Gauge theories

From Scholarpedia

This article has not been peer-reviewed or accepted for publication yet; It may be unfinished, contain inaccuracies, or unapproved changes.

Author: Dr. Gerard ′t Hooft, Institute for Theoretical Physics Universiteit Utrecht

Dr. Gerard 't Hooft accepted the invitation on 19 June 2008 (self-imposed deadline: 19 October 2008).

                   Note: this is a PRELIMINARY VERSION.

Gauge theories refers to a quite general class of quantum field theories used for the description of elementary particles and their interactions. The theories are characterized by the presence of vector fields, and as such are a generalization of the older theory of Quantum Electrodynamics (QED) that is used to describe the electromagnetic interactions of charged elementary particles with spin 1/2. Local gauge invariance is a very central issue. An important feature is that these theories are often renormalizable when used in 3 space- and 1 time dimension.

[edit]

1. Maxwell's equations and gauge invariance

The simplest example of a gauge theory is electrodynamics, as described by the Maxwell equations. The electric field strength $\vec E(\vec x,t)$ and the magnetic field strength $\vec B(\vec x,t)$ obey the homogeneous Maxwell equations (in SI units):

(1)

$\vec\nabla\times\vec E+{\partial \vec B\over\partial t}=0$

(2)

$\vec\nabla\cdot\vec B=0$ .

According to Poincaré's Lemma, Eq. (2) implies that there exists another vector field $\vec A(\vec x,t)$ such that

(3)

$\vec B=\vec\nabla\times \vec A$ .

Since Eq. (1) now reads

(4)

$\vec\nabla\times(\vec E+{\partial\vec A\over\partial t})=0$ ,

we can also conclude that there is a potential field $\Phi(\vec x,t)$ such that

(5)

$\vec E=-\vec\nabla\Phi-{\partial \vec A\over\partial t}$ .

The field $\Phi$ is the electric potential field; the vector field $\vec A$ is called the vector potential field. The strengths of these potential fields are determined by the inhomogeneous Maxwell equations, which are the equations that relate the strengths of the electromagnetic fields to the electric charges and currents that generate these fields. The use of potential fields often simplifies the problem of solving Maxwell's equations.

What turns this theory into a gauge theory is the fact that the values of these potential fields are not completely determined by Maxwell's equations. Consider an electromagnetic field configuration $(\vec E(\vec x,t),\,\vec B(\vec x,t))$ , and suppose that it is described by the potential fields $(\Phi(\vec x,t),\,\vec A(\vec x,t))$ . Then, using any arbitrary scalar function $\Lambda(\vec x,t)$ , one can find a different set of potential fields describing the same electric and magnetic fields, by writing

(6)

$\Phi'=\Phi+{\partial\Lambda\over\partial t}\ ,\quad\vec A'=\vec A-\vec\nabla\Lambda$ .

Inspecting Equations (3) and (5), one easily observes that $\vec E=\vec E'$ and $\vec B=\vec B'$ . Thus, the set ( $\Phi',\,\vec A'$ ) and ( $\Phi,\,\vec A$ ) describe the same physical situation. Because of this, we call the transformation (6) a gauge transformation. Since $\Lambda$ may be chosen to be an arbitrary function of the points $(\vec x,t)$ in space-time, we speak of a local gauge transformation. The fact that the electromagnetic fields are invariant under these local gauge transformations turns Maxwell's theory into a gauge theory.

In relativistic quantum field theory, the field $\psi(\vec x,t)$ of a non-interacting spinless particle would typically obey the equation

(7)

$(\vec\nabla^2-{\partial^2\over\partial t^2})\psi=m^2\psi$ .

This gives the dispersion relation between energy and momentum as dictated by Special Relativity:

(8)

$E=\sqrt{{\vec p}^{\,2}+m^2}$ ,

in units where the velocity of light $c=1$ , and Planck's constant $\hbar=1$ .

Suppose now that the particle in question carries an electric charge $q$ . How is its equation then affected by the presence of electro-magnetic fields? It turns out that one cannot write the correct equations using the fields $\vec E$ and $\vec B$ directly. Here, one can only choose to add terms depending on the (vector) potential fields instead:

(9)

$(\vec\nabla-iq\vec A)^2\psi-({\partial\over\partial t}+iq\Phi)^2\psi=m^2\psi$ .

It can be verified that this equation correctly produces waves that are deflected by the electro-magnetic forces in the way one expects. For instance, the energy $E$ is easily seen to be enhanced by an amount $q\,\Phi(\vec x,t)$ , which is the potential energy of a charged particle in an electric potential field.

However, what happens to this equation when performing a gauge transformation? It appears as if the equation changes, so that the solution for the field $\psi$ should change as well. Indeed, $\psi$ changes in the following way:

(10)

$\psi'=\psi e^{-iq\Lambda}\ ,\quad {\partial\psi'\over\partial t}=({\partial\psi\over\partial t}-iq\psi{\partial\Lambda\over\partial t})e^{-iq\Lambda}$ .

The combinations

(11)

$\vec D\psi=(\vec\nabla-iq\vec A)\psi \quad,\quad D_t\psi=({\partial\over\partial t}+iq\Phi)\psi$ ,

Figure 1: Feynman diagram for electron emitting a photon.

are called covariant derivatives, because they are chosen in such a way that the derivatives of the function $\Lambda(\vec x,t)$ cancel out in a gauge transformation:

(12)

$(\vec D\psi)'=(\vec\nabla-iq(\vec\nabla\Lambda)-iq\vec A+iq(\vec\nabla\Lambda)) \psi e^{-iq\Lambda}= (\vec D\psi)e^{-iq\Lambda}$ ,

and this makes it easy to see that Equation (10) correctly describes the way $\psi$ transforms under a local gauge transformation, obeying the same field equation (9) both before and after the transformation (all terms in the equation are multiplied by the same exponential $e^{-iq\Lambda}$ , so that that factor is immaterial).

The absolute value, $|\psi(\vec x,t)|^2$ does not change at all under a gauge transformation, and indeed this is the quantity that corresponds to something that is physically observable: it is the probability that a particle can be found at $(\vec x,t)$ . A rule of thumb is that local gauge invariance requires all derivatives in our equations to be replaced by covariant derivatives.

[edit]

2. Yang-Mills theory

Figure 2: Feynman diagrams for emission of Yang-Mills photons. Above: electron turning into en electron-neutrino; below: neutron turning into proton.

In the 1950s, it was known that the field equations for the field of a proton, $P(\vec x,t)$ , and the field of a neutron, $N(\vec x,t)$ , are such that one can rotate these fields in a complex two-dimensional space:

(13)

$\left({P'(\vec x,t)\atop N'(\vec x,t)}\right)=\left({a\quad b\atop c\quad d}\right)\left({P(\vec x,t)\atop N(\vec x,t)}\right)$ ,

where the matrix $U=\left({a\quad b\atop c\quad d}\right)$ may contain four arbitrary complex numbers, as long as it is unitary ( $U\,U^\dagger=I$ ), and usually, the determinant of $U$ is restricted to be 1. Since these equations resemble the rotations one can perform in ordinary space, to describe spin of a particle, the symmetry in question here was called isospin.

In 1954, C.N. Yang and R.L. Mills published a very important idea. Could one modify the equations in such a way that these isospin rotations could be regarded as local gauge rotations? this would mean that, unlike the case that was known, the matrices $U$ should be allowed to depend on space and time, just like the gauge generator $\Lambda(\vec x,t)$ in electromagnetism. Yang an Mills were also inspired by the observation that Einstein's theory of gravity, General Relativity, also allows for transformations very similar to local gauge transformations: the replacement of the coordinate frame by other coordinates in an arbitrary, space-time dependent way.

To write down field equations for protons and neutrons, one needs the derivatives of these fields. The way these derivatives transform under a local gauge transformation implies that there will be terms containing the gradients $\vec\nabla U$ of the matrices $U$ . To make the theory gauge-invariant, these gradients would have to be cancelled out, and in order to do that, Yang and Mills replaced the derivatives $\vec\nabla$ by covariant derivatives $\vec D=\vec\nabla -ig\vec A(\vec x,t)$ , as was done in electromagnetism, see Equation (11). Here, however, the fields $\vec A$ had to be matrix-valued, just as the isospin $U$ matrices. Since the $U$ matrices contain four coefficients with one constraint (the determinant has to be 1), one ends up with a set of three new vector fields. At first sight, they appear to be the fields of a vector particle with isospin one. In practice, this should correspond to particles with one unit of spin (i.e., the particle rotates about its axis), and its electric charge could be neutral or one or minus one unit. Yang-Mills theory therefore predicts and describes a new type of particles with spin one that transmit a force not unlike the electro-magnetic force.

The fields that are equivalent to Maxwell's electric and magnetic fields are obtained by considering the commutator of two covariant derivatives:

(14)

$[D_\mu,\,D_\nu]=D_\mu D_\nu-D_\nu D_\mu= -ig(\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,\,A_\nu]) = -ig F_{\mu\nu}$ ,

where the indices take the values $\mu,\ \nu=0,1,2,3$ , with 0 referring to the time-component.

Since $F_{\mu\nu}=-F_{\nu\mu}$ , this tensor has 6 independent components, three forming an electric vector field, and three a magnetic field. Each of these components is also a matrix. The commutator, $[A_\mu,\,A_\nu]$ is a new, non-linear term, which makes the Yang-Mills equations a lot more complicated than the Maxwell system.

In other respects, the Yang-Mills particles, being the energy quanta of the Yang-Mills fields, are similar to photons, the quanta of light. Yang-Mills particles also carry no intrinsic mass, and travel with the speed of light. Indeed, these features were at first reasons to dismiss this theory, because massless particles of this sort should have been detected long ago, whereas they were conspicuously absent.

[edit]

3. The Brout-Englert-Higgs mechanism

The theory was revived when it was combined with spontaneous breakdown of local gauge symmetry, also known as the Brout-Englert-Higgs mechanism. Consider a scalar (spinless) particle described by a field $\phi(\vec x,t)$ . This field is assumed to be a vector field, in the sense that it undergoes some rotation when a gauge transformation is performed. In practice this means that the particle carries one or several kinds of charges that make it sensitive to the Yang-Mills force, and often it has several components, which means there are various species of this particle. Such particles must obey Bose-Einstein statistics, which implies that it can undergo Bose-Einstein condensation. In terms of its field $\phi$ this means the following:

Figure 3: Spontaneous symmetry breaking. An object residing in a rotationally symmetric potential finds a stable, asymmetric position. In the BEH case, it is the Higgs field, $(\phi_1, \phi_2)$ that finds an asymmetric value $(F,\,0)$ .

In the vacuum the field $\phi$ takes a non-vanishing value $F$ .

This is usually written as

(15)

$\langle\phi(\vec x,t)\rangle=F$ .

After a local gauge transformation, this would look like

(16)

$\langle\phi(\vec x,t)\rangle=U(\vec x,t)\,F$ ,

where $U(\vec x,t)$ is a matrix field representing the local gauge transformation.

It is often said that, therefore, the vacuum is not gauge-invariant, but, strictly speaking, this is not correct. The situation described by Equation (16) is the same vacuum as (15); it is only described differently. However, this property of the vacuum does have important consequences. Due to the fact that the rotated field now describes the same situation as the previous value, there is no different physical particle associated to the rotated field. Only the length of the vector $\phi$ has physical significance. This length is gauge-invariant. therefore, only the length of the vector $\phi$ is associated to one type of particle, which must be neutral for the Yang-Mills forces. This particle is now called the Higgs particle.

As the Higgs field is a constant source for the Yang-Mills field strength, the Yang-Mills field equations are modified by it. Due to the Higgs field, the Yang-Mills "photons" described by the Yang-Mills field $A_\mu(\vec x,t)$ get a mass. This can also be explained as follows. Massless photons can only have two helicity states, that is, they can spin only in two directions. This is related to the fact that light can be polarized in exactly two directions. Massive photons (particles with non-vanishing mass and with one unit of spin), can always spin in three directions. This third rotation mode is now provided by the Higgs field, which itself loses several of its physical components. The total number of physical field components stays the same before and after the Higgs mechanism. A further consequence of this effect on the Yang-Mills field is that the force transmitted by the massive photons is a short-range one (the range of the force being inversely proportional to the mass of the photon).

The weak interactions could now be successfully described by a Yang-Mills theory. The set of local gauge transformations forms the mathematical group $SU(2)\times U(1)$ . This group generates 4 species of photons (3 for $SU(2)$ and 1 for $U(1)$ ). The Higgs mechanism breaks this group down in such a way that a subgroup of the form $U(1)$ remains. This is the electromagnetic theory, with just one photon. The other three photons become massive; they are responsible for the weak interactions, which in practice appear to be weak just because these forces have a very short range. With respect to electromagnetism, two of these intermediate vector bosons, $W^\pm$ , are electrically charged, and a third, $Z^0$ , is electrically neutral. When the latter's existence was derived from group theoretical arguments, this gave rise to the prediction of a hitherto unnoticed form of the weak interaction: the neutral current interaction. This theory, that combines electromagnetism and the weak force into one, is called the electro-weak theory, and it was the first fully renormalizable theory for the weak force (see Chapter 5.

[edit]

4. Quantum Chromodynamics

Figure 4: The six flavors and three colors of quarks and their antiparticles. Arrows show the weak and the strong transitions

When it was understood that the weak interactions, together with the electromagnetic ones, can be ascribed to a Yang-Mills gauge theory, the question was asked how to address the strong force, a very strong force with relatively short range of action, which controls the behavior of the hadronic particles such as the nucleons and the pions. It was understood since 1961 that these particles behave as if built from subunits, called quarks. Three varieties of quarks were known (up, down, and strange), and three more would be discovered later (charm, top, and bottom). These quarks have the peculiar property that they permanently stick together either in triplets, or one quark sticks together with one anti-quark. Yet when they approach one another very closely, they begin to behave more freely as individuals.

Figure 5: Feynman diagrams for emission of QCD gluons. Quarks change color, but their flavor stays the same: u stays u and d stays d.

These features we now understand as, again, being due to a Yang-Mills gauge theory. Here, we have the mathematical group $SU(3)$ as local gauge group, while now the symmetry is not affected by any Brout-Englert-Higgs mechanism. Due to the non-linear nature of the Yang-Mills field, it self-interacts, which forces the fields to come in patterns quite different from the electromagnetic case: vortex lines are formed, which form unbreakable bonds between quarks. At close distances, the Yang-Mills force becomes weak, and this is a feature that can be derived in an elementary way using perturbation expansions, but it is a property of the quantized Yang-Mills system that hitherto had been thought to be impossible for any quantum field theory, called asymptotic freedom. The discovery of this feature has a complicated history.

Figure 6: The quantum chromodynamical fields form vortices that keep quarks and antiquarks (left) or three-quark systems (right) permanently confined.

$SU(3)$ implies that every species of quark comes in three types, referred to as color: they are "red", "green" or "blue". The field of a quark is therefore a 3-component vector in an internal 'color' space. Yang-Mills gauge transformations rotate this vector in color space. The Yang-Mills fields themselves form 3 by 3 matrices, with one constraint (since the determinant of the Yang-Mills gauge rotation matrices must be kept equal to one). Therefore, the Yang-Mills field has 8 colored photon-like particles, called gluons. Anti-quarks carry the conjugate colors ("cyan", "magenta" or "yellow"). Quantum chromodynamics ia also a renormalizable theory.

The gluons effectively keep the quarks together in such a way that their colors add up to a total that is color-neutral ("white" or a "shade of gray"). This is why either three quarks or one quark and one anti-quark can sit together to form a physically observable particle (a hadron). This property of the theory is called permanent quark confinement. Because of the strongly non-linear nature of the fields, quark confinement is in fact quite difficult to prove, whereas the property of asymptotic freedom can be demonstrated exactly. Indeed, a mathematically air-tight demonstration of confinement, with the associated phenomenon of a 'mass gap in the theory (the absence of strictly massless hadronic objects) has not yet been given, and is the subject of a $1,000,000,- prize, issued by the Clay Mathematics Institute of Cambridge, Massachusetts.

[edit]

5. Renormalization and Anomalies

According to the laws of quantum mechanics, the energy in a field consists of energy packets, and these energy packets are in fact the particles associated to the field. Quantum mechanics gives extremely precise prescriptions on how these particles interact, as soon as the field equations are known. The theory explains not only how forces are transmitted by the exchange of particles, but it also states that multiple exchanges should occur. In many older theories, these multiple exchange gave rise to difficulties: their effects seem to be unbounded, or infinite. In a gauge theory, however, the small distance structure is very precisely prescribed by the requirement of gauge-invariance. In such a theory one can combine the infinite effects of the multiple exchanges with redefinitions of masses and charges of the particles involved. This procedure is called renormalization. In 3 space and 1 time dimension, most gauge theories are renormalizable. This allows us to compute the effects of multiple particle exchanges to high accuracy, thus allowing for detailed comparison with experimental data.

Figure 7: Feynman diagrams containing loops, due to multiple particle exchanges. The loops often generate infinite expressions.

Renormalization requires that masses and coupling strengths of particles be defined very carefully. Usually, requiring the theory to remain gauge invariant throughout leaves no ambiguity for such definitions. However, it is not obvious that unambiguous, gauge invariant definitions exist at all, since gauge invariance has to hold for all interactions, whereas only a few infinite expressions can be replaced by finite ones.

The proof that showed how and why unambiguous renormalized expressions can be obtained, could be most elegantly obtained by realizing that gauge theories can be formulated in any number of space-time dimensions. It was even possible to define all Feynman diagrams unambiguously for theories in spaces where the dimensions are $3-\epsilon$ , where $\epsilon$ is an infinitesimal quantity. Taking the limit $\epsilon\rightarrow 0$ requires the subtraction of poles of the form $C/\epsilon^n$ from the original, "bare" mass and coupling parameters. The result is a set of unique, finite and gauge invariant expressions. In practice, it was found that this procedure, called dimensional regularization and renormalization is also convenient for carrying out technically complicated calculations of loop diagrams.

Figure 8: The diagram where a fermionic particle forms a closed triangle, coupling to three gauge particles, is the main source of anomalies.

However, there is a special case where extension to dimensions different from the canonical one is impossible. This is when fermionic particles exhibit chiral symmtry. Chiral symmetry is a symmetry that distinguishes left-rotating from right rotating particles, and indeed it plays a crucial role in the Standard Model. Chiral symmetry is only possible if space is 3 dimensional, and so does not allow for dimensional renormalization. Indeed, chiral symmetry cannot be preserved. An anomaly occurs, called chiral anomaly. It was first discovered when a calculation of the $\pi_0\rightarrow\gamma\gamma$ gave answers that did not follow the expected symmetry pattern.

Since the gauge symmetries of the Standard Model do distinguish left rotating from right rotating particles (in particular, only left-rotating neutrinos are produced in a weak interaction), anomalies were a big concern. It so happens, however, that all anomalous amplitudes that would jeopardize gauge invariance and hence the self consistency of our equations, all cancel out. This is related to the fact that certain "grand unified" extensions of the Standard Model are based on anomaly free gauge groups (see Chapter 7).

The anomaly has a direct physical implication. A topologically twisted field configuration called the instanton (because it represents an event at a given instant in time), represents exactly the configuration where the anomaly is maximal. It causes a violation of the conservation of the gauge charges. In the Standard Model, it only contributes when at least one of the charges is not a gauge charge, but something like baryonic charge. Instantons here trigger the violation of the conservation laws of baryons.

[edit]

6. Standard Model

Apart from the weak force, the electromagnetic force and the strong force, there is the gravitational force acting upon elementary particles. No other elementary forces are known. At the level of individual particles, gravity is so weak that it can be ignored in most cases. Suppose now that we take the $SU(2)\times U(1)$ Yang-Mills system, together with the Higgs field, to describe electromagnetism and the weak force, and add to this the $SU(3)$ Yang-Mills theory for the strong force, and we include all known elementary matter fields, being the quarks and the leptons, with their appropriate transformation rules under a gauge transformation; suppose we add to this all possible ways these fields can mix, a feature observed experimentally, which can be accounted for as a basic type of self-interaction of the fields. Then we obtain what is called the Standard Model. It is one great gauge theory that literally represents all our present understanding of the subatomic particles and their interactions.

The Standard Model owes its strength to the fact that it is renormalizable. It has been subject of numerous experimental experiments and observations. It has withstood all these tests remarkably well. One important modification became inevitable around the early 1990's: in the leptonic sector, also the neutrinos carry a small amount of mass, and their fields mix. This was not totally unexpected, but the highly successful neutrino experiments now had made it clear that these effects are really there. They actually implied a further reinforcement of the Standard Model.

One ingredient has not yet been confirmed: the Higgs particle. Observation of this object is expected in the near future, notably by the Large Hadron Collider at CERN, Geneva. The simplest versions of the Standard model only require one single, electrically neutral Higgs particle, but the 'Higgs sector' could be more complicated: the Higgs could be much heavier than presently expected, or there could exist more than one variety, in which case also electrically charged scalar particles would be found.

The Standard Model is not perfect from a mathematical point of view. At extremely high energies (energies much higher than what can be attained today in the particle accelerators), the theory becomes unnatural. In practice, this means that we do not believe anymore that everything will happen exactly as prescribed in the theory; new phenomena are to be expected. The most popular scenario is the emergence of a new symmetry called supersymmetry, a symmetry relating bosons with fermions (particles such as electrons and quarks, which require Dirac fields for their description).

[edit]

7. Grand Unified Theories

It is natural to suspect that the electroweak forces and the strong forces should also be connected by gauge rotations. This would imply that all forces among the subatomic particles are actually related by gauge transformations. There is no direct evidence for this, but there are several circumstances that appear to point in this direction. In the present version of the Standard Model, the $SU(3)$ Yang-Mills fields, describing the strong force, indeed exhibit very large coupling strengths, whereas the $U(1)$ sector, describing the electric (and part of the weak) sector, has a tiny coupling strength. One can now use the mathematics of renormalization, in particular the so-called renormalization group, to calculate the effective strengths of these forces at much higher energies. It is found that the $SU(3)$ forces decrease in strength, due to asymptotic freedom, but that the $U(1)$ coupling strength increases. The $SU(2)$ force varies more slowly. At extremely high energies, corresponding to ultra short distance scales, around $10^{-32}$ cm, the three coupling strengths appear to approach one another, as if that is the place where the forces unite.

It was found that $SU(2)\times U(1)$ and $SU(3)$ fir quite nicely in a group called $SU(5)$ . They indeed form a subgroup of $SU(5)$ . One may then assume that a Brout-Englert-Higgs mechanism breaks this group down to a $SU(2)\times U(1)\times SU(3)$ subgroup. One obtains a so-called Grand Unified Field theory. In this theory, one assumes three generations of fermions, each transforming in the same way under $SU(5)$ transformations (mathematically, they form a $\mathbf{10}$ and a $\overline\mathbf{5}$ representation).

The $SU(5)$ theory, however, predicts that the proton can decay, extremely slowly, into leptons and pions. The decay has been searched for but not found. Also, in this model, it is not easy to account for the neutrino mass and its mixings. A better theory was found where $SU(5)$ is enlarged into $SO(10)$ . This grand unified model puts the neutrinos at the same level as the charged leptons, and it appears to be more promising.

[edit]

8. Final remarks

There are infinite many ways to construct gauge theories. It seems that the models that are most useful to describe observed elementary particles, are the relatively simple ones, based on fairly elementary mathematical groups and representations. One may wonder why Nature appears to be so simple, and whether it will stay that way when new particles and interactions are discovered.

Related subjects are Supersymmetry and Superstring theory. They are newer ideas about particle structure and particle symmetries, where gauge invariance also plays a very basic role.

Invited by:

Dr. Riccardo Guida, Institut de Physique Théorique, CEA, IPhT; CNRS, Gif-sur-Yvette, France

Retrieved from "http://www.scholarpedia.org/article/Gauge_theories"

Categories: Physics | Quantum and Statistical Field Theory | Quantum Field Theory (Foundations)

Gauge theories

From Scholarpedia

1. Maxwell's equations and gauge invariance

2. Yang-Mills theory

3. The Brout-Englert-Higgs mechanism

4. Quantum Chromodynamics

5. Renormalization and Anomalies

6. Standard Model

7. Grand Unified Theories

8. Final remarks

Views

Personal tools

Scholarpedia

Encyclopedia of

For readers

For authors

Toolbox