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Haag's Theorem
While working on the mathematical physics of an interacting, relativistic, quantum field theory, Rudolf Haag developed an argument against the existence of the interaction picture, a result now commonly known as Haag’s theorem. Haag’s original proof relied on the specific form of then-common field theories, but subsequently generalized by a number of authors, notably Hall & Wightman, who concluded that no single, universal Hilbert space representation can describe both free and interacting fields. A generalization due to Reed & Simon shows that applies to free neutral scalar fields of different masses, which implies that the interaction picture is always inconsistent, even in the case of a free field. Introduction Traditionally, describing a quantum field theory requires describing a set of operators satisfying the canonical (anti)commutation relations, and a Hilbert space on which those operators act. Equivalently, one should give a representation of the free algebra o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum State
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as a synonym for the vacuum state of a quantized field which is completely individual. According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space". According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of the quantum field. The QED vacuum of quantum electrodynamics (or QED) was the first vacuum of quantum field 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, who jointly received the Nobel prize for this work in 1965. For a historical discussion, see for exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than \aleph_0. The first three of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass (physics)
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body's inertia, meaning the resistance to acceleration (change of velocity) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The SI base unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wightman Function
In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scattering theory affirmed their significance. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields. Rationale One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension ''n'' of the space, and is commonly denoted E(''n'') or ISO(''n''). The Euclidean group E(''n'') comprises all translations, rotations, and reflections of \mathbb^n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be ''direct'' or ''indirect'', depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(''n''), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Field
In physics a free field is a field without interactions, which is described by the terms of motion and mass. Description In classical physics, a free field is a field whose equations of motion are given by linear partial differential equations. Such linear PDE's have a unique solution for a given initial condition. In quantum field theory, an operator valued distribution is a free field if it satisfies some linear partial differential equations such that the corresponding case of the same linear PDEs for a classical field (i.e. not an operator) would be the Euler–Lagrange equation for some quadratic Lagrangian. We can differentiate distributions by defining their derivatives via differentiated test functions. See Schwartz distribution for more details. Since we are dealing not with ordinary distributions but operator valued distributions, it is understood these PDEs aren't constraints on states but instead a description of the relations among the smeared fields. Beside th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction. In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. History The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Polarization
In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic field. It is also sometimes referred to as the self-energy of the gauge boson (photon). After developments in radar equipment for World War II resulted in higher accuracy for measuring the energy levels of the hydrogen atom, I.I. Rabi made measurements of the Lamb shift and the anomalous magnetic dipole moment of the electron. These effects corresponded to the deviation from the value −2 for the spectroscopic electron ''g''-factor that are predicted by the Dirac equation. Later, Hans Bethe theoretically calculated those shifts in the hydrogen energy levels due to vacuum polarization on his return train ride from the Shelter Island Conference to Cornell. The effects of vacuum polarizatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_2,\cdots,\phi_n\rangle_\nu be a Fock state, composed of single-particle states , \phi_i\rangle drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators a^(\phi_i) and a(\phi_i)\, we define the number operator by :\hat \ \stackrel\ a^(\phi_i)a(\phi_i) and we have :\hat, \Psi\rangle_\nu=N_i, \Psi\rangle_\nu where N_i is the number of particles in state , \phi_i\rangle. The above equality can be proven by noting that :\begin a(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_i,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu \\ a^(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
