1.
Vacuum state
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In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles, zero-point field is sometimes used as a synonym for the vacuum state of an individual quantized field. According to present-day understanding of what is called the state or the quantum vacuum. According to quantum mechanics, the state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into. The QED vacuum of quantum electrodynamics was the first vacuum of quantum theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga and Schwinger, today the electromagnetic interactions and the weak interactions are unified in the theory of the electroweak interaction. The Standard Model is a generalization of the QED work to all the known elementary particles. Quantum chromodynamics is the portion of the Standard Model that deals with strong interactions and it is the object of study in the Large Hadron Collider and the Relativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions. In this case the vacuum value of any field operator vanishes. For quantum field theories in which perturbation theory breaks down at low energies field operators may have non-vanishing vacuum expectation values called condensates. In the Standard Model, the vacuum expectation value of the Higgs field. In many situations, the state can be defined to have zero energy. The vacuum state is associated with a zero-point energy, and this zero-point energy has measurable effects, in the laboratory, it may be detected as the Casimir effect. In physical cosmology, the energy of the cosmological vacuum appears as the cosmological constant, in fact, the energy of a cubic centimeter of empty space has been calculated figuratively to be one trillionth of an erg. An outstanding requirement imposed on a potential Theory of Everything is that the energy of the vacuum state must explain the physically observed cosmological constant. For a relativistic theory, the vacuum is Poincaré invariant. Poincaré invariance implies that only scalar combinations of field operators have non-vanishing VEVs, the VEV may break some of the internal symmetries of the Lagrangian of the field theory. In this case the vacuum has less symmetry than the theory allows, in principle, quantum corrections to Maxwells equations can cause the experimental electrical permittivity ε of the vacuum state to deviate from the defined scalar value ε0 of the electric constant

2.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined