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Higgs lede

The Higgs boson or Higgs particle is an elementary particle in the Standard Model of Partical physics. Its main relevance is that it is the smallest possible excitation of the Higgs field; A field that unlike the more familiar electromagnetic field cannot be "turned off", but instead takes a constant value almost everywhere. Its existence explains why some fundamental particles have mass when the symmetries controlling their interactions should require them to be massless, and why the weak force has a much shorter range than the electromagnetic force.

Despite being present everywhere, the existence of the Higgs field has been very hard to confirm, because it is extremely hard to create excitations (i.e. Higgs particles). The search for this elusive particle has taken more than 40 years and led to the construction of one of the world's most expensive and complex experimental facilities to date, the Large Hadron Collider, able to create Higgs bosons and other particles for observation and study. On 4 July 2012, it was announced that a previously unknown particle with a mass between 125 and 127 GeV/c² (134.2 and 136.3 amu) had been detected; physicists suspected at the time that it was the Higgs boson. By March 2013, the particle had been proven to behave, interact and decay in many of the ways predicted by the Standard Model, and was also tentatively confirmed to have positive parity and zero spin, two fundamental attributes of a Higgs boson. This appears to be the first elementary scalar particle discovered in nature. More data is needed to know if the discovered particle exactly matches the predictions of the Standard Model, or whether, as predicted by some theories, multiple Higgs bosons exist.

References

Big bang lede

The Big Bang theory is the prevailing cosmological model for the early development of the universe. The main idea is that the universe is expanding. Consequently, the universe was denser and hotter in the past. In particular, the Big Bang model suggests that at some moment all matter in the universe was contained in a point. Modern measurements place this moment at approximately 13.82 billion years ago, which is thus considered the age of the universe. After the initial expansion, the universe cooled sufficiently to allow the formation of subatomic particles, including protons, neutrons, and electrons. Though simple atomic nuclei formed within the first three minutes after the Big Bang, thousands of years passed before the first electrically neutral atoms formed. The majority of atoms that were produced by the Big Bang are hydrogen, along with helium and traces of lithium. Giant clouds of these primordial elements later coalesced through gravity to form stars and galaxies, and the heavier elements were synthesized either within stars or during supernovae.

Religious Implications (ec)

Main article: Religious interpretations of the Big Bang theory

The Big Bang is a scientific theory, and as such is dependent on its agreement with observations. But as a theory which addresses the origins of reality, it has always carried theological implications, most notably, those based upon the philosophical concept of creation ex nihilo (a Latin phrase meaning "out of nothing"). Since the acceptance of the Big Bang as the dominant physical cosmological paradigm, there have been a variety of reactions by religious groups as to its implications for their respective religious cosmologies. Some accept the scientific evidence at face value, while others seek to reconcile the Big Bang with their religious tenets, and others completely reject or ignore the evidence for the Big Bang theory.

Tachyon lede

A tachyon /ˈtæki.ɒn/ (or tachyonic particle), is a hypothetical particle that always moves faster than light. Most physicists do not believe that such particles exist, or are consistent with the known laws of physics. Tachyons could be used to send signals faster than light, and (according to special relativity) this leads to violations of causality.

The first hypothesis regarding tachyons is attributed to German physicist Arnold Sommerfeld, and more recent discussions include. The term itself was coined by Gerald Feinberg in a 1967 paper. The name comes from the Greek: ταχύς (tachys, “swift, quick, fast, rapid”). Conventional massive particles that travel slower than the speed of light are sometimes termed "bradyons" or "tardyons" in contrast, although these terms are only used in the context of discussions about tachyons.

In the language of special relativity, a tachyon would have space-like four-momentum and imaginary proper time, and would be constrained to the spacelike portion of the energy-momentum graph. Therefore, it cannot slow down to subluminal speeds.

In his 1967 paper Feinberg proposed that tachyons could occur as the quanta of a field with imaginary mass. It was however later shown that the excitations of this field do not propagate faster than light, but instead represent an instability of the ground state of the field (see tachyon condensation). Today, the fields of the type proposed by Feinberg still play a role in theoretical physics and are referred to as tachyonic fields (or sometimes as simply tachyons). Potentially consistent theories that allow faster-than-light particles include those that break Lorentz invariance, the symmetry underlying special relativity, so that the speed of light is not a barrier.

Despite the theoretical arguments against the existence of faster than light particles, experimental searches have been conducted to search for them. Until recently, no compelling experimental evidence for their existence had been found. Following the results of the September 2011 observation of faster-than-light neutrino velocities, the faster-than-light neutrino anomaly, the value of the neutrino velocity is now an active subject of theoretical and experimental studies.

Motivating Examples (tensor)

Tensors generalize the concept of scalars, vectors, and linear maps. This section reviews the properties of these objects that will be relevant to all tensors.

Vector

See also: Euclidean vector

A vector is a quantity that has a direction and a magnitude. Vectors can be added together through vector addition and can be scaled through scalar multiplication. In three dimensions, if you choose three linearly independent vectors, e₁, e₂, and e₃, any other vector v can be written as a linear combination of these vectors,

v = c¹e₁ + c²e₂ + c³e₃.

Such a set {e₁, e₂, e₃} of linearly independent vectors is called a basis.

When dealing with vectors it is common to fix a choice of basis, e.g. vectors of unit length in the x, y, and z directions, and identify a vector by its coefficients, c^{1, c2, and c3, with respect to that basis, also called its components. This is commonly denoted by placing the components in a column,}

$\mathbf {v} ={\begin{pmatrix}c^{1}\\c^{2}\\c^{3}\end{pmatrix}}$.

A different choice of basis vectors {ê₁, ê₂, ê₃} leads to different components, ĉ¹, ĉ², and ĉ³. The components with respect to new basis are related to the components with respect to the old basis, because the vectors in the new basis can be expressed as linear combinations of the old basis,

ê_i = R_i¹e₁ + R_i²e₂ + R_i³e₃ = R_i^je_j,

where i runs from 1 to 3, and last part introduces a notational shorthand called Einstein notation, which implies that any repeated indices (in this case j) are summed over their range (in this case 1 to 3). This notational shorthand is very convenient when dealing with tensors and is used throughout the rest of this article. It follows from this relation that the coefficients cⁱ can be expressed in terms of the coefficients ĉⁱ as follows,

cⁱ = Rⁱ_jĉ^j,

or conversely,

ĉⁱ = (R^-1)ⁱ_jc^j,

where (R^-1)ⁱ_j is the inverse transformation of Rⁱ_j (which is defined by the condition (R^-1)ⁱ_kR^k_j = Id). The coefficients of a vector does transform with de inverse transformation with respect to the basis transformation. They are said to transform contravariantly.

Covector

See also: Covector

For every vector space V there exists a dual vector space V* defined as the space of all linear functions on V, the elements of which are called covectors. If V has a basis e₁, ..., e_n then there exists a dual basis ε¹, ..., εⁿ of V* defined by the condition,

εⁱ(e_j) = 1 if i = j and 0 otherwise.

If ω is a covector with coefficients ω₁, ..., ω_n then its effect of a vector v with components v¹, ..., vⁿ is given by

ω(v) = ω_ivⁱ,

or expressed as a matrix multiplication,

$\mathbf {\omega } (\mathbf {v} )={\begin{pmatrix}\omega _{1}&\dots &\omega _{n}\end{pmatrix}}{\begin{pmatrix}v^{1}\\\vdots \\v^{n}\end{pmatrix}}.$

For this reason covectors are often referred to as row vectors.

Linear maps

See also: Linear map and Matrix (mathematics)

Notes

References