Free Dirac Field Research Articles

Non-abelian Gauge Symmetry for Fields in Phase Space: a Realization of the Seiberg-Witten Non-abelian Gauge Theory

The Seiberg-Witten formalism has been realized as an electrodynamics in phase space (associated to the Dirac equation written in phase space) and this fact is explored here with non-abelian gauge group. First, a physically heuristic presentation of the Seiberg-Witten approach is carried out for non-abelian gauge in order to guide the calculation procedures. These results are realized by starting with the Lagrangian density for the free Dirac field in phase space. Then a field strength is derived, where the non-abelian gauge group is the SU(2), corresponding to an isospin (non-abelian) field theory in phase space. An application to nucleon is then discussed.

Background potentials and superselection sectors

The basic aspects of the Aharonov–Bohm effect can be summarized by the remark that wavefunctions become sections of a line bundle with a flat connection (that is, a ”flat potential”). Passing at the level of quantum field theory in curved spacetimes, we study the Dirac field interacting with a classical (background) flat potential and show that it can be interpreted as a topological sector of the observable net of free Dirac field. On the converse, starting from a topological sector we reconstruct a classical flat potential, interpreted as an interaction of a Dirac field. This leads to a description of Aharonov–Bohm-type effects in terms of localized observables.

The role of acceleration and vorticity in relativistic hydrodynamics

The role of vorticity and acceleration in relativistic hydrodynamics is investigated. Particular attention is paid to the effects of vorticity, as well as mixed effects, when the medium has both acceleration and vorticity. Quantumfield corrections to the energy density of free Dirac fields are calculated using the Zubarev density operator. The corresponding nonperturbative formulas are proposed and justified in the particular cases of parallel vorticity and acceleration and zero acceleration.

General thermodynamic equilibrium with axial chemical potential for the free Dirac field

We calculate the constitutive equations of the stress-energy tensor and the currents of the free massless Dirac field at thermodynamic equilibrium with acceleration and rotation and a conserved axial charge by using the density operator approach. We carry out an expansion in thermal vorticity to the second order with finite axial chemical potential μA. The obtained coefficients of the expansion are expressed as correlators of angular momenta and boost operators with the currents. We confirm previous observations that the axial chemical potential induces non-vanishing components of the stress-energy tensor at first order in thermal vorticity due to breaking of parity invariance of the density operator with μA ≠ 0. The appearance of these components might play an important role in chiral hydrodynamics.

Circuit complexity for free fermions

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.

What Spatial Geometries do (2+1)-Dimensional Quantum Field Theory Vacua Prefer?

We consider relativistic (2+1)-dimensional quantum field theories (QFTs) on a product of time with a two-space and study the vacuum free energy as a functional of the temperature and spatial geometry. We focus on free scalar and Dirac fields on arbitrary perturbations of flat space, finding that the free energy difference from flat space is finite and always negative to leading order in the perturbation. Thus, free (2+1)-dimensional QFTs appear to always energetically favor a crumpled space on all scales; this is true both as a purely quantum effect at zero temperature and as a purely thermal effect at high temperature. Importantly, we show that this quantum effect is non-negligible for the relativistic Dirac degrees of freedom on monolayer graphene even at room temperature, so we argue that this vacuum energy effect should be included for a proper analysis of the equilibrium configuration of graphene or similar materials.

General equilibrium second-order hydrodynamic coefficients for free quantum fields

We present a systematic calculation of the corrections of the stress-energy tensor and currents of the free boson and Dirac fields up to second order in thermal vorticity, which is relevant for relativistic hydrodynamics. These corrections are non-dissipative because they survive at general thermodynamic equilibrium with non vanishing mean values of the conserved generators of the Lorentz group, i.e. angular momenta and boosts. Their equilibrium nature makes it possible to express the relevant coefficients by means of correlators of the angular-momentum and boost operators with stress-energy tensor and current, thus making simpler to determine their so-called “Kubo formulae”. We show that, at least for free fields, the corrections are of quantum origin and we study several limiting cases and compare our results with previous calculations. We find that the axial current of the free Dirac field receives corrections proportional to the vorticity independently of the anomalous term.

Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions

We study the Fock quantization of a free Dirac field in 2+1-dimensional backgrounds which are conformally ultrastatic, with a time-dependent conformal factor. As it is typical for field theories, there is an infinite ambiguity in the Fock representation of the canonical anticommutation relations. Different choices may lead to unitarily inequivalent theories that describe different physics. To remove this ambiguity one usually requires that the vacuum be invariant under the unitary transformations that implement the symmetries of the equations of motion. However, in non-stationary backgrounds, where time translation is not a symmetry transformation, the requirement of vacuum invariance is in general not enough to fix completely the Fock representation. We show that this problem is overcome in the considered scenario by demanding, in addition, a unitarily implementable quantum dynamics. The combined imposition of these conditions selects a unique family of equivalent Fock representations. Moreover, one also obtains an essentially unique splitting of the time variation of the Dirac field into an explicit dependence on the background scale factor and a quantum evolution of the corresponding creation and annihilation operators.

Local temperatures and local terms in modular Hamiltonians

We show there are analogues to the Unruh temperature that can be defined for any quantum field theory and region of the space. These local temperatures are defined using relative entropy with localized excitations. We show important restrictions arise from relative entropy inequalities and causal propagation between Cauchy surfaces. These suggest a large amount of universality for local temperatures, specially the ones affecting null directions. For regions with any number of intervals in two space-time dimensions the local temperatures might arise from a term in the modular Hamiltonian proportional to the stress tensor. We argue this term might be universal, with a coefficient that is the same for any theory, and check analytically and numerically this is the case for free massive scalar and Dirac fields. In dimensions $d\ge 3$ the local terms in the modular Hamiltonian producing these local temperatures cannot be formed exclusively from the stress tensor. For a free scalar field we classify the structure of the local terms.

Vacuum state of the Dirac field in de Sitter space and entanglement entropy

We compute the entanglement entropy of a free massive Dirac field between two causally disconnected open charts in de Sitter space. We first derive the Bunch-Davies vacuum mode functions of the Dirac field. We find there exists no supercurvature mode for the Dirac field. We then give the Bogoliubov transformation between the Bunch-Davies vacuum and the open chart vacua that makes the reduced density matrix diagonal. We find that the Dirac field becomes more entangled than a scalar field as m2/H2 becomes small, and the difference is maximal in the massless limit.

The Unruh effect interpreted as a quantum noise channel

We make use of the tools of quantum information theory to shed light on the Unruh effect. A modal qubit appears as if subjected to quantum noise that degrades quantum information, as observed in the accelerated reference frame. The Unruh effect experienced by a mode of a free Dirac field, as seen by a relativistically accelerated observer, is treated as a noise channel, which we term the Unruh channel. We characterize this channel by providing its operator-sum representation, and study various facets of quantum correlations, such as, Bell inequality violations, entanglement, teleportation and measurement-induced decoherence under the effect. We compare and contrast this channel from conventional noise due to environmental decoherence. We show that the Unruh effect produces an amplitude-damping-like channel, associated with zero temperature, even though the Unruh effect is associated with a non-zero temperature. Asymptotically, the Bloch sphere subjected to the channel does not converge to a point, as would be expected by fluctuation-dissipation arguments, but contracts by a finite factor. We construct for the Unruh effect the inverse channel, a non-completely-positive map, that formally reverses the effect, and offer some physical interpretation.

Nonlocality and Multipartite Entanglement in Asymptotically Flat Space-Times

We study the Bell's inequality and multipartite entanglement generation for initially maximally entangled states of free Dirac field in a non inertial frame and asymptotically flat Robertson–Walker space-time. For two qubit case, we show that the Bell's inequality always is violated as measured by the accelerated observers which are in the causally connected regions. On the other hand, for those observers in the causally disconnected regions inequality is not violated for any values of acceleration. The generated three qubit state from two qubit state due to acceleration of one parties has a zero 3-tangle. For a three qubit state, the inequality violated for measurements done by both causally connected and disconnected observers. Initially GHZ state with non zero 3-tangle, in accelerated frame, transformed to a four qubit state with vanishing 4-tangle value. On the other hand, for a W-state with zero 3-tangle, in non inertial frame, transformed to a four qubit state with a non-zero 4-tangle acceleration dependent. In an expanding space-time with asymptotically flat regions, for an initially maximally entangled state, the maximum value of violation of Bell's inequality in the far past decreased in the far future due to cosmological particle creation. For some initially maximally entangled states, the generated four qubit state due to expansion of space-time, has non vanishing 4-tangle.

Pure quasifree states of the Dirac field from the fermionic projector

We consider the quantized free massive Dirac field on oriented and globally hyperbolic ultrastatic slab spacetimes with compact spatial section and demonstrate how a gauge invariant, pure and quasifree state on the C*-completion of the self-dual CAR-algebra can be extracted from the fermionic projector construction of Finster and Reintjes. This state is analogous to the ‘SJ-state’ of the free scalar field recently discussed in the literature. We prove that this state generically fails to be Hadamard. However, we also show how a modified version of the construction, inspired by the work of Brum and Fredenhagen, yields states that are Hadamard. We also relate the Hadamard condition to the finiteness of fluctuations of Wick polynomials.

Entanglement entropy as a witness of the Aharonov–Bohm effect in QFT

We study the dependence of the entanglement entropy with a magnetic flux and show that the former quantity witnesses an Aharonov–Bohm-like effect. In particular, we consider free charged scalar and Dirac fields living on a two-dimensional cylinder and study how the entanglement entropy for a strip-like region on the surface of the cylinder is affected by a magnetic field enclosed by it.

Probing the entanglement distillability responses to the Unruh effect and prepared states

How the Unruh---Hawking effect and prepared states influence the entanglement distillability of a free Dirac field is investigated by using the Werner and Horodecki states. It is found that Werner and Horodecki states will be converted from distillable into the separate in the noninertial frame. The parameter $$\alpha $$? of the generic entangled states has a different effect on Werner-like and Horodecki-like states. For the Werner-like states, although the parameter $$\alpha $$? influences the entanglement, it does not change the range of the parameter $$F$$F where the entanglement distillability of the Werner-like states is possible. For the Horodecki-like states, the parameter $$\alpha $$? not only influences the entanglement but also changes the range of the parameter $$P$$P where the entanglement distillability of Horodecki-like states can be achieved.

Hawking Effect for a Toy Model of Interacting Fermions

We consider a toy model of interacting Dirac fermions in a 1 + 1 dimensional space time describing the exterior of a star collapsing to a black hole. In this situation, we give a rigorous proof of the Hawking effect, namely that under the associated quantum evolution, an initial vacuum state will converge when t → + ∞ to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null characteristics and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.

ERRATUM: CONFINEMENT AND QUANTUM ANOMALY IN QUASI-1D SPINLESS FERMION CHAINS

We consider the field renormalization group (RG) of two coupled one-spatial dimension (1D) spinless fermion chains under intraforward, interforward, interbackscattering and interumklapp interactions until two-loops order. Up to this order, we demonstrate the quantum confinement in the RG flow, where the interband chiral Fermi points reduce to single chiral Fermi points and the renormalized interaction couplings have Luttinger liquid fixed points. We show that this quasi-1D system is equivalently described in terms of one and two-color interactions and address the problem of quantum anomaly, inherent to this system, as a direct consequence of coupling 1+1 free Dirac fields to one and two-color interactions.

CONFINEMENT AND QUANTUM ANOMALY IN QUASI-1D SPINLESS FERMION CHAINS

We consider the field renormalization group (RG) of two coupled one-spatial dimension (1D) spinless fermion chains under intraforward, interforward, interbackscattering and interumklapp interactions until two-loops order. Up to this order, we demonstrate the quantum confinement in the RG flow, where the interband chiral Fermi points reduce to single chiral Fermi points and the renormalized interaction couplings have Luttinger liquid fixed points. We show that this quasi-1D system is equivalently described in terms of one- and two-color interactions and address the problem of quantum anomaly, inherent to this system, as a direct consequence of coupling 1+1 free Dirac fields to one- and two-color interactions.

Asymptotic states and the definition of the S-matrix in quantum gravity

Viewing gravitational energy–momentum as equal by observation, but different in essence from inertial energy–momentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space M4. The generalized asymptotic free scalar, Dirac and gauge fields in that theory are canonically quantized, the Fock spaces of stationary states are constructed and the gravitational limit—mapping the gravitational energy–momentum onto the inertial energy–momentum to account for their observed equality—is introduced. Next the S-matrix in quantum gravity is defined as the gravitational limit of the transition amplitudes of asymptotic in- to out-states in the gauge theory of volume-preserving diffeomorphisms. The so-defined S-matrix relates in- and out-states of observable particles carrying gravitational equal to inertial energy–momentum. Finally, generalized Lehmann–Symanzik–Zimmermann reduction formulae for scalar, Dirac and gauge fields are established which allow us to express S-matrix elements as the gravitational limit of truncated Fourier-transformed vacuum expectation values of time-ordered products of field operators of the interacting theory. Together with the generating functional of the latter established in Wiesendanger (2011 arXiv:1103.1012) any transition amplitude can in principle be computed consistently to any order in perturbative quantum gravity.

ESTENSIONE DELLA TEORIA QUANTISTICA DI CAMPO A TEORIA DI AUTOMA CELLULARE QUANTISTICO

In the present note I will briefly report recent results on the novel field theory based on the information-theoretical paradigm. This program continues the previous one on the axiomatic derivation of quantum theory from information-theoretical principles, of which I presented a note on January 31st 2008. The passage from the quantum theory of abstract systems to the theory of fields is achieved by adding to the axioms of quantum theory new general principles for the network of interation among a denumerable set of quantum systems. Such principles can be synthesized with the requirement of minimal complexity of the quantum algorithm describing the physical law. From general principles such as homogeneity, isotropy, locality, linearity, and unitarity of the interaction network, along with minimal dimension of the field vector, one derives the free quantum field theory, namely Weyl, Dirac, and Maxwell fields. The principles lead to a description in terms of a quantum cellular automaton on the Cayley graph of a group. The advantages of the new theory are numerous. For example the theory tautologically solves all the problems originating from the continuum, such as the ultraviolet divergencies, the localization problem, and causality violations. The mechanics itself is not assumed, but it is a consequence of the principles, and the theory is quantum ab initio, namely it does not need quantization of a classical field theory. Lorentz covariance is not assumed, but follows itself from the principles, in the relativistic limit of small wavevectors. The notion of reference system is restated in terms of irreducible representation of the quantum automaton. Finally, the new theory, derived from purely mathematical principles–whence with adimensional variables–contains itself the units of measure for distance, time, and mass, which are given in terms of extremal values of the variables, which are in principle experimentally detectable. Also effects with GR flavor emerge from the automaton theory, such as an upper bound for the Dirac particle rest-mass (from the unitariety condition), where the dispersion relation becomes completely flat, in strict analogy with the notion of mini black hole.