Canonical Commutation Relations Research Articles

Deviating from the canonical: induced noncommutativity

The motion of a quantum particle, constrained to move along a space curve via a confining potential, is governed by a hamiltonian that depends on the geometry of the curve. Adiabatic changes in the shape of the curve are shown to give rise to geometric phases. The effect of the latter on the quantization of the motion of the curve itself can be naturally described by a deformation of the canonical commutation relations, providing an example of a noncommutative effective quantum field theory.

Pseudo-Bosons, So Far

In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called {\em pseudo-bosons}. They arise from a special deformation of the canonical commutation relation $[a,a^\dagger]=\1$, which is replaced by $[a,b]=\1$, with $b$ not necessarily equal to $a^\dagger$. We start discussing some of their mathematical properties and then we discuss several examples.

Angular momentum decomposition of the three-dimensional Wigner harmonic oscillator

In the Wigner framework, one abandons the assumption that the usual canonical commutation relations are necessarily valid. Instead, the compatibility of Hamilton's equations and the Heisenberg equations is the starting point, and no further assumptions are made about how the position and momentum operators commute. Wigner quantization leads to new classes of solutions, and representations of Lie superalgebras are needed to describe them. For the n-dimensional Wigner harmonic oscillator, solutions are known in terms of the Lie superalgebras $\mathfrak {osp}(1|2n)$osp(1|2n) and $\mathfrak {gl}(1|n)$gl(1|n). For n = 3N, the question arises as to how the angular momentum decomposition of representations of these Lie superalgebras is computed. We construct generating functions for the angular momentum decomposition of specific series of representations of $\mathfrak {osp}(1|6N)$osp(1|6N) and $\mathfrak {gl}(1|3N)$gl(1|3N), with N = 1 and N = 2. This problem can be completely solved for N = 1. However, for N = 2 only some classes of representations allow executable computations.

D-foam-induced flavor condensates and breaking of supersymmetry in free Wess-Zumino fluids

Recently we argued that a particular model of string-inspired quantum space-time foam (D-foam) may induce oscillations and mixing among flavoured particles. As a result, rather than the mass-eigenstate vacuum, the correct ground state to describe the underlying dynamics is the flavour vacuum, proposed some time ago by Blasone and Vitiello as a description of quantum field theories with mixing. At the microscopic level, the breaking of target-space supersymmetry is induced in our space-time foam model by the relative transverse motion of brane defects. Motivated by these results, we show that the flavour vacuum, introduced through an inequivalent representation of the canonical (anti-) commutation relations, provides a vehicle for the breaking of supersymmetry (SUSY) at a low-energy effective field theory level; on considering the flavour-vacuum expectation value of the energy-momentum tensor and comparing with the form of a perfect relativistic fluid, it is found that the bosonic sector contributes as dark energy while the fermion contribution is like dust. This indicates a strong and novel breaking of SUSY, of a non-perturbative nature, which may characterize the low energy field theory of certain quantum gravity models.

Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems

A method to construct non-dissipative non-Dirac-Hermitian relativistic quantum system that is isospectral with a Dirac-Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-Hermitian operators, which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-Hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry.

On the axiomatic approach to quasi-classical field theory

We study the connection between quasi-classical field theory and axiomatic statements of the quantum field theory, Schwinger source theory, and the Lehmann-Symanzik-Zimmermann (LSZ) formalism. The classical Schwinger source is connected with the classical field; the LSZ R-function is connected with the quantum field operator. The axioms of the quantum field theory are written in the context of the quasi-classical expansion. In the considered approach, the stationary action principle and canonical commutation relations for field operators are obtained as corollaries and are not postulated as initial statements of the theory.

PATH INTEGRALS FOR QUASI-HERMITIAN HAMILTONIANS

Inner products in quasi-Hermitian quantum theories, and hence probabilities, are defined through a metric that depends on the details of the Hamiltonians themselves. We shall see that the functional integral for quasi-Hermitian theories, and hence Feynman diagrams, for example, can be calculated without needing to evaluate the metric. The reason turns out be that their derivation is based fundamentally on the Heisenberg equations of motion and the canonical equal-time commutation relations, which retain their standard form. As an application, we show how co-ordinate transformations in the path integral can enable us to recover equivalent Hermitian Hamiltonians.

Deconstructing non-Dirac–Hermitian supersymmetric quantum systems

A method to construct a non-Dirac–Hermitian supersymmetric quantum system that is isospectral with a Dirac–Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving both bosonic and fermionic degrees of freedom in terms of non-Dirac–Hermitian operators which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. A pseudo-Hermitian realization of the Clifford algebra for a pre-determined positive-definite metric is used to construct supersymmetric systems with one or many degrees of freedom. It is shown that exactly solvable non-Dirac–Hermitian supersymmetric quantum systems can be constructed corresponding to each exactly solvable Dirac–Hermitian system. Examples of non-Dirac–Hermitian (i) non-relativistic Pauli Hamiltonian, (ii) super-conformal quantum system, and (iii) supersymmetric Calogero-type models admitting entirely real spectra are presented.

An affinity for affine quantum gravity

The main principle of affine quantum gravity is the strict positivity of the matrix $$\{ \hat g_{ab} (x)\} $$ composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.

Quantisation of the free electromagnetic field implies quantisation of its angular momentum

It is shown that when the gauge-invariant Bohr–Rosenfeld commutators of the free electromagnetic field are applied to the expressions for the linear and angular momentum of the electromagnetic field interpreted as operators then, in the absence of electric and magnetic charge densities, these operators satisfy the canonical commutation relations for momentum and angular momentum. This confirms their validity as operators that can be used in quantum mechanical calculations of angular momentum.

Quantum mechanics of a free particle from properties of the Dirac delta function

Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, ⟨x|p⟩=exp(ipx/ℏ)/2πℏ. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of ⟨x|p⟩.

Generalization of the Cooper pairing mechanism for spin-triplet in superconductors

A generalization of the Cooper pairing mechanism is proposed which allows for a triplet state of lower energy. This is achieved by incorporating spin into the canonical commutation relations and by modifying the δ potential contact interaction. The gap equation contain as solutions both singlet and triplet states. It is shown that the triplet state is lower in energy than the singlet state which may explain the spin-triplet superconductivity observed.

Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes

The Fock quantization of fields propagating in cosmological spacetimes is not uniquely determined because of several reasons. Apart from the ambiguity in the choice of the quantum representation of the canonical commutation relations, there also exists a certain freedom in the choice of field: one can scale it arbitrarily absorbing background functions, which are spatially homogeneous but depend on time. Each nontrivial scaling turns out into a different dynamics and, in general, into an inequivalent quantum field theory. In this work we analyze this freedom at the quantum level for a scalar field in a nonstationary, homogeneous spacetime whose spatial sections have ${S}^{3}$ topology. A scaling of the configuration variable is introduced as part of a linear, time dependent canonical transformation in phase space. In this context, we prove in full detail a uniqueness result about the Fock quantization requiring that the dynamics be unitary and the spatial symmetries of the field equations have a natural unitary implementation. The main conclusion is that, with those requirements, only one particular canonical transformation is allowed, and thus only one choice of the field-momentum pair (up to irrelevant constant scalings). This complements another previous uniqueness result for scalar fields with a time varying mass on ${S}^{3}$, which selects a specific equivalence class of Fock representations of the canonical commutation relations under the conditions of a unitary evolution and the invariance of the vacuum under the background symmetries. In total, the combination of these two different statements of uniqueness picks up a unique Fock quantization for the system. We also extend our proof of uniqueness to other compact topologies and spacetime dimensions.

Constructing Exactly Solvable Pseudo-hermitian Many-Particle Quantum Systems by Isospectral Deformation

A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.

Implementation problem for the canonical commutation relation in terms of quantum white noise derivatives

The implementation problem for the canonical commutation relation is reduced to a system of differential equations for Fock space operators containing new type of derivatives. We solve these differential equations systematically by means of quantum white noise calculus, and obtain the solution to the implementation problem.

Modified Landau levels, damped harmonic oscillator, and two-dimensional pseudo-bosons

In a series of recent papers, one of us has analyzed in some details a class of elementary excitations called pseudo-bosons. They arise from a special deformation of the canonical commutation relation [a, a†] = 11, which is replaced by [a, b] = 11, with b not necessarily equal to a†. Here, after a two-dimensional extension of the general framework, we apply the theory to a generalized version of the two-dimensional Hamiltonian describing Landau levels. Moreover, for this system, we discuss coherent states and we deduce a resolution of the identity. We also consider a different class of examples arising from a classical system, i.e., a damped harmonic oscillator.

A new definition of regularity for representation of canonical commutation relations algebras

A new criterion of regularity for the representation of canonical commutation relations algebras is given on the basis of the concept of an analytical vector.

Path Integrals for Pseudo-Hermitian Hamiltonians

Inner products in pseudo-Hermitian quantum theories depend on the details of the Hamiltonians themselves, which makes them difficult to calculate. We shall see that, for some questions, the functional integrals for such theories can be calculated without needing to determine the inner product metric. The reason is that their derivation is based on the Heisenberg equations of motion and the canonical commutation relations, which are unchanged. In particular, this can greatly simplify the derivation of Hermitian theories that are equivalent to these pseudo-Hermitian systems.

On urn models, non-commutativity and operator normal forms

Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg–Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.

Quasi-free semigroups on the CCR algebra

We analyze the general structure of one-parameter semigroups of quasi-free maps on the C*-algebra of canonical commutation relations, a classification of these semigroups under a mild regularity assumption is obtained. Moreover, continuity properties of quasi-free semigroups in representations of the algebra of canonical commutation relations are discussed.