Quantum field theory |
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In the terminology of quantum field theory, a ghost, ghost field, ghost particle, or gauge ghost is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge invariance in theories where the local fields exceed a number of physical degrees of freedom.
If a given theory is self-consistent by the introduction of ghosts, these states are labeled "good". Good ghosts are virtual particles that are introduced for regularization, like Faddeev–Popov ghosts. Otherwise, "bad" ghosts admit undesired non-virtual states in a theory, like Pauli–Villars ghosts that introduce particles with negative kinetic energy.
An example of the need of ghost fields is the photon, which is usually described by a four component vector potential Aμ, even if light has only two allowed polarizations in the vacuum. To remove the unphysical degrees of freedom, it is necessary to enforce some restrictions; one way to do this reduction is to introduce some ghost field in the theory. While it is not always necessary to add ghosts to quantize the electromagnetic field, ghost fields are strictly needed to consistently and rigorously quantize non-Abelian Yang–Mills theory, such as done with BRST quantization. [1] [2]
A field with a negative ghost number (the number of ghosts excitations in the field) is called an anti-ghost.
Faddeev–Popov ghosts are extraneous anticommuting fields which are introduced to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov. [3] [4]
Goldstone bosons are sometimes referred to as ghosts, mainly when speaking about the vanishing bosons of the spontaneous symmetry breaking of the electroweak symmetry through the Higgs mechanism. These good ghosts are artifacts of gauge fixing. The longitudinal polarization components of the W and Z bosons correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2) ⊗ U(1), which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons, or ghosts, are "eaten" by the three gauge bosons (W± and Z) corresponding to the three broken generators; this gives these three gauge bosons a mass, and the associated necessary third polarization degree of freedom. [5]
"Bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm, [6] or fields with the wrong sign of the kinetic term, such as Pauli–Villars ghosts, whose existence allows the probabilities to be negative thus violating unitarity. [7]
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A ghost condensate is a speculative proposal in which a ghost, an excitation of a field with a wrong sign of the kinetic term, acquires a vacuum expectation value. This phenomenon breaks Lorentz invariance spontaneously. Around the new vacuum state, all excitations have a positive norm, and therefore the probabilities are positive definite.
We have a real scalar field φ with the following action
where a and b are positive constants and
The theories of ghost condensate predict specific non-Gaussianities of the cosmic microwave background. These theories have been proposed by Nima Arkani-Hamed, Markus Luty, and others. [8]
Unfortunately, this theory allows for superluminal propagation of information in some cases and has no lower bound on its energy. This model doesn't admit a Hamiltonian formulation (the Legendre transform is multi-valued because the momentum function isn't convex) because it is acausal. Quantizing this theory leads to problems.
The Landau pole is sometimes referred as the Landau ghost. Named after Lev Landau, this ghost is an inconsistency in the renormalization procedure in which there is no asymptotic freedom at large energy scales. [9]
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on quantum field theory.
The Standard Model of particle physics is the theory describing three of the four known fundamental forces in the universe and classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists worldwide, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, proof of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.
String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators, that give a Feynman diagram-like expansion for string scattering amplitudes. In most string field theories, this expansion is encoded by a classical action found by second-quantizing the free string and adding interaction terms. As is usually the case in second quantization, a classical field configuration of the second-quantized theory is given by a wave function in the original theory. In the case of string field theory, this implies that a classical configuration, usually called the string field, is given by an element of the free string Fock space.
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in particle physics within the context of the BCS superconductivity mechanism, and subsequently elucidated by Jeffrey Goldstone, and systematically generalized in the context of quantum field theory. In condensed matter physics such bosons are quasiparticles and are known as Anderson–Bogoliubov modes.
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W+, W−, and Z0 bosons actually have relatively large masses of around 80 GeV/c2. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) which permeates all of space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature. The view of the Higgs mechanism as involving spontaneous symmetry breaking of a gauge symmetry is technically incorrect since by Elitzur's theorem gauge symmetries can never be spontaneously broken. Rather, the Fröhlich–Morchio–Strocchi mechanism reformulates the Higgs mechanism in an entirely gauge invariant way, generally leading to the same results.
In physics, Faddeev–Popov ghosts are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov.
Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special unitary group SU(n), or more generally any compact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of the understanding of the Standard Model of particle physics.
In quantum field theory, the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
In particle physics, little Higgs models are based on the idea that the Higgs boson is a pseudo-Goldstone boson arising from some global symmetry breaking at a TeV energy scale. The goal of little Higgs models is to use the spontaneous breaking of such approximate global symmetries to stabilize the mass of the Higgs boson(s) responsible for electroweak symmetry breaking.
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a certain transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.
In gauge theory, especially in non-abelian gauge theories, global problems at gauge fixing are often encountered. Gauge fixing means choosing a representative from each gauge orbit, that is, choosing a section of a fiber bundle. The space of representatives is a submanifold and represents the gauge fixing condition. Ideally, every gauge orbit will intersect this submanifold once and only once. Unfortunately, this is often impossible globally for non-abelian gauge theories because of topological obstructions and the best that can be done is make this condition true locally. A gauge fixing submanifold may not intersect a gauge orbit at all or it may intersect it more than once. The difficulty arises because the gauge fixing condition is usually specified as a differential equation of some sort, e.g. that a divergence vanish. The solutions to this equation may end up specifying multiple sections, or perhaps none at all. This is called a Gribov ambiguity.
In theoretical physics, the BRST formalism, or BRST quantization denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson with zero spin, even (positive) parity, no electric charge, and no colour charge that couples to mass. It is also very unstable, decaying into other particles almost immediately upon generation.
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, do not change under local transformations according to certain smooth families of operations. Formally, the Lagrangian is invariant.
The 1964 PRL symmetry breaking papers were written by three teams who proposed related but different approaches to explain how mass could arise in local gauge theories. These three papers were written by: Robert Brout and François Englert; Peter Higgs; and Gerald Guralnik, C. Richard Hagen, and Tom Kibble (GHK). They are credited with the theory of the Higgs mechanism and the prediction of the Higgs field and Higgs boson. Together, these provide a theoretical means by which Goldstone's theorem can be avoided. They showed how gauge bosons can acquire non-zero masses as a result of spontaneous symmetry breaking within gauge invariant models of the universe.
In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gauge symmetries of the theory, and which remains valid after renormalization.
This page is a glossary of terms in string theory, including related areas such as supergravity, supersymmetry, and high energy physics.
Daya Shankar Kulshreshtha is an Indian theoretical physicist, specializing in formal aspects of quantum field theory, string theory, supersymmetry, supergravity and superstring theory, Dirac's instant-form and light-front quantization of field theories and D-brane actions. His work on the models of gravity focuses on the studies of charged compact boson stars and boson shells.
In string theory, the Ramond–Neveu–Schwarz (RNS) formalism is an approach to formulating superstrings in which the worldsheet has explicit superconformal invariance but spacetime supersymmetry is hidden, in contrast to the Green–Schwarz formalism where the latter is explicit. It was originally developed by Pierre Ramond, André Neveu and John Schwarz in the RNS model in 1971, which gives rise to type II string theories and can also give type I string theory. Heterotic string theories can also be acquired through this formalism by using a different worldsheet action. There are various ways to quantize the string within this framework including light-cone quantization, old canonical quantization, and BRST quantization. A consistent string theory is only acquired if the spectrum of states is restricted through a procedure known as a GSO projection, with this projection being automatically present in the Green–Schwarz formalism.