Canonical Commutation Relations Research Articles

ON A FAMILY OF DISCRETE ND LADDER-TYPE OPERATORS CONSTRUCTED IN TERMS OF THE HERMITIAN TOEPLITZ COMMUTATOR OPERATOR Z_N

A development of an algebraic system with N-dimensional ladder-type operators associated with the discrete Fourier transform is described, following an analogy with the canonical commutation relations of the continuous case. It is found that a Hermitian Toeplitz matrix Z_N, which plays the role of the identity, is sufficient to satisfy the Jacobi identity and, by solving some compatibility relations, a family of ladder operators with corresponding Hamiltonians can be constructed. The behaviour of the matrix Z_N for large N is elaborated. It is shown that this system can be also realized in terms of the Heun operator W, associated with the discrete Fourier transform, thus providing deeper insight on the underlying algebraic structure.

Quantum geometrodynamics revived II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Time operators of harmonic oscillators and their representations

A time operator T̂ϵ of the one-dimensional harmonic oscillator ĥϵ=12(p2+ϵq2) is rigorously constructed. It is formally expressed as T̂ϵ=121ϵ(arctan(ϵt̂0)+arctan(ϵt̂1)) with t̂0=p−1q and t̂1=qp−1. It is shown that the canonical commutation relation [hϵ,T̂ϵ]=−i1 holds true on a dense domain in the sense of sesqui-linear forms, and the limit of T̂ϵ as ϵ → 0 is shown. Finally a matrix representation of T̂ϵ and its analytic continuation are given.

KMS Dirichlet forms, coercivity and superbounded Markovian semigroups on von Neumann algebras

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non-tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space L2(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(M)$$\\end{document} and the associated noncommutative Lp(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p(M)$$\\end{document} spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein–Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.

Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition

Abstract We consider the quantization of gravity as an SL(2,C) gauge theory in terms of Ashtekar's selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII). We start from a holomorphic phase space formulation. It is then natural to push for a quantization in terms of holomorphic wave functions. Thus we consider holomorphic cylindrical wave functions over SL(2,C) connections. 

We use an overall phase ambiguity of the complex selfdual action to obtain Poisson brackets that mirror those of the real theory. We then show that there is a representation of the corresponding canonical commutation relations the space of holomorphic cylindrical functions. We describe a class of cylindrically consistent measures that implements RCI. We show that spin networks with SU(2) intertwiners form a basis for gauge invariant states. They are still mutually orthogonal, but the normalisation is different than for the Ashtekar-Lewandowski measure for SU(2). 

We do not consider RCII in the present article. Work on RCII is ongoing and will be presented elsewhere.

The operators of stochastic calculus

Abstract We study a family of representations of the canonical commutation relations (CCR)-algebra, which we refer to as “admissible,” with an infinite number of degrees of freedom. We establish a direct correlation between each admissible representation and a corresponding Gaussian stochastic calculus. Moreover, we derive the operators of Malliavin’s calculus of variation using an algebraic approach, which differs from the conventional methods. The Fock-vacuum representation leads to a maximal symmetric pair. This duality perspective offers the added advantage of resolving issues related to unbounded operators and dense domains much more easily than with alternative approaches.

Ladder operators with no vacuum, their coherent states, and an application to graphene

In literature ladder operators of different nature exist. The most famous are those obeying canonical (anti-) commutation relations, but they are not the only ones. In our knowledge, all ladder operators have a common feature: the lowering operators annihilate a non zero vector, the vacuum. This is connected to the fact that operators of these kind are often used in factorizing some positive operators, or some operators which are bounded from below. This is the case, of course, of the harmonic oscillator, but not only. In this paper we discuss what happens when considering lowering operators with no vacua. In particular, after a general analysis of this situation, we propose a possible construction of coherent states, and we apply our construction to graphene.

Quantization of spinor field in the Schwarzschild spacetime and spin sums for solutions of the Dirac equation

We discuss the problem of canonical quantization of a free massive spinor field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the internal region of the black hole, the canonical commutation relations in the resulting theory hold exactly and the Hamiltonian has the standard form. Spin sums are obtained for solutions of the Dirac equation in the Schwarzschild spacetime.

Exact quantisation of U(1)3 quantum gravity via exponentiation of the hypersurface deformation algebroid

Abstract The U(1)3 model for 3+1 Euclidian signature general relativity (GR) is an interacting, generally covariant field theory with two physical polarisations that shares many features of Lorentzian GR. In particular, it displays a non-trivial realisation of the hypersurface deformation algebroid with non-trivial, i.e. phase space dependent structure functions rather than structure constants. In this paper we show that the model admits an exact quantisation. The quantisation rests on the observation that for this model and in the chosen representation of the canonical commutation relations the density unity hypersurface algebra can be exponentiated on non-degenerate states. These are states that represent a non-degenerate quantum metric and from a classical perspective are the relevant states on which the hypersurface algebra is representable. The representation of the algebra is exact, with no ambiguities involved and anomaly free. The quantum constraints can be exactly solved using groupoid averaging and the solutions admit a Hilbert space structure that agrees with the quantisation of a recently found reduced phase space formulation. Using the also recently found covariant action for that model, we start a path integral or spin foam formulation which, due to the Abelian character of the gauge group, is much simpler than for Lorentzian signature GR and provides an ideal testing ground for general spin foam models. The solution of U(1)3 quantum gravity communicated in this paper motivates an entirely new approach to the implementation of the Hamiltonian constraint in quantum gravity.

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group.

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators

Abstract In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.

Characterisations of dilations via approximants, expectations, and functional calculi

We consider characterisations of unitary dilations and approximations of irreversible classical dynamical systems on a Hilbert space. In the commutative case, building on the work in [9], one can express well known approximants (e.g. Hille- and Yosida-approximants) via expectations over certain stochastic processes. Using this, our first result characterises the simultaneous regular unitary dilatability of commuting families of C0-semigroups via the dilatability of such approximants as well as via regular polynomial bounds. This extends the results in [13] to the unbounded setting. We secondly consider characterisations of unitary and regular unitary dilations via two distinct functional calculi. Applying these tools to a large class of classical dynamical systems, these two notions of dilation exactly characterise when a system admits unitary approximations under certain distinct notions of weak convergence. This establishes a sharp topological distinction between the two notions of unitary dilations. Our results are applicable to commutative systems as well as non-commutative systems satisfying the canonical commutation relations (CCR) in the Weyl form.

Bounded perturbations of the Heisenberg commutation relation via dilation theory

We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is finite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d d -tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result of Haagerup and Rørdam [Duke Math. J. 77 (1995), pp. 627–656 ] that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation are a bounded perturbation of a pair of strongly commuting selfadjoint operators. We also recover Gao’s higher-dimensional generalization of Haagerup and Rørdam’s result, and in typical cases we significantly improve control of the bound when the dimension grows.

Discriminating quantum gravity models by gravitational decoherence

Several phenomenological approaches to quantum gravity predict the existence of a minimal measurable length and/or a maximum measurable momentum near the Planck scale. When embedded into the framework of quantum mechanics, such constraints induce a modification of the canonical commutation relations and thus a generalization of the Heisenberg uncertainty relations, commonly referred to as generalized uncertainty principle (GUP). Different models of quantum gravity imply different forms of the GUP. For instance, in the framework of string theory the GUP is quadratic in the momentum operator, while in the context of doubly special relativity it includes an additional linear dependence. Among the possible physical consequences, it was recently shown that the quadratic GUP induces a universal decoherence mechanism, provided one assumes a foamy structure of quantum spacetime close to the Planck length. Along this line, in the present work we investigate the gravitational decoherence associated to the linear-quadratic GUP and we compare it with the one associated to the quadratic GUP. We find that, despite their similarities, the two generalizations of the Heisenberg uncertainty principle yield decoherence times that are completely uncorrelated and significantly distinct. Motivated by this result, we introduce a theoretical and experimental scheme based on cavity optomechanics to measure the different time evolution of nonlocal quantum correlations corresponding to the two aforementioned decoherence mechanisms. We find that the deviation between the two predictions occurs on time scales that are macroscopic and thus potentially amenable to experimental verification. This scenario provides a possible setting to discriminate between different forms of the GUP and therefore different models of quantum gravity.

Hopf Structure for the q-Lévy-Meixner Oscillator Algebra

The main purpose of this paper is to investigate a generalized oscillator algebra, naturally associated with the [Formula: see text]-Lévy-Meixner polynomials. We solve the problem of the Hopf algebraic structure for the [Formula: see text]-deformed Lévy-Meixner oscillator algebra based on the one-parameter deformation of canonical commutation relations.

Efficient representation for simulating U(1) gauge theories on digital quantum computers at all values of the coupling

We derive a representation for a lattice U(1) gauge theory with exponential convergence in the number of states used to represent each lattice site that is applicable at all values of the coupling. At large coupling, this representation is equivalent to the Kogut-Susskind electric representation, which is known to provide a good description in this region. At small coupling, our approach adjusts the maximum magnetic field that is represented in the digitization as in this regime the low-lying eigenstates become strongly peaked around zero magnetic field. Additionally, we choose a representation of the electric component of the Hamiltonian that gives minimal violation of the canonical commutation relation when acting upon low-lying eigenstates. For $(2+1)$ dimensions with 4 lattice sites the expectation value of the plaquette operator can be calculated with only 7 states per lattice site with per-mille level accuracy for all values of the coupling constant.

Generalised uncertainty relations from finite-accuracy measurements

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a regionσ̃g>0in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by settingσg≃ℏG/c3, the Planck length, andσ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.

Doubling of physical states in the quantum scalar field theory for a remote observer in the Schwarzschild spacetime

We discuss the problem of canonical quantization of a free real massive scalar field in the Schwarzschild spacetime. It is shown that a consistent procedure of canonical quantization of the field can be carried out without taking into account the black hole interior, so that in the resulting theory the canonical commutation relations are satisfied exactly, and the Hamiltonian has the standard form. However, unlike some papers, in which the expansion of the quantum field in spherical harmonics is used, here we use an expansion in scatteringlike states for energies larger than the mass of the field. This reveals a strange property of the resulting quantum field theory---doubling of the quantum states, which look as having the same asymptotic momentum to an observer located far away from the black hole. This purely topological effect cannot be eliminated by moving away from the black hole.

Probabilistic Bounds with Quadratic-Exponential Moments for Quantum Stochastic Systems

This paper is concerned with quadratic-exponential moments (QEMs) for dynamic variables of quantum stochastic systems with position-momentum type canonical commutation relations. The QEMs play an important role for statistical “localisation” of the quantum dynamics in the form of upper bounds on the tail probability distribution for a positive definite quadratic function of the system variables. We employ a randomised representation of the QEMs in terms of the moment-generating function (MGF) of the system variables, which is averaged over its parameters using an auxiliary classical Gaussian random vector. This representation is combined with a family of weighted L2 -norms of the MGF, leading to upper bounds for the QEMs of the system variables. These bounds are demonstrated for open quantum harmonic oscillators with vacuum input fields and non-Gaussian initial states.