the classical reduced phase space is finite-dimensional, the quantum reduction procedure is mathematically fascinating, involving firstly the Wiener measure on a loop group, secondly a generalization of the Cameron--Martin theorem to loop groups, and thirdly Hall's coherent states for compact Lie groups. Our approach is based on a quantum analogue of the classical Marsden--Weinstein symplectic reduction process.

math-ph↗

arXiv1998

Quantization of singular systems and incomplete motions

The need for a mathematically rigorous quantization procedure of singular spaces and incomplete motions is pointed out in connection with quantum cosmology. We put our previous suggestion for such a procedure, based on the theory of induced representations of C*-algebras, in the light of L. Schwartz' theory of Hilbert subspaces. This turns out to account for the freedom in the induction procedure, at the same time providing a basis for generalized eigenfunction expansions pertinent to the needs of quantum cosmology. Reinforcing our previous proposal for the wave-function of the Universe, we are now able to add a concrete prescription for its calculation.

gr-qc↗

arXiv1998

Twisted Lie group $C^*$-algebras as strict quantizations

A nonzero 2-cocycle $Γ\in Z^2(\g,\R)$ on the Lie algebra $\g$ of a compact Lie group $G$ defines a twisted version of the Lie-Poisson structure on the dual Lie algebra $\g^*$, leading to a Poisson algebra $C^{\infty}(\g_{(Γ)}^*)$. Similarly, a multiplier $c\in Z^2(G,U(1))$ on $G$ which is smooth near the identity defines a twist in the convolution product on $G$, encoded by the twisted group $C^*$-algebra C^*(G,c)$. Further to some superficial yet enlightening analogies between $C^{\infty}(\g^*_{(Γ)})$ and $C^*(G,c)$, it is shown that the latter is a strict quantization of the former, where Planck's constant $\hbar$ assumes values in $(\Z\backslash\{0\})^{-1}$. This means that there exists a continuous field of $C^*$-algebras, indexed by $\hbar\in 0\cup (\Z\backslash\{0\})^{-1}$, for which $\A^0=C_0(\g^*)$ and $\A_{\hbar}=C^*(G,c)$ for $\hbar\neq 0$, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in $\A_{\hbar}$ to the Poisson bracket on $C^{\infty}(\g^*_{(Γ)})$. Note that the `quantization' of $\hbar$ does not occur for $Γ=0$.

math-ph↗

arXiv1998

Constrained quantization in algebraic field theory

Quantization relates Poisson algebras to $C^*$-algebras. The analysis of local gauge symmetries in algebraic quantum field theory is approached through the quantization of classical gauge theories, regarded as constrained dynamical systems. This approach is based on the analogy between symplectic reduction and Rieffel induction on the classical and on the quantum side, respectively. Thus one is led to a description of e.g. $θ$-angles and gauge anomalies in the algebraic setting.

math-ph↗

arXiv1998

Lecture notes on C-algebras, Hilbert C-modules, and quantum mechanics

This is a graduate-level introduction to C*-algebras, Hilbert C*-modules, vector bundles, and induced representations of groups and C*-algebras, with applications to quantization theory, phase space localization, and configuration space localization. The reader is supposed to know elementary functional analysis and quantum mechanics.

math-ph↗

arXiv1998

Strict quantization of coadjoint orbits

A strict quantization of a compact symplectic manifold $S$ on a subset $I\subseteq\R$, containing 0 as an accumulation point, is defined as a continuous field of $C^*$-algebras $\{A_{\hbar}\}_{\hbar\in I}$, with $A_0=C_0(S)$, and a set of continuous cross-sections $\{Q(f)\}_{f\in C^{\infty}(S)}$ for which $Q_0(f)=f$. Here $Q_{\hbar}(f^*)=Q_{\hbar}(f)^*$ for all $\hbar\in I$, whereas for $\hbar\to 0$ one requires that $i[Q_{\hbar}(f),Q_{\hbar}(g)]/\hbar\to Q_{\hbar}(\{f,g\})$ in norm. We discuss general conditions which guarantee that a (deformation) quantization in a more physical sense leads to one in the above sense. Using ideas of Berezin, Lieb, Simon, and others, we construct a strict quantization of an arbitrary integral coadjoint orbit $O_{\lm}$ of a compact connected Lie group $G$, associated to a highest weight $\lm$. Here $I=0\cup 1/\N$, so that $\hbar=1/k$, $k\in\N$, and $A_{1/k}$ is defined as the $C^*$-algebra of all matrices on the finite-dimensional Hilbert space $V_{k\lm}$ carrying the irreducible representation $U_{k\lm}(G)$ with highest weight $k\lm$. The quantization maps $Q_{1/k}(f)$ are constructed from coherent states in $V_{k\lm}$, and have the special feature of being positive maps.

math-ph↗

arXiv1997

Constrained quantization and $θ$-angles

We apply a new and mathematically rigorous method for the quantization of constrained systems to two-dimensional gauge theories. In this method, which quantizes Marsden-Weinstein symplectic reduction, the inner product on the physical state space is expressed through a certain integral over the gauge group. The present paper, the first of a series, specializes to the Minkowski theory defined on a cylinder. The integral in question is then constructed in terms of the Wiener measure on a loop group. It is shown how $þ$-angles emerge in the new method, and the abstract theory is illustrated in detail in an example.

hep-th↗

arXiv1996

Poisson spaces with a transition probability

The common structure of the space of pure states $P$ of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function $p:P\times P-> [0,1]$, with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formulation intrinsic to the given context). In classical mechanics, where $p(ρ,σ)=\dl_{ρσ}$, unitarity poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of $p$, and by the property that the irreducible components of $P$ as a transition probability space coincide with the symplectic leaves of $P$ as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\em Acta Math.} {\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 82} (1982) 497-509), we give axioms guaranteeing that $P$ is the space of pure states of a unital $C^*$-algebra. We give an explicit construction of this algebra from $P$.

quant-ph↗

arXiv1996

Simple New Axioms for Quantum Mechanics

The space P of pure states of any physical system, classical or quantum, is identified as a Poisson space with a transition probability. The latter is a function p: PxP -> [0,1]; in addition, a Poisson bracket is defined for functions on P. These two structures are connected through unitarity. Classical and quantum mechanics are each characterized by a simple axiom on the transition probability p. Unitarity then determines the Poisson bracket of quantum mechanics up to a multiplicative constant (identified with Planck's constant). Superselection rules are naturally incorporated.

quant-ph↗

arXiv1996

The quantization of constrained systems: from symplectic reduction to Rieffel induction

This is an introduction to the author's recent work on constrained systems. Firstly, a generalization of the Marsden-Weinstein reduction procedure in symplectic geometry is presented - this is a reformulation of ideas of Mikami-Weinstein and Xu. Secondly, it is shown how this procedure is quantized by Rieffel induction, a technique in operator algebra theory. The essential point is that a symplectic space with generalized moment map is quantized by a pre-(Hilbert) C^*-module. The connection with Dirac's constrained quantization method is explained. Three examples with a single constraint are discussed in some detail: the reduced space is either singular, or defined by a constraint with incomplete flow, or unproblematic but still interesting. In all cases, our quantization procedure may be carried out. Finally, we re-interpret and generalize Mackey's quantization on homogeneous spaces. This provides a double illustration of the connection between C^*-modules and the moment map.

dg-ga↗

arXiv1995

Against the Wheeler-DeWitt equation

The ADM approach to canonical general relativity combined with Dirac's method of quantizing constrained systems leads to the Wheeler-DeWitt equation. A number of mathematical as well as physical difficulties that arise in connection with this equation may be circumvented if one employs a covariant Hamiltonian method in conjunction with a recently developed, mathematically rigorous technique to quantize constrained systems using Rieffel induction. The classical constraints are cleanly separated into four components of a covariant momentum map coming from the diffeomorphism group of spacetime, each of which is linear in the canonical momenta, plus a single finite-dimensional quadratic constraint that arises in any theory, parametrized or not. The new quantization method is carried through in a minisuperspace example, and is found to produce a ``wavefunction of the universe". This differs from the proposals of both Vilenkin and Hartle-Hawking for a closed FRW universe, but happens to coincide with the latter in the open case.

gr-qc↗

arXiv1995

Classical behaviour in quantum mechanics: a transition probability approach

A formalism is developed for describing approximate classical behaviour in finite (but possibly large) quantum systems. This is done in terms of a structure common to classical and quantum mechanics, viz. a Poisson space with a transition probability. Both the limit where Planck's constant goes to zero in a fixed finite system and the limit where the size of the system goes to infinity are incorporated. In either case, classical behaviour is seen only for certain observables and in a restricted class of states.

quant-ph↗

arXiv1995

The Stueckelberg-Kibble Model as an Example of Quantized Symplectic Reduction

Recently, it has been observed that a certain class of classical theories with constraints can be quantized by a mathematical procedure known as Rieffel induction. After a short exposition of this idea, we apply the new quantization theory to the Stueckelberg-Kibble model. We explicitly construct the physical state space ${\cal H}_{phys}$, which carries a massive representation of the Poincaré group. The longitudinal one-particle component arises from a particular Bogoliubov transformation of the five (unphysical) degrees of freedom one has started with. Our discussion exhibits the particular features of the proposed constrained quantization theory in great clarity.

hep-th↗

arXiv1994

The infinite unitary group, Howe dual pairs, and the quantization of constrained systems

The irreducible unitary representations of the Banach Lie group $U_0(\H)$ (which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary operators on a separable Hilbert space $\H$, which were found by Kirillov and Ol'shanskii, are reconstructed from quantization theory. Firstly, the coadjoint orbits of this group are realized as Marsden-Weinstein symplectic quotients in the setting of dual pairs. Secondly, these quotients are quantized on the basis of the author's earlier proposal to quantize a more general symplectic reduction procedure by means of Rieffel induction (a technique in the theory of operator algebras). As a warmup, the simplest such orbit, the projective Hilbert space, is first quantized using geometric quantization, and then again with Rieffel induction. Reduction and induction have to be performed with either $U(M)$ or $U(M,N)$. The former case is straightforward, unless the half-form correction to the (geometric) quantization of the unconstrained system is applied. The latter case, in which one induces from holomorphic discrete series representations, is problematic. For finite-dimensional $\H=\C^k$, the desired result is only obtained if one ignores half-forms, and induces from a representation, `half' of whose highest weight is shifted by $k$ (relative to the naive orbit correspondence). This presumably poses a problem for any theory of quantizing constrained systems.

hep-th↗

arXiv1994

Classical and quantum representation theory

These notes present an introduction to an analytic version of deformation quantization. The central point is to study algebras of physical observables and their irreducible representations. In classical mechanics one deals with real Poisson algebras, whereas in quantum mechanics the observables have the structure of a real non-associative Jordan-Lie algebra. The non-associativity is proportional to $\pl^2$, hence for $\pl\raw 0$ one recovers a real Poisson algebra. This observation lies at the basis of `strict' deformation quantization, where one deforms a given Poisson algebra into a $C^*$-algebra, in such a way that the basic algebraic structures are preserved. Our main interest lies in degenerate Poisson algebras and their quantization by non-simple Jordan-Lie algebras. The traditional symplectic manifolds of classical mechanics, and their quantum counterparts (Hilbert spaces and operator algebras which act irreducibly) emerge from a generalized representation theory. This two-step procedure sheds considerable light on the subject. We discuss a large class of examples, in which the Poisson algebra canonically associated to an (integrable) Lie algebroid is deformed into the Jordan-Lie algebra of the corresponding Lie groupoid. A special case of this construction, which involves the gauge groupoid of a principal fibre bundle, describes the classical and quantum mechanics of a particle moving in an external gravitational and Yang-Mills field.

hep-th↗

arXiv1994

Observation and superselection in quantum mechanics

We attempt to clarify the main conceptual issues in approaches to `objectification' or `measurement' in quantum mechanics which are based on superselection rules. Such approaches venture to derive the emergence of classical `reality' relative to a class of observers; those believing that the classical world exists intrinsically and absolutely are advised against reading this paper. The prototype approach (Hepp) where superselection sectors are assumed in the state space of the apparatus is shown to be untenable. Instead, one should couple system and apparatus to an environment, and postulate superselection rules for the latter. These are motivated by the locality of any observer or other (actual or virtual) monitoring system. In this way `environmental' solutions to the measurement problem (Zeh, Zurek) become consistent and acceptable, too. Points of contact with the modal interpretation are briefly discussed. We propose a minimal value attribution to observables in theories with superselection rules, in which only central observables have properties. In particular, the eigenvector-eigenvalue link is dropped. This is mainly motivated by Ockham's razor.

hep-th↗

arXiv1994

Massless particles, electromagnetism, and Rieffel induction

The connection between space-time covariant representations (obtained by inducing from the Lorentz group) and irreducible unitary representations (induced from Wigner's little group) of the Poincaré group is re-examined in the massless case. In the situation relevant to physics, it is found that these are related by Marsden-Weinstein reduction with respect to a gauge group. An analogous phenomenon is observed for classical massless relativistic particles. This symplectic reduction procedure can be (`second') quantized using a generalization of the Rieffel induction technique in operator algebra theory, which is carried through in detail for electro- magnetism. Starting from the so-called Fermi representation of the field algebra generated by the free abelian gauge field, we construct a new (`rigged') sesquilinear form on the representation space, which is positive semi-definite, and given in terms of a Gaussian weak distribution (promeasure) on the gauge group (taken to be a Hilbert Lie group). This eventually constructs the algebra of observables of quantum electro- magnetism (directly in its vacuum representation) as a representation of the so-called algebra of weak observables induced by the trivial representation of the gauge group.

hep-th↗

arXiv1993

Rieffel induction as generalized quantum Marsden-Weinstein reduction

A new approach to the quantization of constrained or otherwise reduced classical mechanical systems is proposed. On the classical side, the generalized symplectic reduction procedure of Mikami and Weinstein, as further extended by Xu in connection with symplectic equivalence bimodules and Morita equivalence of Poisson manifolds, is rewritten so as to avoid the use of symplectic groupoids, whose quantum analogue is unknown. A theorem on symplectic reduction in stages is given. This allows one to discern that the `quantization' of the generalized moment map consists of an operator-valued inner product on a (pre-) Hilbert space (that is, a structure similar to a Hilbert $C^*$-module). Hence Rieffel's far-reaching operator-algebraic generalization of the notion of an induced representation is seen to be the exact quantum counterpart of the classical idea of symplectic reduction, with imprimitivity bimodules and strong Morita equivalence of $C^*$-algebras falling in the right place. Various examples involving groups as well as groupoids are given, and known difficulties with both Dirac and BRST quantization are seen to be absent in our approach.

hep-th↗