Equal-time Commutators Research Articles

Quantization of nonlocal fields via fractional calculus

In this study, we investigate the effect of nonlocality in quantum mechanics and propose a fractional approach the theory of quantized fields. For this purpose, we embedded the fractional calculus to broaden theory of quantum fields since the integral and derivative operators are nonlocal in fractional calculus.Additionally, quantum entanglement is discussed to gain comprehension of nonlocality in the foundation of quantum mechanics. Besides, fractional Lagrangian formalism was presented due to fact that the Lagrangian density is the starting point to establish a field theory.Furthermore, to make fractional field operators quantum mechanical, equal-time commutator have been defined for the these operators in terms of Caputo fractional derivative. Thus, a scheme of quantization of fractional fields is introduced and general aspects of the method is illustrated with the theory of massive scalar fields. This approach laid out to a successful generalization of the quantum field theory which is coherent with the standard formalism. Consequently, we developed promising concept for a quantum field theory by introducing nonlocality into standard mathematical formalism.

Quantum scale invariant gravity in the de Donder gauge

We perform the manifestly covariant quantization of a scale invariant gravity with a scalar field, which is equivalent to the well-known Brans-Dicke gravity via a field redefinition of the scalar field, in the de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance. First, without specifying the expression of a gravitational theory, we write down various equal-time (anti-)commutation relations (ETCRs), in particular, those involving the Nakanishi-Lautrup field, the FP ghost, and the FP antighost only on the basis of the de Donder gauge condition. It is shown that choral symmetry, which is a Poincar${\rm{\acute{e}}}$-like $IOSp(8|8)$ supersymmetry, can be derived from such a general action with the de Donder gauge. Next, taking the scale invariant gravity with a scalar field as a classical theory, we derive the ETCRs for the gravitational sector involving the metric tensor and scalar fields. Moreover, we account for how scale symmetry is spontaneously broken in quantum gravity, thereby showing that the dilaton is a massless Nambu-Goldstone particle.

Quantisation of Lorentz invariant scalar field theory in non-commutative space-time and its consequence

Quantisation of Lorentz invariant scalar field theory in Doplicher-Fredenhagen-Roberts (DFR) space-time, a Lorentz invariant, non-commutative space-time is studied. We use an approach to quantisation that is based on the equations of motion alone and derive the equal time commutation relation between Doplicher-Fredenhagen-Roberts-Amorim (DFRA) scalar field and its conjugate, which has non-commutative dependent modifications, but the corresponding creation and annihilation operators obey usual algebra. We show that imposing the condition that the commutation relation between the field and its conjugate is same as that in the commutative space-time leads to a deformation of the algebra of quantised oscillators. Both these deformed commutation relations derived are valid to all orders in the non-commutative parameter. By analysing the first non-vanishing terms which are θ3 order, we show that the deformed commutation relations scale as 1/λ4, where λ is the length scale set by the non-commutativity of the space-time. We also derive the conserved currents for DFRA scalar field. Further, we analyse the effects of non-commutativity on Unruh effect by analysing a detector coupled to the DFRA scalar field, showing that the Unruh temperature is not modified but the thermal radiation seen by the accelerated observer gets correction due to the non-commutativity of space-time.

A note on commutation relation in conformal field theory

In this note, we explicitly compute the vacuum expectation value of the commutator of scalar fields in a d-dimensional conformal field theory on the cylinder. We find from explicit calculations that we need smearing not only in space but also in time to have finite commutators except for those of free scalar operators. Thus the equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian. We also have the commutator for a conformal field theory on Minkowski space, instead of the cylinder, by taking the small distance limit. For the conformal field theory on Minkowski space, the above statements are also applied.

A criterion to characterize interacting theories in the Wightman framework

We propose that a criterion to characterize interacting theories in a suitable Wightman framework of relativistic quantum field theories which incorporates a “singularity hypothesis”, which has been conjectured for a long time, is supported by renormalization group theory, but has never been formulated mathematically. The (nonperturbative) wave function renormalization Z occurring in these theories is shown not to be necessarily equal to zero, except if the equal time commutation relations (ETCR) are assumed. Since the ETCR are not justified in general (because the interacting fields cannot in general be restricted to sharp times, as is known from model studies), the condition $$Z=0$$ is not of general validity in interacting theories. We conjecture that it characterizes either unstable (composite) particles or the charge-carrying particles, which become infraparticles in the presence of massless particles. In the case of QED, such “dressed” electrons are not expected to be confined, but in QCD, we propose a quark confinement criterion, which follows naturally from lines suggested by the works of Casher, Kogut and Susskind, and Lowenstein and Swieca.

Gauge transformation in macroscopic quantum electrodynamics near polarizable surfaces

To describe charged particles interacting with the quantized electromagnetic field, we point out the differences of working in the so-called generalized and the true Coulomb gauges. We find an explicit gauge transformation between them for the case of the electromagnetic field operators quantized near a macroscopic boundary described by a piece-wise constant dielectric function. Starting from the generalized Coulomb gauge we transform operators into the true Coulomb gauge where the vector potential operator is truly transverse everywhere. We find the generating function of the gauge transformation to carry out the corresponding unitary transformation of the Hamiltonian and show that in the true Coulomb gauge the Hamiltonian of a particle near a polarizable surface contains extra terms due to the fluctuating surface charge density induced by the vacuum field. This extra term is represented by a second-quantised operator on equal footing with the vector field operators. We demonstrate that this term contains part of the electrostatic energy of the charged particle interacting with the surface and that the gauge invariance of the theory guarantees that the total interaction energy in all cases equals the well known result obtainable by the method of images when working in generalized Coulomb gauge. The mathematical tools we have developed allow us to work out explicitly the equal-time commutation relations and shed some light on typical misconceptions regarding issues of whether the presence of the boundaries should affect the field commutators or not, especially when the boundaries are modelled as perfect reflectors.

Radiative decays and the SU(6) Lie algebra

We present research on radiative decays of vector [Formula: see text] to pseudoscalar [Formula: see text] particles ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] quark system) using broken symmetry techniques in the infinite-momentum frame and equal-time commutation relations and the [Formula: see text] Lie algebra, and conducted without ascribing any specific form to meson quark structure or intra-quark interactions. We utilize the physical electromagnetic current [Formula: see text] including its singlet [Formula: see text] term and focus on the [Formula: see text] 35-plet. We derive new relations involving the electromagnetic current (including its singlet — proportional to the [Formula: see text] singlet). Remarkably, we find that the electromagnetic current singlet plays an intrinsic role in understanding the physics of radiative decays and that the charged and neutral [Formula: see text] meson radiative decays into [Formula: see text] are due entirely to the singlet term in [Formula: see text]. Although there is insufficient radiative decay experimental data available at this time, parametrization of possible predicted values of [Formula: see text] is made. For conciseness and self-containment, we compute all [Formula: see text] Lie algebra simple roots, positive roots, weights and fundamental weights which allow the construction of all [Formula: see text] representations. We also derive all nonzero [Formula: see text] generator commutators and anticommutators — useful for further research on grand unified theories.

Axiomatic construction of quantum Langevin equations

A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation–dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a Markovian semi-classical approach, due to Bedeaux and Mazur, from a non-Markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation–dissipation theorem is seen to be incompatible with a Markovian dynamics. Possible applications to the quantum spherical model are discussed.

Relational Hamiltonian for group field theory

Using a massless scalar field as a clock variable, the Legendre transform of a group field theory Lagrangian gives a relational Hamiltonian. In the classical theory, it is natural to define ``equal relational time'' Poisson brackets, where ``equal time'' corresponds to equal values of the scalar field clock. The quantum theory can then be defined by imposing equal relational time commutation relations for the fundamental operators of the theory, with the states being elements of a Fock space with their evolution determined by the relational Hamiltonian operator. A particularly interesting type of states are condensates, as they may correspond to the cosmological sector of group field theory. For the relational Hamiltonian considered in this paper, the coarse-grained dynamics of a simple family of condensate states agree exactly with the Friedmann equations in the classical limit and also include quantum gravity corrections that ensure the big bang singularity is replaced by a bounce.

Characteristics of chiral anomaly in view of various applications

In view of the recent applications of chiral anomaly to various fields beyond particle physics, we discuss some basic aspects of chiral anomaly which may help deepen our understanding of chiral anomaly in particle physics also. It is first shown that Berry's phase (and its generalization) for the Weyl model $H =v_{F} \vec{\sigma}\cdot \vec{p}(t)$ assumes a monopole form at the exact adiabatic limit but deviates from it off the adiabatic limit and vanishes in the high frequency limit of the Fourier transform of $\vec{p}(t)$ for bounded $|\vec{p}(t)|$. An effective action, which is consistent with the non-adiabatic limit of Berry's phase, combined with the Bjorken-Johnson-Low prescription gives normal equal-time space-time commutators and no chiral anomaly. In contrast, an effective action with a monopole at the origin of the momentum space, which describes Berry's phase in the precise adiabatic limit but fails off the adiabatic limit, gives anomalous space-time commutators and a covariant anomaly to the gauge current. We regard this anomaly as an artifact of the postulated monopole and not a consequence of Berry's phase. As for the recent application of the chiral anomaly to the description of effective Weyl fermions in condensed matter and nuclear physics, which is closely related to the formulation of lattice chiral fermions, we point out that the chiral anomaly for each species doubler separately vanishes for a finite lattice spacing, contrary to the common assumption. Instead a general form of pair creation associated with the spectral flow for the Dirac sea with finite depth takes place. This view is supported by the Ginsparg-Wilson fermion, which defines a single Weyl fermion without doublers on the lattice and gives a well-defined index (anomaly) even for a finite lattice spacing.

LOCALITY OF QUANTIZED SCALAR FIELDS FOR GENERATIONS OF PARTICLES

It is shown that an integral corresponding to the contribution of one particle to equal-time commutator of quantized scalar fields diverges in a reality, contrary to usual assumption that this integral vanishes. It means that commutator of scalar fields does not vanish for space-like intervals between the field coordinates. In relation with this divergence the generalization of the Klein-Gordon equation is considered. The generalized equation is presented as products of the operators for the Klein-Gordon equation with different masses. The solutions of derived homogeneous equations are sums of fields, corresponding to particles with the same values of the spin, the electric charge, the parities, but with different masses. Such particles are grouped into the kinds (or families, or dynasties) with members which are the particle generations. The commutator of fields for the kinds of particles can be presented as sum of the products of the commutators for one particle and the definite coefficients. The sums of these coefficients for all the generation equal zero. The sums of the products of these coefficients and the particle masses to some powers equal zero too, i.e., for these coefficients some relations exist. In consequence of these relations the commutators of the fields for the particle generations vanish on space-like intervals. Thus, the locality (the microcausality) is valid for the fields of the particle kinds. It is possible if the number of the generations is greater than two.

λ-Deformations of left–right asymmetric CFTs

We compute the all-loop anomalous dimensions of current and primary field operators in deformed current algebra theories based on a general semi-simple group, but with different (large) levels for the left and right sectors. These theories, unlike their equal level counterparts, possess a new non-trivial fixed point in the IR. By computing the exact in λ two- and three-point functions for these operators we deduce their OPEs and their equal-time commutators. Using these we argue on the nature of the CFT at the IR fixed point. The associated to the currents Poisson brackets are a two-parameter deformation of the canonical structure of the isotropic PCM.

All-loop correlators of integrable λ-deformed σ-models

We compute the 2- and 3-point functions of currents and primary fields of λ-deformed integrable σ-models characterized also by an integer k. Our results apply for any semisimple group G, for all values of the deformation parameter λ and up to order 1/k. We deduce the OPEs and equal-time commutators of all currents and primaries. We derive the currents' Poisson brackets which assume Rajeev's deformation of the canonical structure of the isotropic PCM, the underlying structure of the integrable λ-deformed σ-models. We also present analogous results in two limiting cases of special interest, namely for the non-Abelian T-dual of the PCM and for the pseudodual model.

The Epstein–Glaser causal approach to the light-front [formula omitted]. I: Free theory

In this work we present the study of light-front field theories in the realm of the axiomatic theory. It is known that when one uses the light-cone gauge pathological poles (k+)−n arises, demanding a prescription to be employed in order to tame these ill-defined poles and to have the correct Feynman integrals due to the lack of Wick rotation in such theories. In order to shed a new light on this long standing problem we present here a discussion based on the use of rigorous mathematical machinery of the distributional theory combined with physical concepts, such as causality, to show how to deal with these singular propagators in a general fashion without making use of any prescription. The first step of our development will consist in showing how the analytic representation for propagators arises by requiring general physical properties within the framework of Wightman’s formalism. From that we shall determine the equal-time (anti)commutation relations in the light-front form for the scalar and fermionic fields, as well as for the dynamical components of the electromagnetic field. In conclusion, we introduce the Epstein–Glaser causal method in order to have a mathematical rigorous description of the free propagators of the theory, allowing us to discuss a general treatment for propagators of the type (k+)−n. Afterwards, we show that at given conditions our results reproduce known prescriptions in the literature.

Gauge fields as Goldstone bosons triggered by spontaneously broken supersymmetry

The emergent gauge theories are reconsidered in light of supersymmetry and an appropriate emergence conjecture is formulated. Accordingly, it might be expected that only global symmetries are fundamental symmetries of nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of the underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. We further argue that this breaking, taken in the form of the nonlinear $\ensuremath{\sigma}$-model-type pattern for vector fields or superfields, puts essential restrictions on geometrical degrees of freedom of a physical field system that makes it adjust itself in such a way that its global internal symmetry $G$ turns into the local symmetry ${G}_{\text{loc}}$. Otherwise, a given field system could lose too many degrees of freedom, thus getting unphysical, which would make it impossible to set the required initial conditions in an appropriate Cauchy problem, or to choose self-consistent equal-time commutation relations in quantum theory. Remarkably, this emergence process may naturally be triggered by supersymmetry, as is illustrated in detail by an example of a general supersymmetric QED model which is then extended to the Standard Model and GUTs. The requirement of vacuum stability in such a class of models makes both Lorentz invariance and supersymmetry become spontaneously broken in the visible sector. As a consequence, the massless photon and other gauge bosons appear as the corresponding Goldstone and pseudo-Goldstone zero modes and special local invariance is simultaneously generated. Due to this invariance, all possible Lorentz violations turn out to be completely canceled out among themselves. However, broken supersymmetry effects related to the existence of a light pseudo-Goldstino (being essentially a photino) are still left in the theory. It typically appears in the low-energy particle spectrum as the eV-scale stable lightest supersymmetric particle or the electroweak-scale long-lived next-to-lightest supersymmetric particle, and in both cases it is accompanied by a very light gravitino that could be considered as some observational signature in favor of emergent supersymmetric theories.

VACUUM STABILITY OF THE ${\mathcal{PT}}$-SYMMETRIC (-ϕ4) SCALAR FIELD THEORY

In this paper, we study the vacuum stability of the classical unstable (-ϕ4) scalar field potential. Regarding this, we obtained the effective potential, up to second-order in the coupling, for the theory in 1+1 and 2+1 space–time dimensions. We found that the obtained effective potential is bounded-from-below, which proves the vacuum stability of the theory in space–time dimensions higher than the previously studied 0+1 case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the (-ϕ4) theory at the quantum mechanical level while our work extends the argument to the level of field quantization.

De Sitter QED in Coulomb gauge: First order transition amplitudes

We construct the de Sitter QED in Coulomb gauge assuming that the quantum modes are prepared by a global apparatus which is able to determine a stable and invariant vacuum state, independent of the local coordinates. Then we proceed in a traditional manner, postulating the appropriate equal-time commutators and anticommutators of the interacting fields and deriving the perturbation expansion of the scattering operator. In this approach, we consider the in--out transitions amplitudes of the QED on the de Sitter expanding universe. We define the in and out fields as asymptotic-free fields, without electromagnetic interaction but remaining coupled to the de Sitter gravity, assuming that these are measured by the same global apparatus. Then the transition amplitudes can be calculated exactly by using the reduction formalism and the perturbation procedure as in the flat case but with significant differences due to the de Sitter gravity. A specific feature is that this eliminates the constraints due to the simultaneous momentum-energy conservation giving rise to QED transitions with nonvanishing amplitudes even in the first order of perturbations. Of a special interest could be the first order amplitudes of the electromagnetic particle creation allowed by the expansion of the de Sitter universe. We show that this effect is significant only in the very strong gravity of the early universe.

Källén-Lehmann representation for Lorentz-violating field theory

We consider field-theoretic models, one consisting purely of scalars, the other also involving fermions, that couple to a set of constant background coupling coefficients transforming as a symmetric observer Lorentz two-tensor. We show that the exact propagators can be cast in the form of a K\"all\'en-Lehmann representation. We work out the resulting form of the Feynman propagators and the equal-time field commutators, and derive sum rules for the spectral density functions.

Gauging away physics

We consider the recent argument by Higuchi et al(2011 Class. Quantum Grav. 28 245012) that a nonlocal gauge transformation can be used to eliminate the infrared divergence of the graviton propagator, when evaluated in Bunch–Davies vacuum on the open coordinate submanifold of de Sitter space in the transverse-traceless-synchronous gauge. Because the transformation is not local, the equal time commutator of undifferentiated fields no longer vanishes. From explicit examination of the Wightman function we demonstrate that the transformation adds anti-sources in the far future which cancel the bad infrared behavior but also change the propagator equation. The same problem exists in the localized version of the recent argument. Adding such anti-sources does not seem to be legitimate and could be used to eliminate the infrared divergence of the massless, minimally coupled scalar. The addition of such anti-sources in flat space QED could be effected by an almost identical gauge transformation, and would seem to eliminate the well-known infrared divergences which occur in loop corrections to exclusive amplitudes.

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.