Gauge theory

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https://en.wikipedia.org/wiki/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).

The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.

Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).

Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.

Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields.

Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

Quantum field theory
Feynman diagram
History of quantum field theory

Background

Symmetries

Tools

Equations

Standard Model

Incomplete theories

Scientists

History

The earliest field theory having a gauge symmetry was Maxwell's formulation, in 1864–65, of electrodynamics ("A Dynamical Theory of the Electromagnetic Field") which stated that any vector field whose curl vanishes—and can therefore normally be written as a gradient of a function—could be added to the vector potential without affecting the magnetic field. The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived the Einstein field equations by postulating the invariance of the action under a general coordinate transformation. Later Hermann Weyl, in an attempt to unify general relativity and electromagnetism, conjectured that Eichinvarianz or invariance under the change of scale (or "gauge") might also be a local symmetry of general relativity. After the development of quantum mechanics, Weyl, Vladimir Fock and Fritz London modified gauge by replacing the scale factor with a complex quantity and turned the scale transformation into a change of phase, which is a U(1) gauge symmetry. This explained the electromagnetic field effect on the wave function of a charged quantum mechanical particle. This was the first widely recognised gauge theory, popularised by Pauli in 1941.

In 1954, attempting to resolve some of the great confusion in elementary particle physics, Chen Ning Yang and Robert Mills introduced non-abelian gauge theories as models to understand the strong interaction holding together nucleons in atomic nuclei. (Ronald Shaw, working under Abdus Salam, independently introduced the same notion in his doctoral thesis.) Generalizing the gauge invariance of electromagnetism, they attempted to construct a theory based on the action of the (non-abelian) SU(2) symmetry group on the isospin doublet of protons and neutrons. This is similar to the action of the U(1) group on the spinor fields of quantum electrodynamics. In particle physics the emphasis was on using quantized gauge theories.

This idea later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory. Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. Asymptotic freedom was believed to be an important characteristic of strong interactions. This motivated searching for a strong force gauge theory. This theory, now known as quantum chromodynamics, is a gauge theory with the action of the SU(3) group on the color triplet of quarks. The Standard Model unifies the description of electromagnetism, weak interactions and strong interactions in the language of gauge theory.

In the 1970s, Michael Atiyah began studying the mathematics of solutions to the classical Yang–Mills equations. In 1983, Atiyah's student Simon Donaldson built on this work to show that the differentiable classification of smooth 4-manifolds is very different from their classification up to homeomorphism. Michael Freedman used Donaldson's work to exhibit exotic R⁴s, that is, exotic differentiable structures on Euclidean 4-dimensional space. This led to an increasing interest in gauge theory for its own sake, independent of its successes in fundamental physics. In 1994, Edward Witten and Nathan Seiberg invented gauge-theoretic techniques based on supersymmetry that enabled the calculation of certain topological invariants (the Seiberg–Witten invariants). These contributions to mathematics from gauge theory have led to a renewed interest in this area.

The importance of gauge theories in physics is exemplified in the tremendous success of the mathematical formalism in providing a unified framework to describe the quantum field theories of electromagnetism, the weak force and the strong force. This theory, known as the Standard Model, accurately describes experimental predictions regarding three of the four fundamental forces of nature, and is a gauge theory with the gauge group SU(3) × SU(2) × U(1). Modern theories like string theory, as well as general relativity, are, in one way or another, gauge theories.

See Pickering for more about the history of gauge and quantum field theories.

Description

Global and local symmetries

Global symmetry

In physics, the mathematical description of any physical situation usually contains excess degrees of freedom; the same physical situation is equally well described by many equivalent mathematical configurations. For instance, in Newtonian dynamics, if two configurations are related by a Galilean transformation (an inertial change of reference frame) they represent the same physical situation. These transformations form a group of "symmetries" of the theory, and a physical situation corresponds not to an individual mathematical configuration but to a class of configurations related to one another by this symmetry group.

This idea can be generalized to include local as well as global symmetries, analogous to much more abstract "changes of coordinates" in a situation where there is no preferred "inertial" coordinate system that covers the entire physical system. A gauge theory is a mathematical model that has symmetries of this kind, together with a set of techniques for making physical predictions consistent with the symmetries of the model.

Example of global symmetry

When a quantity occurring in the mathematical configuration is not just a number but has some geometrical significance, such as a velocity or an axis of rotation, its representation as numbers arranged in a vector or matrix is also changed by a coordinate transformation. For instance, if one description of a pattern of fluid flow states that the fluid velocity in the neighborhood of (x=1, y=0) is 1 m/s in the positive x direction, then a description of the same situation in which the coordinate system has been rotated clockwise by 90 degrees states that the fluid velocity in the neighborhood of (x=0, y=-1) is 1 m/s in the negative y direction. The coordinate transformation has affected both the coordinate system used to identify the location of the measurement and the basis in which its value is expressed. As long as this transformation is performed globally (affecting the coordinate basis in the same way at every point), the effect on values that represent the rate of change of some quantity along some path in space and time as it passes through point P is the same as the effect on values that are truly local to P.

Local symmetry

Use of fiber bundles to describe local symmetries

In order to adequately describe physical situations in more complex theories, it is often necessary to introduce a "coordinate basis" for some of the objects of the theory that do not have this simple relationship to the coordinates used to label points in space and time. (In mathematical terms, the theory involves a fiber bundle in which the fiber at each point of the base space consists of possible coordinate bases for use when describing the values of objects at that point.) In order to spell out a mathematical configuration, one must choose a particular coordinate basis at each point (a local section of the fiber bundle) and express the values of the objects of the theory (usually "fields" in the physicist's sense) using this basis. Two such mathematical configurations are equivalent (describe the same physical situation) if they are related by a transformation of this abstract coordinate basis (a change of local section, or gauge transformation).

In most gauge theories, the set of possible transformations of the abstract gauge basis at an individual point in space and time is a finite-dimensional Lie group. The simplest such group is U(1), which appears in the modern formulation of quantum electrodynamics (QED) via its use of complex numbers. QED is generally regarded as the first, and simplest, physical gauge theory. The set of possible gauge transformations of the entire configuration of a given gauge theory also forms a group, the gauge group of the theory. An element of the gauge group can be parameterized by a smoothly varying function from the points of spacetime to the (finite-dimensional) Lie group, such that the value of the function and its derivatives at each point represents the action of the gauge transformation on the fiber over that point.

A gauge transformation with constant parameter at every point in space and time is analogous to a rigid rotation of the geometric coordinate system; it represents a global symmetry of the gauge representation. As in the case of a rigid rotation, this gauge transformation affects expressions that represent the rate of change along a path of some gauge-dependent quantity in the same way as those that represent a truly local quantity. A gauge transformation whose parameter is not a constant function is referred to as a local symmetry; its effect on expressions that involve a derivative is qualitatively different from that on expressions that don't. (This is analogous to a non-inertial change of reference frame, which can produce a Coriolis effect.)

Gauge fields

The "gauge covariant" version of a gauge theory accounts for this effect by introducing a gauge field (in mathematical language, an Ehresmann connection) and formulating all rates of change in terms of the covariant derivative with respect to this connection. The gauge field becomes an essential part of the description of a mathematical configuration. A configuration in which the gauge field can be eliminated by a gauge transformation has the property that its field strength (in mathematical language, its curvature) is zero everywhere; a gauge theory is not limited to these configurations. In other words, the distinguishing characteristic of a gauge theory is that the gauge field does not merely compensate for a poor choice of coordinate system; there is generally no gauge transformation that makes the gauge field vanish.

When analyzing the dynamics of a gauge theory, the gauge field must be treated as a dynamical variable, similar to other objects in the description of a physical situation. In addition to its interaction with other objects via the covariant derivative, the gauge field typically contributes energy in the form of a "self-energy" term. One can obtain the equations for the gauge theory by:

• starting from a naïve ansatz without the gauge field (in which the derivatives appear in a "bare" form);
• listing those global symmetries of the theory that can be characterized by a continuous parameter (generally an abstract equivalent of a rotation angle);
• computing the correction terms that result from allowing the symmetry parameter to vary from place to place; and
• reinterpreting these correction terms as couplings to one or more gauge fields, and giving these fields appropriate self-energy terms and dynamical behavior.

This is the sense in which a gauge theory "extends" a global symmetry to a local symmetry, and closely resembles the historical development of the gauge theory of gravity known as general relativity.

Physical experiments

Gauge theories used to model the results of physical experiments engage in:

• limiting the universe of possible configurations to those consistent with the information used to set up the experiment, and then
• computing the probability distribution of the possible outcomes that the experiment is designed to measure.

We cannot express the mathematical descriptions of the "setup information" and the "possible measurement outcomes", or the "boundary conditions" of the experiment, without reference to a particular coordinate system, including a choice of gauge. One assumes an adequate experiment isolated from "external" influence that is itself a gauge-dependent statement. Mishandling gauge dependence calculations in boundary conditions is a frequent source of anomalies, and approaches to anomaly avoidance classifies gauge theories.

Continuum theories

The two gauge theories mentioned above, continuum electrodynamics and general relativity, are continuum field theories. The techniques of calculation in a continuum theory implicitly assume that:

• given a completely fixed choice of gauge, the boundary conditions of an individual configuration are completely described
• given a completely fixed gauge and a complete set of boundary conditions, the least action determines a unique mathematical configuration and therefore a unique physical situation consistent with these bounds
• fixing the gauge introduces no anomalies in the calculation, due either to gauge dependence in describing partial information about boundary conditions or to incompleteness of the theory.

Determination of the likelihood of possible measurement outcomes proceed by:

• establishing a probability distribution over all physical situations determined by boundary conditions consistent with the setup information
• establishing a probability distribution of measurement outcomes for each possible physical situation
• convolving these two probability distributions to get a distribution of possible measurement outcomes consistent with the setup information

These assumptions have enough validity across a wide range of energy scales and experimental conditions to allow these theories to make accurate predictions about almost all of the phenomena encountered in daily life: light, heat, and electricity, eclipses, spaceflight, etc. They fail only at the smallest and largest scales due to omissions in the theories themselves, and when the mathematical techniques themselves break down, most notably in the case of turbulence and other chaotic phenomena.

Quantum field theories

Other than these classical continuum field theories, the most widely known gauge theories are quantum field theories, including quantum electrodynamics and the Standard Model of elementary particle physics. The starting point of a quantum field theory is much like that of its continuum analog: a gauge-covariant action integral that characterizes "allowable" physical situations according to the principle of least action. However, continuum and quantum theories differ significantly in how they handle the excess degrees of freedom represented by gauge transformations. Continuum theories, and most pedagogical treatments of the simplest quantum field theories, use a gauge fixing prescription to reduce the orbit of mathematical configurations that represent a given physical situation to a smaller orbit related by a smaller gauge group (the global symmetry group, or perhaps even the trivial group).

More sophisticated quantum field theories, in particular those that involve a non-abelian gauge group, break the gauge symmetry within the techniques of perturbation theory by introducing additional fields (the Faddeev–Popov ghosts) and counterterms motivated by anomaly cancellation, in an approach known as BRST quantization. While these concerns are in one sense highly technical, they are also closely related to the nature of measurement, the limits on knowledge of a physical situation, and the interactions between incompletely specified experimental conditions and incompletely understood physical theory. The mathematical techniques that have been developed in order to make gauge theories tractable have found many other applications, from solid-state physics and crystallography to low-dimensional topology.

Classical gauge theory

Classical electromagnetism

Historically, the first example of gauge symmetry discovered was classical electromagnetism. In electrostatics, one can either discuss the electric field, E, or its corresponding electric potential, V. Knowledge of one makes it possible to find the other, except that potentials differing by a constant, ${\displaystyle V\mapsto V+C}$, correspond to the same electric field. This is because the electric field relates to changes in the potential from one point in space to another, and the constant C would cancel out when subtracting to find the change in potential. In terms of vector calculus, the electric field is the gradient of the potential, ${\displaystyle \mathbf {E} =-\nabla V}$. Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A, with

${\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}$

The general gauge transformations now become not just ${\displaystyle V\mapsto V+C}$ but

${\displaystyle {\begin{aligned}\mathbf {A} &\mapsto \mathbf {A} +\nabla f\\V&\mapsto V-{\frac {\partial f}{\partial t}}\end{aligned}}}$

where f is any twice continuously differentiable function that depends on position and time. The fields remain the same under the gauge transformation, and therefore Maxwell's equations are still satisfied. That is, Maxwell's equations have a gauge symmetry.

An example: Scalar O(n) gauge theory

The remainder of this section requires some familiarity with classical or quantum field theory, and the use of Lagrangians.

Definitions in this section: gauge group, gauge field, interaction Lagrangian, gauge boson.

The following illustrates how local gauge invariance can be "motivated" heuristically starting from global symmetry properties, and how it leads to an interaction between originally non-interacting fields.

Consider a set of n non-interacting real scalar fields, with equal masses m. This system is described by an action that is the sum of the (usual) action for each scalar field ${\displaystyle \varphi _{i}}$

${\displaystyle {\mathcal {S}}=\int \,\mathrm {d} ^{4}x\sum _{i=1}^{n}\left[{\frac {1}{2}}\partial _{\mu }\varphi _{i}\partial ^{\mu }\varphi _{i}-{\frac {1}{2}}m^{2}\varphi _{i}^{2}\right]}$

The Lagrangian (density) can be compactly written as

${\displaystyle \ {\mathcal {L}}={\frac {1}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}\partial ^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }$

by introducing a vector of fields

${\displaystyle \ \Phi ^{\mathsf {T}}=(\varphi _{1},\varphi _{2},\ldots ,\varphi _{n})}$

The term ${\displaystyle \partial _{\mu }\Phi }$ is the partial derivative of ${\displaystyle \Phi }$ along dimension ${\displaystyle \mu }$.

It is now transparent that the Lagrangian is invariant under the transformation

${\displaystyle \ \Phi \mapsto \Phi '=G\Phi }$

whenever G is a constant matrix belonging to the n-by-n orthogonal group O(n). This is seen to preserve the Lagrangian, since the derivative of ${\displaystyle \Phi '}$ transforms identically to ${\displaystyle \Phi }$ and both quantities appear inside dot products in the Lagrangian (orthogonal transformations preserve the dot product).

${\displaystyle \ (\partial _{\mu }\Phi )\mapsto (\partial _{\mu }\Phi )'=G\partial _{\mu }\Phi }$

This characterizes the global symmetry of this particular Lagrangian, and the symmetry group is often called the gauge group; the mathematical term is structure group, especially in the theory of G-structures. Incidentally, Noether's theorem implies that invariance under this group of transformations leads to the conservation of the currents

${\displaystyle \ J_{\mu }^{a}=i\partial _{\mu }\Phi ^{\mathsf {T}}T^{a}\Phi }$

where the T^a matrices are generators of the SO(n) group. There is one conserved current for every generator.

Now, demanding that this Lagrangian should have local O(n)-invariance requires that the G matrices (which were earlier constant) should be allowed to become functions of the space-time coordinates x.

In this case, the G matrices do not "pass through" the derivatives, when G = G(x),

${\displaystyle \ \partial _{\mu }(G\Phi )\neq G(\partial _{\mu }\Phi )}$

The failure of the derivative to commute with "G" introduces an additional term (in keeping with the product rule), which spoils the invariance of the Lagrangian. In order to rectify this we define a new derivative operator such that the derivative of ${\displaystyle \Phi '}$ again transforms identically with ${\displaystyle \Phi }$

${\displaystyle \ (D_{\mu }\Phi )'=GD_{\mu }\Phi }$

This new "derivative" is called a (gauge) covariant derivative and takes the form

${\displaystyle \ D_{\mu }=\partial _{\mu }-igA_{\mu }}$

Where g is called the coupling constant; a quantity defining the strength of an interaction. After a simple calculation we can see that the gauge field A(x) must transform as follows

${\displaystyle \ A'_{\mu }=GA_{\mu }G^{-1}-{\frac {i}{g}}(\partial _{\mu }G)G^{-1}}$

The gauge field is an element of the Lie algebra, and can therefore be expanded as

${\displaystyle \ A_{\mu }=\sum _{a}A_{\mu }^{a}T^{a}}$

There are therefore as many gauge fields as there are generators of the Lie algebra.

Finally, we now have a locally gauge invariant Lagrangian

${\displaystyle \ {\mathcal {L}}_{\mathrm {loc} }={\frac {1}{2}}(D_{\mu }\Phi )^{\mathsf {T}}D^{\mu }\Phi -{\frac {1}{2}}m^{2}\Phi ^{\mathsf {T}}\Phi }$

Pauli uses the term gauge transformation of the first type to mean the transformation of ${\displaystyle \Phi }$, while the compensating transformation in ${\displaystyle A}$ is called a gauge transformation of the second type.

Feynman diagram of scalar bosons interacting via a gauge boson

The difference between this Lagrangian and the original globally gauge-invariant Lagrangian is seen to be the interaction Lagrangian

${\displaystyle \ {\mathcal {L}}_{\mathrm {int} }=i{\frac {g}{2}}\Phi ^{\mathsf {T}}A_{\mu }^{\mathsf {T}}\partial ^{\mu }\Phi +i{\frac {g}{2}}(\partial _{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi -{\frac {g^{2}}{2}}(A_{\mu }\Phi )^{\mathsf {T}}A^{\mu }\Phi }$

This term introduces interactions between the n scalar fields just as a consequence of the demand for local gauge invariance. However, to make this interaction physical and not completely arbitrary, the mediator A(x) needs to propagate in space. That is dealt with in the next section by adding yet another term, ${\displaystyle {\mathcal {L}}_{\mathrm {gf} }}$, to the Lagrangian. In the quantized version of the obtained classical field theory, the quanta of the gauge field A(x) are called gauge bosons. The interpretation of the interaction Lagrangian in quantum field theory is of scalar bosons interacting by the exchange of these gauge bosons.

The Yang–Mills Lagrangian for the gauge field

Main article: Yang–Mills theory

The picture of a classical gauge theory developed in the previous section is almost complete, except for the fact that to define the covariant derivatives D, one needs to know the value of the gauge field ${\displaystyle A(x)}$ at all space-time points. Instead of manually specifying the values of this field, it can be given as the solution to a field equation. Further requiring that the Lagrangian that generates this field equation is locally gauge invariant as well, one possible form for the gauge field Lagrangian is

${\displaystyle {\mathcal {L}}_{\text{gf}}=-{\frac {1}{2}}\operatorname {tr} \left(F^{\mu \nu }F_{\mu \nu }\right)=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}$

where the ${\displaystyle F_{\mu \nu }^{a}}$ are obtained from potentials ${\displaystyle A_{\mu }^{a}}$, being the components of ${\displaystyle A(x)}$, by

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\sum _{b,c}f^{abc}A_{\mu }^{b}A_{\nu }^{c}}$

and the ${\displaystyle f^{abc}}$ are the structure constants of the Lie algebra of the generators of the gauge group. This formulation of the Lagrangian is called a Yang–Mills action. Other gauge invariant actions also exist (e.g., nonlinear electrodynamics, Born–Infeld action, Chern–Simons model, theta term, etc.).

In this Lagrangian term there is no field whose transformation counterweighs the one of ${\displaystyle A}$. Invariance of this term under gauge transformations is a particular case of a priori classical (geometrical) symmetry. This symmetry must be restricted in order to perform quantization, the procedure being denominated gauge fixing, but even after restriction, gauge transformations may be possible.

The complete Lagrangian for the gauge theory is now

${\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{loc}}+{\mathcal {L}}_{\text{gf}}={\mathcal {L}}_{\text{global}}+{\mathcal {L}}_{\text{int}}+{\mathcal {L}}_{\text{gf}}}$

An example: Electrodynamics

As a simple application of the formalism developed in the previous sections, consider the case of electrodynamics, with only the electron field. The bare-bones action that generates the electron field's Dirac equation is

${\displaystyle {\mathcal {S}}=\int {\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }\partial _{\mu }-mc^{2}\right)\psi \,\mathrm {d} ^{4}x}$

The global symmetry for this system is

${\displaystyle \psi \mapsto e^{i\theta }\psi }$

The gauge group here is U(1), just rotations of the phase angle of the field, with the particular rotation determined by the constant θ.

"Localising" this symmetry implies the replacement of θ by θ(x). An appropriate covariant derivative is then

${\displaystyle D_{\mu }=\partial _{\mu }-i{\frac {e}{\hbar }}A_{\mu }}$

Identifying the "charge" e (not to be confused with the mathematical constant e in the symmetry description) with the usual electric charge (this is the origin of the usage of the term in gauge theories), and the gauge field A(x) with the four-vector potential of electromagnetic field results in an interaction Lagrangian

${\displaystyle {\mathcal {L}}_{\text{int}}={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)A_{\mu }(x)=J^{\mu }(x)A_{\mu }(x)}$

where ${\displaystyle J^{\mu }(x)={\frac {e}{\hbar }}{\bar {\psi }}(x)\gamma ^{\mu }\psi (x)}$ is the electric current four vector in the Dirac field. The gauge principle is therefore seen to naturally introduce the so-called minimal coupling of the electromagnetic field to the electron field.

Adding a Lagrangian for the gauge field ${\displaystyle A_{\mu }(x)}$ in terms of the field strength tensor exactly as in electrodynamics, one obtains the Lagrangian used as the starting point in quantum electrodynamics.

${\displaystyle {\mathcal {L}}_{\text{QED}}={\bar {\psi }}\left(i\hbar c\,\gamma ^{\mu }D_{\mu }-mc^{2}\right)\psi -{\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }}$

See also: Dirac equation, Maxwell's equations, and Quantum electrodynamics

Mathematical formalism

See also: Gauge theory (mathematics)

Gauge theories are usually discussed in the language of differential geometry. Mathematically, a gauge is just a choice of a (local) section of some principal bundle. A gauge transformation is just a transformation between two such sections.

Although gauge theory is dominated by the study of connections (primarily because it's mainly studied by high-energy physicists), the idea of a connection is not central to gauge theory in general. In fact, a result in general gauge theory shows that affine representations (i.e., affine modules) of the gauge transformations can be classified as sections of a jet bundle satisfying certain properties. There are representations that transform covariantly pointwise (called by physicists gauge transformations of the first kind), representations that transform as a connection form (called by physicists gauge transformations of the second kind, an affine representation)—and other more general representations, such as the B field in BF theory. There are more general nonlinear representations (realizations), but these are extremely complicated. Still, nonlinear sigma models transform nonlinearly, so there are applications.

If there is a principal bundle P whose base space is space or spacetime and structure group is a Lie group, then the sections of P form a principal homogeneous space of the group of gauge transformations.

Connections (gauge connection) define this principal bundle, yielding a covariant derivative ∇ in each associated vector bundle. If a local frame is chosen (a local basis of sections), then this covariant derivative is represented by the connection form A, a Lie algebra-valued 1-form, which is called the gauge potential in physics. This is evidently not an intrinsic but a frame-dependent quantity. The curvature form F, a Lie algebra-valued 2-form that is an intrinsic quantity, is constructed from a connection form by

${\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} +\mathbf {A} \wedge \mathbf {A} }$

where d stands for the exterior derivative and ${\displaystyle \wedge }$ stands for the wedge product. (${\displaystyle \mathbf {A} }$ is an element of the vector space spanned by the generators ${\displaystyle T^{a}}$, and so the components of ${\displaystyle \mathbf {A} }$ do not commute with one another. Hence the wedge product ${\displaystyle \mathbf {A} \wedge \mathbf {A} }$ does not vanish.)

Infinitesimal gauge transformations form a Lie algebra, which is characterized by a smooth Lie-algebra-valued scalar, ε. Under such an infinitesimal gauge transformation,

${\displaystyle \delta _{\varepsilon }\mathbf {A} =[\varepsilon ,\mathbf {A} ]-\mathrm {d} \varepsilon }$

where ${\displaystyle [\cdot ,\cdot ]}$ is the Lie bracket.

One nice thing is that if ${\displaystyle \delta _{\varepsilon }X=\varepsilon X}$, then ${\displaystyle \delta _{\varepsilon }DX=\varepsilon DX}$ where D is the covariant derivative

${\displaystyle DX\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {d} X+\mathbf {A} X}$

Also, ${\displaystyle \delta _{\varepsilon }\mathbf {F} =[\varepsilon ,\mathbf {F} ]}$, which means ${\displaystyle \mathbf {F} }$ transforms covariantly.

Not all gauge transformations can be generated by infinitesimal gauge transformations in general. An example is when the base manifold is a compact manifold without boundary such that the homotopy class of mappings from that manifold to the Lie group is nontrivial. See instanton for an example.

The Yang–Mills action is now given by

${\displaystyle {\frac {1}{4g^{2}}}\int \operatorname {Tr} [*F\wedge F]}$

where * stands for the Hodge dual and the integral is defined as in differential geometry.

A quantity which is gauge-invariant (i.e., invariant under gauge transformations) is the Wilson loop, which is defined over any closed path, γ, as follows:

${\displaystyle \chi ^{(\rho )}\left({\mathcal {P}}\left\{e^{\int _{\gamma }A}\right\}\right)}$

where χ is the character of a complex representation ρ and ${\displaystyle {\mathcal {P}}}$ represents the path-ordered operator.

The formalism of gauge theory carries over to a general setting. For example, it is sufficient to ask that a vector bundle have a metric connection; when one does so, one finds that the metric connection satisfies the Yang–Mills equations of motion.

Quantization of gauge theories

Gauge theories may be quantized by specialization of methods which are applicable to any quantum field theory. However, because of the subtleties imposed by the gauge constraints (see section on Mathematical formalism, above) there are many technical problems to be solved which do not arise in other field theories. At the same time, the richer structure of gauge theories allows simplification of some computations: for example Ward identities connect different renormalization constants.

Methods and aims

The first gauge theory quantized was quantum electrodynamics (QED). The first methods developed for this involved gauge fixing and then applying canonical quantization. The Gupta–Bleuler method was also developed to handle this problem. Non-abelian gauge theories are now handled by a variety of means. Methods for quantization are covered in the article on quantization.

The main point to quantization is to be able to compute quantum amplitudes for various processes allowed by the theory. Technically, they reduce to the computations of certain correlation functions in the vacuum state. This involves a renormalization of the theory.

When the running coupling of the theory is small enough, then all required quantities may be computed in perturbation theory. Quantization schemes intended to simplify such computations (such as canonical quantization) may be called perturbative quantization schemes. At present some of these methods lead to the most precise experimental tests of gauge theories.

However, in most gauge theories, there are many interesting questions which are non-perturbative. Quantization schemes suited to these problems (such as lattice gauge theory) may be called non-perturbative quantization schemes. Precise computations in such schemes often require supercomputing, and are therefore less well-developed currently than other schemes.

Anomalies

Some of the symmetries of the classical theory are then seen not to hold in the quantum theory; a phenomenon called an anomaly. Among the most well known are:

• The scale anomaly, which gives rise to a running coupling constant. In QED this gives rise to the phenomenon of the Landau pole. In quantum chromodynamics (QCD) this leads to asymptotic freedom.
• The chiral anomaly in either chiral or vector field theories with fermions. This has close connection with topology through the notion of instantons. In QCD this anomaly causes the decay of a pion to two photons.
• The gauge anomaly, which must cancel in any consistent physical theory. In the electroweak theory this cancellation requires an equal number of quarks and leptons.

Pure gauge

A pure gauge is the set of field configurations obtained by a gauge transformation on the null-field configuration, i.e., a gauge-transform of zero. So it is a particular "gauge orbit" in the field configuration's space.

Thus, in the abelian case, where ${\displaystyle A_{\mu }(x)\rightarrow A'_{\mu }(x)=A_{\mu }(x)+\partial _{\mu }f(x)}$, the pure gauge is just the set of field configurations ${\displaystyle A'_{\mu }(x)=\partial _{\mu }f(x)}$ for all f(x).

Reticular formation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Reticular_formation 

 

Reticular formation
Coronal section of the pons, at its upper part. (Formatio reticularis labeled at left.)
Traverse section of the medulla oblongata at about the middle of the olive. (Formatio reticularis grisea and formatio reticularis alba labeled at left.)
Details
Location	Brainstem
Identifiers
Latin	formatio reticularis
MeSH	D012154
NeuroNames	1223
NeuroLex ID	nlx_143558
TA98	A14.1.00.021 A14.1.05.403 A14.1.06.327
TA2	5367
FMA	77719

The reticular formation is a set of interconnected nuclei that are located throughout the brainstem. It is not anatomically well defined, because it includes neurons located in different parts of the brain. The neurons of the reticular formation make up a complex set of networks in the core of the brainstem that extend from the upper part of the midbrain to the lower part of the medulla oblongata. The reticular formation includes ascending pathways to the cortex in the ascending reticular activating system (ARAS) and descending pathways to the spinal cord via the reticulospinal tracts.

Neurons of the reticular formation, particularly those of the ascending reticular activating system, play a crucial role in maintaining behavioral arousal and consciousness. The overall functions of the reticular formation are modulatory and premotor, involving somatic motor control, cardiovascular control, pain modulation, sleep and consciousness, and habituation. The modulatory functions are primarily found in the rostral sector of the reticular formation and the premotor functions are localized in the neurons in more caudal regions.

The reticular formation is divided into three columns: raphe nuclei (median), gigantocellular reticular nuclei (medial zone), and parvocellular reticular nuclei (lateral zone). The raphe nuclei are the place of synthesis of the neurotransmitter serotonin, which plays an important role in mood regulation. The gigantocellular nuclei are involved in motor coordination. The parvocellular nuclei regulate exhalation.

The reticular formation is essential for governing some of the basic functions of higher organisms and is one of the phylogenetically oldest portions of the brain.

Structure

A cross section of the lower part of the pons showing the pontine reticular formation labeled as #9

The human reticular formation is composed of almost 100 brain nuclei and contains many projections into the forebrain, brainstem, and cerebellum, among other regions. It includes the reticular nuclei, reticulothalamic projection fibers, diffuse thalamocortical projections, ascending cholinergic projections, descending non-cholinergic projections, and descending reticulospinal projections. The reticular formation also contains two major neural subsystems, the ascending reticular activating system and descending reticulospinal tracts, which mediate distinct cognitive and physiological processes. It has been functionally cleaved both sagittally and coronally.

Traditionally the reticular nuclei are divided into three columns:

• In the median column – the raphe nuclei
• In the medial column – gigantocellular nuclei (because of larger size of the cells)
• In the lateral column – parvocellular nuclei (because of smaller size of the cells)

The original functional differentiation was a division of caudal and rostral. This was based upon the observation that the lesioning of the rostral reticular formation induces a hypersomnia in the cat brain. In contrast, lesioning of the more caudal portion of the reticular formation produces insomnia in cats. This study has led to the idea that the caudal portion inhibits the rostral portion of the reticular formation.

Sagittal division reveals more morphological distinctions. The raphe nuclei form a ridge in the middle of the reticular formation, and, directly to its periphery, there is a division called the medial reticular formation. The medial RF is large and has long ascending and descending fibers, and is surrounded by the lateral reticular formation. The lateral RF is close to the motor nuclei of the cranial nerves, and mostly mediates their function.

Medial and lateral reticular formation

The medial reticular formation and lateral reticular formation are two columns of nuclei with ill-defined boundaries that send projections through the medulla and into the midbrain. The nuclei can be differentiated by function, cell type, and projections of efferent or afferent nerves. Moving caudally from the rostral midbrain, at the site of the rostral pons and the midbrain, the medial RF becomes less prominent, and the lateral RF becomes more prominent.

Existing on the sides of the medial reticular formation is its lateral cousin, which is particularly pronounced in the rostral medulla and caudal pons. Out from this area spring the cranial nerves, including the very important vagus nerve. The lateral RF is known for its ganglions and areas of interneurons around the cranial nerves, which serve to mediate their characteristic reflexes and functions.

Function

The reticular formation consists of more than 100 small neural networks, with varied functions including the following:

1. Somatic motor control – Some motor neurons send their axons to the reticular formation nuclei, giving rise to the reticulospinal tracts of the spinal cord. These tracts function in maintaining tone, balance, and posture—especially during body movements. The reticular formation also relays eye and ear signals to the cerebellum so that the cerebellum can integrate visual, auditory, and vestibular stimuli in motor coordination. Other motor nuclei include gaze centers, which enable the eyes to track and fixate objects, and central pattern generators, which produce rhythmic signals of breathing and swallowing.
2. Cardiovascular control – The reticular formation includes the cardiac and vasomotor centers of the medulla oblongata.
3. Pain modulation – The reticular formation is one means by which pain signals from the lower body reach the cerebral cortex. It is also the origin of the descending analgesic pathways. The nerve fibers in these pathways act in the spinal cord to block the transmission of some pain signals to the brain.
4. Sleep and consciousness – The reticular formation has projections to the thalamus and cerebral cortex that allow it to exert some control over which sensory signals reach the cerebrum and come to our conscious attention. It plays a central role in states of consciousness like alertness and sleep. Injury to the reticular formation can result in irreversible coma.
5. Habituation – This is a process in which the brain learns to ignore repetitive, meaningless stimuli while remaining sensitive to others. A good example of this is a person who can sleep through loud traffic in a large city, but is awakened promptly due to the sound of an alarm or crying baby. Reticular formation nuclei that modulate activity of the cerebral cortex are part of the ascending reticular activating system.

Major subsystems

Ascending reticular activating system

Ascending reticular activating system. Reticular formation labeled near center.

The ascending reticular activating system (ARAS), also known as the extrathalamic control modulatory system or simply the reticular activating system (RAS), is a set of connected nuclei in the brains of vertebrates that is responsible for regulating wakefulness and sleep-wake transitions. The ARAS is a part of the reticular formation and is mostly composed of various nuclei in the thalamus and a number of dopaminergic, noradrenergic, serotonergic, histaminergic, cholinergic, and glutamatergic brain nuclei.

Structure of the ARAS

The ARAS is composed of several neural circuits connecting the dorsal part of the posterior midbrain and anterior pons to the cerebral cortex via distinct pathways that project through the thalamus and hypothalamus. The ARAS is a collection of different nuclei – more than 20 on each side in the upper brainstem, the pons, medulla, and posterior hypothalamus. The neurotransmitters that these neurons release include dopamine, norepinephrine, serotonin, histamine, acetylcholine, and glutamate. They exert cortical influence through direct axonal projections and indirect projections through thalamic relays.

The thalamic pathway consists primarily of cholinergic neurons in the pontine tegmentum, whereas the hypothalamic pathway is composed primarily of neurons that release monoamine neurotransmitters, namely dopamine, norepinephrine, serotonin, and histamine. The glutamate-releasing neurons in the ARAS were identified much more recently relative to the monoaminergic and cholinergic nuclei; the glutamatergic component of the ARAS includes one nucleus in the hypothalamus and various brainstem nuclei. The orexin neurons of the lateral hypothalamus innervate every component of the ascending reticular activating system and coordinate activity within the entire system.

Key components of the ascending reticular activating system

Nucleus type	Corresponding nuclei that mediate arousal
Dopaminergic nuclei	Ventral tegmental area Substantia nigra pars compacta
Noradrenergic nuclei	Locus coeruleus Related noradrenergic brainstem nuclei
Serotonergic nuclei	Dorsal raphe nucleus Median raphe nucleus
Histaminergic nuclei	Tuberomammillary nucleus
Cholinergic nuclei	Forebrain cholinergic nuclei Pontine tegmental nuclei: laterodorsal and pedunculopontine tegmental nucleus
Glutamatergic nuclei	Brainstem nuclei: parabrachial nucleus, precoeruleus, and pedunculopontine tegmental nucleus Hypothalamic nuclei: supramammillary nucleus
Thalamic nuclei	Thalamic reticular nucleus Intralaminar nucleus, including the centromedian nucleus

The ARAS consists of evolutionarily ancient areas of the brain, which are crucial to the animal's survival and protected during adverse periods, such as during inhibitory periods of Totsellreflex, aka, "animal hypnosis". The ascending reticular activating system which sends neuromodulatory projections to the cortex - mainly connects to the prefrontal cortex. There seems to be low connectivity to the motor areas of the cortex.

Functions of the ARAS

Consciousness

The ascending reticular activating system is an important enabling factor for the state of consciousness. The ascending system is seen to contribute to wakefulness as characterised by cortical and behavioural arousal.

Regulating sleep-wake transitions

Further information: Neuroscience of sleep

The main function of the ARAS is to modify and potentiate thalamic and cortical function such that electroencephalogram (EEG) desynchronization ensues. There are distinct differences in the brain's electrical activity during periods of wakefulness and sleep: Low voltage fast burst brain waves (EEG desynchronization) are associated with wakefulness and REM sleep (which are electrophysiologically similar); high voltage slow waves are found during non-REM sleep. Generally speaking, when thalamic relay neurons are in burst mode the EEG is synchronized and when they are in tonic mode it is desynchronized. Stimulation of the ARAS produces EEG desynchronization by suppressing slow cortical waves (0.3–1 Hz), delta waves (1–4 Hz), and spindle wave oscillations (11–14 Hz) and by promoting gamma band (20 – 40 Hz) oscillations.

The physiological change from a state of deep sleep to wakefulness is reversible and mediated by the ARAS. The ventrolateral preoptic nucleus (VLPO) of the hypothalamus inhibits the neural circuits responsible for the awake state, and VLPO activation contributes to the sleep onset. During sleep, neurons in the ARAS will have a much lower firing rate; conversely, they will have a higher activity level during the waking state. In order that the brain may sleep, there must be a reduction in ascending afferent activity reaching the cortex by suppression of the ARAS.

Attention

The ARAS also helps mediate transitions from relaxed wakefulness to periods of high attention. There is increased regional blood flow (presumably indicating an increased measure of neuronal activity) in the midbrain reticular formation (MRF) and thalamic intralaminar nuclei during tasks requiring increased alertness and attention.

Clinical significance of the ARAS

Mass lesions in brainstem ARAS nuclei can cause severe alterations in level of consciousness (e.g., coma). Bilateral damage to the reticular formation of the midbrain may lead to coma or death.

Direct electrical stimulation of the ARAS produces pain responses in cats and elicits verbal reports of pain in humans. Ascending reticular activation in cats can produce mydriasis, which can result from prolonged pain. These results suggest some relationship between ARAS circuits and physiological pain pathways.

Pathologies

Some pathologies of the ARAS may be attributed to age, as there appears to be a general decline in reactivity of the ARAS with advancing years. Changes in electrical coupling have been suggested to account for some changes in ARAS activity: if coupling were down-regulated, there would be a corresponding decrease in higher-frequency synchronization (gamma band). Conversely, up-regulated electrical coupling would increase synchronization of fast rhythms that could lead to increased arousal and REM sleep drive. Specifically, disruption of the ARAS has been implicated in the following disorders:

• Narcolepsy: Lesions along the pedunculopontine (PPT/PPN) / laterodorsal tegmental (LDT) nuclei are associated with narcolepsy. There is a significant down-regulation of PPN output and a loss of orexin peptides, promoting the excessive daytime sleepiness that is characteristic of this disorder.
• Progressive supranuclear palsy (PSP) : Dysfunction of nitrous oxide signaling has been implicated in the development of PSP.
• Parkinson's disease: REM sleep disturbances are common in Parkinson's. It is mainly a dopaminergic disease, but cholinergic nuclei are depleted as well. Degeneration in the ARAS begins early in the disease process.

Developmental influences

There are several potential factors that may adversely influence the development of the ascending reticular activating system:

• Preterm birth: Regardless of birth weight or weeks of gestation, premature birth induces persistent deleterious effects on pre-attentional (arousal and sleep-wake abnormalities), attentional (reaction time and sensory gating), and cortical mechanisms throughout development.
• Smoking during pregnancy: Prenatal exposure to cigarette smoke is known to produce lasting arousal, attentional and cognitive deficits in humans. This exposure can induce up-regulation of α4β2 nicotinic receptors on cells of the pedunculopontine nucleus (PPN), resulting in increased tonic activity, resting membrane potential, and hyperpolarization-activated cation current. These major disturbances of the intrinsic membrane properties of PPN neurons result in increased levels of arousal and sensory gating, deficits (demonstrated by a diminished amount of habituation to repeated auditory stimuli). It is hypothesized that these physiological changes may intensify attentional dysregulation later in life.

Descending reticulospinal tracts

Spinal cord tracts - reticulospinal tract labeled in red, near-center at left in figure

The reticulospinal tracts, also known as the descending or anterior reticulospinal tracts, are extrapyramidal motor tracts that descend from the reticular formation in two tracts to act on the motor neurons supplying the trunk and proximal limb flexors and extensors. The reticulospinal tracts are involved mainly in locomotion and postural control, although they do have other functions as well. The descending reticulospinal tracts are one of four major cortical pathways to the spinal cord for musculoskeletal activity. The reticulospinal tracts works with the other three pathways to give a coordinated control of movement, including delicate manipulations. The four pathways can be grouped into two main system pathways – a medial system and a lateral system. The medial system includes the reticulospinal pathway and the vestibulospinal pathway, and this system provides control of posture. The corticospinal and the rubrospinal tract pathways belong to the lateral system which provides fine control of movement.

Components of the reticulospinal tracts

This descending tract is divided into two parts, the medial (or pontine) and lateral (or medullary) reticulospinal tracts (MRST and LRST).

• The MRST is responsible for exciting anti-gravity, extensor muscles. The fibers of this tract arise from the caudal pontine reticular nucleus and the oral pontine reticular nucleus and project to lamina VII and lamina VIII of the spinal cord.
• The LRST is responsible for inhibiting excitatory axial extensor muscles of movement. It is also responsible for automatic breathing. The fibers of this tract arise from the medullary reticular formation, mostly from the gigantocellular nucleus, and descend the length of the spinal cord in the anterior part of the lateral column. The tract terminates in lamina VII mostly with some fibers terminating in lamina IX of the spinal cord.

The ascending sensory tract conveying information in the opposite direction is known as the spinoreticular tract.

Functions of the reticulospinal tracts

1. Integrates information from the motor systems to coordinate automatic movements of locomotion and posture
2. Facilitates and inhibits voluntary movement; influences muscle tone
3. Mediates autonomic functions
4. Modulates pain impulses
5. Influences blood flow to lateral geniculate nucleus of the thalamus.

Clinical significance of the reticulospinal tracts

The reticulospinal tracts provide a pathway by which the hypothalamus can control sympathetic thoracolumbar outflow and parasympathetic sacral outflow.

Two major descending systems carrying signals from the brainstem and cerebellum to the spinal cord can trigger automatic postural response for balance and orientation: vestibulospinal tracts from the vestibular nuclei and reticulospinal tracts from the pons and medulla. Lesions of these tracts result in profound ataxia and postural instability.

Physical or vascular damage to the brainstem disconnecting the red nucleus (midbrain) and the vestibular nuclei (pons) may cause decerebrate rigidity, which has the neurological sign of increased muscle tone and hyperactive stretch reflexes. Responding to a startling or painful stimulus, both arms and legs extend and turn internally. The cause is the tonic activity of lateral vestibulospinal and reticulospinal tracts stimulating extensor motoneurons without the inhibitions from rubrospinal tract.

Brainstem damage above the red nucleus level may cause decorticate rigidity. Responding to a startling or painful stimulus, the arms flex and the legs extend. The cause is the red nucleus, via the rubrospinal tract, counteracting the extensor motorneuron's excitation from the lateral vestibulospinal and reticulospinal tracts. Because the rubrospinal tract only extends to the cervical spinal cord, it mostly acts on the arms by exciting the flexor muscles and inhibiting the extensors, rather than the legs.

Damage to the medulla below the vestibular nuclei may cause flaccid paralysis, hypotonia, loss of respiratory drive, and quadriplegia. There are no reflexes resembling early stages of spinal shock because of complete loss of activity in the motorneurons, as there is no longer any tonic activity arising from the lateral vestibulospinal and reticulospinal tracts.

History

The term "reticular formation" was coined in the late 19th century by Otto Deiters, coinciding with Ramon y Cajal's neuron doctrine. Allan Hobson states in his book The Reticular Formation Revisited that the name is an etymological vestige from the fallen era of the aggregate field theory in the neural sciences. The term "reticulum" means "netlike structure", which is what the reticular formation resembles at first glance. It has been described as being either too complex to study or an undifferentiated part of the brain with no organization at all. Eric Kandel describes the reticular formation as being organized in a similar manner to the intermediate gray matter of the spinal cord. This chaotic, loose, and intricate form of organization is what has turned off many researchers from looking farther into this particular area of the brain. The cells lack clear ganglionic boundaries, but do have clear functional organization and distinct cell types. The term "reticular formation" is seldom used anymore except to speak in generalities. Modern scientists usually refer to the individual nuclei that compose the reticular formation.

Moruzzi and Magoun first investigated the neural components regulating the brain's sleep-wake mechanisms in 1949. Physiologists had proposed that some structure deep within the brain controlled mental wakefulness and alertness. It had been thought that wakefulness depended only on the direct reception of afferent (sensory) stimuli at the cerebral cortex.

As direct electrical stimulation of the brain could simulate electrocortical relays, Magoun used this principle to demonstrate, on two separate areas of the brainstem of a cat, how to produce wakefulness from sleep. He first stimulated the ascending somatic and auditory paths; second, a series of "ascending relays from the reticular formation of the lower brain stem through the midbrain tegmentum, subthalamus and hypothalamus to the internal capsule." The latter was of particular interest, as this series of relays did not correspond to any known anatomical pathways for the wakefulness signal transduction and was coined the ascending reticular activating system (ARAS).

Next, the significance of this newly identified relay system was evaluated by placing lesions in the medial and lateral portions of the front of the midbrain. Cats with mesencephalic interruptions to the ARAS entered into a deep sleep and displayed corresponding brain waves. In alternative fashion, cats with similarly placed interruptions to ascending auditory and somatic pathways exhibited normal sleeping and wakefulness, and could be awakened with physical stimuli. Because these external stimuli would be blocked on their way to the cortex by the interruptions, this indicated that the ascending transmission must travel through the newly discovered ARAS.

Finally, Magoun recorded potentials within the medial portion of the brain stem and discovered that auditory stimuli directly fired portions of the reticular activating system. Furthermore, single-shock stimulation of the sciatic nerve also activated the medial reticular formation, hypothalamus, and thalamus. Excitation of the ARAS did not depend on further signal propagation through the cerebellar circuits, as the same results were obtained following decerebellation and decortication. The researchers proposed that a column of cells surrounding the midbrain reticular formation received input from all the ascending tracts of the brain stem and relayed these afferents to the cortex and therefore regulated wakefulness.