|
Thermal Quantum Field Theory
In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature. In the Matsubara formalism, the basic idea (due to Felix Bloch) is that the expectation values of operators in a canonical ensemble : \langle A\rangle=\frac may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time \tau = i t(0\leq\tau\leq\beta). One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity \beta = 1/(kT) (we are assuming natural units \hbar = 1). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum Space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all '' position vectors'' r in space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all ''momentum vectors'' p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of asslength] imesup>−1. Mathematically, the duality between position and momentum is an example of ''Pontryagin duality''. In particular, if a function is given in position space, ''f''(r), then its Fourier transform obtains the function in momentum space, ''φ''(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function. These quantities and ideas tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Statistical Mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space ''H'' describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic. Expectation From classical probability theory, we know that the expectation of a random variable ''X'' is defined by its distribution D''X'' by : \mathbb(X) = \int_\mathbb \lambda \, d \, \operatorname_X(\lambda) assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let ''A'' be an observable of a quantum mechanical system. ''A'' is given by a densely defined self-adjoint operator on ''H''. The spectral measure of ''A'' defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Thermodynamics
Quantum thermodynamics is the study of the relations between two independent physical theories: thermodynamics and quantum mechanics. The two independent theories address the physical phenomena of light and matter. In 1905, Albert Einstein argued that the requirement of consistency between thermodynamics and electromagnetism leads to the conclusion that light is quantized obtaining the relation E= h \nu . This paper is the dawn of quantum theory. In a few decades quantum theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics. It differs from quantum statistical mechanics in the emphasis on dynamical processes out of equilibrium. In addition, there is a quest for the theory to be relevant for a single individual quantum system. Dynamical view There is an intimate connection of quantum thermodynamics with the theory of open quantum systems. Quantum mechanics inserts dynamic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyakov Loop
In quantum field theory, the Polyakov loop is the thermal analogue of the Wilson loop, acting as an order parameter for confinement in pure gauge theories at nonzero temperatures. In particular, it is a Wilson loop that winds around the compactified Euclidean temporal direction of a thermal quantum field theory. It indicates confinement because its vacuum expectation value must vanish in the confined phase due to its non-invariance under center gauge transformations. This also follows from the fact that the expectation value is related to the free energy of individual quarks, which diverges in this phase. Introduced by Alexander M. Polyakov in 1975, they can also be used to study the potential between pairs of quarks at nonzero temperatures. Definition Thermal quantum field theory is formulated in Euclidean spacetime with a compactified imaginary temporal direction of length \beta. This length corresponds to the inverse temperature of the field \beta \propto 1/T. Compactifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matsubara Frequency
In thermal quantum field theory, the Matsubara frequency summation (named after Takeo Matsubara) is the summation over discrete imaginary frequencies. It takes the following form :S_\eta = \frac\sum_ g(i\omega_n), where \beta = \hbar / k_ T is the inverse temperature and the frequencies \omega_n are usually taken from either of the following two sets (with n\in\mathbb): :bosonic frequencies: \omega_n=\frac, :fermionic frequencies: \omega_n=\frac, The summation will converge if g(z=i\omega) tends to 0 in z\to\infty limit in a manner faster than z^. The summation over bosonic frequencies is denoted as S_ (with \eta=+1), while that over fermionic frequencies is denoted as S_ (with \eta=-1). \eta is the statistical sign. In addition to thermal quantum field theory, the Matsubara frequency summation method also plays an essential role in the diagrammatic approach to solid-state physics, namely, if one considers the diagrams at finite temperature. iers Coleman ''Introduction to Many- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
KMS State
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin– Schwinger state or, more commonly, a KMS state: a state satisfying the KMS condition. introduced the condition, used it to define thermodynamic Green's functions, and used the condition to define equilibrium states and called it the KMS condition. Overview The simplest case to study is that of a finite-dimensional Hilbert space, in which one does not encounter complications like phase transitions or spontaneous symmetry breaking. The density matrix of a thermal state is given by :\rho_=\frac=\frac where ''H'' is the Hamiltonian operator and ''N'' is the particle number operator (or charge operator, if we wish to be more general) and :Z(\beta,\mu)\ \stackrel\ \mathrm\left e^ \right/math> is the partition function. We assume that ''N'' commutes with ''H,'' or in other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cutkosky Rules
Cutkosky is a surname. Notable people with the surname include: *Ethan Cutkosky (born 1999), American actor and singer *Richard E. Cutkosky Richard Edwin Cutkosky (29 July 1928 – 17 June 1993) was a physicist, best known for the Cutkosky cutting rules in quantum field theory, which give a simple way to calculate the discontinuity of the scattering amplitude by Feynman diagram ... (1928–1993), American physicist {{Short pages monitor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bogoliubov Transformation
In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous system. The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the stationary solutions of the corresponding Schrödinger equation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, pairing effects in nuclear physics, and many other topics. The Bogoliubov transformation is often used to diagonalize Hamiltonians, ''with'' a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keldysh Formalism
In Non-equilibrium thermodynamics, non-equilibrium physics, the Keldysh formalism is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields (electrical field, magnetic field etc.). Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel. Extensions to driven-dissipative open quantum systems is given not only for bosonic systems, but also for fermionic systems. The Keldysh formalism provides a systematic way to study non-equilibrium systems, usually based on the two-point functions corresponding to excitations in the system. The main mathematical object in the Keldysh formalism is the non-equilibrium Green's function (NEGF), which is a two-point function of particle fields. In this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yang–Mills Theory
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics. History and theoretical description In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
