In the terminology of
quantum field theory
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 a ...
, a ghost, ghost field, ghost particle, or gauge ghost is an
unphysical state in a
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
. Ghosts are necessary to keep
gauge invariance in theories where the local fields exceed a number of physical
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
.
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
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are Massless particle, massless ...
, which is usually described by a four component
vector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a '' scalar potential'', which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v, a ''ve ...
, 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 when dealing with non-Abelian
Yang–Mills theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using ...
extensions to the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. I ...
.
A field with a negative ghost number (the number of ghosts excitations in the field) is called an anti-ghost.
Good ghosts
Faddeev–Popov ghosts
Faddeev–Popov ghosts are extraneous
anticommuting
In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
fields which are introduced to maintain the consistency of the
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional ...
. They are named after
Ludvig Faddeev and
Victor Popov
Victor Nikolaevich Popov (russian: Ви́ктор Никола́евич Попо́в; 27 October 1937 – 16 April 1994) was a Russian theoretical physicist known for his contribution to the quantization of non-abelian gauge fields. His work wi ...
.
Goldstone bosons
Goldstone boson
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 part ...
s are sometimes referred to as ghosts. Mainly, when speaking about the vanishing
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s of the
spontaneous symmetry breaking
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 ...
of the
electroweak symmetry
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
through the
Higgs mechanism
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 bei ...
. These ''good'' ghosts are artifacts of gauge fixing. The longitudinal polarization components of the
W and Z bosons
In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , and ...
correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
⊗
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
, which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons, or ghosts, are "eaten" by the three
gauge boson
In particle physics, a gauge boson is a bosonic elementary particle that acts as the force carrier for elementary fermions. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gau ...
s (''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.
Bad ghosts
"Bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm, 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.
Ghost particles could obtain the symmetry or break it in gauge fields. The "good ghost" particles actually obtain the symmetry by unchanging the "
gauge fixing
In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
" in a gauge transformation, while bad ghost particles break the symmetry by bringing in the
non-abelian G-matrix which does change the symmetry, and this was the main reason to introduce the gauge covariant and contravariant derivatives.
Ghost condensate
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
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
. This phenomenon breaks
Lorentz invariance spontaneously. Around the new
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used a ...
, all excitations have a positive norm, and therefore the probabilities are positive definite.
We have a real
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantit ...
φ with the following action
: