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68-555: Fifth Dimension or fifth dimension may refer to: Five-dimensional space , a mathematical concept or construct The 5th Dimension , a pop music vocal group debuting in the 1960s 5th Dimension (album) , a 2013 album by Momoiro Clover Z Fifth Dimension (album) , a 1966 album by the Byrds " 5D (Fifth Dimension) ", a 1966 song by the Byrds Fifth Dimension ,

136-422: 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

204-408: 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

272-415: A concept that postulates an infinite number of mathematical dimensions to allow for a limitless number of quantum states. Einstein , Bergmann , and Bargmann later tried to extend the four-dimensional spacetime of general relativity into an extra physical dimension to incorporate electromagnetism, though they were unsuccessful. In their 1938 paper, Einstein and Bergmann were among the first to introduce

340-488: 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 do not. (This is analogous to a non-inertial change of reference frame, which can produce a Coriolis effect .) 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

408-472: A fundamental modification to Einstein's theory of general relativity . Minkowski space and Maxwell's equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor . In 1993, the physicist Gerard 't Hooft put forward the holographic principle , which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension . For example, holograms are three-dimensional pictures placed on

476-448: A general coordinate transformation. The importance of these symmetry invariances remained unnoticed until Weyl's work. Inspired by Pauli's descriptions of connection between charge conservation and field theory driven by invariance, Chen Ning Yang sought a field theory for atomic nuclei binding based on conservation of nuclear isospin . In 1954, Yang and Robert Mills generalized the gauge invariance of electromagnetism, constructing

544-492: 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,

612-436: A more generalized approach known as M-theory . M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level. The Kaluza–Klein theory today is seen as essentially a gauge theory , with the gauge being the circle group . The fifth dimension

680-415: 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

748-406: A physical entity, rather than an excuse to combine the metric tensor and electromagnetic potential. But they then reneged, modifying the theory to break its five-dimensional symmetry. Their reasoning, as suggested by Edward Witten , was that the more symmetric version of the theory predicted the existence of a new long range field, one that was both massless and scalar , which would have required

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816-462: 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

884-486: A pond can only see shadows of ripples across the surface of the water caused by raindrops. While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The Kaluza–Klein theory experienced a revival in the 1970s due to the emergence of superstring theory and supergravity : the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into

952-610: A program in the Radio Tales series for National Public Radio The 5th Dimension, the fictional home dimension of DC Comics villain Mister Mxyzptlk The 5th Dimension (ride) , an amusement park ride See also [ edit ] Dimension 5 (disambiguation) 5D (disambiguation) Fourth dimension in literature , discusses dimensions 4 and up Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1020-406: A set of techniques for making physical predictions consistent with the symmetries of the model. 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

1088-594: 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

1156-474: 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 . The Yang-Mills theory became the prototype theory to resolve some of the great confusion in elementary particle physics . This idea later found application in the quantum field theory of

1224-403: 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

1292-401: A two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle. 'T Hooft has speculated that the fifth dimension is really the "spacetime fabric". Recent research suggests several alternative interpretations of

1360-400: 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

1428-427: Is a frequent source of anomalies , and approaches to anomaly avoidance classifies gauge 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: Determination of the likelihood of possible measurement outcomes proceed by: These assumptions have enough validity across

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1496-474: 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

1564-526: Is a type of field theory in which the Lagrangian , and hence the dynamics of the system itself, do not change under local transformations according to certain smooth families of operations ( Lie groups ). Formally, the Lagrangian is invariant under these transformations. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of

1632-404: 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

1700-506: Is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain the same under the gauge transformation. 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 {\displaystyle n} non-interacting real scalar fields , with equal masses m . This system

1768-458: Is called a (gauge) covariant derivative and takes the form 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 The gauge field is an element of the Lie algebra, and can therefore be expanded as There are therefore as many gauge fields as there are generators of

1836-545: Is described by an action that is the sum of the (usual) action for each scalar field φ i {\displaystyle \varphi _{i}} The Lagrangian (density) can be compactly written as by introducing a vector of fields The term ∂ μ Φ {\displaystyle \partial _{\mu }\Phi } is the partial derivative of Φ {\displaystyle \Phi } along dimension μ {\displaystyle \mu } . It

1904-471: Is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Physicists theorize that collisions of subatomic particles in turn produce new particles as a result of the collision, including a graviton that escapes from the fourth dimension, or brane , leaking off into a five-dimensional bulk. M-theory would explain

1972-451: Is generally represented using 5 coordinate values (x,y,z,w,v), where moving along the v axis involves moving between different hyper-volumes . In five or more dimensions, only three regular polytopes exist. In five dimensions, they are: An important uniform 5-polytope is the 5-demicube , h{4,3,3,3} has half the vertices of the 5-cube (16), bounded by alternating 5-cell and 16-cell hypercells. The expanded or stericated 5-simplex

2040-447: Is now transparent that the Lagrangian is invariant under the transformation 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 Φ ′ {\displaystyle \Phi '} transforms identically to Φ {\displaystyle \Phi } and both quantities appear inside dot products in

2108-425: 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

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2176-513: Is the vertex figure of the A 5 lattice , [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] . It and has a doubled symmetry from its symmetric Coxeter diagram. The kissing number of the lattice, 30, is represented in its vertices. The rectified 5-orthoplex is the vertex figure of the D 5 lattice , [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] . Its 40 vertices represent

2244-418: 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,

2312-447: The currents where the T 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 spacetime coordinates x . In this case, the G matrices do not "pass through"

2380-521: 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 R s , 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

2448-402: The kissing number of the lattice and the highest for dimension 5. A hypersphere in 5-space (also called a 4-sphere due to its surface being 4-dimensional) consists of the set of all points in 5-space at a fixed distance r from a central point P, that is rotationally symmetrical . The hypervolume enclosed by this hypersurface is: Gauge theory In physics , a gauge theory

2516-402: 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

2584-552: 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. In physics , the mathematical description of any physical situation usually contains excess degrees of freedom ;

2652-434: 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 ,

2720-610: The 5D extension of spacetime , most of them generalizing the earlier Kaluza-Klein theory . The first approach is space-time-matter , which utilizes an unrestricted group of 5D coordinate transforms to derive new solutions of the Einstein's field equations that agree with the corresponding classical solutions in 4D spacetime. Another 5D representation describes quantum physics from a thermal-space-time ensemble perspective and draws connections with classical field theory as limiting cases. Yet another approach, spacekime representation, lifts

2788-532: The Electromagnetic Field " suggested the possibility of invariance, when he 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 . Similarly unnoticed, David Hilbert had derived the Einstein field equations by postulating the invariance of the action under

Fifth Dimension - Misplaced Pages Continue

2856-414: The Lagrangian (orthogonal transformations preserve the dot product). 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

2924-577: 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

2992-417: The Lie algebra. Finally, we now have a locally gauge invariant Lagrangian Pauli uses the term gauge transformation of the first type to mean the transformation of Φ {\displaystyle \Phi } , while the compensating transformation in A {\displaystyle A} is called a gauge transformation of the second type . The difference between this Lagrangian and

3060-456: 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 ,

3128-458: The case of turbulence and other chaotic phenomena. 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

3196-413: The concept provided a basis for further research over the past century. To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10 centimeters. Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in

3264-552: 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, E = − ∇ V {\displaystyle \mathbf {E} =-\nabla V} . Generalizing from static electricity to electromagnetism, we have a second potential, the vector potential A , with The general gauge transformations now become not just V ↦ V + C {\displaystyle V\mapsto V+C} but where f

3332-526: 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 . In order to adequately describe physical situations in more complex theories, it

3400-465: 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: 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 . Gauge theories used to model the results of physical experiments engage in: We cannot express

3468-482: The derivatives, when G = G ( x ), 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 Φ ′ {\displaystyle \Phi '} again transforms identically with Φ {\displaystyle \Phi } This new "derivative"

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3536-453: 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

3604-562: The early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity , and electromagnetism . German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force . Although their approaches were later found to be at least partially inaccurate,

3672-412: 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, V ↦ V + C {\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

3740-483: 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

3808-440: 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

3876-406: 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

3944-420: The modern viewpoint that a four-dimensional theory, which coincides with Einstein–Maxwell theory at long distances, is derived from a five-dimensional theory with complete symmetry in all five dimensions. They suggested that electromagnetism resulted from a gravitational field that is “polarized” in the fifth dimension. The main novelty of Einstein and Bergmann was to seriously consider the fifth dimension as

4012-424: The name of gauge theory derives from the work of Hermann Weyl in 1918. Weyl, in an attempt to generalize the geometrical ideas of general relativity to include 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 replaced

4080-454: 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 . In electrostatics , one can either discuss

4148-511: The ordinary time from an event-ordering positive-real number to complex-time (kime), which effectively transforms longitudinal processes from time-series into 2D manifolds (kime-surfaces). According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. Fifth dimensional geometry

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4216-533: 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. The concept and

4284-509: 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

4352-405: 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

4420-511: The simple 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 . Weyl's 1929 paper introduced the modern concept of gauge invariance subsequently popularized by Wolfgang Pauli in his 1941 review. In retrospect, James Clerk Maxwell 's formulation, in 1864–65, of electrodynamics in " A Dynamical Theory of

4488-471: 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

4556-479: The title Fifth Dimension . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Fifth_Dimension&oldid=1106432178 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Five-dimensional space Much of

4624-403: The weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table—the magnet overcomes the gravitational pull of the entire earth with ease. Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct. These theories make reference to Hilbert space ,

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