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Quartic Interaction
In quantum field theory, a quartic interaction or ''φ''4 theory is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varphi satisfies the Klein–Gordon equation. If a scalar field is denoted \varphi, a quartic interaction is represented by adding a potential energy term (/) \varphi^4 to the Lagrangian density. The coupling constant \lambda is dimensionless in 4-dimensional spacetime. This article uses the (+ - - -) metric signature for Minkowski space. Lagrangian for a real scalar field The Lagrangian density for a real scalar field with a quartic interaction is :\mathcal(\varphi)=\frac partial^\mu \varphi \partial_\mu \varphi -m^2 \varphi^2-\frac \varphi^4. This Lagrangian has a global Z2 symmetry mapping \varphi\to-\varphi. Lagrangian for a complex scalar field The Lagrangian for a complex scalar field can be motivated as follows. For ''two'' scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau Pole
In physics, the Landau pole (or the Moscow zero, or the Landau ghost) is the momentum (or energy) scale at which the coupling constant (interaction strength) of a quantum field theory becomes infinite. Such a possibility was pointed out by the physicist Lev Landau and his colleagues in 1954. The fact that couplings depend on the momentum (or length) scale is the central idea behind the renormalization group. Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or theory—a scalar field with a quartic interaction—such as may describe the Higgs boson. In these theories, the renormalized coupling constant grows with energy. A Landau pole appears when the coupling becomes infinite at a finite energy scale. In a theory purporting to be complete, this could be considered a mathematical inconsistency. A possible solution is that the renormalized charge could go to zero as the cut-off is removed, meaning that the charge is complete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in a wide variety of fields such as biology, neuroscience, computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ..., information theory and sociology. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wick Rotation
In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, , where is time and is the imaginary unit (). More precisely, in statistical mechanics, the Gibbs measure describes the relative probability of the system to be in any given state at temperature , where is a function describing the energy of each state and is the Boltzmann constant. In quantum mechanics, the transformation describes time evolution, where is an operator descri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Expectation Value
In quantum field theory, the vacuum expectation value (VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. One of the most widely used examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect. This concept is important for working with correlation functions in quantum field theory. In the context of spontaneous symmetry breaking, an operator that has a vanishing expectation value due to symmetry can acquire a nonzero vacuum expectation value during a phase transition. Examples are: *The Higgs field has a vacuum expectation value of 246 GeV. This nonzero value underlies the Higgs mechanism of the Standard Model. This value is given by v = 1/\sqrt = 2M_W/g \approx 246.22\, \rm, where ''MW'' is the mass of the W Boson, G_F^0 the reduced Fermi constant, and the weak isospin coupling, in natural units. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time-ordered
In theoretical physics, path-ordering is the procedure (or a meta-operator \mathcal P) that orders a product of operators according to the value of a chosen parameter: :\mathcal P \left\ \equiv O_(\sigma_) O_(\sigma_) \cdots O_(\sigma_). Here ''p'' is a permutation that orders the parameters by value: :p : \ \to \ :\sigma_ \leq \sigma_ \leq \cdots \leq \sigma_. For example: :\mathcal P \left\ = O_4(1) O_2(2) O_3(3) O_1(4) . In many fields of physics, the most common type of path-ordering is time-ordering, which is discussed in detail below. Examples If an operator is not simply expressed as a product, but as a function of another operator, we must first perform a Taylor expansion of this function. This is the case of the Wilson loop, which is defined as a path-ordered exponential to guarantee that the Wilson loop encodes the holonomy of the gauge connection. The parameter ''σ'' that determines the ordering is a parameter describing the contour, and because the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Integral Formulation
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The calculation of probability amplitudes in theoretical particle physics requires the use of large, complicated integrals over a large number of variables. Feynman diagrams instead represent these integrals graphically. Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams apply primarily to quantum field theory, they can be used in other areas of physics, such as solid-state theory. Frank Wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent Nuclear magnetic moment, magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a Graph (abstract data type), graph, usually a lattice (group), lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases.The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. Though it is a highly simplified model of a magnetic material, the Ising model can sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
