Representations Of Canonical Commutation Relations Research Articles

Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

Functional integrals and inequivalent representations in Quantum Field Theory

We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one works with functional integrals, it is not immediately clear how this algebraic feature manifests itself in the formalism. Here we attack this issue by considering the canonical transformations in the context of coherent-state functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functional-integral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinite-volume limit the canonical transformations induce a transition from one representation of canonical commutation relations to another one and under what conditions the representations are unitarily inequivalent. We also consider the partition function and derive the energy gap between statistical systems described in two different representations which, among others, allows to establish a connection with continuous phase transitions. We illustrate the inner workings of the outlined mechanism by discussing two prototypical systems: the van Hove model and the Bogoliubov model of weakly interacting Bose gas.

Inequivalent representations in the functional integral framework

An important feature of Quantum Field Theory is the existence of unitarily inequivalent representations of canonical commutation relations. When one works with the functional integral formalism, it is not clear, however, how this feature emerges. By following the seminal work of M. Swanson on canonical transformations in phase-space path integral, we generalize his approach to coherent-state functional integrals which in turn will lead to a simplified formalism which makes the appearance of the inequivalent representations more transparent.

Bogolyubov transformations for regular representations of canonical commutation relations in space with an indefinite metric

Bogolyubov transformations for regular representations of the algebra of canonical commutation relations (CCRs) are considered in space with an indefinite metric.

Automatic Regularization by Quantization in Reducible Representations of CCR: Point-Form Quantum Optics with Classical Sources

Electromagnetic fields are quantized in a manifestly covariant way by means of a class of reducible “center-of-mass N-representations” of the algebra of canonical commutation relations (CCR). The four-potential Aa(x) transforms in these representations as a Hermitian four-vector field in Minkowski four-position space (without change of gauge), but in momentum space it splits into spin-1 massless photons and two massless scalars. What we call quantum optics is the spin-1 sector of the theory. The scalar fields have physical status similar to that of dark matter (spin-1 and spin-0 particle numbers are separately conserved). There are no negative-norm or zero-norm states. Unitary dynamics is given by the point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary of the Milne universe. SL(2,C) transformations as well as the dynamics are represented unitarily in the Hilbert space corresponding to N four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the N-oscillator gas and is given by any N-oscillator product state annihilated by all annihilation operators. The form of Aa(x) is determined by an analogue of the twistor equation. The same equation guarantees that the set of vacuum states is Poincare invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined even for pointlike charges, with ultraviolet and infrared regularizations occurring automatically as a consequence of the formalism. The probabilities are not Poissonian but of a Renyi type with α=1−1/N; the Shannon limit N→∞ is an ultraviolet/infrared-regularized Poisson distribution. The average number of photons occurring in Bremsstrahlung splits into two parts: The one due to acceleration, and the one that remains nonvanishing even for inertially moving charges. Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. We show in particular that static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is sensitive to both the choice of representation of CCR and the corresponding vacuum state.

P-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrodinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.

Quantum Teleportation and Holomorphic Representation of CCR

For the treatment of teleportation of continuous quantum variables, often the Wigner functions are used. In this paper, we will show that by using the holomorphic representation of the canonical commutation relations (CCR), the teleportation of continuous quantum variables is treated in a parallel way as the teleportation of qubits.

Nonequivalent representations of canonical commutation relations in quantum mechanics: The case of the Aharonov-Bohm effect

We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons.

Cavity-QED Tests of Representations of Canonical Commutation Relations Employed in Field Quantization

Various aspects of dissipative and nondissipative decoherence of Rabi oscillations are discussed in the context of field quantization in alternative representations of CCR. Theory is confronted with experiment, and a possibility of more conclusive tests is analyzed.

Regularization as Quantization in Reducible Representations of CCR

A covariant quantization scheme employing reducible representations of canonical commutation relations with positive-definite metric and Hermitian four-potentials is tested on the example of quantum electrodynamic fields produced by a classical current. The scheme implies a modified but very physically looking Hamiltonian. We solve Heisenberg equations of motion and compute photon statistics. Poisson statistics naturally occurs and no infrared divergence is found even for pointlike sources. Classical fields produced by classical sources can be obtained if one computes coherent-state averages of Heisenberg-picture operators. It is shown that the new form of representation automatically smears out pointlike currents. We discuss in detail Poincar\'e covariance of the theory and the role of Bogoliubov transformations for the issue of gauge invariance. The representation we employ is parametrized by a number that is related to R\'enyi's $\alpha$. It is shown that the ``Shannon limit" $\alpha\to 1$ plays here a role of correspondence principle with the standard regularized formalism.

Degree of entanglement as a physically ill-posed problem: The case of entanglement with vacuum

We analyze an example of a photon in a superposition of different modes, and ask what is the degree of their entanglement with a vacuum. The problem turns out to be ill-posed since we do not know which representation of the algebra of canonical commutation relations (CCR) to choose for the field quantization. Once we make a choice, we can solve the question of entanglement unambiguously. So the difficulty is not with mathematics, but with physics of the problem. In order to make the discussion explicit we analyze from this perspective a popular argument based on a photon leaving a beam splitter and interacting with two two-level atoms. We first solve the problem algebraically in the Heisenberg picture, without any assumption about the form of representation of CCR. Then we take the $\ensuremath{\infty}$-representation and show in two ways that in two-mode states the modes are maximally entangled with the vacuum, but single-mode states are not entangled. Next we repeat the analysis in terms of the representation of CCR taken from Berezin's book and show that two-mode states do not involve the mode-vacuum entanglement. Finally, we switch to a family of reducible representations of CCR recently investigated in the context of field quantization, and show that the entanglement with the vacuum is present even for single-mode states. Still, the degree of entanglement here is difficult to estimate, mainly because there are $N+2$ subsystems, with $N$ unspecified and large.

States of light via reducible quantization

Multi-photon and coherent states of light are formulated in terms of a reducible representation of canonical commutation relations. Standard properties of such states are recovered as certain limiting cases. The new formalism leads to field operators and not operator-valued distributions. The example of radiation fields produced by a classical current shows an automatic regularization of the infrared divergence.

Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials: Canonical commutation relations, quantum algebras, and reduction to lattice quantum systems

Some representation-theoretic aspects of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R2 are investigated. For each vector v in a set V(D)⊂R2\{0}, the projection Pv of the physical momentum operator P≔p−αA to the direction of v is defined by Pv≔v⋅P as an operator acting in L2(R2), where p=(−iDx,−iDy)[(x,y)∈R2] with Dx (resp., Dy) being the generalized partial differential operator in the variable x (resp., y) and α∈R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter unitary group {eitP̄v}t∈R generated by the self-adjoint operator P̄v (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2\D are also considered. Conjugately to Pv and Pw [w∈V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a linear combination of the position operators x and y, such that, if A is flat on R2\D, then πv,wA≔{Qv,w,Qw,v,Pv,Pw} gives a representation of the canonical commutation relations (CCR) with two degrees of freedom. Properties of the representation πv,wA are analyzed. In particular, a necessary and sufficient condition for πv,wA to be unitarily equivalent (or inequivalent) to the Schrödinger representation of CCR is established. The case where πv,wA is inequivalent to the Schrödinger representation corresponds to the Aharonov–Bohm effect. Quantum algebraic structures [quantum plane and the quantum group Uq(sl2)] associated with the pair {P̄v,P̄w} are also discussed. Moreover, for every A in a class of vector potentials having singularities on the infinite lattice L(ω1,ω2)≔{mω1+nω2|m,n∈Z} [the case D=L(ω1,ω2)], where ω1∈R2 and ω2∈R2 are linearly independent, it is shown that the magnetic translations eiP̄ωj, j=1,2, with A replaced by a modified vector potential are reduced by the Hilbert space l2(L(ω1,ω2)) identified with a closed subspace of L2(R2). This result, which may be regarded as one of the most important novel results of the present paper, establishes a connection of continuous quantum systems in vector potentials to lattice ones.

A Phase Space Approach to the Projective Representation of the Canonical Commutation Relations

The projective nature of the representations of canonical commutation relations on the space of states is probed via cohomological considerations. Some interesting geometrical pictures arise, which associate themselves with the phase descriptions of quantum systems.

Gauge theory on a non-simply connected domain and representations of canonical commutation relations

A quantum system of a particle interacting with a (non-Abelian) gauge field on the non-simply connected domain M=R2\{an}Nn=1 is considered, where an, n=1,...,N, are fixed isolated points in R2. The gauge potential A of the gauge field is a p×p anti-Hermitian matrix-valued 1-form on M, and may be strongly singular at the points an, n=1,...,N. If A is flat, then the physical momentum and the position operators {Pj,qj}2j=1 of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert space L2(R2;Cp). A necessary and sufficient condition for this representation to be the Schrödinger 2-system is given in terms of the Wilson loops of the rectangles not intersecting an, n=1,...,N. This also gives a characterization for the representation to be non-Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, Pj is essentially self-adjoint. Moreover, an example, which gives a class of non-Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.

Representation of Canonical Commutation Relations in a Gauge Theory, the Aharonov-Bohm Effect, and the Dirac-Weyl Operator

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sucient condition for the representation to be

On the Instability of an Oscillator in a Field

We discuss the origin of dissipation in a one-dimension model describing the interaction of a microsystem (an oscillator) with a bath (a quantized field). The Hamiltonian is a gauge-type coupling of the oscillator with the field and it is bounded below. Classical and quantum pictures are considered. Our formulation of the problem: what stable states are described by the total Hamiltonian if the excited states of the oscillator are unstable? How can these unstable states arise in a conservative system? The vacua of the free and the interacting system are found in dipole approximation. The theory determines a formfactor which optimizes the contributions of the total Hamiltonian in dipole approximation. These two vacua generate equivalent representations of canonical commutation relations. As a result of the oscillator-field interaction the stable states of this system consist of the vacuum (oscillator ground state) and quanta of the quantized field (bath). It means that the oscillator as a stable state can exist only in the ground state. Any excited oscillator states can be seen as resonances in the field-field scattering.

Representations of canonical commutation relations defined by Gaussian measure and Gaussian cocycle

Representations of canonical commutation relations defined by Gaussian measure and Gaussian cocycle

Momentum operators with gauge potentials, local quantization of magnetic flux, and representation of canonical commutation relations

Commutation properties of two-dimensional momentum operators with gauge potentials are investigated. A notion of local quantization of magnetic flux is introduced to characterize physically the strong commutativity of the momentum operators. In terms of the notion, a necessary and sufficient condition is given for the position and the momentum operators to be equivalent to the Schrödinger representation of the canonical commutation relations.

Hadamard singularity and quantum states in Bianchi type-I space-time.

The construction of physical states for a scalar quantum field in a Bianchi type-I universe is analyzed. A local constraint based upon the Hadamard form of the anticommutator function is implemented. The constraints imposed on the Fock space by this requirement are derived and the unitary equivalence between different representations of canonical commutation relations is shown. Within this formalism we analyze recently proposed Fock-space constructions and we show the uniqueness of the inflationary vacuum.